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- 1. ACI STRUCTURAL JOURNAL TECHNICAL PAPER Title no. 100-S27 Analysis of Fiber-Reinforced Polymer Composite Grid Reinforced Concrete Beams by Federico A. Tavarez, Lawrence C. Bank, and Michael E. Plesha This study focuses on the use of explicit finite element analysis tools to predict the behavior of fiber-reinforced polymer (FRP) composite grid reinforced concrete beams subjected to four-point bending. Predictions were obtained using LS-DYNA, an explicit finite element program widely used for the nonlinear transient analysis of structures. The composite grid was modeled in a discrete manner using beam and shell elements, connected to a concrete solid mesh. The load-deflection characteristics obtained from the simulations show good correlation with the experimental data. Also, a detailed finite element substructure model was developed to further analyze the stress state of the main longitudinal reinforcement at ultimate conditions. Based on this analysis, a procedure was proposed for the analysis of composite grid reinforced concrete beams that accounts for different failure modes. A comparison of the proposed approach with the experimental data indicated that the procedure provides a good lower bound for conservative predictions of load-carrying capacity. Keywords: beam; composite; concrete; fiber-reinforced polymer; reinforcement; shear; stress. INTRODUCTION In recent years, research on fiber-reinforced polymer (FRP) composite grids has demonstrated that these products may be as practical and cost-effective as reinforcements for concrete structures.1-5 FRP grid reinforcement offers several advantages in comparison with conventional steel reinforcement and FRP reinforcing bars. FRP grids are prefabricated, noncorrosive, and lightweight systems suitable for assembly automation and ideal for reducing field installation and maintenance costs. Research on constructability issues and economics of FRP reinforcement cages for concrete members has shown the potential of these reinforcements to reduce life-cycle costs and significantly increase construction site productivity.6 Three-dimensional FRP composite grids provide a mechanical anchorage within the concrete due to intersecting elements, and thus no bond is necessary for proper load transfer. This type of reinforcement provides integrated axial, flexural, and shear reinforcement, and can also provide a concrete member with the ability to fail in a pseudoductile manner. Continuing research is being conducted to fully understand the behavior of composite grid reinforced concrete to commercialize its use and gain confidence in its design for widespread structural applications. For instance, there is a need to predict the correct failure mode of composite grid reinforced concrete beams where there is significant flexural-shear cracking.7 This type of information is critical for the development of design guidelines for FRP grid reinforced concrete members. Current flexural design methods for FRP reinforced concrete beams are analogous to the design of concrete beams using conventional reinforcement.8 The geometrical shape, ductility, modulus of elasticity, and force transfer characteristics of FRP composite grids, however, are likely to be different than 250 conventional steel or FRP bars. Therefore, the behavior of concrete beams with this type of reinforcement needs to be thoroughly investigated. OBJECTIVES The objectives of the present study were: 1) to investigate the ability of explicit finite element analysis tools to predict the behavior of composite grid reinforced concrete beams, including load-deflection characteristics and failure modes; 2) to evaluate the effect of the shear span-depth ratio in the failure mode of the beams and the stress state of the main flexural reinforcement at ultimate conditions; and 3) to develop an alternate procedure for the analysis of composite grid reinforced concrete beams considering multiple failure modes. RESEARCH SIGNIFICANCE The research work presented describes the use of advanced numerical simulation for the analysis of FRP reinforced concrete. These numerical simulations can be used effectively to understand the complex behavior and phenomena observed in the response of composite grid reinforced concrete beams. In particular, this effort provides a basis for the understanding of the interaction between the composite grid and the concrete when large flexural-shear cracks are present. As such, alternate analysis and design techniques can be developed based on the understanding obtained from numerical simulations to ensure the required capacity in FRP reinforced concrete structures. Background Several researchers have studied the viability of threedimensional FRP grids to reinforce concrete members.3,5,9,10 One specific type of three-dimensional FRP reinforcement is constructed from commercially manufactured pultruded FRP profiles (also referred to as FRP grating cages). Figure 1 shows a schematic of the structural members present in a concrete beam reinforced with the three-dimensional FRP reinforcement investigated in this study. A pilot experimental and analytical study was conducted by Bank, Frostig, and Shapira3 to investigate the feasibility of developing three-dimensional pultruded FRP grating cages to reinforce concrete beams. Failure of all beams tested occurred due to rupture of the FRP main longitudinal reinforcement in the shear span of the beam. Experimental results also revealed that most of the deflection at high loads appeared to occur due to localized rotations at large flexural crack widths ACI Structural Journal, V. 100, No. 2, March-April 2003. MS No. 02-100 received March 27, 2002, and reviewed under Institute publication policies. Copyright © 2003, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion will be published in the January-February 2004 ACI Structural Journal if received by September 1, 2003. ACI Structural Journal/March-April 2003
- 2. Federico A. Tavarez is a graduate student in the Department of Engineering Physics at the University of Wisconsin-Madison. He received his BS in civil engineering from the University of Puerto Rico-Mayagüez and his MSCE from the University of Wisconsin. His research interests include finite element analysis, the use of composite materials for structural applications, and the use of discrete element methods for modeling concrete damage and fragmentation under impact. ACI member Lawrence C. Bank is a professor in the Department of Civil and Environmental Engineering at the University of Wisconsin-Madison. He received his PhD in civil engineering and engineering mechanics from Columbia University in 1985. He is a member of ACI Committee 440, Fiber Reinforced Polymer Reinforcement. His research interests include FRP reinforcement systems for structures, progressive failure of materials and structural systems, and durability of FRP materials. Michael E. Plesha is a professor in the Engineering Mechanics and Astronautics Program in the Department of Engineering Physics at the University of WisconsinMadison. He received his PhD from Northwestern University in 1983. His research interests include finite element analysis, discrete element analysis, dynamics of geologic media, constitutive modeling of geologic discontinuity behavior, soil structure interaction modeling, and continuum modeling of jointed saturated rock masses. developed in the shear span near the load points. The study concluded that further research was needed to obtain a better understanding of the stress state in the longitudinal reinforcement at failure to predict the correct capacity and failure modes of the beams. Further experimental tests on concrete beams reinforced with three-dimensional FRP composite grids were conducted to investigate the behavior and performance of the grids when used to reinforce beams that develop significant flexural-shear cracking.7 Different composite grid configurations were designed to study the influence of the FRP grid components (longitudinal bars, vertical bars, and transverse bars) on the load-deflection behavior and failure modes. Even though failure modes of the beams were different depending upon the characteristics of the composite grid, all beams failed in their shear spans. Failure modes included splitting and rupture of the main longitudinal bars and shear-out failure of the vertical bars. Research results concluded that the design of concrete beams with composite grid reinforcements must account for failure of the main bars in the shear span. A second phase of this experimental research was performed by Ozel and Bank5 to investigate the capacity and failure modes of composite grid reinforced concrete beams with different shear span-to-effective depth ratios. Three different shear spandepth ratios (a/d) were investigated, with values of 3, 4.5, and 6, respectively.11 The data obtained from this recently completed experimental study was compared with the finite element results obtained in the present study. Experimental studies have shown that due to the development of large cracks in the FRP-reinforced concrete beams, most of the deformation takes place at a relatively small number of cracks between rigid bodies.12 A schematic of this behavior is shown in Fig. 2. As a result, beams with relatively small shear span-depth ratios typically fail due to rupture of the main FRP longitudinal reinforcement at large flexural-shear cracks, even though they are over-reinforced according to conventional flexural design procedures.5,7,13,14 Due to the aforementioned behavior for beams reinforced with composite grids, especially those that exhibit significant flexural-shear cracking, it is postulated that the longitudinal bars in the member are subjected to a uniform tensile stress distribution, plus a nonuniform stress distribution due to localized rotations at large cracks, which can be of great importance in determining the ultimate flexural strength of the beam. The present study investigates the stress-state at the flexuralshear cracks in the main longitudinal bars, using explicit finite element tools to simulate this behavior and determine the conditions that will cause failure in the beam. ACI Structural Journal/March-April 2003 Fig. 1—Structural members in composite grid reinforced concrete beam. Fig. 2—Deformation due to rotation of rigid bodies. Numerical analysis of FRP composite grid reinforced beams Implicit finite element methods are usually desirable for the analysis of quasistatic problems. Their efficiency and accuracy, however, depend on mesh topology and severity of nonlinearities. In the problem at hand, it would be very difficult to model the nonlinearities and progressive damage/ failure using an implicit method, and thus an explicit method was chosen to perform the analysis.15 Using an explicit finite element method, especially to model a quasistatic experiment as the one presented herein, can result in long run times due to the large number of time steps that are required. Because the time step depends on the smallest element size, efficiency is compromised by mesh refinement. The three-dimensional finite element mesh for this study was developed in HyperMesh16 and consisted of brick elements to represent the concrete, shell elements to represent the bottom longitudinal reinforcement, and beam elements to represent the top reinforcement, stirrups, and cross rods. Figure 3 shows a schematic of the mesh used for the models developed. Beams with span lengths of 2300, 3050, and 3800 mm were modeled corresponding to shear span-depth ratios of 3, 4.5, and 6, respectively. These models are referred to herein as short beam, medium beam, and long beam, respectively. The cross-sectional properties were identical for the three models. As will be seen later, the longitudinal bars play an important role in the overall behavior of the system, and therefore they were modeled with greater detail than the rest of the reinforcement. The concrete representation consisted of 8-node solid elements with dimensions 25 x 25 x 12.5 mm (shortest dimension parallel to the width of the beam), with one-point integration. The mesh discretization was established so that the reinforcement nodes coincided with the concrete nodes. The reinforcement mesh was connected to the concrete mesh by shared nodes between the concrete and the 251
- 3. Fig. 3—Finite element model for composite grid reinforced concrete beam. Fig. 4—Short beam model at several stages in simulation. reinforcement. As such, a perfect bond is assumed between the concrete and the composite grid. The two-node Hughes-Liu beam element formulation with 2 x 2 Gauss integration was used for modeling the top longitudinal bars, stirrups, and cross rods in the finite element models. In this study, each model contains two top longitudinal bars with heights of 25 mm and thicknesses of 4 mm. The models also have four cross rods and three vertical members at each stirrup location, as shown in Fig. 3. The vertical members have a width of 38 mm and a thickness of 6.4 mm. The cross rod elements have a circular cross-sectional area with a diameter of 12.7 mm. To model the bottom longitudinal reinforcement, the four-node BelytschkoLin-Tsay shell element formulation was used, as shown in Fig. 3, with two through-the-thickness integration points. 252 Boundary conditions and event simulation time To simulate simply supported conditions, the beam was supported on two rigid plates made of solid elements. The finite element simulations were displacement controlled, which is usually the control method for plastic and nonlinear behavior. That is, a displacement was prescribed on the rigid loading plates located on top of the beam. The prescribed displacement was linear, going from zero displacement at t = 0.0 s to 60, 75, and 90 mm at t = 1.0 s for the short, medium, and long beams, respectively. The corresponding applied load due to the prescribed displacement was then determined by monitoring the vertical reaction forces at the concrete nodes in contact with the support elements. The algorithm CONTACT_AUTOMATIC_SINGLE_ SURFACE in LS-DYNA was used to model the contact ACI Structural Journal/March-April 2003
- 4. between the supports, load bars, and the concrete beam. This algorithm automatically generates slave and master surfaces and uses a penalty method where normal interface springs are used to resist interpenetration between element surfaces. The interface stiffness is computed as a function of the bulk modulus, volume, and face area of the elements on the contact surface. The finite element analysis was performed to represent quasistatic experimental testing. As the time over which the load is applied approaches the period of the lowest natural frequency of vibration of the structural system, inertial forces become more important in the response. Therefore, the load application time was chosen to be long enough so that inertial effects would be negligible. The flexural frequency of vibration was computed analytically for the three beams using conventional formulas for vibration theory. 17 Accordingly, it was determined that having a load application time of 1.0 s was sufficiently long so that inertial effects are negligible and the analysis can be used to represent a quasistatic experiment. For the finite element simulations presented in this study, the CPU run time varied approximately from 22 to 65 h (depending on the length of the beam) for 1.0 s of load application time on a 600 MHz PC with 512 MB RAM. Material models Material Type 72 (MAT_CONCRETE_DAMAGE) in LS-DYNA was chosen for the concrete representation in the present study. This material model has been used successfully for predicting the response of standard uniaxial, biaxial, and triaxial concrete tests in both tension and compression. The formulation has also been used successfully to model the behavior of standard reinforced concrete dividing walls subjected to blast loads.18 This concrete model is a plasticitybased formulation with three independent failure surfaces (yield, maximum, and residual) that change shape depending on the hydrostatic pressure of the element. Tensile and compressive meridians are defined for each surface, describing the deviatoric part of the stress state, which governs failure in the element. Detailed information about this concrete material model can be found in Malvar et al.18 The values used in the input file corresponded to a 34.5 MPa concrete compressive strength with a 0.19 Poisson’s ratio and a tensile strength of 3.4 MPa. The softening parameters in the model were chosen to be 15, –50, and 0.01 for uniaxial tension, triaxial tension, and compression, respectively.19 The longitudinal bars were modeled using an orthotropic material model (MAT_ENHANCED_COMPOSITE_DAMAGE), which is material Type 54 in LS-DYNA. Properties used for this model are shown in Table 1. Because the longitudinal bars were drilled with holes for cross rod connections, the tensile strength in the longitudinal direction of the FRP bars was taken from experimental tensile tests conducted on notched bar specimens with a 12.7 mm hole to account for stress concentration effects at the cross rod locations. The tensile properties in the transverse direction were taken from tests on unnotched specimens. 11 Values for shear and compressive properties were chosen based on data in the literature. The composite material model uses the Chang/Chang failure criteria. 20 The remaining reinforcement (top longitudinal bars, stirrups, and cross rods) was modeled using two-noded beam elements using a linear elastic material model (MAT_ELASTIC) with the same properties used for the longitudinal direction in the bottom FRP longitudinal bars. A rigid material model ACI Structural Journal/March-April 2003 Fig. 5—Experimental and finite element load-deflection results for short, medium, and long beams. Fig. 6—Typical failure of composite grid reinforced concrete beam (Ozel and Bank5). Table 1—Material properties of FRP bottom bars Ex 26.7 GPa Xt 266.8 MPa 151.0 MPa Ey 14.6 GPa Yt Gxy 3.6 GPa Sc 6.9 MPa νxy 0.26 Xc 177.9 MPa β 0.5 Yc 302.0 MPa (MAT_RIGID) was used to model the supports and the loading plates. FINITE ELEMENT RESULTS AND DISCUSSION Graphical representations of the finite element model for the short beam at several stages in the simulation are shown in Fig. 4. The lighter areas in the model represent damage (high effective plastic strain) in the concrete material model. As expected, there is considerable damage in the shear span of the concrete beam. Figure 4 also shows the behavior of the composite grid inside the concrete beam. All displacements in the simulation graphics were amplified using a factor of 5 to enable viewing. Actual deflection values are given in Fig. 5, which shows the applied load versus midspan deflection behavior for the short, medium, and long beams for the experimental and LS-DYNA results, respectively. The jumps in the LS-DYNA curves in the figure represent the progressive tensile and shear failure in the concrete elements. As shown in this figure, the ultimate load value from the finite element model agrees well with the experimental result. The model slightly over-predicts the stiffness of the beam, however, and under-predicts the ultimate deflection. The significant drop in load seen in the load-deflection curves produced in LS-DYNA is caused by failure in the 253
- 5. Fig. 7—Medium beam model at several stages in simulation. Fig. 8—Long beam model at several stages in simulation. longitudinal bars, as seen in Fig. 4. The deformed shape seen in this figure indicates a peculiar behavior throughout the length of the beam. It appears to indicate that after a certain level of damage in the shear span of the model, localized rotations occur in the beam near the load points. These rotations create a stress concentration that causes the longitudinal bars to fail at those locations. This deflection behavior was also observed in the experimental tests. Figure 6 shows a typical failure in the longitudinal bars from the experiments conducted on these beams. 11 As shown in this figure, there is considerable damage in the shear span of the member. Large shear cracks develop in the beam, causing the member to deform in the same fashion as the one seen in the finite element model. Figure 7 shows the medium beam model at several stages in the simulation. The figure also shows the behavior of the main longitudinal bars. Comparing this simulation with the one obtained for the short beam, it can be seen that the shear damage is not as significant as in the previous simulation. The deflected shape seen in the longitudinal bars shows that this model does not have the abrupt changes in rotation that 254 were observed in the short beam, which would imply that this model does not exhibit significant flexural-shear damage. For this model, the finite element analysis slightly over-predicted both the stiffness and the ultimate load value obtained from the experiment. On the other hand, the ultimate deflection was under-predicted. Failure in this model was also caused by rupture of the longitudinal bars at a location near the load points. In the experimental test, failure was caused by a combination of rupture in the longitudinal bars as well as concrete crushing in the compression zone. This compressive failure was located near the load points, however, and could have been initiated by cracks formed due to stress concentrations produced by the rigid loading plates. 11 Figure 8 shows the results for the long beam model. Comparing this simulation with the two previous ones, it can be seen that this model exhibits the least shear damage, as expected. As a result, the longitudinal bars exhibit a parabolic shape, which would be the behavior predicted using conventional moment-curvature methods based on the curvature of the member. Once again, the stiffness of the beam was slightly over-predicted. However, the ultimate load ACI Structural Journal/March-April 2003
- 6. Table 2—Summary of experimental and finite element results Total load capacity, kN Tensile force in each main bar, kN Finite element analysis Flexural analysis Finite element analysis Beam Short value compares well with the experimental result. Failure in the model was caused by rupture of the longitudinal bars. Failure in the experimental test was caused by a compression failure at a location near one of the load application bars, followed by rupture of the main longitudinal bars. Figure 5 also shows the time at total failure for each beam, which can be related to the simulation stages given in Fig. 4, 7, and 8 for the short, medium, and long beam, respectively. To investigate the stress state of a single longitudinal bar at ultimate conditions, the tensile force and the internal moment of the longitudinal bars at the failed location for the three finite element models was determined, as shown in Fig. 9(a) and (b). It is interesting to note that for the short beam model, the tensile force at failure was approximately 51.6 kN, while for the medium beam model and the long beam model the tensile force at failure was approximately 76.5 kN. On the other hand, the internal moment in the short beam model was approximately 734 N-m, while the internal moment was approximately 339 N-m for both the short beam model and the long beam model. It is clear that the shear damage in the short beam model causes a considerable localized effect in the stress state of the longitudinal bars, which is important to consider for design purposes. According to Fig. 9(a), the total axial load in the longitudinal bars for the short beam model produces a uniform stress of 130 MPa, which is not enough to fail the element in tension at this location. However, the ultimate internal moment produces a tensile stress at the bottom of the longitudinal bars of 141 MPa. The sum of these two components produces a tensile stress of 271 MPa. When this value is entered in the Chang/Chang failure criterion for the tensile longitudinal direction, the strength is exceeded and the elements fail. Using conventional over-reinforced beam analysis formulas, the tensile force in the longitudinal bars at midspan would be obtained by dividing the ultimate moment obtained from the experimental test by the internal moment arm. This would imply that there is a uniform tensile force in each longitudinal bar of 88.1 kN. This tensile force is never achieved in the finite element simulation due to considerable shear damage in the concrete elements. As a result of this shear damage in the concrete, the curvature at the center of the beam is not large enough to produce a tensile force in the bars of this magnitude (88.1 kN). The internal moment in the longitudinal bars shown in Fig. 9(b), however, continues to develop, resulting in a total failure load comparable to the experimental result. As mentioned before, the force in the bars according to the simulation was approximately 51.6 kN, which is approximately half the load predicted using conventional methods. Therefore, the use of conventional beam analysis formulas to analyze this composite grid reinforced beam would not only erroneously predict the force in the longitudinal bars, but it would also predict a concrete ACI Structural Journal/March-April 2003 215.7 196.2 215.3 90.7 51.6 Medium Fig. 9—(a) Tensile force in longitudinal bars; and (b) internal moment in longitudinal bars. Experimental Flexural analysis 143.2 130.8 161.9 90.7 76.5 Long 108.1 97.9 113.0 90.7 76.5 compression failure mode, which was not the failure mode observed from the experimental tests. The curves for the medium beam model and the long beam model, shown in Fig. 9, show that for both cases, the beam shear span-depth ratio was sufficiently large so that the stress state in the longitudinal bars would not be greatly affected by the shear damage produced in the beam. As such, the ultimate axial force obtained in the longitudinal bars for both models was close to the ultimate axial load that would be predicted by using conventional methods. In summary, Table 2 presents the ultimate load capacity for the three models, including experimental results, conventional flexural analysis results, and finite element results. As shown in this table, conventional flexural analysis under-predicts the actual ultimate load carried by the beams and a better ultimate load prediction was obtained using finite element analysis. The tensile load in the bars was computed (analytically) by dividing the experimental moment capacity by the internal moment arm computed by using strain compatibility. Although the finite element results over-predicted the ultimate load for the medium and long beams, the simulations provided a better understanding of the complex phenomena involved in the behavior of the beams, depending on their shear span-depth ratio. The results for tensile load in the bars reported in this table suggest that composite grid reinforced concrete beams with values of shear span-depth ratio greater than 4.5 can be analyzed by using the current flexural theory. It is important to mention that the concrete material model parameters that govern the post-failure behavior of the material played a key role in the finite element results for the three finite element models. In the concrete material formulation, the elements fail in an isotropic fashion and, therefore, once an element fails in tension, it cannot transfer further shear. Because the concrete elements are connected to the reinforcement mesh, this behavior causes the beam to fail prematurely as a result of tensile failure in the concrete. Therefore, the parameters that govern the post-failure behavior in the concrete material model were chosen so that when an element fails in tension, the element still has the capability to transfer shear forces and the stresses will gradually decrease to zero. Because the failed elements can still transfer tensile stresses, however, the modifications caused an increase in the stiffness of the beam. In real concrete behavior, when a crack opens, there is no tension transfer between the concrete at that location, causing the member to lose stiffness as cracking progresses. Regarding shear transfer, factors such as aggregate interlock and dowel action would contribute to transfer shear forces in a concrete beam, and tensile failure in the concrete would not affect the response as directly as in the finite element model. 255
- 7. Stress analysis of FRP bars As discussed previously, failure modes observed in experimental tests performed on composite grid reinforced concrete beams suggest that the longitudinal bars are subjected to a uniform tensile stress plus a nonuniform bending stress due to localized rotations at locations of large cracks. This section presents a simple analysis procedure to determine the stress conditions at which the longitudinal bars fail. As a result of this analysis, a procedure is presented to analyze/design a composite grid reinforced concrete beam, considering a nonuniform stress state in the longitudinal bars. A more detailed finite element model of a section of the longitudinal bars was developed in HyperMesh16 using shell elements, as shown in Fig. 10. A height of 50.8 mm was specified for the bar model, with a thickness of 4.1 mm. The length of the bar and the diameter of the hole were 152 and 12.7 mm, respectively. The material formulation and properties were the same as the ones used for the longitudinal bars in the concrete beam models, with the exception that now the unnotched tensile strength of the material (Xt = 521 MPa) was used as an input parameter because the hole was incorporated in the model. The finite element model was first loaded in tension to establish the tensile strength of the notched bar. The load was applied by prescribing a displacement at the end of the bar. Figure 10 shows the simulation results for the model at three stages, including elastic deformation and ultimate failure. As expected, a stress concentration developed on the boundary of the hole causing failure in the web of the model, followed by ultimate failure of the cross section. A tensile strength of 274 MPa was obtained for the model. A value of 267 MPa was obtained from experimental tests conducted on notched bars (tensile strength used in Table 2), demonstrating good agreement between experimental and finite element results. A similar procedure was performed to establish the strength of the bar in pure bending. That is, displacements were prescribed at the end nodes to induce bending in the model. Figure 11 shows the simulation results for the model at three stages, showing elastic bending and ultimate failure caused by flexural failure at the tension flange. As shown in this figure, the width of the top flange was modified to prevent buckling in the flange (which was present in the original model). Because buckling would not be present in a longitudinal bar due to concrete confinement, it was decided to modify the finite element model to avoid this behavior. To maintain an equivalent cross-sectional area, the thickness of the flange was increased. A maximum pure bending moment of 2.92 kN-m was obtained for the model. Knowing the maximum force that the bar can withstand in pure tension and pure bending, the model was then loaded at different values of tension and moment to cause failure. This procedure was performed several times to develop a tensionmoment interaction diagram for the bar, as shown in Fig. 12. The discrete points shown in the figure are combinations of tensile force and moment values that caused failure in the finite element model. This interaction diagram can be used to predict what combination of tensile force and moment would cause failure in the FRP longitudinal bar. Considerations for design The strength design philosophy states that the flexural capacity of a reinforced concrete member must exceed the flexural demand. The design capacity of a member refers 256 Fig. 10—Failure on FRP bar subjected to pure tension. Fig. 11—Failure on FRP bar subjected to pure bending. to the nominal strength of the member multiplied by a strengthreduction factor φ, as shown in the following equation φ Mn ≥ Mu (1) For FRP reinforced concrete beams, a compression failure is the preferred mode of failure, and, therefore, the beam should be over-reinforced. As such, conventional formulas are used to ensure that the selected cross-sectional area of the longitudinal bars is sufficiently large to have concrete compression failure before FRP rupture. Considering a concrete compression failure, the capacity of the beam is computed using the following8 a M n = A f f f d – -- 2 (2) Af ff a = -------------β1 fc b ′ (3) β1 d – a f f = E f ε cu ----------------a (4) Experimental tests have shown, however, that there is a critical value of shear Vscrit in a beam where localized rotations due to large flexural-shear cracks begin to occur. The ultimate moment in the beam is assumed to be related to this shear-critical value and it is determined according to the following equation Mn = n ⋅ ( t ⋅ i e + m ) (5) where n is the number of longitudinal bars. Once the beam has reached the shear-critical value, it is assumed (conservatively) that the tensile force t, which is the force in each bar at the shear-critical stage, remains constant and any additional load is carried by localized internal moment m in the longitudinal bars. Furthermore, it is assumed that at this stage the concrete is still in its elastic range, and, therefore, the internal moment arm ie can be determined by equilibrium and elastic strain compatibility. The tensile force t in Eq. (5) is computed ACI Structural Journal/March-April 2003
- 8. Table 3—Summary of results for three beams using proposed approach Beam Experimental ultimate Theoretical shear shear, kN critical, kN Total load capacity, kN Equation for moment capacity Experimental Analytical Tension in each Pn = Mn /as main bar, kN Short 108.1 88.1 Mn = t · ie + m 216 199 70.7 Medium 71.6 88.1 Mn = Af f f (d – a/ 2) 143 131 90.7 88.1 Mn = Af f f (d – a/2) 109 99 90.7 Long 54.7 according to the following equation for a simply supported beam in four-point bending crit V s ⋅ as t = --------------------ni e (6) where as is the shear span of the member. The obtained value for the tension t in each bar is then entered in Eq. (7), which is the equation for the interaction diagram, to determine the ultimate internal moment m in Eq. (5) that causes the bar to fail. In this equation, tmax and mmax are known properties of the notched composite bar. t- 2 m = m max 1 – -------- for t > 0 ; m > 0 t max (7) The aforementioned procedure is a very simplified analysis to determine the capacity of a composite grid reinforced concrete beam, and, as can be seen, it depends considerably on the shearcritical value Vscrit established for the beam. This value is somewhat difficult to determine. Based on experimental data, a value given by Eq. (8) (analogous to Eq. (9-1) of ACI 440.1R-01) can be considered to be a lower bound for FRP reinforced beams with shear reinforcement. crit Vs 7 ρf Ef 1 - ′ = ----------------- -- f c bd 90 β 1 f c 6 ′ (8) where fc′ is the specified compressive strength of the concrete in MPa. In summary, the ultimate moment capacity in the beam is determined according to one of the following equations crit M n = A f f f d – a for V ult < V s - 2 crit M n = n ⋅ ( t ⋅ i e + m ) for V ult > V s (9) (10) According to Eq. (9), if the ultimate shear force computed analytically based on conventional theory does not exceed the shear-critical value Vscrit, the moment capacity can be computed from flexural analysis. On the other hand, if the computed ultimate shear force is greater than Vscrit, Eq. (10) is used. Table 3 presents a summary showing the load capacity for the three beams obtained experimentally and analytically using the present approach. As shown in this table, the equation used to determine the flexural capacity depends on the ultimate shear obtained for each beam. As seen in this procedure, the only difficulty in applying these formulas is the fact that an equation needs to be determined ACI Structural Journal/March-April 2003 Fig. 12—Tension-moment interaction diagram for longitudinal bar. to compute the maximum moment that the bar can carry as a function of the tensile force acting in the bar. If a specific bar is always used, however, this difficulty is eliminated, and if the flexural demand is not exceeded, a higher capacity can be obtained by increasing the number of longitudinal bars in the section. According to the results obtained for the three beams analyzed herein, the proposed procedure will under-predict the capacity of the composite grid reinforced concrete beam, but it will provide a good lower bound for a conservative design. Furthermore, it will ensure that the longitudinal bars will not fail prematurely as a result of the development of large flexural-shear cracks in the member, and thus the member will be able to meet and exceed the flexural demand for which it was designed. CONCLUSIONS Based on the explicit finite element results and comparison with experimental data, the following conclusions can be made: 1. Failure in the FRP longitudinal bars occurs due to a combination of a uniform tensile stress plus a nonuniform stress caused by localized rotations at large flexural-shear cracks. Therefore, this failure mode has to be accounted for in the analysis and design of composite grid reinforced concrete beams, especially those that exhibit significant flexuralshear cracking; 2. The shear span for the medium beam and the long beam studied was sufficiently large so that the stress state in the longitudinal bars was not considerably affected by shear damage in the beam. Therefore, the particular failure mode observed by the short beam model is only characteristic of 257
- 9. beams with a low shear span-depth ratio. Moreover, according to the proposed analysis for such systems, both the medium beam and the long beam could be designed using conventional flexural theory because the shear-critical value was never reached for these beam lengths; 3. Numerical simulations can be used effectively to understand the complex behavior and phenomena observed in the response of composite grid reinforced concrete beams and, therefore, can be used as a complement to experimental testing to account for multiple failure modes in the design of composite grid reinforced concrete beams; and 4. The proposed method of analysis for composite grid reinforced concrete beams considering multiple failure modes will under-predict the capacity of the reinforced concrete beam, but it will provide a good lower bound for a conservative design. These design considerations will ensure that the longitudinal bars will not fail prematurely (or catastrophically) as a result of the development of large flexural-shear cracks in the member, and thus the member can develop a pseudoductile failure by concrete crushing, which is more desirable than a sudden FRP rupture. ACKNOWLEDGMENTS This work was supported by the National Science Foundation under Grant. No. CMS 9896074. Javier Malvar and Karagozian & Case are thanked for providing information regarding the concrete material formulation used in LS-DYNA. Jim Day, Todd Slavik, and Khanh Bui of Livermore Software Technology Corporation (LSTC) are also acknowledged for their assistance in using the finite element software, as well as Strongwell Chatfield, MN, for producing the custom composite grids. NOTATION a as b d = = = = Ef Ex Ey Gxy f ′c ff ie Mn m n Sc t Vscrit = = = = = = = = = = = = = Vult Xc Xt Yc Yt β β1 = = = = = = = εcu ρf νxy = = = 258 depth of equivalent rectangular stress block length of shear span in reinforced concrete beam width of rectangular cross section distance from extreme compression fiber to centroid of tension reinforcement modulus of elasticity for FRP bar modulus of elasticity in longitudinal direction of FRP grid material modulus of elasticity in transverse direction of FRP grid material shear modulus of FRP grid members specified compressive strength of concrete stress in FRP reinforcement in tension internal moment arm in the elastic range nominal moment capacity internal moment in longitudinal FRP grid bars number of longitudinal FRP grid bars shear strength of FRP grid material tensile force in a longitudinal bar at the shear critical stage critical shear resistance provided by concrete in FRP grid reinforced concrete ultimate shear force in reinforced concrete beam longitudinal compressive strength of FRP grid material longitudinal tensile strength of FRP grid material transverse compressive strength of FRP grid material transverse tensile strength of FRP grid material weighting factor for shear term in Chang/Chang failure criterion ratio of the depth of Whitney’s stress block to depth to neutral axis concrete ultimate strain FRP reinforcement ratio Poisson’s ratio of FRP grid material REFERENCES 1. Sugita, M., “NEFMAC—Grid Type Reinforcement,” Fiber-ReinforcedPlastic (FRP) Reinforcement for Concrete Structures: Properties and Applications, Developments in Civil Engineering, A. Nanni, ed., Elsevier, Amsterdam, V. 42, 1993, pp. 355-385. 2. Schmeckpeper, E. R., and Goodspeed, C. H., “Fiber-Reinforced Plastic Grid for Reinforced Concrete Construction,” Journal of Composite Materials, V. 28, No. 14, 1994, pp. 1288-1304. 3. Bank, L. C.; Frostig, Y.; and Shapira, A., “Three-Dimensional FiberReinforced Plastic Grating Cages for Concrete Beams: A Pilot Study,” ACI Structural Journal, V. 94, No. 6, Nov.-Dec. 1997, pp. 643-652. 4. Smart, C. W., and Jensen, D. W., “Flexure of Concrete Beams Reinforced with Advanced Composite Orthogrids,” Journal of Aerospace Engineering, V. 10, No. 1, Jan. 1997, pp. 7-15. 5. Ozel, M., and Bank, L. C., “Behavior of Concrete Beams Reinforced with 3-D Composite Grids,” CD-ROM Paper No. 069. Proceedings of the 16th Annual Technical Conference, American Society for Composites, Virginia Tech, Va., Sept. 9-12, 2001. 6. Shapira, A., and Bank, L. C., “Constructability and Economics of FRP Reinforcement Cages for Concrete Beams,” Journal of Composites for Construction, V. 1, No. 3, Aug. 1997, pp. 82-89. 7. Bank, L. C., and Ozel, M., “Shear Failure of Concrete Beams Reinforced with 3-D Fiber Reinforced Plastic Grids,” Fourth International Symposium on Fiber Reinforced Polymer Reinforcement for Reinforced Concrete Structures, SP-188, C. Dolan, S. Rizkalla, and A. Nanni, eds., American Concrete Institute, Farmington Hills, Mich., 1999, pp. 145-156. 8. ACI Committee 440, “Guide for the Design and Construction of Concrete Reinforced with FRP Bars (ACI 440.1R-01),” American Concrete Institute, Farmington Hills, Mich., 2001, 41 pp. 9. Nakagawa, H.; Kobayashi. M.; Suenaga, T.; Ouchi, T.; Watanabe, S.; and Satoyama, K., “Three-Dimensional Fabric Reinforcement,” FiberReinforced-Plastic (FRP) Reinforcement for Concrete Structures: Properties and Applications, Developments in Civil Engineering, V. 42, A. Nanni, ed., Elsevier, Amsterdam, 1993, pp. 387-404. 10. Yonezawa, T.; Ohno, S.; Kakizawa, T.; Inoue, K.; Fukata, T.; and Okamoto, R., “A New Three-Dimensional FRP Reinforcement,” FiberReinforced-Plastic (FRP) Reinforcement for Concrete Structures: Properties and Applications, Developments in Civil Engineering, V. 42, A. Nanni, ed., Elsevier, Amsterdam, 1993, pp. 405-419. 11. Ozel, M., “Behavior of Concrete Beams Reinforced with 3-D Fiber Reinforced Plastic Grids,” PhD thesis, University of WisconsinMadison, 2002. 12. Lees, J. M., and Burgoyne, C. J., “Analysis of Concrete Beams with Partially Bonded Composite Reinforcement,” ACI Structural Journal, V. 97, No. 2, Mar.-Apr. 2000, pp. 252-258. 13. Shehata, E.; Murphy, R.; and Rizkalla, S., “Fiber Reinforced Polymer Reinforcement for Concrete Structures,” Fourth International Symposium on Fiber Reinforced Polymer Reinforcement for Reinforced Concrete Structures, SP-188, C. Dolan, S. Rizkalla, and A. Nanni, eds., American Concrete Institute, Farmington Hills, Mich., 1999, pp. 157-167. 14. Guadagnini, M.; Pilakoutas, K.; and Waldron, P., “Investigation on Shear Carrying Mechanisms in FRP RC Beams,” FRPRCS-5, FibreReinforced Plastics for Reinforced Concrete Structures, Proceedings of the Fifth International Conference, C. J. Burgoyne, ed., V. 2, Cambridge, July 16-18, 2001, pp. 949-958. 15. Cook, R. D.; Malkus, D. S.; and Plesha, M. E., Concepts and Applications of Finite Element Analysis, 3rd Edition, John Wiley & Sons, N.Y., 1989, 832 pp. 16. Altair Computing, HyperMesh Version 2.0 User’s Manual, Altair Computing Inc., Troy, Mich., 1995. 17. Thompson, W. T., and Dahleh, M. D., Theory of Vibration with Applications, 5th Edition, Prentice Hall, N.J., 1998, 524 pp. 18. Malvar, L. J.; Crawford, J. E.; Wesevich, J. W.; and Simons, D., “A Plasticity Concrete Material Model for DYNA3D,” International Journal of Impact Engineering, V. 19, No. 9/10, 1997, pp. 847-873. 19. Tavarez, F. A., “Simulation of Behavior of Composite Grid Reinforced Concrete Beams Using Explicit Finite Element Methods,” MS thesis, University of Wisconsin-Madison, 2001. 20. Hallquist, J. O., LS-DYNA Keyword User’s Manual, Livermore Software Technology Corporation, Livermore, Calif., Apr. 2000. ACI Structural Journal/March-April 2003
- 10. ACI STRUCTURAL JOURNAL TECHNICAL PAPER Title no. 100-S64 Compression Field Modeling of Reinforced Concrete Subjected to Reversed Loading: Formulation by Daniel Palermo and Frank J. Vecchio Constitutive formulations are presented for concrete subjected to reversed cyclic loading consistent with a compression field approach. The proposed models are intended to provide substantial compatibility to nonlinear finite element analysis in the context of smeared rotating cracks in both the compression and tension stress regimes. The formulations are also easily adaptable to a fixed crack approach or an algorithm based on fixed principal stress directions. Features of the modeling include: nonlinear unloading using a Ramberg-Osgood formulation; linear reloading that incorporates degradation in the reloading stiffness based on the amount of strain recovered during the unloading phase; and improved plastic offset formulations. Backbone curves from which unloading paths originate and on which reloading paths terminate are represented by the monotonic response curves and account for compression softening and tension stiffening in the compression and tension regions, respectively. Also presented are formulations for partial unloading and partial reloading. Keywords: cracks; load; reinforced concrete. RESEARCH SIGNIFICANCE The need for improved methods of analysis and modeling of concrete subjected to reversed loading has been brought to the fore by the seismic shear wall competition conducted by the Nuclear Power Engineering Corporation of Japan.1 The results indicate that a method for predicting the peak strength of structural walls is not well established. More important, in the case of seismic analysis, was the apparent inability to accurately predict structure ductility. Therefore, the state of the art in analytical modeling of concrete subjected to general loading conditions requires improvement if the seismic response and ultimate strength of structures are to be evaluated with sufficient confidence. This paper presents a unified approach to constitutive modeling of reinforced concrete that can be implemented into finite element analysis procedures to provide accurate simulations of concrete structures subjected to reversed loading. Improved analysis and design can be achieved by modeling the main features of the hysteresis behavior of concrete and by addressing concrete in tension. INTRODUCTION The analysis of reinforced concrete structures subjected to general loading conditions requires realistic constitutive models and analytical procedures to produce reasonably accurate simulations of behavior. However, models reported that have demonstrated successful results under reversed cyclic loading are less common than models applicable to monotonic loading. The smeared crack approach tends to be the most favored as documented by, among others, Okamura and Maekawa2 and Sittipunt and Wood.3 Their approach, assuming fixed cracks, has demonstrated good correlation to experimental results; 616 however, the fixed crack assumption requires separate formulations to model the normal stress and shear stress hysteretic behavior. This is at odds with test observations. An alternative method of analysis, used herein, for reversed cyclic loading assumes smeared rotating cracks consistent with a compression field approach. In the finite element method of analysis, this approach is coupled with a secant stiffness formulation, which is marked by excellent convergence and numerical stability. Furthermore, the rotating crack model eliminates the need to model normal stresses and shear stresses separately. The procedure has demonstrated excellent correlation to experimental data for structures subjected to monotonic loading.4 More recently, the secant stiffness method has successfully modeled the response of structures subjected to reversed cyclic loading,5 addressing the criticism that it cannot be effectively used to model general loading conditions. While several cyclic models for concrete, including Okamura and Maekawa;2 Mander, Priestley, and Park;6 and Mansour, Lee, and Hsu,7 among others, have been documented in the literature, most are not applicable to the alternative method of analysis used by the authors. Documented herein are models, formulated in the context of smeared rotating cracks, for reinforced concrete subjected to reversed cyclic loading. To reproduce accurate simulations of structural behavior, the modeling considers the shape of the unloading and reloading curves of concrete to capture the energy dissipation and the damage of the material due to load cycling. Partial unloading/reloading is also considered, as structural components may partially unload and then partially reload during a seismic event. The modeling is not limited to the compressive regime alone, as the tensile behavior also plays a key role in the overall response of reinforced concrete structures. A comprehensive review of cyclic models available in the literature and those reported herein can be found elsewhere.8 It is important to note that the models presented are not intended for fatigue analysis and are best suited for a limited number of excursions to a displacement level. Further, the models are derived from tests under quasistatic loading. CONCRETE STRESS-STRAIN MODELS For demonstrative purposes, Vecchio5 initially adopted simple linear unloading/reloading rules for concrete. The formulations were implemented into a secant stiffness-based finite element algorithm, using a smeared rotating crack ACI Structural Journal, V. 100, No. 5, September-October 2003. MS No. 02-234 received July 2, 2002, and reviewed under Institute publication policies. Copyright © 2003, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published in the JulyAugust 2004 ACI Structural Journal if the discussion is received by March 1, 2004. ACI Structural Journal/September-October 2003
- 11. Daniel Palermo is a visiting assistant professor in the Department of Civil Engineering, University of Toronto, Toronto, Ontario, Canada. He received his PhD from the University of Toronto in 2002. His research interests include nonlinear analysis and design of concrete structures, constitutive modeling of reinforced concrete subjected to cyclic loading, and large-scale testing and analysis of structural walls. ACI member Frank J. Vecchio is Professor and Associate Chair in the Department of Civil Engineering, University of Toronto. He is a member of Joint ACI-ASCE Committee 441, Reinforced Concrete Columns, and 447, Finite Element Analysis of Reinforced Concrete Structures. His interests include nonlinear analysis and design of concrete structures. approach, to illustrate the analysis capability for arbitrary loading conditions, including reversed cyclic loading. The models presented herein have also been formulated in the context of smeared rotating cracks, and are intended to build upon the preliminary constitutive formulations presented by Vecchio.5 A companion paper 9 documenting the results of nonlinear finite element analyses, incorporating the proposed models, will demonstrate accurate simulations of structural behavior. Compression response First consider the compression response, illustrated in Fig. 1, occurring in either of the principal strain directions. Figure 1(a) and (b) illustrate the compressive unloading and compressive reloading responses, respectively. The backbone curve typically follows the monotonic response, that is, Hognestad parabola 10 or Popovics formulation,11 and includes the compression softening effects according to the Modified Compression Field Theory. 12 The shape and slope of the unloading and reloading responses p are dependent on the plastic offset strain εc , which is essentially the amount of nonrecoverable damage resulting from crushing of the concrete, internal cracking, and compressing of internal voids. The plastic offset is used as a parameter in defining the unloading path and in determining the degree of damage in the concrete due to cycling. Further, the backbone curve for the tension response is shifted such that its origin coincides with the compressive plastic offset strain. Various plastic offset models for concrete in compression have been documented in the literature. Karsan and Jirsa13 were the first to report a plastic offset formulation for concrete subjected to cyclic compressive loading. The model illustrated the dependence of the plastic offset strain on the strain at the onset of unloading from the backbone curve. A review of various formulations in the literature reveals that, for the most part, the models best suit the data from which they were derived, and no one model seems to be most appropriate. A unified model (refer to Fig. 2) has been derived herein considering data from unconfined tests from Bahn and Hsu14 and Karsan and Jirsa,13 and confined tests from Buyukozturk and Tseng.15 From the latter tests, the results indicated that the plastic offset was not affected by confining stresses or strains. The proposed plastic offset formulation is described as ε 2c 2 ε 2c p ε c = ε p 0.166 ------ + 0.132 ------ εp εp Fig. 1—Hysteresis models for concrete in compression: (a) unloading; and (b) reloading. (1) where εcp is the plastic offset strain; εp is the strain at peak stress; and ε2c is the strain at the onset of unloading from the backbone curve. Figure 2 also illustrates the response of other plastic offset models available in the literature. The plot indicates that models proposed by Buyukozturk and Tseng15 and Karsan and Jirsa13 represent upper- and ACI Structural Journal/September-October 2003 lower-bound solutions, respectively. The proposed model (Palermo) predicts slightly larger residual strains than the lower limit, and the Bahn and Hsu14 model calculates progressively larger plastic offsets. Approximately 50% of the datum points were obtained from the experimental results of Karsan and Jirsa;13 therefore, it is not unexpected that the Palermo model is skewed towards the lower-bound Karsan and Jirsa13 model. The models reported in the literature were derived from their own set of experimental data and, thus, may be affected by the testing conditions. The proposed formulation alleviates dependence on one set of experimental data and test conditions. The Palermo model, by predicting Fig. 2—Plastic offset models for concrete in compression. 617
- 12. relatively small plastic offsets, predicts more pinching in the hysteresis behavior of the concrete. This pinching phenomenon has been observed by Palermo and Vecchio8 and Pilakoutas and Elnashai16 in the load-deformation response of structural walls dominated by shear-related mechanisms. In analysis, the plastic offset strain remains unchanged unless the previous maximum strain in the history of loading is exceeded. The unloading response of concrete, in its simplest form, can be represented by a linear expression extending from the unloading strain to the plastic offset strain. This type of representation, however, is deficient in capturing the energy dissipated during an unloading/reloading cycle in compression. Test data of concrete under cyclic loading confirm that the unloading branch is nonlinear. To derive an expression to describe the unloading branch of concrete, a RambergOsgood formulation similar to that used by Seckin17 was adopted. The formulation is strongly influenced by the unloading and plastic offset strains. The general form of the unloading branch of the proposed model is expressed as f c ( ∆ε ) = A + B ∆ε + C ∆ε N stress point on the reloading path that corresponded to the maximum unloading strain. The new stress point was assumed to be a function of the previous unloading stress and the stress at reloading reversal. Their approach, however, was stress-based and dependent on the backbone curve. The approach used herein is to define the reloading stiffness as a degrading function to account for the damage induced in the concrete due to load cycling. The degradation was observed to be a function of the strain recovery during unloading. The reloading response is then determined from f c = f ro + E c1 ( ε c – ε ro ) (6) where fc and εc are the stress and strain on the reloading path; f ro is the stress in the concrete at reloading reversal and corresponds to a strain of εro ; and Ec1 is the reloading stiffness, calculated as follows ( β d ⋅ f max ) – f ro E c1 = ----------------------------------ε 2c – ε ro (7) (2) where where fc is the stress in the concrete on the unloading curve, and ∆ε is the strain increment, measured from the instantaneous strain on the unloading path to the unloading strain, A, B, and C are parameters used to define the general shape of the curve, and N is the Ramberg-Osgood power term. Applying boundary conditions from Fig. 1(a) and simplifying yields 1 β d = ----------------------------------------------0.5 1 + 0.10 (ε rec ⁄ ε p ) for ε c < ε p (8) 1 β d = -------------------------------------------------0.6 1 + 0.175 (ε rec ⁄ ε p ) for ε c > ε p (9) and N ( E c3 – E c2 )∆ε f c ( ∆ε ) = f 2c + E c2 ( ∆ε ) + -------------------------------------N–1 p N ( ε c – ε 2c ) (3) where and ∆ε = ε – ε 2c (4) and p ( E c2 – E c3 ) ( εc – ε 2c ) N = --------------------------------------------------p f c2 + E c2 ( ε c – ε 2c ) (5) ε is the instantaneous strain in the concrete. The initial unloading stiffness Ec2 is assigned a value equal to the initial tangent stiffness of the concrete Ec, and is routinely calculated as 2fc′ / ε′c . The unloading stiffness Ec3, which defines the stiffness at the end of the unloading phase, is defined as 0.071 E c, and was adopted from Seckin. 17 f2c is the stress calculated from the backbone curve at the peak unloading strain ε 2c. Reloading can sufficiently be modeled by a linear response and is done so by most researchers. An important characteristic, however, which is commonly ignored, is the degradation in the reloading stiffness resulting from load cycling. Essentially, the reloading curve does not return to the backbone curve at the previous maximum unloading strain (refer to Fig. 1 (b)). Further straining is required for the reloading response to intersect the backbone curve. Mander, Priestley, and Park6 attempted to incorporate this phenomenon by defining a new 618 ε rec = ε max – ε min (10) βd is a damage indicator, fmax is the maximum stress in the concrete for the current unloading loop, and εrec is the amount of strain recovered in the unloading process and is the difference between the maximum strain εmax and the minimum strain εmin for the current hysteresis loop. The minimum strain is limited by the compressive plastic offset strain. The damage indicator was derived from test data on plain concrete from four series of tests: Buyukozturk and Tseng,15 Bahn and Hsu,14 Karsan and Jirsa,13 and Yankelevsky and Reinhardt.18 A total of 31 datum points were collected for the prepeak range (Fig. 3(a)) and 33 datum points for the postpeak regime (Fig. 3(b)). Because there was a negligible amount of scatter among the test series, the datum points were combined to formulate the model. Figure 3(a) and (b) illustrate good correlation with experimental data, indicating the link between the strain recovery and the damage due to load cycling. βd is calculated for the first unloading/reloading cycle and retained until the previous maximum unloading strain is attained or exceeded. Therefore, no additional damage is induced in the concrete for hysteresis loops occurring at strains less than the maximum unloading strain. This phenomenon is further illustrated through the partial unloading and partial reloading formulations. ACI Structural Journal/September-October 2003
- 13. It is common for cyclic models in the literature to ignore the behavior of concrete for the case of partial unloading/ reloading. Some models establish rules for partial loadings from the full unloading/reloading curves. Other models explicitly consider the case of partial unloading followed by reloading to either the backbone curve or strains in excess of the previous maximum unloading strain. There exists, however, a lack of information considering the case where partial unloading is followed by partial reloading to strains less than the previous maximum unloading strain. This more general case was modeled using the experimental results of Bahn and Hsu.14 The proposed rule for the partial unloading response is identical to that assumed for full unloading; however, the previous maximum unloading strain and corresponding stress are replaced by a variable unloading strain and stress, respectively. The unloading path is defined by the unloading stress and strain and the plastic offset strain, which remains unchanged unless the previous maximum strain is exceeded. For the case of partial unloading followed by reloading to a strain in excess of the previous maximum unloading strain, the reloading path is defined by the expressions governing full reloading. The case where concrete is partially unloaded and partially reloaded to a strain less than the previous maximum unloading strain is illustrated in Fig 4. Five loading branches are required to construct the response of Fig. 4. Unloading Curve 1 represents full unloading from the maximum unloading strain to the plastic offset and is calculated from Eq. (3) to (5) for full unloading. Curve 2 defines reloading from the plastic offset strain and is defined by Eq. (6) to (10). Curve 3 represents the case of partial unloading from a reloading path at a strain less than the previous maximum unloading strain. The expressions used for full unloading are applied, with the exception of substituting the unloading stress and strain for the current hysteresis loop for the unloading stress and strain at the previous maximum unloading point. Curve 4 describes partial reloading from a partial unloading branch. The response follows a linear path from the load reversal point to the previous unloading point and assumes that damage is not accumulated in loops forming at strains less than the previous maximum unloading strain. This implies that the reloading stiffness of Curve 4 is greater than the reloading stiffness of Curve 2 and is consistent with test data reported by Bahn and Hsu.14 The reloading stiffness for Curve 4 is represented by the following expression f max – f ro E c1 = ---------------------ε max – ε ro f c = f max + E c1 ( ε c – ε max ) (13) The proposed constitutive relations for concrete subjected to compressive cyclic loading are tested in Fig. 5 against the experimental results of Karsan and Jirsa.13 The Palermo model generally captures the behavior of concrete under cyclic compressive loading. The nonlinear unloading and linear loading formulations agree well with the data, and the plastic offset strains are well predicted. It is apparent, though, that the reloading curves become nonlinear beyond the point of intersection with the unloading curves, often referred to as the Fig. 3—Damage indicator for concrete in compression: (a) prepeak regime; and (b) postpeak regime. (11) The reloading stress is then calculated using Eq. (6) for full reloading. In further straining beyond the intersection with Curve 2, the response of Curve 4 follows the reloading path of Curve 5. The latter retains the damage induced in the concrete from the first unloading phase, and the stiffness is calculated as β d ⋅ f 2c – f max E c1 = ------------------------------ε 2c – ε max (12) The reloading stresses are then determined from the following ACI Structural Journal/September-October 2003 Fig. 4—Partial unloading/reloading for concrete in compression. 619
- 14. common point. The Palermo model can be easily modified to account for this phenomenon; however, unusually small load steps would be required in a finite element analysis to capture this behavior, and it was thus ignored in the model. Furthermore, the results tend to underestimate the intersection of the reloading path with the backbone curve. This is a direct result of the postpeak response of the concrete and demonstrates the importance of proper modeling of the postpeak behavior. Tension response Much less attention has been directed towards the modeling of concrete under cyclic tensile loading. Some researchers consider little or no excursions into the tension stress regime and those who have proposed models assume, for the most Fig. 5—Predicted response for cycles in compression. part, linear unloading/reloading responses with no plastic offsets. The latter was the approach used by Vecchio5 in formulating a preliminary tension model. Stevens, Uzumeri, and Collins19 reported a nonlinear response based on defining the stiffness along the unloading path; however, the models were verified with limited success. Okumura and Maekawa2 proposed a hysteretic model for cyclic tension, in which a nonlinear unloading curve considered stresses through bond action and through closing of cracks. A linear reloading path was also assumed. Hordijk 20 used a fracture mechanics approach to formulate nonlinear unloading/reloading rules in terms of applied stress and crack opening displacements. The proposed tension model follows the philosophy used to model concrete under cyclic compression loadings. Figure 6 (a) and (b) illustrate the unloading and reloading responses, respectively. The backbone curve, which assumes the monotonic behavior, consists of two parts adopted from the Modified Compression Field Theory12: that describing the precracked response and that representing postcracking tension-stiffened response. A shortcoming of the current body of data is the lack of theoretical models defining a plastic offset for concrete in tension. The offsets occur when cracked surfaces come into contact during unloading and do not realign due to shear slip along the cracked surfaces. Test results from Yankelevsky and Reinhardt21 and Gopalaratnam and Shah22 provide data that can be used to formulate a plastic offset model (refer to Fig. 7). The researchers were able to capture the softening behavior of concrete beyond cracking in displacementcontrolled testing machines. The plastic offset strain, in the proposed tension model, is used to define the shape of the unloading curve, the slope and damage of the reloading path, and the point at which cracked surfaces come into contact. Similar to concrete in compression, the offsets in tension seem to be dependent on the unloading strain from the backbone curve. The proposed offset model is expressed as p 2 ε c = 146ε1c + 0.523 ε 1c (14) where εcp is the tensile plastic offset, and ε1c is the unloading strain from the backbone curve. Figure 7 illustrates very good correlation to experimental data. Observations of test data suggest that the unloading response of concrete subjected to tensile loading is nonlinear. The accepted approach has been to model the unloading branch as linear and to ignore the hysteretic behavior in the concrete Fig. 6—Hysteresis models for concrete in tension: (a) unloading; and (b) reloading. 620 Fig. 7—Plastic offset model for concrete in tension. ACI Structural Journal/September-October 2003
- 15. due to cycles in tension. The approach used herein was to formulate a nonlinear expression for the concrete that would generate realistic hysteresis loops. To derive a model consistent with the compression field approach, a Ramberg-Osgood formulation, similar to that used for concrete in compression, was adopted and is expressed as fc = D + F∆ε + G∆εN (15) where fc is the tensile stress in the concrete; ∆ε is the strain increment measured from the instantaneous strain on the unloading path to the unloading strain; D, F, and G are parameters that define the shape of the unloading curve; and N is a power term that describes the degree of nonlinearity. Applying the boundary conditions from Fig. 6(a) and simplifying yields concrete due to load cycling. Limited test data confirm that linear reloading sufficiently captures the general response of the concrete; however, it is evident that the reloading stiffness accumulates damage as the unloading strain increases. The approach suggested herein is to model the reloading behavior as linear and to account for a degrading reloading stiffness. The latter is assumed to be a function of the strain recovered during the unloading phase and is illustrated in Fig. 8 against data reported by Yankelevsky and Reinhardt.21 The reloading stress is calculated from the following expression f c = β t ⋅ tf max – E c4 ( ε1c – ε c ) ( β t ⋅ tf max ) – tf ro E c4 = -------------------------------------ε 1c – t ro (16) where ∆ε = ε 1c – ε (17) (22) where N ( E c5 – E c6 )∆ε f c ( ∆ε ) = f 1c – E c5 ( ∆ε ) + -------------------------------------p N–1 N ( ε 1c – ε c ) (21) fc is the tensile stress on the reloading curve and corresponds to a strain of εc. Ec4 is the reloading stiffness, βt is a tensile damage indicator, tf max is the unloading stress for the current hysteresis loop, and tfro is the stress in the concrete at reloading reversal corresponding to a strain of tro. The damage parameter βt is calculated from the following relation 1 β t = ---------------------------------------0.25 1 + 1.15 ( ε rec ) (23) ε rec = ε max – ε min and (24) p ( E c5 – E c6 ) ( ε 1c – ε c ) N = --------------------------------------------------p E c5 ( ε 1c – ε c ) – f 1c (18) f1c is the unloading stress from the backbone curve, and Ec5 is the initial unloading stiffness, assigned a value equal to the initial tangent stiffness Ec. The unloading stiffness Ec6, which defines the stiffness at the end of the unloading phase, was determined from unloading data reported by Yankelevsky and Reinhardt.21 By varying the unloading stiffness Ec6, the following models were found to agree well with test data E c6 = 0.071 ⋅ E c ( 0.001 ⁄ ε 1c ) ε 1c ≤ 0.001 (19) E c6 = 0.053 ⋅ E c ( 0.001 ⁄ ε 1c ) ε 1c > 0.001 (20) The Okamura and Maekawa2 model tends to overestimate the unloading stresses for plain concrete, owing partly to the fact that the formulation is independent of a tensile plastic offset strain. The formulations are a function of the unloading point and a residual stress at the end of the unloading phase. The residual stress is dependent on the initial tangent stiffness and the strain at the onset of unloading. The linear unloading response suggested by Vecchio5 is a simple representation of the behavior but does not capture the nonlinear nature of the concrete and underestimates the energy dissipation. The proposed model captures the nonlinear behavior and energy dissipation of the concrete. The state of the art in modeling reloading of concrete in tension is based on a linear representation, as described by, among others, Vecchio5 and Okamura and Maekawa.2 The response is assumed to return to the backbone curve at the previous unloading strain and ignores damage induced to the ACI Structural Journal/September-October 2003 where εrec is the strain recovered during an unloading phase. It is the difference between the unloading strain εmax and the minimum strain at the onset of reloading εmin, which is limited by the plastic offset strain. Figure 8 depicts good correlation between the proposed formulation and the limited experimental data. Following the philosophy for concrete in compression, βt is calculated for the first unloading/reloading phase and retained until the previous maximum strain is at least attained. The literature is further deficient in the matter of partial unloading followed by partial reloading in the tension stress regime. Proposed herein is a partial unloading/reloading Fig. 8—Damage model for concrete in tension. 621
- 16. model that directly follows the rules established for concrete in compression. No data exist, however, to corroborate the model. Figure 9 depicts the proposed rules for a concrete element, lightly reinforced to allow for a post-cracking response. Curve 1 corresponds to a full unloading response and is identical to that assumed by Eq. (16) to (18). Reloading from a full unloading curve is represented by Curve 2 and is computed from Eq. (21) to (24). Curve 3 represents the case of partial unloading from a reloading path at a strain less than the previous maximum unloading strain. The expressions for full unloading are used; however, the strain and stress at unloading, now variables, replace the strain and stress at the previous peak unloading point on the backbone curve. Reloading from a partial unloading segment is described by Curve 4. The response follows a linear path from the reloading strain to the previous unloading strain. The model explicitly assumes that damage does not accumulate for loops that form at strains less than the previous maximum unloading strain in the history of loading. Therefore, the reloading stiffness of Curve 4 is larger than the reloading stiffness for the first unloading/reloading response of Curve 2. The partial reloading stiffness, defining Curve 4, is calculated by the following expression tf max – tf ro E c4 = -----------------------ε max – t ro (25) and the reloading stress is then determined from f c = tf ro + E c4 ( ε c – t ro ) (26) As loading continues along the reloading path of Curve 4, a change in the reloading path occurs at the intersection with Curve 2. Beyond the intersection, the reloading response follows the response of Curve 5 and retains the damage induced to the concrete from the first unloading/reloading phase. The stiffness is then calculated as β t ⋅ f 1c – tf max E c4 = -------------------------------ε 1c – ε max (27) The reloading stresses can then be calculated according to f c = tf max + E c4 ( ε c – ε max ) (28) The previous formulations for concrete in tension are preliminary and require experimental data to corroborate. The models are, however, based on realistic assumptions derived from the models suggested for concrete in compression. CRACK-CLOSING MODEL In an excursion returning from the tensile domain, compressive stresses do not remain at zero until the cracks completely close. Compressive stresses will arise once cracked surfaces come into contact. The recontact strain is a function of factors such as crack-shear slip. There exists limited data to form an accurate model for crack closing, and the preliminary model suggested herein is based on the formulations and assumptions suggested by Okamura and Maekawa. 2 Figure 10 is a schematic of the proposed model. The recontact strain is assumed equal to the plastic offset strain for concrete in tension. The stiffness of the concrete during closing of cracks, after the two cracked surfaces have come into contact and before the cracks completely close, is smaller than that of crack-free concrete. Once the cracks completely close, the stiffness assumes the initial tangent stiffness value. The crack-closing stiffness Eclose is calculated from f close E close = ----------p εc (29) fclose = –Ec(0.0016 ⋅ ε1c + 50 × 10–6) Fig. 9—Partial unloading/reloading for concrete in tension. (30) where fclose , the stress imposed on the concrete as cracked surfaces come into contact, consists of two terms taken from the Okamura and Maekawa2 model for concrete in tension. The first term represents a residual stress at the completion of unloading due to stress transferred due to bond action. The second term represents the stress directly related to closing of cracks. The stress on the closing-of-cracks path is then determined from the following expression p Fig. 10—Crack-closing model. 622 f c = E close ( ε c – ε c ) (31) ACI Structural Journal/September-October 2003
- 17. After the cracks have completely closed and loading continues into the compression strain region, the reloading rules for concrete in compression are applicable, with the stress in the concrete at the reloading reversal point assuming a value of fclose. For reloading from the closing-of-cracks curve into the tensile strain region, the stress in the concrete is assumed to be linear, following the reloading path previously established for tensile reloading of concrete. In lieu of implementing a crack-closing model, plastic offsets in tension can be omitted, and the unloading stiffness at the completion of unloading Ec6 can be modified to ensure that the energy dissipation during unloading is properly captured. Using data reported by Yankelevsky and Reinhardt,21 a formulation was derived for the unloading stiffness at zero loads and is proposed as a function of the unloading strain on the backbone curve as follows E c6 = – 1.1364 ( ε 1c – 0.991 ) (32) Implicit in the latter model is the assumption that, in an unloading excursion in the tensile strain region, the compressive stresses remain zero until the cracks completely close. REINFORCEMENT MODEL The suggested reinforcement model is that reported by Vecchio,5 and is illustrated in Fig. 11. The monotonic response of the reinforcement is assumed to be trilinear. The initial response is linear elastic, followed by a yield plateau, and ending with a strain-hardening portion. The hysteretic response of the reinforcement has been modeled after Seckin,17 and the Bauschinger effect is represented by a Ramberg-Osgood formulation. The monotonic response curve is assumed to represent the backbone curve. The unloading portion of the response follows a linear path and is given by fs ( ε i ) = f s – 1 + Er ( ε i – εs – 1 ) (33) where fs(εi) is the stress at the current strain of εi , fs – 1 and εs – 1 are the stress and strain from the previous load step, and Er is the unloading modulus and is calculated as Er = Es if ( ε m – ε o ) < ε y ( Em – Er ) ( εm – εo ) N = -------------------------------------------fm – Er ( εm – εo ) (38) fm is the stress corresponding to the maximum strain recorded during previous loading; and Em is the tangent stiffness at εm. The same formulations apply for reinforcement in tension or compression. For the first reverse cycle, εm is taken as zero and fm = fy, the yield stress. IMPLEMENTATION AND VERIFICATION The proposed formulations for concrete subjected to reversed cyclic loading have been implemented into a two-dimensional nonlinear finite element program, which was developed at the University of Toronto.23 The program is applicable to concrete membrane structures and is based on a secant stiffness formulation using a total-load, iterative procedure, assuming smeared rotating cracks. The package employs the compatibility, equilibrium, and constitutive relations of the Modified Compression Field Theory.12 The reinforcement is typically modeled as smeared within the element but can also be discretely represented by truss-bar elements. The program was initially restricted to conditions of monotonic loading, and later developed to account for material prestrains, thermal loads, and expansion and confinement effects. The ability to account for material prestrains provided the framework for the analysis capability of reversed cyclic loading conditions. 5 For cyclic loading, the secant stiffness procedure separates the total concrete strain into two components: an elastic strain and a plastic offset strain. The elastic strain is used to compute an effective secant stiffness for the concrete, and, therefore, the plastic offset strain must be treated as a strain offset, similar to an elastic offset as reported by Vecchio.4 The plastic offsets in the principal directions are resolved into components relative to the reference axes. From the prestrains, free joint displacements are determined as functions of the element geometry. Then, plastic prestrain nodal forces can be evaluated using the effective element stiffness matrix due to the concrete component. The plastic offsets developed in (34) ε m – εo E r = E s 1.05 – 0.05 ---------------- if ε y < ( ε m – ε o ) < 4 ε y (35) εy Er = 0.85Es if (εm – εo) > 4εy (36) where Es is the initial tangent stiffness; εm is the maximum strain attained during previous cycles; εo is the plastic offset strain; and εy is the yield strain. The stresses experienced during the reloading phase are determined from Em – Er N f s ( ε i ) = E r ( ε i – ε o ) + -------------------------------------- ⋅ ( ε i – ε o ) N–1 N ⋅ ( εm – εo ) where ACI Structural Journal/September-October 2003 (37) Fig. 11—Hysteresis model for reinforcement, adapted from Seckin (1981). 623
- 18. each of the reinforcement components are also handled in a similar manner. The total nodal forces for the element, arising from plastic offsets, are calculated as the sum of the concrete and reinforcement contributions. These are added to prestrain forces arising from elastic prestrain effects and nonlinear expansion effects. The finite element solution then proceeds. The proposed hysteresis rules for concrete in this procedure require knowledge of the previous strains attained in the history of loading, including, amongst others: the plastic offset strain, the previous unloading strain, and the strain at reloading reversal. In the rotating crack assumption, the principal strain directions may be rotating presenting a complication. A simple and effective method of tracking and defining the strains is the construction of Mohr’s circle. Further details of the procedure used for reversed cyclic loading can be found from Vecchio.5 A comprehensive study, aimed at verifying the proposed cyclic models using nonlinear finite element analyses, will be presented in a companion paper.9 Structures considered will include shear panels and structural walls available in the literature, demonstrating the applicability of the proposed formulations and the effectiveness of a secant stiffnessbased algorithm employing the smeared crack approach. The structural walls will consist of slender walls, with heightwidth ratios greater than 2.0, which are heavily influenced by flexural mechanisms, and squat walls where the response is dominated by shear-related mechanisms. The former is generally not adequate to corroborate constitutive formulations for concrete. CONCLUSIONS A unified approach to constitutive modeling of reversed cyclic loading of reinforced concrete has been presented. The constitutive relations for concrete have been formulated in the context of a smeared rotating crack model, consistent with a compression field approach. The models are intended for a secant stiffness-based algorithm but are also easily adaptable in programs assuming either fixed cracks or fixed principal stress directions. The concrete cyclic models consider concrete in compression and concrete in tension. The unloading and reloading rules are linked to backbone curves, which are represented by the monotonic response curves. The backbone curves are adjusted for compressive softening and confinement in the compression regime, and for tension stiffening and tension softening in the tensile region. Unloading is assumed nonlinear and is modeled using a Ramberg-Osgood formulation, which considers boundary conditions at the onset of unloading and at zero stress. Unloading, in the case of full loading, terminates at the plastic offset strain. Models for the compressive and tensile plastic offset strains have been formulated as a function of the maximum unloading strain in the history of loading. Reloading is modeled as linear with a degrading reloading stiffness. The reloading response does not return to the backbone curve at the previous unloading strain, and further straining is required to intersect the backbone curve. The degrading reloading stiffness is a function of the strain recovered during unloading and is bounded by the maximum unloading strain and the plastic offset strain. The models also consider the general case of partial unloading and partial reloading in the region below the previous maximum unloading strain. 624 NOTATION Ec = Eclose = Ec1 = Ec2 = Ec3 = Ec4 = Ec5 = Ec6 = Em = = Er = Es Esh = f1c = f2c = = fc = f ′c fclose = = fcr = fm fmax = = fp fro = = fs fs – 1 = = fy tfmax = tfro = tro = βd = βt = ∆ε = ε = ε0 = ε1c = ε2c = εc = ε′c = p εc = εcr = ε i , εs = εm = εmax = εmin = εp = εrec = εro = εsh = εs – 1 = εy = initial modulus of concrete crack-closing stiffness modulus of concrete in tension compressive reloading stiffness of concrete initial unloading stiffness of concrete in compression compressive unloading stiffness at zero stress in concrete reloading stiffness modulus of concrete in tension initial unloading stiffness modulus of concrete in tension unloading stiffness modulus at zero stress for concrete in tension tangent stiffness of reinforcement at previous maximum strain unloading stiffness of reinforcement initial modulus of reinforcement strain-hardening modulus of reinforcement unloading stress from backbone curve for concrete in tension unloading stress on backbone curve for concrete in compression normal stress of concrete peak compressive strength of concrete cylinder crack-closing stress for concrete in tension cracking stress of concrete in tension reinforcement stress corresponding to maximum strain in history maximum compressive stress of concrete for current unloading cycle peak principal compressive stress of concrete compressive stress at onset of reloading in concrete average stress for reinforcement stress in reinforcement from previous load step yield stress for reinforcement maximum tensile stress of concrete for current unloading cycle tensile stress of concrete at onset of reloading tensile strain of concrete at onset of reloading damage indicator for concrete in compression damage indicator for concrete in tension strain increment on unloading curve in concrete instantaneous strain in concrete plastic offset strain of reinforcement unloading strain on backbone curve for concrete in tension compressive unloading strain on backbone curve of concrete compressive strain of concrete strain at peak compressive stress in concrete cylinder residual (plastic offset) strain of concrete cracking strain for concrete in tension current stress of reinforcement maximum strain of reinforcement from previous cycles maximum strain for current cycle minimum strain for current cycle strain corresponding to maximum concrete compressive stress strain recovered during unloading in concrete compressive strain at onset of reloading in concrete strain of reinforcement at which strain hardening begins strain of reinforcement from previous load step yield strain of reinforcement REFERENCES 1. Nuclear Power Engineering Corporation of Japan (NUPEC), “Comparison Report: Seismic Shear Wall ISP, NUPEC’s Seismic Ultimate Dynamic Response Test,” Report No. NU-SSWISP-D014, Organization for Economic Co-Operation and Development, Paris, France, 1996, 407 pp. 2. Okamura, H., and Maekawa, K., Nonlinear Analysis and Constitutive Models of Reinforced Concrete, Giho-do Press, University of Tokyo, Japan, 1991, 182 pp. 3. Sittipunt, C., and Wood, S. L., “Influence of Web Reinforcement on the Cyclic Response of Structural Walls,” ACI Structural Journal, V. 92, No. 6, Nov.-Dec. 1995, pp. 745-756. 4. Vecchio, F. J., “Finite Element Modeling of Concrete Expansion and Confinement,” Journal of Structural Engineering, ASCE, V. 118, No. 9, 1992, pp. 2390-2406. 5. Vecchio, F. J., “Towards Cyclic Load Modeling of Reinforced Concrete,” ACI Structural Journal, V. 96, No. 2, Mar.-Apr. 1999, pp. 132-202. 6. Mander, J. B.; Priestley, M. J. N.; and Park, R., “Theoretical StressStrain Model for Confined Concrete,” Journal of Structural Engineering, ASCE, V. 114, No. 8, 1988, pp. 1804-1826. 7. Mansour, M.; Lee, J. Y.; and Hsu, T. T. C., “Cyclic Stress-Strain Curves of Concrete and Steel Bars in Membrane Elements,” Journal of Structural Engineering, ASCE, V. 127, No. 12, 2001, pp. 1402-1411. 8. Palermo, D., and Vecchio, F. J., “Behaviour and Analysis of Reinforced Concrete Walls Subjected to Reversed Cyclic Loading,” Publication No. 2002-01, Department of Civil Engineering, University of Toronto, Canada, 2002, 351 pp. ACI Structural Journal/September-October 2003
- 19. 9. Palermo, D., and Vecchio, F. J., “Compression Field Modeling of Reinforced Concrete Subjected to Reversed Loading: Verification,” ACI Structural Journal. (accepted for publication) 10. Hognestad, E.; Hansen, N. W.; and McHenry, D., “Concrete Stress Distribution in Ultimate Strength Design,” ACI JOURNAL, Proceedings V. 52, No. 12, Dec. 1955, pp. 455-479. 11. Popovics, S., “A Numerical Approach to the Complete Stress-Strain Curve of Concrete,” Cement and Concrete Research, V. 3, No. 5, 1973, pp. 583-599. 12. Vecchio, F. J., and Collins, M. P., “The Modified Compression-Field Theory for Reinforced Concrete Elements Subjected to Shear,” ACI JOURNAL, Proceedings V. 83, No. 2, Mar.-Apr. 1986, pp. 219-231. 13. Karsan, I. K., and Jirsa, J. O., “Behaviour of Concrete Under Compressive Loadings,” Journal of the Structural Division, ASCE, V. 95, No. 12, 1969, pp. 2543-2563. 14. Bahn, B. Y., and Hsu, C. T., “Stress-Strain Behaviour of Concrete Under Cyclic Loading,” ACI Materials Journal, V. 95, No. 2, Mar.-Apr. 1998, pp. 178-193. 15. Buyukozturk, O., and Tseng, T. M., “Concrete in Biaxial Cyclic Compression,” Journal of Structural Engineering, ASCE, V. 110, No. 3, Mar. 1984, pp. 461-476. 16. Pilakoutas, K., and Elnashai, A., “Cyclic Behavior of RC Cantilever Walls, Part I: Experimental Results,” ACI Structural Journal, V. 92, No. 3, May-June 1995, pp. 271-281. ACI Structural Journal/September-October 2003 17. Seckin, M., “Hysteretic Behaviour of Cast-in-Place Exterior BeamColumn Sub-Assemblies,” PhD thesis, University of Toronto, Toronto, Canada, 1981, 266 pp. 18. Yankelevsky, D. Z., and Reinhardt, H. W., “Model for Cyclic Compressive Behaviour of Concrete,” Journal of Structural Engineering, ASCE, V. 113, No. 2, Feb. 1987, pp. 228-240. 19. Stevens, N. J.; Uzumeri, S. M.; and Collins, M. P., “Analytical Modelling of Reinforced Concrete Subjected to Monotonic and Reversed Loadings,” Publication No. 87-1, Department of Civil Engineering, University of Toronto, Toronto, Canada, 1987, 201 pp. 20. Hordijk, D. A., “Local Approach to Fatigue of Concrete,” Delft University of Technology, The Netherlands, 1991, pp. 210. 21. Yankelevsky, D. Z., and Reinhardt, H. W., “Uniaxial Behaviour of Concrete in Cyclic Tension,” Journal of Structural Engineering, ASCE, V. 115, No. 1, 1989, pp. 166-182. 22. Gopalaratnam, V. S., and Shah, S. P., “Softening Response of Plain Concrete in Direct Tension,” ACI JOURNAL, Proceedings V. 82, No. 3, MayJune 1985, pp. 310-323. 23. Vecchio, F. J., “Nonlinear Finite Element Analysis of Reinforced Concrete Membranes,” ACI Structural Journal, V. 86, No. 1, Jan.-Feb. 1989, pp. 26-35. 625
- 20. ACI STRUCTURAL JOURNAL TECHNICAL PAPER Title no. 104-S45 Cyclic Load Behavior of Reinforced Concrete Beam-Column Subassemblages of Modern Structures by Alexandros G. Tsonos The seismic performance of four one-half scale exterior beam-column subassemblages is examined. All subassemblages were typical of new structures and incorporated full seismic details in current building codes, such as a weak girder-strong column design philosophy. The subassemblages were subjected to a large number of inelastic cycles. The tests indicated that current design procedures could sometimes result in excessive damage to the joint regions. Keywords: beam-column frames; connections; cyclic loads; reinforced concrete; structural analysis. INTRODUCTION The key to the design of ductile moment-resisting frames is that the beam-to-column connections and columns must remain essentially elastic throughout the load history to ensure the lateral stability of the structure. If the connections or columns exhibit stiffness and/or strength deterioration with cycling, collapse due to P-Δ effects or to the formation of a story mechanism may be unavoidable.1,2 Four one-half scale beam-column subassemblages were designed and constructed in turn, according to Eurocode 23 and Eurocode 8,4 according to ACI 318-055 and ACI 352R-02,6 and according to the new Greek Earthquake Resistant Code7 and the new Greek Code for the Design of Reinforced Concrete Structures.8 The subassemblages were subjected to cyclic lateral load histories so as to provide the equivalent of severe earthquake damage. The results indicate that current design procedures could sometimes result in severe damage to the joint, despite the use of a weak girder-strong column design philosophy. RESEARCH SIGNIFICANCE Experimental data and experience from earthquakes indicate that loss of capacity might occur in joints that are part of older reinforced concrete (RC) frame structures.9-12 There is scarce experimental evidence and insufficient data, however, about the performance of joints designed according to current codes during strong earthquakes. This research provides structural engineers with useful information about the safety of new RC frame structures that incorporate seismic details from current building codes. In some cases, safety could be jeopardized during strong earthquakes by premature joint shear failures. The joints could at times remain the weak link even for structures designed in accordance with current model building codes. DESCRIPTION OF TEST SPECIMENS— MATERIAL PROPERTIES Four one-half scale exterior beam-column subassemblages were designed and constructed for this experimental and analytical investigation. Reinforcement details of the subassemblages are shown in Fig. 1(a) and (b). All the 468 subassemblages (A1, E1, E2, and G1) had the same general and cross-sectional dimensions, as shown in Fig. 1. Subassemblages E1, E2, and G1 had the same longitudinal column reinforcement, eight bars with a diameter of 14 mm, while the longitudinal column reinforcement of A1 consisted of eight bars with a diameter of 10 mm (0.4 in.). The longitudinal column reinforcement of A1 was lower than that of the other three subassemblages (E1, E2, and G1) due to the restrictions of ACI 352R-026 for the column bars passing through the joint. Subassemblages E1 and G1 had the same percentage of longitudinal beam reinforcement (ρE1 = ρG1 = 7.7 × 10–3) and Subassemblages A1 and E2 also had the same percentage of longitudinal beam reinforcement (ρA1 = 5.23 × 10–3 and ρE2 = 5.2 × 10–3), but different from the percentage of E1 and G1. The longitudinal beam reinforcement of A1 consisted of four bars with a diameter of 10 mm, while the beam reinforcement of E2 consisted of two bars with a diameter of 14 mm. Subassemblage A1 had smaller beam reinforcing bars than Subassemblage E2 due to the restrictions of ACI 352R-026 for the beam bars passing through the joint. The joint shear reinforcements of the subassemblages used in the experiments, are as follows: Ø6 multiple hoop at 5 cm for Subassemblage A1 (Fig. 1(a)), Ø6 multiple hoop at 5 cm for Subassemblage E1, (Fig. 1(b)), Ø6 multiple hoop at 4.8 cm for Subassemblage E2 (Fig. 1(a)) and Ø8 multiple hoop at 10 cm for Subassemblage G1 (Fig. 1(b)). All subassemblages incorporated seismic details. The purpose of Subassemblages A1, E1, E2, and G1 was to represent details of new structures. As is clearly demonstrated in Fig. 1(a) and (b), all the subassemblages had high flexural strength ratios MR. The purpose of using an MR ratio (sum of the flexural capacity of columns to that of beam(s)) significantly greater than 1.00 in earthquake-resistant constructions is to push the formation of the plastic hinge in the beams, so that the safety (that is, collapse prevention) of the structure is not jeopardized.1,2,4-7,9,10,13 Thus, in all these subassemblages, the beam is expected to fail in a flexural mode during cyclic loading. The concrete 28-day compressive strength of both Subassemblages A1 and E2 was 35 MPa (5075 psi), while the concrete 28-day compressive strength of both Subassemblages E1 and G1 was 22 MPa (3190 psi). Reinforcement yield strengths are as follows: Ø6 = 540 MPa (78 ksi), Ø10 = 500 MPa (73 ksi), and Ø14 = 495 MPa (72 ksi) (note: Ø6 [No. 2]), Ø10 [No. 3], and Ø14 [No. 4]) are bars with a diameter of 6, 10, and 14 mm). ACI Structural Journal, V. 104, No. 4, July-August 2007. MS No. S-2006-230.R1 received June 21, 2006, and reviewed under Institute publication policies. Copyright © 2007, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published in the MayJune 2008 ACI Structural Journal if the discussion is received by January 1, 2008. ACI Structural Journal/July-August 2007
- 21. ACI member Alexandros G. Tsonos is a Professor of reinforced concrete structures, Department of Structural Engineering, the Aristotle University of Thessaloniki, Thessaloniki, Greece. He received his PhD from the Aristotle University of Thessaloniki in 1990. His research interests include the inelastic behavior of reinforced concrete structures, structural design, fiber-reinforced concrete, seismic repair and rehabilitation of reinforced concrete structures, and the seismic repair and restoration of monuments. Approximately 10 electrical-resistance strain gauges were bonded on the reinforcing bars of each subassemblage of the program. EXPERIMENTAL SETUP AND LOADING SEQUENCE The general arrangement of the experimental setup is shown in Fig. 2(a). All subassemblages were subjected to 11 cycles applied by slowly displacing the beam’s free end according to the load history shown in Fig. 2(b) without reaching the actuator stroke limit. The amplitudes of the peaks in the displacement history were 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, and 65 mm. One loading cycle was performed at each displacement amplitude. An axial load equal to 200 kN was applied to the columns of the subassemblages and kept constant throughout the test. The experimental loading sequence used is a typical one, commonly used in previous studies.1,11,13 It was not the objective of this study to investigate the effect of other, nonstandard loading histories on the response of the subassemblages. As previously mentioned, all the subassemblages were loaded slowly. The strain rate of the load applied corresponded to static conditions. In the case of seismic loading, the strain · rate ε is higher than the rate corresponding to static conditions. · Soroushian and Sim14 showed that an increase in ε with respect to static conditions leads to a moderate increase in the strength of concrete 2 · · f c, dyn = [ 1.48 + 0.160 × log ε + 0.0127 ( log ε ) ] × f c, stat (1) Scott et al.15 tested column subassemblages with various amounts of hoop reinforcement under strain rates ranging from 0.33 × 10–5 sec–1 (static loading), to 0.0167 sec–1 (seismic loading). Their test results conformed with the results obtained from Eq. (1). Using the aforementioned expression, it is estimated that for · a strain rate of ε = 0.0167 sec–1, concrete strengths increase by approximately 20% (compared with the static one). An expression similar to Eq. (1) can be found in the CEB code.16 Thus, the strengths exhibited by Subassemblages A1, E1, E2, and G1 during the tests are somewhat lower than the strengths they would exhibit if subjected to load histories similar to actual seismic events. EXPERIMENTAL RESULTS Failure mode of Subassemblages A1, E1, E2, and G1 The failure mode of Subassemblages A1 and E2, as expected, involved the formation of a plastic hinge in the beam at the column face. The formation of plastic hinges caused severe cracking of the concrete near the fixed beam end of each subassemblage (Fig. 3). The behavior of Subassemblages A1 and E2 was as expected and as documented in the seismic design philosophy of the modern codes as will be explain in the following.4-7 Fig. 1—Dimensions and cross-sectional details of: (a) Subassemblages A1 and E2; and (b) Subassemblages E1 and G1. (Note: dimensions are in cm. 1 cm = 0.0394 in.) ACI Structural Journal/July-August 2007 469
- 22. Fig. 2—(a) General arrangement of experimental setup and photograph of test setup (dimensions are in m; 1 m = 3.28 ft); and (b) lateral displacement history. (Note: 1 mm = 0.039 in.) Fig. 4—Applied shear versus strain in beam-column joint hoop reinforcement of: (a) Subassemblages A1 and E2; and (b) Subassemblages E1 and G1. (Note: 1 kN = 0.225 kip.) Fig. 3—Views of collapsed Subassemblages A1, E1, E2, and G1. Significant inelastic deformations occurred in the beams’ longitudinal reinforcement in both Subassemblages A1 and E2 (strains of over 40.000με were obtained in the beams’ longitudinal bars), while joint shear reinforcement remained elastic. Figure 4(a) shows strain gauge data of joint hoop reinforcement for both Subassemblages A1 and E2. As is clearly shown in Fig. 4(a), the maximum strain recorded in 470 the joint hoop reinforcement for both subassemblages was below the yield strain of 2.500με, which was in agreement with the observed failure modes of Subassemblages A1 and E2.17 One difference between the failure modes of Subassemblages A1 and E2 was that hairline cracks appeared in the joint region of E2, and partial loss of the concrete cover in the rear face of the joint of E2 took place during the three last cycles of loading (ninth, tenth, and eleventh) when drift Angle R ratios exceeded 4.5 while the joint region of Subassemblage A1 remained intact at the conclusion of the test (refer to Fig. 3). The connections of both Subassemblages E1 and G1, contrary to expectations, exhibited shear failure during the ACI Structural Journal/July-August 2007
- 23. Fig. 5—Maximum strain during each cycle of loading in beam longitudinal reinforcement of Subassemblages A1, E1, E2, and G1. Fig. 6—Gradual cracking configuration of Subassemblage E1 during test. early stages of cyclic loading. Damage occurred both in the joint area and in the columns’ critical regions. Figure 4(b) shows strain gauge data for the joint hoop reinforcement for Subassemblages E1 and G1. As shown in Fig. 4(b), the maximum strain recorded in the joint hoop reinforcement of both Subassemblages E1 and G1 was significantly higher than the yield strain 2.500 με. Joint shear damage has been shown to occur after yielding of the joint hoop reinforcement, which is in agreement with the damage observed in the joints of these subassemblages.18 The maximum strain recorded in the longitudinal bars of the beams of both Subassemblages E1 and G1 was below 2.500με (refer to Fig. 5). In Fig. 6, the progression of cracking of Subassemblage E1 during the test is demonstrated. Load-drift angle curves Plots of applied shear force versus drift angles for all the Subassemblages (A1, E1, E2, and G1) are shown in Fig. 7. The beam calculated flexural capacities of the subassemblages are shown as dashed lines in Fig. 7. ACI Structural Journal/July-August 2007 Fig. 7—Hysteresis loops of Subassemblages A1, E1, E2, and G1. (Note: 1 kN = 0.225 kip.) A major concern in the seismic design of RC structures is the ability of members to develop their flexural strength before failing in shear. This is especially true for members framing at a beam column joint (beams and columns), where it is important to develop their flexural strengths before joint shear failure. Moreover, by designing the flexural strengths of columns in RC frame structures to meet the strong-column weak-beam rule, all members against premature shear failure, and by detailing plastic hinge (critical) regions for ductility, RC frame structures have been shown to exhibit a controlled and very ductile inelastic response.2,4,9 471
- 24. As can be seen in Fig. 7, the beam of Subassemblage A1 developed maximum shear forces higher than those corresponding to its ultimate flexural strength until the sixth cycle of loading. This is an indication of the flexural response of this beam because it developed its flexural strength until a drift Angle R ratio of 4.0 was reached and exceeded. Also, a flexural failure was observed for this beam, caused by crushing of the concrete cover of the longitudinal reinforcement, and subsequent inelastic buckling of the longitudinal bars. The beam of Subassemblage E2 also developed maximum shear forces higher than those corresponding to its ultimate flexural strength until the eleventh upper half cycle of loading and until the seventh lower half cycle of loading. In particular, during the final cycles of loading beyond drift Angle R ratios of 4.5 when large displacements were imposed, crushing of the concrete cover of the reinforcement took place and the beam’s hoops could not provide adequate support to the longitudinal reinforcement. As a result, buckling of the beam longitudinal reinforcement in Subassemblages A1 and E2 occurred after the sixth and seventh cycles of loading, respectively. The beam of Subassemblage E1 developed maximum shear forces very close to those corresponding to its ultimate flexural strength only during the second and third cycle of loading. For the remaining cycles (four through 11), the premature joint shear failure did not allow the beam in this subassemblage to develop its flexural capacity (Fig. 6 and 7). The premature joint shear failure of Subassemblage G1 also did not allow the beam in this subassemblage to develop its flexural capacity. As can be seen in Fig. 7, the beam of Subassemblage G1 developed maximum shear forces significantly lower than those corresponding to its ultimate flexural strength. One of the basic provisions of all modern structural codes is to provide the structures with sufficient strength and sufficient ductility to undergo post-elastic deformations without losing a large percentage of their strength.2,4,7,9 As can be seen in Fig. 7, this criterion is fulfilled for Subassemblies A1 and E2. By contrast, it is not fulfilled for Subassemblies E1 and G1 because they exhibited significant loss of strength during cyclic loading. The beam-column Subassemblages A1, E1, E2, and G1 are similar to real modern frame structures. If the sequence in the breakdown of the chain of resistance of these real frame structures follows the desirable hierarchy during a catastrophic earthquake, the formation of plastic hinges in the beams of these structures would be expected, because the use of a weak girder-strong column design philosophy is adopted by the modern codes.2,4,5,7,9 The aforementioned desirable failure mode (with formation of a plastic hinge in the beam) was developed by Subassemblages A1 and E2. Thus, the magnitude of loads resisted by Subassemblages A1 and E2 are consistent with the expected values from actual events. Story drifts allowed by modern codes are on the order of 2% of the story height.4,7,8 While it was reassuring that story drifts of as much as 4% of the story height were achieved in most reported tests referring to the seismic response of beamcolumn specimens, it should be remembered that drifts in excess of 2% are not likely to be readily accommodated in high rise frames. This is due to significant and detrimental influence of P-Δ phenomena on both lateral load resistance and dynamic response.19 Subassemblages A1 and E2, which developed plastic hinges in their beams (Fig. 3 and 7), showed stable hysteretic behavior up to drift Angle R ratios of 4.0. They showed a 472 Table 1—Comparison of joint of Subassemblage A1 design parameters with ACI 318-055 and ACI 352R-026 Subassemblage A1 γ ldh, cm Ash, cm 2 hbeam/ column bar sh, cm diameter 5.0 0.67 < 17 (15.65)* 0.95 (0.66)*† (5.0)*† (1.0)*† (17)† MR 30 1.72 (23.80)† (1.20)*† * Numbers inside parentheses are required values of ACI 318-05.5 †Numbers inside parentheses are required values of ACI 352R-02.6 Note: γ is shear strength factor reflecting confinement of joint by lateral members, ldh is development length of hooked beam bars, Ash is total cross-sectional area of transverse steel in joint, and sh is spacing of transverse reinforcement in joint. Numbers outside parentheses are provided values. 1 cm = 0.394 in. Table 2—Comparison of joints of Subassemblages E1 and E2 design parameters with Eurocode 84 and Eurocode 23 Ash, Subassemblage Vjh, kN cm2 Asv , cm2 dbl , mm MR lb,net, cm sw , cm E1 14 2.60 126 < 6.85 3.08 45 (43)† 5 (5)* (168)* (2.85)* (1.06)* (9.15)* (1.20)* E2 75.6 < 6.85 3.08 14 3.30 45 (32)† 5 (5)* (222)* (2.85)* (1.06)* (11.20)* (1.20)* *Numbers inside parentheses are required values of Eurocode 8.4 Numbers inside parentheses are required values of Eurocode 2.3 Note: Vjh is horizontal joint shear force, Ash is total cross-sectional area of transverse steel of joint, Asv is vertical joint shear reinforcement, dbl is diameter of hooked beam bars (in both E1 and E2 setup recommended by EC8 and shown in Fig. 5 was applied), lb,net is development length of hooked beam bars, and sw is spacing of transverse reinforcement of joint. Numbers outside parentheses are provided values. 1 m = 0.394in.; 1 mm = 0.039 in.; 1 kN = 0.225 kip. † considerable loss of strength, stiffness, and unstable hysteretic behavior, but beyond drift Angle R ratios of 4.5 (Fig. 7). Subassemblages E1 and G1, which exhibited premature joint shear failure (refer to Fig. 3 and 7) showed a considerable loss of strength, stiffness, and unstable degrading hysteresis beyond drift Angle R ratios of 2.5 and 2.0%, respectively (Fig. 7). CODE REQUIREMENTS Despite the fact that all the subassemblages were designed according to their corresponding modern codes, two developed failure modes dominated by joint shear failure (Fig. 3). For this reason, it is discussed how requirements of these codes used for the design of the joints of Subassemblages A1,5,6 E1, E23,4 (for DC”M” structures), and G17,8 were satisfied. Table 1 clearly indicates that the joint of A1 satisfied the design requirements of ACI 318-055 and ACI 352R-026 for exterior beam-column joints for seismic loading. Table 2 indicates that the joints of both E1 and E2 satisfied the design provisions for exterior beam-column joints of Eurocode 23 and Eurocode 84 for DC”M” structures. In both subassemblages, two 8 mm diameter short bars were placed and were tightly connected on the top bends of the beam reinforcing bars and two on the bottom, running in the transverse direction of the joint, as shown in Fig. 5. This is the setup recommended by Eurocode 8 when the requirement of limitation of beam bar diameter (dbl) to ensure appropriate anchorage through the joint is not satisfied (refer to Table 2). It was considered worthwhile, however, to determine the beam bar pull-out. Strain gauge measurements were used to determine beam bar pull-out. If the maximum strains in a beam’s longitudinal bar during each two consecutive cycles of loading remained the same or decreased, as long as buckling ACI Structural Journal/July-August 2007
- 25. Fig. 8—(a) Exterior beam-column joint; (b) internal forces around exterior beam-column joint as result of seismic actions;10,12 (c) two mechanisms of shear transfer (diagonal concrete strut and truss mechanism);10,12,19 and (d) forces acting in joint core concrete through Section I-I from two mechanisms.27,28 Table 3—Comparison of joint of Subassemblage G1 design parameters with ERC-19957 and CDCS-19958 Subassemblage G1 Ash, cm2 2.01 (2.01) lb,net, cm * 45 (43) * MR 2.60 (1.40)† *Numbers inside parentheses are required values of CDCS-1995.8 † Numbers inside parentheses are required values of ERC-1995.7 Note: Ash is total cross-sectional area of transverse steel of joint and lb,net is development length of hooked beam bars. Numbers outside parentheses are provided values. 1 cm = 0.394 in. of this bar had not taken place, it was concluded that a pullout of this bar had occurred.13,18 As shown in Fig. 5, the beam’s longitudinal reinforcement in Subassemblages E1 and E2 maintained adequate anchorage throughout the tests due to the short bars placed and tightly connected under the bends of a group of reinforcing bars (refer to Fig. 5). Table 3 also clearly indicates that the joint of G1 satisfied the design provisions for exterior beam-column joints of both the new Greek codes.7,8 The codes prescribe minimum MR values. So, as can be seen from Tables 1 through 3, the minimum value for the MR ratio according to ACI 318-05 and ACI 352R-02, as well as according to Eurocode 8 (DC”M”), is 1.20.4-6 The minimum value for the MR ratio according to the new Greek Earthquake Resistant Code is 1.40.7 Thus, a good target MR for most structures is between 1.20 and 1.40. Neither the New Greek Code for the Design of RC Structures8 nor the new Greek Earthquake Resistant Code7 require limitations for the joint shear stress. Of course both of these codes need to add requirements to limit joint shear stress. ACI Structural Journal/July-August 2007 THEORETICAL CONSIDERATIONS A new formulation published in recent studies20-26 predicts the beam-column joint ultimate shear strength and was used in the present study to predict the failure modes of Subassemblages A1, E1, E2, and G1. A summary of this formulation is presented. Figure 8(a) shows an RC exterior beam-column joint for a moment resisting frame and Fig. 8(b) shows the internal forces around this joint.10,12 The shear forces acting in the joint core are resisted partly by a diagonal compression strut that acts between diagonally opposite corners of the joint core (refer to Fig. 8(c)) and partly by a truss mechanism formed by horizontal and vertical reinforcement and concrete compression struts.10,12,19 The horizontal and vertical reinforcement is normally provided by horizontal hoops in the joint core around the longitudinal column bars and by longitudinal column bars between the corner bars in the side faces of the column.10,12,27 Both mechanisms depend on the core concrete strength. Thus, the ultimate concrete strength of the joint core under compression/tension controls the ultimate strength of the connection. After failure of the concrete, strength in the joint is limited by gradual crushing along the cross-diagonal cracks and especially along the potential failure planes (Fig. 8(a)). For instance, consider Section I-I in the middle of the joint height (Fig. 8(a)). In this section, the flexural moment is almost zero. The forces acting in the concrete are shown in Fig. 8(d).27,28 Each force acting in the joint core is analyzed into two components along the x and y axes (Fig. 8(d)). The values of Ti are the tension forces acting on the longitudinal column bars between the corner bars in the side faces of the column. Their resultant is ΣTi. An equal and opposing 473
- 26. compression force (–ΣTi) must act in the joint core to balance the vertical tensile forces generated in the reinforcement. This compression force was generated by the resultant of the vertical components of the truss mechanism’s diagonal compression forces D1, D2 …Dv.27 Thus, D1y + D2y + … + Dvy = ΣTi = T1 + T2 + T3 + T4.27 The column axial load is resisted by the compression strut mechanism.12 The summation of vertical forces equals the vertical joint shear force Vjv D cy + ( T 1 + ... + T 4 ) = D cy + D sy = V jv ↓ ↓ (2a) compression strut truss model fc = K × fc ′ Also, f ′c is the concrete compressive strength and K is a parameter of the model15 expressed as ρ s × f yh K = 1 + ---------------fc ′ D cy + D sy V jv σ = ---------------------- = ------------------hc ′ × bc ′ hc ′ × bc ′ (3) V jh τ = ------------------hc ′ × bc ′ (4) where h ′c and b ′c are the length and the width of the joint core, respectively. It is now necessary to establish a relationship between the average normal compressive stress σ and the average shear stress τ. From Eq. (3) and (4) V jv σ = ------ × τ V jh (5) It has been shown that V jv hb ------ = ---- = α V jh hc (6) where α is the joint aspect ratio.4,10,12 The principle (σI = maximum, σII = minimum) stresses are calculated 2 σ σ 4τ - σ I, II = -- ± -- 1 + ------2 2 2 σ αγ 4 ---------- ⎛ 1 + 1 + -----⎞ 2⎠ ⎝ 2 fc α (2b) The vertical normal compressive stress σ and the shear stress τ uniformly distributed over Section I-I are given by Eq. (3) and (4) (7) Equation (8)29 was adopted for the representation of the concrete biaxial strength curve30 by a fifth-degree parabola (9b) where ρs is the volume ratio of transverse reinforcement and fyh is its yield strength. Substituting Eq. (5) through (7) into Eq. (8) and using τ = γ f c gives the following expression The summation of horizontal forces equals the horizontal joint shear force Vjh D cx + ( D 1x + D 2x … + D vx ) = D cx + D sx = V jh (9a) 5 5αγ 4 + --------- ⎛ 1 + ----- – 1⎞ = 1 2 ⎝ ⎠ fc α (10) Assume herein that αγ x = ---------2 fc (11) 4 αγ ψ = ---------- 1 + ----2 2 fc α (12) and Then Eq. (10) is transformed into 5 ( x + ψ ) + 10ψ – 10x = 1 (13) The solution of the system of Eq. (11) to (13) gives the beam-column joint ultimate strength τult = γult f c (MPa). This system is solved each time for a given value of the joint aspect ratio using standard mathematical analysis. The joint ultimate strength τult depends on the increased joint concrete compressive strength due to confining fc and on the joint aspect ratio α. Thus, typical values of τult for comparison with the values of ACI 318-05,5 ACI 352R-02,6 and Eurocode 84 are not possible to derive. A particular value, however, for each joint would be calculated as in the following example. Example for Subassemblage A1 The value α = 1.5 and the solution of the system of Eq. (11) to (13) gives x = 0.1485 and y = 0.248; f ′c(A1) = 35 MPa, K(A1) = 1.558 according to the Scott et al.15 model and fc(A1) = K(A1) × f ′c(A1) = 54.53 MPa. Equation (11) gives 2 ( 0.1458 ) 54.53 γ ult ( A1 ) = ----------------------------------------- = 1.46 1.5 (8) and finally τult(A1) = 1.46 54.53 MPa = 10.78 MPa (refer to Table 4). where fc is the increased joint concrete compressive strength due to confinement by joint hoop reinforcement, which is given by the model of Scott et al.15 according to the equation COMPARISON OF PREDICTIONS AND EXPERIMENTAL RESULTS The proposed shear strength formulation can be used to predict the failure mode of the subassemblages and thus the σ σ II – 10 ----I + -----fc fc 474 5 = 1 ACI Structural Journal/July-August 2007
- 27. Table 4—Joint ultimate strength and ratios τpred /τexp and γcal /γult for Subassemblages A1, E1, E2, and G1 According to Park and Paulay10 According to proposed shear strength formulation Subassemblage τult , MPa τpred /τexp γcal /γult τult, MPa τpred /τexp γcal /γult A1 6.05 1.19 1.0 10.78 1.17 0.47 E1 8.94 1.31 1.0 6.92 1.19 1.08 E2 5.96 1.24 1.0 10.78 1.20 0.46 G1 8.34 1.28 1.0 6.60 1.19 1.04 Note: 1 MPa = 144.93 psi. actual values of connection shear stress. Therefore, when the calculated joint shear stress τcal is greater or equal to the joint ultimate strength (τcal = γcal f c ≥ τult = γult f c ), then the predicted actual value of connection shear stress will be near τult(τult = γult f c ). This is because the connection fails earlier than the adjacent beam(s). When the calculated joint shear stress τcal is lower than the connection ultimate strength (τcal = γcal f c < τult = γult f c ), then the predicted actual value of connection shear stress will be near τcal because the connection permits its adjacent beam(s) to yield. τult = γult f c is calculated from the solution of the system of Eq. (11) to (13). The value of τcal is calculated from the horizontal joint shear force assuming that the top reinforcement of the beam yields (Fig. 8(a)). In this case, the horizontal joint shear force is expressed as V jhcal = 1.25A s1 × f y – V col (14) where As1 is the top longitudinal beam reinforcement (Fig. 8(a)), fy is the yield stress of this reinforcement, and Vcol is the column shear force (Fig. 8(a)). For Type 2 joints, the design forces in the beam according to ACI 352R-026 should be determined using a stress value of α × fy for beam longitudinal reinforcement, where α = 1.25. The improved retention of strength in the beam-column subassemblages, as the values of the ratio τcal/τult = γcal/γult decrease was also demonstrated. For τcal/τult = γcal/γult ≤ 0.50, the beam-column joints of the subassemblages performed excellently during the tests and remained intact at the conclusion of the tests.20-26 The validity of the formulation was checked using test data from more than 120 exterior and interior beam-column subassemblages that were tested in the Structural Engineering Laboratory at the Aristotle University of Thessaloniki,20-26 as well as data from similar experiments carried out in the U.S., Japan, and New Zealand.1,12,13,31-36 A part of this verification is presented in Table 5 where the comparison is shown between experimental and predicted results by the preceding methodology for 39 exterior and interior beam-column joint subassemblages from the literature. A very good correlation is observed (Table 5). In Table 5, the limiting values of joint shear stress according to ACI 318-055 and ACI 352R-026 (1.0 f c ′ MPa for exterior beam-column joints and 1.25 f c ′ MPa for interior beam-column joints) are included for each reference subassemblage. In Table 5, the limiting values of joint shear stress according to Eurocode 84 (15τR MPa for exterior beam-column joints and 20τR MPa for interior beam-column joints) are also included. The shear capacities of the connections of Subassemblages A1, E1, E2, and G1 were also computed using the aforementioned methodology. One of the motivations behind this ACI Structural Journal/July-August 2007 study was the verification of the shear strength formulation presented herein for beam-column joints designed according to modern codes. The horizontal joint shear stresses are mainly produced by the longitudinal beam reinforcement as clearly described by Eq. (14). The longitudinal beam reinforcement of Subassemblages A1 and E2 was purposely chosen to produce low joint shear stresses during the tests, that is, a ratio τcal/τult = γcal/γult less than 0.5. Table 6 shows that γcal/γult is equal to 0.47 in Subassemblage A1 (that is, lower than 0.5) and γcal/γult is equal to 0.46 in Subassemblage E2 (that is, lower than 0.5). Thus, the formation of a plastic hinge in the beams near the columns is expected without any serious damage in the joint regions and, as a result, there will be satisfactory performance for both Subassemblages A1 and E2. As predicted, both subassemblages failed in flexure, exhibiting remarkable seismic performance (Fig. 3 and 7). Values τpred of A1 and E2, which are shown in Table 6, are equal to their τcal values (because γcal < γult) and are significantly different from their τult values, which are shown in Table 4. The percentage of longitudinal beam reinforcement of Subassemblages E1 and G1 was purposely chosen to be higher than that of Subassemblages A1 and E2 to produce higher joint shear stresses than those corresponding to their ultimate capacities. The joint region of E1, however, satisfied all the design requirements of Eurocode 23 and Eurocode 84 and the joint regions of G1 satisfied all the design requirements of the two Greek codes.7,8 Table 6 also shows that for both Subassemblages E1 and G1, the calculated joint shear stress τcal = γcal f c when the beams reach their ultimate strength is higher than the joint ultimate capacity τult = γult f c . Therefore, the joints of both these subassemblages will fail earlier than their beams according to the aforementioned methodology, because the joints of both E1 and G1 reach their ultimate shear strength during the tests before the beams reach their ultimate strength. Thus, according to the aforementioned methodology, a joint shear failure is expected for both Subassemblages E1 and G1 without any serious damage in their beams and, as a result, the performance of both subassemblages will not be satisfactory. As expected, both Subassemblages E1 and G1 demonstrated premature joint shear failure starting from the early stages of seismic loading and damage concentrated mostly in this region (Fig. 3). As also predicted, both Subassemblages E1 and G1 exhibited poor seismic performance, which was characterized by significant loss of strength, stiffness, and energy dissipation capacity during the tests. Furthermore, the volume ratios of joint transverse reinforcement for Subassemblages E1 and G1 were 0.025 and 0.017, respectively. Thus, the joint of Subassemblage E1 was more confined than the joint of Subassemblage G1, which explains why the hysteretic response of the former was better than that of the latter (Fig. 7). The concrete compressive strength significantly increases the joint ultimate strength τult. Thus, if the Subassemblages E1 and G1 had higher values with concrete compressive strengths, they would have behaved as well as Subassemblages A1 and E2. This would have happened for values with concrete compressive strength of approximately 50 MPa, which would have resulted in values of ratio γcal/γult lower than 0.5. The value of concrete 28-day compressive strengths of 22 MPa for both Subassemblages E1 and G1, however, is acceptable for Eurocode 2,3 Eurocode 8,4 and for both Greek codes.7,8 475
- 28. Table 5—Experimental verifications Joint Concrete Type of Longitudinal Joint aspect compressive subassem- ratio strength f ′ , τACI, τEC8, beam bar fy , hoop fy , Subc * α = hb/hc MPa MPa MPa MPa MPa γcal γexp Reference assemblage blage γult Observed Predicted shear strength τpred, shear strength μ = τpred / τexp τexp, MPa MPa No. 1 1.00 31.10 5.58 7.80 391 250 0.78 0.88 0.92 4.46 5.03 0.89 No. 2 E 1.00 41.70 6.46 9.45 391 250 0.68 0.74 1.06 4.50 4.90 0.92 No. 3 E 1.00 41.70 6.46 9.45 391 250 0.68 0.67 1.06 4.50 4.43 1.01 No. 4 E 1.00 44.70 6.69 9.90 391 281 0.66 0.67 1.08 4.43 4.50 0.99 No. 5 E 1.00 36.70 6.06 8.63 391 281 0.74 0.69 0.99 4.50 4.20 1.07 No. 6 E 1.00 40.40 6.35 9.30 391 281 0.70 0.69 1.03 4.47 4.41 1.01 No. 7 34 E E 1.00 32.20 5.67 7.95 391 250 0.77 0.82 0.93 4.47 4.76 0.94 No. 8 E 1.00 41.20 6.42 9.40 391 250 0.68 0.72 1.06 4.47 4.74 0.94 No. 9 E 1.00 40.60 6.37 9.30 391 250 0.69 0.67 1.05 4.51 4.40 1.03 No. 10 E 1.00 44.40 6.65 9.83 391 281 0.67 0.69 1.08 4.49 4.62 0.97 No. 11 E 1.00 41.90 6.47 9.48 391 281 0.69 0.70 1.05 4.49 4.55 0.99 No. 12 E 1.00 35.10 5.92 8.34 391 281 0.75 0.74 0.96 4.47 4.40 1.01 No. 13 E 1.00 46.40 6.81 10.16 391 250 0.64 0.64 1.12 4.47 4.47 1.00 No. 14 E 1.00 41.00 6.40 9.36 391 281 0.70 0.69 1.03 4.50 4.44 1.01 No. 15 E 1.00 30.70 5.54 7.74 391 281 0.71 0.74 1.02 3.95 4.12 0.96 No. 16 E 1.00 37.40 6.11 8.76 391 250 0.72 0.76 1.01 4.51 4.76 0.95 A1 I 1.14 40,20 7.93 12.33 1070 291 4.62 1.34 1.11 7.21 8.70 0.83 A2 33 I 1.14 40.20 7.93 12.33 409 291 1.76 1.23 1.11 7.21 7.99 0.90 A3 I 1.14 40.20 7.93 12.33 1070 291 4.62 1.34 1.11 7.21 8.70 0.83 A4 I 1.14 40.20 7.93 12.33 1070 291 4.48 1.33 1.14 7.62 8.88 0.86 1.14 30.00 5.48 7.65 1070 291 2.68 0.93 0.96 5.39 5.22 1.03 E 1.14 30.00 5.48 7.65 409 291 1.02 0,83 0.96 5.39 4.66 1.16 B3 E 1.14 30.00 5.48 7.65 1070 291 2.68 1.03 0.96 5.39 5.78 0.93 B4 E 1.14 30.00 5.48 7.65 1070 291 2.60 1.05 0.99 5.71 6.06 0.94 UNIT1 I 1.126 41.30 8.03 12.54 315 320 1.20 1.13 1.26 8.96 8.44 1.06 UNIT2 I 1.126 46.90 8.56 13.65 307 320 1.31 1.08 1.33 10.23 8.43 1.20 UNIT3 E† 1.126 38.20 6.18 8.70 473 321 1.17 0.90 1.09 7.06 5.85 1.21 UNIT4 E 1.126 38.90 6.23 8.55 473 321 2.32 0.90 1.11 7.29 5.91 1.23 SHC1 36 E B2 12 B1 I 1.14 56.50 9.39 15.9 413 551 1.00 0.91 1.31 7.81 7.11 1.10 SHC2 I 1.14 59.50 9.64 16.5 413 551 0.97 0.91 1.36 7.90 7.41 1.07 SOC3 I 1.14 47.10 8.58 13.71 413 551 1.06 1.00 1.22 7.70 7.26 1.06 SP1 30.70 5.54 7.74 347 0 0.90 0.78 1.03 4.99 4.32 1.15 E‡ 1.33 31.10 5.58 7.80 349 0 0.90 0.77 1.04 5.02 4.30 1.17 SP3 E § 1.33 27.00 5.20 7.11 350 427 0.94 0.83 1.00 5.17 4.56 1.13 SP4 E§ 1.33 31.00 5.57 7.79 349 379 0.86 0.87 1.09 5.13 5.19 0.90 SP5 E‡§ 1.33 32.00 5.66 7.92 347 0 0.88 0.75 1.05 4.97 4.24 1.17 SP6 E 1.33 36.20 6.02 8.55 352 357 0.78 0.78 1.20 5.16 5.16 1.00 SP7 E 1.33 30.70 5.54 7.74 352 365 0.87 0.83 1.08 5.16 4.93 1.05 SP8 Total 1.33 SP2 35 E ‡§ E 1.33 26.30 5.13 7.00 352 365 1.19 1.02 1.11 6.44 5.92 1.09 39 Average 1.02 COV 0.10 * I equals interior beam-column subassemblage; E equals exterior beam-column subassemblage. Beam bars of UNIT3 were anchored in beam stub at far face of column. joints. §Subassemblages with one transverse beam for γ cal < γult, γpred = γcal, τpred = τcal and for γcal ≥ γult, γpred = γult, τpred = τult. Notes: τACI is the limiting values of joint stress according to ACI 318-055 and ACI 352R-02;6 τEC8 is the limiting values of joint shear stress according to Eurocode 8.4 Neither relevant Greek codes7,8 provide information regarding limiting values for joint shear stress. All subassemblages have flexural strength ratios MR higher than 1.0. Overstrength factor a = 1.25 for † ‡Unreinforced beam steel is included in computations of joint shear stress τcal = γcal f c MPa. 1 MPa = 144.93 psi; 1.0 f c MPa = 12.05 f c psi. 476 ACI Structural Journal/July-August 2007
- 29. A question arises regarding how concrete slabs, which are typical in buildings, affect the performance of the joints of subassemblages such as A1, E1, E2, and G1. Ehsani and Wight31 found that “the flexural strength ratio MR at the connections is reduced significantly due to the contribution of the slab longitudinal reinforcement.” They recommended that, to ensure flexural hinging in the beam, flexural strength ratios should be no less than 1.20.31 The flexural strength ratios of all the Subassemblages A1, E1, E2, and G1 tested in this study were significantly higher than 1.20 (refer to Fig. 1(a) and (b)); thus, the presence of a concrete slab would not have had any influence on the response of these subassemblages. It would be of interest to learn whether simpler procedures for arriving to the beam-column joint ultimate strength such as that proposed by Park and Paulay,10 would lead to similar findings as those derived from the solution of the system of Eq. (11) to (13). To this end, Table 4 presents the joint ultimate strength and ratios, τpred/τexp and γcal/γult for Subassemblages A1, E1, E2, and G1 according to the aforementioned procedures. The ultimate joint shear strengths of Subassemblages A1, E1, E2, and G1 derived from the solution of the system of Eq. (11) to (13) depend on the increased joint concrete compressive strength due to confining fc, as well as on the joint aspect ratio α. These values differ significantly from those of Park and Paulay,10 which mainly depend on the percentage of top longitudinal beam reinforcement. Thus, Table 4 shows that the values of ultimate joint shear strengths of Subassemblages A1 and E2 derived from the solution of the system of Eq. (11) to (13) are higher than those of Subassemblages E1 and G1 derived by the same methodology. This clearly explains why the Park and Paulay10 values of ultimate joint shear strength in Table 4 are larger than the values from Eq. (11) to (13) for E1 and G1 and less than the values from Eq. (11) to (13) for A1 and E2. Finally, as can be seen from Table 4, the proposed shear strength formulation predicted the failure mode for Subassemblages A1, E1, E2, and G1 with significant accuracy, while the Park and Paulay10 procedure predicted only the failure mode of Subassemblages A1 and E2. CONCLUSIONS Based on the test results described in this paper, the following conclusions can be drawn. 1. The behavior of Subassemblages A1 and E2 was as expected and as documented in the seismic design philosophy of ACI 318-05,5 ACI 352R-02,6 and Eurocode 8.4 The beamcolumn joints of both Subassemblages A1 and E2 performed satisfactorily during the cyclic loading sequence to failure, allowing the formation of plastic hinges in their adjacent beams. Both subassemblages showed high strength without any appreciable deterioration after reaching their maximum capacity; 2. Despite the fact that Subassemblages E1 and G1 represented beam-column subassemblages of contemporary structures, they performed poorly under reversed cyclic lateral deformations. The joints of both Subassemblages E1 and G1, contrary to expectations based on Eurocode 2,3 Eurocode 8,4 and the two Greek codes7,8 exhibited shear failure during the early stages of cyclic loading. This happened because, for both Subassemblages E1 and G1, the calculated joint shear stress τcal was higher than the joint ultimate strength τult (Table 6). Damage occurred both in the joint area and in the columns’ critical regions. This effect cannot be underestimated as it may lead to premature lateral instability in ductile momentresisting frames of modern structures; and ACI Structural Journal/July-August 2007 Table 6—Experimental and predicted values of strength of Subassemblages A1, E1, E2, and G1 Joint aspect ratio Subassem- α = blage hb/hc K γcal Predicted Observed μ= shear shear strength strength τpred/ γcal/ γexp γult τpred, MPa τexp, MPa τexp γult A1 1.50 1.558 0.685 0.584 1.46 5.05 E1 1.50 1.593 1.26 0.98 1.17 6.92 5.80 1.19 1.08 E2 1.50 1.558 0.675 0.554 1.46 5.00 4.10 1.20 0.46 G1 1.50 6.60 5.56 1.19 1.04 1.50 1.20 0.96 1.15 4.31 1.17 0.47 Notes: For γcal < γult, γpred = γcal, τpred = τcal and for γcal ≥ γult, γpred = γult, τpred = τult. 1 MPa = 144.93 psi; 1.0 f c MPa = 12.05 f c psi. Overstrength factor a = 1.25 for beam steel is included incomputations of joint shear stress τcal = γcal f c MPa. 3. It was demonstrated that the design assumptions of Eurocode 2,3 Eurocode 8,4 and those in the Greek codes7,8 did not avoid premature joint shear failures because the resulting design can not ensure that the joint shear stress will be significantly lower than the joint ultimate strength τult and did not ensure the development of the optimal failure mechanism with plastic hinges occurring in the beams while columns remained elastic, according to the requisite strong column-weak beam. Thus, provisions in Eurocode 23 and Eurocode 84 and those in the two Greek codes7,8 related to the design of beam-column joints need improvement. ∅ a b ′c f ′c hb h ′c hc MR N Vjh Vjv α γcal γexp γult τ NOTATION = = = = = = = = = = = = = = = bar diameter overstrength factor width of joint core compressive strength of concrete total depth of beam length of joint core total depth or width of square column sum of flexural capacity of columns to that of beam applied column axial load during test horizontal joint shear force vertical joint shear force hb/hc design values of parameter [γcal = (τcal / f c )] actual values of parameter [γexp = (τexp/ f c )] values of parameter γ at ultimate capacity of connection [γult = (τult/ f c )] = joint shear stress REFERENCES 1. Leon, R. T., “Shear Strength and Hysteretic Behavior of Interior BeamColumn Joints,” ACI Structural Journal, V. 87, No. 1, Jan.-Feb. 1990, pp. 3-11. 2. Penelis, G. G., and Kappos, A. J., Earthquake-Resistant Concrete Structures, E&FN Spon, London, 1997, 572 pp. 3. CEN Technical Committee 250/SC2, “Eurocode 2: Design of Concrete Structures—Part 1: General Rules and Rules for Buildings (ENV 1992-1-1),” CEN, Berlin, Germany, 1991, 61 pp. 4. CEN Technical Committee 250/SC8, “Eurocode 8: Earthquake Resistant Design of Structures—Part 1: General Rules and Rules for Buildings (ENV 1998-1-1/2/3),” CEN, Berlin, Germany, 1995, 192 pp. 5. ACI Committee 318, “Building Code Requirements for Structural Concrete (ACI 318-05) and Commentary (318R-05),” American Concrete Institute, Farmington Hills, Mich., 2005, 430 pp. 6. Joint ACI-ASCE Committee 352, “Recommendations for Design of Beam-Column Connections in Monolithic Reinforced Concrete Structures (ACI 352R-02),” American Concrete Institute, Farmington Hills, Mich., 2002, 37 pp. 7. “New Greek Earthquake Resistant Code (ERC-1995),” Athens, Greece, 1995, 145 pp. (in Greek) 8. “New Greek Code for the Design of Reinforced Concrete Structures (CDCS-1995),” Athens, Greece, 1995, 167 pp. (in Greek) 9. Hakuto, S.; Park, R.; and Tanaka, H., “Seismic Load Tests on Interior and Exterior Beam-Column Joints with Substandard Reinforcing Details,” ACI Structural Journal, V. 97, No. 1, Jan.-Feb. 2000, pp. 11-25. 477
- 30. 10. Park, R., and Paulay, T., Reinforced Concrete Structures, John Wiley Publications, New York, 1975, 769 pp. 11. Park, R., “A Summary of Results of Simulated Seismic Load Tests on Reinforced Concrete Beam-Column Joints, Beams and Columns with Substandard Reinforcing Details,” Journal of Earthquake Engineering, V. 6, No. 2, 2000, pp. 147-174. 12. Paulay, T., and Park, R., “Joints of Reinforced Concrete Frames Designed for Earthquake Resistance,” Research Report 84-9, Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand, 1984, 71 pp. 13. Ehsani, M. R., and Wight, J. K., “Exterior Reinforced Concrete Beam-to-Column Connections Subjected to Earthquake-Type Loading,” ACI JOURNAL, Proceedings V. 82, No. 4, July-Aug. 1985, pp. 492-499. 14. Soroushian, P., and Sim., J., “Axial Behavior of Reinforced Concrete Columns under Dynamic Loads,” ACI JOURNAL, Proceedings V. 83, No. 6, Nov.-Dec. 1986, pp. 1018-1025. 15. Scott, B. D.; Park, R.; and Priestley, M. J. N., “Stress-Strain Behavior of Concrete Confined by Overlapping Hoops at Low and High Strain Rates,” ACI JOURNAL, Proceedings V. 79, No. 1, Jan.-Feb. 1982, pp. 13-27. 16. CEB-FIP, “Model Code 1990,” Bulletin d’ Information, CEB, Lausanne, Switzerland, 1993, 490 pp. 17. Mitchel, D., “Controversial Issues in the Seismic Design of Connections in Reinforced Concrete Frames,” Recent Developments in Lateral Force Transfer in Buildings, SP-157, N. Priestley, M. P. Collins, and F. Seible, eds., American Concrete Institute, Farmington Hills, Mich., 1995, pp. 75-96. 18. Ehsani, M. R.; Moussa, A. E.; and Vallenilla, C. R., “Comparison of Inelastic Behavior of Reinforced Ordinary- and High-Strength Concrete Frames,” ACI Structural Journal, V. 84, No. 2, Mar.-Apr. 1987, pp. 161-169. 19. Paulay, T., “Seismic Behavior of Beam-Column Joints in Reinforced Concrete Space Frames, State-of-the Art Report,” Proceeding of the Ninth World Conference on Earthquake Engineering, V. VIII, Tokyo, Japan, 1988, pp. 557-568. 20. Tsonos, A. G., “Towards a New Approach in the Design of R/C Beam-Column Joints,” Technika Chronika, Scientific Journal of the Technical Chamber of Greece, V. 16, No. 1-2, 1996, pp. 69-82. 21. Tsonos, A. G., “Shear Strength of Ductile Reinforced Concrete Beam-to-Column Connections for Seismic Resistant Structures,” Journal of European Association for Earthquake Engineering, No. 2, 1997, pp. 54-64. 22. Tsonos, A. G., “Lateral Load Response of Strengthened Reinforced Concrete Beam-to-Column Joints,” ACI Structural Journal, V. 96, No. 1, Jan.-Feb. 1999, pp. 46-56. 23. Tsonos, A. G., “Seismic Retrofit of R/C Beam-to-Column Joints using Local Three-Sided Jackets,” Journal of European Earthquake 478 Engineering, No. 1, 2001, pp. 48-64. 24. Tsonos, A. G., “Seismic Rehabilitation of Reinforced Concrete Joints by the Removal and Replacement Technique,” Journal of European Earthquake Engineering, No. 3, 2001, pp. 29-43. 25. Tsonos, A. G., “Seismic Repair of Exterior R/C Beam-to-Column Joints using Two-Sided and Three-Sided Jackets,” Structural Engineering and Mechanics, V. 13, No. 1, 2002, pp. 17-34. 26. Tsonos, A. G., “Effectiveness of CFRP-Jackets and RC-Jackets in PostEarthquake and Pre-Earthquake Retrofitting of Beam-Column Subassemblages,” Final Report, Grant No. 100/11-10-2000, Earthquake Planning and Protection Organization (E.P.P.O.), Sept. 2003, 167 pp. (in Greek). 27. Paulay, T., “Equilibrium Criteria for Reinforced Concrete Beam-Column Joints,” ACI Structural Journal, V. 86, No. 6, Nov.-Dec. 1989, pp. 635-643. 28. Park, R., “The Paulay Years,” Recent Developments in Lateral Force Transfer in Buildings, SP-157, N. Priestley, M. P. Collins, and F. Seible, eds., American Concrete Institute, Farmington Hills, Mich., 1995, pp. 1-30. 29. Tegos, I. A., “Contribution to the Study and Improvement of Earthquake-Resistant Mechanical Properties of Low Slenderness Structural Elements,” PhD thesis, Appendix 13, V. 8, Aristotle University of Thessaloniki, 1984, pp. 185. (in Greek) 30. Kupfer, H.; Hilsdorf, H. K.; and Rusch, H., “Behavior of Concrete under Biaxial Stresses,” ACI JOURNAL, Proceedings V. 66, No. 8, Aug. 1969, pp. 656-667. 31. Ehsani, M. R., and Wight, J. K., “Effect of Transverse Beams and Slab on Behavior of Reinforced Concrete Beam-to-Column Connections,” ACI JOURNAL, Proceedings V. 82, No. 2, Mar.-Apr. 1985, pp. 188-195. 32. Durrani, A. J., and Wight, J. K., “Behavior of Interior Beam-to-Column Connections under Earthquake-Type Loading,” ACI JOURNAL, Proceedings V. 82, No. 3, May-June 1985, pp. 343-349. 33. Fujii, S., and Morita, S., “Comparison Between Interior and Exterior RC Beam-Column Joint Behavior,” Design of Beam-Column Joints for Seismic Resistance, SP-123, J. O. Jirsa, ed., American Concrete Institute, Farmington Hills, Mich., 1991, pp. 145-166. 34. Kaku, T., and Asakusa, H., “Ductility Estimation of Exterior BeamColumn Subassemblages in Reinforced Concrete Frames,” Design of Beam-Column Joints for Seismic Resistance, SP-123, J. O. Jirsa, ed., American Concrete Institute, Farmington Hills, Mich., 1991, pp. 167-185. 35. Uzumeri, S. M., “Strength and Ductility of Cast-in-Place Beam-Column Joints,” Reinforced Concrete Structures in Seismic Zones, SP-53, American Concrete Institute, Farmington Hills, Mich., 1977, pp. 293-350. 36. Attaalla, S. A., and Agbabian, M. S., “Performance of Interior BeamColumn Joints Cast from High Strength Concrete Under Seismic Loads,” Journal of Advances in Structural Engineering, V. 7, No. 2, 2004, pp. 147-157. ACI Structural Journal/July-August 2007
- 31. ACI STRUCTURAL JOURNAL TECHNICAL PAPER Title no. 104-S44 Eccentric Reinforced Concrete Beam-Column Connections Subjected to Cyclic Loading in Principal Directions by Hung-Jen Lee and Jen-Wen Ko Cyclic loading responses of five reinforced concrete corner beamcolumn connections with one concentric or eccentric beam framing into a rectangular column in the strong or weak direction are reported. The specimen variables are the shear direction and the eccentricity between the beam and column centerlines. Experimental results showed that two joints connecting a beam in the strong direction were capable of supporting adjacent beam plastic mechanisms. The other three joints connecting a beam in the weak direction, however, exhibited significant damage and loss of strength after beam flexural yielding. Eccentricity between beam and column centerlines had detrimental effects on the strength, energy dissipation capacity, and displacement ductility of the specimens. Experimental verification shows that the current ACI design procedures are acceptable for seismic design purposes; however, it could not prevent the failure of corner connections at large drift levels. Keywords: beam-column connections; joints; shear strength. INTRODUCTION Shear failure in beam-column connections, leading to the collapse of reinforced concrete (RC) buildings, has been observed in the post-earthquake reconnaissance.1-3 The cause of collapse has been attributed to the lack of joint confinement, especially for the exterior and corner beamcolumn connections without beams framing into all four sides. Since the late 1960s, amounts of experimental investigations on the seismic performance of RC beam-column connections have been extensively studied. The majority of the experimental programs have concentric beam-column connections isolated from a lateral-force-resisting frame at the nearest inflection points in the beams and columns framing into the joint. Since 1976, Joint ACI-ASCE Committee 352 has issued design recommendations for RC beam-column joints.4,5 Throughout the years, these guidelines evolved into state-of-the-art reports6,7 by integrating results of new experimental programs. Finally, a number of these design recommendations for beam-column connections have been adopted in Chapter 21 of the ACI 318 Building Code8 for seismic design. Current ACI design provisions are primarily developed from test results of concentric beam-column connections, whereas eccentric beam-column connections are rather common in practice. Relatively few tests of eccentric RC beam-column connections have been reported in the literature to date.9-19 To clarify the effect of eccentric beams on the behavior of connections, Joint ACI-ASCE Committee 352 has called for additional research on this topic over the past two decades,5-7 and appointed a task group to review and summarize previous research on eccentric RC beamcolumn connections.20 In the early 1990s, Joh et al.,9 Lawrance et al.,10 and Raffaelle and Wight11 tested six cruciform eccentric beamACI Structural Journal/July-August 2007 column connections with square columns. Early deterioration of strength and ductility was observed in these eccentric connections. The measured strains in joint hoop reinforcement and the joint shear deformations on the side near the beam centerline were larger than those on the side away from the beam centerline. Raffaelle and Wight11 suggested a formula for reducing the effective joint width for shear resistance of eccentric connections, and indicated that further study of eccentric beam-column connections with rectangular columns is needed. Chen and Chen12 first tested five T-shaped eccentric corner beam-column connections in the late 1990s, while Vollum and Newman13 also tested 10 corner connections with two beams (one concentric and one eccentric) framing in from two perpendicular directions. Chen and Chen12 concluded that the performance of eccentric corner connections was inferior to that of concentric corner connections, and tapered width beams could eliminate the detrimental effect of eccentric beams. On the other hand, Vollum and Newman13 tested specimens with combined loading in various load paths to investigate the behavior of eccentric beam-column connections and to verify a previously proposed design method. The researchers concluded that the performance of corner connections improved significantly when reducing joint eccentricity. Notably, the aforementioned corner connections had square columns. In the early 2000s, Teng and Zhou14 also tested four cruciform eccentric beam-column connections with rectangular columns in aspect ratios of 2 and 1.33, and concluded that joint eccentricity slightly reduced the strength and stiffness of the connections. Based on their analysis, Teng and Zhou14 also proposed an empirical equation for calculating the nominal shear strength of eccentric joints by reducing the effective joint width. Because floor slabs were typically not included in previous tests of eccentric connections, Burak and Wight15 as well as Shin and LaFave16 tested five eccentric beamcolumn-slab connections. Each subassembly consisted of eccentric edge beams, one concentric transverse beam, floor slabs, and rectangular columns with aspect ratios varied from 1.0 to 1.5. Burak and Wight’s15 three specimens were tested under sequential loading in two principal directions in which lateral loading was first applied in the edge beam direction and then in the transverse beam direction. Shin and LaFave’s16 two specimens were tested under lateral loading in the edge beam direction to simulate the behavior of an ACI Structural Journal, V. 104, No. 4, July-August 2007. MS No. S-2006-226 received June 2, 2006, and reviewed under Institute publication policies. Copyright © 2007, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published in the May-June 2008 ACI Structural Journal if the discussion is received by January 1, 2008. 459
- 32. ACI member Hung-Jen Lee is an Assistant Professor in the Department of Construction Engineering and a Research Engineer of the Service Center for Construction Technology and Materials in the National Yunlin University of Science and Technology, Yunlin, Taiwan. He received his PhD from the National Taiwan University of Science and Technology, Taipei, Taiwan, in 2000. His research interests include seismic design of reinforced concrete structures, behavior of beam-column connections, reinforcement detailing, and strut-and-tie models. Jen-Wen Ko is a PhD Student in the Department of Construction Engineering at the National Taiwan University of Science and Technology. He received his MS from the National Yunlin University of Science and Technology in 2005. Fig. 1—Illustration of test specimens. edge connection in an exterior moment-resisting frame. The researchers15,16 reported that the damage in the joint region of these eccentric beam-column-slab connections was not as severe as that of previous tests without floor slabs.9,11 Including floor slabs significantly improves the overall performance of eccentric connections and delays the deterioration of joint stiffness and strength. LaFave et al.20 pointed out that including floor slabs in cruciform eccentric connections would not only raise the joint shear demand but would also reduce the effect of joint eccentricity and enhance the joint shear-resisting mechanisms. Test and analytical results of another nine cruciform eccentric beam-column connections were presented in the 13th World Conference on Earthquake Engineering.17-19 Based on experimental results and finite element analysis of three cruciform eccentric connections, Goto and Joh17 concluded that the joint shear strength decreases as the joint eccentricity increases due to the stress concentration on the eccentric side. Similar observations were also concluded by Kusuhara et al.18 who tested two cruciform eccentric connections (one with additional U-shaped reinforcement in the eccentric side). Finally, Kamimura et al.19 tested four cruciform eccentric connections (three deep beam-wide column connections) and proposed an equation combining shear and torsion to evaluate the joint shear strength. Beam-column joints in RC buildings are probably subjected to lateral loading in two principal directions during an earthquake. Nevertheless, current ACI design procedures7,8 require that the joint shear strength be evaluated in each direction independently and implicitly assume an elliptical interaction relationship for biaxial loading. Notably, only one value of permissible shear stress is selected for a joint according to the effective confinement on the vertical faces of the joint, even though the column cross section is rectangular. Current ACI design procedures consider the effects of the column’s aspect ratio and eccentric beam on joint shear 460 strength by limiting and reducing the effective joint width. More experimental results are needed to verify the effective joint width in eccentric connections.5-7 Thus, this experimental program focuses on the behavior of eccentric corner connections with rectangular columns because they have not been experimentally verified. RESEARCH SIGNIFICANCE Current ACI design provisions for estimating joint shear strength of eccentric beam-column connections are established based on few experimental investigations. The effects of a column’s aspect ratio and eccentric beam on joint shear strength are evaluated by the effective joint width. Additional experimental verification of the design provisions for eccentric connections is needed, especially for eccentric corner connections with rectangular columns. This paper presents experimental results for five corner connections with one concentric or eccentric beam framing into a rectangular joint in the strong or weak direction. Experimental verifications on the ACI approach provided contribution to the understanding of beam-column connections. EXPERIMENTAL PROGRAM Five RC corner beam-column connections were designed, constructed, and tested under reversed cyclic loading. A T-shaped assembly was used to represent the essential components of a corner beam-column connection in a two-way building frame subjected to lateral loading in each principal direction. The primary test variables were the lateral loading directions and the eccentricity between the beam and column centerlines. Neither transverse beams nor floor slabs were constructed to ease testing. As a result, each subassembly had only one beam framing into one corner column in each principal direction. For a corner, interstory connections, floor slabs, and transverse beams could not only introduce additional demand on joint shear force but also reduce the effect of joint eccentricity. The enhancement on the joint shear capacity from confinement of floor slabs and transverse beams is questionable because a corner joint is only confined on two adjacent faces and it is likely to sustain biaxial loading. Further study on the behavior of corner beam-column-slab connections subjected to biaxial loading is needed. Specimen geometry and reinforcement The experimental program was designed using a concrete compressive strength f ′c of 30 MPa (4.35 ksi) and a reinforcement yield stress fy of 420 MPa (60.9 ksi). Cross sections and reinforcement details of the five specimens, designated as S0, S50 (Series S), W0, W75, and W150 (Series W), are shown in Fig. 1. The first character (S or W) of the designation represents one south or west beam framing into the rectangular column in the strong or weak direction. The subsequent numerals denote the eccentricity between the beam and column centerlines in mm. Thus, two concentric (S0 and W0) and three eccentric (S50, W75, W150) connections were tested in total. The corner column had a cross section of 400 x 600 mm (16 x 24 in.) and used 12 D22 (No. 7) longitudinal bars (gross reinforcement ratio of 1.9%) and D10 (No. 3) hoops with crossties at a spacing of 100 mm (4 in.) throughout the column. The total cross-sectional area of the lateral reinforcement for each direction of the column was approximately equal to the minimum amount required by ACI 318-05, Section ACI Structural Journal/July-August 2007
- 33. Table 1—Connection design parameters Specimen S0 S50 W0 W75 W150 Column width bc, mm (in.) 400 (16) 600 (24) Column depth hc, mm (in.) 600 (24) 400 (16) 5.10 3.46 24 15 Moment strength ratio Mr * Provided embedment length db† Joint shear demand Vu, kN (kips) 699 (157.1) Joint eccentricity e, mm (in.) Effective joint width bj318, 706 (158.7) 0 (0) 50 (2) 0 (0) 75 (3) 150 (6) 400 mm (in.) (16) 300 (12) 600 (24) 450 (18) 300 (12) 0.53 0.71 0.54 0.72 1.07 Vu -------------------------318 γ f c ′b j h c Effective joint width b352, mm (in.) j 350 (14) 450 (18) 360 (14.4) Vu -------------------------352 γ f c ′b j h c 0.61 0.72 0.90 *M r † = ΣMn(columns)/ΣMn(beams). Embedment lengths required by ACI 318-05 and ACI 352R-02 are 14.2db and 16.8db, respectively. Note: All values are computed with fc′ = 30 MPa (4.35 ksi) and fy = 420 MPa (60.9 ksi). 21.4.4.1.8 To control the demand of shear force acting on the joint, the loading beam had a cross section of 300 x 450 mm (12 x 18 in.) and used four D22 (No. 7) longitudinal bars (steel ratio of 1.29%) at both top and bottom. To avoid beam shear failure and ensure adequate confinement in the beam plastic hinge region, closed overlapping hoops were provided through the length of the beam. Figure 2 illustrates the overall geometry of the specimens. The lengths of the beam and column that were chosen to simulate the nearest inflection points in the beam and column framing into the joint. In general, the five specimens were nominally identical except for the joint shear direction, the embedment lengths of the hooked beam bars, and the eccentricity between the beam and column centerlines. Connection design parameters Table 1 shows the main design parameters for the specimens. Due to column bending in the strong or weak direction, the ratios of column-to-beam flexural strength Mr at the connections of Series S and W were equal to 5.10 and 3.46, respectively. Because both Mr values were much greater than the specified value of 1.2, flexural hinging in the beam was anticipated. To ensure the anchorage of beam longitudinal bars and to promote the development of a diagonal compression strut within the joint, the beam longitudinal bars were anchored using a 90-degree standard hook bent into the joint and embedded as close as possible to the back of the column (Fig. 2). Leaving a 70 mm (2.8 in.) back cover behind the hook, the provided embedment lengths within the joint were 24db for Series S and 15db for Series W. The required development lengths of hooked beam bars, measured from the critical section, are given in ACI 318-05, Section 21.5.4.1, and ACI 352R-02, Section 4.5.2.4, for Type 2 connections. Per ACI 318-05,8 the critical section is taken at the beam-column interface. Per ACI 352R-02,7 for Type 2 connections, it is taken at the outside edge of the column core. For fy of 420 MPa (60.9 ksi) and f ′c of 30 MPa (4.35 ksi), the embedment lengths required by ACI 318-05 and ACI 352R-02 are 14.2db and 16.8db, respectively. As shown in Table 1, the provided embedment length within the joints in Series W is ACI Structural Journal/July-August 2007 Fig. 2—Overall geometry of test specimens. 105% of that required by ACI 318-05 but only 89% of that required by ACI 352R-02. Based on the capacity design concept, the demand of the joint shear force Vu is dominated by the flexural capacity of the beam. When computing Vu values, a probable strength of 1.25fy for the beam longitudinal reinforcement was included. Due to small differences in beam lengths, the value of Vu is equal to 699 kN (157.1 kips) for the specimens in Series S and 706 kN (158.7 kips) for the specimens in Series W, respectively. The current ACI design procedures for joint shear strength are based on Eq. (1) φV n = φγ f c ′b j h c ≥ V u (1) where φ is the strength reduction factor of 0.85; Vn is the nominal joint shear strength; γ f c′ is the nominal joint shear stress of 1.0 f c′ MPa (12 f c′ psi) for corner, interstory connections; hc is the column depth (mm or in.) in the direction of joint shear to be considered; and bj is the effective joint width (mm or in.) calculated using the following equations 8 ACI 318-05 : 318 bj ⎧ b b + 2x ⎪ = the smaller of ⎨ b b + h c ⎪ ⎩ bc (2) 461
- 34. sheets and wet-cured for 1 week. For each batch of concrete, 12 150 x 300 mm (6 x 12 in.) concrete cylinders were cast and cured together with the beam-column assemblies. Three cylinders were tested at 28 days and the rest were tested at the testing date of each beam-column assembly. Table 2 summarizes the concrete compressive strengths at 28 days and the testing date. The average of concrete compressive strengths at the testing date are used for analytical f ′c in this paper, because the variation of concrete compressive strengths within each batch of concrete is small. Fig. 3—Test setup for Series W (similar setup for Series S). Table 2—Concrete compressive strengths Specimen S0 Concrete batch S50 W0 1 28-day f ′c, MPa (psi) W75 W150 2 28.5 (4133) 25.2 (3655) Test days 49 67 53 57 60 Test day f ′c, MPa (psi) 32.6 (4728) 34.2 (4960) 28.9 (4191) 30.4 (4409) 29.1 (4220) Analytical f ′c, MPa (psi) 33.2 (4815) 352 ACI 352R-027: b j 29.5 (4278) ⎧ (b + b ) ⁄ 2 c ⎪ b ⎪ mh = the smaller of ⎨ b + Σ --------c (3) 2 ⎪ b ⎪ ⎩ bc where bb is the beam width (mm or in.); x is the smaller distance between the beam and column edges (mm or in.); bc is the column width (mm or in.); and m is 0.3 when e is greater than bc/8, otherwise m is 0.5. The summation term is applied on each side of the joint where the column edge extends beyond the beam edge. The joint eccentricity e was designed to be bc/8 for Specimen S50 and W75, and to be bc/4 for Specimen W150. As shown in Table 1, only Specimen W150 had a target joint shear stress exceeding the nominal value of 1.0 f c′ MPa (12 f c′ psi) for the effective joint width per ACI 318-05.8 The other four specimens satisfied the requirement on the joint shear stress when following ACI design procedures with a strength reduction factor of 0.85. Construction and material properties Two sizes of standard reinforcement meeting ASTM A 706 were used for longitudinal and transverse reinforcement in all specimens. The D22 (No. 7) longitudinal reinforcement had an average yield stress of 455 MPa (66 ksi) and an average ultimate strength of 682 MPa (99 ksi). The average yield and ultimate strengths were 471 and 715 MPa (68 and 104 ksi) for D10 (No. 3) transverse reinforcement, respectively. Each specimen was cast in a wood form with the beam and column lying on the ground and the exterior column side (east side for Series S and north side for Series W) facing up. Concrete was supplied by a local ready mix plant using normal concrete aggregate and delivered by pump using a 125 mm (5 in.) diameter hose. Series S was cast at one time using a single batch of concrete, and then Series W was cast using another batch of concrete with the same mixture proportions. The fresh concrete was covered with plastic 462 Test setup and loading sequence Figure 3 shows the elevation views of the test setup. To restrain the column for twisting about the column axis, each beam-column assembly was rotated 90 degrees and tied down to a strong floor with reaction steel beams, cover plates, and rods. In addition, four one-dimensional rollers were seated beside the column to allow in-plane rotation at both ends of the column. This arrangement was chosen to provide stability against torsional action. The actuator load was applied at the beam centerline while the column axial load was applied along the column longitudinal axis. Thus, a twist of the column about its longitudinal axis was applied for the eccentric connections. To simulate the displacement reversal of beam-column connections during earthquake events, the specimens were subjected to reversed cyclic lateral displacements. Axial load was applied at the beginning of a test and held at a level of 0.10Ag fc′ during testing. A typical lateral displacement history consisting of three cycles at monotonically increasing drift levels (0.25, 0.50, 0.75, 1.0, 1.5, 2, 3, 4, 5, 6, and 7%) was used for all specimens. The actuator applied each target displacement in a quasi-static manner at a speed ranging from 0.05 to 1.40 mm/s (0.002 to 0.056 in./s). Target displacement amplitudes at the beam tip Δ were computed using the following equation Δ Drift ratio θ = -----------------------L b + 0.5h c (4) where drift ratio θ is the angular rotation of the beam chord with respect to the column chord; Lb + 0.5hc is the vertical distance between the actuator and column centerlines, and it is equal to 2.15 m (86 in.) for Series W and 2.075 m (83 in.) for Series S (Fig. 2). EXPERIMENTAL RESULTS Experimental results showed that two joints of Series S were capable of supporting the complete formation of a beam plastic hinge. In contrast, three joints of Series W exhibited significant damage and strength degradation after the beam flexural yielding. Measured responses are summarized and discussed in the following subsections. Results presented include: 1) beam flexural failure for Series S; 2) joint failure after beam yielding for Series W; 3) discussion of joint shear capacity; and 4) effect of joint eccentricity. The results are used to evaluate the influence of joint eccentricity and loading directions on the seismic performance of corner beam-column connections. Beam flexural failure for Series S Figure 4 depicts the normalized load-displacement hysteretic curves for the test specimens. The actuator load P was ACI Structural Journal/July-August 2007
- 35. Fig. 5—Final damage states for test specimens. Fig. 4—Normalized load versus displacement response. normalized to the nominal yield load Pn that was calculated at a given strain of 0.004 for extreme compression fiber of the critical beam section. When analyzing the beam section, the measured material properties were used to model the concrete and reinforcing bars. In addition, the beam-tip displacement Δ was also normalized to the drift ratio and displacement ductility ratio. As shown in Fig. 4, the nominal yield displacement Δy was determined by extrapolation from measured displacement at 0.75Pn in the 1% drift cycle. Table 3 reports the nominal yield load and displacement for each specimen. The load-displacement responses for Specimens S0 and S50, as shown in Fig. 4, are very similar in stiffness, strength, and ductility. Beam bars initiated yielding in the 1.0% drift cycle and maximum load was recorded at 5% drift level. The hysteretic curves show relatively little pinching, which is typical for a flexure-dominated system. The failure mechanisms for specimens of Series S were core concrete crushing and subsequent buckling of longitudinal bars in the beam plastic hinge region. The buckling of the beam bars in eccentric Specimen S50 appeared earlier than that of concentric Specimen S0. The failure mode for the specimens in Series S was classified as beam flexure failure (Mode B) due to buckling of the beam bars. Figure 5 shows the final damage states for test specimens. For Specimens S0 and S50, only hairline shear (diagonal) cracks were observed on the east and west face of the joint during testing. Concrete crushing in the beam plastic region was evident, but only minor cover concrete spalling appeared on the east face of the joint adjacent to the beam-column interfaces. Further, the readings of shear deformations ACI Structural Journal/July-August 2007 Table 3—Test results Specimen Nominal yield load Pn, kN (kips) S0 S50 W0 W75 W150 158 158 147 147 147 (35.5) (35.5) (33.0) (33.0) (33.0) Nominal yield displacement Δ y , mm (in.) 18.9 20.1 23.5 23.5 24.8 (0.74) (0.79) (0.93) (0.93) (0.98) Over strength factor Pmax /Pn 1.22 1.20 1.11 1.11 1.05 Ductility ratio Δmax /Δy 5.41 5.12 4.58 4.60 3.41 Maximum joint shear Vj,max, kN (kips) 827 814 778 781 739 (186) (183) (175) (176) (166) V j, max -------------------------318 γ f c ′b j h c 0.60 0.78 0.60 0.80 1.13 V j, max -------------------------352 γ f c ′b j h c 0.68 0.67 0.80 0.80 0.94 BJ BJ Failure mode* B B BJ *Failure Mode B means beam flexural failure and BJ means joint shear failure after beam yielding. Note: All values are computed with analytical f c (refer to Table 2) of concrete and ′ measured strengths of reinforcement. measured on the east face of the joints remained in elastic range during testing. Accordingly, it was concluded that both joints of Series S were capable of maintaining joint integrity and remaining elastic during the formation of adjacent beam plastic hinges. Joint failure after beam yielding for Series W As shown in Fig. 4, the load-displacement responses for the specimens in Series W were similar up to 4% drift cycles after yielding of the beam bars (1% drift cycle) and joint transverse reinforcement (2 to 3% drift cycle). All three 463
- 36. Fig. 8—Measurement of joint shear deformation on north face of Specimen W150: (a) cracking patterns on north face at 5% drift; and (b) load versus joint shear deformation. Fig. 6—Strain profiles of hooked beam bars for Specimens W0 and W150. Fig. 7—Strain histories of Gauge 9 on hooked beam bars for Specimens W0 and W150. joints were capable of supporting beam flexural yielding up to 4% drift; however, a considerable strength degradation was observed after the maximum loads recorded at the 4% drift level (Specimen W150) or 5% drift levels (Specimens W0 and W75). Eventually, specimens in Series W exhibited significant pinching curves in Fig. 4, which were typical responses of the shear or bond-slip mechanism. The beam bar strains were measured using electrical resistance strain gauges attached to reinforcing bars at selected locations. Figure 6 shows the strain distributions along the beam bars at peak drift values for Specimens W0 and W150. The hooked beam bars initiated yielding at the critical section (Gauge 10) during the 1% drift cycle, and then spread plasticity into the plastic hinge region (Gauges 11 and 12) during the 2 and 3% drift cycle. Meanwhile, the strain readings of Gauge 9 within the joint remained elastic up to the 3% drift level. This denoted that some bond still existed along the straight part of the bar embedded within the joint. Figure 7 depicts the available strain histories of Gauge 9 for Specimens W0 and W150 during testing. Both gauge 464 readings remained elastic in the 3% drift cycles and then went into yielding plateau in the first or second cycle of the 4% drift level. It is evident that the beam bar was adequately developed up to 4% drift. The bond along the straight portion of the bar was lost at this stage, and therefore the bearing inside the bent portion of the hook resisted most of the tension force. The stress of the bar would begin to drop after crushing of the diagonal strut within the joint. In this paper, this type of failure is classified as diagonal shear compression failure of the joint rather than premature anchorage failure of the beam bars. Figure 8 shows the cracking pattern and measurement of joint shear deformation on the north (flush) face of the joint for Specimen W150. The initial joint shear cracks appeared diagonally during the 0.5% drift cycle, followed by propagation of diagonal cracks up to a 4% drift level. After strength degradation commenced at 4% drift, however, no new joint shear cracks appeared while crushing and spalling of concrete started on the north face of the joint. The measured joint shear deformation rapidly increased after the maximum load recorded at 4% drift, followed by significant degradation on strength and stiffness. The joint shear failure after beam yielding (Mode BJ) was evident due to the nonlinear shear deformation, wide-opened diagonal shear cracks, and visible expansion from crushing of concrete in the joint region. Specimens W0 and W75, which had similar behavior with Specimen W150, also failed in Mode BJ. Visible cracking, crushing, and spalling of concrete in Specimens W0 and W75 were less than those in Specimen W150 (Fig. 5). Due to the distance between beam and column edges (Fig. 1), the appearance of initial joint shear cracks on the north face of the joint was delayed to the 1.0 and 1.5% drift cycle for Specimens W75 and W0, respectively. Strength degradation after the 5% drift cycle was attributed to the crushing of concrete within the joints, followed by extensive pushout cracks distributed on the east face of the joint behind the hooked beam bars (Fig. 5). Due to crushing of the concrete within the joint, the hooked beam bars might gradually lose its bond and anchorage within the joint. As a result, the pushout movement of the beam compression bars induced the pushout cracks on the east face of the joints in Series W. The joint failure and subsequent pushout cracks were observed at a drift level of 5% or more, which is large for a welldesigned building system. Therefore, the observed behavior appears to be acceptable for the seismic design purpose. Discussion of joint shear capacity Paulay et al.12 first discussed that there are two shear-resisting mechanisms exiting in joints, the truss mechanism and the diagonal strut mechanism. The truss mechanism transfers the ACI Structural Journal/July-August 2007
- 37. forces uniformly from the beam and column bars through the bond mechanism. Adequate bond must exist between the reinforcement and concrete to necessitate a truss mechanism, which also requires considerable amounts of horizontal and vertical tie forces in the truss panel to be in equilibrium. Figure 9 illustrates a conceptual model for the degradation of joint shear capacity under increasing drift or ductility ratio. Joints subjected to inelastic displacement reversals often undergo significant bond deterioration along the reinforcing bars from the adjacent beam plastic hinge. At this stage, a part of the joint shear is transferred through the horizontal hoops with fan-shaped struts, while the remainder is carried by the diagonal strut. As the drift or ductility ratio increases, the horizontal hoops would yield progressively, the joint concrete may crack excessively, and the bond of the reinforcing bars within the joint might be lost. Eventually, the joint shear force is directly transferred by the diagonal strut mechanism. Real shear-transferring mechanisms in joints may be a combination of the diagonal strut and the truss mechanism, with the bond deterioration being at a certain degree of longitudinal reinforcement during cyclic loading (Fig. 9). Hence, the joint shear capacity decreases as the cyclic inelastic loading increases, which is referred to as the degradation of the joint shear capacity. When the joint shear capacity falls below the shear demand from beam hinging, the joint will fail in the shear after beam yielding (Mode BJ). If the joint shear capacity is greater than the demand, the maximum strength is limited by the beam flexure capacity (Mode B). Three levels of strength and ductility ratios for the test specimens are shown in Table 3. Because the maximum strengths of Specimens S0 and S50 were dominated by the beam flexure capacity rather than the joint shear capacity, Specimens S0 and S50 had over-strength factors of approximately 1.2 and ductility ratios greater than 5. In contrast, Specimens W0 and W75 had over-strength factors of approximately 1.1 and ductility factors of approximately 4.6 due to the joint shear failure at 5% drift level. Further, the largejoint-eccentricity Specimen W150 barely reached the nominal yield load and deteriorated at a ductility ratio of only 3.4. Corresponding to the maximum actuator load, the maximum shear force acting on the horizontal cross section within the joint can be estimated by L b ( L b + 0.5h c ) V j, max = T max – V col = P max ⎛ ---- – -----------------------------⎞ ⎝ jd ⎠ Lc (5) where Tmax is the maximum force in the tension reinforcement of the beam (N or lb); Vcol is the column shear in equilibrium with the applied loading (N or lb); and jd is the internal level arm of the beam section (mm or in.). From standard momentcurvature analysis for each specimen, jd is approximately 7/8 of the effective depth of the beam section. Thus, jd is simply assumed to be 350 mm (13.8 in.) for the following evaluation of maximum joint shear forces. Table 3 compares the maximum joint shear force with the nominal joint shear strength following the methods in ACI 318-058 or ACI 352R-02.7 When following ACI 318-05,8 Specimens S0 and W0 had equal effective joint area. Thus, the maximum joint shear forces were only 60% of the nominal strength for concentric Specimens S0 and W0 (Table 3), but different failure modes occurred during testing (Fig. 4). For the flexure-dominated Specimen S0, ACI Structural Journal/July-August 2007 Fig. 9—Conceptual model for degradation of joint shear capacity. the maximum shear force acting on the joint was less than the joint shear capacity (Fig. 9). In contrast, Specimen W0 failed in Mode BJ when the joint shear force reached the joint shear capacity at 5% drift. Clearly, the joint shear capacity in the strong direction of the rectangular joint (Specimen S0) was greater than that in the weak direction (Specimen W0). Comparing eccentric Specimens S50 and W75 can also find similar observation. This point cannot be rationally reflected on the calculation of a cross-sectional approach within the joint, especially for the effective joint width given by Eq. (2). When following ACI 352R-02,7 the maximum joint shear forces were approximately 70% of the nominal strengths for Series S, 80% of those for Specimens W0 and W75, and 94% of that for Specimen W150. Three levels of demand-to-capacity ratios reasonably reflected three levels of performance on strength and ductility ratios shown in Table 3. This shows that the effective joint width bj352 is more rational than bj318 for test specimens. Although following the ACI 352R-027 procedures could not avoid joint shear failure at a large drift level of 4 or 5%, it is considered acceptable in a real structural system. In this experimental program, each specimen was able to carry the applied column axial load of 0.10Ag f c over the ′ entire displacement history. Strain readings of gauges confirmed that all column longitudinal bars remained elastic during testing. For a building frame during earthquake events, however, the axial load in a corner column may be higher than 0.10Ag f c′ , or even in tension, due to overturning moment from lateral loads. Therefore, more research on the behavior of eccentric beam-column connections under high axial loads is still needed. Effect of joint eccentricity The relative energy dissipation ratio β and the equivalent viscous damping ratio ξeq, as shown in Fig. 10(a), were used to evaluate the energy dissipation capacities of the test specimens. The first index β represents a fatter or narrower hysteretic curve (pinching) with respect to an elastic perfectly plastic model. Another quantitative index ξeq describes the hysteretic damping (or energy dissipation per cycle) with respect to an equivalent linear elastic system. Average β and ξeq of three cycles at each drift level for the test specimens are compared in Fig. 10(b) and (c). Three performance levels of energy dissipation capacities 465
- 38. Fig. 12—Strain profiles at central layer of joint shear reinforcement in Series S. Fig. 10—Normalized energy dissipation at each drift level for test specimens. Fig. 11—Strain profiles at central layer of joint shear reinforcement in Series W. are evident. The flexure-dominated Specimens S0 and S50 had a highest performance while Specimen W150 had lowest performance. A small joint eccentricity of bc /8 (Specimens S0 and W75) had a slight influence on this experimental program. Obviously, the large joint eccentricity of bc/4 had significant detrimental effects on the seismic performance of Specimen W150. Strain histories for the joint hoops and crossties were used to plot the strain distribution along the joint width at peak 466 drift values. There were three layers of transverse reinforcement at a spacing of 100 mm (4 in.) in each joint. Only the strain profiles of the hoop legs and crossties in the direction of shear and at the central layer of the transverse reinforcement were compared in Fig. 11 and 12. For the corresponding drift ratios shown in Fig. 11, the strain readings of Gauge 24 in Specimens W75 and W150 were larger than those in Specimen W0. These profiles confirm the observations of more extensive shear or torsion cracks on the north side of the eccentric joints. On the south side, the strain readings of Gauge 20 in Specimens W75 and W150 were less than those in Specimen W0 because the shear and torsional stresses counteract each other.11 The effective joint width bj352 is also displayed in Fig. 11. For the joints of Series W, strain gauges on hoop legs and crossties within bj352 yielded during the 2 or 3% drift cycles while the outside strain gauges remained elastic at the same drift level. During testing of Series W, crushing of joint concrete was observed within bj352 on the west side of the joint. These observations agreed well with the strain profiles shown in Fig. 11. For the specimens in Series S, all joint hoops and crossties remained elastic over the entire displacement history. Figure 12 shows the strain distributions of hoop legs and crossties along the joint width. Due to torsional stresses from joint eccentricity, Specimen S50 had asymmetric strain distribution with respect to concentric Specimen S0. It should be noted that the total cross-sectional area of joint transverse reinforcement in two principal directions was different. Although the maximum joint shear forces in ACI Structural Journal/July-August 2007
- 39. Series S and W were similar (Table 3), the joint shear forces transferring by the lateral joint reinforcement in Series S were obviously less than those in Series W. These profiles agree well with Hwang and Lee’s model,22 which proposed that the fraction of shear carried by the joint transverse reinforcement depends on the inclination of the diagonal strut. Due to a deeper joint depth, the joints in Series S had a flatter diagonal strut that can resist horizontal joint shear more efficiently.23 As a result, the shear forces transferring by the lateral joint reinforcement was reduced and then the lateral joint reinforcement remained elastic during testing. CONCLUSIONS Based on the evaluation of the cyclic loading responses of five reinforced concrete beam-column corner connections in this experimental program, the conclusions are as follows: 1. The joint shear capacity in the strong direction of a rectangular joint is greater than that in the weak direction. In this experimental program, two joints subjected to lateral loading in the strong direction were capable of supporting the complete formation of a beam plastic hinge. The other three joints exhibited significant damage at the joints with the joint shear acting along the weak direction of the column; 2. Joint eccentricity between the beam and column centerlines had detrimental effects on the seismic performance of beam-column connections. Slight influence on the connection performance was found when the joint eccentricity was equal to half-quarter width of the column. As the joint eccentricity increasing to one-quarter of the column width, significant reductions in the strength, ductility, and energy dissipation capacity was observed; and 3. Compared with seismic performance levels, strain distributions, joint damage of the test specimens, the effective joint width recommend by ACI 352R-02 is a better choice than that given in the ACI 318 code. Experimental verifications show that the current ACI design procedures are acceptable for seismic design purposes but could not prevent the failure of corner connections at a large drift level of 4 or 5%. ACKNOWLEDGMENTS The authors are grateful to the funding support (NSC 93-2211-E-224-010) of the National Science Council in Taiwan. The assistance of graduate students for the construction and testing of the beam-column connections in the structural laboratory of the National Yunlin University of Science and Technology is also acknowledged. REFERENCES 1. Moehle, J. P., and Mahin, S. A., “Observations on the Behavior of Reinforced Concrete Buildings during Earthquakes,” Earthquake-Resistant Concrete Structures—Inelastic Response and Design, SP-127, S. K. Ghosh, ed., American Concrete Institute, Farmington Hills, Mich., 1991, pp. 67-89. 2. Sezen, H.; Whittaker, A. S.; Elwood K. J.; and Mosalam, K. 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K., “Seismic Behavior of Eccentric R/C Beam-Column-Slab Connections under Sequential Loading in Two Principal Directions,” ACI Fifth International Conference on Innovations in Design with Emphasis on Seismic, Wind and Environmental Loading, Quality Control, and Innovation in Materials/Hot Weather Concreting, SP-209, V. M. Malhotra, ed., American Concrete Institute, Farmington Hills, Mich., 2002, pp. 863-880. 16. Shin, M., and LaFave, J. M., “Seismic Performance of Reinforced Concrete Eccentric Beam-Column Connections with Floor Slabs,” ACI Structural Journal, V. 101, No. 3, May-June 2004, pp. 403-412. 17. Goto, Y., and Joh, O., “Shear Resistance of RC Interior Eccentric Beam-Column Joints,” Proceedings of the 13th World Conference on Earthquake Engineering, Paper No. 649, Vancouver, British Columbia, Canada, 2004, 13 pp. 18. 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- 40. ACI STRUCTURAL JOURNAL TECHNICAL PAPER Title no. 104-S41 Flexural Behavior of Concrete Beams Strengthened with Near-Surface-Mounted CFRP Strips by Joseph Robert Yost, Shawn P. Gross, David W. Dinehart, and Jason J. Mildenberg Flexural strengthening using near-surface mounted (NSM) fiberreinforced polymer (FRP) materials is a promising technology. As NSM reinforcement, the FRP is surrounded by concrete on three sides so the bond and damage problems associated with externally bonded FRP strengthening systems are reduced or eliminated. This paper presents experimental results from 12 full-scale concrete beams strengthened with NSM carbon FRP (CFRP) strips. Three companion unstrengthened specimens were also tested to serve as a control. Experimental variables include three different ratios of steel reinforcement ρs and two different ratios of CFRP reinforcement ρfrp . Yield and ultimate strengths, flexural failure modes, and ductility are discussed based on measured load, deflection, and strain data. Test results show measurable increases in yield and ultimate strengths; predictable nominal strengths and failure modes; and effective force transfer between the CFRP, epoxy grout, and surrounding concrete. Also, strengthening with CFRP resulted in a decrease in both energy ductility and deflection ductility. Keywords: beam; polymer; reinforcement; strength. INTRODUCTION In-service steel-reinforced concrete flexural members may require strengthening due to material decay of the internal reinforcement and surrounding concrete, errant design and construction practice, increased service loads, and unforeseen settlement and structural damage. These conditions require structural retrofit to increase the flexural strength of the section. A popular method of increasing the flexural strength of beams, walls, and slabs is through external bonding of fiber-reinforced polymer (FRP) plates and sheets. FRP materials are characterized by high tensile strength and low unit weight, and they are noncorrosive when exposed to chloride environments. An excellent summary of research in this area is available by Teng et al. (2002) and ACI has published a design guide for strengthening concrete structures with externally-bonded FRP materials (ACI Committee 440 2002). Premature failure of externally-bonded FRP plates and sheets can occur before the ultimate flexural capacity of the strengthened section is achieved. This is typically due to bond failure between the FRP and concrete or tensile peeling of the cover concrete. Available research documenting this behavior is abundant. Brena et al. (2003) reported debonding of longitudinal carbon FRP (CFRP) sheets at deformation levels less than half the deformation capacity of control specimens. Nguyen et al. (2001) observed only a limited increase in flexural capacity for beams strengthened with partial length longitudinal CFRP sheets due to premature delamination, or ripping, of the concrete cover surrounding the steel reinforcement. For beams strengthened with CFRP plate and fabric systems, Grace et al. (2002) identified brittle failure by shear tension and debonding, respectively. Shin and Lee (2003) reported failure of beams held under sustained load and strengthened with CFRP laminates due to 430 Fig. 1—Concrete member strengthened in flexure with NSM FRP. rip-off type failure of the CFRP at loads well below the ultimate flexural capacity of the sections. Similar results have been reported by Rahimi and Hutchinson (2001), Bencardino et al. (2002), Arduini and Nanni (1997), Sharif et al. (1994), Saadatmanesh (1994), and Mukhopadhyaya and Swamy (1999). In addition to problems associated with bond failure, external FRP plates are vulnerable to mechanical, thermal, and environmental damage. It should be noted, however, that mechanical anchors can be used to improve the peel resistance of externally bonded FRP. In response to the detrimental conditions associated with externally bonded FRP, engineers have proposed relocating the strengthening FRP material from the unprotected exterior of the concrete to the protected interior. This technology is referred to as near-surface mounted (NSM) strengthening and is shown in Fig. 1. The surrounding concrete now protects the FRP so that mechanical and thermal damage is unlikely. Other advantages of using NSM FRP technology include improved bond and force transfer with the surrounding concrete and the ability to increase the negative bending strength of bridge decks, pavements, and other structural riding surfaces. RESEARCH SIGNIFICANCE This paper documents behavior of full-scale test beams strengthened in flexure with NSM CFRP strips and tested to failure in four-point bending. The parameters of steel and FRP reinforcement ratios are investigated. Concrete strength, shear span-to-depth ratio, and steel reinforcement ratios were selected as typical for concrete flexural components in the civil infrastructure. Theory related to failure modes and strength models are evaluated based on comparison with the test data. It is expected that the conclusions reported will ultimately contribute to the development of a design guide for using NSM FRP for flexural strengthening of concrete beams and slabs. ACI Structural Journal, V. 104, No. 4, July-August 2007. MS No. S-2006-212 received May 25, 2006, and reviewed under Institute publication policies. Copyright © 2007, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published in the MayJune 2008 ACI Structural Journal if the discussion is received by January 1, 2008. ACI Structural Journal/July-August 2007
- 41. Joseph Robert Yost is an Associate Professor of Civil and Environmental Engineering at Villanova University, Villanova, Pa. His research interests include the use of innovative materials in transportation infrastructure, nondestructive methods for health monitoring of structures, and seismic design and analysis of bridges. ACI member Shawn P. Gross is an Associate Professor in the Department of Civil and Environmental Engineering at Villanova University. He is Secretary of Joint ACI-ASCE Committee 423, Prestressed Concrete, and a member of ACI Committees 213, Lightweight Aggregate and Concrete; 363, High-Strength Concrete; 435, Deflection of Concrete Building Structures; 440, Fiber Reinforced Polymer Reinforcement; and E803, Faculty Network Coordinating Committee. His research interests include the design and behavior of reinforced and prestressed concrete structures, including the use of high-strength concrete and fiber-reinforced polymer reinforcement. David W. Dinehart is an Associate Professor of Civil and Environmental Engineering at Villanova University. His research interests include seismic evaluation of wood structures, passive damping systems, and the design and behavior of concrete and steel structures. Jason J. Mildenberg is a Structural Engineer with Schoor De Palma of Brick, Manalapan, N.J. He received an MS in civil engineering from Villanova University. BACKGROUND AND LITERATURE REVIEW Nanni (2000) compared the behavior of full-scale simply supported highway bridge deck panels strengthened in flexure with either externally bonded CFRP laminates or internally placed NSM CFRP bars. Failure of the CFRP laminate reinforced deck spans was through a combination of rupture and peeling of the CFRP laminates. The NSM CFRP-reinforced span failed by tensile rupture of the CFRP bars. Relative to the capacity of an unstrengthened control deck, moment strength increases of 17 and 29% were reported for decks retrofitted with externally bonded CFRP laminates and internally placed NSM CFRP bars, respectively. DeLorenzis et al. (2000) tested three steel-reinforced concrete T-beams strengthened in flexure with NSM glass FRP (GFRP) and NSM CFRP bars. The CFRP retrofitted beams experienced increases in strength of 30% (two No. 3 CFRP bars) and 44% (two No. 4 CFRP bars) over an unstrengthened control specimen. Both CFRP strengthened beams failed due to debonding of the NSM rods. The specimen strengthened with two No. 4 GFRP bars also failed due to debonding of the NSM GFRP bars at a load 26% higher than the control specimen. The authors reported that bond is critical to using this technology effectively. Bond failure of the NSM FRP bars was also identified by DeLorenzis and Nanni (2001) as in need of further investigation. Debonding of the NSM FRP bars due to splitting of the epoxy used for holding the rod in place was reported. It was suggested that this failure limit-state could possibly be avoided by increasing bond lengths or anchoring the NSM rods in the flange. Significantly, the authors reported that, where debonding of the NSM FRP bars is prevented, splitting of the concrete cover surrounding the longitudinal steel bars might become the controlling ultimate limit-state. Loss of anchorage was observed in several of their test specimens. In a related experimental bond study, DeLorenzis et al. (2004) state that epoxy is superior to cement paste as the groove filler material, a groove size-to-bar diameter of 2.0 is optimal, and a smooth grove surface yields slightly lower local bond strengths, but is preferable because it yields a more ductile bond-slip behavior. Taljsten and Carolin (2001) evaluated four rectangular concrete beams subjected to four-point bending and monotonically loaded in deformation control. Three of the test beams were strengthened with NSM CFRP strips and the fourth served as a control specimen. Two of the three strengthened beams used an epoxy for bonding the FRP and the third used a cement grout. Test results showed that two of the three retrofitted beams failed due to anchorage loss ACI Structural Journal/July-August 2007 Fig. 2—Test setup. between the NSM FRP strips and concrete. The third strengthened beam failed due to tensile rupture of the FRP strip. Predicted failure loads overestimated measured strengths. El-Hacha and Rizkalla (2004) compared the behavior of beams strengthened on an equal axial stiffness basis using NSM FRP bars and strips and externally bonded FRP laminates. Their research showed that higher ultimate strengths and increased ductility were achieved by the NSM strengthened specimens. They also noted that bond integrity of NSM FRP bars was less effective than for NSM FRP strips. Together, these research findings demonstrate that bond integrity can not be taken for granted and that bond related limit states must also be considered for NSM FRP. DeLorenzis and Nanni (2002) suggest that bond performance will be influenced by multiple factors including bond length, NSM FRP bar diameter and surface characteristic, material characteristics of the FRP, groove geometry, and properties of the epoxy grout. Their experimental bond tests showed three bond related failure modes, namely, splitting of the epoxy cover, cracking of the concrete surrounding the grove, and pullout of the NSM FRP rod. EXPERIMENTAL PROGRAM This experimental investigation consisted of testing 15 simply supported full-scale concrete beams in flexure and material characterization of the CFRP, steel reinforcement, and concrete. All test beams had a shear-span-to-steel-reinforcement-depth ratio av /ds of 8.4. This ratio was intentionally selected so that ultimate strength would be controlled by flexural failure and not shear failure. The test setup and associated specimen details are shown in Fig. 2. The 15 test beams were separated into three groups of five beams, with all beams in a given group having the same cross section and steel reinforcement ratio ρs. Within each group of five beams, two beams had one CFRP strip (designated 6-1Fa&b, 9-1Fa&b, and 12-1Fa&b), two beams had two CFRP strips (designated 6-2Fa&b, 9-2Fa&b, and 12-2Fa&b), and one beam acted as a control with no CFRP (designated 6-C, 9-C, and 12-C). Note that beams identified as a and b are replicate specimens. Thus, the two parameters investigated in the study are the amount of steel and CFRP reinforcements. Table 1 presents the unstrengthened steel reinforcement ratio ρs relative to a balanced design ρs /ρsb. 431
- 42. Table 1—Specimen design and predicted strength parameters Specimen ρs/ρsb* Afb, mm2 (in.2) 0.684 –38.86 (–0.060) Failure type†‡ NA 6-C 6-1Fa&b ff-ult, MPa (ksi) Mn, kN-mm (kip-in.) Pn , kN (kip)§ Af /Afb SY/CC NA 23,068 (204.2) 18.92 (4.25) 1 –0.85 CC 810 (117.4) 26,606 (235.5) 21.82 (4.91) 1.15 1.26 Pn/PnC 6-2Fa&b –1.69 CC 709 (102.8) 29,168 (258.2) 23.92 (5.38) 9-C NA SY/CC NA 25,104 (222.2) 20.59 (4.63) 1 –31.66 CC 1276 (185) 31,221 (276.3) 25.61 (5.76) 1.24 1.41 9-1Fa&b 0.470 –1.04 (–0.0016) 9-2Fa&b –63.31 CC 1091 (158) 35,415 (313.5) 29.05 (6.53) 12-C NA SY/CC NA 25,790 (228.3) 21.15 (4.76) 1 0.84 TR 1648 (239) 34,071 (301.6) 27.95 (6.28) 1.32 1.69 CC 1436 (208.2) 40,023 (354.2) 32.83 (7.38) 1.55 12-1Fa&b 0.353 38.94 (0.060) 12-2Fa&b *ρ s † = As /bds and ρsb = 0.85(fc′/fy)β1(εcu)/(εcu + εsy) is unstrengthened balanced reinforcement ratio. SY = steel yield, CC = concrete compression failure, TR = tensile rupture of FRP. ‡For all samples with CC failure, steel has yielded at ultimate as per analysis of Eq. (4). §P = M /1219 mm (M /48 in.). n n n Fig. 3—CFRP and tensile test results. Fig. 4—Specimen preparation. The ratios of 0.353, 0.470, and 0.684 were selected as typical for existing structures. All specimens were instrumented with a concrete strain gauge located on the top compression fiber at the center span. Strengthened Specimens 6-1Fb, 6-2Fb, 9-1Fb, 9-2Fb, 12-1Fb, and 12-2Fb had an additional strain gauge bonded to the CFRP at the center span. Linear variable displacement transducers (LVDTs) were used to measure displacement at the center span. Concrete for the test specimens was delivered to the laboratory by a concrete supplier. The concrete was in accordance with Pennsylvania Department of Transportation (PennDOT) Class AAA, Concrete for Bridge Decks, with design specifications and properties given in BD-601M 432 (PennDOT 2001). The mixture design was selected as typical for bridge decks and is given as follows: water 1530 N/m3 (263 lb/yd3), cement 3967 N/m3 (682 lb/yd3), coarse aggregate 1784 lb/yd3, fine aggregate 7242 N/m3 (1245 lb/yd3), air entrainment 30 N/m3 (3 oz/yd3), and retarder 196 N/m3 (20 oz/yd3). The slump at specimen casting was 101.6 mm (4 in.), and the 33-day compressive strength as determined by ASTM C 684-99 (ASTM 1999) using 100 mm (4 in.) diameter by 200 mm (8 in.) high cylinders was 37.2 MPa (5.4 ksi) for all beams. Yield strength of the steel reinforcement was determined from uniaxial coupon testing to be 510 MPa (74 ksi) for No. 4 bars and 490 MPa (71 ksi) for the No. 5 bars. Elastic modulus Es is taken as 200 GPa (29,000 ksi). The CFRP strips have a thin rectangular cross section that measures approximately 15 x 2.5 mm (0.60 x 0.10 in.), and the surface of the wide face is roughened to enhance force transfer with the concrete epoxy grout. A photo of the CFRP reinforcement with associated instrumentation detail can be seen in Fig. 3(a). The material composition is 60% 4137 MPa (600 ksi) carbon fiber by volume in a bisphenol epoxy vinylester resin matrix. The CFRP elastic modulus Ef and ultimate tensile strength ffu were determined from testing uniaxial coupon specimens according to ACI Committee 440 (2004). Test results are shown in Fig. 3(b) from which Ef and ffu were determined to be 136 GPa and 1648 MPa (19,765 and 239 ksi), respectively. Installation of the NSM CFRP strips is shown in Fig. 4 and described as follows. First, the beams were rotated 180 degrees about the long axis so that the steel reinforcement was at the top of the beam. Next a rectangular groove approximately 6.4 mm (1/4 in.) wide by 19 mm (3/4 in.) deep was cut longitudinally in the concrete where the CFRP was to be installed. The groove was cut using a hand-held circular with an 18 cm (7 in.) diameter diamond-tooth, abrasive cutting blade. The saw was fitted with a rip guide, so that the distance from the edge of the beam to the blade could be set and maintained during cutting. The depth of the blade was set to 19 mm (3/4 in.) by adjusting the saw. The saw blade was just over 3.2 mm (1/8 in.) wide so that two passes were made to achieve the required width. For test specimens having one CFRP strip, the longitudinal groove was located at the center of the cross section; and for specimens having two CFRP strips, the grooves were located at the 1/3 points in the cross section. Next, the groove was thoroughly cleaned of debris with compressed air and then partially filled with a structural epoxy material that bonds with the concrete and FRP to ACI Structural Journal/July-August 2007
- 43. provide a mechanism for force transfer. The epoxy grout used was a two-part epoxy. Finally, the FRP was depressed into the groove, where care was taken to ensure that no air voids were trapped within the epoxy gel. Excess epoxy gel was then cleaned from the concrete surface and curing was done for a minimum of 2 weeks. All beams were tested monotonically from an uncracked condition. Two 90 kN (10 ton) hydraulic cylinders, located 152 mm (6 in.) on either side of center span and controlled by a manually-operated pump, were used to apply load at an approximate rate of 4.5 kN/minute (1 kip/minute). A load cell was located under each hydraulic cylinder to measure applied load. Electronic signals from the strain gauges (concrete and CFRP), LVDTs, and load cell were recorded by a 16-bit data acquisition system at a frequency of 1 Hz. ANALYTICAL STRENGTH Figure 5 illustrates the assumed basic analytical conditions of internal strain, stress, and resultant force for a cracked section at ultimate that is under-reinforced with steel (ρs < ρsb) and strengthened with FRP. From Fig. 5, the following assumptions are implicit: strain varies linearly through the cross section, the section is initially uncracked, perfect bond exists between the steel and FRP reinforcements and concrete, the concrete strain at compression failure is 0.003, the Whitney rectangular stress block in the compression zone is a valid substitution for a nonlinear stress distribution at ultimate, and the steel stress-strain behavior is assumed to be elastic-plastic. Also noted in Fig. 5, because the section is initially uncracked and df > ds, the FRP strain εf will slightly exceed the steel strain εs. The theoretical nominal flexural strength Mn of an initially uncracked beam that is under-reinforced with steel (ρs < ρsb) and strengthened with FRP is dependent on the amount of FRP provided (Af) relative to the FRP area corresponding to a balanced-strengthened strain condition (Afb). In this context, balanced-strengthened represents simultaneous tensile rupture of the FRP and compression failure of the concrete. Again, for an initially uncracked section with df > ds and εf = εfu in Fig. 5, by default the steel for a balanced-strengthened design will have yielded (εs > εsy). Using these assumptions and strain limits, and considering compatibility and equilibrium, the theoretical balanced-strengthened area of FRP is A fb ⎧ ε cu ⎫ 0.85f′ c bβ 1 d f ⎨ ------------------- ⎬ – A s f y ⎩ ε cu + ε fu ⎭ = -----------------------------------------------------------------------f fu ACI Structural Journal/July-August 2007 M n = Af f fu ⎛ d f – a⎞ + As f y ⎛ d s – a⎞ for A f < A fb --⎝ ⎝ 2⎠ 2⎠ (2b) For sections controlled by concrete crushing, the stress level in the steel is initially unknown, as is shown in Fig. 5(b). It can be determined by fixing the steel and concrete strains at yield εsy and crushing εcu, respectively, calculating the steel area corresponding to yield Asy, and comparing this with the area of steel present As. From Fig. 5(b), this is as follows A sy ε cu df 0.85f c ′bβ 1 d s ⎛ ------------------- ⎞ – A f E f ε sy ⎛ ----⎞ ⎝ d s⎠ ⎝ ε cu + ε sy⎠ = -----------------------------------------------------------------------------------------fy (3) Accordingly, for As ≤ Asy, the steel stress is equal to fy. Likewise, for As > Asy, the steel stress is less than fy and must be determined from compatibility and equilibrium. Using this procedure, the steel stress at ultimate for all specimens controlled by concrete failure in this study was equal to yield. With the steel stress at yield, the compression block a, stress in the FRP reinforcement ff, and nominal moment capacity Mn for sections controlled by concrete failure are found from compatibility and equilibrium as follows 2 ( A f E f ε cu – A s f y ) + 4 ( 0.85 )f c ′bβ 1 A f E f ε cu d f – ( A f E f ε cu – A s f y ) a = -----------------------------------------------------------------------------------------------------------------------------------------------------( 2 )0.85f c ′b (4a) ( df – α ⁄ β1 ) f f = E f ε cu --------------------------- ≤ f fu α ⁄ β1 (4b) M u = A f f f ⎛ d f – a⎞ + A s fy ⎛ d s – a⎞ --⎝ ⎝ 2⎠ 2⎠ (4c) (1) Using Eq. (1) as a theoretical FRP reinforcement limit, failure will be tensile rupture of the FRP when Af > Afb , or compression failure of the concrete, when Af < Afb. It is noted that Afb can be either positive or negative, depending on the existing amount of steel reinforcement present (As). For a negative result from Eq. (1), Af provided will always be greater than Afb, indicating a compression failure of the concrete. Strain distributions for FRP failure, balanced-strengthened, and compression failure are shown in Fig. 5(b). For sections controlled by FRP failure, the compression block depth a and nominal moment strength at ultimate Mn are calculated from equilibrium as follows Af f fu + As f y a = -------------------------- for A f < A fb 0.85f c ′b Fig. 5—Analytical model at ultimate. (2a) The preceding analysis is offered as an alternative to the trial and error procedure set forth by ACI Committee 440 (2002) and yields identical results as would be obtained using the ACI 440.2R procedure. Table 1 summarizes relevant design and strength parameters. Moment strength Mn was calculated using the measured material strengths for the steel, CFRP, and concrete. It is evident from Table 1 that, for a given area of FRP Af , the relative increase in strength Pn/PnC is inversely proportional to the amount of steel reinforcement. TEST RESULTS Load-deflection and load-strain results are shown in Fig. 6 and summarized in Table 2. Typical photos at failure are shown in Fig. 7. The applied cylinder loads plotted in Fig. 6 and recorded in Table 2 have been corrected to include the self-weight bending effects of the beam. Moment equivalence at center span 433
- 44. Fig. 6—Load-deflection and load-strain results. Fig. 7—Test specimens at failure. Table 2—Summary of test results Measured Theory Yield Ultimate Comparison Pn, kN (kip) Py, kN (kip) Mechanism type* Pmax, kN (k) Py /PyC 6-C (control) 18.9 (4.25) 19 (4.28) SY/CC 21.12 (4.75) 1 Sample ID 6-1Fa 6-1Fb 6-2Fa 6-2Fb 21.8 (4.91) 23.9 (5.38) 20.9 (4.69) CC 24.83 (5.58) 1.10 21.3 (4.78) CC 23.24 (5.23) 1.12 24.4 (5.48) CC 24.99 (5.62) 1.28 24.7 (5.56) CC 26.94 (6.06) 1.30 SY/CC 25.29 (5.69) 1 9-C (control) 20.6 (4.63) 22.4 (5.03) 9-1Fa 9-1Fb 9-2Fa 9-2Fb 25.6 (5.76) 29.0 (6.53) 25.3 (5.70) CC 28.22 (6.34) 1.13 24.5 (5.50) CC 27.93 (6.28) 1.09 27.7 (6.22) CC 37.05 (8.33) 1.24 25.0 (5.63) CC 35.82 (8.05) 1.12 SY/CC 23.52 (5.29) 1 12-C (control) 21.2 (4.76) 21.5 (4.84) 12-1Fa 12-1Fb 12-2Fa 12-2Fb 27.9 (6.28) 32.8 (7.38) 24.7 (5.56) TR 29.59 (6.65) 1.15 25.9 (5.81) TR 31.01 (6.97) 1.20 26.5 (5.97) CC 33.80 (7.60) 1.23 28.0 (6.30) CC 41.77 (9.39) 1.30 Average Pmax/PmaxC Average Pmax/Py Average 1 1 — 1.11 — 1.11 1.29 1 1.11 1.18 1 1.18 1.27 1.18 1.10 1.18 1.28 1 1.12 1.10 1.47 1.42 1 1.26 1.32 1.44 1.78 1.14 1.23 — 1.11 1.44 — 1.29 1.61 1.19 1.09 1.02 1.09 1.13 1.11 1.14 1.34 1.43 1.09 1.20 1.20 1.27 1.49 1.14 1.06 — 1.13 1.38 — 1.20 1.38 Pmax/Pn 1.12 1.14 1.06 1.04 1.13 1.23 1.10 1.09 1.28 1.23 1.11 1.06 1.11 1.03 1.27 * SY = steel yield, CC = concrete crushing, TR = CFRP tensile rupture. was used to calculate an equivalent concentrated force Peq that was added to all laboratory measured load data. Moment equivalence at center span is expressed as {1/8wbeamL2} = {Peqav}. From Fig. 3, Peq for the 152, 230, and 305 mm (6, 9, and 12 in.) wide specimens is calculated to be 0.50, 0.77, and 1.0 kN (0.115, 0.172, and 0.230 kips), respectively. From Fig. 6, the physical effects of supplemental strengthening with CFRP are clearly evident when strengthened specimens are compared with companion control (unstrengthened) specimens. All specimens strengthened with CFRP showed 434 a significant increase in ultimate strength when compared with the companion control specimens. To a lesser degree, strengthening with CFRP increased stiffness and yield load. Detailed discussions of the test results for control and strengthened specimens are presented in the following sections. Control specimens: 6-C, 9-C, and 12-C Referring to the load-deflection behavior of control Specimens 6-C, 9-C, and 12-C, the ductile behavior characteristic of under-reinforced steel flexural (ρs < ρsb) members ACI Structural Journal/July-August 2007
- 45. is apparent. Initially, all sections are uncracked and gross section properties apply (Ig). At the cracking load Pcr, behavior changes from uncracked to cracked-elastic. As load is increased further, the section responds elastically until the yield strength of the steel reinforcement fy is reached. At the yield load Py, behavior changes from cracked-elastic to inelastic. For Specimens 6-C, 9-C, and 12-C, steel yield occurred at 19, 22.4, and 21.5 kN (4.28, 5.03, and 4.84 kips), respectively. The yield load corresponds to a flattening of the load-deflection trace and simultaneous inflection in the concrete load-strain response. Yield is followed by a load plateau where the moment capacity of the section remains roughly constant. The load plateau is clearly visible for Specimens 9-C and 12-C, and to a lesser degree for Specimen 6-C. At the ultimate load Pmax, failure occurred by concrete crushing. Ultimate load for Specimens 6-C, 9-C, and 12-C was 21.1, 25.3, and 23.5 kN (4.75, 5.69, and 5.29 kips), respectively. For all control specimens, the ultimate load Pmax was approximately 12% greater than the yield load Py. The measured failure loads for Specimens 6-C, 9-C, and 12-C were 12, 23, and 11%, respectively, greater than the theoretical nominal capacity Pn. Specimens strengthened with one CFRP strip: 6-1Fa&b, 9-1Fa&b, and 12-1Fa&b For specimens strengthened with one CFRP strip, the change from cracked-elastic to inelastic behavior (yield point) is less abrupt and the associated reduction in the slope of the loaddeflection curve is less than for the control specimens. This is especially true for specimens with a large relative amount of steel reinforcement ρs/ρsb. Referring to Fig. 6, for Specimens 6-1Fa&b, the change in stiffness at ensuing nonlinear loaddeflection response associated with steel yielding is negligible. These specimens have the largest relative area of steel reinforcement equal to 0.68ρsb. For Specimens 9-1Fa&b and 12-1Fa&b, however, the change in stiffness after steel yield is more apparent. These specimens were reinforced with 0.47ρsb and 0.34ρsb, respectively. The mechanism of failure at ultimate for all specimens in this group is consistent with that predicted using the theory outlined previously and summarized in Table 1. As can be seen in Table 2, all 152 and 230 mm (6 and 9 in.) wide specimens strengthened with one CFRP strip failed by crushing of the concrete. For these specimens, the CFRP did not rupture prior to concrete crushing, indicating that the strain level was less than the ultimate material strength. For the 305 mm (12 in.) wide specimens with one CFRP strip, however, the CFRP reinforcement did rupture at ultimate. This was followed by compression failure in the concrete. Thus, the bond between the CFRP and concrete for Specimens 12-1Fa&b was able to develop the tensile strength of the CFRP strip. Also, for all samples in this group, no debonding or slip between the CFRP strip and concrete was observed (refer to Fig. 7(b)). When compared with control specimens, the average yield and ultimate loads for Specimens 6-1Fa&b, 9-1Fa&b, and 12-1Fa&b increased by 11%, 11, and 18%, and 14%, 11, and 29%, respectively. Thus, the relative increase in yield Py and ultimate Pmax loads for the 152 and 230 mm (6 and 9 in.) wide specimens strengthened with one CFRP strip relative to the respective control specimens (PyC and PmaxC) was roughly the same and taken approximately as 11%. For the 305 mm (12 in.) wide specimens, the yield load increased by 18% and the ultimate load increased by 29%. Therefore, a greater increase in both yield and ultimate load capacities was ACI Structural Journal/July-August 2007 achieved for the 305 mm (12 in.) wide specimens than for the 152 and 230 mm (6 and 9 in.) wide specimens. This is verification that the increase in strength is inversely proportional to the relative area of steel reinforcement (ρs/ρbs). For the strengthened specimens in this group, the average ultimate loads Pmax were between 13 and 20% greater than the average yield loads Py. Thus, the strength increase between yield and ultimate limit states is slightly greater for these specimens than for the control specimens (which was approximately 12%). This is expected and represents the additional tensile capacity provided by the CFRP after steel yield, which is not available for the control specimens. All specimens failed at loads slightly in excess of their respective predicted nominal flexural strength Pn. Referring to Table 2, the measured failure loads Pmax were between 6% (6-1Fb) and 14% (6-1Fa) greater than the theoretical strength Pn. The magnitude and range of this comparison suggest that the analytical model and associated assumptions used in Eq. (2) and (4) are acceptable for predicting the flexural capacity of these four test specimens. Specimens strengthened with two CFRP strips: 6-2Fa&b, 9-2Fa&b, and 12-2Fa&b Referring to Fig. 6, the change from cracked-elastic to inelastic behavior for the 230 and 305 mm (9 and 12 in.) wide specimens reinforced with two CFRP strips can still be seen. For the 152 mm (6 in.) wide specimens strengthened with two CFRP strips, however, this change from elastic to inelastic behavior is much less obvious from the load-deflection graphs. The load-strain curve for Specimen 6-2Fb, however, shows a clear redistribution of tensile force to the CFRP as a result of steel yield. It is therefore concluded that the steel did yield for these specimens (6-2Fa&b). Failure of all 152, 230, and 305 mm (6, 9, and 12 in.) wide specimens reinforced with two CFRP strips occurred by concrete crushing. This is consistent with the failure mode predicted in Table 1. After concrete crushing, the 305 mm (12 in.) wide specimens were further deformed until rupture of the CFRP occurred. This rupture is significant in that it again confirmed that force transfer is sufficient to develop the full tensile capacity of the CFRP strip. For all specimens, there was a significant increase in yield load Py relative to the respective companion control specimens PyC. Referring to Table 2, the yield loads for 152, 230, and 305 mm (6, 9, and 12 in.) wide specimens reinforced with two CFRP strips increased by 29, 18, and 27% over the control, respectively. Comparing results, the yield load increase for specimens with two CFRP strips was significantly higher than for specimens with one CFRP strip. Relative to the control specimens, the increase in ultimate load Pmax for the 152, 230, and 305 mm (6, 9, and 12 in.) wide specimens was 23, 44, and 61%, respectively. The trend in these values is consistent with those listed in Table 1, where the gain in ultimate strength increases with decreasing steel reinforcement ratio. Thus, in design, the expected additional strength from the CFRP must consider the existing relative amount of steel in the unstrengthened condition. For the 152 mm (6 in.) wide specimens with two CFRP strips, the average ultimate load was only 6% greater than the yield load. This indicates that at steel yield, the concrete strain was near ultimate so that any increase in strength is limited by the threshold level corresponding to concrete compression failure. For the 230 and 305 mm (9 and 12 in.) wide specimens, the average ultimate loads increased by 435
- 46. Table 3—Ductility results Yield Sample ID 6-C 6-1Fa 6-1Fb 6-2Fa 6-2Fb 9-C 9-1Fa 9-1Fb 9-2Fa 9-2Fb 12-C 12-1Fa 12-1Fb 12-2Fa 12-2Fb * E = † Δy , mm (in.) 22.17 (0.87) 19.51 (0.77) 23.06 (0.91) 24.66 (0.97) 25.26 (0.99) 21.05 (0.83) 21.14 (0.83) 24.16 (0.95) 20.76 (0.82) 22.15 (0.87) 17.55 (0.69) 19.50 (0.77) 20.56 (0.81) 20.23 (0.80) 19.90 (0.78) Ey*, Ultimate *, Deflection ductility kN-mm (kip-in.) Δu, mm (in.) Eu kN-mm (kip-in.) μd = Δu/Δy Ratio† 233 (2.07) 235 (2.08) 2823 (2.50) 353 (3.12) 354 (3.13) 280 (2.48) 323 (2.86) 331 (2.93) 344 (3.05) 323 (2.86) 228 (2.02) 296 (2.62) 317 (2.80) 334 (2.96) 334 (2.95) 30.23 (1.19) 28.98 (1.14) 29.30 (1.15) 26.19 (1.03) 31.04 (1.22) 47.03 (1.85) 36.80 (1.45) 44.45 (1.75) 40.81 (1.61) 47.87 (1.88) 44.68 (1.76) 44.09 (1.74) 47.36 (1.86) 46.10 (1.81) 58.55 (2.31) 395 (3.50) 455 (4.02) 423 (3.74) 389 (3.45) 503 (4.45) 909 (8.05) 729 (6.46) 863 (7.64) 989 (8.75) 1125 (9.96) 845 (6.80) 976 (8.64) 1081 (9.50) 1147 (10.15) 1732 (15.33) 1.36 1.49 1.27 1.06 1.23 2.23 1.74 1.84 1.97 2.16 2.55 2.26 2.30 2.28 2.94 1.00 1.09 0.93 0.78 0.90 1.00 0.78 0.82 0.88 0.97 1.00 0.89 0.90 0.89 1.16 Energy ductility μE = Eu/Ey 1.69 1.93 1.50 1.10 1.42 3.24 2.26 2.61 2.87 3.49 3.70 3.29 3.42 3.43 5.19 Ratio† 1.00 1.14 0.88 0.65 0.84 1.00 0.70 0.80 0.88 1.08 1.00 0.89 0.92 0.93 1.40 ∫ P dΔ . Ratio = {strengthened sample}/{control sample}. 38% over the yield loads. This is expected and represents the increased available capacity in the concrete at steel yield. This behavior is reflective of the relative amounts of both steel and CFRP reinforcement and how these reinforcement areas compare with that required for a balanced-strengthened design. Predicted flexural strength of all specimens with two CFRP strips was less than measured values, indicating the analytical model is conservative. Referring to Table 2, the measured loads were between 3 and 28% greater than predicted strengths. Thus, the model is an acceptable analytical tool for strength prediction in design. Ductility and energy The reported effect of flexural strengthening with external FRP reinforcement is a reduction in flexural ductility relative to the unstrengthened condition (ACI Committee 440 2002, Bencardino et al. 2002). Typically, ductility is calculated in terms of dimensionless deflection or energy ratios. Using these parameters ductility μ relative to the yield condition is defined as Deflection ductility: μd = Δu /Δy (5a) Energy ductility: μE = Eu /Ey (5b) In Eq. (5) Δu and Δy are the ultimate and yield center-span deflections, respectively, and Eu and Ey are the areas under the load-deflection diagrams at ultimate and yield, respectively. Numerical integration of the measured load-deflection diagrams was used to determine Eu and Ey. Ductility results are summarized in Table 3 where it is observed that most specimens experience a decrease in both deflection ductility and energy ductility relative to the control beams. The exceptions are Specimens 6-1Fa and 12-2Fb, which experienced an increase in both deflection and energy ductilities, and Specimen 9-2Fb, which experienced a slight increase in energy ductility. Under closer scrutiny, Specimen 12-2Fb, experienced a major crack at approximately 35 kN (7.84 kips). It could be argued that in a load controlled test this would have been the ultimate limit state for which Δu, Eu, μd , and μE are 32.3 mm (1.27 in.), 724.2 kN-mm (6.41 k-in.), 1.62, and 2.17, respectively. This reduces the deflection ductility and 436 energy ductility ratios to 0.64 and 0.60, respectively, resulting in a decrease in both ductility indexes. The experimental ductility analysis presented previously is subjective for two reasons. First, for some specimens, the yield limit state is not an instantaneous condition that occurs at a clearly defined load, deflection, or strain. Secondly, the ultimate limit state is also subject to interpretation. Thus, depending on the selection for the yield and ultimate limit states, a range of ductility results can be expected that may be slightly different from those reported in Table 3. The general conclusion, however, must be that ductility is decreased relative to the unstrengthened condition. Further parametric investigation of ductility using theoretical modeling to calculate deflection and strain is recommended. CONCLUSIONS The research presented in this study evaluated strength and ductility of steel reinforced concrete beams strengthened with near surface mounted CFRP strips. Experimental variables were the amount of steel and CFRP reinforcements. Steel reinforcement ratios ρs and concrete strength were selected as typical for existing concrete flexural members that would be found in nonprestressed bridge and building flexural members. The conclusions reported are restricted to the material properties (for concrete and CFRP), reinforcement ratios (ρs and ρf), type of CFRP (thin rectangular strips), and testing procedures that were used in this study. From the data presented, the following conclusions are made. 1. The strengthened beams failed in flexure as predicted according to the amounts of steel and CFRP reinforcement. All 152 and 230 mm (6 and 9 in.) wide specimens, and 305 mm (12 in.) wide specimens with two CFRP strips failed by steel yield followed by concrete crushing. The CFRP remained intact at concrete failure and no debonding was detected. These beams were predicted to fail in compression. The 305 mm (12 in.) wide specimens strengthened with one CFRP strip failed by steel yield followed by CFRP rupture. These beams were predicted to fail by CFRP rupture. In all cases, no debonding of the CFRP was detected; 2. All beams strengthened with CFRP failed at loads greater than their respective control beams. Relative to control specimen capacity, CFRP strengthened specimens ACI Structural Journal/July-August 2007
- 47. had measured increases in yield strength ranging from 9 to 30%, and measured increases in ultimate strength ranging from 10 to 78%. In general, the increase in strength was inversely proportional to the relative amount of steel reinforcement normalized to a balanced design ρs /ρsb; 3. The measured ultimate capacity of CFRP strengthened beams was between 6 and 28% greater than the respective predicted nominal strength. Nominal strength was calculated using a simplified closed-form analysis that yields identical results to the trial and error procedure given in ACI 440.2R-02. For unstrengthened beams, the measured ultimate strength was between 11 and 23% greater than the section’s predicted nominal strength. These ratios suggest that the CFRP strengthened section nominal flexural capacity is appropriately predicted using the simplified closed-form or ACI 440.2R-02 methodologies; 4. Force transfer between the CFRP, epoxy grout, and surrounding concrete was able to develop the full tensile strength of the CFRP strips. Tensile rupture of the single CFRP strip was achieved in the 305 mm (12 in.) wide specimens with no apparent slip or damage to the concrete cover or epoxy grout. For all other specimens where the CFRP did not fail, there was no apparent loss in force transfer between the CFRP, epoxy grout, and surrounding concrete. Thus, the CFRP strip’s thin rectangular cross section and roughened surface provide an effective mechanism of force transfer with this epoxy; and 5. For the specimens tested, there was no discernable trend between the change in ductility (energy and deflection) and the relative amount of steel reinforcement ρs/ρsb or CFRP strengthening reinforcement Afrp. With the exception of two strengthened beam, energy and deflection ductilities were reduced for CFRP strengthened beams. The authors suggest that additional research is required to study the strength and ductility behavior of a beam strengthened with wider range of combinations of steel and FRP reinforcement ratios. Furthermore, NSM FRP splice and bond behavior, appropriate code mandated design limitations for strength, deflection, and ductility need to be investigated. ACKNOWLEDGMENTS The authors wish to thank Hughes Brothers, Inc., for donating the CFRP reinforcement and the Office of Research and Sponsored Projects at Villanova University for providing financial support for this research. NOTATION Af , As Afb Asy = area of CFRP and steel reinforcement, respectively = balanced-strengthened area of CFRP = steel area corresponding to simultaneous concrete crushing and steel yielding a, av = depth of compression block at ultimate and shear span, respectively b, c = beam width and depth on neutral axis, respectively df , ds = depth to CFRP and steel reinforcement, respectively Ef , fc′ = FRP elastic modulus and concrete strength, respectively ff , fs = stress in CFRP and steel, respectively = ultimate strength of FRP (1648 MPa [239 ksi]) and steel ffu, fy yield strength, respectively ff-ult = calculated CFRP stress at sections theoretical moment strength = theoretical nominal moment strength Mn Pn = theoretical applied load corresponding to Mn PnC = theoretical applied load for control specimens corresponding to Mn Py , Pmax = measured load at steel yield and ultimate, respectively PyC, PmaxC = measured load for control specimen at steel yield and ultimate, respectively Tf , T s = tensile force in CFRP and steel, respectively wbeam = self-weight of beam = ratio of a/c β1 εf, εs = strain in CFRP and steel, respectively = ultimate strain of concrete (0.003) and FRP (0.012), respectively εcu, εfu ACI Structural Journal/July-August 2007 ρs, ρf ρsb = steel As/bds and CFRP Af /bdf reinforcement ratio, respectively = balanced steel reinforcement ratio for unstrengthened section REFERENCES ACI Committee 440, 2002, “Design and Construction of Externally Bonded FRP Systems for Strengthening Concrete Structures (ACI 440.2R-02),” American Concrete Institute, Farmington Hills, Mich., 45 pp. ACI Committee 440, 2004, “Guide Test Methods of Fiber-Reinforced Polymers (FRPs) for Reinforcing or Strengthening Concrete Structures (ACI 440.3R-04),” American Concrete Institute, Farmington Hills, Mich., 40 pp. ASTM C 684-99, 1999, “Standard Test Method for Making, Accelerated Curing, and Testing Concrete Compression Test Specimens,” ASTM International, West Conshohocken, Pa., 10 pp. Arduini, M., and Nanni, A., 1997, “Behavior of Precracked RC Beams Strengthened with Carbon FRP Sheets,” Journal of Composites for Construction, ASCE, V. 1, No. 2, pp. 63-70. Bencardino, F.; Spadea, G.; and Swamy, R., 2002, “Strength and Ductility of Reinforced Concrete Beams Externally Reinforced with Carbon Fiber Fabric,” ACI Structural Journal, V. 99, No. 2, Mar.-Apr., pp. 163-171. Brena, S. F.; Bramblett, R. M.; Wood, S. L.; and Kreger, M. E., 2003, “Increasing Flexural Capacity of Reinforced Concrete Beams Using Carbon Fiber-Reinforced Polymer Composites,” ACI Structural Journal, V. 100, No. 1, Jan.-Feb., pp. 36-46. DeLorenzis, L. A.; Nanni, A.; and Tegila, A. L., 2000, “Flexural and Shear Strengthening of Reinforced Concrete Structures with Near Surface Mounted FRP Bars,” Proceedings of the 3rd International Conference on Advanced Composite Materials in Bridges and Structures, Ottawa, Canada, Aug. 15-18, pp. 521-528. DeLorenzis, L., and Nanni, A., 2001, “Shear Strengthening of Reinforced Concrete Beams with Near-Surface Mounted Fiber-Reinforced Polymer Rods,” ACI Structural Journal, V. 98, No. 1, Jan.-Feb., pp. 60-68. DeLorenzis, L., and Nanni, A., 2002, “Bond between Near-Surface Mounted Fiber-Reinforced Polymer Rods and Concrete in Structural Strengthening,” ACI Structural Journal, V. 99, No. 2, Mar.-Apr., pp. 123-132. DeLorenzis, L.; Lundgren, K.; and Rizzo, A., 2004, “Anchorage Length of Near-Surface Mounted Fiber-Reinforced Polymer Bars for Concrete Strengthening—Experimental Investigation and Numerical Modeling,” ACI Structural Journal, V. 101, No. 2, Mar.-Apr., pp. 269-278. El-Hacha, R., and Rizkalla, S., 2004, “Near-Surface-Mounted FiberReinforced Polymer Reinforcements for Flexural Strengthening of Concrete Structures,” ACI Structural Journal, V. 101, V. 5, Sept.-Oct., pp. 717-726. Grace, N.; Abdel-Sayed, G.; and Ragheb, W., 2002, “Strengthening of Concrete Beams Using Innovative Ductile Fiber-Reinforced Polymer Fabric,” ACI Structural Journal, V. 99, No. 5, Sept.-Oct., pp. 692-700. Mukhopadhyaya, P., and Swamy, R. N., 1999, “Critical Review of Plate Anchorage Stresses in Premature Debonding Failures of Plate Bonded Reinforced Concrete Beams,” Fourth International Symposium on Fiber Reinforced Polymer Reinforcement for Reinforced Concrete Structures, SP-188, C. W. Dolan, S. H. Rizkalla, and A. Nanni, eds., American Concrete Institute, Farmington Hills, Mich., pp. 359-368. Nanni, A., 2000, “FRP Reinforcement for Bridge Structures,” Proceedings, Structural Engineering Conference, University of Kansas, Lawrence, Kans., Mar. 16, pp. 1-5. Nguyen, D.; Chan, T.; and Cheong, H., 2001, “Brittle Failure and Bond Development Length of CFRP-Concrete Beams,” Journal of Composites for Construction, ASCE, V. 5, No. 1, pp. 12-17. Pennsylvania Department of Transportation (PennDOT), 2001, “The Bridge Design Specification Sheet, BD-601M,” Specifications for the Concrete, Class AAA. Rahimi, H., and Hutchinson, A., 2001, “Concrete Beams Strengthened with Externally Bonded FRP Plates,” Journal of Composites for Construction, ASCE, V. 5, No. 1, Jan., pp. 44-55. Saadatmanesh, H., 1994, “Fiber Composites for New and Existing Structures,” ACI Structural Journal, V. 91, No. 3, May-June, pp. 346-354. Sharif, A.; Al-Sulaimani, G. J.; Basunbul, I. A.; Baluch, M. H.; and Ghaleb, B. N., 1994, “Strengthening of Initially Loaded Reinforced Concrete Beams Using FRP Plates,” ACI Structural Journal, V. 91, No. 2, Mar.-Apr., pp. 160-168. Shin, Y. S.; and Lee, C., 2003, “Flexural Behavior of Reinforced Concrete Beams Strengthened with Carbon Fiber-Reinforced Polymer Laminates at Different Levels of Sustaining Load,” ACI Structural Journal, V. 100, No. 2, Mar.-Apr., pp. 231-239. Taljsten, B., and Carolin, A., 2001, “Concrete Beams Strengthened with Near Surface Mounted CFRP Laminates,” Proceedings of the Non-Metallic Reinforcement for Concrete Structures, FRP RCS-5 Conference, July 16-18, Cambridge, UK, pp. 107-116. Teng, J. G.; Chen, J. F.; Smith, S. T.; and Lam, L., 2002, FRP-Strengthened RC Structures, John Wiley & Sons, West Sussex, UK, 266 pp. 437
- 48. ACI STRUCTURAL JOURNAL TECHNICAL PAPER Title no. 104-S40 Influence of Shear Reinforcement on Reinforced Concrete Continuous Deep Beams by Keun-Hyeok Yang, Heon-Soo Chung, and Ashraf F. Ashour Test results of 24 reinforced concrete continuous deep beams are reported. The main variables studied were concrete strength, shear span-to-overall depth ratio (a/ h) and the amount and configuration of shear reinforcement. The results of this study show that the load transfer capacity of shear reinforcement was much more prominent in continuous deep beams than in simply supported deep beams. For beams having an a/ h of 0.5, horizontal shear reinforcement was always more effective than vertical shear reinforcement. The ratio of the load capacity measured and that predicted by the strutand-tie model recommended by ACI 318-05 dropped against the increase of a/h. This decrease rate was more remarkable in continuous deep beams than that in simple deep beams. The strut-and-tie model recommended by ACI 318-05 overestimated the strength of continuous deep beams having a/ h more than 1.0. Keywords: beams; load; shear reinforcement; strut-and-tie model. INTRODUCTION Reinforced concrete deep beams are used in structures as load distribution elements such as transfer girders, pile caps, and foundation walls in tall buildings. Although these members commonly have several supports, extensive experimental investigations have brought simple deep beams into focus. The behavior of continuous deep beams is significantly different from that of simply supported deep beams. The coexistence of high shear and high moment within the interior shear span in continuous deep beams has a considerable effect on the development of cracks, leading to a significant reduction in the effective strength of the concrete strut, which is the main load transfer element in deep beams.1 Indeed, few experiments1-3 were carried out on continuous deep beams of shear span-to-overall depth ratio (a/h) greater than 1.08. The results of simple deep beams tested by Tan et al.4 and Smith and Vantsiotis,5 however, showed that the relative effectiveness of horizontal and vertical shear reinforcement on controlling diagonal cracks and enhancing load capacity reversed for deep beams having an a/h less than 1.0, that is, horizontal shear reinforcement was more effective for an a/h below 1.0, whereas vertical shear reinforcement was more effective for an a/h lager than 1.0. Therefore, a reasonable evaluation of the influence of shear reinforcement on continuous deep beams having an a/h less than 1.0 requires further investigation. The current codes6-8 and several researchers9-12 have recommended the design of deep beams using the strut-andtie model. In these strut-and-tie models, the main function of shear reinforcement is to restrain diagonal cracks near the ends of bottle-shaped struts and to give some ductility to struts. ACI 318-05, Section A.3.3, allows the use of an effectiveness factor of 0.75 when computing the effective concrete compressive strength of bottle-shaped struts with reinforcement satisfying ACI 318-05, Section A.3.3. The value of the effectiveness factor drops to 0.6 if shear reinforce420 ment as recommended by ACI 318-05, Section A.3.3, is not provided. This implies that shear reinforcement satisfying ACI 318-05, Section A.3.3, would increase the ultimate strength of beams predicted by the strut-and-tie model by 25%. Studies on the validity of the strut-and-tie model recommended by ACI 318-05, however, are very rare even in simple deep beams.12-14 This paper presents test results of 24 two-span reinforced concrete deep beams. The main variables included concrete strength, a/h, and the amount and configuration of shear reinforcement. The influence of shear reinforcement on the ultimate shear strength in continuous deep beams was compared with that in the corresponding simple ones. The load capacity predictions of reinforced concrete continuous deep beams by the strut-and-tie model of ACI 318-05 were evaluated by comparison with test results. RESEARCH SIGNIFICANCE A great deal of research has focused on simply supported deep beams. Even the few tests on continuous deep beams were carried out on beams having an a/h exceeding 1.0 and concrete strength less than 35 MPa (5.0 ksi). Test results in this study clearly showed the influence of shear reinforcement on the structural behavior of continuous deep beams according to the variation of concrete strength and a/h. The ultimate shear strength of continuous deep beams and load transfer capacity of shear reinforcement were compared with those of the corresponding simple deep beams and the predictions obtained from the strut-and-tie model recommended in ACI 318-05. EXPERIMENTAL INVESTIGATION The details of geometrical dimensions and reinforcement of test specimens are shown in Table 1 and Fig. 1. The main variables studied were compressive strength of concrete fc′ , a/h, and the amount and configuration of shear reinforcement. Beams tested were classified into two groups according to the concrete compressive strength: L-series for design concrete strength of 30 MPa (4350 psi) and H-series for design concrete strength of 60 MPa (8700 psi). The a/h were initially designed to be 0.5 and 1.0 to allow comparison with current results with those reported by Yang13 for simple deep beams. The value of a/h in H-series, however, was increased from 0.5 to 0.6, as the capacity of beams having fc′ of 60 MPa (8700 psi) and an a/h of 0.5 had exceeded the capacity of the loading machine in the pilot test. The ACI Structural Journal, V. 104, No. 4, July-August 2007. MS No. S-2006-206.R1 received May 20, 2006, and reviewed under Institute publication policies. Copyright © 2007, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published in the May-June 2008 ACI Structural Journal if the discussion is received by January 1, 2008. ACI Structural Journal/July-August 2007
- 49. Table 1—Details of test specimens Keun-Hyeok Yang is a Visiting Research Fellow at the University of Bradford, UK, and an Assistant Professor at Mokpo National University, Korea. He received his MSc and PhD from Chungang University, Korea. His research interests include ductility, strengthening, and shear of reinforced, high-strength concrete structures. Heon-Soo Chung is a Professor at Chungang University, Korea. He received his MSc and PhD from Tokyo Institute of Technology, Japan. His research interests include flexure, shear, and bond behavior of reinforced, high-strength concrete members. Ashraf F. Ashour is a Senior Lecturer at the University of Bradford, UK. He received his BSc and MSc from Mansoura University, Egypt, and his PhD from Cambridge University, UK. His research interests include shear, plasticity, and optimization of reinforced concrete and masonry structures. Details of shear reinforcement fc′ , Specimen MPa Horizontal a/h a/jd L, mm sh, mm ρh Vertical sv , mm ρv L5NN — — — — L5NS — — 120 0.003 0.006 L5NT — — 60 120 0.003 — — L5SS 120 0.003 120 0.003 — L5SN 32.4 0.5 0.58 600 L5TN 60 0.006 — L10NN — — — — L10NS — — 120 0.003 0.006 L10NT — — 60 120 0.003 — — L10SS 120 0.003 120 0.003 L10TN 60 0.006 — — H6NN — — — — H6NS — — 120 0.003 0.006 L10SN H6NT 32.1 1.0 1.17 1200 — — 60 120 0.003 — — H6SS 120 0.003 120 0.003 — H6SN 65.1 0.6 0.7 720 H6TN 60 0.006 — H10NN — — — — H10NS — — 120 0.003 0.006 H10NT — — 60 120 0.003 — — H10SS 120 0.003 120 0.003 H10TN 60 0.006 — — H10SN 68.2 1.0 1.17 1200 Note: 1 MPa = 145 psi; 1 mm = 0.039 in. Fig. 1—Geometrical dimensions and reinforcement of test specimens. (Note: all dimensions are in mm and • indicates locations of strain gauges. 1 mm = 0.039 in.) configuration of shear reinforcement included four different arrangements as shown in Fig. 1: none, only vertical, only horizontal, and orthogonal reinforcement. The spacing of shear reinforcement was chosen to be 60 and 120 mm (2.36 and 4.72 in.) and the corresponding shear reinforcement ratios, ρ (= Aw /bw s, where Aw equals the area of shear reinforcement at spacing s, and bw equals the beam width), were 0.003 and 0.006, respectively, to satisfy the maximum spacing specified in ACI 318-05, Section 11.8, and the minimum amount recommended in ACI 318-05, Section A.3.3.2. The beam notation given in Table 1 includes four parts. The first part refers to the concrete design strength: L for low compressive strength and H for high compressive strength. The second part is used to identify the a/h. The third and fourth parts give the amount of horizontal and vertical shear reinforcement, ACI Structural Journal/July-August 2007 respectively: N for no shear reinforcement, and S and T for shear reinforcement ratios of 0.003 and 0.006, respectively. For example, L5-SS is a continuous deep beam having design concrete strength of 30 MPa (4350 psi), an a/h of 0.5, and both horizontal and vertical shear reinforcement ratios of 0.003. All beams tested had the same section width bw of 160 mm (6.3 in.) and overall section depth h of 600 mm (23.6 in.). Both longitudinal top, ρs = (As′ /bwd), and bottom, ρs = (As /bwd), ′ reinforcement ratios were kept constant in all beams as 1%, which were calculated from nonlinear FE analysis,15 to ensure no flexural yielding of longitudinal reinforcement prior to failure of concrete struts. The length of each span L varied according to a/h, as given in Table 1. The clear covers to longitudinal top and bottom reinforcement, and shear reinforcement were 35 and 29 mm (1.38 and 1.14 in.), respectively. The longitudinal bottom reinforcement was continuous over the full length of the beam and welded to 160 x 100 x 10 mm (6.3 x 3.9 x 0.39 in.) end plates, whereas longitudinal top reinforcement was anchored in the outside of the exterior supports by 90-degree hooks according to ACI 318-05. The vertical shear reinforcement was closed stirrups and the horizontal shear reinforcement with 90 degree hooks was arranged along the longitudinal axis in both sides of the beams. Material properties The mechanical properties of reinforcement are given in Table 2. All longitudinal and shear reinforcing bars were deformed bars of a 19 mm (0.75 in.) diameter, having a nominal area of 287 mm2 (0.44 in.2) and yield strength of 562 MPa (81.6 ksi) and a 6 mm (0.23 in.) diameter, having a nominal area of 28.2 mm2 (0.04 in.2) and yield strength of 421
- 50. Fig. 2—Test setup. (Note: all dimensions are in mm. 1 mm = 0.039 in.) Table 2—Mechanical properties of reinforcement fy, MPa εy fsu, MPa Es, GPa * 6 483 0.0044 549 199 19 562 0.00284 741 198 Diameter, mm * Yield stress of 6 mm diameter reinforcement was obtained by 0.2% offset method. Note: 1 mm = 0.039 in.; 1 MPa = 145 psi. 483 MPa (70 ksi), respectively. The yield strength of 6 mm (0.23 in.) diameter reinforcement was obtained by 0.2% offset method. The ingredients of ready mixed concrete were ordinary portland cement, fly ash, irregular gravel of a maximum size of 25 mm (0.98 in.), and sand. The water-binder ratios of the L-series added with fly ash of 12% and of the H-series added with fly ash of 20% were 0.41 and 0.27, respectively. All specimens were cast in a vertical position in the same wooden mold. Control specimens, which were 100 mm (3.94 in.) diameter by 200 mm (7.87 in.) high cylinders, were cast and cured simultaneously with beams to determine the compressive strength. They were tested soon after the beam test. The results of the cylinder compressive strength given in Table 1 are the average value from testing nine cylinders. Test setup Loading and instrumentation arrangements are shown in Fig. 2. All beams having two spans were tested to failure under a symmetrical two-point top loading system with a loading rate of 30 kN/minute (6.7 kip/minute) using a 3000 kN (675 kip) capacity universal testing machine (UTM). Each span was identified as E-span or W-span, as shown in Fig. 1. The two exterior end supports were designed to allow horizontal and rotational movements, whereas the intermediate support prevented horizontal movement but allowed rotation. To evaluate the shear force and loading distribution, 1000 kN (225 kip) capacity load cells were installed in both exterior end supports. At the location of loading or support point, a steel plate of 100, 150, or 200 mm (3.94, 5.9, or 7.88 in.) wide was provided to prevent premature crushing or bearing failure, as shown in Fig. 2. All steel plates were 50 mm (1.97 in.) thick and 300 mm (11.8 in.) long to cover the full width of test specimen. All beams were preloaded up to a total load of 150 kN (33.7 kip) before testing, which wouldn’t produce any cracks, to assure a similar loading distribution to supports according to the result of the linear two-dimensional finite element (2-D FE) analysis. Vertical deflections at a distance of 0.45L to 0.47L from the exterior support, which is the location of the maximum 422 Fig. 3—Crack patterns and failure of concrete strut. Numbers indicate total load in kN at which crack occurred. (Note: 1 kN = 0.2248 kips.) deflection predicted by the linear 2-D FE analysis, and at the midspan of each span were measured using linear variable differential transformers (LVDTs). Both surfaces of the beams tested were whitewashed to aid in the observation of crack development during testing. The inclined crack width of concrete struts joining the edges of load and support plates was monitored by the π-shape displacement transducers (PI gauges) as shown in Fig. 2. The strains of shear reinforcement were measured by 5 mm (0.2 in.) electrical resistance strain gauges (ERS) at the region crossing the line joining the edges of load and intermediate support plates as shown in Fig. 1. At each load increment, the test data were captured by a data logger and automatically stored. Support settlements Continuous deep beams are sensitive to differential support settlements causing additional moment and shear. To assess the effect of differential settlements on the beams tested, a linear 2-D FE analysis considering shear deformation effect was performed on the beams shown in Fig. 1. For the beams tested, sources of relative support settlements were the elastic shortening of the load cell and plates and elastic deformation of the bed of the testing machine. The second moment of area of the testing machine bed cross section about the bending axis was 3.2 × 1010 mm4 (7.69 × 104 in.4), then the elastic deformation under a point load R (in kN) at a distance 1500 mm (59 in.) from the center of the testing machine is 0.000176R mm. The amount of elastic shortening due to a load at the exterior and intermediate supports involving the load cell and plates was considered in designing the support size as follows. When a/h is 0.5, the reactions of the exterior and intermediate supports due to the total applied load P, from the linear 2-D FE analysis, are 0.2P and 0.6P, respectively. As the height of the intermediate support was equal to that of the exterior load cell, the contact area of the intermediate support with the bed of the testing machine was designed to be three times wider than that of the load cell at the exterior support to produce the same elastic shortening. The pilot test results showed that the maximum settlement of the exterior support relative to the intermediate support was in order of L/25,000. For a differential settlement between the exterior and intermediate supports of L/25,000, the maximum additional shear forces obtained from linear 2-D FE analysis are 25 and 7 kN (5.62 and 1.57 kip) for beams ACI Structural Journal/July-August 2007
- 51. Table 3—Details of test results and predictions obtained from ACI 318-05 Failure load Pn and ultimate shear force (Vn)I at interior shear spans, kN Load Pcr and shear force Vcr at first diagonal crack, kN W-span Interior Specimen (Pcr)I (Vcr)I E-span Exterior (Pcr)E (Vcr)E Interior (Pcr)I (Vcr)I ACI 318-05 (Vn)I Exterior (Pcr)E (Vcr)E Pn W-span E-span (Pn)Exp./ (Vn)I-Exp./ Pn, kN (Vn)I, kN (Pn)ACI (Vn)I-ACI L5NN 852 255 902 180 816 244 937 187 1635 473 456* 1298 342 1.260 1.334 L5NS 849 247 1028 210 857 262 1330 281 1710 486 475* 1298 342 1.317 1.389 L5NT 1017 278 1380 284 850 230 1260 262 1789 512* 494 1298 342 1.378 1.498 L5SN 864 255 1268 252 867 257 927 179 1887 537* 546 1298 342 1.454 1.571 * L5SS 814 247 990 192 980 293 1020 202 2117 607 583 1623 427 1.305 1.420 L5TN 912 266 1130 230 910 278 966 185 2317 655 * 640 1298 342 1.785 1.872 L10NN 537 173 — — 537 171 — — 880 264* 262 1000 265 0.880 0.997 L10NS 477 156 — — 596 195 — — 1153 349 * 348 1000 265 1.153 1.314 L10NT 635 206 1023 230 647 208 — — 1541 446* 439 1000 265 1.541 1.684 L10SN 498 153 — — 490 151 782 146 884 266 265* 1000 265 0.884 1.000 L10SS 521 166 — — 452 148 713 129 1177 357 352* 1250 331 0.942 1.063 287 288* 1000 265 0.935 1.087 L10TN 538 175 — — 621 193 775 143 935 * H6NN 1046 305 1562 321 1236 303 1960 407 2248 633 634 2520 668 0.892 0.950 H6NS 1261 379 1646 316 978 300 2280 457 2289 684 683* 2520 668 0.908 1.023 H6NT 1116 324 2550 550 915 264 2480 531 2625 757 757* 2520 668 1.042 1.134 H6SN 1322 393 2420 517 1022 297 2420 513 2427 703* 708 2520 668 0.963 1.053 H6SS 1207 367 2630 548 825 256 2630 542 2763 792 799* 3150 834 0.877 0.958 H6TN 1442 439 — — 980 297 2648 540 2966 854 852* 2520 668 1.177 1.276 H10NN 690 228 868 149 690 228 840 143 1276 373 372* 2124 563 0.601 0.661 H10NS 759 237 — — 751 234 — — 1443 413* 414 2124 563 0.679 0.734 H10NT 788 251 — — 717 224 — — 2116 638 637* 2124 563 0.996 1.132 H10SN 757 255 — — 757 252 — — 1309 387* 378 2124 563 0.616 0.688 H10SS 718 232 — — 768 244 — — 1575 492* 484 2655 703 0.593 0.699 393 * 2124 563 0.606 0.689 H10TN 754 234 — — 704 220 — — 1287 388 * Failure occurred in this shear span. Note: 1 kN = 0.2248 kips. having an a/h of 0.5 and 1.0, respectively. This indicates that the differential settlement had no significant effect on the test arrangement. EXPERIMENTAL RESULTS AND DISCUSSION Crack propagation and failure mode The crack propagation was significantly influenced by the a/h as shown in Fig. 3 and Table 3. The crack pattern in the L-series was similar to that in the H-series; therefore, it is not shown in Fig. 3. For beams with a/h = 0.5, the first crack suddenly developed in the diagonal direction at approximately 40% of the ultimate strength at the middepth of the concrete strut within the interior shear span, and then a flexural crack in the sagging region immediately followed. The first flexural crack over the intermediate support generally occurred at approximately 80% of the ultimate strength, and was less than 0.2h deep at failure. As the load increased, more flexural and diagonal cracks were formed and a major diagonal crack extended to join the edges of the load and intermediate support plates. A diagonal crack within the exterior shear span occurred suddenly near the failure load. Cracks in beams with a/h = 1.0 developed in a different order from that described previously for beams with a/h = 0.5. In those beams, the first crack occurred vertically in the hogging ACI Structural Journal/July-August 2007 zone, followed by a diagonal crack in the interior shear span, and then a vertical crack took place in the sagging zone, but diagonal cracks within exterior shear spans were seldom developed. The influence of shear reinforcement on the first flexural and diagonal crack loads was not significant (refer to Table 3) as also observed in simple deep beams given in Appendix A. Just before failure, the two spans showed nearly the same crack patterns. All beams developed the same mode of failure as observed in other experiments.3 The failure planes evolved along the diagonal crack formed at the concrete strut along the edges of the load and intermediate support plates. Two rigid blocks separated from original beams at failure due to the significant diagonal cracking. An end block rotated about the exterior support leaving the other block fixed over the other two supports as shown in Fig. 3. Load versus midspan deflection The beam deflection at midspan was less than that measured at 0.45L to 0.47L from the exterior support until the occurrence of the first diagonal crack as predicted by the 2-D FE analysis. After the first diagonal crack, however, the midspan deflection was higher. Therefore, the midspan 423
- 52. Fig. 5—Total applied load versus support reactions for L-series beams tested having a/ h of 0.5. (Note: 1 kN = 0.2248 kips.) This stiffness reduction was prominent in case of lower concrete strength and higher a/h. Support reaction Figure 5 shows the amount of the load transferred to the end and intermediate supports against the total applied load in the L-series beams having a/h = 0.5. On the same figure, the support reactions obtained from the linear 2-D FE analysis are also presented. The end and intermediate support reactions of the L-series beams having a/h = 1.0 and the H-series beams were similar to those of the L-series beams having an a/h = 0.5; therefore, not presented herein. Before the first diagonal crack, the relationship of the end and intermediate support reactions against the total applied load in all beams tested shows good agreement with the prediction of the linear 2-D FE analysis. The amount of loads transferred to the end support, however, was slightly higher than that predicted by the linear 2-D FE analysis after the occurrence of the first diagonal crack within the interior shear span. At failure, the difference between the measured end support reaction and prediction of the linear 2-D FE analysis was in order of 7 and 12%, for beams with a/h = 0.5 and a/h = 1.0, respectively. The distribution of applied load to supports was independent of the amount and configuration of shear reinforcement. This means that, although after the occurrence of diagonal cracks the beam stiffness has reduced, as shown in Fig. 4, the internal redistribution of forces is limited. Fig. 4—Total load versus midspan deflection. (Note: 1 kN = 0.2248 kips.) deflection of the failed span for different beams tested are only presented in Fig. 4 against the total applied load: Fig. 4(a) for beams in the L-series and Fig. 4(b) for beams in the H-series. The initial stiffness of beams tested increased in accordance with the increase of concrete strength and the decrease of the a/h, but it seems to be independent of the amount and configuration of shear reinforcement. The development of flexural cracks in sagging and hogging zones has little influence on the stiffness of beams tested. But the occurrence of diagonal cracks in the interior shear span caused a sharp decrease in the beam stiffness and an increase of the beam deflection. 424 Width of diagonal crack Figure 6 shows the variation of the diagonal crack width in the interior shear span according to the configuration of shear reinforcement: Fig. 6(a) at the first diagonal cracking load and Fig. 6(b) at the same load as the ultimate failure load of the corresponding deep beam without shear reinforcement. For the same concrete compressive strength, the larger the a/h, the wider the diagonal crack width. Shear reinforcement had an important role in restraining the development of the diagonal crack width, which significantly depended on the a/h. A more prominent reduction of diagonal crack width appeared in beams with horizontal shear reinforcement only or orthogonal shear reinforcement than in beams with vertical shear reinforcement only when a/h was 0.5. On the other hand, for beams with a/h = 1.0, a smaller diagonal crack width was observed in beams with vertical shear reinforcement only than in beams with orthogonal shear reinforcement, even though the total shear reinforcement ratio in these beams was the same (ρv + ρh = 0.006). It seems possible to reduce the diagonal crack width by more than twice if shear reinforcement is suitably arranged according to the variation of a/h. ACI Structural Journal/July-August 2007
- 53. Fig. 7—Total load versus strains in shear reinforcement for beams in H-series. (Note: 1 kN = 0.2248 kips.) Fig. 6—Configuration of shear reinforcement versus diagonal crack width. (Note: 1 mm = 0.039 in.) Figure 7 shows the strain in shear reinforcement against the total applied load in the H-series beams: Fig. 7(a) for vertical shear reinforcement in beams having either vertical or orthogonal shear reinforcement, and Fig. 7(b) for horizontal shear reinforcement in beams having either horizontal or orthogonal shear reinforcement. The relation between strains in shear reinforcement and the total applied load in the L-series beams was similar to that in the H-series beams; therefore, not presented herein. The strains of shear reinforcement were recorded by ERS gauges at different locations, as shown in Fig. 1. Shear reinforcement was not generally strained at initial stages of loading. However, strains suddenly increased with the occurrence of the first diagonal crack. In beams with a/h = 0.6, only horizontal reinforcing bars yielded, whereas in beams with a/h = 1.0, only vertical reinforcing bars yielded. This indicates that the reinforcement ability to transfer tension across cracks strongly depends on the angle between the reinforcement and the axis of the strut. Ultimate shear stress The normalized ultimate shear strength, λ = Vn/bwd f c′ , plotted against a/h, is given in Fig. 8: Fig. 8(a) for simply supported deep beams given in Appendix A, and Fig. 8(b) for continuous deep beams including the test results of Rogowsky et al.1 and Ashour.2 It can be seen that the ultimate shear strength of all beams without or with shear reinforcement dropped due to the increase of a/h. The reduction of the ultimate shear strength was also dependent on the configuration of shear reinforcement. For deep beams without shear reinforcement, the normalized ultimate shear strength λ in continuous deep beams was less than that in simply supported ones by an average of 26% due to higher transverse tensile strains produced by the tie action of longitudinal top and bottom ACI Structural Journal/July-August 2007 Fig. 8—Normalized ultimate shear strength versus shear span-to-overall depth ratio. reinforcement. When shear reinforcement is provided, the normalized ultimate shear strength λ in continuous deep beams matched that of the corresponding simply supported ones. The influence of the horizontal and vertical shear reinforcement on the ultimate shear strength is influenced by the a/h. The lower the a/h, the more effective the horizontal shear reinforcement and the less effective the vertical shear reinforcement. When a/h was below 0.6, the shear strength 425
- 54. Vn – ( Vn )W ⁄ O Vs ----- = ------------------------------Vn Vn (2) The variations of Vs /Vn at the failed shear span against the increase of a/h are given in Fig. 9: Fig. 9(a), (b), and (c) for beams with vertical shear reinforcement only, with horizontal shear reinforcement only, and with orthogonal shear reinforcement, respectively. On the same figure, the test results of simply supported deep beams given in Appendix A, which had the same material and geometrical properties as continuous deep beams tested in the current study, are also presented. The load transfer capacity of shear reinforcement is more pronounced in continuous deep beams than that in simple ones. The load transfer capacity of shear reinforcement is dependent on a/h. The load transfer capacity of vertical shear reinforcement was higher in beams having a/h = 1.0 than those having a/h = 0.5 as shown in Fig. 9(a). On the other hand, the load transfer capacity of horizontal shear reinforcement was higher in beams having a/h = 0.5 than those having a/h = 1.0, as shown in Fig. 9(b). Existing test results of continuous deep beams carried out by Rogowsky et al.1 and Ashour,2 and the comments of ACI 318-05, Section 11.8, have suggested that horizontal shear reinforcement has little influence on the shear strength improvement and crack control. In the current tests, horizontal shear reinforcement is more effective than vertical shear reinforcement for beams with a/h of 0.5, as shown in Fig. 8 and 9. Comparison with current codes It has been shown by several researchers,1,2,4 that the shear capacity prediction of reinforced concrete deep beams obtained from ACI 318-9916 (unchanged since 1983) was unconservative. For the design of deep beams, ACI 318-05 requires the use of either nonlinear analysis or strut-and-tie model. Figure 10 shows a schematic strut-and-tie model of continuous deep beams in accordance with ACI 318-05, Appendix A. The strut-and-tie model shown in Fig. 10 identifies two main load transfer systems: one of which is the strut-and-tie action formed with the longitudinal bottom reinforcement acting as a tie and the other is the strut-and-tie action due to the longitudinal top reinforcement. As the applied loads in the two-span continuous deep beams are carried to supports through concrete struts of exterior and interior shear spans (refer to Fig. 10), the total load capacity of two-span continuous deep beams Pn due to failure of concrete struts is Fig. 9—Shear reinforcement ratios versus Vs / Vn. Pn = 2(FE – FI)sinθ of deep beams with minimum horizontal shear reinforcement had an average value of 150% higher than the upper bound value, 0.83 f c′ bwd, specified in ACI 318-05, Section 11.8.3. Load transfer capacity of shear reinforcement The shear strength of deep beams Vn can be described as follows Vn = Vc + Vs where FE and FI equal the load capacities of exterior and interior concrete struts, respectively, and θ equals the angle between the concrete strut and the longitudinal axis of the deep beam, which can be expressed as tan–1(jd/a). The distance between the center of top and bottom nodes jd could be approximately assumed as the distance between the center of longitudinal top and bottom reinforcing bars as (1) jd = h – c – c′ where Vc and Vs equal the load capacity of concrete and load transfer capacity of shear reinforcement, respectively. As the load capacity of concrete is usually regarded as the strength of beams without shear reinforcement, (Vn)W/O, the ratio of the load transfer capacity of shear reinforcement to the shear strength of beams Vs/Vn is 426 (3) (4) where h equals the overall section depth and c and c′ equal the cover of longitudinal bottom and top reinforcement, respectively, as shown in Fig. 10. The nodes at the applied load point could be classified as a CCC type, which is a hydrostatic node connecting both ACI Structural Journal/July-August 2007
- 55. exterior and interior compressive struts in sagging zone and a CCT type for longitudinal top reinforcement in the hogging zone. It was proved by Marti10 that the width of the strut at a CCC node is in proportion to the principal stress normal to the node face to make the state of stress in the whole node region constant. To accommodate both CCC type and CCT type, the loading plate width can be assumed to be subdivided into two parts in accordance with the ratio of the exterior reaction to the applied load β, each to form the node connecting the exterior and the interior struts, respectively. The β values of tested beams are 0.4 and 0.346 when a/h ratios are 0.5 and 1.0, respectively, as estimated from the linear 2-D FE analysis. If enough anchorage of longitudinal reinforcement is provided, average widths of concrete struts in interior (ws)I and exterior shear spans (ws)E are ( w t ′ + 2c′ ) cos θ + [ 0.5 ( l p ) I + ( 1.0 – β ) ( l p ) P ] sin θ ( w s ) I = ----------------------------------------------------------------------------------------------------------------------2 (5a) ( w t ′ + 2c′ ) cos θ + [ ( l p ) E + β ( l p ) P ] sin θ ( w s ) E = --------------------------------------------------------------------------------------------2 Fig. 10—Qualitative strut-and-tie model of continuous deep beams according to ACI 318-05. (5b) where (lp)P, (lp)E, and (lp)I equal the widths of loading, exterior support, and interior support plates, respectively, and wt′ equals the smaller of the height of the plate anchored to longitudinal bottom reinforcement wt and twice of the cover of longitudinal bottom reinforcement 2c as shown in Fig. 10. The load transfer capacity of the concrete strut depends on the area of the strut and the effective concrete compressive strength. Hence, the load capacities of the exterior and interior concrete struts are FE = ve f ′cbw(ws)E (6a) FI = ve f ′cbw(ws)I (6b) where ve equals the effectiveness factor of concrete. The shear capacity at the interior shear span (Vn)I, where the failure is expected to occur in continuous deep beams, can be calculated from FI sinθ. The minimum amount of shear reinforcement required in bottle-shaped struts, which is recommended to be placed in two orthogonal directions in each face, is suggested by ACI 318-05 as follows A si ∑ ----------i sin αi ≥ 0.003 bw s (7) Fig. 11—Comparison of test results and predictions by ACI 318-05. where Asi and si equal the total area and spacing in the i-th layer of reinforcement crossing a strut, respectively, and αi equals the angle between i-th layer of reinforcement and the strut. The effectiveness factors for concrete strength not exceeding 40 MPa (5.8 ksi) in ACI 318-05 are suggested as 0.75 and 0.6 when shear reinforcement satisfying Eq. (7) is arranged and is not provided, respectively. The truss model representing the load transfer mechanism of horizontal and vertical shear reinforcement is not included in ACI 318-05. This implies that shear reinforcement satisfying Eq. (7) enables the strength of beams to be increased by 25%. Comparisons between test results and predictions obtained from the strut-and-tie model recommended by ACI 318-05 as developed previously are shown in Table 3 and Fig. 11: Fig. 11(a) for simple deep beams given in Appendix A and Fig. 11(b) for continuous deep beams including Rogowsky et al.’s and Ashour’s test results. In simple deep beams, the width of the strut can be calculated from wt′cosθ + (lp)Esinθ, and the total load capacity is 2FEsinθ. Although Eq. (7) proposed by ACI 318-05 is recommended for deep beams having concrete strength of less than 40 MPa, the load capacity of the H-series beams were also predicted using this equation to evaluate its conservatism in case of high-strength concrete deep beams. The mean and standard deviation of the ratio, (Pn)Exp./(Pn)ACI, between the experimental and predicted load capacities are 1.229 and 0.326, respectively, for simply supported deep beams, and 0.969 and 0.306, ACI Structural Journal/July-August 2007 427
- 56. respectively, for two-span continuous deep beams as shown in Fig. 11. The ratio of the test result to prediction generally dropped with the increase of a/h. This decrease rate was more remarkable in continuous deep beams than that in simple ones. In particular, the predictions for several continuous deep beams having a/h exceeding 1.0 were unconservative, even though the effectiveness factor used in the beams with either horizontal or vertical shear reinforcement was 0.6 regardless of the amount of shear reinforcement. In addition, for high-strength concrete continuous deep beams having a/h = 1.0, the ratio, (Vn)I-Exp./(Vn)I-ACI, between the experimental and predicted shear capacities in the interior shear span was generally below 1.0 as given in Table 3; namely, the strutand-tie model recommended by ACI 318-05 overestimated the shear capacity of high-strength concrete continuous deep beams having a/h = 1.0. CONCLUSIONS Tests were performed to study the influence of the amount and configuration of shear reinforcement on the structural behavior of continuous deep beams according to the variation of concrete strength and a/h. The following conclusions are drawn: 1. In beams having a/h of 0.6, only horizontal shear reinforcement reached its yield strength with a sharp increase of stress after the first diagonal crack. On the other hand, only vertical shear reinforcement yielded in beams with a/h of 1.0; 2. For deep beams without shear reinforcement, the normalized ultimate shear strength was 26% lower in continuous beams than that in simple ones. When shear reinforcement was provided, however, the normalized ultimate shear strength in continuous deep beams matched that in simply supported deep beams; 3. The load transfer capacity of all shear reinforcement was much more prominent in continuous deep beams than that in simple ones. Horizontal shear reinforcement was always more effective than vertical shear reinforcement when the a/h was 0.5. However, vertical shear reinforcement was more effective for a/h higher than 1.0; 4. In deep beams with a/h not exceeding 0.6, the critical upper bound on shear strength suggested in ACI 318-05, 0.83 f c ′ bwd, highly underestimated the actual measured shear strength, as if it was a lower limit; and 5. The ratios of measured load capacity to that obtained from the strut-and-tie model recommended by ACI 318-05 dropped with the increase of the a/h. This decrease rate was more remarkable in continuous deep beams than that in simple ones. The strut-and-tie model recommended by ACI 318-05 overestimated the shear capacity of high-strength concrete continuous deep beams having a/h more than 1.0. ACKNOWLEDGMENTS This work was supported by the Korea Research Foundation Grant (KRF-2003-041-D00586) and the Regional Research Centers Program (Bio-housing Research Institute), granted by the Korean Ministry of Education and Human Resources Development. The authors wish to express their gratitude for financial support. NOTATION Ah As As′ Aw a bw 428 = = = = = = area of horizontal shear reinforcement area of longitudinal bottom reinforcement area of longitudinal top reinforcement area of shear reinforcement shear span width of beam section c c′ d h Es FE FI fc′ fsu fy jd L lp Pcr Pn sh sv T Vc Vcr Vn Vs ve ws wt α β εy λ θ ρh ρst ρst ′ ρv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = cover of longitudinal bottom reinforcement cover of longitudinal top reinforcement effective depth of beam section overall depth of beam section elastic modulus of steel load capacity of concrete strut in exterior shear span load capacity of concrete strut in interior shear span concrete compressive strength tensile strength of reinforcement yield strength of reinforcement distance between center of top and bottom nodes span length width of loading plate diagonal crack load ultimate load at failure spacing of horizontal shear reinforcement spacing of vertical shear reinforcement tensile force in longitudinal reinforcement load capacity of concrete diagonal crack shear force ultimate shear force at failure load transfer capacity of shear reinforcement effectiveness factor of concrete width of concrete strut height of plate anchored to longitudinal reinforcement angle between shear reinforcement and axis of concrete strut ratio of exterior reaction to applied load yield strain of reinforcement normalized ultimate shear strength angle between concrete strut and longitudinal axis of beam horizontal shear reinforcement ratio (Ah/bwsh) longitudinal bottom reinforcement ratio (As/bwd) longitudinal top reinforcement ratio (As /bwd) ′ vertical shear reinforcement ratio (Av /bwsv) REFERENCES 1. Rogowsky, D. M.; MacGregor, J. G.; and Ong, S. Y., “Tests of Reinforced Concrete Deep Beams,” ACI JOURNAL , Proceedings V. 83, No. 4, JulyAug. 1986, pp. 614-623. 2. Ashour, A. F., “Tests of Reinforced Concrete Continuous Deep Beams,” ACI Structural Journal, V. 94, No. 1, Jan.-Feb. 1997, pp. 3-12. 3. Subedi, N. K., “Reinforced Concrete Two-Span Continuous Deep Beams,” Proceedings of the Institution of Civil Engineers, Structures & Buildings, V. 128, Feb. 1998, pp. 12-25. 4. Tan, K. H.; Kong, F. K.; Teng, S.; and Weng, L. W., “Effect of Web Reinforcement on High-Strength Concrete Deep Beams,” ACI Structural Journal, V. 94, No. 5, Sept.-Oct. 1997, pp. 572-582. 5. Smith, K. N., and Vantsiotis, A. S., “Shear Strength of Deep Beams,” ACI JOURNAL, Proceedings V. 79, No. 3, May-June 1982, pp. 201-213. 6. ACI Committee 318, “Building Code Requirements for Structural Concrete (ACI 318-05) and Commentary (318R-05),” American Concrete Institute, Farmington Hills, Mich., 2005, 430 pp. 7. Canadian Standards Association (CSA), “Design of Concrete Structures,” A23.3-94, Canadian Standards Association, Rexdale, Ontario, Canada, Dec. 1994, 199 pp. 8. FIP Recommendations: Practical Design of Structural Concrete. 1999. 9. MacGregor, J. G., Reinforced Concrete: Mechanics and Design, PrenticeHall International, Inc., 1997. 10. Marti, P., “Basic Tools of Reinforced Concrete Beam Design.” ACI JOURNAL, Proceedings V. 82, No. 1, Jan.-Feb. 1985, pp. 46-56. 11. Schlaich, J.; Schafer, K.; and Jennewein, M., “Toward a Consistent Design of Structural Concrete,” Journal of the Prestressed Concrete Institute, V. 32, No. 3, May-June 1987, pp. 74-150. 12. Tjhin, T. N., and Kuchma, D. A., “Example 1b: Alternative Design for the Non-Slender Beam (Deep Beam),” Strut-and-Tie Models, SP-208, K.-H. Reineck, ed., American Concrete Institute, Farmington Hills, Mich., 2002, pp. 81-90. 13. Yang, K. H., “Evaluation on the Shear Strength of High-Strength Concrete Deep Beams,” PhD Thesis, Chungang University, Korea, Feb. 2002, 120 pp. 14. ACI Committee 445, “Shear and Torsion,” Strut-and-Tie Bibliography, ACI Bibliography No. 16, American Concrete Institute, Farmington Hills, Mich., Sept. 1997, 50 pp. 15. Cervenka, V.; Jendele, L.; and Cervenka, J., “ATENA Computer Program Documentation: Part 1,” Cervenka Consultant, 2003, 106 pp. 16. ACI Committee 318, “Building Code Requirements for Structural Concrete (ACI 318-99) and Commentary (318R-99),” American Concrete Institute, Farmington Hills, Mich., 1999, 369 pp. ACI Structural Journal/July-August 2007
- 57. APPENDIX A Table A1—Details and test results of simple deep beams13 Pn , kN f c , MPa ′ a/h a/jd No.1 ρh ρv Vcr , kN Exp. ACI 318-05 (Pn)Exp./(Pn)ACI 0 Simple 0 254.0 958.0 684.1 1.400 No. 2 0 0.006 259.0 992.0 684.1 1.450 No. 3 0 0.012 260.0 1111.3 684.1 1.624 0.006 0 249.9 1042.7 684.1 1.524 No. 5 0.006 0.006 262.6 1323.0 855.2 1.547 No. 6 0.012 0 270.5 1391.6 684.1 2.034 No. 7 0 0.006 188.0 876.1 624.7 1.402 0.5 No. 4 No. 8 0.7 No. 9 0.59 0.82 0.006 0 215.6 993.7 624.7 1.591 0.006 31.4 0.006 205.8 1044.7 780.9 1.338 No. 10 0 0 173.5 750.7 520.0 1.444 No. 11 0 0.006 172.5 762.4 520.0 1.466 0 0.012 195.0 1107.4 520.0 2.130 No. 12 1.0 No. 13 1.18 0.006 0 178.4 601.7 520.0 1.157 No. 14 0.006 0.006 181.0 905.5 650.0 1.393 No. 15 0.012 0 185.0 707.6 520.0 1.361 No. 16 0 0 107.8 409.6 378.8 1.081 0.006 0.006 142.1 721.3 473.5 1.523 1.5 0.59 1.0 No. 18 No. 19 1.76 0.5 No. 17 1.18 52.9 No. 20 No. 21 0 0 290.0 1540.6 1154.5 1.334 0.006 0.006 318.5 1775.8 1443.1 1.230 0 0 225.4 952.6 877.4 1.086 0.006 0.006 245.0 1129.0 1096.8 1.029 No. 22 0 0 347.9 1646.4 1710.4 0.963 No. 23 0 0.006 357.7 1789.5 1710.4 1.046 No. 24 0 0.012 347.9 1934.5 1710.4 1.131 0.006 0 392.0 1962.0 1710.4 1.147 No. 26 0.006 0.006 345.0 2061.9 2138.0 0.964 No. 27 0.012 0 401.8 2269.7 1710.4 1.327 No. 28 0 0.006 289.1 1622.9 1561.8 1.039 0.006 0 303.8 1395.5 1561.8 0.894 0.006 0.006 308.7 1701.3 1952.2 0.871 0.5 No. 25 No. 29 0.7 No. 30 0.59 0.82 78.4 No. 31 0 0 254.8 1146.6 1299.9 0.882 No. 32 0 0.006 240.1 1356.3 1299.9 1.043 No. 33 0 0.012 294.0 1558.2 1299.9 1.199 0.933 1.0 No. 34 1.18 0.006 0 249.9 1213.2 1299.9 No. 35 0.006 0.006 281.3 1295.6 1624.9 0.797 No. 36 0.012 0 291.1 1215.2 1299.9 0.935 No. 37 1.5 No. 38 1.76 0 0 173.5 656.6 947.0 0.693 0.006 0.006 181.3 836.9 1183.7 0.707 Mean 1.229 Standard deviation 0.326 Note: 1 MPa = 145 psi; 1 kN = 0.2248 kips. ACI Structural Journal/July-August 2007 429
- 58. ACI STRUCTURAL JOURNAL TECHNICAL PAPER Title no. 104-S47 Reliability of Transfer Length Estimation from Strand End Slip by José R. Martí-Vargas, César A. Arbeláez, Pedro Serna-Ros, and Carmen Castro-Bugallo An experimental program on 12 series of specimens with different embedment lengths to determine the transfer length was conducted. Transfer length test results of seven-wire strand on twelve different concrete mixtures were analyzed. A testing technique based on the analysis of bond behavior by means of measuring the force supported by the tendon has been used. The specimens had been instrumented with slip measurement devices at each end of the specimen. A sequence of slip values at each end of the specimen after release versus the embedment length has been analyzed. The expressions relating the transfer length to the tendon end slip are presented. A value of Guyon’s factor for tendon stress distribution shape has been obtained. Two criteria to determine the transfer length from the slip sequences at both ends of the specimens have been analyzed. Table 1—Proposed α coefficient values from Guyon’s formula Reference Guyon8 FIP Olesniewicz9 FIP10 IRANOR5 LCPC By hypothesis Adopted value 2.86 Experimental 2.8 Adopted value den Uijl 12 Jonsson13 Guyon 2/(1 – b)† 2.67 By theoretical studies 2.3 to 2.6 Experimental value and by theoretical studies 2.5 Assumed value 2 By hypothesis 1.5 Indicated value for linear ascending bond stress distribution 8 Brooks et al.14 Balogh15 Russell and Burns16 Logan17 Steinberg et al.18 Oh and Kim19 Wan et al.20 CEB-FIP21* Rose and Russell22 den Uijl12 (1) fib23 where Lt is the transfer length, δ is the strand end slip at the free end of a prestressed concrete member, εpi is the initial strand strain, and the α coefficient represents the shape factor of the bond stress distribution along the transfer zone. Two hypotheses were considered8: α = 2 for uniform bond stress distribution (linear variation in strand stress); and α = 3 for linear descending bond stress distribution (parabolic variation in strand stress) Equation (1) can be rewritten as follows (2) where Ep is the modulus of elasticity of the prestressing strand and fpi is the strand stress immediately before release. Several researchers have proposed different values of α for the bond stress distribution along the transfer zone from ACI Structural Journal/July-August 2007 3 6 Balázs11 INTRODUCTION The force in a prestressing strand is transferred by bond to the concrete in the release operation. At this stage, strand stress varies from zero at the free end of the member to a maximum value (effective stress). Transfer length is defined as the distance required to develop the effective stress in the prestressing strand.1 Variation in strand stress along the transfer length involves slip between the strand and the concrete. The measurement of the strand end slip is an indirect method to determine the transfer length.2 Most experimental standards3-6 are based on this method, and it has been proposed as a simple nondestructive assurance procedure by which the quality of bond can be monitored within precasting plants.7 Guyon8 proposed the following expression from a theoretical analysis δE p L t = α -------f pt Indicated value when stress in prestressing strand is rapidly increasing RILEM3 Keywords: bond; precast concrete; prestressing; pretensioning; slip; strand; transfer length. δ L t = α ----ε pi 4* Origin of value 4 FIP4 Coefficient Lopes and do Carmo24 *Substituting fpi by effective stress in strand immediately after release. b is experimental constant value that must be fixed for each type of prestressing strand according to its bond characteristics (for 12.7 mm [0.5 in.] seven-wire strand, b = 0.25 and α = 2.67). † experimental results and theoretical studies. Table 1 indicates the different assigned values of α. Table 2 shows other expressions that relate the transfer length to the strand end slip at the free end of a pretensioned concrete member, where db is the diameter of prestressing strand, and fci is the compressive strength of concrete at the ′ time of prestress transfer. ACI Structural Journal, V. 104, No. 4, July-August 2007. MS No. S-2006-250 received June 16, 2006, and reviewed under Institute publication policies. Copyright © 2007, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published in the May-June 2008 ACI Structural Journal if the discussion is received by January 1, 2008. 487
- 59. José R. Martí-Vargas is an Associate Professor of civil engineering in the Department of Construction Engineering and Civil Engineering Projects, Polytechnic University of Valencia (UPV), Valencia, Spain. He is member of the Institute of Science and Concrete Technology (ICITECH) at UPV. He received his degree in civil engineering and his PhD from UPV. His research interests include bond behavior of reinforced and prestressed concrete structural elements, durability of concrete structures, and strut-and-ties models. César A. Arbeláez is a PhD Assistant Researcher in the Department of Construction Engineering and Civil Engineering Projects at Polytechnic University of Valencia. He is member of ICITECH at UPV. He received his civil engineering degree from Quindío University, Armenia, Quindío, Colombia, and his PhD from UPV. His research interests include bond properties of prestressed concrete structures and the use of advance cement-based materials in structural applications. Pedro Serna-Ros is a Professor of civil engineering in the Department of Construction Engineering and Civil Engineering Projects at Polytechnic University of Valencia. He is a member of ICITECH at UPV. He received his degree in civil engineering from UPV and his PhD from l’Ecole National des Ponts et Chaussées, Paris, France. His research interests include self-consolidating concrete, fiber-reinforced concrete, and bond behavior of reinforced and prestressed concrete. Carmen Castro-Bugallo is a PhD candidate in the Department of Construction Engineering and Civil Engineering Projects at Polytechnic University of Valencia. She is member of ICITECH at UPV. She received her degree in civil engineering from UPV. Her research interests include bond properties of reinforced concrete and prestressed concrete structures and strut-and-ties models. Table 2—Proposed equations for transfer length from strand end slip Equation no. Equation (U.S. units) Reference Equation (SI units) δ δ --L t = --K K K = 0.00009 in.–1 for K = 0.0000035 mm–1 0.5 in. seven-wire for 12.7 mm sevenstrand wire strand Lt = Marshall and Krishnamurthy25 (3) Balázs26 (4) δ Lt = 218db 4 -------f ci ′ 11 Balázs (5) 24.7δ L t = ----------------------------0.4 0.15 ⎛ f pi ⎞ f ci ′ ---⎝ E p⎠ 111δ L t = ----------------------------0.4 0.15 ⎛ f pi ⎞ f ci ′ ---⎝ E p⎠ Rose and Russell22 (6) E L t = 2δ ----p + 5.4 f pi E L t = 2δ ----p + 137.16 f pi 3⁄2 0.625 3⁄2 δ Lt = 105db 4 -------f ci ′ 0.625 Notes: For U.S. units: fpi, f ′ci , and Ep in ksi; db, δ, and Lt in inches; for SI units: fpi, f ′ci, and Ep in MPa; db, δ, and Lt in mm. 1 in. = 25 mm; 1 MPa = 0.145 ksi. Some researchers conducted experimental studies to obtain the transfer length from the strand end slip at the free end in hollow-core slabs,7,13-15 in beams,16,18,19,22,27-29 in piles,20,30 in prisms,31 and in specimens to simulate bond behavior along transfer length.32 Several authors7,14,20,30 have established an allowable free end slip as the strand end slip which results in a transfer length equal to that computed by the ACI provisions for transfer length (Eq. (7)).1 By setting Eq. (2) to be equal to the Eq. (7) and substituting α = 2 and α = 3 in Eq. (2), the implied allowable value of end slip can be calculated by Eq. (8) and (9), respectively. 1 1L t = -- f se d b (U.S. units) L t = --------- f se d b (SI units) 3 20.7 1 f pi δ all2 = -- -----f se d b (U.S. units) 6 Ep δ all2 488 1 - f pi = --------- -----f se d b (SI units) 41.4 E p (7) (8) 1 f pi δ all3 = -- -----f se d b (U.S. units) 9 Ep δ all3 (9) 1 f pi = --------- -----f se d b (SI units) 62.1 E p where fse is the effective stress in the prestressing strand after allowance for all prestress losses, db is the nominal diameter of prestressing strand, δall2 is the implied allowable value of free end slip when α = 2, and δall3 is the implied allowable value of free end slip when α = 3 (Lt , fpi , and Ep, as previously described). To apply Guyon’s end slip theory to determine transfer length is easy, but the measurements of slips are affected by the local bond loss at the ends. Equations (1) to (6) are not applicable to elements of a poor bond quality.14 In this case, greater slips are measured resulting in incorrect transfer length estimation. The other disadvantages of Guyon’s method are larger scatter of experimental results,15 difficulty to measure accurately smaller slips,13 breakage of gauges to measure the strand end slip when a flame cutting process is applied,27 and excessive free end slip in prestressed members with poor concrete consolidation around the strand.7 RESEARCH SIGNIFICANCE This research study provides information on the transfer length of a seven-wire prestressing strand in twelve concretes of different compositions and properties. A test method based on the measurement and analysis of the force supported by the strand has been used. This paper analyzes the reliability of transfer length determination from free end slips according with proposed expressions in the literature. Findings of the research are presented in procedures for the experimental determination of transfer length measuring forces or slips. The information is valuable for all parties involved in the precast/prestressed concrete industry: manufacturers, producers, designers, builders, and owners. EXPERIMENTAL INVESTIGATION An experimental program has been conducted to determine the transfer length of prestressing strands: the ECADA* test method33-34 (*Ensayo para Caracterizar la Adherencia mediante Destesado y Arrancamiento [Test to Characterize the Bond by Release and Pull-out]). Materials Twelve different concretes with a range of water-cement ratios (w/c) from 0.3 to 0.5, cement content from 590 to 843 lb/yd3 (350 to 500 kg/m3) and a compressive strength at the time of testing fci from 3.5 to 8 ksi (24 to 55 MPa) were tested. ′ Concrete components were a) cement CEM I 52.5 R;35 b) crushed limestone aggregate (0.275 to 0.472 in. [7 to 12 mm]); c) washed rolled limestone sand (0 to 0.157 in. [0 to 4 mm]); and d) policarboxilic ether high-range water-reducing additive. The mixtures of the tested concretes are shown in Table 3. The prestressing strand was a low-relaxation seven-wire strand specified as UNE 36094:97 Y 1860 S7 13.036 with a guaranteed ultimate strength of 270 ksi (1860 MPa). The main characteristics were adopted from the manufacturer: diameter 0.5 in. (12.7 mm), cross-sectional area 0.154 in.2 (99.69 mm2), ultimate strength 43.3 kips (192.60 kN), yield stress at 0.2% 40 kips (177.50 kN), and modulus of elasticity 28,507 ksi (196,700 MPa). The prestressing strand was ACI Structural Journal/July-August 2007
- 60. Table 3—Concrete mixtures from test program f ci (at time of ′ Gravel/sand testing, 24 hours), ratio ksi (MPa) Cement, Designation lb/yd3 (kg/m3) w/c M-350-0.50 0.50 3.8 (26.1) 0.45 5.4 (37.3) M-350-0.45 590 (350) M-350-0.40 0.40 6.8 (46.7) M-400-0.50 0.50 3.5 (24.2) 0.45 4.1 (28.3) M-400-0.45 M-400-0.40 674 (400) M-400-0.35 M-450-0.40 M-450-0.35 0.35 758 (450) M-500-0.30 1.14 0.40 6.0 (41.4) 843 (500) Fig. 1—Test equipment layout. 6.6 (45.3) 5.3 (36.3) 0.35 6.7 (46.6) 0.40 M-500-0.40 M-500-0.35 0.40 4.5 (30.8) 0.35 6.8 (46.6) 0.30 7.9 (54.8) tested in the as-received condition (free of rust and free of lubricant). The strand was no treated in any special manner. The strand was stored indoors, and care was taken not to drag the strand on the floor. Testing technique The ECADA test method is based on the measurement and analysis of the force supported by the strand in a series of pretensioned concrete specimens with different embedment lengths. Figure 1 shows the test equipment layout. An anchorage-measurement-access (AMA) system is placed at one end (stressed end) of a pretensioning frame to simulate the sectional stiffness of the specimen. The AMA system is made up of a sleeve in the final stretch of the specimen to prevent the influence of the confinement caused by the end frame plate, the stressed end frame plate, and an anchorage plate supported on the frame by two separators. The step-by-step test procedure was described in detail in Martí-Vargas et al.,34 and may be summarized as follows: Preparation stage— 1. The strand is placed in the frame; 2. Strand tensioning; 3. Strand anchorage by an adjustable strand anchorage; 4. The concrete is mixed, placed into the formwork in the frame, and consolidated; and 5. After concrete placement, the specimen is cured to achieve the desired concrete properties at the time of testing. Testing stage— 1. The adjustable strand anchorage is relieved using the hydraulic jack; and 2. Strand release is produced at a controlled speed, and the prestressing force transfer to the concrete is performed. The strand is completely released. The specimen is supported at the stressed end frame plate. Stabilization period—The level of force during this time is zero at the free end. The force in the strand at the stressed end depends on the strain compatibility with the concrete specimen. This force requires a stabilization period to guarantee its measurement. The strand force in the AMA system is recorded continuously during the test. Although it is not included in this study, the test can continue with the pull-out operation positioning the hydraulic jack at the stressed end to increase the force in the strand, separating the anchorage plate of the AMA system from the frame. ACI Structural Journal/July-August 2007 Fig. 2—LVDT at free end of specimen. Fig. 3—LVDT at stressed end of specimen. Test parameters The specimens had a 4 x 4 in.2 (100 x 100 mm2) cross section with a concentrically located single strand at a prestress level before release of 75% of guaranteed ultimate strand strength. All specimens were subjected to the same consolidation and curing conditions. Release was gradually performed 24 hours after concreting at a controlled speed of 0.18 kips/s (0.80 kN/s). A stabilization period of 2 hours from release was established. With these test parameters, visible splitting cracks have not happened in any of the tested specimens. Instrumentation The instrumentation used was a hydraulic jack pressure sensor to control tensioning and release operations; a hollow force transducer included in the AMA system to measure the force supported by the strand; and two linear variable differential transducers (LVDTs), one at the free end (Fig. 2) to measure the draw-in (δ, free end slip), and another at the stressed end (Fig. 3) to measure the strand slip to the last embedment concrete cross-section of the specimen (δl, stressed end slip). No internal measuring devices were used in the test specimens so as to not distort the bond phenomenon. Criterion to determine transfer length With the ECADA test method, the transfer length is obtained with a series of specimens with different embedment 489
- 61. Fig. 4—Determination of transfer length through ECADA test method. Fig. 5—Force loss versus embedment length for Concrete M-350-0.50. lengths. For each specimen, the strand force loss in the AMA system directly after the stabilization period is measured. The force loss values are arranged according to the specimen embedment length (Fig. 4). The obtained curve shows a bilinear tendency. The transfer length corresponds to the smallest specimen embedment length that marks the beginning of the horizontal branch.33,34 The resolution in the determination of the transfer length will depend on the sequence of lengths of the specimens tested. For an embedment length sequence of 2 in. (50 mm), the transfer length obtained by the ECADA test method is repeated when a same concrete mixture is tested.34 Transfer length over-estimation The ideal AMA system must have the same sectional rigidity as the specimen. This rigidity depends on the concrete properties, the age of the concrete at the time of testing, and the specimen cross section. It would not really be feasible to design a system for each specific test conditions. For this reason, in this experimental work, the rigidity of the AMA system designed is slightly greater than the sectional rigidity of the specimens. A discontinuity section is generated in the border between the specimen and the AMA system. In these conditions, the strand force measured in the AMA system after release will be slightly higher than the effective prestressing force of the strand in the specimen. This difference of forces gives rise to a small over-estimation of the real transfer length.34 Consequently, even if the specimen embedment length is greater than the transfer length, a small slip of the strand at the stressed end is registered. 490 EXPERIMENTAL RESULTS AND DISCUSSION Determination of transfer length Transfer length is determined for each concrete mixture in accordance with the exposed criterion. As an example, Fig. 5 shows the results of force loss versus the embedment length for the concrete M-350-0.50 (designation according with Table 3). Two curves are shown, one with the force losses registered just after release (ΔP), and another with the force losses registered after the stabilization period (ΔP). Both curves present a bilinear tendency with a descendent initial branch with a strong slope and a practically horizontal branch starting from 21.7 in. (550 mm) embedment length. The transfer length determined by the ECADA test method for this concrete mixture is 21.7 in. (550 mm). The difference between the two curves corresponds to the increment of force loss registered during the stabilization period. When specimens have an embedment length below 21.7 in. (550 mm), the force loss after the stabilization period is greater than the force loss registered just after release. When specimens have an embedment length equal to or greater than 21.7 in. (550 mm), however, the force loss is similar at both points of time. As it can be observed in Fig. 5, for this concrete, the beginning of the horizontal branches coincides at both points of time. In some cases, however, increases of force loss have taken place during the stabilization period in the first point of the resulting horizontal branch just after release. For this reason, the transfer length must be always determined on the curve measured after the stabilization period. Comparison of test results with Guyon’s formula Figure 6 shows the transfer length results obtained by the ECADA test method for each concrete mixture, as well as the transfer length obtained from the free end slips by applying Guyon’s formula (Eq. (2)). This formula has been applied to free end slips registered after the stabilization period in specimens with an embedment length equal to or greater than the transfer length. Between four and 18 specimens for each concrete mixture, with a total of 121 specimens, have been considered. Two intervals are drawn for each concrete mixture. The interior interval corresponds to the extreme transfer length values obtained by applying Guyon’s formula with α = 2.8 (adopted by RILEM3) to the minimum and maximum free end slips. The exterior interval corresponds to the extreme transfer length values according to the hypotheses by Guyon obtained as follows: the lower limit was calculated by applying α = 2 to the minimum free end slip, and the upper limit was calculated by applying α = 3 to the maximum free end slip. The amplitude of the transfer length intervals is very variable for the different concrete mixtures, as shown in Fig. 6. The results obtained by the ECADA test method are located within both intervals in all cases except for the M-500-0.30 concrete mixture for the interior interval. Figure 7 shows the transfer length results obtained by the ECADA test method in the corresponding series versus the free end slip registered after the stabilization period in each specimen. Only the specimens with an embedment length equal to or greater than the transfer length have been included. The predicted transfer lengths by Guyon’s formula are also plotted in Fig. 7. It is shown that 38.8% of the experimental results fall outside the limits (33.0% show a transfer length greater than the predicted maximum values, ACI Structural Journal/July-August 2007
- 62. Fig. 6—Graphical comparison between experimental transfer length and predicted transfer lengths from Guyon’s formula and RILEM provisions. Fig. 8—Comparison between results of present tests and those of other researchers. Fig. 9—Comparison of measured transfer lengths with calculated values according to Eq. (6). Fig. 7—Transfer length versus free end slip for specimens with embedment length equal to or greater than transfer length. Table 4—Comparison between measured and calculated transfer lengths Equation no. Average Lt (calculated)/L t (measured) Coefficient of correlation R2 (3) and 5.8% show a transfer length smaller than the predicted minimum values). A value of α = 2.44 from the regression analysis of the test results has been obtained. Figure 8 shows the experimental transfer lengths versus the registered free end slips obtained in beams by several authors. The predicted transfer length according to ACI 318-051 (LtACI) and the allowable free end slips δall2 (Eq. (8)) and δall3 (Eq. (9)) are also plotted in Fig. 8. The LtACI, δall2, and δall3 values have been calculated by considering that fpi = 202 ksi, fse = 0.8fpi = 162 ksi, Ep = 28,528 ksi and db = 0.5 in. (fpi = 1395 MPa, fse = 0.8fpi = 1116 MPa, Ep = 196,700 MPa and db = 12.7 mm). The percentages of results included in each sector delimited by LtACI, δall2, and δall3 are indicated in Fig. 8. The range of free end slip registered is very ample for one same transfer length, as observed in Fig. 8. Also the range of transfer length values is very variable for one same free end slip. Figure 8 also shows that when a transfer length is smaller than LtACI, the δall2 limit is exceeded in 2.8% of the cases, and the δall3 is exceeded in 32.3% of the cases (2.8% + 29.5%). On the other hand, for registered free end slips smaller than δall3 or δall2, transfer lengths greater than LtACI are measured in some cases (2.3 and 4.6%, respectively). Consequently, the use of an assurance procedure for bond quality based on a limit value for the allowable free end slip is not completely reliable. ACI Structural Journal/July-August 2007 1.18 0.07 (4) 1.17 0.54 (5) 1.11 0.35 (6) 1.01 0.21 (2) with α = 2.44 0.95 0.20 Comparison of test results with other expressions The experimental results obtained with both the ECADA test method and the theoretical predictions from Eq. (3) to (6) have been compared. As an example, Fig. 9 illustrates the comparison with Eq. (6). Table 4 summarizes these comparisons. Besides, the comparison with Eq. (2) by substituting α = 2.44 (obtained value from the experimental results of this study) is included. It can be observed that the expressions based on Guyon’s formula (Eq. (6) and Eq. (2) with α = 2.44) show a good prediction of the average measured transfer length. The coefficient of correlation improves when the expressions include, in addition to the slips, other parameters like the concrete compressive strength. Use of end slips sequences to determine transfer length The possibility of determining the transfer length from the sequences of end slip values at both ends versus the embedment length of specimens was considered. 491
- 63. Fig. 10—Free end slip versus embedment length for Concrete M-350-0.50. Fig. 11—Stressed end slip versus embedment length for Concrete M-350-0.50. Table 5—Transfer length obtained from three sequences of results (ΔP, δ, and δl) Transfer length, in. (mm) Designation ECADA test method ΔP Free end slip δ Stressed end slip δl M-350-0.50 21.7 (550) 21.7 (550) 21.7 (550) M-350-0.45 21.7 (550) 21.7 (550) 21.7 (550) M-350-0.40 21.7 (550) 21.7 (550) 21.7 (550) M-400-0.50 25.6 (650) — 25.6 (650) M-400-0.45 21.7 (550) — 21.7 (550) M-400-0.40 21.7 (550) 21.7 (550) 21.7 (550) M-400-0.35 19.7 (500) 19.7 (500) 17.7 (450) M-450-0.40 21.7 (550) — 21.7 (550) M-450-0.35 19.7 (500) 19.7 (500) 19.7 (500) M-500-0.40 23.6 (600) — 23.6 (600) M-500-0.35 17.7 (450) 17.7 (450) 17.7 (450) M-500-0.30 15.7 (400) 15.7 (400) 15.7 (400) Figure 10 shows the free end slip results versus the embedment length for the concrete mixture M-350-0.50. Two curves are shown, one with the free end slip just after release δ, and the other with the free end slip registered after the stabilization period δ. Both curves present a bilinear tendency, with a descendent initial branch and a practically horizontal branch starting from 21.7 in. (550 mm) embedment length. This embedment length coincides with the result obtained by the ECADA test method (refer to Fig. 5). The free end slip increases during the stabilization period in all the specimens. 492 Fig. 12—Ratios ΔP/ΔPAVE , δ/δAVE , and δl/δlAVE versus embedment length for Concrete M-350-0.50. Similarly, Fig. 11 shows the stressed end slip just after release δl, and the stressed end slip after the stabilization period δl versus the embedment length for the same concrete mixture. Both curves present a bilinear tendency. The beginning of the horizontal branch coincides with the result obtained by the ECADA test method (21.7 in. [550 mm]). In regard to the force losses, the stressed end slip only increases during the stabilization period in specimens whose embedment length is smaller than the transfer length. Figure 12 summarizes the results of the three variables (force loss and slip at both ends) for the concrete M-350-0.50 after the stabilization period. The shown ratios are the quotient between each specimen test result (ΔP, δ, and δl), and the average test results (ΔPAVE, δAVE, and δlAVE) of specimens with an embedment length equal to or greater than the transfer length. Again, a bilinear tendency is observed with a descendent initial branch and a perceptibly horizontal branch from 21.7 in. (550 mm) embedment length. Although the slope of the descendent initial branch is very pronounced in the cases of force loss and stressed end slip, it is very weak in the case of free end slip. Consequently, the beginning of the horizontal branch is more easily identifiable by analyzing the force loss and stressed end slip than the free end slip. This procedure of test results analysis for each concrete mixture has been applied. The transfer lengths from the three sequences of results obtained from the test instrumentation (ΔP, δ, and δl) versus the embedment length have been determined. Table 5 summarizes the obtained results. The transfer lengths obtained from the stressed end slip and by the ECADA test method coincide in 11 out of the 12 concrete mixtures, and only a 2 in. (50 mm) difference is observed in the concrete M-400-0.35. The transfer lengths obtained from the free end slip coincide in eight out of the 12 concrete mixtures. Given the wide dispersion of the measured free end slip, no bilinear behavior was detected in the remaining cases (see range of free end slip to one same transfer length in Fig. 7). It was not possible to determine the transfer length if the beginning of the horizontal branch was not clearly defined. These cases correspond to concrete mixtures with greater water content in their mixture. CONCLUSIONS Based on the results of this experimental investigation, the following conclusions are drawn: 1. The feasibility of applying the ECADA test method to determine the transfer length of prestressing strands has been verified, even in concretes with a low compressive strength; ACI Structural Journal/July-August 2007
- 64. 2. An average value of α = 2.44 for Guyon’s formula has been obtained from the experimental results of this study. An ample range of free end slip values has been obtained for one same transfer length. Furthermore, the range of transfer length values for one same free end slip is very variable; 3. Consequently, a great variability of results for one same concrete mixture has been observed in transfer length estimation from the experimental free end slips when Guyon’s formula was applied; 4. The prediction range of transfer lengths from expressions proposed by several authors relating the transfer length to the free end slip is very ample; 5. Determining transfer length from the free end slip is relatively easy, although it can lead to a false perception that the transfer length value is very variable; 6. Using a limit value for the allowable free end slip as an assurance procedure for bond quality may give rise to uncertain situations; 7. In relation to the results from the ECADA test method, the sequence of stressed end slip values versus the embedment length is a reliable assurance procedure for the experimental determination of transfer length; and 8. The sequence of free end slip values versus the embedment length is not a reliable assurance procedure for the experimental determination of transfer length. The beginning of the horizontal branch is not clearly defined when the dispersion of measured free end slip is wide. This particularly occurs when concrete has a low compressive strength. ACKNOWLEDGMENTS The contents of this paper are within the framework of a line of research that is currently being carried out by the Concrete Technology and Science Institute (ICITECH) of the Polytechnic University of Valencia, Valencia, Spain, in collaboration with the companies PREVALESA and ISOCRON. Financial support provided by the Ministry of Education and Science and FEDER funds (Project MAT2003-07157 and Project BIA2006-05521) made this research possible. The authors appreciate the collaboration of the aforementioned companies and organizations, as well as the participation of the technical staff of the Concrete Structures Laboratory at the Polytechnic University of Valencia for their assistance in preparing and testing specimens. NOTATION db Ep fci ′ = nominal diameter of prestressing strand = modulus of elasticity of prestressing strand = compressive strength of concrete at time of prestress transfer (cylinder) fpi = strand stress immediately before release fse = effective stress in the prestressing strand after allowance for all prestress losses Lt = transfer length LtACI = predicted transfer length according ACI 318-05 α = coefficient to take into account assumed shape of bond stress distribution δ = strand end slip at free end δall2 = allowable free end slip when α = 2 δall3 = allowable free end slip when α = 3 δ = free end slip just after release δAVE = average free end slip after stabilization period obtained in all specimens of series with embedment length equal to or greater than transfer length δl = stressed (loaded) end slip δl = stressed end slip just after release δlAVE = average stressed end slip after stabilization period obtained in all specimens of series with embedment length equal to or greater than transfer length ΔP = force loss after stabilization period ΔPAVE = average force loss after stabilization period obtained in all specimens of series with embedment length equal to or greater than transfer length ΔP = force loss just after release εpi = initial strand strain ACI Structural Journal/July-August 2007 REFERENCES 1. ACI Committee 318, “Building Code Requirements for Structural Concrete (ACI 318-05) and Commentary (318R-05),” American Concrete Institute, Farmington Hills, Mich., 2005, 430 pp. 2. Thorsen, N., “Use of Large Tendons in Pretensioned Concrete,” ACI JOURNAL , Proceedings V. 53, No. 6, Feb. 1956, pp. 649-659. 3. RILEM RPC6, “Specification for the Test to Determine the Bond Properties of Prestressing Tendons,” Réunion Internationale des Laboratoires et Experts des Matériaux, Systèmes de Constructions et Ouvrages, Bagneux, France, 1979, 7 pp. 4. Fédération Internationale de la Précontrainte, “Report on Prestressing Steel: 7. Test for the Determination of Tendon Transfer Length under Static Conditions,” FIP, London, UK, 1982, 19 pp. 5. IRANOR UNE 7436, “Bond Test of Steel Wires for Prestressed Concrete,” Instituto Nacional de Racionalización y Normalización, Madrid, Spain, 1982, 13 pp. 6. Laboratoire Central des Ponts et Chaussées, “Transfer Length Determination. Test Method Applicable for Prestressed Reinforcement,” Techniques et Méthodes, No. 53, LCPC, Paris, France, 1999, pp. 45-55. 7. Anderson, A. R., and Anderson, R. G., “An Assurance Criterion for Flexural Bond in Pretensioned Hollow Core Units,” ACI JOURNAL, Proceedings V. 73, No. 8, Aug. 1976, pp. 457-464. 8. Guyon, Y., Pretensioned Concrete: Theoretical and Experimental Study, Paris, France, 1953, 711 pp. 9. Olesniewicz, A., “Statistical Evaluation of Transfer Length of Strand,” Research and Design Centre for Industrial Building (BISTYP), Warsaw, Poland, 1975. 10. Fédération Internationale de la Précontrainte, “Report on Prestressing Steel: 2. Anchorage and Application of Pretensioned 7-Wire Strands,” FIP, London, UK, 1978, 45 pp. 11. Balázs, G., “Transfer Length of Prestressing Strand as a Function of Draw-In and Initial Prestress,” PCI Journal, V. 38, No. 2, Mar.-Apr. 1993, pp. 86-93. 12. den Uijl, J. A., “Bond Modelling of Prestressing Strand,” Bond and Development of Reinforcement, SP-180, R. Leon, ed., American Concrete Institute, Farmington Hills, Mich., 1998, pp. 145-169. 13. Jonsson, E., “Anchorage of Strands in Prestressed Extruded HollowCore Slabs,” Proceedings of the International Symposium Bond in Concrete: From Research to Practice, Riga Technical University and CEB, eds., Riga, Latvia, 1992, pp. 2.20-2.28. 14. Brooks, M. D.; Gerstle, K. H.; and Logan, D. R., “Effect of Initial Strand Slip on the Strength of Hollow-Core Slabs,” PCI Journal, V. 33, No. 1, Jan.-Feb. 1988, pp. 90-111. 15. Balogh, T., “Statistical Distribution of Draw-in of Seven-Wire Strands,” Proceedings of the International Symposium Bond in Concrete: From Research to Practice, Riga Technical University and CEB, eds., Riga, Latvia, 1992, pp. 2.10-2.19. 16. Russell, B. W., and Burns, N. H., “Measured Transfer Lengths of 0.5 and 0.6 in. Strands in Pretensioned Concrete,” PCI Journal, V. 44, No. 5, Sept.-Oct. 1996, pp. 44-65. 17. Logan, D. R., “Acceptance Criteria for Bond Quality of Strand for Pretensioned Prestressed Concrete Applications,” PCI Journal, V. 42, No. 2, Mar.-Apr. 1997, pp. 52-90. 18. Steinberg, E.; Beier, J. T.; and Sargand, S., “Effects of Sudden Prestress Force Transfer in Pretensioned Concrete Beams,” PCI Journal, V. 46, No. 1, Jan.-Feb. 2001, pp. 64-75. 19. Oh, B. H., and Kim, E. S., “Realistic Evaluation of Transfer Lengths in Pretensioned, Prestressed Concrete Members,” ACI Structural Journal, V. 97, No. 6, Nov.-Dec. 2000, pp. 821-830. 20. Wan, B.; Harries, K. A.; and Petrou, M. F., “Transfer Length of Strands in Prestressed Concrete Piles,” ACI Structural Journal, V. 99, No. 5, Sept.-Oct. 2002, pp. 577-585. 21. Comité Euro-International du Béton-Fédération Internationale de la Précontrainte, “Model Code for Concrete Structures,” CEB-FIP, Lausanne, Switzerland, 1993, 437 pp. 22. Rose, D. R., and Russell, B. W., “Investigation of Standardized Tests to Measure the Bond Performance of Prestressing Strand,” PCI Journal, V. 42, No. 4, July-Aug. 1997, pp. 56-80. 23. Fédération Internationale du Béton, “Bond of Reinforcement in Concrete: State-of-Art Report,” Bulletin d’Information, No. 10, fib, Lausanne, Switzerland, 2000, 427 pp. 24. Lopes, S. M., and do Carmo, R. N., “Bond of Prestressed Strands to Concrete: Transfer Rate and Relationship between Transfer Length and Tendon Draw-in,” Structural Concrete, V. 3, No. 3, 2002, pp. 117-126. 25. Marshall, W. T., and Krishnamurthy, D., “Transfer Length of Prestressing Tendons from Concrete Cube Strength at Transfer,” The Indian Concrete Journal, V. 43, No. 7, July 1969, pp. 244-275. 26. Balázs, G., “Transfer Control of Prestressing Strands,” PCI Journal, 493
- 65. V. 37, No. 6, Nov.-Dec. 1992, pp. 60-71. 27. Kahn, L. F.; Dill, J. C.; and Reutlinger, C. G., “Transfer and Development Length of 15-mm Strand in High-Performance Concrete Girders,” Journal of Structural Engineering, V. 128, No. 7, July 2002, pp. 913-921. 28. Cousins, T. E.; Johnston, D. W.; and Zia, P., “Transfer Length of Epoxy-Coated Prestressing Strand,” ACI Materials Journal, V. 87, No. 3, May-June 1990, pp. 193-203. 29. Cousins, T. E.; Stallings, J. M.; and Simmons, M. B., “Reduced Strand Spacing in Pretensioned, Prestressed Members,” ACI Structural Journal, V. 91, No. 3, May-June 1994, pp. 277-286. 30. Petrou, M. F.; Wan, B.; Joiner, W. S.; Trezos, C. G.; and Harries, K. A., “Excessive Strand End Slip in Prestressed Piles,” ACI Structural Journal, V. 97, No. 5, Sept.-Oct. 2000, pp. 774-782. 31. Mahmoud, Z. I.; Rizkalla, S. H.; and Zaghloul, E. R., “Transfer and Development Lengths of Carbon Fiber Reinforcement Polymers Prestressing Reinforcing,” ACI Structural Journal, V. 96, No. 4, July-Aug. 1999, pp. 594-602. 494 32. Abrishami, H. H., and Mitchell, D., “Bond Characteristics of Pretensioned Strand,” ACI Materials Journal, V. 90, No. 3, May-June 1993, pp. 228-235. 33. Martí, J. R., “Experimental Study on Bond of Prestressing Strand in High-Strength Concrete,” PhD thesis, Polytechnic University of Valencia, ProQuest Information and Learning Co., UMI number 3041710, Mich., 2003, 355 pp. (in Spanish) 34. Martí-Vargas, J. R.; Serna-Ros, P.; Fernández-Prada, M. A.; MiguelSosa, P. F.; and Arbeláez, C. A., “Test Method for Determination of the Transmission and Anchorage Lengths in Prestressed Reinforcement,” Magazine of Concrete Research, V. 58, No. 1, Feb. 2006, pp. 21-29. 35. Comité Européen de Normalisation, “Cement. Part 1: Compositions, Specifications and Conformity Criteria for Common Cements,” European Standard EN 197-1:2000, CEN, Brussels, Belgium, 2000, 30 pp. 36. Asociación Española de Normalización y Certificación, “UNE 36094: Steel Wire and Strand for Prestressed Concrete,” AENOR, Madrid, Spain, 1997, 21 pp. ACI Structural Journal/July-August 2007
- 66. ACI STRUCTURAL JOURNAL TECHNICAL PAPER Title no. 104-S43 Seismic Design Criteria for Slab-Column Connections by Mary Beth D. Hueste, JoAnn Browning, Andres Lepage, and John W. Wallace Two-way slabs without beams are popular floor systems because of their relatively simple formwork and the potential for shorter story heights. Earthquakes, however, have demonstrated that slab-column frames are vulnerable to brittle punching shear failures in the slabcolumn connection region and dropping of the slab, which are costly to repair. This paper focuses on the behavior and design of slab-column connections under combined gravity and lateral loading and reviews current design procedures, performance-based design approaches, and relevant experimental data. An equation relating the gravity shear ratio at a slab-column connection to drift capacity is presented. Finally, practical recommendations are provided for defining specific performance objectives. Keywords: deformation capacity; effective slab width; performance-based design; punching shear. INTRODUCTION Two-way slabs without beams are popular floor systems because of their relatively simple formwork and the potential for shorter story heights due to their shallow profile. This structural system is common in regions of low to moderate seismic risk, where it is allowed as a lateral-force-resisting system (LFRS), as well as in regions of high seismic risk for gravity systems where moment frames or shear walls are provided as the main LFRS. Earthquakes, however, have demonstrated that slab-column frames are not suitable as a main LFRS in regions of high seismic risk because they are relatively flexible and because of the potential for brittle punching shear failures in the slab-column connection region. In the last 40 years, a significant number of experiments have been conducted to evaluate the performance of slabcolumn connections under cyclic lateral loading. This information has formed the basis of current code provisions and guidelines for the design of slab-column connections under combined gravity and lateral loading. As performance-based seismic design (PBSD) becomes more common in structural engineering practice, it is important to evaluate the recommended limits for various structural systems with respect to the latest experimental data and post-earthquake observations. This paper focuses on the behavior and design of interior slab-column connections under combined gravity and lateral loading and serves to review current design procedures, PBSD approaches, and relevant experimental data. Equation (23), for drift capacity of these systems in terms of the gravity shear ratio, is derived using the collected experimental data. Finally, practical recommendations are provided for the PBSD of slab-column connections. RESEARCH SIGNIFICANCE The objectives of this paper, developed by a task group within ACI Committee 374, Performance-Based Seismic Design of Concrete Buildings, are: 1) to review the current state of practice and PBSD approaches for slab-column connections; 2) to summarize experimental data for slab-column connections tested under combined gravity and lateral loads; and 3) to 448 present a practical approach for PBSD of slab-column connections. The PBSD material is presented in a format consistent with the limit states suggested in FEMA 356 (ASCE 2000) and is intended to provide guidance primarily for new construction. The criteria, however, could also be applied to existing structures that contain subpar seismic details where a moderate seismic demand is expected. As a significant benefit for design approaches outside the PBSD framework, a practical equation that relates drift capacity to gravity shear ratio is presented (Eq. (23)). SLAB-COLUMN FRAMES AND CONNECTIONS Slab-column frame construction can deliver several desirable architectural features, including larger open space, lower building heights for a given number of stories, and efficient construction. The FEMA 356, “Prestandard and Commentary for the Seismic Rehabilitation of Buildings” (ASCE 2000) classifies slab-column moment frames as frames that meet the following conditions: 1. Framing components shall be slabs (with or without beams in the transverse direction), columns, and their connections; 2. Frames shall be of monolithic construction that provides for moment transfer between slabs and columns; and 3. Primary reinforcement in slabs contributing to lateral load resistance shall include nonprestressed reinforcement, prestressed reinforcement, or both. This classification includes both frames that are or are not intended to be part of the LFRS for new, existing, and rehabilitated structures. The connections between the slab and a column can be accomplished in several ways including direct connection (whether from solid or waffle slab construction), with column drop panels, and with column or shear capitals. Shear capitals are provided to increase the shear capacity at the slab-column connection and are defined by Joint ACIASCE Committee 352 (1989) as a thickened portion of the slab around a column that does not meet the ACI 318 plan dimension requirements for drop panels. A column capital is defined as a flared portion of the column below the slab that is cast monolithically with the slab. Slab-column connections in structures subjected to earthquake or wind loading must transfer forces due to both gravity and lateral loads. This combination can create large shear and unbalanced moment demands at the connection. Without proper detailing, the connection can be susceptible to two-way (punching) shear failure during response to lateral loads. The flexibility of a slab-column frame can lead to large lateral deformations, which may increase the potential ACI Structural Journal, V. 104, No. 4, July-August 2007. MS No. S-2006-222 received June 1, 2006, and reviewed under Institute publication policies. Copyright © 2007, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published in the May-June 2008 ACI Structural Journal if the discussion is received by January 1, 2008. ACI Structural Journal/July-August 2007
- 67. ACI member Mary Beth D. Hueste is an Associate Professor in the Department of Civil Engineering at Texas A&M University, College Station, Tex. She is a member of ACI Committees 374; Performance-Based Seismic Design of Concrete Buildings; 375, Performance-Based Design of Concrete Buildings for Wind Loads; E803, Faculty Network Coordinating Committee; and Joint ACI-ASCE Committee 352, Joints and Connections in Monolithic Concrete Structures. Her research interests include earthquake-resistant design of reinforced concrete structures, structural rehabilitation including seismic retrofitting, performance-based seismic design, and design and evaluation of prestressed concrete bridge structures. ACI member JoAnn Browning is an Associate Professor in the Department of Civil, Environmental, and Architectural Engineering at the University of Kansas, Lawrence, Kans. She is a member of ACI Committees 314, Simplified Design of Concrete Buildings; 318-D, Flexure and Axial Loads; Beams, Slabs, and Columns; 341, EarthquakeResistant Concrete Bridges; 374, Performance-Based Seismic Design of Concrete Buildings; and 408, Bond and Development of Reinforcement. Her research interests include the performance of reinforced concrete structures under seismic loads, design and analysis of concrete structures, and durability of concrete structures. Andres Lepage, FACI, is an Assistant Professor in the Department of Architectural Engineering at Pennsylvania State University, University Park, Pa. He is a member of ACI Committees 318-H, Seismic Provisions; 335, Composite and Hybrid Structures; 369, Seismic Repair and Rehabilitation; 374, Performance-Based Seismic Design of Concrete Buildings; and 375, Performance-Based Design of Concrete Buildings for Wind Loads. His research interests include the design of concrete, steel, and hybrid structural systems subjected to extreme events. John W. Wallace, FACI, is a Professor of civil engineering at the University of California-Los Angeles, Los Angeles, Calif. He is a member of ACI Committee 318-H, Seismic Provisions; 335, Composite and Hybrid Structures; 369, Seismic Repair and Rehabilitation; 374, Performance-Based Seismic Design of Concrete Buildings; E803, Faculty Network Coordinating Committee; and Joint ACI-ASCE Committee 352, Joints and Connections in Monolithic Concrete Structures. His research interests include response and design of buildings and bridges to earthquake actions, laboratory and field testing of structural components and systems, and structural health monitoring and use of sensor networks. for punching failures; therefore, in regions of high seismic risk, slab-column frames are used in conjunction with beamcolumn moment frames or shear walls. Compatibility of lateral deformations between the slab-column frame and the LFRS, however, must be considered to determine the demands on the connections. The seismic performance of reinforced concrete structures with flat-slab construction has demonstrated the vulnerabilities of the system. For example, following the 1985 Mexico City earthquake, punching shear failures were noted in a 15-story building with waffle flat-plate construction (Rodriguez and Diaz 1989). This failure was partly attributed to a high flexibility combined with low-ductility capacities of the waffle slab-to-column connection. In a department store during the 1994 Northridge earthquake, discontinuous flexural reinforcement at slab-column connections led to punching failures at column drop panels (Holmes and Somers 1996). Punching failures around shear capitals were also noted in the post-tensioned floor slabs of a four-story building during the same event (Hueste and Wight 1997). CURRENT DESIGN APPROACH General The shear strength of slabs in the vicinity of columns is governed by the more severe of two conditions, either beam action or two-way action. In beam action, the slab acts as a wide beam with the critical section for shear extending across the entire width of the slab. This critical section is assumed to be located at a distance d (effective slab depth) from the face of the column or shear capital. For this condition, conventional beam theory applies and will not be discussed in detail herein. For the condition of two-way action, the critical section is assumed to be located at a distance d/2 from the perimeter of the column or shear capital, with potential diagonal tension cracks occurring along a truncated cone or pyramid passing through the critical section (refer to Fig. 1, where d1 ACI Structural Journal/July-August 2007 Fig. 1—Critical sections for two-way shear for interior slab-column connection with shear capital. is the effective slab depth within the thickened shear capital region and d2 is the effective slab depth). Existing methods for calculating the shear strength of slabcolumn connections include applications of elastic plate theory, beam analogies, truss analogies, strip design methods, and others. The design method specified by ACI 318-05 (ACI Committee 318 2005) provides acceptable estimates of shear strength with reasonable computational effort. The procedure is based on the results of a significant number of experimental tests involving slab-column specimens. The eccentric shear stress model is the basis of the general design procedure embodied in ACI 318 for determining the shear strength of slab-column connections transferring shear and moment. The model was adopted by the 1971 version of the ACI 318 and only minor modifications have been included in subsequent versions. Recently, ACI 318-05 has incorporated special provisions related to the lateral-load capacity of slab-column connections in structures located in regions of high seismic risk or structures assigned to high seismic performance or design categories. The design approach presented in this section of the paper is based on the design procedures given in ACI 318-05 complemented by ACI 421.1R-99 (Joint ACI-ASCE Committee 421 1999) and 352.1R-89 (Joint ACI-ASCE Committee 352 1989). ACI 318 eccentric shear stress model Slab-column connections experience very complex behavior when subjected to lateral displacements or unbalanced gravity loads. This involves transfer of flexure, shear, and torsion in the portion of the slab around the column. Combined flexural and diagonal cracking are coupled with significant in-plane compressive forces in the slab induced by the restraint of the surrounding unyielding slab portions. Relatively simple design equations have been derived by considering the critical section to be located at d/2 away from the face of the column and by assuming that shear stress on the critical perimeter varies linearly with distance from the centroidal axis. This eccentric shear stress model is based on the work by DiStasio and Van Buren (1960) and reviewed by Joint ACI-ASCE Committee 326 (1962). For a slab-column connection transferring shear and moment, the ACI 318-05 design equations for limiting the shear stresses vu are given by vu ≤ φvn (1) Vu γv Mu c vu = ------- ± -------------J bo d (2) 449
- 68. where vu is the factored shear stress; φ is the strength reduction factor for shear; vn is the nominal shear stress; Vu is the factored shear force acting at the centroid of the critical section; Mu is the factored unbalanced bending moment acting about the centroid of the critical section; d is the distance from the extreme compression fiber to the centroid of the longitudinal tension reinforcement; bo is the length of the perimeter of the critical section; c is the distance from the centroidal axis of the critical section to the point where shear stress is being computed; J is a property of the critical section analogous to the polar moment of inertia; and γv is the fraction of the unbalanced moment considered to be transferred by eccentricity of shear, defined by 1 γ v = 1 – --------------------2 b1 1 + -- ---- 3 b2 (3) where b1 and b2 are the widths of the critical section measured in the direction of the span for which Mu is determined (Direction 1) and in the perpendicular direction (Direction 2). For an interior column and a critical section of rectangular shape, bo and J are determined by bo = 2(b1 + b2) 3 3 (4) 2 db 1 b 1 d db 2 b 1 J = ------- + ---------- + ------------6 6 2 (5) The first term of Eq. (2), the shear stresses due to direct shear, is assumed uniformly distributed on the critical section, and the fraction γv Mu is assumed to be resisted by linear variation of shear stresses on the critical section. The portion of the moment not carried by eccentric shear is to be carried by slab flexural reinforcement placed within lines 1.5h on either side of the column (h is the slab thickness, including drop panel, if any). This flexural reinforcement is also used to resist slab design moments within the column strip. The provisions of the ACI 318 specify that in absence of shear reinforcement, the nominal shear strength (in stress units) carried by the concrete vc in nonprestressed slabs is given by ⎧ 4 f c′ (psi) ⎪ ⎪ 4 ⎪ ⎛ 2 + ---- ⎞ f c′ (psi) v c = min ⎨ ⎝ β c⎠ ⎪ ⎪ ⎛ αs d ⎞ ⎪ ⎝ -------- + 2⎠ f c′ (psi) bo ⎩ For prestressed slabs without shear reinforcement, Eq. (6) is replaced by ⎧ ⎪ ⎪ v c = min ⎨ ⎪ ⎪ ⎩ Vp 3.5 f c′ (psi) + 0.3f pc + -------bo d Vp ⎛ α s d + 1.5⎞ f ′ (psi) + 0.3f + --------------pc ⎝b ⎠ c b d o o (7) ⎧ Vp ⎪ 0.29 f c′ (MPa) + 0.3f pc + -------⎪ bo d v c = min ⎨ Vp αs d ⎪ ⎪ 0.083 ⎛ -------- + 1.5⎞ f c′ (MPa) + 0.3f pc + -------⎝b ⎠ bo d ⎩ o where αs equals 40, 30, and 20 for interior, edge, and corner columns, respectively; bo and d are defined previously; βc is the ratio of long side to short side of column; fc′ is the specified concrete compressive strength (psi units); fpc is the average compressive stress in two vertical slab sections in perpendicular directions, after allowance for all prestress losses; and Vp is the vertical component of all effective prestress forces crossing the critical section. The use of Eq. (7) is restricted to cases where fc′ is less than 5000 psi (35 MPa); fpc ranges between 125 and 500 psi (0.9 and 3.5 MPa) in each direction; and no portion of the column cross section is closer than four times the slab thickness to a discontinuous edge. If these conditions are not satisfied, the slab should be treated as nonprestressed and Eq. (6) applies. When vu > φvn, the slab shear capacity can be increased by: (a) thickening the slab in the vicinity of the column with a column capital, shear capital, or drop panel; (b) adding shear reinforcement; (c) increasing the specified compressive strength of concrete; or (d) increasing the column size. In a flat slab with shear capitals or drop panels, stresses must be checked at all critical locations—both at the thickened portion of the slab near the face of the column and at the section outside the shear capital or drop panels (refer to Fig. 1). Shear reinforcement, which can be in the form of bars or wires and single- or multiple-leg stirrups properly anchored, increases both the shear strength and the ductility of the connection when transferring moment and shear. Shear reinforcement consisting of structural steel shapes (shearheads) is also effective in increasing the shear strength and ductility of slab-column connections. Design procedures for shearhead reinforcement are presented in Corley and Hawkins (1968) and are not discussed in this paper. For members with shear reinforcement other than shearheads, the nominal shear strength (in stress units) is calculated using (6) ⎧ 0.33 f c′ (MPa) ⎪ ⎪ 2 ⎪ 0.17 ⎛ 1 + ---- ⎞ f c′ (MPa) ⎝ or v c = min ⎨ β c⎠ ⎪ ⎪ ⎛ αs d ⎞ ⎪ 0.083 ⎝ -------- + 2⎠ f c′ (MPa) bo ⎩ 450 v c = v c + v s ≤ 6 f c′ (psi) or 0.5 f c′ (MPa) (8) v c = 2 f c′ (psi) or 0.17 f c′ (MPa) (9) A v f yv v s = ----------bo s (10) ACI Structural Journal/July-August 2007
- 69. where vs is the nominal shear stress provided by shear reinforcement; Av is the area of shear reinforcement; fyv is the specified yield strength of shear reinforcement; s is the spacing of shear reinforcement; and vc, fc′ , and bo are defined previously. When lightweight aggregate concrete is used, the value of f c′ in Eq. (6) through (9) is multiplied by 0.75 for all lightweight concrete or by 0.85 for sand-lightweight concrete. The extent of the shear-reinforced zone is determined to ensure that punching shear failure does not occur immediately outside this region for the design actions. The nominal ultimate concrete shear stress along the critical section acting with shear reinforcement is taken as 2 f c′ (psi) ( 0.17 f c′ [MPa] ) because at approximately this stress, diagonal tension cracks begin to form and cracking is needed to mobilize the shear reinforcement. The shear reinforcement or shear capital must be extended for a sufficient distance until the critical section outside the reinforced region satisfies Eq. (9). In nonprestressed slabs, the maximum spacing of shear reinforcement is 0.5d. In prestressed slabs, the spacing of shear reinforcement is allowed to reach 0.75h but not to exceed 24 in. (0.61 m). For both prestressed and nonprestressed slabs, ACI 318 mandates continuity reinforcement to give the slab some residual capacity following a single punching shear failure at a single support. Thus, in nonprestressed slabs, all bottom bars within the column strip shall be continuous and at least two of the column strip bottom bars in each direction shall pass through the column core (ACI Committee 318 2005, Section 13.3.8.5). In prestressed slabs, a minimum of two tendons shall be provided in each direction through the critical shear section over columns (ACI Committee 318 2005, Section 18.12.4). ACI 421.1R-99 refinements ACI 318 sets out the principles of design for slab shear reinforcement but does not make specific reference to mechanically anchored shear reinforcement, also referred to as shear studs. ACI 421.1R-99 (Joint ACI-ASCE Committee 421 1999) gives recommendations for the design of shear reinforcement using shear studs in slabs. This report also includes equations for calculating shear stresses on nonrectangular critical sections. Shear studs have proven to be effective in increasing the strength and ductility of slab-column connections. ACI 421.1R-99 suggests treating a shear stud as the equivalent of a vertical branch of a stirrup and to use higher limits on some of the design parameters used in ACI 318. In particular, ACI 421.1R-99 suggests higher allowable values for vn, vc, s, and fyv, as follows v c = v c + v s ≤ 8 f c′ (psi) or 0.66 f c′ (MPa) (11) v c = 3 f c′ (psi) or 0.25 f c′ (MPa) (12) ⎧ vu ⎪ 0.75d when ---- ≤ 6 f ′ (psi) or 0.5 f ′ (MPa) c c ⎪ φ s≤⎨ (13) vu ⎪ ⎪ 0.5d when ---- > 6 f c′ (psi) or 0.5 f c′ (MPa) φ ⎩ ACI Structural Journal/July-August 2007 fyv ≤ 72,000 psi (500 MPa) (14) The justification for these higher values is mainly due to the almost slip-free anchorage of the studs and that the mechanical anchorage at the top and bottom of the stud is capable of developing forces in excess of the specified yield strength at all sections of the stud stem. ACI 352.1R-89 recommendations ACI 352.1R-89 (Joint ACI-ASCE Committee 352 1989) includes recommendations for the determination of connection proportions and details to ensure adequate performance of monolithic, reinforced concrete slab-column connections. The recommendations address connection strength, ductility, and structural integrity for resisting gravity and lateral forces. ACI 352.1R-89 only applies to nonprestressed slabcolumn connections with fc′ less than 6000 psi (42 MPa), with or without drop panels or shear capitals, and without slab shear reinforcement. The provisions are limited to connections where severe inelastic load reversals are not expected, and do not apply to slab-column connections that are part of a primary LFRS in regions of high seismic risk because slab-column frames are generally considered to be inadequate for multi-story buildings in these areas. ACI 352.1R-89 classifies slab-column connections as one of two types: 1) Type 1—connections not expected to undergo deformations into the inelastic range; and 2) Type 2— connections requiring sustained strength under moderate deformations into the inelastic range. In structures subjected to high winds or seismic loads, a slab-column connection should be classified as Type 2 even though it is not designated as part of the primary LFRS. To ensure a minimal level of ductility, ACI 352.1R-89 references the work by Pan and Moehle (1989) and recommends that for all Type 2 connections—without shear reinforcement—the direct factored shear Vu acting on the connection, for which inelastic moment transfer is anticipated, must satisfy Vu ≤ 0.4Vc = 0.4vcbod (15) where vc is determined by either Eq. (6) or (7). The limitation defined by Eq. (15) was based on a review of test data that revealed that the deformation capacity of interior connections without shear reinforcement is inversely related to the direct shear on the connection. Connections not complying with Eq. (15) exhibit virtually no post-yield deformation capacity under lateral loading. Pan and Moehle (1989) found that when the stress due to direct shear approaches 0.4vc, the connection experiences a brittle failure for story drift ratios of approximately 1.5%. No additional statements are made in ACI 352.1R-89 regarding other combinations of shear stress and story drift ratio. The report states that Eq. (15) may be waived if calculations demonstrate that the imposed displacement will not induce yield in the slab system. For example, the use of structural walls may adequately limit the imposed drifts on slab-column frames such that yield at the slab-column connection may not occur. The approach by ACI 352.1R-89 suggests that the deformation capacity of slab-column connections may be defined as a function of the shear stress due to direct shear only. This approach has been developed further by Moehle (1996) and Megally and Ghali (2000). ACI 318-05 has incorporated this 451
- 70. Vu VR = ---------------φv c b o d Fig. 2—ACI 318-05 relationship for determining adequacy of slab-column connections in seismic regions. Fig. 3—Design steps when adding shear reinforcement. concept into a general approach for addressing the deformation capacity of slab-column connections not designated as part of the LFRS. Requirements of ACI 318-05, Section 21.11.5 Model building codes (SEI/ASCE 2005) have deformation compatibility requirements for members that are not designated as part of the LFRS. These members should be able to resist the gravity loads at lateral displacements corresponding to the design level earthquake. ACI 318-05, Section 21.11.5, has incorporated a design provision to account for the deformation compatibility of slab-column connections. Instead of calculating the induced effects under the design displacement, ACI 318-05 describes a prescriptive approach. The connection is evaluated based on a simple relationship between the design story drift ratio (DR) and the shear stress due to factored gravity loads. The design DR (story drift divided by story height) should be taken as the largest value for the adjacent stories above and below the connection. The maximum DR (in percent) that a slab-column connection can tolerate, in the absence of shear reinforcement, is given by the following relationship and illustrated in Fig. 2. ⎧ DR = ⎨ 3.5 – 5.0VR ( for VR < 0.6 ) ( for VR ≥ 0.6 ) ⎩ 0.5 where VR is the shear ratio, defined as 452 (16) (17) The term vc is calculated using Eq. (6) or (7). The factored shear force Vu on the slab critical section for two-way action is determined for the load combination 1.2D + 1.0L + 0.2S, where D, L, and S are the dead, live, and snow loads. If the DR exceeds the limit given by Eq. (16), shear reinforcement must be provided (or the connection can be redesigned). When adding shear reinforcement, ACI 318-05 prescribes that the term vs, defined by Eq. (10), must exceed 3.5 f c′ (psi) ( 0.29 f c′ [MPa] ) and the shear reinforcement must extend at least four times the slab thickness from the face of the support. Given that this approach is relatively simple, and that the added cost of providing shear reinforcement at connections is not significant for structures designed for high seismic performance categories, use of this prescriptive approach is likely to be common. The representative design steps are shown in Fig. 3. If shear capitals, column capitals, or drop panels are used, all potential critical sections must be investigated. ACI 318-05 does not prescribe a minimum extension of shear capitals. Wey and Durrani (1992), however, recommend a minimum length equal to two times the slab thickness from the face of the column. ANALYTICAL MODELING The shear stresses due to the combined factored shear and moment transferred between the slab and the column under the design displacement can be determined by creating an appropriate analytical model of the slab-column frame and directly assessing the potential for punching. Recommendations by Hwang and Moehle (2000) may be used to establish the effective stiffness of the slab and to include the impact of cracking. Hwang and Moehle (2000) recommend that the uncracked effective stiffness for a model with rigid joints, for ratios of c2/c1 from 1/2 to 2 and a slab aspect ratio l2/l1 greater than 2/3, be determined using an effective beam width represented as l1 b int = 2c 1 + --3 (18) where bint is the effective width for interior frame connections (interior connections and edge connections with bending perpendicular to the edge); c1 and l1 are the column dimension and slab span parallel to the direction of load being considered; and c2 and l2 correspond to the orthogonal direction. For exterior frame connections (corner connections and edge connections with bending parallel to the edge), half the width defined in Eq. (18) is used. Effects of cross section changes, such as slab openings, are to be considered. One way to accomplish this is to vary the width of the effective beam along the span (Hwang and Moehle 1990). To account for cracking, a stiffness reduction factor β has been proposed by Hwang and Moehle (2000) for nonprestressed slabs and is given by c 1 β = 4 - > -- l 3 (19) ACI Structural Journal/July-August 2007
- 71. where c and l are the column dimension and slab span parallel to the load direction. Kang and Wallace (2005) recommend β = 0.5 for post-tensioned floor systems with approximate values for span-to-slab thickness ratios of 40, c1/l1 of 1/14, and precompression of 200 psi (1.4 MPa). The analytical model of the slab-column frame should capture the potential for both slab yielding and connection failure due to punching as recommended in FEMA 356. Figure 4 shows an approach where yielding within the slab column strip is modeled using slab-beam elements (in this case, an elastic slab-beam with stiffness properties defined by the effective beam width model, and zero-length plastic hinges on either side of the connection). Further details of this model are described by Kang et al. (2006). Punching failures can occur if the capacity of the connection element is reached or if a limiting story drift ratio is reached for a given gravity shear ratio. Hueste and Wight (1999) suggested an approach for incorporating this behavior into a nonlinear analysis program, where, after prediction of a punching shear failure, the member behavior is modified to account for the significant reduction in stiffness and strength. Kang and Wallace (2005) suggest a direct approach by employing a limit state model. The FEMA 356 guidelines note that the analytical model for a slab-column frame should consider all potential failures including flexure, shear, shear-moment transfer, and reinforcement development at any section. The modeling information mentioned previously gives a convenient and relatively straightforward approach to modeling the behavior of slabcolumn frames for nonlinear static and dynamic analysis. PERFORMANCE-BASED DESIGN CRITERIA A review of current practice with respect to performance-based design is needed to provide context to the material presented subsequently on performance objectives for slab-column connections. The FEMA 356 prestandard (ASCE 2000) provides analytical procedures and criteria for the performancebased evaluation of existing buildings and for designing seismic rehabilitation alternatives. This prestandard includes recommended limits for deformation capacities based on the calculated gravity shear ratio, as well as a general framework for creating performance levels and objectives. In FEMA 356, performance levels describe limitations on the maximum damage sustained during a ground motion, while performance objectives define the target performance level to be achieved for a particular intensity of ground motion. Structural performance levels in FEMA 356 include immediate occupancy, life safety, and collapse prevention. Structures at collapse prevention are expected to remain standing, but with little margin against collapse. Structures at life safety may have sustained significant damage, but still provide an appreciable margin against collapse. Structures at immediate occupancy should have only minor damage. In FEMA 356, the Basic Safety Objective is defined as life safety-performance for the basic safety earthquake 1 (BSE-1) earthquake hazard level and collapse prevention performance for the BSE-2 earthquake hazard level. BSE-1 is the smaller event corresponding to 10% probability of exceedance in 50 years (10% in 50 years) and 2/3 of the BSE-2 (2% in 50 years) event. For a given design event and a target performance level, FEMA 356 provides acceptance criteria when using either static or dynamic analysis based on linear and nonlinear procedures. To evaluate acceptability using linear procedures, ACI Structural Journal/July-August 2007 Fig. 4—Modeling of slab-column connection (adapted from Kang et al. 2006). an action is classified as either deformation-controlled or force-controlled. Deformation-controlled actions are applicable for components that have the capacity to undergo deformations into the inelastic range without failure. Based on the demand to capacity ratio (DCR), calculated using the linear static or dynamic analysis procedures, components are classified as having low (DCR < 2), moderate (2 ≤ DCR ≤ 4), or high (DCR > 4) ductility demands. The acceptance criteria based on linear analysis procedures are expressed in terms of m-factors. The factor m is intended to provide an indirect measure of the total deformation capacity of a structural element or component. As such, the factor m is only used to evaluate the acceptability of deformation-controlled actions mκQCE ≥ QUD (20) where κ is the knowledge factor used to reduce the strength of existing components based on quality of information, QCE is the expected strength of a component or element at the deformation level considered, and QUD is the deformation-controlled design action. Equation (20) can be rearranged for direct comparison of the DCR to m to determine acceptability Q UD DCR ≤ m = ------------κQ CE (21) The FEMA 356 limiting values for m-factors for two-way slabs and slab-column connections are provided in Table 1. The m-factors for slab-column connections range from 1 to 4 and depend on several parameters: the gravity-shear ratio, the presence of continuity reinforcement through the column cage, the development of reinforcement, and the selected performance level. The connections must also be classified as primary or secondary elements to determine the limits for life safety and collapse prevention. Secondary elements are those typically not considered to provide resistance to earthquake effects. For nonlinear static and dynamic analysis procedures, FEMA 356 restricts inelastic response values determined from the analytical model in terms of maximum plastic rotations. Generally, plastic rotation is computed as the 453
- 72. Table 1—Acceptance criteria for linear procedures— two-way slabs and slab-column connections (adapted from FEMA 356 [ASCE 2000]) m-factors by performance level* Component type Primary Conditions IO LS Secondary CP LS 1. Slab controlled by flexure and slab-column connections CP † Vg /Vo‡ ≤ 0.2 Yes 2 2 3 3 4 ≥ 0.4 Yes 1 1 1 2 3 ≤ 0.2 No 2 2 3 2 3 ≥ 0.4 Fig. 5—Test data for interior slab-column connection specimens with no shear reinforcement. Continuity reinforcement§ No 1 1 1 1 1 2. Slabs controlled by inadequate development or splicing along span† — — — 3 4 3. Slabs controlled by inadequate embedment into slab-column joint† 2 2 3 3 4 *IO = immediate occupancy; LS = life safety; and CP = collapse prevention. When more than one of Conditions 1, 2, and 3 occurs for given component, use minimum appropriate numerical value from table. ‡ Vg = gravity shear acting on slab critical section and Vo = direct punching shear strength as defined by ACI 318. §Under heading “Continuity reinforcement,” use “Yes” where at least one of the main bottom bars in each direction is effectively continuous through column cage. Where that slab is post-tensioned, use “Yes” where at least one of post-tensioning tendons in each direction passes through column cage. Otherwise, use “No.” † Fig. 6—Test data for interior slab-column connection specimens with shear reinforcement. difference between the maximum rotation during analysis and the yield rotation at the member end. Therefore, it is critical for the nonlinear model to represent the maximum plastic rotation for a certain level of demand. The plastic rotation limits in FEMA 356 range from 0.0 to 0.02 radians for primary slab-column connections and from 0.0 to 0.05 radians for secondary slab-column connections. These limits are based on the gravity-shear ratio, the presence of continuity reinforcement through the column cage, the development of reinforcement, and the selected performance level (immediate occupancy, life safety, or collapse prevention). EXPERIMENTAL DATA Over the past 40 years, experimental studies have been conducted by researchers at a number of universities. Much of the earlier data has been summarized by Pan and Moehle (1989), Megally and Ghali (1994), and Luo and Durrani (1995). Tables 2 and 3 provide information on interior slabcolumn connection test specimens, with and without shear reinforcement. Limited tests have been conducted for nonductile slab-column connections where the bottom slab reinforcement is discontinuous at the interior slab-column connection (Durrani et al. 1995; Dovich and Wight 1996; Robertson and Johnson 2006) and available data is included in Table 2. The failure mode for each specimen is provided, when available, as either: punching shear P, flexure F, or a 454 combination of flexure and punching shear (F-P) where a punching shear failure occurred at a higher drift level following yielding of the slab reinforcement. The gravity shear ratio and peak drift are also provided for each specimen. The peak drift is defined as the drift corresponding to the peak lateral load. Therefore, the maximum drift attained for a particular specimen may be larger than the reported peak drift. The maximum drift at which an interior connection will fail can be estimated from the gravity shear ratio Vg/Vo (Pan and Moehle 1989; Luo and Durrani 1995). The gravity shear ratio represents the unfactored vertical gravity shear Vg divided by the theoretical punching shear strength without moment transfer Vo determined using Vo = vc bo d (22) The term vc is calculated using Eq. (6) or (7). A similar ratio can be computed for slabs with shear reinforcement by replacing vc with vn defined by Eq. (8) through (10). Figure 5 provides a plot of peak drift as a function of Vg/Vo for interior slab-column connection specimens with no shear reinforcement. The figure shows the direct influence of the gravity shear ratio on the lateral drift capacity of slabcolumn connections. It may be observed that punching shear occurs for a large range of Vg/Vo values (approximately 0.1 to 0.9), while flexural failures primarily occur for Vg/Vo values of 0.3 or less. Figure 6 provides a similar plot for interior slab-column connection specimens with shear reinforcement. The experimental data indicates that larger drift ratios are possible when shear reinforcement is used. In particular, a number of slab-column specimens with stud-shear reinforcement (SSR) attained story drift ratios well over 3% before failure. The data from slab-column connection tests, with and without shear reinforcement, are compared in Fig. 7, along ACI Structural Journal/July-August 2007
- 73. Table 2—Test data for interior slab-column connection test specimens with no shear reinforcement Label Vg/Vo Peak drift, % 0.85 0.90 NA I.I 0.08 5.00 F CD 2 0.65 1.20 NA INT1 0.43 NA P 0.52 1.40 NA DNY 1* 0.20 3.00 F INT2 0.50 NA P DNY 2* 0.30 2.00 P MG-2A 0.58 1.17 P * 0.24 2.00 F MG-7 0.29 3.10 F-P DNY 4* Durrani et al. (1995) Vg/Vo CD 8 Dilger and Cao (1991) Label CD 1 Source 0.28 2.60 F-P MG-8 0.42 2.30 F-P 1 0.46 NA P MG-9 0.36 2.17 F-P DNY 3 Elgabry and Ghali (1987) Peak drift, % Mode Source Luo and Durrani (1995) Megally and Ghali (2000) Mode 1 S1 0.03 4.70 F F S2 0.03 2.80 F 3 0.26 3.56 P S3 0.03 4.20 F 0.30 2.40 P S4 0.07 4.50 F 0.31 6.00 F S5 0.15 4.80 F SM 1.0 0.33 2.70 F-P AP 1 0.37 1.60 F-P SM 1.5 0.30 2.70 F-P A12 0.29 NA P A13L 0.29 NA P AP 4 B16 Hanson and Hanson (1968) F 4.04 SM 0.5 Ghali et al. (1976) 4.81 0.00 4 Farhey et al. (1993) 0.00 2 0.29 NA P 1 2 0.35 1.50EW/0.79NS P 3 0.22 3.10 F-P P Morrison and Sozen (1983) Pan and Moehle (1989) AP 2 0.36 1.50 F-P AP 3 0.18 3.70 F-P 0.19 3.50 F-P 0.35 1.50 P F-P NA F-P 0.05 5.80 F 4 0.22 3.20EW/1.75NS 0.33 3.75 P 1 0.21 2.75 F S2 0.45 2.00 P 2C 0.22 3.50 F-P S3 0.45 2.00 P S4 0.40 2.60 P 4 Int. Joints 0.24 4.00 1 0.25 3.67 2 0.23 3.33 P 3C Islam and Park (1976) 3.80 0.24 S1 Hwang and Moehle (1990) 0.04 C8 Hawkins et al. (1974) B7 C17 0.23 4.00 F-P * Pan and Moehle (1992) 3SE 0.19 3.50 F 5SO 0.21 3.50 F NA 6LL 0.54 0.85 P P 7L 0.40 1.45 P Robertson and Durrani (1990) 8I Robertson et al. (2002) 0.18 3.50 F-P 1C 0.17 3.50 P S6 0.86 1.10 P S7 0.81 1.00 P ND1C 0.23 3.00 to 5.00 F-P ND4LL* 0.28 3.00 F-P * 0.47 1.50 P SC 0 0.25 3.50 P * ND6HR 0.29 3.00 P SC 2 0.18 6.00 F NC7LR* 0.26 3.00 F-P ND5XL Robertson and Johnson (2006) ND8BU* 0.26 3.00 F-P Symonds et al. (1976) Wey and Durrani (1992) SC 4 Zee and Moehle (1984) 0.15 6.00 F SC 6 0.15 5.00 P INT 0.21 3.30 F-P *Bottom slab reinforcement is discontinuous at interior connection. Note: EW = east-west lateral load for biaxial test; NS = north-south lateral load for biaxial test; F = flexural failure; P = punching shear failure; and F-P = flexural and punching shear failure. NA: Not available. with the ACI 318-05 limits for assessing the need for shear reinforcement. The line defined by ACI 318-05 is a reasonable lower-bound limit for the data corresponding to specimens without shear reinforcement. A strength reduction factor of φ = 1 is used when determining Vg/Vo for the test data. PERFORMANCE-BASED SEISMIC DESIGN RECOMMENDATIONS Research studies and past structural performance have shown that slab-column frames provide lateral stiffness contributions to the overall LFRS and, as such, they do resist lateral loads during a seismic event even if they were designed for gravity loads only. For this reason, compatibility of deformations must be considered to calculate the demands at the slab-column connections. Likewise, the analytical model should include the strength and stiffness of the slab-column ACI Structural Journal/July-August 2007 frames to ensure an accurate representation of the overall building stiffness and allow an evaluation of the magnitude of the lateral load that must be resisted by the slab-column frame members. The appropriate parameters that should be included in such a model were highlighted previously (effective slab width for equivalent beams, cracked section properties, and hysteretic behavior for nonlinear models). Performance-based seismic design (PBSD) criteria are suggested in the following. The criteria are based on experimental data of interior slab-column connections under combined gravity and lateral load. The suggested criteria reference FEMA 356 performance levels (immediate occupancy, life safety, and collapse prevention) and seismic design requirements for slab-column connections that are adopted in ACI 318-05. As noted previously, in regions of high seismic risk, the slab-column connections of two-way 455
- 74. Table 4—Key points for recommended PBSD criteria for interior slab-column connections Table 3—Test data for interior slab-column connection test specimens with shear reinforcement Source Label Shear Vg /Vo Peak drift, % reinforcement Mode SJB-1 0.48 5.50 SSR S1 SJB-2 0.47 5.70 SSR S1 SJB-3 SSR 0.47 7.60 SSR 5.70 SSR 7.10 SSR 0.91 3.50 SSR 4.80 SSR 0.64 5.40 SSR 0.51 5.60 SSR 0.47 NA SSR 0.87 NA SSR 0.85 NA SSR P 5 1.20 NA SSR P SS1 0.49 3.50 Stirrups C3 SS2 0.47 3.43 Stirrups P SS3 0.48 4.10 Stirrups F SS4 0.47 5.50 Stirrups NA SS5 0.42 4.90 Stirrups F 4S 4.33 Bent up P 0.23 4.17 Shear head 6CS 0.24 4.00 Stirrups P 0.24 3.70 Stirrups P 8CS 0.27 5.00 Stirrups 0.16 4.50 Closed hoop F 4HS 0.15 5.00 Headed stud F 3SL 4.50 Single leg F 0.60 5.20 SSR NA MG-3 0.56 5.40 SSR NA MG-4 0.86 4.60 SSR F-P MG-5 0.31 6.50 SSR F-P MG-6 Robertson and Durrani (1990) 0.10 MG-10 Megally and Ghali (2000) 0.59 6.00 SSR F-P 4S 0.19 3.50 Closed hoop F 1 Note: SSR = stud-shear reinforcement; S = shear failure outside shear reinforced zone; S2 = shear failure in shear in zones without shear reinforcement; C3 = crushing failure at column face without apparent punching shear failure; F = flexural failure; P = punching shear failure; and F-P = flexural and punching shear failure. NA: not available. slabs without beams must be checked for the induced effects caused by the lateral displacement expected for the designbasis earthquake. It is important to note the direct influence of the gravity shear ratio on the lateral drift capacity of slabcolumn connections without shear reinforcement illustrated by the test data in Fig. 5. As suggested by the FEMA 356 limits for slab-column connections, this relationship is critical to the development of appropriate PBSD criteria for slab-column connections. The ACI 318-05 seismic design limits for slabcolumn connections given in Eq. (16) also underscore the direct relationship between these two parameters. Linear regression analysis on the experimental data for slab-column connections without shear reinforcement and having a gravity shear ratio Vg/Vo less than 0.6, results in a line defined by a slope of –6.95 and a zero intercept of 4.97. 456 Fig. 7—Comparison of recommended performance-based seismic design limits with slab-column connection test data. Thus, the mean for the data gives the following expression for the maximum story drift ratio (in percent) P 2CS Note: IO = immediate occupancy; LS = life safety; and CP = collapse prevention. P 7CS Robertson et al. (2002) 0.23 5S Islam and Park (1976) 0.75 P 4 0.75 0.5 P 3 0.5 0.25 NA 2 0.25 1.0 NA CD 7 0.6 NA CD 6 5.0 NA 0.62 3.5 S2 CD 4 CP 1.75 S2 0.49 LS 0.0 S1 0.46 IO S2 CD 3 Hawkins et al. (1975) SSR 6.40 SJB-9 Elgabry and Ghali (1987) 5.00 0.43 SJB-8 Dilger and Cao (1991) 0.48 SJB-4 SJB-5 Dilger and Brown (1995) S2 Drift ratio, %, by performance level Gravity shear ratio (Vg /Vo) Vg DR = 5 – 7 ----Vo (23) The PBSD criteria suggested herein use Eq. (23) as a reference for selecting the collapse prevention performance level limits. The life safety performance level was initially defined as 2/3 of the values used for collapse prevention; and for immediate occupancy, 1/3 of the values for collapse prevention was used. The drift limits determined using the aforementioned parameters were the basis for finalizing the key points of the graphed PBSD criteria. Table 4 summarizes the key points for the recommended PBSD criteria and the values are shown graphically in relationship to the test data in Fig. 7. For the suggested PBSD criteria, the drift limits for the immediate occupancy performance level are relatively low so that the slab-column frame members remain at or near the elastic range of behavior. The suggested line for life safety corresponds to the ACI 318-05 design limits (refer to Fig. 2). The life safety performance level includes the combination of Vg/Vo = 0.4 and a drift of 1.5%, which is consistent with the recommendation in ACI 352.1R-89 that the gravity shear ratio should be kept below 0.4 to ensure some minimal ductility with the availability of approximately 1.5% drift capacity. The collapse prevention limits correspond to approximately the mean of the experimental data for specimens without shear reinforcement. For all performance levels, a constant story drift ratio capacity is assigned for gravity shear ratios in excess of 0.6. As the approximate mean of the data for specimens without shear reinforcement (Fig. 7), the collapse prevention limits correspond to a 50% probability of failure (without considering the load and resistance factors provided in the ACI Structural Journal/July-August 2007
- 75. code). Assuming a normal distribution, the life safety limits, defined as 2/3 of collapse prevention, correspond to approximately 5% probability of failure, and the immediate occupancy limits, defined as 1/3 of collapse prevention, correspond to less than 1% probability of failure. When the story drift limit corresponding to the acting gravity shear ratio is exceeded for the performance level considered, various options exist, including: 1) reduce the gravity shear ratio by thickening the slab, adding shear capitals, or adding drop panels; 2) reduce the story drift ratio to be within the allowable limit by stiffening the lateral system; or 3) add shear reinforcement as prescribed by ACI 318-05. For Options 1 and 2, consideration must be given to increased lateral forces resulting from the structural modification. For Option 3, the experimental data indicates that larger drift ratios are possible when shear reinforcement is used (refer to Fig. 6). The data for the shear reinforced specimens are included in Fig. 7 for comparison. A direct comparison of the suggested PBSD criteria to the FEMA 356 acceptance criteria is not simply accomplished because FEMA limits are in terms of plastic rotations rather than drift ratios. FEMA 356 is intended for assessing existing structures and also addresses cases involving several possible deficiencies, including: 1) inadequate development or splicing along the slab span; 2) inadequate embedment into the slab-column joint; and 3) lack of continuity reinforcement through the column cage. In addition, a distinction is made between primary and secondary components. In general, the proposed PBSD limits appear to be in the range of the corresponding FEMA 356 limits. One exception is that FEMA 356 does not allow plastic rotation in primary components when the gravity shear ratio is above 0.4. The aforementioned PBSD criteria are intended primarily for new construction. The criteria, however, could also be applied to existing structures that contain subpar seismic details where a moderate seismic demand is expected. For assessing the expected performance of a structure, the value of Vo should be computed with φ = 1.0, whereas for new building design, Vo should include a strength reduction factor for shear, currently φ = 0.75 in ACI 318-05. SUMMARY AND CONCLUSIONS This paper focuses on the behavior and design of interior slab-column connections under combined gravity and lateral loading and serves to review current design procedures, performance-based seismic design (PBSD) approaches, and relevant experimental data. Practical recommendations are provided for PBSD of slab-column connections under seismic loading conditions that can be readily implemented into design practice. An assessment of the experimental data versus the ACI 318-05 recommendations for slab-column connections indicate that the limits for determining the necessity of slab shear reinforcement are a reasonable lower bound of the test data. Very few reports for slab-column connection specimens include plastic rotation data. FEMA 356, however, provides limits in terms of plastic rotations for nonlinear analysis procedures that are determined in part by the gravity shear ratio at the slab-column connections. The recommended PBSD criteria in this paper use two key parameters for assessing slab-column connections: the gravity shear ratio at the connection and the maximum story drift ratio. The use of story drift ratio allows a direct comparison to the experimental data and is readily available when conducting a structural ACI Structural Journal/July-August 2007 analysis. A relationship between drift capacity and gravity shear ratio is provided in Eq. (23), representing an average of the collected experimental data. Three performance levels are used to match those in FEMA 356: immediate occupancy, life safety, and collapse prevention. The proposed limits correlate well with the ACI 318-05 seismic design provisions for slab-column connections and provide a practical approach for conducting PBSD for slab-column connections. ACKNOWLEDGMENTS The authors wish to thank the members of ACI Committee 374, PerformanceBased Seismic Design of Concrete Buildings, for their input. The contribution of Y.-H. Kim, a graduate student at Texas A&M University, College Station, Tex., is also appreciated. REFERENCES ACI Committee 318, 2005, “Building Code Requirements for Structural Concrete (ACI 318-05) and Commentary (318R-05),” American Concrete Institute, Farmington Hills, Mich., 430 pp. American Society of Civil Engineers (ASCE), 2000, “Prestandard and Commentary for the Seismic Rehabilitation of Buildings (FEMA Publication 356),” Federal Emergency Management Agency, Washington, D.C., 528 pp. Corley, W. G., and Hawkins, N. M., 1968, “Shearhead Reinforcement for Slabs,” ACI JOURNAL, Proceedings V. 65, No. 10, Oct., pp. 811-824. Dilger, W., and Brown, S. J., 1995, “Earthquake Resistance of SlabColumn Connection,” Festschrift Professor Dr. Hugo Bachmann Zum 60, Geburtstag, Institut FÜR Baustatik Und Konstruktion, Eth ZÜRich, Switzerland, pp. 22-27. Dilger, W., and Cao, H., 1991, “Behaviour of Slab-Column Connections under Reversed Cyclic Loading,” Proceedings of the Second International Conference of High-Rise Buildings, China, 10 pp. DiStasio, J., and Van Buren, M. P., 1960, “Transfer of Bending Moment between Flat Plate Floor and Column,” ACI JOURNAL, Proceedings V. 57, No. 3, Mar., pp. 299-314. Dovich, L., and Wight, J. K., 1996, “Lateral Response of Older Flat Slab Frames and Economic Effect on Retrofit,” Earthquake Spectra, V. 12, No. 4, pp. 667-691. Durrani, A. J.; Du, Y.; and Luo, Y. H., 1995, “Seismic Resistance of Nonductile Slab-Column Connections in Existing Flat-Slab Buildings,” ACI Structural Journal, V. 92, No. 4, July-Aug., pp. 479-487. Elgabry, A., and Ghali, A., 1987, “Tests on Concrete Slab-Column Connections with Stud-Shear Reinforcement Subjected to Shear-Moment Transfer,” ACI Structural Journal, V. 84, No. 5, Sept.-Oct., pp. 433-442. Farhey, D. N.; Adin, M. A.; and Yankelevsky, D. Z., 1993, “RC Flat Slab-Column Subassemblages under Lateral Loading,” Journal of the Structural Division, V. 119, No. 6, ASCE, pp. 1903-1916. Ghali, A.; Elmasri, M. Z.; and Dilger, W., 1976, “Punching of Flat Plates under Static and Dynamic Horizontal Forces,” ACI JOURNAL, Proceedings V. 73, No. 10, Oct., pp. 566-572. Hanson, N. W., and Hanson, J. M., 1968, “Shear and Moment Transfer between Concrete Slabs and Columns,” Journal, V. 10, No. 1, PCA Research and Development Laboratories, pp. 1-16. Hawkins, N. M.; Mitchell, D.; and Sheu, M. S., 1974, “Cyclic Behavior of Six Reinforced Concrete Slab-Column Specimens Transferring Moment and Shear,” Progress Report 1973-74 on NSF Project GI-38717, Section II, University of Washington, Seattle, Wash., 50 pp. Hawkins, N. M.; Mitchell, D.; and Hanna, S. N., 1975, “Effects of Shear Reinforcement on the Reversed Cyclic Loading Behavior of Flat Plate Structures,” Canadian Journal of Civil Engineering, V. 2, pp. 572-582. Holmes, W. T., and Somers, P., 1996, “Northridge Earthquake of January 17, 1994: Reconnaissance Report,” Earthquake Spectra, V. 11, Supplement C, pp. 224-225 Hueste, M. D., and Wight, J. K., 1997, “Evaluation of a Four-Story Reinforced Concrete Building Damaged During the Northridge Earthquake,” Earthquake Spectra, V. 13, No. 3, pp. 387-414. Hueste, M. D., and Wight, J. K., 1999, “Nonlinear Punching Shear Failure Model for Interior Slab-Column Connections,” Journal of Structural Engineering, ASCE, V. 125, No. 9, pp. 997-1008. Hwang, S. J., and Moehle, J. P., 1990, “An Experimental Study of FlatPlate Structures Under Vertical and Lateral Loads,” Report No. UCB/ SEMM-90/11, University of California-Berkeley, Berkeley, Calif., 271 pp. Hwang, S. J., and Moehle, J. P., 2000, “Models for Laterally Loaded SlabColumn Frames,” ACI Structural Journal, V. 97, No. 2, Mar.-Apr., pp. 345-353. Islam, S., and Park, R., 1976, “Tests on Slab-Column Connections with Shear and Unbalanced Flexure,” Journal of the Structural Division, V. 102, No. ST3, ASCE, pp. 549-568. 457
- 76. Joint ACI-ASCE Committee 326, 1962, “Shear and Diagonal Tension, Slabs,” ACI JOURNAL, Proceedings V. 59, No. 3, Mar., pp. 353-396. Joint ACI-ASCE Committee 352, 1989, “Recommendations for Design of Slab-Column Connections in Monolithic Reinforced Concrete Structures (ACI 352.1R-89),” American Concrete Institute, Farmington Hills, Mich., 22 pp. Joint ACI-ASCE Committee 421, 1999, “Shear Reinforcement for Slabs (ACI 421.1R-99),” American Concrete Institute, Farmington Hills, Mich., 15 pp. Kang, T. H. K., and Wallace, J. W., 2005, “Dynamic Response of Flat Plate Systems with Shear Reinforcement,” ACI Structural Journal, V. 102, No. 5, Sept.-Oct., pp. 763-773. Kang, T. H. K.; Elwood, K. J.; and Wallace, J. W., 2006, “Dynamic Tests and Modeling of RC and PT Slab-Column Connections,” Paper 0362, Proceedings of the 8th U.S. National Conference on Earthquake Engineering, San Francisco, Calif., 10 pp. (CD-ROM) Luo, Y., and Durrani, A. J., 1995, “Equivalent Beam Model for Flat-Slab Buildings—Part 1: Interior Connections,” ACI Structural Journal, V. 92, No. 1, Jan.-Feb., pp. 115-124. Megally, S., and Ghali, A., 1994, “Design Considerations for Slab-Column Connections in Seismic Zones,” ACI Structural Journal, V. 91, No. 3, MayJune, pp. 303-314. Megally, S., and Ghali, A., 2000, “Punching Shear Design of Earthquake-Resistant Slab-Column Connections,” ACI Structural Journal, V. 97, No. 5, Sept.-Oct., pp. 720-730. Moehle, J. P., 1996, “Seismic Design Considerations for Flat Plate Construction,” Mete A. Sozen Symposium: A Tribute from his Students, SP-162, J. K. Wight and M. E. Kreger, eds., American Concrete Institute, Farmington Hills, Mich., pp. 1-35. Morrison, D. G., and Sozen, M. A., 1983, “Lateral Load Tests of R/C Slab-Column Connections,” Journal of the Structural Division, ASCE, V. 109, No. 11, pp. 2699-2714. 458 Pan, A., and Moehle, J. P., 1989, “Lateral Displacement Ductility of Reinforced Concrete Flat Plates,” ACI Structural Journal, V. 86, No. 3, May-June, pp. 250-258. Pan, A., and Moehle, J. P., 1992, “An Experimental Study of SlabColumn Connections,” ACI Structural Journal, V. 89, No. 6, Nov.-Dec., pp. 626-638. Robertson, I., and Durrani, A. J., 1990, “Seismic Response of Connections in Indeterminate Flat-Slab Subassemblies,” Report No. 41, Department of Civil Engineering, Rice University, Houston, Tex., 266 pp. Robertson, I.; Kawai, T.; Lee, J.; and Enomoto, B., 2002, “Cyclic Testing of Slab-Column Connections with Shear Reinforcement,” ACI Structural Journal, V. 99, No. 5, Sept.-Oct., pp. 605-613. Robertson, I., and Johnson, G., 2006, “Cyclic Lateral Loading of Nonductile Slab-Column Connections,” ACI Structural Journal, V. 103, No. 3, MayJune, pp. 356-364. Rodriguez, M., and Diaz, C., 1989, “Analysis of the Seismic Performance of a Medium Rise, Waffle Flat Plate Building,” Earthquake Spectra, V. 5, No. 1, pp. 25-40. SEI/ASCE, 2005, “Minimum Design Loads for Buildings and other Structures (SEI/ASCE 7-05),” Structural Engineering Institute, ASCE, Reston, Va., 376 pp. Symonds, D. W.; Mitchell, D.; and Hawkins, N. M., 1976, “Slab-Column Connections Subjected to High Intensity Shears and Transferring Reversed Moments,” Progress Report on NSF Project GI-38717, Department of Civil Engineering, University of Washington, Seattle, Wash., 80 pp. Wey, E. H., and Durrani, A. J., 1992, “Seismic Response of Interior Slab-Column Connections with Shear Capitals,” ACI Structural Journal, V. 89, No. 6, Nov.-Dec., pp. 682-691. Zee, H. L., and Moehle, J. P., 1984, “Behavior of Interior and Exterior Flat Plate Connections Subjected to Inelastic Load Reversals,” Report No. UCB/EERC-84/07, Earthquake Engineering Research Center, University of California-Berkeley, Berkeley, Calif., 130 pp. ACI Structural Journal/July-August 2007
- 77. ACI STRUCTURAL JOURNAL TECHNICAL PAPER Title no. 104-S37 Shake Table Studies of Bridge Columns with Double Interlocking Spirals by Juan F. Correal, M. Saiid Saiidi, David Sanders, and Saad El-Azazy The California Department of Transportation (Caltrans) code provides the only guidelines in the U.S. for the design of columns with interlocking spirals. Previous studies have shown that columns with interlocking spirals have a satisfactory behavior, but none of them have addressed the Caltrans upper limit on horizontal spacing between centers of the spirals in detail and none used dynamic testing. Six large-scale column models were designed and tested on a shake table at the University of NevadaReno to study the effects of the shear level, spiral distance, and crossties. The observed damage progression, load-displacement responses, reinforcement strains, and the apparent plastic hinge lengths were examined to evaluate the response. The results revealed that the Caltrans upper spiral distance limit of 1.5 times the spiral radius is satisfactory. However, supplementary crossties are needed to prevent premature vertical shear cracking and strength degradation in columns with relatively high shear. Keywords: bridge; columns; interlocking spirals; seismic behavior. INTRODUCTION The current seismic design philosophy for reinforced concrete structures relies on confinement of concrete to provide the necessary ductility and energy dissipation capacity of structural members. Confinement is mainly provided by the transverse reinforcement, which in columns usually consists of spirals in members with circular or square shape and ties in those with square or rectangular cross sections. Spirals confine concrete more effectively than rectilinear ties because they counteract the dilation of concrete through hoop action instead of a combination of bending and hoop action that takes place in rectilinear ties. As a result, to provide the same level of confinement, the amount of tie reinforcement is greater than that provided by spirals. Another advantage of spirals is that they are generally easier to construct. The circular shape of spirals makes them suitable for circular and square columns. To use the benefits of spirals in rectangular columns, two or more sets of interlocking spirals are used. The Caltrans Bridge Design Specifications (BDS)1 and Seismic Design Criteria (SDC)2 are currently the only codes in the U.S. that include provisions for the design of columns with interlocking spirals. Because the amount of research on interlocking spirals has been limited, the design provisions are driven mainly by research on single spirals. Studies3-5 were conducted on the effect of several design parameters, including a comparison between interlocking spirals and ties, horizontal distance between centers of the spirals, quantity of transverse reinforcement, variation of the axial load ratios, appropriate size and spacing of longitudinal bars in the interlocking region, and cross section shape. These studies generally concluded that flexural and shear capacities of columns with interlocking spirals can be conservatively estimated using current procedures. Conflicting ACI Structural Journal/July-August 2007 recommendations exist, however, with respect to the distance between spiral sets and uncertainties about the need for supplemental crossties between adjacent spiral sets. For example, the BDS upper limit on the distance between the centers of adjacent spirals is 1.5 times the radius of the spiral R, whereas the study in Reference 3 places an upper limit of 1.2R. To address these issues, a study was undertaken using large-scale testing of bridge column models on one of the shake tables of the University of Nevada-Reno. The study included both experimental and analytical components to evaluate the seismic performance of bridge columns with double interlocking spirals with different parameters, including the spread between the spiral sets, the level of shear, and crossties. The focus of this paper is on the experimental phase of the investigation. Details of all aspects of the study are presented in Reference 6. RESEARCH SIGNIFICANCE Interlocking spirals are used in the columns of many bridges. The spirals are designed based on provisions that have yet to be verified and, in part, are in conflict with some of the recommendations that are based on the limited available past studies. The research presented in this paper was used to: 1) evaluate the dynamic performance of bridge columns that are designed based on the current Caltrans provisions; 2) determine if the limits in the provisions are satisfactory; and 3) identify cases and limit states in which supplemental crossties are needed. EXPERIMENTAL STUDIES Test specimens Six large-scale specimens were designed, constructed, and tested. The limit of 1.2R on the horizontal distance of the centers of the spirals, di, recommended in Reference 3 is to ensure sufficient vertical shear transfer between adjacent spiral sets. Because vertical shear is a function of horizontal shear, the test parameters were selected to capture the effect of a range of realistic horizontal shear stresses. The test variables were: 1) the level of average shear stress; 2) the horizontal distance between the centers of the spirals, di; and 3) supplementary horizontal crossties. The test variables are listed in Table 1. The effect of other parameters such as axial load and material strength was not considered because the variation of these parameters in real bridges is relatively small. The average horizontal shear stress was calculated as the lateral load divided by the effective shear area taken equal to ACI Structural Journal, V. 104, No. 4, July-August 2007. MS No. S-2005-200 received August 8, 2005, and reviewed under Institute publication policies. Copyright © 2007, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published in the May-June 2008 ACI Structural Journal if the discussion is received by January 1, 2008. 393
- 78. Juan F. Correal is an Assistant Professor of civil and environmental engineering at the University of Los Andes, Colombia, where he received his BS and MSCE. He received his PhD in 2004 from the University of Nevada-Reno, Reno, Nev. His research interests include the seismic design of bridges and applications of innovative materials for design, repair, and rehabilitation of structures. M. Saiid Saiidi, FACI, is a Professor of civil and environmental engineering and is the Director of the Office of Undergraduate Research at the University of Nevada-Reno. He is a Past Chair and a member of ACI Committee 341, Earthquake-Resistant Concrete Bridges, and is member of ACI Committees 342, Bridge Evaluation; E803, Faculty Network Coordinating Committee; and Joint ACI-ASCE Committee 352, Joints and Connections in Monolithic Concrete Structures. His research interests include analysis and shake table studies of reinforced concrete bridges and application of innovative materials. David Sanders, FACI, is an Associate Professor of civil and environmental engineering at the University of Nevada-Reno. He is Chair of ACI Committee 445, Shear and Torsion, is Past Chair of ACI Committee 341, Earthquake-Resistant Concrete Bridges, and is a member of the ACI Technical Activities Committee; ACI Committees 318, Structural Concrete Building Code; 369, Seismic Repair and Rehabilitation; 544, Fiber Reinforced Concrete; E803, Faculty Network Coordinating Committee; E804, Educational Awards Nomination Committee; and Joint ACI-ASCE Committee 423, Prestressed Concrete. His research interests include shake table studies of reinforced concrete bridges. Table 1—Test variables Steel reinforcement Specimen ISL1.0 ISL1.5 Scale factor 0.25 Shear index Aspect ratio di (× R) ρl , % ρs ,* % 3.0 3.3 1.0 2.0 1.1 3.0 3.6 1.5 2.0 1.1 ISH1.0 7.0 2.0 1.0 2.9 0.6 ISH1.25 7.0 2.0 1.25 2.8 0.9 7.0 2.1 1.5 2.9 0.9 7.0 2.1 1.5 2.9 0.9* ISH1.5 ISH1.5T 0.2 * Steel ratio from additional crossties is not included. Note: ρl = ratio of longitudinal reinforcement and ρs = ratio of transversal reinforcement. Saad EI-Azazy is the Seismic Research Program Manager at the California Department of Transportation (Caltrans). He received his BS from Cairo University, Giza, Egypt, and his MS and PhD from Ohio State University, Columbus, Ohio. His research interests include bridge seismic retrofit and performance of new bridges. 80% of the gross area (SDC).2 A shear stress index was defined as the average shear stress divided by 0.083√f ′c (MPa) (√f ′c [psi]). This index represents the level of shear in the column. In this study, two levels of shear were selected: low index equal to 3 and high index equal to 7. These indexes represent column shear stresses in real bridges. Actual bridge columns are designed to be ductile and the load capacity is controlled by flexure, although shear damage is expected to increase as the shear index increases. The Caltrans BDS1 states that when more than one cage is used to confine an oblong column core, the spirals must be interlocked or the pier must be designed as though it consists of multiple single columns. A maximum limit of 1.5 times the radius of the spirals, R, (where R is measured to the outside edge of the spiral) for the horizontal distance of the spirals, measured center-to-center of the spirals, di, is specified. A minimum distance of 1.0R is recommended to avoid overlaps of more than two spirals in multiple spiral cases. Of the six models used in this study, two were designed with a di of 1.0R, one with a di of 1.25R, and three with a di of 1.5R. Three alphabetical characters followed by a number were used to identify the test specimens. The initials I and S represented interlocking and spirals, respectively. The third initial L or H was for the shear index of low or high, respectively. A numeral indicated the fraction of R used for di. In one specimen an initial T was added at the end of the specimen, designation to indicate the presence of supplementary crossties (Fig. 1). The experimental program was developed to use one of the shake tables at the Large-Scale Structures Laboratory at the University of Nevada-Reno. Scale factors of 1/4 for the specimens with low shear and 1/5 for the columns with high shear were selected. These were the largest scales that could be used without exceeding shake table capacity. The displacement-based design procedure in the SDC2 was used for a target displacement ductility capacity of 5. In the SDC,2 the displacement ductility is defined as the displacement divided by the effective yield displacement excluding bond slip and shear deformations. Typical steel ratios of 2.0% and 394 Fig. 1—Test specimens cross sections. 2.8% were selected for the longitudinal reinforcement. The transverse steel ratio was designed to provide sufficient confinement for the columns to reach the target displacement ductility capacity. Additional crossties with the same bar size as the spirals and spacing of two times the spacing of the spirals were used based on a design recommendation described in Reference 6. An axial load index, defined as the axial load divided by the product of the gross cross-sectional area and the specified concrete compressive strength of 10%, was used to represent the axial load level in real bridge columns. The details of the cross section and the elevations of the specimens are shown in Fig. 1 and 2, respectively. The spirals were continuous with constant pitch throughout the height of the specimens. The spirals were extended along the height of the footing and the top loading head. The longitudinal reinforcement was continuous with 90-degree standard hooks at the ends. In the specimens with low shear, the height was taken from the top of the footing to the center of the lateral loading head because these columns were tested in single curvature cantilever mode. The height for others was taken as the clear distance between the top of the footing and the bottom of the loading head because these columns were tested in double curvature. The specified concrete compressive strength was 34.5 MPa (5000 psi) with 9.52 mm (3/8 in.) maximum aggregate size. The average measured concrete strength of the standard cylindrical ACI Structural Journal/July-August 2007
- 79. samples on the day of testing was 36.8 MPa (5339 psi) for Specimens ISL1.0 and ISL1.5, 31.1 MPa (4514 psi) for Specimens ISH1.0 and ISH1.5, and 45.1 MPa (6542 psi) for Specimens ISH1.25 and ISH1.5T. The specified yield stress for all the reinforcement was 420 MPa (60 ksi). The average measured yield stress of the steel samples was 462 MPa (67 ksi) for Specimens ISL1.0 and ISL1.5, 443 MPa (64 ksi) for Specimens ISH1.0 and ISH1.5, and 431 MPa (63 ksi) for Specimens ISH1.25 and ISH1.5T. Test setup, instrumentation, and testing procedure Figure 3 shows the shake table setup for the high shear specimens. The test setup for the low-shear specimens was similar but with only one link between the mass rig and the column to achieve single-curvature testing. All specimens were tested in the strong direction. The lateral load was applied through an inertial mass system off the table for better stability. Two sets of swiveled links were used to connect the inertial mass system to the specimens. One set consisted of one link connected at the column loading head to test the specimens as a cantilever member with single curvature. The other set consisted of two links connected at the top of the column, allowing the specimens to be tested in double curvature. The double-link system was designed to prevent rotation of the loading head. The specimens with low shear (ISL1.0 and ISL1.5) were tested in single curvature whereas the specimens with high shear (ISH1.0, ISH1.25, ISH1.5, and ISH1.5T) were tested in double curvature. The total equivalent weight of the inertia mass was 445 and 356 kN (100 and 80 kips) for specimens tested in single and double curvature, respectively. The axial load was applied through a steel spreader beam by prestressed bars connected to hydraulic jacks and an accumulator to limit axial load fluctuation. Electrical strain gauges were attached to the longitudinal and transverse steel to measure strain variation. A series of curvature measurement instruments were installed in the plastic hinge zones. Additional displacement transducers forming panels were placed along the height of the columns with high shear. Load cells were used to measure both the axial and lateral forces. The acceleration at the top of the columns was measured using an accelerometer placed on the link connecting the mass rig to the specimens. Wire potentiometers were used to measure the lateral displacements of the columns. Preliminary moment-curvature analysis was performed to estimate the lateral load and displacement capacities of the specimens. Once the capacity was estimated, a series of dynamic analyses were conducted to select the input motion to be simulated in the shake table tests. The 1994 Northridge earthquake, recorded at the Sylmar Hospital (0.606g peak ground acceleration [PGA]) was selected as the input motion based on the maximum displacement ductility demand placed on the columns without exceeding the shake table capacity. The earthquake record is referred to as “Sylmar” hereafter. The time axis of the input record was compressed to account for the scale effect and the minor differences between the axial load and the effective mass. Each column was subjected to multiple simulated earthquakes, each referred to as a “run.” The amplitude of the motions was increased in subsequent runs. Small increments of the Sylmar record (10 to 20% of the full Sylmar amplitude) were initially applied to the specimens to determine the initial stiffness, the elastic response, and the effective yield point. Once the effective yield was reached, the amplitude of the input record was increased until failure. ACI Structural Journal/July-August 2007 Fig. 2—Test specimens elevations. Fig. 3—Double cantilever test setup. Intermittent free vibration tests were conducted to measure the change in frequency and damping ratio of the columns. EXPERIMENTAL RESULTS Important aspects of the seismic performance of the test columns were evaluated. The observed damage progression, load-displacement response, and strains were used to judge the behavior of the columns. Additional response parameters, the curvature and plastic hinge length, were computed based on the measured data and used in performance evaluation. Observed response Specimens with low shear—The observed performance was correlated with the displacement ductility μd, which represents the displacement divided by the effective measured yield displacement. Only flexural cracks were observed during the first three runs (displacement ductility of up to 0.8) in Specimen ISL1.0 and during the first six runs (μd of up to 1.5) in Specimen ISL1.5. Most of these cracks were located in the lower 1/3 of the column height. First spalling and shear cracks were seen in Specimen ISL1.0 after 0.5 × Sylmar (μd = 1.5) and in Specimen ISL1.5 after 1.25 × Sylmar (μd = 2.4). The shear cracks were located in the interlocking region in the lower 1/3 of the height of the column and were extensions of the flexural cracks. Considerable spalling at the bottom of the column, as well as propagation of flexural and shear cracks, was observed after 1.25 × Sylmar (μd = 2.8) in Specimen ISL1.0 and 1.5 × Sylmar (μd = 3.1) in Specimen ISL1.5. Spirals were visible 395
- 80. Fig. 4—Specimen ISL1.0 after failure. Fig. 6—Specimen ISH1.25 after failure. Fig. 5—Vertical crack (µd = 0.7) Specimen ISH1.5. Fig. 7—Specimen ISH1.5T after failure. after 1.5 × Sylmar (μd = 4.1) and longitudinal bars were exposed after 1.75 × Sylmar (μd = 5.6) in Specimen ISL1.0. The spirals were visible in Specimen ISL1.5 after 1.75 × Sylmar (μd = 4.5) and became exposed over a large area after 2.0 × Slymar (μd = 7.5). There was no visible core damage in either specimen. Specimens ISL1.0 and ISL1.5 failed during 2.0 × Sylmar (1.21g PGA and μd = 9.6) and 2.125 × Sylmar (1.29g PGA and μd = 10.4), respectively. Figure 4 shows the damage after failure in Specimen ISL1.0. The failure in both columns was similar and was due to rupture of the spirals and buckling of the longitudinal bars at the bottom of the column in the plastic hinge zone. Specimens with high shear—Even though these columns had a relatively high shear index, they were flexural members and, hence, only flexural cracks were formed during the initial three or four runs. The measured displacement ductilities associated with initial flexural cracks were 0.4, 0.6, 0.7, and 0.6 in Specimens ISH1.0, ISH1.25, ISH1.5, and ISH1.5T, respectively. The flexural cracks were located in the plastic hinge zones near the top and bottom of the columns. These cracks were concentrated mainly at the top and bottom 1/3 of the column height. A vertical crack in the interlocking region extending from the top of the column to the midheight was observed after 0.4 × Slymar (μd = 0.7) in Specimen ISH1.5 (Fig. 5). Diagonal cracks were formed in the interlocking region in the plastic hinge zones of all the specimens. These cracks began to form starting with 0.5 × Sylmar (μd = 0.6) and became noticeable under 0.75 × Sylmar (μd = 0.9) in Specimen ISH1.0, and 1.0 × Sylmar (μd = 1.4) in Specimen ISH1.25. In Specimen ISH1.5, shear cracks were visible starting with 0.75 × Sylmar (μd = 1.0) and in Specimen ISH1.5T under 1.0 × Sylmar (μd = 1.2). Localized small vertical cracks were observed in Specimen ISH1.5T under 1.0 × Sylmar. After 1.0 × Sylmar (μd = 1.4), first spalling at the top and bottom of the column was observed in Specimens ISH1.0 and ISH1.5, whereas in Specimens ISH1.25 and ISH1.5T, the first spalling was observed during 1.25 × Sylmar (μd = 1.6 in Specimen ISH1.25 and 1.7 in Specimen ISH1.5T). Flexural and shear cracks propagated and more concrete spalled during 1.5 × Sylmar (μd = 2.5) in Specimen ISH1.0, 1.75 × Sylmar (μd = 2.2) in Specimen ISH1.25, 1.25 × Sylmar (μd = 1.7) in Specimen ISH1.5, and 1.75 × Sylmar (μd = 2.5) in Specimen ISH1.5T. The spirals were visible at the top and bottom of the column after 2.125 × Sylmar (μd = 2.9) in Specimen ISH1.25. The longitudinal bars were exposed after 1.75 × Sylmar (μd = 3.6) in Specimen ISH1.0, 2.25 × Sylmar (μd = 3.7) in Specimen ISH1.25, 1.5 × Sylmar (μd = 2.2) in Specimen ISH1.5, and 2.0 × Sylmar (μd = 2.8) in Specimen ISH1.5T. Specimens ISH1.0 and ISH1.25 (Fig. 6) failed in flexure/shear during 2.0 × Sylmar (μd = 4.7) near the bottom and 2.375 × Sylmar (μd = 4.7) near the top, respectively. Damage in the core was observed in Specimen ISH1.5 after 2.125 × Sylmar (μd = 4.7) and in Specimen ISH1.5T after 2.25 × Sylmar (μd = 3.0). The longitudinal bars buckled at the bottom of the column during 2.25 × Sylmar (μd = 3.4) in Specimen ISH1.5 and 2.5 × Sylmar (μd = 3.4) in Specimen ISH1.5T Specimens ISH1.5 and ISH.5T (Fig. 7) failed during 2.375 × Sylmar (μd = 4.0) and 2.625 × Sylmar (μd = 3.8), respectively. Failure in Specimen ISH1.5, was due to fracture of the spirals and buckling of the longitudinal bars, whereas in Specimen ISH1.5T, failure was due to fracture of the spirals and one of the longitudinal bars. 396 ACI Structural Journal/July-August 2007
- 81. Fig. 8—Hysteretic curves and envelopes for low-shear specimens. Fig. 9—Hysteretic curves and envelopes for high-shear specimens. Force-displacement relationships The accumulated measured hysteresis curves for the ISL and ISH groups are plotted in Fig. 8 and 9, respectively. For each column, a backbone force-displacement envelope was developed based on the peak forces with corresponding displacements for all the motions before failure. The failure point in the backbone curve was assumed either at the point of maximum displacement or at a point with 80% of the peak force with the corresponding displacement. The latter was used when the force at the maximum displacement dropped more than 20% of the pick force (Fig. 8 and 9). The backbone curves for the predominant direction of the motion were idealized by elasto-plastic curves to quantify the ductility capacity. The force corresponding to the first reinforcement yield and the corresponding displacement on the measured envelope was used to define the elastic portion of the idealized curve. Once the elastic portion was defined, the yield level was established by equalizing the area between the measured backbone and the idealized curves. Figures 8 and 9 show the idealized curves for specimens with ACI Structural Journal/July-August 2007 low and high shear, respectively. Based on the elasto-plastic curves, displacement ductility capacities of 9.5 and 10.4 were obtained for Specimens ISL1.0 and ISL1.5, respectively. In Specimens ISH1.0, ISH1.25, ISH1.5, and ISH1.5T, the measured displacement ductility capacities were 4.7, 5.0, 4.0, and 3.8, respectively. The column section total depths were different within each specimen group due to different distances between the spiral sets. As a result, the lateral load capacity varied among the columns. To compare the performance of the specimens, forces were normalized with respect to the effective yield force of each specimen and the normalized forcedisplacement envelopes were compared (Fig. 10). The effect of a large distance between the spiral sets in low-shear columns can be seen in Fig. 10(a). The overall ductility capacity of the two low-shear specimens was comparable. The strength of the specimen with di of 1.5R (Specimen ISL1.5), however, degraded starting with displacement ductility of 7.4, whereas the strength of the column with di of 1.0R 397
- 82. Fig. 10—Normalized lateral force-displacement envelopes. Fig. 11—Measured displacement ductility capacity versus average shear stress index. Fig. 12—Measured curvature for ISL group. (Specimen ISL1.0) did not drop until failure. At a displacement ductility of 9, the strength degradation in the column with di of 1.5R was 10% whereas it was 4% when di was 1.0R. Nevertheless, degradation started at a relatively high ductility and hence is not of concern. Note that the target design displacement ductility for the columns was 5. In specimens with high shear, the displacement ductility capacity was comparable in the two columns with di of 1.0R or 1.25R. The ductility capacity dropped by approximately 20% when di was increased to 1.5R. The slightly lower ductility of Specimen ISH1.5T versus Specimen ISH1.5 (3.8 versus 4) suggests that the addition of crossties had little effect on the ductility capacity. A comparison of Fig. 9(c) and (d) indicates that the response of Specimen ISH1.5 contained limited excursions into the negative displacement range, whereas the Specimen ISH1.5T response was somewhat symmetric. Variations of concrete strength properties, column stiffness, and the shake table response are attributed 398 to the difference in the column responses. Symmetric cyclic displacements tend to place higher demands on reinforced concrete members. It is hence concluded that, had the displacements in the two columns been identical, Specimen ISH1.5T would have shown a higher ductility capacity. Nonetheless, the ductility capacity of approximately 4 measured in Specimens ISH1.5 and ISH1.5T was considered to be satisfactory. The displacement ductility at which strength degradation began in columns with di of 1.0R and 1.25R was approximately 3.7, and in those with di of 1.5R was approximately 3. The larger spread of the spirals clearly shows some effect on the overall load-displacement response. The addition of the crossties reduced the slope of the degradation part of the responses (Fig. 10(b)). The displacement ductility capacity versus the average shear stress index is shown in Fig. 11. The measured concrete compressive strengths were used in this graph. In general, the displacement ductility capacity decreased when the average shear stress index increased. This was because columns subjected to high shear failed in shear/flexural mode, whereas those with low shear failed in flexure with no significant shear damage. Measured curvatures Displacement transducers were used to measure curvature in the plastic hinge regions at the bottom of the ISL group and at the top and bottom of the ISH group. The strain on each side of the column was calculated from the vertical displacement measured by each external transducer divided by the gauge length. The average curvature over the gauge length was computed as the difference between the strains on the opposite sides of the column divided by the horizontal distance between the instruments. This procedure assumes that sections remained plane. The curvature profiles for the predominant direction of motion are shown in Fig. 12 and 13 for specimens with low and high shear, respectively. High values of curvature were measured in the plastic hinges, as expected. The curvatures at the ends are influenced by the localized longitudinal reinforcement bond slip and are not purely due to flexural deformation of the plastic hinge. The maximum ultimate curvatures in Specimens ISL1.0 and ISL1.5 were comparable, indicating that the change in distance of the spiral sets did not affect the curvature performance. This observation was in agreement with the displacement ductility capacities of the two models. The maximum curvatures in the columns with high shear were also comparable within the group, but were approximately 2/3 of the curvatures of the ISL group. The lower curvatures are consistent with the smaller displacement ductility capacities that were observed for this group. The peak top and bottom curvatures in Specimens ISH1.0 and ISH1.25 were comparable, confirming that the loading mechanism to bend the columns in double-curvature fixed-fixed mode was successful. In Specimens ISH1.5 and ISH1.5T, the peak top curvatures were 20 to 25% lower than the bottom curvature due to slight rotation of the loading head that occurred under high loads and prevented fully fixed response at the top. Measured strains The strain gauges on the longitudinal reinforcement were placed at the potential plastic hinge regions of all the columns and the footings, and in the loading heads of the ISH group. In all specimens, the longitudinal bars yielded ACI Structural Journal/July-August 2007
- 83. Fig. 13—Measured curvature for ISH group. extensively and flexural deformations dominated the response. Higher strains were measured at or near the base of all the columns and also at the top of the ISH group. Because the response in all the specimens was dominated by flexure, the test variables did not significantly affect the trends in the longitudinal and spiral bar strains except as noted in the following. The correlation between the apparent damage and the longitudinal bar strains was studied. Five damage states were selected representing an increasing level of damage: 1) flexural cracks; 2) first spalling and shear cracks; 3) extensive cracking and spalling; 4) visible spirals and longitudinal bars; and 5) imminent failure. The fifth damage state refers to the case where core damage is observed or is about to occur and some of the longitudinal bars show signs of bending that might lead to buckling and failure in subsequent runs. This damage state corresponded to the run before the failure run in the shake table tests. Figure 14 shows the average of the highest three strain data in the longitudinal bars versus the damage states in each model. The average data for three gauges, rather than the maximum strain, were used because local bar strains are influenced by cracks and present erratic patterns. The data for all specimens were averaged and shown on the graph. It can be seen in Fig. 14 that, within each damage state, the longitudinal bar strains were generally higher in the ISL group. This is because the moment gradient in the high-shear columns is relatively high, making the strain more localized and the average strains lower. The larger distance between the spiral sets in Specimen ISL1.5 led to higher bar strains in the first three damage states. Within the ISH group, the bar strains did not seem to be sensitive to the distance between the spiral sets. The average bar strains in all specimens increased especially during the first three damage states. Average strains of approximately 3.5 times the yield strain were recorded when flexural cracks were observed in the columns. When first spalling and shear cracks were visible, the strain in the longitudinal bars increased to approximately 7.5 times the yield strain. An average strain of 14.5 times the ACI Structural Journal/July-August 2007 Fig. 14—Longitudinal bars strain versus observed damage. yield strain was recorded when extensive cracking and spalling was observed in the columns. Average strains of 18 and 19 times the yield strain were recorded for the last two damage states. The correlation between the spiral bar strains and different damage states was also reviewed. It was found that spiral bar strains remain small (generally less than 2/3 of the yield strain) until the run before failure. These data are presented and discussed in more detail in Reference 7. It was determined that it would be more useful if the trends in spiral bar strains are studied as a function of displacement ductilities. The average of peak spiral strains is plotted against displacement ductilities in Fig. 15. It can be seen that average strain was below yield until higher ductilities were reached. The larger distance between the spiral sets in Specimen ISL1.5 led to higher strains than those of Specimen ISL1.0 under large ductilities. The higher spiral strains are attributed to the slight degradation of the load capacity (Fig. 10) observed in Specimen ISL1.5. In addition, Fig. 15 shows slightly smaller strains in Specimen ISH1.0 compared with the rest of the high-shear specimens until the last motion. The average maximum spiral strains in 399
- 84. Fig. 15—Maximum average strain in the spirals. Table 2—Data for plastic hinge length Specimen Variables ISL1.0 ISL1.5 ISH1.0 ISH1.25 ISH1.5 ISH1.5T φp , Rad/mm (Rad/in.) 0.204 (0.008) 0.159 (0.006) 0.124 0.116 (0.005) (0.005) 0.101 (0.004) 0.074 (0.003) Δy, mm (in.) 16.901 (0.67) 18.172 (0.72) 21.1 (0.83) 21.1 (0.83) 32.1 (1.26) 26.7 (1.05) Δu, mm (in.) 161 (6.34) 188 (7.4202) 98.6 (3.88) 105 (4.15) 127 (5.02) 102 (4.00) L, mm (in.) 1473 (58) 1828 (72) 1473 (58) 1600 (63) 1753 (69) 1753 (69) lp , mm (in.) 351 (13.8) 428 (16.84) 363 (14.3) 384 (15.1) 480 (18.9) 541 (21.3) Specimens ISH1.25 and ISH1.5T were nearly the same, and the average maximum spiral strain in Specimen ISH1.5 was the highest until a displacement ductility of approximately 1.6 was reached. Plastic hinge length The plastic hinge length (PHL) is used to estimate postyield lateral displacements based on the moment curvature properties of the plastic hinge while empirically taking into account displacements due to bond slip and shear deformation. To determine the sensitivity of PHL to the spiral set distance and the level of shear, the PHL for each column was estimated using the measured plastic curvatures and displacements. The moment area method was used to relate displacements and curvatures assuming that the plastic rotation θp over the equivalent PHL, lp, is defined by θ p = ( φ u – φ y )l p (1) where φu equals the ultimate curvature capacity, and φy equals the idealized yield curvature. The center of rotation was assumed to be at the center of the plastic hinge. Equation (2) was assumed to relate plastic rotation and plastic displacements. The PHL was determined using this equation. In the ISH group, two plastic hinges were formed and, hence, the average measured curvatures at the top and bottom were used. lp Δ p = θ p ⎛ L – ---⎞ ⎝ 2⎠ (2) where L equals the distance from point of maximum moment to the point of contraflexure. 400 In Eq. (1), the average of the measured curvatures over the extreme two gauge lengths (203 mm [8 in.] in low-shear columns and 254 mm [10 in.] in high-shear columns) was used because most of the plastic deformation was concentrated over that region according to the measured curvatures and strains. Table 2 lists the data used to determine the measured lp for Specimens ISL1.0 and ISL 1.5. The values of lp of 0.75 and 0.83 times the total depth of the column were found for Specimens ISL1.0 and ISL1.5, respectively. It can be seen that the larger spiral distance in Specimen ISL1.5 led to an increase in the ratio of the PHL over the column depth by approximately 10%. The aspect ratio (column height divided by the column section depth in the loading direction) of Specimen ISL1.5 was approximately 10% larger than the Specimen ISL1.0 aspect ratio. Under equal conditions, Specimen ISL1.5 would experience a smaller shear deformation. The larger spread of the spirals in Specimen ISL1.5, however, appear to have led to higher shear deformations that necessitated a larger PHL to match the measured displacement. The values of lp of 0.98, 0.96, 1.12, and 1.27 times the total depth of the columns were found for Specimens ISH1.0, ISH1.25, ISH1.5, and ISH1.5T, respectively. The aspect ratios for these columns were nearly the same. In high-shear columns, the increase in the distance between the spirals from 1.0R to 1.5R appears to have increased displacement due to shear, thus increasing the apparent plastic hinge length by approximately 20%. CONCLUSIONS Based on the observations and the experimental results of this study, the following conclusions are drawn: 1. The seismic performance of columns with relatively low shear with spiral distance di of 1.0R and 1.5R was similar and satisfactory with displacement ductility capacities of near 10. The strength degradation was slightly larger when di was 1.5R. This degradation began at a displacement ductility of 7.4, however, which exceeded the target design displacement ductility of 5; 2. Because the low-shear column with di of 1.5R did not experience significant shear cracking, and based on the satisfactory displacement ductility capacity, it appears that the Caltrans provision of allowing a di value of up to 1.5R is appropriate for columns with low shear; 3. The seismic performance of column models with di of 1.0R and 1.25R subjected to high shear was similar and satisfactory. Even though the columns failed in shear/flexure mode, they were ductile and achieved the design displacement ductility capacity of 5; 4. Vertical cracks in the interlocking region were observed under small earthquakes in the column with high shear and di of 1.5R. The large area of plain concrete in the interlocking zone is susceptible to cracking when di is 1.5R and the column shear is relatively high. The addition of horizontal crossties connecting the interlocking hoops not only reduced and delayed vertical cracks in the interlocking region, but also reduced the strength degradation; 5. The measured displacement ductility capacity was approximately 4 in columns with high shear and a di of 1.5R. Even though the desired ductility capacity was 5, the column is considered to be sufficiently ductile for most applications. Crossties are recommended to reduce premature vertical cracking in these columns; and ACI Structural Journal/July-August 2007
- 85. 6. The plastic hinge length to match the measured plastic lateral displacement increased as the distance of the spirals sets increased from 1.0R to 1.5R by 10 to 20%, depending on the level of shear. ACKNOWLEDGMENTS The research presented in this paper was sponsored by the California Department of Transportation. The dedicated assistance of P. Laplace, J. Pedroarena, and P. Lucas of the University of Nevada-Reno bridge laboratory is gratefully acknowledged. Specials thanks are expressed to N. Wehbe of South Dakota State University for developing a moment-curvature analysis program for interlocking spiral columns. REFERENCES 1. California Department of Transportation, “Bridge Design Specifications,” Engineering Service Center, Earthquake Engineering Branch, Calif., July 2000, 250 pp. 2. California Department of Transportation, “Seismic Design Criteria ACI Structural Journal/July-August 2007 Version 1.2,” Engineering Service Center, Earthquake Engineering Branch, Calif., Dec. 2001, 133 pp. 3. Tanaka, H., and Park, R., “Seismic Design and Behavior of Reinforced Concrete Columns with Interlocking Spirals,” ACI Structural Journal, V. 90, No. 2, Mar.-Apr. 1993, pp. 192-203. 4. Buckingham, G. C., “Seismic Performance of Bridge Columns with Interlocking Spirals Reinforcement,” MS thesis, Washington State University, Pullman, Wash., 1992, 146 pp. 5. Benzoni, G.; Priestley, M. J. N.; and Seible, F., “Seismic Shear Strength of Columns with Interlocking Spiral Reinforcement,” 12th World Conference on Earthquake Engineering, Auckland, New Zealand, 2000, 8 pp. 6. Correal, J.; Saiidi, M.; and Sanders, D., “Seismic Performance of RC Bridge Columns Reinforced with Two Interlocking Spirals,” Report No. CCEER-04-6, Center for Civil Engineering Earthquake Research, Department of Civil Engineering, University of Nevada-Reno, Reno, Nev., Aug. 2004, 438 pp. 7. Correal, J., and Saiidi, M., “Lessons Learned from Shake Table Testing of RC Columns in Relation to Health Monitoring,” IMAC-XXIII—A Conference & Exposition on Structural Dynamics—Structural Health Monitoring, Orlando, Fla., 2005, 9 pp. 401
- 86. DISCUSSION Disc. 103-S67/From the Sept.-Oct. 2006 ACI Structural Journal, p. 656 Shear Strength of Reinforced Concrete T-Beams without Transverse Reinforcement. Paper by A. Koray Tureyen, Tyler S. Wolf, and Robert J. Frosch Discussion by Himat Solanki Professional Engineer, Building Dept., Sarasota County Government, Sarasota, Fla. The authors have presented an interesting paper on the shear strength of reinforced concrete T-beams without transverse reinforcement. However, the discusser would like to offer the following comments: 1. The authors have mentioned the basic outline of a derivation of Eq. (1), but Eq. (1) appears to be based on the neutral axis (NA) located at the center of the beam in a typical homogeneous rectangular concrete beam. The authors’ Eq. (2) was a simplification of Eq. (1) based on the experimental database of reinforced concrete beams, which is inconsistent with the Rankine’s failure criteria of a plain homogeneous concrete beam. Based on ft = 6 f c′ (or 0.1fc′ ) and assuming the Rankine’s failure criteria of plain concrete, and by considering various strength ratios of flexural stress σm versus concrete compressive stress fc′ (σm/f ′c = 14.2%,10 the flexural stress σm of a plain homogeneous concrete beam equals to 114.2%10 of the tensile strength of plain concrete ft . Based on the aforementioned assumptions, the discusser arrived at the authors’ Eq. (2) without considering the experimental database of reinforced concrete beams. Another simplified approach is that Eq. (2) can also be derived from the current ACI Building Code9 (that is, authors’ Eq. (5)) by assuming an average depth of NA equals 0.4d11,12 and by substituting c = 0.4d in the authors’ Eq. (5), which would result in authors’ Eq. (2). Based on the aforementioned two approaches, the discusser believes that there is no need to have a reinforced concrete beam database, that is, Fig. 1 and 2. Is this consistent with the shear strength of reinforced concrete T-beams without transverse reinforcement plain concrete? 2. The authors’ concept on shear funnel (Fig. 8 and 10) is somewhat unclear. Please note that there is no reinforcement within the compression and/or flange area. Based on Fig. 8, considering a simplified approach, a portion of the crosssectional area above NA in the T-beam could be converted into an equivalent rectangular section, but not the entire section of the T-beam when a shear force is computed. The discusser has computed over 100 specimens of T-beams from Reference 1 by assuming the flange depth as one unit and the overall depth and web width were transferred into the flange depth units with varying depths of NA (that is, NA was assumed within the flange and within the web of the T-beams) and found that approximately 20% of the crosssectional area increases above NA as compared with its equivalent rectangular section and approximately 10% of the cross-sectional area increases to its equivalent rectangular section, if the entire beam was compared with the rectangular section. These values are somewhat inconsistent in the authors’ Table 2. ACI Structural Journal/July-August 2007 REFERENCES 10. Kato, K., Concrete Engineering Data Book, Nihon University, Koriyama-City, Fukushima Prefecture, Japan, 2000. 11. Eurocode No. 2, “Design of Concrete Structures, Part 1: General Rules and Rules of Buildings,” ENV 1992-1-1, Commission of the European Communities,1991. 12. British Standard Institution, “Structural Use of Concrete, Part 1: Code of Practice for Design and Construction,” BS 8110:Part 1:1997, London, UK, 1997. AUTHORS’ CLOSURE The authors thank the discusser for his interest in this paper. The comments are addressed in the same order as presented by the discusser. The detailed derivation of Eq. (1) and its simplification into Eq. (2) are presented in Reference 2 of the paper. This derivation was not based on the neutral axis located at the center of a beam, but rather based on the location of the neutral axis as calculated based on a cracked section analysis. The discusser is referred to Reference 2 for further clarification. As noted in Reference 2, Eq. (1) was derived considering that failure initiates when the principal stress in the compression zone reaches the tensile strength of concrete ft. It was shown that this equation could be simplified for an assumed tensile strength (6 f c′ ) and considering the flexural stress σm. The experimental results, however, were considered so that a complete perspective of the performance of the simplified expression could be accessed. The discusser notes that Eq. (2) can be derived from the ACI code. It appears that the discusser is referring to ACI Eq. (11-3) rather than (11-5). Perhaps a better view is that Eq. (11-3) is a subset of Eq. (2). For k = 0.4, Eq. (2) simplifies it to 2 f c′ bwd. The neutral axis depth, c = kd varies according to the flexural reinforcement ratio ρ and the modular ratio n. Therefore, Eq. (2) accounts for the reinforcement ratio and the concrete compressive strength, whereas ACI 318 Eq. (11-3) is insensitive except with respect to its inclusion in the term f c′ . Unfortunately, the discusser’s question “Is this consistent with the shear strength of reinforced concrete T-beams without transverse reinforcement plain concrete?” is unclear and cannot be addressed. The results presented in Fig. 10 are based on an angled approach using a 45-degree angle. Simplification can be achieved using an effective flange width approach. Based on the area achieved from the 45-degree shear funnel, an effective overhanging flange width of 0.5 times the flange depth on each side of the web can be considered for shear. If the neutral axis falls within the thickness of the flange, this effective width approach is conservative. It should be noted that in either the shear funnel or equivalent flange width approach, the neutral axis depth should be computed using an effective flange width that is based on flexural behavior 503
- 87. and that is different from the flange width considered effective for shear. Table 2 presents a statistical comparison of the performance of the various design methods considering the ratio of Vtest/ Vcalc. Therefore, it is unclear what inconsistencies the discusser is referring to. However, as emphasized in the paper, for the evaluation of the shear area when the flanges were ignored, the neutral axis depth was calculated ignoring the flanges while the shear funnel approach computed the neutral axis depth with the flanges considered. This may explain the perceived inconsistencies in the discusser’s analysis if he was directly comparing the results provided in Table 2. Regardless, the main premise is that additional shear area beyond that bounded by the web can be considered as effective in shear transfer. The percentage of flange area considered will vary depending on the section considered and the location of the neutral axis. Disc. 103-S71/From the Sept.-Oct. 2006 ACI Structural Journal, p. 693 Shear Strength of Reinforced Concrete T-Beams. Paper by Ionanis P. Zararis, Maria K. Karaveziroglou, and Prodromos D. Zararis Discussion by Himat Solanki Professional Engineer, Building Dept., Sarasota County Government, Sarasota, Fla. The authors have presented an interesting concept in their paper on shear strength of reinforced concrete T-beams. However, the discusser would like to offer the following comments: 1. The authors have considered εco = 0.002 and fct = 0.30f ′ 2/3 (Eq. (4)) based on Reference 10, but no consideration c was given to the depth of compression zone equals 0.8c value as suggested in Reference 10. Also, the authors have not thoroughly explained the assumption of 0.667c value (in the Appendix) other than the test result values versus their theoretical values. Please note that BS 8110:Part 1:199715 considers the depth of compression zone equal a value of 0.9c. 2. Based on Eq. (7) and Fig. 7, the authors assumption for a 45-degree projection angle from web to flange appears to be inconsistent with Fig. 6(b) and other researchers. The 45-degree projection angle was a simplified assumption based on the depth of compression equals the depth of flange, that is, the neutral axis (NA) is located at the interface of the bottom of flange and the top of web. 3. In conclusion, the authors’ statement “An increase of stirrups does not give any advantage to T-beams over the rectangular beams” is a little confusing without thorough explanation, because the authors have converted a T-beam into a rectangular beam with bef web width in lieu of bw web width. Let’s consider beam pairs from Table 1: Beam Pair TA11-TA12 of Reference 2; Beam Pair T2-T3 and Beam Pair T15-T16 of Reference 4; Beam Pair T3a-T3b of Reference 5; and Beam Pair A00-A75 of Reference 7. These beam pairs all have test parameters such as concrete strength fc′ , longitudinal reinforcement ρ%, and shear span-todepth ratio a/d approximately identical, except for the shear reinforcement ρv fvy; but the shear strength increases with an increase in shear reinforcement ρv fvy. This means the shear reinforcement ρv fvy does have some influence on the shear strength. 4. The authors’ Eq. (9) and the calculated values of A s of ′ the depth of compression block in Fig. A (of the authors’ Appendix) are unclear. It appears that the authors have considered a routine rectangular beam with compression reinforcement but have not considered the reinforcement within the flange width when a T-beam section was converted into a rectangular beam section above NA. The 504 reinforcement in the flange would improve the value of c (depth of NA) as well as the value of Vcr in Eq. (8) and Vu in Eq. (10). 5. The discusser has calculated all T-beams except Beam ET1, which is a rectangular beam from References 1 and 2, as outlined in the authors’ Table 1, by considering the reinforcement in the flange width and by using authors’ Eq. (10) for a calculation of NA, c, and then Vcr and Vu were calculated. Based on this concept, a mean value of Vu,exp /Vu,th of 1.006 and a standard deviation value of 0.05 were found. It was also noticed from Table 11,2,4-8 that the thinner web width with higher reinforcement ratios (both longitudinal and shear reinforcement ratios) do not have any advantage over wider web width with lower reinforcement ratio in T-beams. REFERENCES 15. British Standard Institution, “Structural Use of Concrete, Part 1: Code of Practice for Design and Construction,” BS 8110:Part 1:1997, London, UK, 1997. AUTHORS’ CLOSURE The authors would like to thank the discusser for his interest in the paper and his kind comments. The authors would like to reply to his comments in the order they are asked. In the case of rectangular or T-section beams, the true distribution of stresses in the compressive zone is usually replaced for simplification by an equivalent rectangular stress block. In the ultimate limit state, that is, when the compressive strain in concrete at extreme fiber is εc = 0.0035, the true distribution of stresses in the compressive zone follows a parabola-rectangular diagram. Then, the compressive force of concrete, as a resultant of stresses, is Fc = 0.81bcfc′ . Thus, the equivalent rectangular stress distribution has an approximate height equal to 0.8c. In this case, however, the authors choose a state where the strain of concrete at extreme fiber is εco = 0.002. This strain corresponds in a true, exactly parabolic distribution of stresses in the compressive zone. In this case, the corresponding compressive force of concrete is Fc = 0.667bcfc′ . Thus, the equivalent rectangular stress distribution (shown in Fig. A) has a height equal to 0.667c. There has never been made a 45-degree projection angle by the authors. As it is written in the text of the paper, the failure occurs due to a splitting of concrete that takes place in the compression zone of the T-beam. Taking into account Fig. 2 ACI Structural Journal/July-August 2007
- 88. and 6, the splitting takes place in an inclined area, the projection of which, on a cross section of the beam, is approximately defined from the shaded part of the section in Fig. 7. Equation (7), giving the effective width, results simply from the area of this shaded part of cross section of the T-beam. This statement means that the contribution of stirrups in the shear strength is the same for T-beams and rectangular beams, as it results from the second part of Eq. (10). The increase in the strength of the beams that the discusser has mentioned is due to an increase of the first part of Eq. (10). The compression reinforcement As′ within the flange width has been considered and takes part in Eq. (9) with the ratio ρ′ = As′ bwd. Nevertheless, an increase of As′ does not increase the shear strength of a beam; on the contrary, it decreases the shear strength, exactly because the reinforcement As′ improves the value of c. Equations (8) and (10) show that a decrease of the depth c decreases the strength. This has been observed both in T-beams and rectangular test beams. The compression reinforcement A s has not been considered ′ in the calculations of Table 1, because of the lack of data regarding this reinforcement for all the test beams. As it results from the discusser’s calculations, however, the small ratios of ρ′ have only a small effect on the shear strength. Disc. 103-S76/From the Sept.-Oct. 2006 ACI Structural Journal, p. 736 Effect of Reinforced Concrete Members Prone to Shear Deformations: Part I—Effect of Confinement. Paper by Suraphong Powanusorn and Joseph M. Bracci Discussion by Himat Solanki Professional Engineer, Building Dept., Sarasota County Government, Sarasota, Fla. Though the authors have presented an interesting concept on shear deformations in their paper, they have not fully explained all necessary assumptions other than the use of Mander et al.’s methodology. Also, the authors have not provided the details as outlined by Mander et al. (1988). Without a detailed explanation and information, particularly of the test specimens supplemented by the associated assumptions, it is very difficult to verify the author’s results as well as published results available elsewhere; therefore, the discusser has the following comments: 1. The discusser has tried to understand the authors’ methodology, and has described the authors’ methodology to the best as follows. In the following concept, there are several assumptions that were neither mentioned by Mander et al. (1988), nor by the authors. The authors’ Eq. (12) is expressed as 0.0024 + (0.0024 + 0.002) cot2 35 degrees = 0.0114. Now, εcc = εco[1 + R((f ′cc/f ′co) – 1)]. Based on the test results of Mander et al. (1988) and Scott et al. (1980), f ′cc/f ′co ≈ 1.75 and εco ≈ 0.002 (Richart et al. 1928). In the previous equation, the R value varies from 3 to 6 (Park and Paulay 1990). Based on the authors’ Fig. 1 and 2, the transverse reinforcement details with respect to the longitudinal reinforcement, R = 5, as suggested by the authors in their Eq. (14) appears to be on the low side. Therefore, R = 6 was appropriate and was assumed in the aforementioned equation by the discusser. That is, εcc = 0.002 [1 + 6((1.75) – 1)] = 0.011. Based on the Mander et al. (1988) and Scott et al. (1980) test results, εcc ≈ 0.0115. Based on an average value of εcc = 0.01125 βf ′cc xr σ c = --------------------r r–1+x 1 β = -------------------------------------------- ≤ 1 0.8 + 0.34 ( ε 1 ⁄ ε cc ) where 1 β = -------------------------------------------- ≤ 1 0.8 + 0.34 ( ε 1 ⁄ ε cc ) in which Also, based on an average value of εcc = 0.01125 and εc ≈ 0.0048 was chosen due to lateral expansion (biaxial tensioncompression) x = εc/εcc = 0.0048/0.01125 = 0.425 Esec = f ′co/εcc = 1.75f ′co /3.52εco ε1 = εs + (εs + 0.002)cot2α Ferguson (1964) suggested that the stress in steel develops from 1.15fy to 1.20fy. Therefore, an average value of 1.175fy was considered. That is, εs = 1.175fy/Es, where Es = 29,000 ksi. Furthermore, it was assumed that the tensile strain is causing approximately a 35-degree skew angle crack to the strut’s axis. The 35 degrees falls within the range from 25 to 45 degrees, and this angle is consistent with Cusson and Paultre (1994) and Fig. 5 and 13 of Ferguson (1964): ε1 = ACI Structural Journal/July-August 2007 ≈ 0.5Ec Ec Ec Now, r = -------------------- = ------------------------ = 2.0Ec E c – E sec E c – 0.5E c βf ′ xr cc Now σc = --------------------r r–1+x 505
- 89. Substituting β, x, and r values in the previous equation ( 0.8737 )f ′cc ( 0.425 ) ( 2.0 ) σ c = ------------------------------------------------------------ = 0.629f ′cc 2.0 – 1 + 0.425 ( 2.0 ) Because f ′cc ≈ 1.75fc′ σc = 1.1008fc′ or ≈ 1.10fc′ This means approximately 10% compressive stress increases due to the confinement. This value is consistent with Vecchio’s (1992) concept as well as the authors’ tests results as shown in Tables 1 through 3. Based on Vecchio’s study (Vecchio 1992), an average stress in shear panels was increased by approximately 5.6%, while an average stress in shearwalls was increased by approximately 13.4%, that is, an overall average value increased in stress would be 9.5%. Is this consistent with the methodology/concept/logic used in this paper? 2. Based on Fig. 1(a), the authors have considered a symmetrical loading case, but the symmetrical loading case may not be the case for all structures in the practice. Because asymmetrical loading conditions would create unbalanced loading, it would require some additional reinforcement per truss analogy in the dark area, as shown by the authors in Fig. 9(a) and (b), depending on the unbalanced load due to the asymmetrical loading condition. 3. It is unclear how the theoretical values stated in Tables 1 through 3 were calculated. Was any correction for variable depth considered? Or was a uniform depth considered? Though the authors stated the advantage of overlapping stirrups versus single stirrups, the effectiveness of stirrups as compared with the longitudinal reinforcement was unclear from Table 1 through 3. 4. The discusser would like to point out that because the shear strength and shear deformations relate to the strength of concrete, a simplified method proposed by Muttoni (2003) could be extended to the authors’ specimens. 5. Using the aforementioned concept outlined in this discussion and Muttoni’s (2003) methodology, the discusser has also analyzed other test specimens available in the literature elsewhere (Rodrigues and Muttoni 2004; Fukui et al. 2001; Ferguson 1964). The results are found to be in good agreement with the test results. Due to brevity, the results are not included in the discussion. ACKNOWLEDGMENTS The discusser gratefully appreciates S. Unjoh, Leader, Earthquake Engineering Team, Public Works Research Institute, Tokyo, Japan; A. Muttoni, Institut de Structures, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland; and N. Pippin and A. Wards, TTI, Texas A&M University, College Station, Tex., for providing publications related to the shear strength of beams. REFERENCES Fukui, J.; Shirato, M.; and Umebara, T., 2001, “Study of Shear Capacity of Deep Beams and Footing,” Technical Memorandum No. 3841, Public Works Research Institute, Tokyo, Japan. (in Japanese) Cusson, D., and Paultre, P., 1994, “High Strength Concrete Columns Confined by Rectangular Ties,” Journal of Structural Engineering, ASCE, V. 120, No. 3, Mar., pp. 783-804. Mander, J. B.; Priestley, M. J. N.; and Park, R., 1988, “Observed Stress and Strain Behavior of Confined Concrete,” Journal of Structural Engineering, ASCE, V. 114, No. 8, Aug., pp. 1827-1849. Muttoni, A., 2003, “Schubfestigkeit und Durchstanzen von Platten ohne Querkraftbewehrung,” Beton und Stahlbetonbau, V. 98, No. 2, Feb., pp. 74-84. Park, R., and Paulay, T., 1990, “Bridge Design and Research Seminar: 506 V. I, Strength and Ductility of Concrete Substructures of Bridges,” RR Bulletin 84, Transit New Zealand, Wellington, New Zealand. Richart, F. E.; Brandtzaeg, A.; and Brown, R. L., 1928, “A Study of Failure of Concrete under Combined Compressive Stresses,” Bulletin 185, University of Illinois Engineering Experimental Station, Champaign, Ill. Rodrigues, R. V., and Muttoni, A., 2004, “Influence des Déformations Plastiques de l’Armature de Flexion sur la Résistance a l’Effort Trenchant des Pouters sans étriers: Rappart d’essai,” Laboratoire de Construction en Béton (IS-BETON), Istitut de Structures, Ecole Polytechnique Fédérale de Lausanne, Oct. Scott, B. D.; Park, R.; and Priestley, M. J. N., 1980, “Stress-Strain Relationships for Confined Concrete: Rectangular Sections,” Research Report 80-6, Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand, Feb. AUTHORS’ CLOSURE The authors would like to express a sincere gratitude to the discusser for comments that give the authors an opportunity to clarify certain issues in the article. The authors’ response to the discusser is as follows: General The purpose of the article under discussion was to present an alternative method that incorporates the effects of confinement into the constitutive equations of the Modified Compression Field Theory (MCFT), first proposed by Vecchio and Collins (1986). In essence, the extension of the MCFT proposed by the authors is based on two-dimensional stress and strain analysis. All necessary assumptions were stated at the beginning of the article under the section Proposed analytical model. Response to discusser comments 1. The discusser demonstrates the application of Eq. (12) on the constitutive relationship of concrete in compression taken into account the effect of confinement given in the paper with assumptions on a few parameters shown in the equation. It was concluded that the results from applying Eq. (12) led to an approximate 10% increase in compressive strength of concrete, which was compared with a study by Vecchio (1992) on shearwalls and panels and also by the authors’ reinforced concrete (RC) bent cap tests. From the authors’ point of view, however, the application of Eq. (12) alone to obtain an increase in strength is only part of the comparative study. It is the force-deformation behavior that is important for comparative purposes, especially for members prone to shear deformations near ultimate loading. MCFT is generally developed on the basis of: 1) twodimensional states of stress and strain; 2) the superposition of stresses in the concrete and reinforcing steel as shown in Eq. (1); and 3) the compatibility of strains in the concrete and reinforcing steel as shown in Eq. (2). The model can be categorized into the so-called rotating crack model to maintain the coaxiality between the concrete principal stresses and principal directions. For two-dimensional states of stress and strain, three components of stresses and strains, which are εx, εy, and γxy and σx, σy, and τxy, are required to define a state of stress and strain at a given point within the member. The constitutive relationships under the context of MCFT, however, have been defined in the principal stress and strain components (σ1, σ2) and (ε1, ε2). The general state of stress and strain, εx, εy, and γxy and σx, σy, and τxy, are related to the principal stress and strain components (σ1, σ2) and (ε1, ε2) using Mohr’s circle of stress and strain. The concrete constitutive equation in compression defined in the principal stress and strain directions are given in Eq. (4) through (8) and (11) through (13). The special emphasis of the article is ACI Structural Journal/July-August 2007
- 90. on the incorporation of the beneficial effects of lateral confinement of the transverse reinforcement on the concrete stress-strain relationship in the principal compressive direction using an approach adopted by Mander et al. (1988) using the five-parameter failure surface derived by Willam and Warnke (1974). Due to space limitations, the authors did not include the complete development of five-parameter failure surface in the article. Interested readers should consult the original paper by Willam and Warnke (1974) or books by Chen (1982), Chen and Han (1988), and Chen and Saleeb (1982) for further details. Regarding the discusser’s comments on the R value for determining the peak strain corresponding to the peak concrete stress, additional studies by the authors have shown that the use of R = 6 led to only a marginal change in the strength prediction. 2. The MCFT was formulated on the basis of three fundamental principles of structural mechanics, which are: 1) equilibrium; 2) compatibility; and 3) material constitutive relationships. The rationality and generality of the MCFT should make the theory applicable to any loading pattern. The case of unsymmetric loading, however, was not considered ACI Structural Journal/July-August 2007 in this work and would require further experimental and analytical research to justify recommendations. 3. To justify the proposed model, the authors implemented the proposed model into a finite element code using a userdefined material subroutine. It is the results from FEM analysis that are summarized in Tables 1 through 3. 4 and 5. The authors agree with the discusser that the shear strength and deformation are related to the compressive strength of concrete and would like to look into further details on the article by Muttoni (2003). REFERENCES Chen, W.-F., 1982, Plasticity in Reinforced Concrete, McGraw-Hill, New York, 474 pp. Chen, W.-F., and Han, D. J., 1988, Plasticity for Structural Engineers, Springer-Verlag, New York, 606 pp. Chen, W.-F., and Saleeb, A. F., 1982, “Constitutive Equations for Engineering Materials,” Elasticity and Modeling, V. 1, John Wiley & Sons, New York. Mander, J. B.; Priestley, M. J. N.; and Park, R., 1988, “Theoretical StressStrain Model for Confined Concrete,” Journal of Structural Engineering, ASCE, V. 114, No. 8, pp. 1804-1826. Willam, K. J., and Warnke, E. P., 1974, “Constitutive Model for the Triaxial Behavior of Concrete,” Concrete Structures Subjected to Triaxial Stresses, Paper III-1, International Association of Bridge and Structural Engineers Seminar, Bergamo, Italy, pp. 1-30. 507
- 91. ACI STRUCTURAL JOURNAL TECHNICAL PAPER Title no. 104-S42 Simplified Punching Shear Design Method for Slab-Column Connections Using Fuzzy Learning by Kyoung-Kyu Choi, Mahmoud M. Reda Taha, and Alaa G. Sherif An alternative approach for predicting the punching shear strength of concentrically loaded interior slab-column connections using fuzzy learning from examples is presented. A total of 178 experimental datasets obtained from concentric punching shear tests of reinforced concrete slab-column connections from the literature are used in training and testing of the fuzzy system. The fuzzy-based model is developed to address the interaction between various punching shear modeling parameters and the uncertainties between them, which might not be properly captured in classical modeling approaches. The model is trained using 82 datasets and verified using 96 datasets that are not used in the training process. The punching shear strength predicted by the fuzzy-based model is compared with those predicted by current punching shear strength models widely used in the design practice, such as ACI 318-05, Eurocode 2, CEB-FIP MC 90, and CSA A23.3-04 codes. It is noted that the fuzzy-based model yields a significant enhancement in the prediction of the punching shear strength of concentrically loaded interior slab-column connections while still respecting the fundamental failure mechanisms in punching shear of concrete. Keywords: fuzzy systems; punching shear; slab-column connections. INTRODUCTION Flat plates consist of slabs directly supported on the columns without beams. For this simple appearance, flat plate systems have various economic and functional advantages over other floor systems such as fast construction, low story height, and irregular column layout. From a viewpoint of structural mechanics, however, flat plates are structures of complex behavior. Moreover, flat plates usually fail in a brittle manner by punching at the slab-column connections within the discontinuity region known as the D-region.1,2 At these connections, three-dimensional stresses are developed due to the combined high shear and normal stresses creating a stress state that is complex to analyze accurately.3 For the last three decades, a significant amount of research has been performed to investigate this complex problem of concentric punching shear of reinforced concrete flat plates by using various methods ranging from mechanical models up to purely empirical models. In early models including Yitzhaki4 and Long and Rankin,5 punching shear strength was defined considering the flexural capacity of reinforced concrete slabs. This was based on the experimental observation that the punching shear strength was close to the flexural capacities of the concrete slabs. Pralong6 and Nielsen7 derived lower bound and upper bound values for punching shear strength based on the theory of plasticity. These formulations did not consider the effect of flexural reinforcement on the punching shear strength. Kinnunen and Nylander8 developed the first mechanical model for punching shear strength using failure criteria based on the observation of shear cracks in the experiments. In this model, the failure criteria were defined by the inclined radial compressive stress and the tangential 438 compressive strain at the shear crack. Even though Kinnunen and Nylander’s model8 did not provide high accuracy in punching shear strength predictions, it significantly contributed to a better understanding of the failure mechanism of the slab-column connections and enabled visualizing a rational flow of forces in such connections. Alexander and Simmonds2 proposed a strut-and-tie model with concrete ties to describe the load transfer in the slab-column connections. Bažant and Cao9 developed a punching shear strength model considering size effect of concrete based on principles of fracture mechanics. The size-effect model was able to explain the experimental observations of decreasing punching failure shear stresses of slab-column connections without reinforcement with increasing slab thickness. Numerous models suggested modifications to these general directions outlined previously (flexure, combined stressstrength criteria, plasticity, strut and tie, and size effect). A recent review of such models can be found elsewhere.10 In spite of the importance of these models in understanding the failure mechanism of slab-column connections, there is considerable difficulty in using these models in the daily design practice. Moreover, the level of complexity encountered in using these models for design might be difficult to justify given the fact that most of these models do not usually show high accuracy in the prediction of punching shear strength.11 To develop simple strength equations, most design codes use the so-called control perimeter approach12-15 depicted in Fig. 1. The applied punching shear stress is calculated at a defined critical perimeter and compared with an allowed value based on the calibration of existing test results. The various design codes show significant difference in defining the location of the critical section as well as the allowed punching shear stress. It becomes apparent that the complexity of the punching problem and the dependence of the punching shear strength on a number of interacting variables necessitate the use of empirical modeling approach to estimate the punching shear strength. While classical empirical techniques used by many design codes show limited accuracy, a more robust empirical modeling technique that respects fundamental failure mechanisms of the punching shear is needed. RESEARCH SIGNIFICANCE The present study introduces a new approach for predicting the punching shear strength of concentrically loaded interior slab-column connections using fuzzy learning from examples. The proposed approach incorporates the control perimeter ACI Structural Journal, V. 104, No. 4, July-August 2007. MS No. S-2006-214 received May 27, 2006, and reviewed under Institute publication policies. Copyright © 2007, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published in the May-June 2008 ACI Structural Journal if the discussion is received by January 1, 2008. ACI Structural Journal/July-August 2007
- 92. ACI member Kyoung-Kyu Choi is a Research Assistant Professor at the University of New Mexico, Albuquerque, N. Mex. He received his BE, MS, and PhD in architecture from Seoul National University, Seoul, Korea. He is an associate member of ACI Committees 440, Fiber Reinforced Polymer Reinforcement; 548, Polymers in Concrete; and Joint ACI-ASCE Committee 445, Shear and Torsion. His research interests include shear strength and seismic design of reinforced concrete structures and application of artificial intelligence in structural engineering. ACI member Mahmoud M. Reda Taha is an Assistant Professor in the Department of Civil Engineering at the University of New Mexico. He received BSc and MSc from Ain Shams University, Cairo, Egypt, and his PhD from the University of Calgary, Calgary, Alberta, Canada, in 2000. He is a member of ACI Committees 209, Creep and Shrinkage in Concrete; 235, Electronic Data Exchange; 440, Fiber Reinforced Polymer Reinforcement; 548, Polymers in Concrete; and E803, Faculty Network Coordinating Committee. His research interests include structural monitoring, using artificial intelligence in structural modeling, and fiber-reinforced polymers. ACI member Alaa G. Sherif is an Associate Professor in the Civil Engineering Department, Helwan University, Mataria-Cairo, Egypt. He received his BSc from Cairo University, Cairo, Egypt, and his MSc and PhD from the University of Calgary. He is an associate member of Joint ACI-ASCE Committee 352, Joints and Connections in Monolithic Concrete Structures. His research interests include the behavior and serviceability of reinforced concrete structures and systems for multi-span cable-stayed bridges. approach and targets predicting the punching shear strength of the slab-column connections based on various geometric and material parameters. The proposed fuzzy-based model presented in a simple form respects the failure mechanics of punching shear by learning its rules from the experimental database with the ability to address the interaction between the modeling variables and the uncertainty in these variables. The fuzzy-based model shows high accuracy in predicting punching shear strength. Fig. 1—Current design codes for punching shear. FUZZY LEARNING OF PUNCHING SHEAR DATABANK Fuzzy systems have been widely used in the last decade for modeling complex engineering systems (for example, modeling robots16 and in assessing concrete durability17) and their feasibility as universal approximators has been proven.18 The capability of the fuzzy systems to model complex systems is attributed to their inherent ability to accommodate a tolerance for uncertainty in the modeling parameters.19,20 While probabilistic empirical models are limited to random uncertainties, fuzzy systems have the ability to consider random and nonrandom types of uncertainties that arise due to vagueness and/or ambiguity in the modeling parameters/process.18-20 The fundamental concept in modeling complex phenomena using fuzzy systems is to establish a fuzzy rule-base that is capable of describing the relationship between the input parameters and the output parameters while considering uncertainty bounds.19 This fuzzy rule-base captures individual and group relationships that distinguish the internal complex relations between the system parameters.20 As such, system nonlinearity is not recognized by using nonlinear equations but through establishing a number of fuzzy rules (that could use linear relations) such that the fuzzy system becomes capable of describing the phenomena to a pre-specified level of accuracy.20 A group of successful techniques to establish a fuzzy rule-base using exemplar observations was recently developed.20,21 Here, the use of the fuzzy set theory to model the punching shear strength of a slab-column connection is demonstrated. Preliminary investigations using Bayesian analysis of significance22 have been performed to identify the most primary input parameters that have a significant influence on the punching shear strength. Possible parameters included concrete compressive strength, slab thickness and effective depth, span length, column geometry, punching shear perimeter, and compression and tension reinforcement ratios. Assuming the geometry of punching shear perimeter to be known a priori, the Bayesian analysis showed that for circular and square columns (c1/c2 ratio equals to 1.0), the most significant parameters that affect the punching shear strength are concrete compressive strength fc′, slab thickness h, and tension reinforcement ratio ρ. The assumption of the punching shear perimeter to be known a priori is based on the fact that the punching shear databank does not include detailed information about the failure pattern and the punching shear perimeter. This hinders the ability to learn the failure patterns of slab-column connections as part of the new model. It is also noted that the results of Bayesian analysis showed that the compression reinforcement does not have a significant effect on the maximum punching shear strength. This finding is in agreement with the literature8,23 showing that the primary effect of compression reinforcement is on post-punching behavior providing a membrane action. Hereafter, these three parameters have been used as input parameters to the fuzzy-based model for predicting the punching shear strength. By considering these three parameters, the fuzzy-based model considers the major criteria on punching shear examined by many researchers.24-28 These include shear strength and cracking capacity conventionally represented by the cubical or square root of the compressive strength,6,24,27 size effect related to slab thickness,9 and membrane effect28 represented by the flexural reinforcement ratio. While the ratio of the column dimensions of rectangular columns and the perimeter-to-depth ratio (bo/d) have been reported to affect the punching shear strength of slab-column connections,24,29 the experimental database for rectangular columns or for slabs with significantly large perimeter-to- ACI Structural Journal/July-August 2007 439
- 93. Fig. 2—Cross section of slab-column connection showing critical section at distance d/2 from column face to intersect most plausible failure planes (angle θ ranges between 30 and 45 degrees). Choice of d/2 allows obtaining good estimate of average ultimate punching shear strength vc. failure as observed by many researchers.6-9 The choice of the critical perimeter to be considered at a distance d/2 from the column face is attributed to the possible use of this location to estimate the average ultimate shear strength vc for usually intersecting most plausible failure planes, as shown in Fig. 2, which is similar to the value (h/2) proposed in Nielsen.7 The modeling is started by defining N fuzzy sets A over the domain of each input parameter x. This definition˜ provides each value of the parameter x with N membership values representing its level of belonging to the N fuzzy set A . The ˜ concept of membership or degree of belonging represents the 18,20,21 The basis in the formulation of fuzzy set theory. membership denoted μ A(x) ranges between 0.0 and 1.0. ˜ μ A(x) does not express probability of x but characterizes the ˜ extent to which x belongs to fuzzy set A .20 Several methods for establishing membership functions˜with different levels of complexity exist. While simplified methods can be used according to expert opinion, complex automated methods using artificial neural networks or inductive reasoning are usually considered to be efficient for modeling complex phenomena.20,30 A technique is adopted herein that is based on providing an initial definition of the fuzzy sets using k-means clustering31 followed by the automated update of the fuzzy sets during the learning process.20,21 The modeling process depends on fuzzifying all three input domains and constructing a fuzzy rule-base, which describes the relationship between the fuzzy sets defined on the input domains and the punching shear strength using a group of linear equations. Exemplar rule in the fuzzy rulebase can be defined as k Fig. 3—Pictorial representation of bell-shaped membership function used to represent fuzzy sets defined over input domains. depth ratio (bo/d > 15) is insufficient to develop the knowledge rule base that is necessary for the fuzzy-based model to consider both effects on the punching shear strength. Therefore, first, the fuzzy-based model is trained by using the experimental data with square and circular columns only and with perimeter to depth ratio (bo/d) < 15. Based on this fact, prediction of the fuzzy-based model will be modified to consider the effect of rectangularity of columns or high perimeter-to-depth ratios in excess of that used in the training (bo/d > 15) as shown in the Results and discussion section. In the present study, the punching shear failure load of slab-column connections without shear reinforcement Vc is defined as k k If f ′ ∈ A f , h ∈ A h , and ρ∈ A p , c ˜ ˜ ˜ then vi = ai f c′ + bih + ciρ + di k k (2) k where A f , A h , and A p are the k-th fuzzy set (k = 1, 2, … Nj) ˜ defined˜ on ˜ fuzzy domains of compressive strength f ′c, the slab thickness h, and tension reinforcement ratio ρ, respectively. The value of Nj is the total number of fuzzy sets defined over the j-th input parameter. In the present study, ρ is defined with respect to effective depth. Equation (2) represents the i-th rule in the fuzzy rule-base. The values ai, bi, ci, and di are known as the consequent coefficients that define the output side of the i-th rule in the fuzzy rule-base. A bell-shape membership function is employed to represent the fuzzy sets defined on the input domains. The use of other membership functions (for example, gaussian and triangular) is possible, but constrained by having a differentiable membership function.21 The bell-shape membership function to represent the k-th fuzzy set of the j-th input parameter xj can be k described as μ A (x). ˜ Vc = vcbod (1) where Vc equals the punching failure load and bo equals the critical perimeter at a distance d/2 from the column face; bo = (2c1 + 2c2 + 4d) for a square column and bo = π(D + d) for a circular column. The values c1 and c2 equal the short and long sizes of a rectangular column, D equals the diameter of a circular column, and vc represents the average ultimate punching shear strength, which is defined with respect to defective depth. Equation (1), although simplified, has been adopted by almost all current design codes and respects the fundamental mechanics governing the slab-column punching 440 k 1 μ A ( x j ) = ----------------------------------k ˜ k 2q j x j – x cj 1 + --------------k wj (3) k where x cj , w jk, and q k represents the center, the top width, j and the shape parameters of the membership function defining the k-th fuzzy set defined over the j-th input parameter. A pictorial representation of the bell-shaped membership function is shown in Fig. 3. By considering the T-norm (product) operator (Π) to capture the influence of the interaction ACI Structural Journal/July-August 2007
- 94. between the input parameters32 on the output, the weight of the i-th rule (λi) in the fuzzy rule-base can be computed as T 1 Π j =1 ---------------------------------k k 2q x j – x cj j 1 + --------------k wj λ i = ------------------------------------------------------------ for i = 1...R R T 1 Σ i =1 Π j =1 ----------------------------------k k 2q j x j – x cj 1 + --------------k wj Table 1—Dimensions and properties of specimens Investigator*† No. of specimens f ′, c Training Verification MPa h, mm ρ, % Hallgren and Kinnunen (1993a), Hallgren and Kinuunen (1993b), Hallgren (1996) 3 3 79.5 to 239 to 0.6 to 108.8 245 1.2 Tomaszewicz (1993) 7 6 64.3 to 120 to 1.5 to 119.0 320 2.6 Ramdane (1996), Regan et al. (1993) 4 4 28.9 to 74.2 Marzouk and Hussein (1991) 6 8 30.0 to 90 to 0.4 to 80.0 150 2.1 Lovrovich and McLean (1990) 2 2 (4) Factors affecting the choice of the implication operator are discussed in the following. The value T represents the total number of input parameters (herein, T = 3). The number of fuzzy rules R is a function of the number of input variables T and the number of fuzzy sets Nj defined over each input domain. The punching shear strength vc can then be computed as ⎛ ⎞ vc = ⎜ λ i v i⎟ ⎝i = 1 ⎠ R ∑ ⎛ ⎞ ⁄⎜ λ i⎟ ⎝i = 1 ⎠ 1.7 Tolf (1988) 4 3 Regan (1986) 11 11 8.4 to 37.5 80 to 0.8 to 250 2.4 Swamy and Ali (1982) 1 1 37.4 to 40.1 125 Marti et al. (1977), Pralong et al. (1979) 1 1 23.1 to 180 to 1.2 to 30.4 191 1.5 Schaefers (1984) 1 1 23.1 to 143 to 0.6 to 23.3 200 0.8 Ladner et al. (1977), Schaeidt et al. (1970), Ladner (1973) 2 3 24.6 to 110 to 1.2 to 29.5 280 1.8 Corley and Hawkins (1968) 1 1 44.4 146 1.0 to 1.5 Bernaert and Puech (1996) 9 9 14.0 to 41.4 140 1.0 to 1.9 Manterola (1966) 4 4 24.2 to 39.7 125 0.5 to 1.4 Yitzhaki (1966) 5 6 8.6 to 19.0 102 0.7 to 2.0 Moe (1961) 7 7 20.5 to 35.2 152 1.1 to 2.6 Kinnunen and Nylander (1960) 6 6 21.6 to 149 to 0.5 to 27.7 158 2.1 Elstner and Hognestad (1956) 8 9 9.0 to 35.6 Hawkins et al.34 0 6 25.9 to 138 to 0.77 to 32.0 142 1.12 Teng et al.29 0 4 33.0 to 40.2 Criswell35 0 1 96 (5) where vi is the output of the i-th rule in the fuzzy rule-base and λi represents the weight of the i-th rule in the fuzzy rulebase as computed using Eq. (4). The process for learning from example aims at extracting a knowledge rule-base from a group of input-output datasets. This knowledge rule-base can be used later to model the behavior of the system (herein the punching shear of slabcolumn connections) for input datasets not used in the training process. While other techniques capable of building similar learning systems were reported in the literature (for example, artificial neural networks), the advantage of fuzzy systems is being able to consider nonrandom uncertainty in the modeling process and thus yields robust modeling systems.20 The learning process starts by initializing the premise parameters (parameters describing the membership functions x k , w k , and q k ) using the k-means clustering technique.31 cj j j This is followed by computing the consequence coefficients (ai, bi, ci, and di) using least square techniques33 such that the root mean square prediction error E of the punching shear strength does not exceed a target root mean square prediction error, herein 1.0 × 10–5. The root mean square prediction error E is defined as 100 1.0 to 1.3 20.1 to 120 to 0.4 to 25.1 240 0.8 R ∑ 39.3 125 Total 82 152 0.6 to 0.7 1.2 to 3.7 150 1.24 35.4 146 1.24 8.4 to 119.0 80 to 0.4 to 320 3.7 * Reference to investigators work, unless otherwise noted, can be found in Reference 3. Properties and dimensions of these test specimens were collected from fib Bulletin 12.3 Note: 1 MPa = 0.145 ksi; 1 mm = 0.04 in. † Nd ∑ ( vpn – vdbn ) E = 2 n=1 --------------------------------------- Nd (6) where vpn is the predicted punching shear strength for the n-th dataset, vdbn is the punching shear of the n-th dataset from the database, and Nd is the total number of training datasets. As the target mean square prediction error will not be achieved from the first learning trial (using the initial fuzzy sets and consequence coefficients), the premise parameters describing the fuzzy sets can be updated using the gradient descent method as k k ∂E ( m )x cj ( m ) = x cj ( m – 1 ) + η -----------------∂x cj ( m ) ACI Structural Journal/July-August 2007 (7) k k ∂E ( m )w j ( m ) = w j ( m – 1 ) + η ----------------∂w j ( m ) (8) k k ∂E ( m ) q j ( m ) = q j ( m – 1 ) + η ---------------∂q j ( m ) (9) where x k (m), w k (m), and q k (m) are the center, the top width, j j j and the shape of the membership function, respectively, defining the k-th fuzzy set defined over the j-th input parameter in the m-th learning epoch (trial). The values x k (m – 1), j w k (m – 1), and q k (m – 1) are the center, the top width, and j j the shape of the membership function, respectively, defining the k-th fuzzy set defined over the j-th input parameter in the (m – 1) learning epoch. The value η is the learning rate and ∂E(m)/∂xj(m), ∂E(m)/∂wj(m), and ∂E(m)/∂qj(m) are components 441
- 95. Fig. 4—Fuzzy sets used to describe concrete compressive strength, slab thickness, and tension reinforcement ratio. Before training (left) and after training (right): MF1 (Membership Function 1), MF2 (Membership Function 2), and MF3 (Membership Function 3). Table 2—Parameters describing premise parameters (membership functions)* Compressive strength f ′ c xc, MPa (ksi) 1 Af ˜ 2 Af ˜ w, MPa (ksi) q –23.83 (–3.40) 29.9 (4.34) 1.98 78.30 (11.40) 78.2 (11.30) 2.02 Slab thickness h xc, mm (in.) w, mm (in.) q 1 42.05 (1.66) 68.0 (2.68) 1.982 2 127.07 (5.00) 89.4 (3.52) 2.011 3 272.58 (10.73) 126.7 (4.99) 1.994 Ah ˜ Ah ˜ Ah ˜ Tension reinforcement ratio ρ xc w q 1 Aρ -0.001 0.012 1.997 2 0.035 0.018 2.005 ˜ Aρ ˜ *For compressive strength f c , slab thickness h, and reinforcement ratio ρ. ′ of the gradient vector of the mean square prediction error with respect to the premise parameters of the j-th input parameter evaluated at the m-th learning epoch. The updated premise parameters are then used to recompute a new set of consequence parameters and a new root mean square prediction error. The process continues and the fuzzy rule-base parameters (premise and consequent parameters) are updated in each training epoch until the target root mean square prediction error 442 or a maximum number of training epochs is reached. The update process therefore allows the fuzzy-based model to reduce the root mean square prediction error and thus learn from examples in a much more robust manner compared with any other empirical techniques. For training and testing of the fuzzy-based model, 178 test specimens performed by 21 researchers as reported in the fib bulletin3 and other reports in the literature29,34,35 were used. Only specimens that were reported to fail in pure punching shear (no flexural shear failure) were considered. A specimen reported by Lovrovich and McLean36 was excluded in this study because its span length was extremely short (l1/c1 = 2). Also, six specimens by Yitzchaki,4 Elstner and Hognestad,37 and Tolf38 were also excluded because their tension reinforcement ratios were extremely beyond practical design range (ρ ≥ 6.9%). The specimens had two types of boundary geometries (circular and rectangular flat plates) and two types of column shapes (circular and square columns). The dimensions and properties of the specimens are summarized in Table 1. The test specimens had a broad range of design parameters: 8.4 ≤ f′c ≤ 119.0 MPa (1.2 ≤ f ′c ≤ 17.3 ksi), 80 ≤ h ≤ 320 mm (3.1 ≤ h ≤ 15.6 in.), 0.4 ≤ ρ ≤ 3.7%, and 5.5 ≤ bo/d ≤ 24. These data cover a wide range of the material and geometric properties of slab-column connections. Eighty-two specimens were used for training of the fuzzy-based model while 96 specimens were used for testing the model. All specimens used in the testing were not used in training the fuzzy-based model. All modeling parameters were normalized to their maximum values determined from the database (178 data sets). The normalization process is necessary to avoid the influence of numerical weights on the learning process.39 The fuzzy rulebase that achieved the lowest root mean square error during training was used for testing and verification of the model capability to predict punching shear strength in slab-column connections. The optimum number of fuzzy sets for each modeling parameter was developed using the k-means clustering technique.31 The number of membership functions defined on the domain of any variable x can be used to indicate the sensitivity of the model to this variable x. The higher the sensitivity of the model to the variable x, the larger the number of membership functions used to describe the variable x. It is worth noting, however, that increasing the number of membership functions does not guarantee enhancing the model accuracy.20,21 It was found that the best learning represented by the lowest root mean square prediction error was achieved while using two fuzzy sets to represent the compressive strength and the tension reinforcement ratio. Three fuzzy sets were necessary for describing the slab thickness (N1 = N3 = 2, N2 = 3). The initial and final fuzzy sets, as established by the learning algorithm, are shown in Fig. 4 and Table 2. The total number of rules in the rule-base can be computed by multiplying the number of membership functions of the three variables as R = N1N2N3. Thus, 12 rules (R = 12) were needed to describe the relationship between the input parameters: concrete compressive strength, slab thickness, tension reinforcement ratio, and the punching shear strength. While reduction of the total number of rules in the fuzzy rule-base is possible for limiting combinatorial explosion,20 researchers showed that the efficient reduction of the number of rules shall be performed considering both accuracy and robustness of the model. Exemplar methods for rule reduction in the fuzzy rule-base include the Combs and Andrews40 method and the method suggested by Lucero41 but are beyond the scope of this work. ACI Structural Journal/July-August 2007
- 96. RESULTS AND DISCUSSION The fuzzy-based model was trained using test results with specific geometrical limits: circular and square columns and slabs with perimeter-to-slab-depth ratio (bo/d) ranging between 5.8 and 14.9. Therefore, the punching shear strength of any slab-column connection within the geometrical limitations listed previously can be computed using Eq. (10) to (12). Equation (10) can be used to compute the weight λ for each rule in the rule-base using the premise parameters listed in Table 1. Equation (11) presents the 12 rules forming the fuzzy knowledge rule-base. 3 1 Π j =1 ---------------------------------k k 2q x j – x cj j 1 + --------------k wj λ i = ------------------------------------------------------------ for i = 1...12 12 3 1 Σ i =1 Π j =1 ----------------------------------k k 2q j x j – x cj 1 + --------------k wj 1 1 1 h 2 h 0.83 to 1.00 0.86 to 0.97 0.93 to 1.02 0.96 to 1.07 1.39 to 1.64 0.80 to 1.17 0.94 to 1.29 1.41 to 1.64 0.70 to 1.23 Ramdane (1996), Regan et al. (1993) 1.46 to 1.66 1.15 to 1.31 1.20 to 1.37 1.27 to 1.47 1.25 to 1.41 Marzouk and Hussein (1991) 0.71 to 1.61 1.13 to 1.84 0.97 to 1.64 0.63 to 1.40 0.91 to 1.42 1.18 to 1.26 0.73 to 0.78 0.79 to 0.85 1.02 to 1.10 0.87 to 0.94 Tolf (1988) 0.88 to 1.21 0.92 to 1.34 0.82 to 1.15 0.77 to 1.05 0.91 to 1.02 Regan (1986) 1.17 to 1.78 0.97 to 1.47 1.04 to 1.47 1.02 to 1.54 0.60 to 1.29 Swamy and Ali (1982) 1.10 1.19 1.12 0.96 1.00 Marti et al. (1977), Pralong et al. (1979) 1.32 0.97 1.00 1.15 0.77 Schaefers (1984) 1.19 1.14 1.05 1.04 1.00 Ladner et al. (1977), Schaeidt et al. (1970), Ladner (1973) 1.48 to 1.79 1.22 to 1.34 1.26 to 1.47 1.29 to 1.56 0.89 to 1.26 Corley and Hawkins (1968) 0.87 0.85 0.85 0.75 0.72 Bernaert and Puech (1996) 0.88 to 1.93 0.80 to 1.28 0.81 to 1.43 0.76 to 1.68 0.70 to 1.45 Manterola (1966) 0.88 to 1.36 0.81 to 0.96 0.85 to 0.98 0.76 to 1.18 0.65 to 0.92 Yitzhaki (1966) 1.51 to 1.98 1.01 to 1.54 1.01 to 1.53 1.31 to 1.72 0.80 to 1.16 Moe (1961) 1.24 to 1.65 0.70 to 1.38 0.83 to 1.40 1.07 to 1.43 0.68 to 1.12 Kinnunen and Nylander (1960) 0.83 to 1.75 0.93 to 1.23 0.92 to 1.23 0.72 to 1.52 0.85 to 1.36 Elstner and Hognestad (1956) 1.19 to 2.23 0.88 to 1.20 1.05 to 1.30 1.03 to 1.94 0.79 to 1.27 Hawkins et al.34 0.90 to 1.05 0.87 to 1.10 0.89 to 1.05 0.78 to 0.91 0.88 to 1.19 Teng et al.29 1 ρ 0.88 to 0.98 Lovrovich and McLean (1990) (10) Hallgren and Kinnunen (1993a), Hallgren and Kinuunen (1993b), Hallgren (1996) 0.88 to 1.15 0.89 to 1.15 0.92 to 1.19 0.76 to 1.00 1.02 to 1.49 R = 1: if f c ′ ∈ A f , h ∈ A h, and ρ ∈ A ρ ˜ ˜ ˜ then v 1 = 0.247f c ′ + 0.008h + 153.7ρ + 4.90 R = 2: if f c ′ ∈ A , h ∈ A , and ρ ∈ A ˜ ˜ ˜ then v 2 = – 0.506f c ′ + 0.026h + 835.4ρ – 11.42 R = 3: if f c ′ ∈ A , h ∈ A , and ρ ∈ A ˜ ˜ ˜ then v 3 = 0.174f c ′ + 0.028h + 63.9ρ – 8.12 1 2 2 1 3 1 R = 4: if f c ′ ∈ A f , h ∈ A h, and ρ ∈ A ρ ˜ ˜ ˜ then v 4 = 0.149f c ′ + 0.031h – 136.65ρ – 3.49 R = 5: if f c ′ ∈ A f , h ∈ A h, and ρ ∈ A ρ ˜ ˜ ˜ then v 5 = – 0.248 f c ′ + 0.001h – 236.32ρ + 3.91 1 3 2 2 1 1 2 f 1 h 2 ρ 2 2 1 R = 6: if f c ′ ∈ A f , h ∈ A h, and ρ ∈ A ρ ˜ ˜ ˜ then v 6 = 0.243f c ′ – 0.006h – 53.35ρ + 3.16 (11) R = 7: if f c ′ ∈ A f , h ∈ A h, and ρ ∈ A ρ ˜ ˜ ˜ then v 7 = – 0.005 f c ′ – 0.031h – 84.38ρ + 3.14 R = 8: if f c ′ ∈ A , h ∈ A , and ρ ∈ A ˜ ˜ ˜ then v 8 = 0.006f c ′ + 0.116h – 136.57ρ – 2.67 R = 9: if f c ′ ∈ A f , h ∈ A h, and ρ ∈ A ρ ˜ ˜ ˜ then v 9 = 0.006f c ′ – 0.002h – 30.73ρ + 0.05 2 2 2 2 f 3 h 1 ρ 2 3 FuzzyCSA based ACI 318-05 CEB-FIP Eurocode 2 A23.3-04 model VTest/ VTest/ VTest/ VTest/ VTest/ Vpredicted† Vpredicted† Vpredicted† Vpredicted† Vpredicted† Tomaszewicz (1993) 2 ρ 1 f Investigator* 1 1 f Table 3—Testing to predicted punching shear strength ratio using existing design codes and fuzzy-based model 2 R = 10: if f c ′ ∈ A f , h ∈ A h, and ρ ∈ A ρ ˜ ˜ ˜ then v 10 = 0.021f c ′ + 0.043h + 19.19ρ – 7.58 R = 11: if f c ′ ∈ A , h ∈ A , and ρ ∈ A ˜ ˜ ˜ then v 11 = 0.001f c ′ – 0.006h + 49.86ρ + 1.96 Criswell35 ⎛ 12 ⎞ ⎛ 12 ⎞ λ i⎟ v cf = ⎜ λ i v i⎟ ⁄ ⎜ ⎝i = 1 ⎠ ⎝i = 1 ⎠ ∑ ∑ (12) It is important to emphasize the fact that several implication operators exist.42 The selection of the implication operator is governed by three main issues: the needed logical implication ACI Structural Journal/July-August 2007 0.89 0.96 0.82 0.86 1.375 1.098 1.139 1.219 1.019 Standard deviation where vi, f ′c, and h are in MPa, MPa, and mm, respectively. The punching shear strength vcf can be computed using Eq. (11) and (12) 0.94 Mean R = 12: if f c ′ ∈ A f , h ∈ A h, and ρ ∈ A ρ ˜ ˜ ˜ then v 12 = 0.004f c ′ + 0.018h + 36.19ρ – 6.36 0.314 0.207 0.198 0.280 0.189 * Reference to investigators work, unless otherwise noted, can be found in Reference 3. † Strength ratio (= VTest/Vpredicted), where VTest equals actual strengths (test results), and Vpredicted equals predicted strengths by current design methods (ACI 318-05, CEB-FIP, Eurocode 2, and CSA A23.3-04) or fuzzy-based model, respectively. of information, the influence of the fused output on the model prediction, and the effect of the fusion method on the computational efficiency of the learning algorithm. The product implication Π was selected herein for three reasons. First, to perform the fuzzy and operation as indicated by Eq. (4). Second, the product implication tends to dilute the influence of joint membership values that are small and 443
- 97. Fig. 5—Strength prediction by current design method and fuzzy-based model. (Note: 1 MPa = 0.145 ksi; 1 mm = 0.04 in.) therefore magnify the contribution of the rules associated with high membership values in computing the shear strength (Eq. (4) and (5)). This fact promoted the use of the 444 product operator in artificial neural networks as an efficient Hebbian-type learning algorithm.20 Finally, the choice of the product implication was also controlled by the need to ACI Structural Journal/July-August 2007
- 98. produce a continuous and differentiable error function (Eq. (6)) to enable efficient computation of the error gradients during the learning process. Table 3 presents a summary of punching shear strength of the specimens predicted by the fuzzy-based model. In the verification, the 96 specimens, which were not used in the learning process, were used. Figure 5(a) shows the ratios between the actual test to the fuzzy-based model predicted strength (Vtest /Vpredicted) to have a mean value 1.019 and a standard deviation of 18.9%. Figures 5(b) to (e) show the ratios between actual to predicted strength (Vtest /Vpredicted) using the CEB-FIP MC 90,12 the Eurocode 2,13 ACI 318-05,14 and CSA A23.3-04,15 to have mean values of 1.098, 1.139, 1.375, and 1.219, respectively, with standard deviations of 20.7, 19.8, 31.4, and 28.0%, respectively (refer to Table 3). The results show that the fuzzy-based model can be used to predict the punching shear strength of slab-column connections with various slab thicknesses, reinforcement ratios, and circular and square columns. Moreover, higher prediction accuracy of the fuzzy-based model can be observed compared with predication accuracies for all existing design codes. It is interesting to note that, except for Eurocode 2,13 current design methods show a considerable scatter represented by high standard deviations of test-prediction ratios. Moreover, observing Fig. 5(c), the CEB-FIP MC 90 code underestimates the punching shear strength of specimens with low tension reinforcement ratios while it overestimates the punching shear strength of specimens with high tension reinforcement ratios. The Eurocode 213 shows good accuracy in predicting the punching shear strength at different reinforcement ratios. Finally, ACI 318-0514 and CSA A23.3-0415 underestimate the punching shear strength of specimens with high reinforcement ratios while they overestimate the punching shear strength of specimens with low reinforcement ratios. This is attributed to the fact that ACI 318-05 and CSA A23.3-04 codes do not account for the effect of the tension reinforcement ratio on the punching shear strength. It is also evident from Fig. 5(a) that the fuzzy-based model predicts punching shear strength at both low and high reinforcement ratios with consistent accuracy. It is worth noting that the slab thickness and the tension reinforcement ratio in addition to the compressive strength are found to have a significant influence on modeling punching shear strength using the fuzzy-based model. These parameters have also been promoted by other researchers before because of their influence on the size effect43 and their possible role in developing shear friction.44 To consider other rectangularity ratios c2/c1 (>1) and high perimeter to depth ratios bo/d (>15.0), a design approach based on the fuzzy-based model is proposed as ⎧ v cf ⎪ ⎪ 1 ⎛ 0.5 + ---- ⎞ v ⎪ n⎠ cf ⎝ v c = min ⎨ βc ⎪ ⎪ ⎛ 10 ⎛ ----------- ⎞ n⎞ ⎪ ⎝ 0.5 + ⎝ b ⁄ d⎠ ⎠ v cf o ⎩ (13) where βc = c2/c1, c1 and c2 equal the short and long sizes of rectangular columns, vcf is the fuzzy-based shear strength estimated using Eq. (12), and n is a power coefficient. Equation (13) is modeled in a format similar to that of the ACI equation for predicting the punching shear strength. If ACI Structural Journal/July-August 2007 Fig. 6—Variation of strength-prediction by fuzzy-based model according to bo /d. Fig. 7—Variation of strength-prediction by fuzzy-based model according to c2/c1 higher than 1. n = 1.0 as similar to the ACI equation is used, the model will significantly overestimate the punching slab-column connections with rectangular columns and with bo/d higher than 15. A mean value and a standard deviation of the strength-prediction ratios (Vtest /Vpredicted) of the specimens (Table 3) using n = 1 are 0.977 and 0.193, respectively, while those using n = 2 are 1.019 and 0.189. Therefore, the authors recommend the use of n = 2. The model prediction with n = 2 for a wide range of bo/d and for rectangular columns are shown in Fig. 6 and 7. The choice of n = 2 for the second and third components of Eq. (13) was based on examining each component separately. It has become evident that refinement in the value of n for each part would not yield any enhancement in the prediction accuracy of the model. Figure 6 demonstrates the fact that the modified fuzzybased model using a modification factor (Eq. (13)) can accurately predict the punching shear strength of slabcolumn connections with various bo/d (5.8 ≤ bo/d ≤ 24.0) even though the fuzzy-based model (Eq. (12)) was developed within the geometrical limits (5.8 ≤ bo/d ≤ 14.9) due to the lack of test data. This is attributed to the fact that the fuzzybased model was developed by using the average ultimate shear strength vc considering bo and d (Eq. (1)). It is evident that the modified fuzzy-based model can properly consider the interaction between bo/d and vc in its strength equation (Eq. (13)). In Fig. 7, the fuzzy-based model also accurately predicts the punching shear strength of slab-column connections with rectangular columns (c2/c1 > 1). From this result, it is noted that the modified fuzzy-based model properly considers the effect of rectangularity of columns in practical design range (1 ≤ c2/c1 ≤ 5). It is worth noting that, if enough experimental data with high bo/d ratios and rectangular columns were available in the literature, the use of modification factors for 445
- 99. Fig. 8—Design chart for punching shear strength using fuzzy-based model. (Note: 1 MPa = 0.145 ksi; 1 mm = 0.04 in.) Fig. 9—Strength variation according to primary design parameters.25,37,44-47 addressing these issues can be completely omitted. This indicates the fact that a refined fuzzy-based model would always be possible to develop, once experimental data beyond these geometrical limitations becomes available. PROPOSED DESIGN CHART For design purposes, the direct use of the fuzzy-based model as an empirical method using Eq. (10) to (13) and the premise parameters from Table 2 might not be feasible for designers. To avoid such complexity and to make use of the demonstrated ability and relative high accuracy of the fuzzybased model in design of slab-column connections without shear reinforcement, the authors suggest a simplified design model that is developed based on a set of design charts that are developed using the fuzzy-based model. Following a format similar to that used in ACI 318-05, the design strength for punching shear of slab-column connections is defined as φVn = φvcbod (14) where vc is calculated according to Eq. (13) using n = 2, and φ is the strength reduction factor taken equal to 0.6. The punching shear strength vcf can be estimated using Fig. 8. Figures 8(a) to (d) show a group of design charts to estimate the punching shear strength vcf of slab-column connections using the fuzzy-based model. The design charts are developed 446 for a wide range of primary design parameters: 20 ≤ f ′c ≤ 100 MPa (2.9 ≤ f ′c ≤ 14.5 ksi), 100 ≤ h ≤ 300 mm (3.9 ≤ h ≤ 11.8 in.), and 0.8 ≤ ρ ≤ 2.0%. For space limitations, only four design charts are developed herein covering the aforementioned range of parameters. Additional design charts can be developed using the model equations described previously. The φ factor of 0.6 corresponds conservatively to the lowest bound shown in Fig. 5(a). Obtaining a refined shear strength reduction factor (higher than 0.6) can be done using principles of load and resistance factor design (LRFD),45 but is beyond the scope of this study. It can be observed from Fig. 8(a) to (d) that the punching shear strength decreases as slab thickness increases, which respects previous findings of the size effect by Bažant and Cao9 and Eurocode 2.13 In cases with high reinforcement ratios, however, this size effect is disturbed by the combined effect of size and membrane force generated by the tension reinforcement. As observed in Fig. 8(c) and (d), for high tension reinforcement ratios and low concrete compressive strength, the punching shear strength increases as the slab thickness increases. This can be attributed to the possibility that the increase in the slab thickness with high reinforcement ratios results in an increase in the axial membrane force,24,26,27 which contributes to punching shear strength due to the increase in the shear friction effect.44 This possible shear friction contribution to the punching shear strength has been argued by other researchers in shear analysis.44,46 This phenomenon is due to the combined effect of the primary parameters (compressive strength, slab thickness, and tension reinforcement ratio) and can be also observed in previous test results from the punching shear database.3 Figure 9 shows the punching shear strength reported in existing test results. For this study, Elstner and Hognestad,37 Shaeidt el al.,47 Regan,48 Marzouk and Hussein,25 Hallgren and Kinnunen,49 and Tomaszewicz’s50 specimens were used. Each data set itself has similar dimension and property. The dimensions and properties of the specimens are summarized in Table 1. As expected, for all data sets with high concrete compressive strength, the punching shear strength of thick slabs is always less than that of thin slabs due to the size effect24,38,43 (see Fig. 9(a)). In Fig. 9(b), however, for low concrete compressive strength and high reinforcement ratios (ρ ≥ 0.012), the punching shear strength of thick slabs may be greater than that of thin slabs, which indicates the trade-off between size effect and shear friction effect. These combined effects can be successfully described by the fuzzy-based model. CONCLUSIONS A new alternative design method and a set of design charts based on fuzzy learning from examples are proposed. The new method can accurately predict the punching shear strength of simply supported interior slab-column connections without shear reinforcement. One hundred and seventy eight test specimens from the punching shear databank were used for training and testing the proposed model (82 for training and 96 for testing). The training and testing data sets cover a wide range of the material and geometric properties. The testing data set was not used in the training process. Investigations for developing a model with good accuracy showed that concrete compressive strength, slab thickness, and tension reinforcement ratio are the primary parameters that dominate the punching behavior of slab-column connections. This finding is limited to circular and rectangular columns ACI Structural Journal/July-August 2007
- 100. and slabs with perimeter-to-slab-depth ratios (bo/d) ranging between 5.8 and 24.0 and column size ratios (c2/c1) ranging between 1.0 and 5.0. The fuzzy-based model demonstrates higher prediction accuracy compared with all current design codes including ACI 318-05, Eurocode 2, CEB-FIP MC 90, and CSA A23.3-04 in predicting the punching shear strength of slab-column connections. The proposed model, while addressing uncertainty and interactions between modeling parameters, was shown to respect the fundamental mechanics of punching shear as described by many researchers. ACKNOWLEDGMENTS The financial support by the Defense Threat Reduction Agency (DTRA) University Strategic Partnership to the University of New Mexico is greatly appreciated. REFERENCES 1. Schlaich, J.; Schäfer, K.; and Jennewein, M., “Toward a Consistent Design of Structural Concrete,” Journal of the Prestressed Concrete Institute, V. 32, No. 3, 1987, pp. 74-150. 2. Alexander, S. D. B., and Simmonds, S. H., “Ultimate Strength of Slab-Column Connections,” ACI Structural Journal, V. 84, No. 3, MayJune 1987, pp. 255-261. 3. CEB-FIP Task Group, “Punching of Structural Concrete Slabs,” fib 12, Lausanne, Switzerland, 2001, 314 pp. 4. Yitzhaki, D., “Punching Strength of Reinforced Concrete Slabs,” ACI JOURNAL , Proceedings V. 63, No. 5, May 1966, pp. 527-542. 5. Rankin, G. I. B., and Long, A. E., “Predicting the Punching Strength of Conventional Slab-Column Specimens,” Proceedings of the Institution of Civil Engineers, V. 82, 1987, pp. 327-346. 6. Pralong, J., “Poinçonnement Symétrique des Plachers-Dalles,” IBKBericht No. 131, Insitut für Baustatik und Konstruktion, ETH Zürish, 1982. 7. Nielsen, M. P., Limit Analysis and Concrete Plasticity, 2nd Edition, CRC Press, New York, 1999, 463 pp. 8. Kinnunen, S., and Nylander, H., “Punching of Concrete Slabs without Shear Reinforcement,” Transactions No. 158, Royal Institute of Technology, Stockholm, Sweden, 1960, 112 pp. 9. Bažant, Z. P., and Cao, Z., “Size Effect in Punching Shear Failure of Slabs,” ACI Structural Journal, V. 84, No. 1, Jan.-Feb. 1987, pp. 44-53. 10. Birkle, G., “Flat Slabs: The Influence of the Slab Thickness and the Stud Layout,” PhD dissertation, Department of Civil Engineering, University of Calgary, Calgary, Alberta, Canada, 2004. 11. Theodorakopoulos, D. D., and Swamy, R. N., “Ultimate Punching Shear Strength Analysis of Slab-Column Connections,” Cement and Concrete Composites, V. 24, 2002, pp. 509-521. 12. CEB-FIP MC 90, “Design of Concrete Structures,” CEB-FIP Model Code 1990, Thomas Telford, 1993, 437 pp. 13. EC 2, “Design of Concrete Structures Part I: General Rules and Rules for Buildings,” European Committee for Standardization, Brussels, 2002, 230 pp. 14. ACI Committee 318, “Building Code Requirements for Structural Concrete (ACI 318-05) and Commentary (318R-05),” American Concrete Institute, Farmington Hills, Mich., 2005, 430 pp. 15. CSA Technical Committee on Reinforced Concrete Design, “A23.3-04 Design of Concrete Structures,” Canadian Standards Association, Rexdale, Ontario, Canada, 2004, 250 pp. 16. Chatterjee, A., and Watanabe, K., “An Adaptive Fuzzy Strategy for Motion Control of Robot Manipulators,” Soft Computing, V. 9, No. 3, 2005, pp. 185-193. 17. Zongjin, L.; Chau, C. K.; and Zhou, X., “Accelerated Assessment and Fuzzy Evaluation of Concrete Durability,” Journal of Materials in Civil Engineering, ASCE, V. 17, No. 3, 2005, pp. 257-263 18. Kosko, B., “Fuzzy Systems as Universal Approximators,” IEEE Trans. Comp, V. 43, No. 11, 1993, pp. 1329-1333. 19. Klir, G. J., Uncertainty and Information: Foundations of Generalized Information Theory, John Wiley and Sons, N. J., 2006, 499 pp. 20. Ross, T. J., Fuzzy Logic with Engineering Applications, 2nd Edition, Wiley & Sons, UK, 2004, 650 pp. 21. Jang, J. S. R.; Su, C. T.; and Mizutani, E., Neuro-Fuzzy and Soft Computing, A Computational Approach to Learning and Machine Intelligence, Prentice Hall, N. J., 1997, 614 pp. 22. Carlin, B. P., and Chib, S., “Bayesian Model Choice via Markov Chain Monte Carlo Methods,” Journal of the Royal Statistical Society, Series B, V. 57, No. 3, 1995, pp. 473-484. 23. Pan, A. D., and Moehle, J. P., “An Experimental Study of Slab-Column ACI Structural Journal/July-August 2007 Connections,” ACI Structural Journal, V. 89, No. 6, Nov.-Dec. 1992, pp. 626-638. 24. Sherif, A. G., and Dilger, W. H., “Critical Review of the CSA A23.3-94 for Punching Shear Strength Provisions for Interior Columns,” Canadian Journal of Civil Engineering, V. 23, 1996, pp. 998-1011. 25. Marzouk, H., and Hussein, A., “Experimental Investigation on the Behavior of High-Strength Concrete Slabs,” ACI Structural Journal, V. 88, No. 6, Nov.-Dec. 1991, pp. 701-713. 26. Hawkins, N. M., and Mitchell, D., “Progressive Collapse of Flat Plate Structure,” ACI JOURNAL, Proceedings V. 76, No. 7, July 1979, pp. 775-808. 27. Regan, P. E., and Braestrup, M. W., “Punching Shear in Forced Concrete: A State of the Art Report,” Bulletin d’information, Comité EuroInternational du Béton, Lausanne, Switzerland, Jan. 1985. 28. Rankin, G. I. B., and Long, A. E., “Predicting the Enhanced Punching Strength of Interior Slab-Column Connections,” Proceedings of the Institution of Civil Engineers, V. 82, 1987, pp. 1165-1186. 29. Teng, S.; Cheong, H. K.; Kuang, K. L.; and Geng, J. Z., “Punching Shear Strength of Slabs with Openings and Supported on Rectangular Columns,” ACI Structural Journal, V. 101, No. 5, Sept.-Oct. 2004, pp. 678-687. 30. Laviolette, M.; Seaman, J. W.; Barrett, J. D.; and Woodall, W. H., “A Probabilistic and Statistical View of Fuzzy Methods,” Technometrics, V. 37, 1995, pp. 249-261. 31. Duda, R. O.; Hart, P. E.; and Stork, D. G., Pattern Classification, 2nd Edition, John Wiley & Sons, 2001, 680 pp. 32. Gupta, M. M., and Qi, J., “Theory of T-Norms and Fuzzy Inference Methods,” Fuzzy Sets and Systems, V. 40, No. 3, 1991, pp. 431-450. 33. Fan, J. Y., and Yuan, Y. X., “On the Convergence of a New LevenbergMarquardt Method,” Technical Report, AMSS, Chinese Academy of Sciences, 2001, 11 pp. 34. Hawkins, N. M.; Fallsen, H. B.; and Hinojosa, R.C., “Influence of Column Rectangularity on the Behaviour of Flat Plate Structures,” Cracking, Deflection and Ultimate Load of Concrete Slab Systems, SP-30, American Concrete Institute, Farmington Hills, Mich., 1971, pp. 127-146. 35. Criswell, M. E., “Strength and Behaviour of Reinforced Concrete Slab-Column Connections Subjected to Static and Dynamic Loading,” Technical Report N-70-1, U.S. Army Engineer Waterways Experiment Station, Vicksburg, Miss., Dec. 1970. 36. Lovrovich, J., and McLean, D., “Punching Shear Behaviour of Slab with Varying Span-Depth Ratios,” ACI Structural Journal, V. 87, No. 5, Sept.-Oct. 1990, pp. 507-511. 37. Elstner, R. C., and Hognestad, E., “Shearing Strength of Reinforced Concrete Slabs,” ACI JOURNAL, Proceedings V. 53, No. 7, July 1956, pp. 29-58. 38. Tolf, P., “Plattjocklekens Inverkan På Betongplattors Hållfasthet vid Genomstansning,” Försök med Cikulåra Plattor, TRITA-BST Bull, 146, KTH Stockholm, Sweden, 1988, 64 pp. 39. Berenji, H. R., and Khedkar, P., “Learning and Tuning Fuzzy Logic Controllers Through Reinforcements,” IEEE Transactions on Neural Networks, V. 3, No. 5, 1992, pp. 724-740. 40. Combs, W. E., and Andrews, J. E., “Combinatorial Rule Explosion Eliminated by a Fuzzy Rule Configuration,” IEEE Transactions on Fuzzy Systems, V. 6, No. 1, pp. 1-11. 41. Lucero, J., “Fuzzy Systems Methods in Structural Engineering,” PhD dissertation, University of New Mexico, Department of Civil Engineering, Albuquerque, N. Mex., 2004. 42. Yager, R., “On a General Class of Fuzzy Connectives,” Fuzzy Sets and Systems, V. 4, 1980, pp. 235-242. 43. Bažant, Z. P., Fracture and Size Effect in Concrete and Other Quasi Brittle Materials, CRC Press, New York, 1997, 280 pp. 44. Loov, R. E., “Review of A23.3-94 Simplified Method for Shear Design and Comparison with Results Using Shear Friction,” Canadian Journal of Civil Engineering, V. 25, No. 3, 1998, pp. 437-450. 45. Nowak, A. S., “Calibration of LRFD Bridge Code,” Journal of Structural Engineering, ASCE, V. 121, No. 8, 1995, pp. 1245-1251. 46. Kani, G. N. J., “The Riddle of Shear Failure and Its Solutions,” ACI JOURNAL, Proceedings V. 61, No. 4, Apr. 1964, pp. 441-468. 47. Schaeidt, W.; Ladner, M.; and Rösli, A., “Berechnung von Flachdecken auf Durchstanzen,” Eidgenössische Materialprüfungs-und Versuchsanstalt, Dübendort, 1970. 48. Regan, P., “Symmetric Punching of Reinforced Concrete Slabs,” Magazine of Concrete Research, V. 38, 1986, pp. 115-128. 49. 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- 101. ACI STRUCTURAL JOURNAL TECHNICAL PAPER Title no. 104-S46 Tensile-Headed Anchors with Large Diameter and Deep Embedment in Concrete by Nam Ho Lee, Kang Sik Kim, Chang Joon Bang, and Kwang Ryeon Park This paper presents test results for large cast-in-place anchor bolts in concrete. The tests were performed to evaluate the tensile performance of large anchors, that is, anchors with a diameter greater than 2 in. (50 mm) or an embedment depth greater than 25 in. (635 mm), which are not addressed by ACI 318, Appendix D, and ACI 349, Appendix B. The tests were also intended to investigate the safety of such anchors for use in nuclear power plants and the effects of regular (conventional) and special reinforcement on the strength of such anchors. The test results are used to assess the applicability of existing design formulas valid for smaller anchors to large anchors. Suggestions are made for incorporating the effects of deep embedment or large diameter in existing design provisions for cast-in-place tensile anchor bolts under tension load. Keywords: anchor; anchor bolt; cast-in-place; embedment; tension test. INTRODUCTION Current anchorage designs for nuclear power plants in Korea use large anchor bolts with diameters exceeding 2 in. (50 mm), embedment depths exceeding 25 in. (635 mm), a specified yield strength of 140 ksi (980 MPa), and a specified ultimate strength of 155 ksi (1085 MPa). Whereas the tensile behavior of smaller anchors has been studied extensively, large anchors have not been adequately addressed. In the research described herein, large anchors were tested in tension to develop design criteria for anchors that are not addressed by ACI 318-05, Appendix D,1 or ACI 349-01, Appendix B,2 and to evaluate the applicability of capacity-prediction methods developed for smaller anchors. To evaluate the tensile behavior of anchors with large diameters and embedment depths, various anchors, with diameters from 2.75 to 4.25 in. (69.9 to 108 mm) and embedment depths from 25 to 45 in. (635 to 1143 mm) were tested. method (CCD method),3 which is a derivative of the Kappa method4 described in Reference 5. According to the CCD method, the average concrete breakout capacity of headed anchors in uncracked concrete is given by Eq. (1). This equation is valid for anchors with a relatively small head (mean bearing pressure at breakout load of approximately 13fc′ ).3 In ACI 318, Appendix D,1 the 5%-fractile of the concrete cone breakout loads are predicted, which is assumed as 0.75 times the mean value. This leads to Eq. (2). ACI 318-05, Appendix D,1 allows the use of Eq. (4) for calculating the nominal breakout capacity of headed anchors with an embedment depth hef ≥ 11 in. (279 mm) in uncracked concrete. Equation (4) modifies the CCD method slightly by changing the exponent on the embedment depth hef from 1.5 to 1.67. The mean concrete capacity may be calculated according to Eq. (3). In ACI 349-97,6 a 45-degree cone model is used to calculate the concrete breakout capacity (Eq. (5)). Because Eq. (5) was used in design, it may be considered to predict approximately the 5%-fractile of test results. A summary of the proposed predictors are given as Equation number Predictor 1.5 (1) N u, m = 40 f c′ h ef ( lb ) (2) N u = 30 f c′ h ef ( lb ) (3) N u, m = 26.7 f c′ h ef ( lb ) 1.5 1.67 1.67 (4) RESEARCH SIGNIFICANCE The research described herein is the first experimental information on the tensile behavior of very large headed anchor bolts (hef ≥ 21 in. [525 mm]). It is important because although such anchor bolts are commonly used in power plants and for the anchorage of tanks, no design provisions validated by tests exist for them. EXISTING FORMULAS FOR PREDICTING TENSILE CAPACITY OF ANCHOR BOLTS IN CONCRETE Presuming the head of the anchor is large enough to prevent pull-out failure (refer to ACI 318, Appendix D), the tensile capacity of large anchor bolts is governed by tensile yield and fracture of the anchor steel or by tensile breakout of the concrete in which the anchor is embedded. Steel yield and fracture are well understood. The breakout formulas of current U.S. design provisions (ACI 318-051 and ACI 349-012) are based on the concrete capacity design (CCD) ACI Structural Journal/July-August 2007 N u = 20 f c′ h ef ( lb ) (5) N u = 4 f c ′πhef ( 1 + d k ⁄ h ef ) ( lb ) 2 Remark Mean breakout strength, CCD-method with exponent 1.5 on hef Nominal breakout strength, ACI 318-05, Appendix D Mean breakout strength for anchors with hef ≥ 10 in. (254 mm), CCD-method with exponent 1.67 on hef Nominal breakout strength for anchors with hef ≥ 10 in. (254 mm) according to ACI 318-05, Appendix D Nominal breakout strength, ACI 349-97 (45-degree cone model) Note: fc = specified concrete compressive strength (psi); hef = effective embedment ′ (in.); and db = diameter of anchor head (in.). DESCRIPTION OF EXPERIMENTAL PROGRAM Test specimens To evaluate the effects of embedment depth, anchor diameter, and supplementary reinforcement patterns on the tensile capacity of large anchors, five different test configurations were selected and four test replicates with each configuration ACI Structural Journal, V. 104, No. 4, July-August 2007. MS No. S-2006-232 received June 6, 2006, and reviewed under Institute publication policies. Copyright © 2007, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published in the May-June 2008 ACI Structural Journal if the discussion is received by January 1, 2008. 479
- 102. ACI member Nam Ho Lee is a Senior Research Engineer in the Civil Engineering Department of the Korea Power Engineering Co. He received his BS from Seoul National University and his MS and PhD from the Korea Advanced Institute of Science & Technology. He is a member of ACI Committees 349, Concrete Nuclear Structures, and 355, Anchorage to Concrete, and Joint ACI-ASME Committee 359, Concrete Components for Nuclear Reactors. His research interests include the nonlinear behavior of concrete structures and anchorage to concrete. Kang Sik Kim is a Senior Researcher, Environment and Structure Laboratory, Korea Electric Power Research Institute, Daejeon, Korea. His research interests include the behavior of concrete-filled steel plate structures and anchorage to concrete. Chang Joon Bang is a Project Engineer at Korea Hydro & Nuclear Power Co. Ltd., Seoul, Korea. He is currently a Graduate Student of civil engineering, Lehigh University, Bethlehem, Pa. Kwang Ryeon Park is a Research Engineer, Civil Engineering Department, Korea Power Engineering Co. were performed giving 20 specimens in total. The test program is summarized in Table 1. The test specimens are shown in Fig. 1. All anchors were fabricated of ASTM A540 Gr. B23 Class 2 steel (equivalent to ASME SA 549 Gr. B23 Class 2 used in Korean nuclear power plants) with fy = 140 ksi (980 MPa) and fu = 155 ksi (1085 MPa). The anchor head consisted of a round thick plate which was fixed to the bolt by clamping nuts (Fig. 2). The diameter of the round plate was dh = 6 in. (152.4 mm) (db = 2.75 in. [69.9 mm]), dh = 8.5 in. (215.9 mm) (db = 3.75 in. [95.3 mm]), and dh = 10 in. (254.0 mm) (db = 4.25 in. [108.0 mm]). The size of the concrete test block was large enough to avoid splitting failure. The concrete volume (width/length/depth) available for each anchor is shown in Table 1. Furthermore, to minimize the width of eventual shrinkage cracks, the top and bottom of the test member were reinforced in both directions with No. 10 bars at 16, 10, and 10 in. (406.4, 254, and 254 mm) spacing for Specimens T1, T2, and T3, respectively. This surface reinforcement does not significantly influence the concrete breakout load. As shown in Fig. 1, wooden and steel frames were constructed to suspend the cast-in-place anchors in the correct position and at the correct embedment depth. The concrete mixture for the test specimens is shown in Table 2(a). The concrete used in the test specimens was comparable to the concrete used in the Korean Nuclear Plant, except that 20% by weight of the Type I cement was substituted by fly ash and 1 in. (25 mm.) crushed aggregate was used instead of 3/4 in. (19 mm). The target concrete strength at 42 days was fc′ = 5500 psi (37.9 MPa). The actual concrete strength at the time of testing is given in Table 2(b). The concrete for the specimens of one test series was placed from one batch. Whereas in test Series T1 to T3, no special reinforcement was used to resist the applied tension load, in test Series T4 and T5, supplementary reinforcement (refer to Fig. 3) was used to increase the ultimate load. The supplementary reinforcements consisted of vertical stirrups (eight No. 8 bars and 16 No. 8 bars for test Series T4 and T5, respectively), as shown in Fig. 3. Table 1—Description of tension test specimens Specimen Concrete Diameter volume Anchor of anchor Effective available for diameter, head, embedment each anchor hef , in. (width/length/ dh , in. Reinforce- db, in. ment (mm) (mm) (mm) depth) T1-A,B,C,D None 2.75 (69.9) 6.0 (152.4) 25 (635) 5.9hef /5.0hef / 2.9hef T2-A,B,C,D None 3.75 (95.3) 8.5 (215.9) 35 (889) 5.4hef/4.7hef / 2.0hef T3-A,B,C,D None 4.25 (108.0) 10.0 (254.0) 45 (1143) 5.0hef /3.6hef / 2.0hef T4-A,B,C,D Supp. No. 1 2.75 (69.9) 6.0 (152.4) 25 (635) 5.9hef /5.0hef / 2.9hef T5-A,B,C,D Supp. No. 2 2.75 (69.9) 6.0 (152.4) 25 (635) 5.9hef /5.0hef / 2.9hef Fig. 1—Tension test Specimens T1, T2, T3, T4, and T5. Table 2(a)—Concrete mixture proportioning Nominal strength, psi, at W/ S/a, W, FA, WRA,* AEA,† mL 42 days (C + FA) % lb C, lb lb S, lb G, lb mL 5500 0.44 44 525 514 128 1257 1617 474 26 *Water-reducing admixture. † Air-entraining admixture. Table 2(b)—Concrete strength at time of testing Test specimen 58/50/44/42 5771 (39.8)/5630 (38.8)/ 5508 (38.0)/5464 (37.7) T2-A/B/C/D 41/45/47/49 5177 (35.7)/5248 (36.2) / 5291 (36.5)/5320 (36.7) T3-A/B/C/D 61/56/54/50 5448 (37.6)/5348 (36.9) / 5305 (36.6)/5220 (36.0) T4-A/B/C/D 480 Compressive strength, psi (MPa) T1-A/B/C/D Fig. 2—Details of anchor head. Curing ages, days 57/55/54/50 5945 (41.0)/5917 (40.8)/ 5903 (40.7)/5817 (40.1) T5-A/B/C/D 71/70/69/68 6144 (42.4)/6130 (42.3)/ 6130 (42.3)/6116 (42.2) ACI Structural Journal/July-August 2007
- 103. Test setup The test setup consisted of a loading frame, loading plate, jack assembly, load cell, and other items, as shown in the schematic and photo in Fig. 4. The load was applied to the anchor under force-control in an increment of approximately 3.5% of ultimate steel strength of the anchor bolt (Fu = 925, 1683, and 2192 kips [4114.6, 7486.4, and 9750.5 kN], for bolts with a diameter of 2.75, 3.25, and 4.25 in. [69.90, 82.55, and 107.95 mm], respectively), that is, 30, 60, 77, 68, and 48 kips (133.4, 266.9, 342.5, 302.5, and 213.5 kN) for Series T1, T2, T3, T4, and T5, respectively. It was reacted in two directions by a stiff frame to minimize the bending moment in the test specimen. The clear distance between the supports was 4.0 hef for Specimens T1 through T5, thus allowing for an unrestricted formation of a concrete cone. The applied load was measured by a load cell. Additionally, the strain along the embedment length of the anchor bolt was measured (Fig. 5). Furthermore, the displacement of the top end of the anchor was measured by LVDTs (Fig. 5). surface varied from α = 20 to 30 degrees, following the typical crack profiles shown in Fig. 6(b). In general, test Specimens T4 and T5, with supplementary reinforcement (Fig. 3), were not tested to failure. At the applied peak load, the measured steel strains exceeded the yield strain and because of safety concerns a sudden rupture of the bolt was avoided. Only Specimen T4-A was tested to TEST RESULTS Failure loads, failure modes and load displacement behavior The average failure loads are summarized in Table 3(a) (Series T1 to T3) and Table 3(b) (Series T4 and T5). The values given in the tables are normalized to fc′ = 5500 psi (37.9 MPa) by multiplying the measured peak load of each test with the factor (5500/fc,test)0.5. In test Series T1 to T3, failure was caused by concrete cone breakout well below the anchor bolt steel capacities (Fu = 925, 1683, and 2192 kips [4114.6, 7486.4, and 9750.5 kN] for bolts with diameters of 2.75, 3.25, and 4.25 in. [69.90, 82.55, and 107.95 mm], respectively). The cracking patterns in the specimen after the test are depicted in Fig. 6(a). Generally, one major longitudinal crack was observed, centered approximately on the sides of the block, in combination with a horizontal crack and some transverse cracks. On the top surface of the block, the cracks formed a circular pattern around the anchor. To identify the internal crack propagation defining the roughly conical breakout body, one replicate of each specimen type was selected, and the concrete was cored on two orthogonal planes whose intersection coincided with the axis of the anchor. The cores confirmed a breakout cone whose angle with the concrete Fig. 4—Tension test setup: (a) schematic; and (b) photo. Fig. 3—Supplementary reinforcement in Specimens T4 and T5. ACI Structural Journal/July-August 2007 Fig. 5—Location of LVDTs and strain gauges (Specimen T1). 481
- 104. Table 3(a)—Tension test results and predictions for unreinforced Specimens T1, T2, and T3 Table 3(b)—Tension test results and predictions for reinforced Specimens T4 and T5 Concrete breakout capacities, kips (kN), by embedment Classification Reference Concrete breakout capacities, kips (kN), by embedment Specimen T1 Specimen T2, Specimen T3, 25 in. 35 in. 45 in. (635 mm) (889 mm) (1143 mm) Specimen T4 Specimen T5, Specimen T1, 25 in. 25 in. 25 in. (635 mm) (635 mm) (635 mm) ACI 349-97, Eq. (5) 1305 (5804) ACI 318-05, Eq. (4) 320 (1423) CCD method with 1.5 h ef Eq. (1) Reference 2138 (9510) ACI 349-97, Eq. (5) 676 (3006) 676 (3006) 676 (3006) 562 (2499) 855 (3803) ACI 318-05, Eq. (4) 320 (1423) 320 (1423) 320 (1423) 371 (1650) 614 (2731) 895 (3981) CCD method with 1.5 h ef Eq. (1) 371 (1650) 371 (1650) 371 (1650) CCD method with 428 (1903) 1.67 h ef Eq. (3) Predictions 676 (3006) Classification 750 (3336) 1142 (5079) CCD method with 1.67 h ef Eq. (3) 428 (1903) 428 (1903) 428 (1903) Mean 733 (3260) 725 (3224) 509 (2264) COV, % 1.7 3.5 5.8 5%-fractile 685 (3047) 625 (2780) 393 (1748) 5%-fractile/ mean 0.93 0.86 0.77 Mean 744 (3309) 1242 (5524) COV, % Tests 509 (2264) 5.8 2.8 6.1 5%-fractile 393 (1748) 5%-fractile/ mean 662 (2944) 0.77 944 (4199) 0.89 Ratios of observed to predicted capacities Mean of test results (I) Nu,5%/ Eq. (5) 0.58 0.51 0.44 0.51 (II) Nu,5%/ Eq. (4) 1.24 1.19 1.12 1.18 (III) Mean/ Eq. (1) 1.37 1.21 1.39 (IV) Mean/ Eq. (3) 1.19 0.99 1.09 Symbol in Classification Fig. 9 5% fractile of test results Mean of test results Ratio of observed to predictions (hef = 25 in. [635 mm]) Comparison T4 T5 T1 T4/T1 (I) Nu,5%/Eq. (5) 1.01 0.92 0.58 1.74 (II) Nu,5%/Eq. (5) 2.16 1.97 1.24 1.74 (III) Mean/Eq. (1) 1.98 1.96 1.37 1.45 (IV) Mean/Eq. (3) 1.71 1.70 1.19 1.44 1.32 1.09 failure. Failure of this specimen was caused by forming a concrete cone. From the load-displacement curves (Fig. 7), it can be concluded that in test Series T4, the applied maximum loads were almost identical with the failure loads. In test Series T5, however, the failure load of the anchors was not reached. Because Specimens T4 and T5 showed no cracking at the concrete surface, no cores were taken to check whether a cone had begun to form. The load-displacement curves for Specimens T1, T2, T3, T4, and T5 are shown in Fig. 7(a) through 7(e), using the displacement measured at the top of each anchor. The load-displacement relationship for each test replicate varied based on the concrete strength at the time of testing. The projecting lengths of the anchor shafts from the concrete surface to the top of the anchor for Specimens T1, T2, T3, T4, and T5 were 41.7, 48.6, 53.1, 41.7, and 41.7 in. (1059, 1234, 1348, 1059, and 1059 mm), respectively. Because the measured displacements shown in Fig. 7 include the steel elongation of the projecting anchor length, the actual anchor displacements at the top of the concrete surface, which are accumulated along the embedded portion of the anchor, are much smaller than shown in Fig. 7. In Fig. 8, the relationship between load and anchor displacement at the surface of the concrete (calculated from the displacements measured at the anchor top end subtracting the steel elongation of the projecting length) are plotted for test Series T1 to T5. In some tests, the calculated displacements at the concrete surface are negative for low loads. It is believed that this is 482 Tests 0.76 SymSpecimen T1 Specimen T2, Specimen T3, Classi- bol in Com25 in. 35 in. 45 in. fication Fig. 9 parison (635 mm) (889 mm) (1143 mm) Mean 5% fractile of test results Predictions (a) (b) Fig. 6—(a) Cracking pattern for four test replicates (A, B, C, and D) of Specimens T1, T2, and T3; and (b) typical internal crack profile in Specimen T1. ACI Structural Journal/July-August 2007
- 105. Fig. 7—Measured load-displacement relationships. caused by bending of the anchors if they were not installed perfectly perpendicular to the concrete surface. It can be seen that the anchor displacements at peak load of Specimens T1 to T3 (concrete cone failure) are rather small. This can be explained by the rather large anchor heads that, due to the low concrete stresses, did not slip much. For head sizes allowed by ACI 318-05, Appendix D, the breakout failure loads increase approximately proportional to hef1.5. With much larger heads, the power on the embedment depths is greater than 1.5.7 In the present tests, at failure, the related pressure under the head was on average p/fc′ = 4.37, 3.36, and 5.31 for test Series T1, T2, and T3. It was much smaller than the pressure allowed by ACI 318-05 for uncracked concrete (pn = 10fc′ ). Comparison of predicted and tested tensile breakout capacities In Table 3(a), tension test results for unreinforced Specimens T1, T2, and T3, and results in Table 3(b) for reinforced Specimens T4 and T5, are compared with predicted capacities. The measured mean failure loads are compared with the predicted mean capacities according to Eq. (1) and (3), respectively, and the 5%-fractiles of the measured failure loads calculated by assuming an unknown standard deviation are compared with the values according to Eq. (4) and (5). In Fig. 9, the ratios of measured capacities to predicted values are plotted. Figure 10 shows the measured failure loads of each test compared with the values predicted according to Eq. (5), Fig. 10(a); Eq. (1), Fig. 10(b); and Eq. (3), Fig. 10(c), as a function of the embedment depth. In Fig. 11, the measured concrete breakout loads, as well as the failure loads according to best fit equations using the current test results and Eq. (1), (2), (3), and (5), are plotted as a function of the embedment depth. ACI Structural Journal/July-August 2007 Fig. 8—Relation between load and anchor displacement at concrete surface. EVALUATION OF TEST RESULTS FOR UNREINFORCED SPECIMENS T1, T2, AND T3 According to the 45-degree cone model (Eq. (5)), the 2 breakout capacities increase in proportion to hef . The predicted capacities Nu,calc are much higher than the measure values Nu,test and the ratio Nu,test /Nu,calc decreases with increasing embedment depth (Fig. 10(a)). On average, the 5%-fractiles of the observed capacities are approximately half the capacities predicted by ACI 349-97 (Table 3(a)). This demonstrates that the 45-degree cone model is unconservative for deep anchors. This agrees with the findings by Fuchs et al.3 and Shirvani et. al.8 In contrast, the predictions according to the CCD method are conservative. The measured average breakout loads are approximately 30% higher than the values predicted according to Eq. (1) (Nu proportional to hef1.5) with no significant influence of the embedment depth (Fig. 10(b)). On average, the ratio of measured failure loads to the values predicted by Eq. (3) (Nu 1.67 proportional to hef ) is 1.09 (Table 3(a)). It decreases slightly with increasing embedment depth (Fig. 10(c)). In Fig. 10(d) to 10(f), the breakout failure loads of headed anchors with an embedment depth hef ≥ 8 in. (200 mm) measured in the present tests and taken from other sources3,8 are compared with values predicted by the CCD method. According to Fig. 10(d), the prediction according to Eq. (1) is conservative for large embedment depths. The failure loads predicted by Eq. (3) agree quite well with the measured values (Fig. 10(e)). Figure 10(f) shows that the CCD method changing the exponent on hef from 1.5 to 1.67 at an effective embedment depth of 10 in. (250 mm) predicts the failure loads of anchors with hef ≥ 8 in. (200 mm) best. Only two 483
- 106. test points at hef = 8 in. (200 mm) fall below the assumed 5%-fractile, which is equal to 75% of the average value. The 5%-fractiles of the capacities observed in the present tests average approximately 120% of the values predicted by ACI 318-05, Appendix D (Eq. (4)) (refer to Table 3(a)). The higher ratio Nu,test /Nu,calc when comparing the 5% fractiles with each other instead of the average values is due to the rather low scatter of test results. On average, the coefficient of variation (COV) was approximately 5%. This results in an average ratio Nu,5%/Nu,m of 0.81, whereas in ACI 349-01, a ratio of 0.75 is assumed. In actual structures, the concrete strength, and thus the concrete cone resistance, might vary more than in the present test specimens. Therefore, the ratio Nu,5%/Nu,m assumed in ACI 318-05, Appendix D, should be maintained. Numerical investigations by Ozbolt et al.7 using a sophisticated three-dimensional nonlinear finite element model demonstrates that the concrete breakout capacity of headed anchors is influenced by the head size, that is, the pressure under the head, related to the concrete compressive strength as described previously. Based on the previous evaluations, it is recommended to predict the nominal concrete breakout capacities of anchors with an embedment depth hef ≥ 10 in. (250 mm) in uncracked concrete by Eq. (4). Equation (4) is valid, however, only if the head size is large so that the pressure under the head at the nominal capacity is pn ≤ 3fc′ . This limiting value is deduced from the results of the test Series T1 to T3. In these tests, the pressure under the head was pn/fc′ = 3.4 to 5.3, on average 4.3. The nominal capacity is approximately 75% of the mean capacity (compare Eq. (4) with Eq. (3)). When applying this reduction factor, one gets pn/fc′ = 3.2 ~ 3.0. This limiting value is supported also by the numerical analysis results.7 For smaller heads, for which the nominal pressure under the head is pn > 3fc′ , the breakout capacities in uncracked concrete should be predicted by Eq. (2). In cracked reinforced concrete, lower breakout capacities than in uncracked concrete are observed.9 Therefore, ACI 318-05, Appendix D, reduces the nominal breakout capacities of headed anchors in cracked reinforced concrete by a factor 0.8 compared with uncracked concrete. Therefore, in cracked concrete Eq. (4) with hef1.67, multiplied by the factor 0.8, should only be used for deep anchors if the pressure under the head is pn ≤ 2.4fc′. Fig. 10—Ratios of observed to predicted concrete tensile breakout capacities as function of embedment depth. Fig. 9—Ratios of test results (5% fractile and mean) to predicted capacities; compare with Table 3. 484 Fig. 11—Test results and comparison with predicted capacities. ACI Structural Journal/July-August 2007
- 107. EFFECT OF SUPPLEMENTARY REINFORCEMENT Reinforced Specimen T4 Test Specimens T4, with supplementary reinforcement, are shown in Fig. 3. The mean tested failure load (733 kips [3260 kN]) is close to the sum (806 kips [3585 kN]) of the calculated reinforcement strength (378 kips [1681 kN]) and the unreinforced concrete strength (428 kips [1904 kN]) by Eq. (3). It can be inferred that the adopted reinforcement pattern effectively acted in the anchorage system to resist tension load. The tested breakout strength of the unreinforced test Specimen T1 with the same embedment depth as Specimen T4 was 509 kips (2264 kN). Comparison of the mean tested strengths of Specimens T1 and T4 shows that the effective increase in capacity due to supplementary reinforcement is roughly 224 kips (996 kN), or approximately 60% of the calculated yield strength of the supplementary reinforcement. The loading on Specimen T4-A was increased to the expected total yield force of the supplementary reinforcement so that the load distribution to each of the two reinforcement groups could be estimated. The load resisted by the supplementary reinforcement in the inner concentric circle (4.2 in. [106 mm] from the axis of the anchor) was 2.2 times the load resisted by the equal area of supplementary reinforcement in the outer concentric circle (8.5 in. [216 mm] from the axis of the anchor). According to the measured strains in the strain gauges attached to reinforcing bars, the reinforcing bars close to the anchor were more effective in increasing the tensile capacity and their maximum stress was measured close to the anchor head. Reinforced Specimen T5 The mean tested capacity (725 kips [3225 kN]) of the four replicates of test Specimen T5, with supplementary reinforcement as shown in Fig. 3 was much smaller than the sum (1129 kips [5021 kN]) of the calculated reinforcement strength, 16 x 60 ksi x 0.79 in.2 = 758 kips (3371 kN) and concrete breakout strength per the CCD method given by Eq. (1), 371 kips (1650 kN). These test results indicate that this layout of supplementary reinforcement contributes with a low level of effectiveness to the capacity of the anchor. This conclusion is corroborated by measured strains in the gauges attached to the reinforcing bars, which indicates little strain in the reinforcement. As noted previously, however, Specimen T5 were not fully loaded up to failure due to safety concerns. As a consequence, the results of Series T5 are judged to not be useful in verifying the absolute effectiveness of the supplementary reinforcement. By comparing results from Specimens T4 with those of Specimens T5, however, it is still possible to judge the relative effectiveness of the different supplementary reinforcement patterns. For a given applied load, stresses in the supplementary reinforcement of Specimens T5 along the outer circles are less than half of those along the inner circle. The relative trends of stress distribution are similar for each reinforcement in both Series T4 and T5. Therefore, it can be inferred that the increase in tensile capacity is approximately proportional to the amount of supplementary reinforcement. The load-displacement curves of Series T4 show that the peak load was nearly reached in the tests. In Series T5, the load could still be increased. In Series T4, the supplementary reinforcement was not strong enough to resist the concrete breakout load. In Series T5, the loading was stopped before the supplementary reinforcement could be fully activated. ACI Structural Journal/July-August 2007 Therefore, it is not possible to formulate a general model from the test results. The results, however, show that with supplementary reinforcement arranged as in Specimens T4 and dimensioned for about 80 to 100% of the expected ultimate concrete breakout capacity, the failure load was increased by approximately 50% over the unreinforced case. This result can reasonably be used in the calculation of ultimate strength. SUMMARY AND CONCLUSIONS Tensile load-displacement behavior of large anchors without supplementary reinforcement The test results show that ACI 349-97 (Eq. (5)) significantly overestimates the tensile breakout capacity of large anchors. The ratio Nu,test /Nu, calc decreases with increasing embedment depth (Fig. 10(a)). Furthermore, the slope of the concrete cone was much flatter than 45 degrees. Therefore, the overestimation of the failure loads would be even larger for anchors at an edge or for anchor groups. For these reasons, this formula in ACI 349-97 should not be used in design. 1.5 The CCD method with hef (Eq. (1)) is conservative for large anchors (Fig. 10(b)). This is probably due to the fact that this method is based on linear fracture mechanics, which is valid only for anchors with high bearing pressure, that is, anchors with small heads. The tested anchors, however, had rather large heads. The test results can best be predicted by the CCD method with (Eq. (3)) (refer to Fig. 9 and 10(e)). On average, the measured failure loads are approximately 10% higher than the predicted values. If all available results are taken into account (refer to Fig. 10(f)), however, a change of Eq. (3) seems not to be justified. It is proposed to calculate the characteristic resistance of single anchor bolts with hef ≥ 10 in. (250 mm) and low bearing pressure (pressure under the head at nominal breakout load pn ≤ 3fc′ [uncracked concrete] or pn ≤ 2.4fc′ [cracked concrete]) according to ACI 318-05, Appendix D, or ACI 349-01, Appendix B, using the equation with hef1.67). According to the test results, however, the average cone angle was not 35 degrees (as assumed in the CCD method) but only approximately 25 to 30 degrees. Therefore, the characteristic spacing scr,N and characteristic edge distance ccr,N are probably larger than scr,N = 2ccr,N = 3hef as assumed in ACI 318-05. Therefore, it seems prudent to calculate the resistance of anchorages at an edge or corner, or of group anchorages, according to ACI 318-05, but with scr,N = 4.0 hef instead of scr,N = 3.0 hef as given in ACI 318-05. Tensile load-displacement behavior of large anchors with supplementary reinforcement In Series T4, the supplementary reinforcement was not strong enough to resist the applied load. Even in Test T4-A, in which the supplementary reinforcement yielded, only approximately 1/3 (246/759 ≈ 0.33) of the applied peak load was resisted by the reinforcement. In Series T5, which had a stronger reinforcement, the tests had to be stopped because of tensile yielding of the anchors before the supplementary reinforcement had been fully mobilized. Therefore, the results of these tests cannot be used to develop a general design model for anchors with supplementary reinforcement. Nevertheless, the results of test Series T4 showed that the peak load could be increased by approximately 50% compared with the results from test Series T1 without supplementary reinforcement. Therefore, it is proposed to increase the concrete breakout resistance calculated as described 485
- 108. previously by a factor of 1.5 if supplementary reinforcement is present around each anchor of an anchor group. The supplementary reinforcement must be arranged as in Tests T4 (four U-shaped stirrups at a distance ≤ 4 in. (100 mm) or ≤ 0.15hef from the anchor) and dimensioned for the characteristic concrete breakout resistance according to Eq. (4)). In a more general model, the supplementary reinforcement should be dimensioned to take up 100% of the applied load, thus neglecting the contribution of the concrete. The supplementary reinforcement should be designed using a strut-and-tie model. The characteristic resistance of the supplementary reinforcement is given by the bond capacity of the supplementary reinforcement in the anticipated concrete cone, which should be assumed to radiate from the head of the anchor at an angle of 35 degrees. The bond capacity should be calculated according to codes of practice (for example, ACI 318-051 or Eurocode 210). The design strength is limited by the yield capacity of the bars. This model is described in detail in References 11 and 12. ACKNOWLEDGMENTS The authors would like to acknowledge the financial and technical help of Korea Hydro & Nuclear Power Co. Ltd. and Korea Electric Power Research Institute for financing this research work and several on-going research projects related to the capacity of anchorage to concrete structures. The authors are also grateful for the valuable advice of R. Eligehausen, University of Stuttgart, Stuttgart, Germany; R. Klingner, University of Texas at Austin, Austin, Tex.; and members of ACI Committee 355, Anchorage to Concrete. 486 REFERENCES 1. ACI Committee 318, “Building Code Requirements for Structural Concrete (ACI 318-05) and Commentary (318R-05),” American Concrete Institute, Farmington Hills, Mich., 2005, 430 pp. 2. ACI Committee 349, “Code Requirements for Nuclear Safety-Related Concrete Structures (ACI 349-01),” American Concrete Institute, Farmington Hills, Mich., 2001, 134 pp. 3. Fuchs, W.; Eligehausen, R.; and Breen, J. E., “Concrete Capacity Design (CCD) Approach for Fastening to Concrete,” ACI Structural Journal, V. 92, No. 1., Jan.-Feb. 1995, pp. 73-94. 4. Rehm, G.; Eligehausen, R.; and Mallée, R., “Befestigungstechnik” (Fastening Technique), Betonkalender 1995, Ernst & Sohn, Berlin, Germany, 1995. 5. Comité Euro-International du Beton, “Fastening to Reinforced Concrete and Masonry Structures,” State-of-the-Art Report, CEB, Thomas Telford, London, 1991, pp. 205-210. 6. ACI Committee 349, “Code Requirements for Nuclear Safety Related Concrete Structures (ACI 349-97),” American Concrete Institute, Farmington Hills, Mich., 1997, 123 pp. 7. Ozbolt, J.; Eligehausen, R.; Periskic, G.; and Mayer, U., “3D FE Analysis of Anchor Bolts with Large Embedment Depths,” Fracture Mechanics of Concrete Structures, V. 2, No. 5, Apr. 2004, Vail, Colo., pp. 845-852. 8. Shirvani, M.; Klingner, R. E.; and Graves III, H. L., “Behavior of Tensile Anchors in Concrete: Statistical Analysis and Design Recommendations,” ACI Structural Journal, V. 101, No. 6, Nov.-Dec. 2004, pp. 812-820. 9. Eligehausen, R., and Balogh, T., “Behavior of Fasteners Loaded in Tension in Cracked Reinforced Concrete,” ACI Structural Journal, V. 92, No. 3, May-June 1995, pp. 365-379. 10. Eurocode 2, “Design of Concrete Structures, Part 1: General Rules and Rules for Buildings,” 2004. 11. Technical Committee CEN/TC 250, “Design of Fastening for Use in Concrete, Part 2: Headed Fasteners,” Final Draft, CEN Technical Specifications, 2004. 12. Comité Euro-International du Beton (CEB), Design Guide for Anchorages to Concrete, Thomas Telford, London, 1997. ACI Structural Journal/July-August 2007

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