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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

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