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- 1. State-of-the-Art Review on Nonlinear Inelastic Analysis for Steel Structures NRL Steel Lab., Sejong University
- 2. CONTENTS1. INTRODUCTION · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· · · · · · · ·12. NONLIEAR INELASTIC ANALYSIS · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ······3 2.1 Plastic-Zone Analysis· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· · · · · · · ·4 2.2 Quasi-Plastic Hinge Analysis· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·······6 2.3 Elastic-Plastic Hinge Analysis· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· · · · · · ·7 2.4 Notional-Load Hinge Analysis· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·······8 2.5 Refined-Plastic Hinge Analysis· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· · · · · ·93. NONLINEAR INELASTIC EXPERIMENTS· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· · · ·11 3.1 Kanchanalai’s Two-Bay Frames· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· · · · ·12 3.2 Yarimci’s Three-Story Frames· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· · · · · 12 3.3 Avery and Mahendran’s Large-scale testing of Steel Frame Structures· · · · · · · · · · · · · ·13 3.4 Wakabayashi’s One-Quarter Scaled Test of Portal Frames· · · · · · · · · · · · · · · · · · · · · · ·· 13 3.5 Harrison’s Space Frame Test· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· · · · ·14 3.5 Kim’s 3D Frame Test· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· · · · · 144. DESIGN USING NONLIEAR INELASTIC ANALYSIS· · · · · · · · · · · · · · · · · · · · · · · ·· ·15 4.1 Design Format· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· · · · · · · 15 4.2 Modeling Consideration· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· · · · · · 16 4.2.1 Sections· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· · · · · · · ·16 4.2.2 Structural members· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· · · · · 17 4.2.3 Geometric imperfection· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · i
- 3. · · · · · 17 4.2.4 Load· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· · · · · · · · 17 4.3 Design Consideration· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· · · · · · ·18 4.3.1 Load-carrying capacity· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· · · · · ·18 4.3.2 Resistance factor· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· · · · · ·19 4.3.3 Serviceability limit· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· · · · · · 19 4.3.4 Ductility requirement· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· · · · · · 20REFERENCES · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 211. INTRODUCTION The steel design methods used in the U.S. are Allowable Stress Design (ASD), Plastic Design(PD), and Load and Resistance Factor Design (LRFD). In ASD, the stress computation is based on afirst-order elastic analysis, and the geometric nonlinear effects are implicitly accounted for in themember design equations. In PD, a first-order plastic-hinge analysis is used in the structural analysis.Plastic design allows inelastic force redistribution throughout the structural system. Since geometricnonlinearity and gradual yielding effects are not accounted for in the analysis of plastic design, theyare approximated in member design equations. In LRFD, a first-order elastic analysis withamplification factors or a direct second-order elastic analysis is used to account for geometricnonlinearity, and the ultimate strength of beam-column members is implicitly reflected in the designinteraction equations. All three design methods require separate member capacity checks includingthe calculation of the K-factor. This design approach is marked in Fig. 1 as the indirect analysis and ii
- 4. design method. In the current AISC-LRFD Specification (AISC, 1994), first-order elastic analysis or second-order elastic analysis is used to analyze a structural system. In using first-order elastic analysis, thefirst-order moment is amplified by B1 and B2 factors to account for second-order effects. In theSpecification, the members are isolated from a structural system, and they are then designed by themember strength curves and interaction equations as given by the Specifications, which implicitlyaccount for the effects of second-order, inelasticity, residual stresses, and geometric imperfections(Chen and Lui, 1986). The column curve and beam curve were developed by a curve-fit to boththeoretical solutions and experimental data, while the beam-column interaction equations weredetermined by a curve-fit to the so-called "exact" plastic-zone solutions generated by Kanchanalai(1977). In order to account for the influence of a structural system on the strength of individualmembers, the effective length factor is used as illustrated in Fig. 2. The effective length method generally provides a good design of framed structures.However, several difficulties are associated with the use of the effective length method as follows:(1) The effective length approach cannot accurately account for the interaction between thestructural system and its members. This is because the interaction in a large structural system is toocomplex to be represented by the simple effective length factor K. As a result, this method cannotaccurately predict the actual required strengths of its framed members.(2) The effective length method cannot capture the inelastic redistributions of internal forces in astructural system, since the first-order elastic analysis with B1 and B2 factors accounts only forsecond-order effects but not the inelastic redistribution of internal forces. The effective lengthmethod provides a conservative estimation of the ultimate load-carrying capacity of a large structuralsystem.(3) The effective length method cannot predict the failure modes of a structural system subject to agiven load. This is because the LRFD interaction equation does not provide any information about 2
- 5. failure modes of a structural system at the factored loads.(4) The effective length method is not user-friendly for a computer-based design.(5) The effective length method requires a time-consuming process of separate member capacity checks involving the calculation of K-factors. With the development of computer technology, two aspects, the stability of separate members,and the stability of the structure as a whole, can be treated rigorously for the determination of themaximum strength of the structures. This design approach is marked in Fig. 1 as the direct analysisand design method (Kim and Chen, 1996a-b). The development of the direct approach to design iscalled “Advanced Analysis” or more specifically, “Second-Order Inelastic Analysis for FrameDesign.” In this direct approach, there is no need to compute the effective length factor, sinceseparate member capacity checks encompassed by the specification equations are not required. Withthe current available computing technology, it is feasible to employ nonlinear inelastic analysistechniques for direct frame design. This method has been considered impractical for design officeuse in the past. Over the past 20 years, extensive research has been made to develop and validate severalnonlinear inelastic analysis methods. The purpose of this paper is to review recent efforts to developvarious nonlinear inelastic analyses ranging from a simple elastic-plastic to rigorous plastic-zoneanalysis for frame design. Emphasis in this review is design application of nonlinear inelasticanalysis. This paper also summarizes reports of experimental studies to provide inelastic nonlinearbehavior of framed structures. The analysis and design principle using nonlinear inelastic analysisare also addressed.2. NONLINEAR INELASTIC ANALYSIS 3
- 6. Five different types of nonlinear inelastic analysis methods are discussed in the following: (1) Plastic-zone method (2) Quasi-plastic hinge method (3) Elastic-plastic hinge method (4) Notional-load plastic hinge method (5) Refined-plastic hinge method These different methods are based on the degree of refinement in representing the plasticyielding effects. The plastic-zone method uses the greatest refinement while the elastic-plastic hingemethod allows a drastic simplification. The quasi-plastic hinge method is somewhere in betweenthese two methods. The notional-load plastic hinge method and the refined-plastic hinge method arean improvement on the elastic-plastic hinge method for approximating real behavior of structures.The load-deformation characteristics of the plastic analysis methods are illustrated in Fig. 3, while thespread of plasticity is illustrated schematically in Fig. 4.2.1 Plastic-Zone Method In the plastic-zone method, frame members are discretized into finite elements, and the cross-section of each finite element is subdivided into many fibers shown in Fig. 5. The deflection at eachdivision point along a member is obtained by numerical integration. The incremental load-deflectionresponse at each loading step, which updates the geometry, captures the second-order effects. Theresidual stress in each fiber is assumed constant since the fibers are small enough. The stress state ateach fiber can be explicitly traced so the gradual spread of yielding can be captured. The plastic-zoneanalysis eliminates the need for separate member capacity checks since it explicitly accounts forsecond-order effects, spread of plasticity, and residual stress. As a result, the plastic-zone solution isknown as an "exact solution." The AISC-LRFD beam-column equations were established in partbased upon a curve-fit to the "exact" strength curves obtained from the plastic-zone analysis byKanchanalai (1977). 4
- 7. There are two types of plastic-zone analyses. The first involves the use of three-dimensionalfinite shell elements in which the elastic constitutive matrix in the usual incremental stress-strainrelations, is replaced by an elastic-plastic constitutive matrix when yielding is detected. Based on adeformation theory of plasticity, the effects of combined normal and shear stresses may be accountedfor. This analysis requires modeling of structures using a large number of finite three-dimensionalshell elements and numerical integration for the evaluation of the elastic-plastic stiffness matrix.The three-dimensional spread-of-plasticity analysis when combined with second-order theory whichdeals with frame stability is computational intensive and, therefore, best suited for analyzing small-scale structures, or if the detailed solutions for member local instability and yielding behavior arerequired. Since a detailed analysis of local effects in realistic building frames is not commonpractice in engineering design, this approach is considered too expensive for practical use. The second approach for second-order plastic-zone analysis is based on the use of beam-column theory, in which the member is discretized into line segments, and the cross-section of eachsegment is subdivided into finite elements. Inelasticity is modeled considering normal stress only.When the computed stress at the centroid of any fiber reaches the uniaxial normal strength of thematerial, the fiber is considered to have yielded. Also, compatibility is treated by assuming that fullcontinuity is retained throughout the volume of the structure in the same manner as elastic rangecalculations. Although quite sharp curvature may exist in the vicinity of inelastic portions of thestructure, “plastic hinges” can never develop. In plastic-zone analysis, the calculation of forces anddeformations in the structure after yielding requires an iterative trial-and-error process because of thenonlinearity of the load-deformation response, and the change in cross-section effective stiffness ininelastic regions associated with the increase in the applied loads and the change in structuralgeometry. Although most plastic-zone analysis methods have been developed for planar analyses(Clarke et al., 1992; White, 1985; Vogel, 1985; El-Zanaty et al., 1980; Alvarez and Birnstiel, 1967)three-dimensional plastic-zone techniques are also available (Wang, 1988; Chen and Atsuta, 1977). 5
- 8. A plastic-zone analysis that includes the spread of plasticity, residual stresses, initialgeometric imperfections, and any other significant second-order effects, would eliminate the need forchecking individual member capacities in the frame. Therefore, this type of method is classified asnonlinear inelastic inelastic analysis in which the checking of beam-column interaction equations isnot required. In fact, the member interaction equations in modern limit-states specifications weredeveloped, in part, by curve-fit to results from this type of analysis. In reality, some significantbehaviors such as joint and connection’s performances tend to defy precise numerical and analyticalmodeling. In such cases, a simpler method of analysis that adequately represents the significantbehavior would be sufficient for engineering application. Whereas the plastic-zone solution is regarded as an "exact solution," the method may not beused in daily engineering design, because it is too intensive in computation. Its applications arelimited to (ECCS, 1984): (1) The study of detailed structural behavior (2) Verifying the accuracy of simplified methods (3) Providing comparison with experimental results (4) Deriving design methods or generating charts for practical use (5) Applying for special design problems2. 2 Quasi-Plastic Hinge Method The quasi-plastic hinge method developed by Attala (1994) is an intermediate approachbetween the plastic-zone and the elastic-plastic hinge methods. It requires less computation but itsresults are very similar to those of plastic-zone method. For this reason, it is called a quasi-plastichinge method. An element, developed from equilibrium, kinematic, and constitutive relationships, accountsfor gradual plastification under combined bending and axial force. Inelastic force-strain model of 6
- 9. the cross-section is developed by fitting nonlinear equations to data of the moment-axial force-curvature response. Using the inelastic cross-section model, flexibility coefficients for the fullmember are obtained by successive integrations along its length. An inelastic-element stiffnessmatrix is obtained by the use of the incremental flexibility relationships. Initial yield and full plastification surface are used to analytically represent gradual yieldingeffect of the cross-section. Ketter’s residual stress pattern (1955) is used to determine an initial yieldsurface. Ketter’s pattern has peak compressive residual stresses at the flange tips equal to 0.3Fy witha linear transition of stress from the flange tips to the web-joint and constant tensile stress through theweb. A fully plastic surface is generated by calibration to a plastic-zone solution (Sanz-Picon, 1992).The parameters of the full plastification equation are determined by a curve-fit procedure. This method predicts strengths with an error less than 5% compared with the plastic-zonemethod for a wide range of case studies. The accuracy of this method is thus compatible with theplastic-zone method and less computational effort is necessary. However, it is difficult to extend this method to three-dimensional analysis since theformulation is based on flexibility relationships. As a result, it does not meet one of therequirements of Αnonlinear inelastic analysis≅ of the SSRC task force report (1993), which statesΑThe model should be readily extensible to three-dimensional analysis. That is, the framework ofthe model should accommodate the formulation of three-dimensional elements.≅ Moreover, thismodel does eliminate the necessity of the refined model through the cross-section but still requiresmany elements along the member.2. 3 Elastic-Plastic Hinge Method A more simple and efficient approach for representing inelasticity in frames is the elastic-plastic hinge method. It assumes that the element remains elastic except at its ends where zero-length plastic hinges form. This method accounts for inelasticity but not the spread of yielding or 7
- 10. plasticity at sections nor the residual stress effect between two plastic hinges. The elastic-plastic hinge methods may be divided into; first-order and second-order plasticanalyses. For first-order elastic-plastic hinge analysis, the nonlinear geometric effects are neglected,and not considered in the formulation of the equilibrium equations. As a result, the method predictsthe same ultimate load as conventional rigid-plastic analyses. In second-order elastic-plastic hinge analysis, the deformed structural geometry is considered.The simple way to account for the geometric nonlinearity is to use the stability function which enablesonly one beam-column element per a member to capture the second-order effect. This provides anefficient and economical method of frame analysis, and has a clear advantage over the plastic-zonemethod. This is particularly true for structures in which the axial force in component members issmall and the dominated behavior is bending. In such cases, second-order elastic-plastic hingeanalysis may be used to describe the inelastic behavior sufficiently, assuming that lateral-torsional andlocal buckling modes of failure are not prevented (Liew, 1992). The second-order elastic-plastic hinge analysis is only an approximate method. When usedto analyze a single beam-column element subject to combined axial load and bending moment, it mayoverestimate the strength and stiffness of the element in the inelastic range. Although elastic-plastichinge approaches provide essentially the same load-displacement predictions as plastic-zone methodsfor many frame problems, they may not be classified as nonlinear inelastic analysis methods ingeneral (Liew et al., 1994; Liew and Chen, 1991; White, 1993). However, research by Ziemian (Ziemian et al., 1990; Ziemian, 1990) has shown that theelastic-plastic hinge analysis can be classified as an advanced inelastic analysis since it is accurate formatching the strength and load-displacement response of several building frames from plastic-zoneanalysis. Many cases considered in Ziemian=s work, especially when the axial load is less than0.5Py, are not sensitive benchmarks for determining the accuracy and the possible limitations of theelastic-plastic hinge method. Therefore, suitable benchmark problems should be used to provide a 8
- 11. more in-depth study of the qualities and limitations of second-order elastic-plastic hinge methodbefore it can be accepted as a legitimate tool in the design of steel structures.For slender members whose dominant mode of failure is elastic instability, the method provides goodresults when compared with plastic-zone solutions. However, for stocky members with significantyielding, the plastic-hinge method over-predicts the actual strength and stiffness of members due tothe gradual stiffness reduction as the spread of plasticity increases in an actual member (Liew andChen, 1991; Liew et al., 1991; White et al., 1991). As a result, considerable refinements must bemade before it can be used for analysis of a wide range of framed structures.2. 4 Notional-Load Plastic-Hinge Method One approach to advance the use of second-order elastic-plastic hinge analysis for framedesign is to specify artificially large values of frame imperfections (i.e., initial out-of-plumbness).This is the approach adopted by EC3 (1990) for frame design using second-order analysis. Inaddition to accounting for the standard erection tolerance for out-of-plumbness, these artificial largeimperfections intend to account for the effect of residual stresses, frame imperfections, and distributedplasticity not considered in frame analysis. The geometric imperfections adopted by EC3 are amaximum out-of-plumbness of Ψ0 = 1/200 for an unbraced frame, but no maximum out-of-straightness value recommended for a braced member as shown in Fig. 6. The notional load plastic hinge approach is similar in concept to the “enlarged” geometricimperfection approach of the EC3. The ECCS (1984, 1991), the Canadian Standard (1989, 1994),and the Australian Standard (1990) allow to use this technique. The notional-load approach usesequivalent lateral loads to approximate the effect of member imperfections and distributed plasticity.In the ECCS, the exaggerated notional loads of 0.5 % times gravity loads are used to avoid over-predicting the strength of the member as does the elastic-plastic hinge method. The application ofthese notional loads to several example frames is illustrated in Fig. 7. Liew s research (1992) shows 9
- 12. that this method under-predicts the strength by more than 20% in the various leaning column framesand over-predicts the strength up to 10% in the isolated beam-columns subject to the axial forces andbending moments. As a result, modification of this approach is required before it may be used indesign applications.2. 5 Refined Plastic-Hinge Method In recent work by Abdel-Ghaffar et al. (1991), Al-Mashary and Chen (1991), King, et al.(1991), Liew and Chen (1991), Liew et al. (1993a-b), White et al. (1991), Kim (1996), Kim and Chen(1996), Chen and Kim (1997), Kim and Chen (1997), Kim et al (2000) and among others, an inelasticanalysis approach, based on simple refinements of the elastic-plastic hinge model, has been proposedfor plane frame analysis. It represents the effect of distributed plasticity through the cross-section,assuming that the plastic hinge stiffness degradation is smooth. The inelastic behavior of themember is modeled in terms of member force instead of the detailed level of stresses and strains asused in the plastic-zone analysis model. The principal merits of the refined-plastic hinge model arethat it is as simple and efficient as the elastic-plastic hinge analysis approach, and it is sufficientlyaccurate for the assessment of strength and stability of a structural system and its component members. The refined plastic-hinge method is based on simple modifications of the elastic-plastic hingemethod. Two modifications are made to account for the gradual section stiffness degradation at theplastic hinge locations as well as gradual member stiffness degradation between the two plastic hinges.Herein, the section stiffness degradation function is adopted to reflect the gradual yielding effect informing plastic hinges. Then, the tangent modulus concept is used to capture the residual stresseffect along the member between two plastic hinges. As a result, the refined plastic-hinge methodretains the efficiency and simplicity of the plastic hinge method without overestimating the strengthand stiffness of a member. In the recent work by Liew (1992), the LRFD tangent modulus is used to account for both the 10
- 13. effect of residual stresses and geometric imperfections. This model does not account for geometricimperfections when P/Py is less than 0.39, because the LRFD tangent modulus is identical to theelastic modulus in this range. As a result, the approach over-predicts the column strength by morethan 5% when KL/r of the column is greater than 85 for yield stresses at 36 ksi, and when KL/r of thecolumn is greater than 70 for yield stresses at 50 ksi. The LFRD Et may not be an appropriate modelto be used for nonlinear inelastic analysis (Kim, 1996; Kim and Chen, 1996). The CRC tangent modulus in Liews work (1992) only accounts for the effect of residualstresses. It over-predicts the strength of members by about 20% compared to the conventionalLRFD solutions, because the modulus does not account for the effect of geometric imperfections.However, in the CRC tangent modulus model, different members with different residual stresses canbe incorporated since the effect of geometric imperfections is considered separately. As a result,CRC tangent modulus is used in refined plastic analyses. Second-order inelastic analysis methods for the three-dimensional structure have beendeveloped by Orbison (1982), Prakash and Powell (1993), Liew and Tang (1998), Kim et al (2001),Kim and Choi (2001) and Kim et al (2001). Orbisons method is an elastic-plastic hinge analysiswithout considering shear deformations. The material nonlinearity is considered by the tangentmodulus Et and the geometric nonlinearity is by a geometric stiffness matrix. Orbisons method,however, underestimates the yielding strength up to 7% in stocky members subjected to axial forceonly. DRAIN-3DX developed by Prakash and Powell is a modified version of plastic hinge methods.The material nonlinearity is considered by the stress-strain relationship of the fibers in a section. Thegeometric nonlinearity caused by axial force is considered by the use of the geometric stiffness matrix,but the nonlinearity caused by the interaction between the axial force and the bending moment is notconsidered. This method overestimates the strength and stiffness of the member subjected tosignificant axial force. Liew and Tangs method is a refined plastic hinge analysis. The effect ofresidual stresses is taken into account in conventional beam-column finite element modelling. 11
- 14. Nonlinear material behavior is taken into account by calibration of inelastic parameters describing theyield and bounding surfaces. Liew and Tangs method, however, underestimates the yielding strengthup to 7% in stocky member subjected to axial force only. Against this background, it can be concluded that the refined-plastic hinge method strikes abalance between the requirements for realistic representation of frame behavior and for ease of use.It is considered that in both theses respects, the method is satisfactory for general practical use.3. NONLINEAR INELASTIC EXPERIMENTS Experimental studies to capture inelastic nonlinear behavior of framed structures aresummarized. The frames riviewed herein were tested by Kanchanalai(1977), Yarimci(1966),Avery(1999), Wakabayashi(1972), Harrison(1964) and Kim and Kang(2001).3.1 Kanchanalai’s Two-Bay Frames Three two-bay full-size frames were tested to verify the Plastic-zone analysis(Kanchanalai,1977). The dimensions and members of Frame 2 among these frames are shown in Fig. 8. Thematerial properties of the members are summerized in Table 1. The frames were designed to behaveequivalently to a one-story two-bay and could be tested on the floor. Supports were provided only atthe top and bottom of the interior column member. All frames were bent with respect to the weekaxis in order to avoid out-of-plane buckling. In Frame 2, all columns were loaded simultaneously upto about 70kips, corresponding to points 2-11 in Fig. 9. Then, only the axial load on the interiorcolumn was increased up to point 17, where the frame reached its instability limit load of 233.6 kips.Comparisons of the test results with the plastic zone theory are shown in Fig. 9. In general, goodagreements are observed. 12
- 15. 3.2 Yarimci’s Three-Story Frames An experimental research study was conducted at Lehigh University for three full-size frames(Yarimci, 1966). Fig. 10 shows dimensions and loads conditions of Frame C among the three frames.To investigate and compare the mechanical properties of the members with nominal values, Yarimciconducted a series of seven beam tests. The results of these tests are summarized in Table 2. Thebeams were welded to the columns and designed so as to behave elastically in the worst loadingcondition: the flexibility of the connections was eliminated from a factor which affects the strength ofthe frames. The frames were sandwiched and supported laterally by two parallel auxiliary framespreventing out-of-plane buckling. All members were bent in strong axis. The result of test isshown in Fig. 11 for Frame C. The load deflection behavior at the first and third story is shown inFig. 11.3.3 Avery and Mahendran’s Large-Scale Testing of Steel Frame Structures A series of four tests was conducted by Avery and Mahendran(1999). Each of the fourframes could be classified as a two-dimensional, single-bay, single-story, large-scale sway frame withfull lateral restraint and rigid joints, as shown in Fig. 12. In Frame 2, Non-compact I-sections(310UB32.0) of Grade 300 steel(nominal yield stress=320MPa)was used. This section wasselected as one of the standard hot-rolled I-sections mostly affected by local buckling. Thedimensions, material properties, and section properties used in Frames 2 are listed in Table 3. Thevertical and horizontal loads were applied simultaneously in a ratio of approximately four timesgreater than the horizontal reaction measured by the load cell. The frame failed by in-plane instabilitydue to a reduced stiffness caused by yielding and P-Δ effect. The horizontal reaction force and themeasured relative in-plane horizontal displacement of the right hand column for test Frame 2 arerelated in Fig. 13. 13
- 16. 3.4 Wakabayashi’s One-Quarter Scaled Test of Portal Frames Two-series of test were conducted for a one-story frame and a two-story frame byWakabayashi et al(1972). Configurations of the two-story frame are shown in Fig. 14. Thenominal dimensions of members are H-100×100×6×8 for columns and H-100×50×4×6 for beams.The specimens consist of rolled H-shapes. The connections were welded and stiffened to preventlocal buckling in the joint panels. To prevent the out-of-plane buckling, two of the same specimenswere set in parallel and connected at the joints and the mid-length of the members. In the otherwords, twin specimens were tested simultaneously. Measured Material and sectional properties ofmembers are listed in Table 4. The vertical load was first applied at the top of four columns by a fixed testing machine.The parallel twin specimens were loaded simultaneously. Then, the horizontal load at the top offrame was increased gradually. When the frame swayed by the horizontal loading jack followed ahorizontal movement so that vertical loading points could be kept on the center of the columns. Theloads were measured by the load cells which were installed between the hydraulic jacks and thespecimen. The load-deflection curves of the two-story frames are shown in Fig. 15. Comparisons of aseries of test show the effects of axial force and stiffness of the beam on the frame behavior. Thelarger the axial force in columns and the smaller the stiffness of the beam, the more unstable theframes become.3.5 Harrison’s Space Frame Test The equilateral triangular space frame depicted in Fig. 3 was tested by Harrison(1964) in theJ.W.Roderick Laboratory for Materials and Structures at the University of Sydney. Configuration of theframe is shown in Fig. 16. Measured dimensions and material properties are listed in Table 5. A 14
- 17. horizontal load(H) is applied on the top of the column and a vertical load of 1.3H is applied at midspan of the beam. It can be seen from Fig. 17 that, compared to the experimental results, the plastic-zoneanalysis predicted a slightly stiffer response of the space frame under the applied loads. As thecolumn bases of the space frame were welded to steel plates clamped to steel joists(Harrison 1964),the more flexible response measured in the laboratory test might have been caused by the flexibility ofthe joist flanges.3.6 Kim’s 3D Frame Test Two-series of test were conducted for space steel frame subjected proportional loads shownin Fig 18 and space steel frame subjected proportional loads shown in Fig. 19 by Kim and Kang(2001). Hot-rolled I-section was used for all three frames. Nominal dimension ofthe section was H-150×150×7×10 commonly used in Korea. The dimensions and properties of thesection are listed in Table 6. The section is compact so that it is not susceptible to local buckling. For proportional loads test, The vertical loads were applied on the top of the four columns,and the horizontal loads were applied on the column ② and ④ at the second floor level of the testframe. The vertical loads were slowly increased until the system could not resist any more loads.The horizontal loads were automatically increased according to the specified load ratio for each testframe controlled by the computer system. For non-proportional loads test, The vertical loads were applied on the top of the fourcolumns, and the horizontal load was applied on the column ② at the second floor level of the testframe. The vertical loads were first increased 680 kN and maintained during the experiment. Thehorizontal load was slowly increased until the test frame could not resist any more loads. Fig. 20. and Fig. 21. show load-displacement curve for test frames. The obtained resultsfrom 3D non-linear analysis and AISC-LRFD method were compared with experimental data.ABAQUS, one of mostly widely used and accepted commercial finite element analysis program, was 15
- 18. used. Load carrying capacities obtained by the experiment and AISC-LRFD method are comparedin Table 7 and 8. The results showed that the AISC-LRFD capacities were approximately 25 percentconservative for frame subjected to proportional loads test and 28 percent conservative for non-proportional loads test. This difference is derived from the fact that the AISC-LRFD approach doesnot consider the inelastic moment redistribution, but the experiment includes the inelasticredistribution effect.4. DESIGN USING NONLINEAR INELASTIC ANALYSIS4.1 Design Format Nonlinear inelastic analysis follows the format of Load and Resistance Factor Design. InAISC-LRFD(1994), the factored load effect does not exceed the factored nominal resistance ofstructure. Two kinds of factors are used: one is applied to loads, the other to resistances. The loadand resistance factor design has the format η ∑ γ iQi ≤ φ Rn (1)where Rn = nominal resistance of the structural member, Qi = force effect, φ = resistancefactor, γ i = load factor corresponding to Qi , η = a factor relating to ductility, redundancy, andoperational importance.The main difference between current LRFD method and nonlinear inelastic analysis method is that theright side of Eq. (1), ( φ Rn ) in the LRFD method is the resistance or strength of the component of astructural system, but in the nonlinear inelastic analysis method, it represents the resistance or the 16
- 19. load-carrying capacity of the whole structural system. In the nonlinear inelastic analysis method, theload-carrying capacity is obtained from applying incremental loads until a structural system reachesits strength limit state such as yielding or buckling. The left-hand side of Eq. (1), ( η ∑γ Q ) i irepresents the member forces in the LRFD method, but the applied load on the structural system in thenonlinear inelastic analysis method.4.2 Modeling Consideration4.2.1 Sections The AISC-LRFD Specification uses only one column curve for rolled and welded sections ofW, WT, and HP shapes, pipe, and structural tubing (AISC, 1994). The Specification also uses sameinteraction equations for doubly and singly symmetric members including W, WT, and HP shapes,pipe and structural tubing, even though the interaction equations were developed on the basis of Wshapes by Kanchanalai (1977). The proposed analysis was developed by calibration with the LRFD column curve. To thisend, it is concluded that the proposed methods can be used for various rolled and welded sectionsincluding W, WT, and HP shapes, pipe, and structural tubing without further modifications.4.2.2 Structural members An important consideration in making this nonlinear inelastic analysis practical is therequired number of elements for a member in order to predict realistically the behavior of frames. Asensitivity study of nonlinear inelastic analysis for two-dimensional frames was performed on therequired number of elements (Kim and Chen, 1998). Two-element model adequately predicted the 17
- 20. strength of a two-dimensional member. This rule may be used for modeling a three-dimensionalmember.4.2.3 Geometric imperfection The magnitudes of geometric imperfections are selected as ψ = 2 1,000 for unbracedframes and ψ = 1 1, 000 for braced frames. To model a parabolic out-of-straightness in the member,two-element model with maximum initial deflection at the mid-height of a member adequatelycaptures imperfection effects. It is concluded that practical nonlinear inelastic analysis iscomputationally efficient. The pattern of geometric imperfections is assumed to be the same as theelastic first order deflected shape.4.2.4 Load1) Proportional loading In the proposed nonlinear inelastic analysis, the gravity and lateral loads should be appliedsimultaneously, since it does not account for unloading. As a result, the method under-predicts thestrength of frames subjected to sequential loads, large gravity loads first and then lateral loads. It is,however, justified for the practical design since the development of the LRFD interaction equationswas also based on strength curves subjected to simultaneous loading and the current LRFD elasticanalysis uses the proportional loading rather than the sequential loading.2) Incremental loading It is necessary, in an nonlinear inelastic analysis, to input each increment load (not the totalloads) to trace nonlinear load-displacement behavior. The incremental loading process can beachieved by scaling down the combined factored loads by a number between 20 and 50. For a 18
- 21. highly redundant structure, dividing by about 20 is recommended and for a nearly staticallydeterminate structure, the incremental load may be factored down by 50. One may choose a numberbetween 20 and 50 to reflect the redundancy of a particular structure. Since a highly redundantstructure has the potential to form many plastic hinges and the applied load (i.e. the smaller scalingnumber) may be used.4.3 Design Consideration4.3.1 Load-carrying capacity The elastic analysis method does not capture the inelastic redistribution of internal forcesthroughout a structural system, since the first-order forces, even with the B1 and B2 factors,account for the second-order geometric effect but not the inelastic redistributions of internal forces.The method may provide a conservative estimation of the ultimate load-carrying capacity. Nonlinearinelastic analysis, however, directly considers force redistribution due to material yielding and thusallows smaller member sizes to be selected. This is particularly beneficial in highly indeterminatesteel frames. Because consideration at force redistribution may not always be desirable, the twoapproaches (including and excluding inelastic force redistribution) can be used. First, the load-carrying capacity, including the effect of inelastic force redistribution, is obtained from the finalloading step (limit state) given by the computer program. Secondly, the load-carrying capacitywithout the inelastic force redistribution is obtained by extracting that force sustained when the firstmember yield or buckled. Generally, nonlinear inelastic analysis predicts the same member size as theLRFD method when force redistribution is not considered.4.3.2 Resistance factor 19
- 22. AISC-LRFD specifies the resistance factors of 0.85 and 0.9 for axial and flexural strength ofa member, respectively. The proposed method uses a system-level resistance which is different fromAISC-LRFD specification using member level resistance factors. When a structural system collapsesby forming plastic mechanism, the resistance factor of 0.9 is used since the dominent behavior isflexure. When a structural system collapses by member buckling, the resistance factor of 0.85 is usedsince the dominent behavior is compression.4.3.3 Serviceability limit According to the ASCE Ad Hoc Committee on Serviceability report (Ad Hoc Committee,1986), the normally accepted range of overall drift limits for building is 1 750 to 1 250 times thebuilding height, H , with a typical value of H 400 . The general limits on the interstory drift are1 500 to 1 200 times the story height. Based on the studies by the Ad Hoc Committee (1986), andby Ellingwood (1989), the deflection limits for girder and story are selected as• Floor girder live load deflection : H 360• Roof girder deflection : H 240• Lateral drift : H 400 for wind load• Interstory drift : H 300 for wind loadAt service load levels, no plastic hinges are allowed to occur in order to avoid permanentdeformations under service loads.4.3.4 Ductility requirement Adequate rotation capacity is required for members to develop their full plastic momentcapacity. This is achieved when members are adequately braced and their cross-sections are compact. 20
- 23. The limits for lateral unbraced lengths and compact sections are explicitly defined in AISC-LRFD(1994).REFERENCESAbdel-Ghaffar, M., White, D. W., and Chen, W. F. (1991). “Simplified second-order inelasticanalysis for steel frame design.” Special Volume of Session on Approximate Methods andVerification Procedures of Structural Analysis and Design, Proceedings at Structures Congress 91,ASCE, New York, 47-62.Ad Hoc Committee on Serviceability, Structural serviceability (1986). A critical appraisal andresearch needs, ASCE, J. Struct. Eng., 112(12); 2646-2664.Al-Mashary, F. and Chen, W. F. (1991). “Simplified second-order inelastic analysis for steel frames.”J. Inst. Struct. Eng., 69(23), 395-399.AISC (1994). Load and Resistance Factor Design Specification, American Institute of SteelConstruction, 2nd Ed., Chicago.Alvarez, R. J. and Birnstiel, C. (1967). “Elasto-plastic analysis of plane rigid frames, school ofengineering and science.” Department of Civil Engineering, New York University, New York. 21
- 24. Attala, M. N., Deierlein, G. G., and McGuire, W. (1994). “Spread of plasticity: quasi-plastic-hingeapproach.” J. Struct. Engrg., ASCE, 120(8), 2451-2473.Avery, P. and Mahendran, M. (2000). “Large-scale testing of steel frame structures comprising non-compact sections.” Engrg. Struct., 22, 920-936.Chen, W. F. and Atsuta, T. (1977). “Theory of beam-columns, vol. 2, space behavior and design.”McGraw-Hill, New York, 732 pp.Chen W.F. and Kim, S. E.(1997). “LRFD steel design using advanced analysis.”, CRC Press, BocaRaton, Florida.Chen, W.F. and Lui, E. M.(1986). “Structural stability-theory and implementation.” Elsevier, NewYork, 490pp.Clarke, M. J., Bridge, R. Q., Hancock, G. J., and Trahair, N. S. (1992). benchmarking and verificationof second-order elastic and inelastic frame analysis programs in SSRC TG 29 workshop andmonograph on plastic hinge based methods for advanced analysis and design of steel frames, White,D.W. and Chen, W.F., Eds., SSRC, Lehigh University, Bethlehem, PA.CSA (1989). Limit States Design of Steel Structures, CAN/CSA-S16.1-M89, Canadian StandardsAssociation.CSA (1994). Limit States Design of Steel Structures, CAN/CSA-S16.1-M94, Canadian StandardsAssociation.Ellingwood (1989). “Limit states design of steel structures.”, AISC Engineering Journal, 26, 1stQuarter, 1-8.EC3 (1990). Design of Steel Structures: Part I - General Rules and Rules for Buildings, Vol. 1,Eurocode edited draft, Issue 3.ECCS (1984). Ultimate Limit State Calculations of Sway Frames with Rigid Joints, TechnicalCommittee 8 - Structural Stability Technical Working Group 8.2 - System, Publication No. 33, 20 pp.ECCS (1991). Essentials of Eurocode 3 Design Manual for Steel Structures in Buildings, ECCS-Advisory Committee 5, No. 65, 60 pp. 22
- 25. El-Zanaty, M., Murray, D., and Bjorhovde, R. (1980). “Inelastic behavior of multistory steel frames.”Structural Engineering Report No. 83, University of Alberta, Alberta, Canada.Harrison, H. B. (1964). “The Application of the principles of plastic analysis to three dimentionalsteel structures.”, Ph.D thesis, Department of Civil Engineering, University of sydney.Kanchanalai, T. (1977). “The design and behavior of beam-columns in unbraced steel frames.” AISIProject No. 189, Report No. 2, Civil Engineering/Structures Research Lab., University of Texas atAustin, 300 pp.Ketter, R. L., Kaminsky, E.L., and Beedle, L.S. (1955). “Plastic deformation of wide-flange beamcolumns.” Transactions, ASCE, 120, 1028-1069.Kim, S. E. (1996). “Practical advanced analysis for steel frame design.” Ph.D Dissertation, School ofCivil Engineering, Purdue University, West Lafayette, IN, May, 271 pp.Kim, S. E. and Chen, W. F. (1996). “Practical advanced analysis for steel frame design.” The ASCEStructural Congress XIV Special Proceedings Volume on Analysis and Computation, Chicago,IL,April, 19-30.Kim, S.E. and Chen, W.F. (1996a) "Practical advanced analysis for braced steel frame design", ASCEJ. Struct. Eng., 122(11): 1266-1274.Kim, S.E. and Chen, W.F. (1996b) "Practical advanced analysis for unbraced steel frame design",ASCE J. Struct. Eng., ASCE, 122(11): 1259-1265.Kim, S.E. and Chen, W.F. (1997) "Further studies of practical advanced analysis for weak-axisbending", Engrg. Struct., Elsevier, 19(6): 407-416.Kim, S.E. and Chen, W.F. (1998). "A sensitivity study on number of elements in refined plastic-hingeanalysis", Computers and Structures, 66(5), 665-673.Kim, S. E., Park, M. H., Choi, S. H. (2000). "Improved refined plastic-hinge analysis accounting forstrain reversal.", Engineering Structures, 22(1), 15-25.Kim, S.E. and Choi, S.H.(2001). "Practical advanced analysis for semi-rigid space frames.", Solidsand Structures, Elsevier Science, 38(50-51), 9111-9131. 23
- 26. Kim, S.E., Park, M.H., Choi, S.H. (2001) "Direct design of three-dimensional frames using practicaladvanced analysis", Engineering Structures, 23(11), 1491-1502.Kim, S.E., Kim, Y. and Choi, S.H.(2001) “Nonlinear analysis of 3-D steel frames.”, Thin-walledStructures, Elsevier Science, 39(6), 445-461.Kim, S.E. and Kang, K.W.(2001). “Large-scale testing of space steel frame subjected to non-proportional loads, Solids and Structures, Submitted.Kim, S.E. and Kang, K.W.(2001). “Large-scale testing of space steel frame subjected to proportionalloads, Solids and Structures, Engrg. Struct., Elsevier, AcceptedKing, W. S., White, D. W., and Chen, W. F. (1991). “On second-order inelastic methods for steelframe design.” J. Struct. Engrg, ASCE, 118(2), 408-428.Liew, J. Y. R. (1992). “Advanced analysis for frame design.” Ph.D. Dissertation, School of CivilEngineering, Purdue University, West Lafayette, IN, May, 393 pp.Liew, J. Y. R. and Chen, W. F. (1991). “Refining the plastic hinge concept for advancedanalysis/design of steel frames.” Journal of Singapore Structural Steel Society, Steel Structure, 2(1),13-30.Liew, J. Y. R., White, D. W., and Chen, W. F.(1991). “Beam-column design in steel frameworks-insight on current methods and trends.” J. Constr. Steel Res., 18, 269-308.Liew, J. Y. R., White, D. W., and Chen, W. F. (1993a). “Second-order refined plastic hinge analysisfor frame design: Part I.” J. Struct. Engrg., ASCE, 119 (11), 3196-3216.Liew, J. Y. R., White, D. W., and Chen, W. F. (1993b). “Second-order refine plastic hinge analysisfor frame design: Part II.” J. Struct. Engrg., ASCE, 119 (11), 3217-3237Liew, J.Y.R. and Tang, L.K. (1998) "Nonlinear refined plastic hinge analysis of space framestructures", Research Report No. CE027/98, Department of Civil Engineering, National University ofSingapore, Singapore.Orbison, J.G. (1982) "Nonlinear static analysis of three-dimensional steel frames", Report No. 82-6,Department of Structural Engineering, Cornell University, Ithaca, New York. 24
- 27. Prakash, V. and Powell, G.H. (1993) "DRAIN-3DX: Base program user guide, version 1.10", AComputer Program Distributed by NISEE / Computer Applications, Department of Civil Engineering,University of California, Berkeley.Sanz-Picon, C.F.(1992). “Behavior of composite column cross sections under biaxial bending.” MSThesis, Cornell University, Ithaca, New York.SSRC (1993). Plastic hinge based methods for advanced analysis and design of steel frames, AnAssessment of the State-of-the-Art, White, D.W. and Chen, W.F., Eds., SSRC, Lehigh University,Bethlehem, PA, 299pp.Standards Australia (1990). AS4100-1990, Steel Structures, Sydney, Australia.Vogel, U. (1985). “Calibrating frames” Stahlbau, 10, 1-7.Wakabayashi, M.,and Matsui, C. (1972a). “Elastic-plastic behaviors of full size steel frame.”, Trans.Arch. Inst. Jpn., 198, 7-17Wang, Y. C. (1988). “Ultimate strength analysis of 3-D beam columns and column subassemblageswith flexible connections.” Ph.D. Thesis, University of Sheffield, England.White, D.W., Liew, J. Y. R., and Chen, W. F.(1991). “Second-order inelastic analysis for framedesign.”, A report to SSRC Task Group 29 on Recent Research and the Percieved State-of-art,Structural Engineering Report, CE-STR-91-12, Purdue University, West Lafayette, IN. 116pp.White, D. W. (1993). “Plastic hinge methods for advanced analysis of steel frames.”, J. Constr. SteelRes., 24(2), 121-152pp.White, D. W. (1985). “Material and geometric nonlinear analysis of local planar behavior in steelframes using iterative computer graphics.” M.S. Thesis, Cornell University, Ithaca, NY, 281 pp.Yarimci, E. (1966). “Incremental inelastic analysis of framed structures and some experimentalverification”, Ph.D. dissertation, Department of Civil Engineering, Lehigh University, Bethlehem, PA.Ziemian, R. D.(1990). “Advanced methods of inelastic analysis in the limit states design of steelstructures.”, Ph.D. Dissertation, School of Civil and Environmental Engineering, Cornell University,Ithaca, NY 265pp. 25
- 28. Ziemian, R. D., White, D.W., Deierlein, G. G., and Mcquire, W.(1990). “One approach to inelastic analysis and design, Proceedings of the 1990 National Steel Coferences.”, AISC, Chicago, 19.1-19. TABLE 1. Summary of Tension Coupon Tests Member σy ,* y ,st Est σult Elongation Section Specimen number ksi ×10-5 ×10-5 ksi ksi in 8 in, % W8×17 Flange 37.9 128 1140 442 62.4 28.2 C1A (A36- Flange 37.7 127 1378 356 - 29.7 C1C 70A) Web 40.6 137 2450 345 61.7 32.9 M4×13 Flange 48.5 164 1203 406 69.6 26.6 B1,B2 (A572- Flange 48.6 164 1062 399 69.9 27.2 B3,B4 73) Web 50.1 169 2228 323 69.5 26.7C1B and C2B were not tested,*y= Φy/E(E=29,500ksi) TABLE 2. Measured Properties of Beam and Column Section Handbook Measured Handbook Measured Frame Section EI EI Mp MP 2 4 (kip-in ×10 ) (kip-in2×104) (kip-in) (kip-in) C 12B16.5 310 271 742 845 C 10B15 203 190 576 635 C 6WF15 158 165 686 760 TABLE 3. Dimensions and Properties of Members Test Section D br tr tw r1 Ag I S σy 26
- 29. frame (mm) (mm) (mm) (mm) (mm) (mm2) (106mm4) (104mm3) Flange Web 2 310UB32 298 149 8.0 5.5 13.0 4080 63.2 475 360 395TABLE 4. Actual Section Properties of One-Quarter Scaled Frames A I Z Zp σy (cm2) (cm4) (cm3) (cm3) (t/cm2) Column 21.8 391 77.4 88.5 2.64 Beam 10.6 177 35.0 40.6 3.04TABLE 5. Dimensions and Material Properties of Equilateral Triangular Space Frame L D T E G σy(ksi) (in) (in) (in) (ksi) (ksi) Column Beam All members 48 1.682 0.176 28800 11520 30.6 31.1TABLE 6. Dimensions and Properties of Section H-150 × 150 × 7 × 10 Used in the Frame Moment of Moment of Thickness Thickness Radius of Axial Area Height Width Inertia about X Inertia about Y of Flange of Web Fillet Axis Axis H B tf Ag tw r1 (mm ) IX IY (mm) (mm) (mm) (mm) (mm) (10 mm ) (10 ) 2 6 4 6 mm 4Nominal 150 150 10 7 11 4014 16.40 5.63Measured Column 152.3 149.9 10.2 6.75 - 4053 17.20 5.74 Beam 149.1 150.0 9.2 6.50 - 3713 15.14 5.18TABLE 7. Comparison of Experimental and Design Load Carrying Capacity (a) Experiment (b) Analysis (c) AISC-LRFD design (b)/(a) (c)/(a) P 612.0 612.0 443.5 1.0000 0.7247 H 169.2 175.5 122.6 1.0372 0.7246TABLE 8. Comparison of Experimental and Design Load Carrying Capacities 27
- 30. (a) Experiment (b) Analysis (c) AISC-LRFD design (b)/(a) (c)/(a) P 681.8 680.9 510.2 0.9985 0.7483Test frame H1 136.4 136.2 102.0 0.9984 0.7481 3 H2 67.5 68.1 51.0 1.0083 0.7556 FIG. 1. Analysis and Design Method 28
- 31. FIG. 2. Interaction between A Structural System and Its Component Members FIG. 3. Load-Deformation Characteristics of Plastic Analysis Methods 29
- 32. FIG. 4. Concept of Spread of Plasticity for Various Advanced Analysis Methods FIG. 5. Model of Plastic-Zone Analysis 30
- 33. FIG. 6. Explicit Imperfection Model for Elastic-Plastic Analysis Recommended By ECCSFIG. 7. Examples on Application of Notional Loads for Second-Order Elastic-Plasic Hinge Analysis 31
- 34. FIG. 8. Two-Bay FrameFIG. 9. Axial Load-Deflection Behavior of Specimen 32
- 35. FIG. 10. Specimen for Three-Story FrameFIG. 11. Lateral Load-Sway Behaviour of Frame C 33
- 36. FIG. 12. Schematic Diagram of Test ArrangementFIG. 13. Sway Load-Deflection Curve for Test Frame 2 34
- 37. FIG. 14. One-Quarter Scaled Frames.(From Wakabayashi, M. And Matsui, C., Trans. Arch. Inst. Jpn. 193,17,1972, With Permission) 35
- 38. FIG. 15. Horizontal Force-Displacement Behaviours of One-Quarter Scaled Frame.(Two Story).(From Wakabayashi, M. And Matsui, C., Trans. Arch.Inst. Jpn. 193,17,1972, With Permission) 36
- 39. FIG. 16. Harrison’s Space Frame(Harrison 1964)FIG. 17. Load-Deflection for Harrison’s Space Frame 37
- 40. P P ad Vertical lo P P Roof 2.20m H1 tal load Horizon 2nd floor Z H2 ① 1.76m Y ② ③ X 2.5 Base m ④ 3.0m FIG. 18. Dimension and Loading Condition of Test Frame P P ad Vertical lo P P Roof 2.20m H ta l load Horizon 2nd floor Z ① 1.76m Y ② ③ X 2.5 Base m ④ 3.0mFIG. 19. Dimensions and Loading Conditions of Test Frame in Main Test 38
- 41. 200 160 Horizontal load (kN) 120 80 Experiment(H1) Analysis(H1) Experiment(H2) 40 Analysis(H2) 0 0 10 20 30 40 Horizontal displacement (mm)FIG. 20. Comparison of Horizontal Load-Displacement Curves for Space Test Frame 2 FIG. 21. Horizontal Load-Displacement Curve for Test Frame (Column ②) 39

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