3-D second-order plastic-hinge analysis accounting for local buckling

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1 Engineering Structures 25 (2003) D second-order plastic-hinge analysis accounting for local buckling Seung-Eock Kim, Jaehong Lee, Joo-Soo Park Department of Civil and Environmental Engineering, Construction Tech. Research Institute, Sejong University, Seoul , South Korea Received 15 March 2002; received in revised form 10 July 2002; accepted 17 July 2002 Abstract In this paper, 3-D second-order plastic-hinge analysis accounting for local buckling is developed. This analysis accounts for material and geometric non-linearities of the structural system and its component members. The problem associated with conventional second-order plastic-hinge analyses, which do not consider the degradation of the flexural strength caused by local buckling, is overcome. Efficient ways of assessing steel frame behavior including gradual yielding associated with residual stresses and flexure, second-order effect, and geometric imperfections are presented. In this study, a model containing the width thickness ratio is used to account for local buckling. The proposed analysis is verified by the comparison with other analyses and Load Resistance Factor Design results. A case study shows that local buckling is a very crucial element to be considered in second-order plastic-hinge analysis. The proposed analysis is shown to be an efficient, reliable tool ready to be implemented into design practice Elsevier Science Ltd. All rights reserved. Keywords: Plastic hinge; Second-order analysis; Local buckling; Steel frame; Load resistance factor design 1. Introduction In current engineering practice, the interaction between the structural system and its component members is represented by the effective length factor. The effective length method generally provides a good design for framed structures. However, despite its popular use in the past and present as a basis for design, the approach has its major limitations. The first of these is that it does not give an accurate indication of the factor against failure, because it does not consider the interaction of strength and stability between the member and structural system in a direct manner. It is a well-recognized fact that the actual failure mode of the structural system often does not have any resemblance whatsoever to the elastic buckling mode of the structural system that is the basis for the determination of the effective length factor K. The second and perhaps the most serious limitation is probably the rationale of the current two-stage process in design: elastic analysis is used to determine Corresponding author. address: sekim@sejong.ac.kr (S.-E. Kim). the forces acting on each member of a structural system, whereas inelastic analysis is used to determine the strength of each member treated as an isolated member. There is no verification of the compatibility between the isolated member and the member as part of a frame. The individual member strength equations as specified in specifications are unconcerned with system compatibility. As a result, there is no explicit guarantee that all members will sustain their design loads under the geometric configuration imposed by the framework. In order to overcome the difficulties of the conventional approach, second-order plastic-hinge analysis should be directly performed. With the currently available computing technology and with the advancements in computer hardware and software, it is feasible to employ second-order plastic analysis techniques for direct frame design. Most of second-order plastic analyses can be categorized into one of two types: (1) Plastic zone; or (2) Plastic hinge based on the degree of refinements used to represent yielding. The plastic zone method uses the highest refinements while the elasticplastic hinge method allows for significant simplifications. One of the second-order plastic analyses called the plastic-zone method discretizes frame members /03/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S (02)

2 82 S.-E. Kim et al. / Engineering Structures 25 (2003) Nomenclature A, L area and length of beam column element E modulus of elasticity E t CRC tangent modulus F L F y F R, in ksi F R compressive residual stress in flange, in ksi G shear modulus of elasticity of steel h depth of composite column section h c twice the distance from the centroid to the inside face of the compression flange I y,i z moment of inertia with respect to y and z axes J torsional constant k c,k s coefficients accounting for situation where a large number of columns in a storey and storeys in a frame would reduce the total magnitude of geometric imperfections k iiy, k ijy, k jjy stiffness accounting for h A, h B with respect to y axis k iiz, k ijz, k jjz stiffness accounting for h A, h B with respect to z axis M n local buckling strength M y,m z second-order bending moment with respect to y and z axes M yp,m zp plastic moment capacity with respect to y and z axes M p plastic moment capacity M ya, M yb, M za, M zb end moments with respect to y and z axes P second-order axial force P y squash load r 1,r 2 factors which account for the length and number of columns r y radius of gyration about y axis S x section modulus about x axis S xc elastic section modulus referred to the compression side S 1,S 2,S 3,S 4 stability functions with respect to y and z axes T torsional force t f flange thickness t w web thickness α force-state parameter δ axial displacement η, h A, h B stiffness degradation function at element end A and B, respectively q ya, q yb, q za, q zb the joint rotations f angle of twist into several finite elements. Also, the cross-section of each finite element is further subdivided into many fibers [20,10,5]. Although the plastic zone solution is known as an exact solution, it has yet to be used for practical design purposes. The applicability of the method is limited by its complexity, requiring intensive computational time and cost. The real challenge in our endeavor is to make this type of analysis competitive in present construction engineering practices. A more simple and efficient way to represent inelasticity in frames is the second-order plastic-hinge method. Until now, several second-order plastic-hinge analyses for space structures were developed by Ziemian et al. [21], Prakash and Powell [18], Liew and Tang [14], and Kim et al. [13]. The benefit of the second-order plastichinge analyses is that they are efficient and sufficiently accurate for the assessment of strength and stability of structural systems and their component members. These conventional 3-D second-order plastic-hinge analyses assume the section to be compact, and do not account for the degradation of the flexural strength caused by local buckling. Since the sections of real structures are not always compact, the analysis should be improved to consider local buckling. When the conventional 3-D second-order plastic-hinge analyses does account for local buckling, it will be a considerable contribution to present engineering practices. The objective of this paper is to achieve the accuracy of a plastic zone solution with the ease of the plastic hinge model in capturing the effect of local buckling. In this paper, warping torsion is ignored since the warping stiffness of rolled sections is not significant compared with bending stiff-

3 S.-E. Kim et al. / Engineering Structures 25 (2003) ness. Axial stiffness degradation associated with bending moments, i.e., bowing effect, is ignored since it is not significant D second-order plastic-hinge analysis 22cos(pr y )pr y sin(pr y ) prysin(pry)p2rcos(pry) p 2 r y cosh(pr y )pr y sinh(pr y ) 22cosh(pr y ) pr y sinh(pr y ) 2.1. Stability functions accounting for second-order effect To capture second-order (large displacement) effects, stability functions are used to minimize modeling and solution time. Generally only one or two elements are needed per member. The simplified stability functions for the two-dimensional beam column element were reported by Chen and Lui [3]. The force-displacement equation using the stability functions may be extended for three-dimensional beam-column elements as P T M ya M yb M za M zb (1) EA L EI y 0 S 1 L S EI y 2 L EI y 0 S 2 L S EI y 1 L EI z S 3 L S EI z 4 L 0 EI z S 4 L S EI z 3 L GJ L d q ya q yb q za q zb f where P = axial force, M ya, M yb, M za, M zb end moments with respect to y and z axes, T = torsional force, δ=axial displacement, q ya, q yb, q za, q zb joint rotations, f= angle of twist, S 1,S 2,S 3,S 4 = stability functions with respect to y and z axes, A, L = area and length of beam-column element, I y,i z =moment of inertia with respect to y and z axes, E = modulus of elasticity, G = shear modulus of elasticity, J = torsional constant. The stability functions given by Eq. (1) may be written as S 1 (2a) S 2 S 3 p 2 r y pr y sin(pr y ) 22cos(pr y )pr y sin(pr y ) pr y sinh(pr y )p 2 r y 22cosh(pr y ) pr y sin(pr y ) 22cos(pr z )pr z sin(pr z ) przsin(prz)p2rcos(prz) p 2 rcosh(pr z )pr z sinh(pr z ) S 4 22cosh(pr z ) pr z sinh(pr z ) p 2 r z pr z sin(pr z ) 22cos(pr z )pr z sin(pr z ) pr z sinh(pr z )p 2 r z 22cosh(pr z ) pr z sin(pr z ) ifp 0 (2b) (2c) (2d) where r y P/(p 2 EI y /L 2 ), r z P/(p 2 EI z /L 2 ), and P is positive in tension. The numerical solutions obtained from Eqs. (2a), (2b), (2c) and (2d) are indeterminate when the axial force is zero. To circumvent this problem and to avoid the use of different expressions for S 1,S 2, S 3, and S 4 for a different sign of axial forces, Lui and Chen [15] have proposed a set of expressions that makes use of power-series expansions to approximate the stability functions. The power-series expressions have been shown to converge to a high degree of accuracy within the first ten terms of the series expansions. Alternatively, if the axial force in the member falls within the range 2.0 ρ 2.0, the following simplified expressions may be used to closely approximate the stability functions: S 1 4 2p2 r y 15 (0.004r y 0.285)r 2 y r y (0.01r y 0.543)r 2 y 4 r y (3a) S 2 2 p2 r y 30 (0.01r y 0.543)r 2 y (3b) 4 r y S 3 (0.004r y 0.285)r 2 y r y 4 2p2 r z 15 (0.01r z 0.543)r 2 z 4 r z (3c)

4 84 S.-E. Kim et al. / Engineering Structures 25 (2003) (0.004r z 0.285)r 2 z r z S 4 2 p2 r z 30 (0.01r z 0.543)r 2 z (3d) 4 r z (0.004r z 0.285)r 2 z r z Eqs. (3a), (3b), (3c) and (3d) are applicable for members in tension (positive P) and compression (negative P). For most practical applications, Eqs. (3a), (3b), (3c) and (3d) give an excellent correlation to the exact expressions given by Eqs. (2a), (2b), (2c) and (2d). However, for ρ other than the range of 2.0 ρ 2.0, the conventional stability functions (Eqs. (2a), (2b), (2c) and (2d)) should be used. The stability function approach uses only one element per member and maintains accuracy in the element stiffness terms and in the recovery of element end forces for all ranges of axial loads. In this formulation, all members are assumed to be adequately braced to prevent out-of-plane buckling, and their cross-sections are compact Plastic strength of cross-section Based on the AISC LRFD bilinear interaction equations, a cross-section s plastic strength can be taken as [1] P 8 M y 8 M z 1.0 for P 0.2 (4a) P y 9M yp 9M zp P y P M y M z 1.0 for P 0.2 (4b) 2P y M yp M zp P y where P = second-order axial force, P y = squash load, M y,m z = second-order bending moment with respect to y and z axes, M yp,m zp = plastic moment capacity with respect to y and z axes CRC tangent modulus model associated with residual stresses The Column Research Council (CRC) tangent modulus concept is used to account for gradual yielding (due to residual stresses) along the length of axially loaded members between plastic hinges [3]. The elastic modulus E, instead of moment of inertia I, is hereby reduced. Although it is really the elastic portion of the cross-section (thus I) that is being reduced, changing the elastic modulus is easier than changing the moment of inertia for different sections. The rate of reduction in stiffness is different in the weak and strong directions, but this is not considered since the dramatic degradation of weakaxis stiffness is compensated for by the substantial weakaxis plastic strength [4]. This simplification makes the present methods practical. From Chen and Lui (1992), the CRC E t is written as E t 1.0E for P0.5P y E t 4 P P y E(1 P P y ) for P 0.5P y Parabolic function for gradual yielding due to flexure (5a) (5b) The tangent modulus model is suitable for the member subjected to axial force, but not adequate for cases of both axial force and bending moment. A gradual stiffness degradation model for a plastic hinge is required to represent the partial plastification effects associated with bending. When softening plastic hinges are active at both ends of an element, the force-deflection equation may be expressed as P T M ya M yb M za M zb where EtA L k iiy k ijy k ijy k jjy k iiz k ijz k ijz k jjz 0 d GJ f L k iiy h A (S 1 S2 2 S 1 (1h B )) E ti y L, k ijy h A h B S 2 E t I y L, k jjy h B (S 1 S2 2 S 1 (1h A )) E ti y L, k iiz h A (S 3 S2 4 S 3 (1h B )) E ti z L, k ijz h A h B S 4 E t I z L, q ya q yb q za q zb (6) (7a) (7b) (7c) (7d) (7e) k jjz h B (S 3 S2 4 (1h S A )) E ti z 3 L. (7f) The terms h A and h B are scalar parameters that allow for gradual inelastic stiffness reduction of the element associated with plastification at end A and B. This term is equal to 1.0 when the element is elastic, and zero when a plastic hinge is formed. The parameter h is assumed to vary according to the parabolic function: h 1.0 for a0.5 (8a)

5 S.-E. Kim et al. / Engineering Structures 25 (2003) h 4a(1a) for a 0.5 (8b) where a is a force-state parameter that measures the magnitude of axial force and bending moment at the element end. Herein, a is the function of the AISC LRFD interaction equations written in Eqs. (4a) and (4b). The assumed behavior of plastic hinge on crosssection is shown in Fig Geometric imperfection modeling Braced frame The proposed analysis implicitly accounts for the effects of both residual stresses and spread of yielded zones. To this end, the proposed analysis may be regarded as equivalent to the plastic-zone analysis. As a result, geometric imperfections are necessary only to consider fabrication error. For braced frames, member out-of-straightness, rather than frame out-of-plumbness, needs to be used for geometric imperfections. This is because the P- effect due to the frame out-of-plumbness is diminished by braces. The ECCS [8,9], AS [19], and CSA [6,7] specifications recommend an initial crookedness of column equal to 1/1000 times the column length. The AISC Code recommends the same maximum fabrication tolerance of L c /1000 for member out-of-straightness. In this study, a geometric imperfection of L c /1000 is adopted. The ECCS, AS, and CSA specifications recommend the out-of-straightness varying parabolically with a maximum in-plane deflection at the mid-height. They do not, however, describe how the parabolic imperfection should be modeled in analysis. Ideally, many elements are needed to model the parabolic out-of-straightness of a beam column member, but it is not practical. In this study, two elements with a maximum initial deflection at the mid-height of a member are found adequate for capturing the imperfection. Fig. 2 shows the out-ofstraightness modeling for a braced beam column member. It may be observed that the out-of-plumbness is equal to 1/500 when the half segment of the member is considered. This value is identical to that of sway frames as discussed in recent papers by Kim and Chen [11,12]. Thus, it may be stated that the imperfection values are essentially identical for both sway and braced frames. Fig. 2. Explicit imperfection modeling of braced member Unbraced frame Since the proposed analysis implicitly accounts for both residual stresses and the spread of yielding, it may be considered equivalent to the plastic-zone analysis. Thus, modeling the out-of-plumbness for erection tolerances is used here without the out-of-straightness for the column, so that the same ultimate strength can be predicted for mathematically identical braced and unbraced members. The Canadian Standard [6,7] and the AISC Code of Standard Practice [2] set the limit of erection out-of-plumbness L c /500. The maximum erection tolerances in the AISC are limited to 1 in. toward the exterior of buildings and 2 in. toward the interior of buildings less than 20 storeys. Considering the maximum permitted average lean of 1.5 in. in the same direction of a storey, the geometric imperfection of L c /500 can be used for buildings up to six storeys with each storey approximately 10 feet high. For taller buildings, this imperfection value of L c /500 is conservative since the accumulated geometric imperfection calculated by 1/ 500 times building height is greater than the maximum permitted erection tolerance. In this study, we shall use L c /500 for the out-ofplumbness without any modification because the system strength is often governed by a weak storey which has an out-of-plumbness equal to L c /500 [16] and a constant imperfection has the benefit of simplicity in practical design. The explicit geometric imperfection modeling for an unbraced frame is illustrated in Fig. 3. Fig. 1. Assumed behavior of plastic hinge on cross-section. 3. Model to account for local buckling When the width thickness ratio of a section is greater than a limit, local buckling of a flange or web will occur as the applied load increases. While most rolled sections are compact, welded sections are often non-compact and susceptible to local buckling. Local buckling causes degradation of the flexural strength. The conventional

6 86 S.-E. Kim et al. / Engineering Structures 25 (2003) The buckling stress F cr (in ksi) at the compression flange is controlled either by lateral torsional buckling or flange local buckling. Since this study ignores lateral torsional buckling, F cr is governed by flange local buckling as when ll p F cr F yf (14) when l p ll r F cr C b F y1 1 2 l l p l r l y (15) pf when l l r Fig. 3. Explicit imperfection modeling of unbraced frame. refined plastic-hinge analyses assume all sections are compact, hence they do not consider the degradation of the flexural strength caused by the local buckling. The analyses should be improved to consider the local buckling, since the sections of the real structures are not always compact. The width thickness ratio λ of the flange and web is an important factor influencing the local buckling strength of I-shaped sections. Since the refined plastichinge analysis uses only a line model to represent an element, a rigorous model using volume elements to account for local buckling effect is not applicable to this analysis. In this study, the practical LRFD equations given by Eqs. (9), (10) and (11) are used to determine the local buckling strength [2] when ll p M n M p (9) when l r ll p M n M p (M p M r ) l l p l r l p (10) when l l r M n M cr SF cr M p. (11) The notations of Eqs. (9) and (11) are defined in Table 1. When λ of the web is greater than λ r the flexural strength is given by M n S xc R PG F cr (12 where the reduction factor R PG is expressed as a r R PG a r h c 970 (13) t w F cr1.0 where a r = A w /A f # 10, h c = twice the distance from the centroid to the inside face of the compression flange, (h c = h for girders with equal flanges), S xc = elastic section modulus referred to the compression side. F cr C PG l. (16) 2 The limit states of flange local buckling are given by l b f 2t f (17) l p 65 F y (18) l r 230 F y /k c (19) where k c 4/h/t w and 0.35k c (20) C PG 26,200k c C b 1.0 (21) When the plastic moment M zp of Eqs. (4a) and (4b) is replaced with the local buckling strength M n of Eqs. (9), (10) and (11), Eqs. (4a) and (4b) are revised as P 8 M y 8 M z 1.0 for P 0.2 (22a) P y 9M yp 9M n P y P M y M z 1.0 for P 0.2. (22b) 2P y M yp M n P y Using Eqs. (22a) and (22b) in the 3-D second-order plastic-hinge analysis program, the effect of local buckling can be considered. The proposed analysis allows the inelastic moment redistribution in the structural system. Thus, adequate rotational capacity is required. This is achieved when members are adequately braced and their cross-sections are compact. When local buckling occurs at a section of a member the stiffness of the section is assumed to be zero so that the additional moment is not sustained by the section. This approximation is deemed appropriate for tracing the non-linear behavior of the frame including local buckling effect, since the proposed analysis aims to determine only the ultimate strength of

7 S.-E. Kim et al. / Engineering Structures 25 (2003) Table 1 Flexural strength and limit of width thickness ratio for I-sections Item Flange buckling Web buckling M p F y Z x F y Z x M r F L S x F y S x F cr (ksi) 20,000 Eqs. (14, 15, 16) : compression beam l 2 26,200k c l 2 : plate girder l b f / 2t f h / t w l p 65 F y when P u f b P y, l r 162 F L / k c : plate girder 141 F L : compression beam when P u 0.125, f b P y P u F y1 f b P y 640 F y1 2.75P u f b P y 191 F y2.33 P u f b P y 253 F y where:s x = elastic section modulus about major axis, in 3,F L = F y F R, ksi, F y = specified minimum yield stress, ksi, F R = compressive residual stress in flange, 10 ksi for rolled shapes, 16.5 ksi for welded shapes, b f = flange width of rolled beam or plate girder, t f = flange thickness, t w = web thickness, h = depth of composite column section, f b = 0.9. the whole structural system rather than to examine the local buckling behavior of a component member. 4. Verification study Verifications are performed for the following two cases: (1) Orbison s six-storey frame consisting of compact sections; (2) a single-storey frame comprising noncompact sections. The first is to verify how well the proposed analysis predicts geometric and material non-linear behavior of frames. The second is to show how the proposed analysis captures local buckling strength accurately Orbison s six-storey space frame ignoring local buckling Fig. 4 shows Orbison s six-storey space frame [17]. The yield strength of all members is 250 MPa (36 ksi) and Young s modulus is 206,850 MPa (30,000 ksi). Uniform floor pressure of 4.8 kn/m 2 (100 psf) is converted into equivalent concentrated loads on the top of the columns. Wind loads are simulated by point loads of 26.7 kn (6 kips) in the Y direction at every beam column joints. The load displacement results calculated by the proposed analysis compare well with those of Liew and Tang s (considering shear deformations) and Orbison s (ignoring shear deformations) results (Table 2, Table 3, and Fig. 5). The ratios of load-carrying capacities (calculated from the proposed analysis) over the applied loads are and These values are nearly equi- Fig. 4. Six-storey space frame. Table 2 Result of analysis considering shear deformation Method Proposed Liew s Plastic strength surface LRFD Orbison Orbison Ultimate load factor Displacement at A in Y direction 208 mm 219 mm 250 mm

8 88 S.-E. Kim et al. / Engineering Structures 25 (2003) Table 3 Result of analysis ignoring shear deformation Method Proposed Orbison s Plastic strength surface LRFD Orbison Orbison Ultimate load factor Displacement at A in Y direction 199 mm 208 mm 247 mm a 200,000 MPa (29,000 ksi) elastic modulus. The dimensions of the section are listed in Table 4. The vertical and horizontal loads are applied simultaneously. When the applied load reaches KN (45.28 kips), Nodal point (1) fails by local buckling. At that moment, the member forces of Element A are P = KN (18.25 kips), M y = KN m (838.1 in. k), and M z = KN m (2,707 in. k). The unit value calculated by using Eqs. (22a) and (22b) is Thus, it is verified that the proposed analysis can capture local buckling strength accurately. The additional loads can be sustained until the whole structural system encounters a limit state. The frame collapses when the applied load P u is equal to KN (51.86 kips). The additional load of KN (6.58 kips) is carried by the structural system after local buckling occurs at Node (1). It is the benefit of the proposed second-order plastic-hinge analysis allowing inelastic force redistribution. 5. Case study Fig. 5. Comparison of load displacement of six-storey space frame. valent to and calculated by Liew and Tang and Orbison, respectively Single-storey frame comprising local buckling Fig. 6 shows a single-bay, single-storey space frame. The stress strain relationship is assumed to be elastic perfectly plastic with 250MPa (36 ksi) yield stress and Fig D single-bay, single-storey frame. A three dimensional, one-bay, two-storey frame was selected for the case study. Fig. 7 shows a sideways uninhibited frame subjected to combined lateral and vertical loads. The stress strain relationship was assumed to be elastic perfectly plastic with a 250 Mpa (36 ksi) yield stress and a 200,000 Mpa (29,000 ksi) elastic modulus. The section listed in Table 4 was used for all the members. Out-of-plumbness of H/500 was explicitly modeled. Two analyses were compared in this case study: the proposed and the conventional 3-D secondorder plastic-hinge analysis. In the proposed analysis, the structure collapsed by the local buckling at Nodal points (1) to (5) in sequence. The load-carrying capacity P u in terms of the applied load of the structural system was evaluated to be KN (89.34 kips). If local buckling was ignored, the frame failed by global flexural buckling. The load-carrying capacity P u of the structural system was calculated to be 634 KN ( kips). As a result, the conventional analysis overpredicts the load-carrying capacity of the frame by 1.6 times. The vertical load-displacements at Nodal point (1) of the proposed and the conventional analysis are compared in Fig. 8. The proposed analysis predicts reasonably well the degradation of flexural strength caused by local buckling. The load-carrying capacity determined by the proposed in the case study is directly evaluated through the analysis, so separated member capacity checks encompassed by the specification equations are not required. As a result, the proposed method is time-effective in design process. The proposed analysis captures the limit state strength and stability of the structural system including its individual members, while the current LRFD and ASD method evaluate the strength of the individual

9 S.-E. Kim et al. / Engineering Structures 25 (2003) Table 4 Cross-section dimensions and properties Member d (mm) b (mm) t w (mm) t f (mm) A (mm 2 ) I x (10 6 mm 4 ) I y (10 6 mm 4 ) Z x (10 6 mm 3 ) nominal 1, ,388 10, Fig D one-bay, two-storey frame. 3. When the local buckling effect at the case study is ignored, the analysis overestimates the strength by more than 1.6 times. Thus, local buckling is a very crucial element to be considered in 3-D second-order plastic-hinge analysis. 4. Compared to LRFD and ASD, the proposed method provides more information on structural behavior through a direct second-order plastic-hinge analysis of the entire system. 5. The proposed analysis can capture the factor of safety for the structural system. It is more advanced than the current LRFD and ASD evaluating the strength of the individual members only. 6. The proposed analysis can be used in lieu of costly plastic zone analysis, and it can be a powerful tool for use in daily design. Acknowledgements This work presented in this paper was supported by funds of the National Research Laboratory Program (Grant No N-NL-01-C-162) from the Ministry of Science & Technology in Korea. The author appreciates their financial support. References Fig. 8. Load displacements of 3-D one-bay, two-storey frame. member only. As a result, the proposed method can capture the factor of safety for the whole structural system. 6. Conclusions Second-order plastic-hinge analysis accounting for the effect of local buckling has been developed. The conclusions of this study are as follows. 1. The proposed method appropriately traces the inelastic non-linear behavior including local buckling effect. 2. The errors of the proposed analysis are less than 1% when compared with the other analyses and LRFD results. [1] AISC (1993). Load and Resistance Factor Design Specification for steel buildings, American Institute of Steel Construction, Chicago. [2] AISC (1994). Load and resistance factor design specification, American Institute of Steel Construction, 2nd Ed., Chicago. [3] Chen WF, Lui EM. Stability design of steel frames. Boca Raton, FL: CRC Press, [4] Chen WF, Kim SE. LRFD steel design using advanced analysis. Boca Raton FL: CRC Press, [5] Clarke MJ, Bridge RQ, Hancock GJ, Trahair NS. Benchmarking and verification of second-order elastic and inelastic frame analysis programs in SSRC TG 29 workshop and monograph 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 [6] CSA (1989). Limit states design of steel structures, CAN/CAS- S16.1-M89, Canadian Standards Association. [7] CSA (1994). Limit states design of steel structures, CAN/CAS- S16.1-M94, Canadian Standards Association. [8] ECCS (1984). Ultimate limit state calculation of sway frames with rigid joints, Technical Committee 8 Structural stability technical working group 8.2 System publication No. 33, 20 pp. [9] ECCS (1991). Essentials of Eurocode 3 design manual for steel

10 90 S.-E. Kim et al. / Engineering Structures 25 (2003) structures in buildings, ECCS-Advisory Committee 5, No. 65, 60 pp. [10] El-Zanaty M, Murray D, Bjorhovde R. Inelastic behavior of multistory steel frames. Structural engineering report No. 83, University of Alberta, Alberta (Canada), [11] Kim SE, Chen WF. Practical advanced analysis for braced steel frame design. ASCE Journal of Structural Engineering 1996;122(11): [12] Kim SE, Chen WF. Practical advanced analysis for unbraced steel frame design. ASCE Journal of Structural Engineering 1996;122(11): [13] Kim SE, Park MH, Choi SH. Direct design of three-dimensional frames using practical advanced analysis. Engineering Struct 2001;23: [14] Liew JY, Tang LK. Non-linear refined plastic hinge analysis of space frame structures. Research Report No. CE027/98, Department of Civil Engineering, National University of Singapore, Singapore, [15] Lui EM, Chen WF. Analysis and behavior of flexibly jointed frames. Engineering Struct 1986;8: [16] Maleck AE, White DW, Chen WF. Practical application of advanced analysis in steel design. Proceedings 4th Pacific Structural Steel Conference. Steel Structures 1995;1: [17] Orbison JG. Nonlinear static analysis of three-dimensional steel frames. Report No. 82-6, Department of Structural Engineering, Cornell University, Ithaca, New York, [18] Prakash V, Powell GH. DRAIN-3DX: Base program user guide, version A Computer Program Distributed by NISEE / Computer Applications, Department of Civil Engineering, University of California, Berkeley, [19] Standards Australia. AS , Steel Structures, Sydney, Australia [20] Vogel U. Calibrating frames. Stahlbau 1985;10:1 7. [21] Ziemian RD, McGuire W, Dierlein GG. Inelastic limit states design part II: three-dimensional frame study. ASCE Struct Eng 1992;118(9):