Second-order distributed plasticity analysis of space steel frames

Engineering Structures 24 (2002) 735 744 www.elsevier.com/locate/engstruct Second-order distributed plasticity analysis of space steel frames Seung-Eock Kim *, Dong-Ho Lee Department of Civil and Environmental Engineering, Construction Tech. Research Institute, Sejong University, Seoul 143-747, South Korea Received 20 August 2001; accepted 4 December 2001 Abstract This paper provides benchmark solutions of space steel frames using second-order distributed plasticity analysis. The majority of available benchmark solutions of steel frames in the past were only of two-dimensional frames. Therefore, three-dimensional benchmark solutions are needed to extend the knowledge of this field. Details of the modeling including element type, mesh discretization, material model, residual stresses, initial geometric imperfections, boundary conditions, and load applications are presented. Case studies of Vogel s portal frame and space steel frames are performed. The ultimate loads obtained from the proposed analysis and Vogel agree well within 1% error. The ultimate loads of the space steel frames obtained from the proposed analysis and experiment compare well within 3 5% error. The benchmark solutions of the space steel frames are useful for the verification of various simplified second-order inelastic analyses. It is observed that the load carrying capacities calculated by the AISC-LRFD method are 25 31% conservative when compared with those of the proposed analysis. This difference is attributed to the fact that the AISC-LRFD approach does not consider the inelastic moment redistribution, but the analysis includes the inelastic redistribution effect. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Second-order analysis; Plasticity analysis; Steel frames; Inelastic analysis 1. Introduction The second-order inelastic analysis enables designers to directly evaluate the ultimate strength and behavior of a structural system. Over the past 30 years, researchers have developed and validated various second-order inelastic analyses for steel frames. Most of these studies can be categorized into one of two types: second-order concentrated plastic-hinge analysis; and second-order distributed plasticity analysis. The secondorder concentrated plastic-hinge analyses use the simplified plastic hinge [1 5]. While these analyses are regarded as practical for design use, they must be verified by second-order distributed plasticity analyses. The second-order distributed plasticity analyses use the highest refinement and thus are considered accurate. The analyses generally use fiber or shell elements. The analyses using fiber elements are based on beam column theory. The member is discretized into line segments, * Corresponding author. Tel.: +82-2-3408-3291; fax: +82-2-3408-3332. E-mail address: sekim@sejong.ac.kr (S.-E. Kim). and the cross section of each segment is subdivided into fiber elements. Only normal stress is considered for inelasticity [6,7]. The analyses using shell elements are based on deformation theory of plasticity. Since the analyses consider the combined effect of both normal and shear stresses they may be regarded as more accurate [8 10]. These analyses require a large number of finite three-dimensional shell elements in modeling structures. Commercial finite element analysis programs such as ABAQUS, ANSYS, ADINA, and NASTRAN belong to this analysis category [11 14]. Various benchmarks have been provided for planar steel frames but not yet for space steel frames [15 20]. The purpose of this paper is to provide benchmarks of space steel frames. ABAQUS, one of the mostly widely used and accepted commercial finite element analysis programs, is used. The numerical results obtained by the analysis are compared with experimental results. 2. Modeling and analysis methods Modeling and analysis methods in using ABAQUS are presented in order to provide a general guide for 0141-0296/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S0141-0296(01)00136-5

736 S.-E. Kim, D.-H. Lee / Engineering Structures 24 (2002) 735 744 second-order distributed plasticity analysis of space steel frames. 2.1. Element Analysis results are very sensitive to the types of elements used. The element library in ABAQUS contains many different types of elements for performing various analyses. Stress/displacement shell elements among many different elements are appropriate for carrying out second-order distributed plasticity analysis. ABAQUS provides several stress/displacement shell elements including STRI3, S3, S3R, STRI35, STRI65, S4, S4R, S4R5, S8R, S8R5, and S9R5. The elements may be grouped into the quadrilateral elements (S4, S4R, S4R5, S8R, S8R5, and S9R5) and the triangular elements (STRI3, S3, S3R, STRI35, and STRI65). An appropriate stress/displacement shell element accounting for material, geometric nonlinearity, spread of plasticity, and residual stresses should be selected. The element case studies have indicated that it would be adequate to use the thin, shear flexible, and isoparametric shell element of S4R5, S4R, and STRI35 for modeling steel frames. S4R5 is a quadrilateral shell element with four nodes and five degrees of freedom per node. S4R is a quadrilateral shell element with four nodes and six degrees of freedom per node. STRI35 is a triangular shell element with three nodes and five degrees of freedom per node. The flange and web plates may be modeled by using S4R5 or S4R. The interfacing zone between the web and the x-stiffener may be modeled by using STRI35. 2.2. Mesh discretization The computational time and accuracy largely depend on the number of elements and integration points. Mesh studies have indicated that it would be adequate to use eight elements through the depth of the web and across the width of the flange for hot-rolled or welded sections. The mesh size with an aspect ratio close to 1.0 is preferable. The default integration method is based on Simpson s rule with five integration points through the thickness of the element, which is appropriate for the typical steel section. available they may be simply converted to the true stress and logarithmic plastic strain as s true s nom (1 e nom ) e pl ln ln(1 e nom ) s true E where s true, s nom, e pl ln, e nom, and E are true stress, nominal stress, logarithmic plastic strain, nominal strain, and Young s modulus, respectively. The measured stress strain curve may be idealized as a multi-linear stress strain curve. If measured yield stresses for the web and the flange are different from each other, they should be defined. The fillet at the joint of the flange and the web may be considered by equivalently increasing the thickness of the flanges. 2.4. Residual stresses The uneven cooling during the fabrication creates a set of self-equilibrating stresses in the cross section. These are residual stresses. Only the membrane component in the longitudinal direction is considered for residual stresses. The residual stress distributions recommended by ECCS Technical Committee 8 [21] may be used (Figs. 1 and 2). For modeling residual stress distributions, *INITIAL CONDITIONS, TYPE=STRESS, USER option is used. The residual stresses are defined by using the SIGINI FORTRAN user subroutine. The user subroutine SIGINI is called at the start of the analysis for each applicable material calculation point. SIGINI defines residual stress components as the function of the coordinate, element number, or integration point number. The pre-analysis is recommended to check the residual stress distribution. 2.3. Material model ABAQUS provides the classical metal plasticity model using a standard Mises yield surface with associated plastic flow. Perfect plasticity and isotropic hardening definitions are both available in the classical metal plasticity model. The elastic part is defined by Young s modulus and Poisson s ratio. The plastic part is defined as the true stress and logarithmic plastic strain. If the nominal stress strain data obtained by a uniaxial test are Fig. 1. [21]. Assumed residual stress distribution for hot-rolled I-sections

S.-E. Kim, D.-H. Lee / Engineering Structures 24 (2002) 735 744 737 Fig. 3. Member imperfections [21 26]. Fig. 2. [21]. Assumed residual stress distribution for welded I-sections 2.5. Initial geometric imperfections Two types of geometric imperfections to be considered are member and local imperfections. 2.5.1. Member imperfections The typical member imperfections are out-of-plumbness and out-of straightness imperfections. For braced frames, out-of-straightness rather than out-of-plumbness needs to be modeled for geometric imperfections. This is because the P effect due to the out-of-plumbness is diminished by braces. The ECCS [21,22], AS [23], and CSA [24,25] 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 out-of-straightness. As a result, a geometric imperfection of L c /1000 is recommended. The ECCS [21,22], AS [23], and CSA [24,25] specifications recommend the out-of-straightness varying parabolically with a maximum in-plane deflection at mid-height (Fig. 3). For unbraced frames, out-of-plumbness rather than out-of-straightness needs to be modeled for geometric imperfections. This is because the P d effect due to the out-of-straightness is not dominant. The Canadian Standard [24,25] and the AISC Code of Standard Practice [26] set the limit of erection out-of-plumbness of L c /500 (Fig. 3). If the measured data of member imperfections exist, they may be used in the analysis. The member imperfections can be modeled by using one of following two approaches. One is to superpose weighted eigenmodes. The eigenmodes are obtained by elastic bifurcation analysis using the *BUCKLE option. Any number of appropriate eigenmodes can be selected and scaled by the *IMPERFECTION option. Another approach is to move directly each coordinate of nodal points, but this requires tedious work. 2.5.2. Local imperfections The steel sections are subdivided into two categories: compact and non-compact. If the section is non-compact, it is necessary to model local imperfections. The magnitudes of the local imperfections should be chosen appropriately. Fig. 4 illustrates the local imperfections [20]. If the measured data of local imperfections exist, they may be used in the analysis. Fig. 4. Local imperfections [20].

738 S.-E. Kim, D.-H. Lee / Engineering Structures 24 (2002) 735 744 There are two methods to model local imperfections. One is to scale appropriately the eigenvectors selected from an elastic bifurcation analysis of the structure. Note that local imperfections consisting of a single buckling mode tend to yield non-conservative results. Another method is to simply revise each nodal coordinate, but this requires tedious work. Local imperfections are not necessarily modeled for compact sections. 2.6. Boundary conditions Three typical boundaries are fixed, hinged, and spring conditions. The fixed boundary can be achieved by simply restraining all the degrees of freedom of the nodal points concerned. The ideal hinge connection can be achieved using the following single-point constraint method. First, the translational and rotational degrees of freedom of all the nodal points on the boundary surface concerned are made equal to those of a master node selected among the nodal points, using the *MPC type BEAM or TIE option. Next, the translational degrees of freedom of the master node are made to be restrained. The ideal fixed boundary is very difficult to construct in a real situation. The flexible boundary condition can be simulated by the horizontal and vertical springs attached at each nodal point on a boundary surface. The spring is modeled by using the *SPRING option. The stiffness of the spring can be determined by analysis or experiment. 2.7. Load applications Concentrated or distributed loads may be applied. When panel-zone deformation of steel frames is ignored, concentrated loads should be applied on a master node, which slaves other nodes in the panel zone using the *MPC type BEAM option. Distributed loads may be applied by pressure on elements or equivalent concentrated loads on the nodal points. 3. Case study Case studies were performed for the following: (1) Vogel s portal frame; and (2) space frames. The first case study is to verify the proposed modeling and analysis method. The second is to provide a benchmark solution for space frames. The case studies follow the modeling and analysis guide presented in the previous section. 3.1. Vogel s portal frame Fig. 5 shows Vogel s portal frame. The dimensions and properties of the section used are listed in Table 1. The flange and web plates were modeled by using the S4R5 shell element. The eight elements were used through the depth of the web and across the width of the flange. An aspect ratio close to 1.0 in discretization of the mesh was used along the member. The total number of elements used was 8952. Five integration points were used through the thickness of the element. The geometry and the finite element mesh are shown in Figs. 6 and 7. The yield stress of all members was 235 MPa and Young s modulus was 205,000 MPa. Poisson s ratio was 0.3. Fig. 8 shows a stress strain relationship used for the analysis. The residual stresses were defined by using the function of element numbers in the SIGINI subroutines. The maximum residual stress was selected as 50% of the yield stress. Out-of-plumbness imperfections were modeled by moving each coordinate of the nodal points. Fig. 5 shows the magnitudes of out-of-plumbness imperfections. Local imperfections were not modeled since the frame was composed of compact sections. The fixed boundary was modeled by restraining all the degrees of freedom of the nodal points at the base of the 2.8. Types of analysis There are two types of nonlinear static analysis: *STATIC and *STATIC, RIKS. The *STATIC option traces up to the limit state (i.e., the ascending branch) of a structure. The Newton Raphson solution technique is used for solving the equilibrium equation. The *STATIC, RIKS option captures both the ascending branch and descending branch (i.e., post-buckling behavior). The arc-length method is used for solving the equilibrium equation. The tolerance must be set as less than 0.5% of the time-averaged force in order to maintain accuracy. Fig. 5. [18]. Dimension and loading condition of Vogel s portal frame

S.-E. Kim, D.-H. Lee / Engineering Structures 24 (2002) 735 744 739 Table 1 Dimensions and properties of the section used in the Vogel s portal frame Section d (mm) b f (mm) t w (mm) t f (mm) A (mm 2 ) I (10 6 mm 4 ) S (10 3 mm 3 ) HEA340 330 300 9.5 16.5 13,300 276.9 1850 HEB300 300 300 11.0 19.0 14,900 251.7 1869 Fig. 8. Stress strain relationship for Vogel s portal frame. Fig. 6. Finite element modeling for Vogel s portal frame. loads were applied on the nodal point at the center of the beam column joint. A non-linear static analysis using the *STATIC option was conducted. The horizontal load deflection calculated by the proposed analysis and that of Vogel compare well, as shown in Fig. 9. The ultimate load factor calculated from the proposed analysis was 1.03, which was very close to 1.02 calculated by Vogel. Fig. 7. Finite element mesh for Vogel s portal frame. columns. The beam column joint area was modeled by using the *MPC type BEAM option in order to eliminate panel-zone deformation, since Vogel s analysis did not consider panel-zone deformation. The out-of-plane translational degrees of freedom of all the nodal points located at the intersection of the web and flange were restrained. The concentrated vertical and horizontal Fig. 9. Comparison of horizontal load displacement curves of Vogel s portal frame.

740 S.-E. Kim, D.-H. Lee / Engineering Structures 24 (2002) 735 744 Fig. 10. Dimension and loading condition of test frame. 3.2. Space frames Fig. 11. Three-dimensional finite element modeling of space frame. This case study is to provide benchmark solutions of space steel frames, shown in Fig. 10. The proposed analysis results were compared with test results. Three frames (frames 1, 2, and 3) subjected to proportional loads and one frame (frame 4) subjected to non-proportional loads were tested [27,28]. The dimensions and properties of the section H 150 150 7 10 used in the test frames are listed in Table 2. The flange and web plates of frames 1, 2, 3, and 4 were modeled by using the S4R shell element. The interfacing zone between the web and the x-stiffener in the panel zone of frame 4 was modeled by using STRI35. The eight elements were used through the depth of the web and across the width of the flange. An aspect ratio close to 1.0 in discretization of the mesh was used along the member. The total number of elements used was 50,480. Five integration points were used through the thickness of the element. The geometry and the finite element mesh are shown in Figs. 11 and 12. The measured elastic moduli and yield stresses listed in Table 3 were used for the test frame analyses. Poisson s ratio was 0.3. The measured stress strain curves were idealized as multi-linear stress strain curves listed in Table 4. The residual stresses were defined by using Fig. 12. Finite element mesh of space frame. the function of element numbers in the SIGINI subroutines. The maximum residual stress was selected as 50% of the yield stress. The measured magnitudes of out-ofplumbness imperfections listed in Table 5 were modeled by moving each coordinate of the nodal points. Out-ofstraightness imperfections were not included since the Table 2 Dimensions and properties of section H 150 150 7 10 used in the test frames Test frame H (mm) B (mm) t f (mm) t w (mm) r 1 (mm) A g (mm 2 ) I x (10 6 I y (10 6 mm 4 ) mm 4 ) 1, 2, 3, 4 Nominal 150.0 150.0 10.0 7.00 11 4014 16.4 5.63 Measured column 152.3 149.9 10.2 6.75 4053 17.2 5.74 (average) beam 149.1 150.0 9.2 6.50 3713 15.1 5.19

S.-E. Kim, D.-H. Lee / Engineering Structures 24 (2002) 735 744 741 Table 3 Yield stress and elastic modulus Yield stress Elastic modulus Column Flange 320 217,771 Web 311 214,504 Beam Flange 344 221,313 Web 327 194,814 P d effect in unbraced frames is not dominant. Local imperfections were not modeled since the frame was composed of compact sections. The base plate connection was modeled by the horizontal and vertical springs attached at each nodal point at the column bases to account for the flexibility of the bolted connection. The *SPRING option was used. The spring stiffness of 67,322 kn/m was determined by the pre-test of the frames. While the boundary condition of the second floor level was free to move, the roof level was fixed in displacement by using the single-nodalpoint constraint. The cover plates of the test frame were modeled. The x-stiffeners constructed at the beam column joints of frame 4 were also explicitly modeled. The vertical jack loads were distributed at the seven nodal points. For frames 1, 2 and 3, the horizontal jack load was applied to the outside flange of columns (2) and (4) at the second floor level (Fig. 10). The horizontal load was uniformly distributed by using the equivalent concentrated loads applied on the 25 nodal points. For frame 4, the horizontal jack load was applied to the outside flange of column (2) at the second floor level (Fig. 10). The load cases of the test frames are listed in Table 6. A non-linear static analysis using the *STATIC option was conducted for frames 1, 2, and 3, while the * STATIC, RIKS option was used for frame 4 in order to capture post-buckling behavior. The horizontal load deflection curves obtained from the analysis and the experiment are compared in Figs. 13 16. The ultimate loads obtained from the analysis and the experiment Table 4 Multi-linear stress strain curves for H 150 150 7 10 steel Column Flange Stress 0 320 320 360 388 426 446 453 455 Strain 0 0.00147 0.02190 0.03375 0.04708 0.07450 0.11123 0.14548 0.19320 Web Stress 0 311 311 367 408 436 445 442 Strain 0 0.00145 0.02063 0.04230 0.07055 0.11640 0.16485 0.21863 Beam Flange Stress 0 344 344 397 434 455 463 464 Strain 0 0.00155 0.02190 0.04708 0.07450 0.11123 0.14548 0.19320 Web Stress 0 327 327 366 406 435 445 443 Strain 0 0.00168 0.02063 0.04230 0.07055 0.11640 0.16485 0.21863 Table 5 Measured out-of-plumbness imperfections Test frame Level Imperfections (mm) Column (1) Column (2) Column (3) Column (4) X Y X Y X Y X Y 1 Roof 11.1 4.5 11.4 5.5 6.6 8.2 12 4.3 Second floor 6.9 1.4 6.8 0.7 2.1 5.1 6.2 4 Base 0 0 0 0 0 0 0 0 2 Roof 6.8 5.1 12.1 2.5 2.5 1.5 8.7 0.3 Second floor 5.8 2.9 5.1 0.2 1.4 1.3 0.5 0.3 Base 0 0 0 0 0 0 0 0 3 Roof 2.4 0.8 1.4 0.7 0.7 9.6 6.4 10.8 Second floor 1.8 2 2.9 0.1 0.2 3.9 0.7 5 Base 0 0 0 0 0 0 0 0 4 Roof 1.6 6.8 2.91 3.5 13.5 10.7 3.84 7.21 Second floor 2.8 3.1 0.3 2.4 11.89 8.43 3.39 4.95 Base 0 0 0 0 0 0 0 0

742 S.-E. Kim, D.-H. Lee / Engineering Structures 24 (2002) 735 744 Table 6 Load case of test frame Test frame Vertical load Horizontal load Horizontal load (H 1 ) (H 2 ) 1 P P/3 P/6 2 P P/4 P/8 3 P P/5 P/10 4 680 kn Displacement control Fig. 14. Comparison of horizontal load displacement curves of space test frame 2. Fig. 13. Comparison of horizontal load displacement curves of space test frame 1. were nearly the same, within 3 5% error as shown in Table 7. The horizontal displacements were shown to differ. This difference can be attributed to possible experimental errors (i.e., boundary conditions, eccentric loading, variations in the material properties, and residual stresses) and analytical approximations (i.e., nominal residual stresses, imperfection distributions, and simplified stress strain curves). 4. Comparison of analysis and AISC-LRFD capacities Load carrying capacities obtained by the present analysis and the AISC-LRFD method are compared in Tables 7 and 8. The AISC-LRFD capacities were obtained using the average measured yield stress of the flange and the web specimens [26]. A resistance factor of 0.9 was used for the present analysis capacity, while factors of 0.85 for columns and 0.9 for beams were used for the AISC-LRFD capacity. The AISC-LRFD Fig. 15. Comparison of horizontal load displacement curves of space test frame 3. capacities were evaluated to be approximately 25 31% conservative compared to the analysis results. The difference is attributed to the fact that the AISC-LRFD approach does not consider the inelastic moment redistribution, but the second-order distributed plasticity analysis includes the inelastic redistribution effect. This comparison provides concrete reasons for using inelastic nonlinear analysis, which is quite effective in reducing member sizes.

S.-E. Kim, D.-H. Lee / Engineering Structures 24 (2002) 735 744 743 2. The ultimate loads obtained from the proposed analysis and Vogel agree well within 1% error. 3. The ultimate loads of the space steel frames obtained from the proposed analysis and experiment are found to be well within 3 5% error. 4. The benchmarking solutions of the space steel frames have been provided, and they may be used for the verification of simplified three-dimensional secondorder inelastic analyses. 5. The load carrying capacities calculated by the AISC- LRFD method are 25 31% conservative compared to the analysis results. This is because the AISC-LRFD approach does not consider the inelastic moment redistribution, but the present analysis includes the inelastic redistribution effect. Acknowledgements Fig. 16. Comparison of horizontal load displacement curves of space test frame 4. 5. Conclusions The conclusions of this study are as follows: 1. A general guide for modeling and analysis methods using ABAQUS has been provided for second-order distributed plasticity analysis of space steel frames. The work presented in this paper was supported by funds of the National Research Laboratory Program (grant no. 2000-N-NL-01-C-162) from the Ministry of Science & Technology in Korea. The authors wish to acknowledge the financial support. References [1] Liew JYR, Chen WF. Refining the plastic hinge concept for advanced analysis/design of steel frames. J. Singapore Struct. Steel Soc., Steel Struct. 1991;2(1):13 30. Table 7 Comparison of capacities of analysis, experiment, and AISC-LRFD (a) Analysis (b) Experiment (c) AISC-LRFD (a)/(b) (c)/(a) design Test frame 1 P 534.9 549.4 410.6 0.9736 0.7676 H 1 178.3 183.5 136.9 0.9717 0.7460 H 2 89.1 91.8 68.4 0.9706 0.7677 Test frame 2 P 618.1 607.7 474.2 1.0171 0.7672 H 1 154.5 152.2 118.5 1.0151 0.7670 H 2 77.3 76.1 59.3 1.0158 0.7671 Test frame 3 P 680.9 681.8 510.2 0.9987 0.7493 H 1 136.2 136.4 102.0 0.9985 0.7489 H 2 68.1 67.5 51.0 1.0089 0.7489 Test frame 4 P 612.0 612.0 443.5 1.0000 0.7247 H 1 177.8 169.2 122.6 1.0508 0.6895 H 2 Table 8 Comparison of capacities of analysis, Vogel s, and AISC-LRFD (a) Analysis (b) Vogel (c) AISC-LRFD (a)/(b) (c)/(a) design Portal frame P 2888.0 2856.0 1990.6 1.0112 0.6893 H 36.1 35.7 24.9 1.0112 0.6898

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