This Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left Blank THREE-DIMENSIONAL HYSTERETIC MODELING OF THIN-WALLED CIRCULAR STEEL COLUMNS Lizhi Jiang and Yoshiaki Goto Department of Civil Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya, 466-8555, Japan ABSTRACT An empirical hysteretic model is presented to simulate the three-dimensional cyclic behavior of cantilever-type thin-walled circular steel columns subjected to seismic loading. This steel column is modeled into a rigid bar with multiple vertical springs at its base. Nonlinear hysteretic behavior of thin-walled columns is expressed by the springs. As the hysteretic model for the spring, we modify the Dafalias and Popov's bounding-line assumption in order to take into account the degradation caused by the local buckling. The material properties for the vertical spring are determined by using curve- fitting technique, based on the in-plane restoring force-displacement hysteretic relation at the top of the column obtained by FEM analysis. By properly increasing the number of springs, the homogeneity of thin-walled circular columns is maintained. Finally this model is used in three-dimensional earthquake response analysis. KEYWORDS Hysteretic model, Three-dimensional behavior, Local buckling effect, Steel, Thin-walled column, Empirical model, Earthquake response analysis INTRODUCTION In the three-dimensional earthquake response analysis for thin-walled steel columns used as elevated highway piers shown in Fig. 1, FEM analysis using shell elements is the only direct procedure that can consider both axial force and biaxial bending interaction and local buckling effect. However, it requires a large amount of computing. Herein, we propose a simple three-dimensional hysteretic model for thin-walled circular steel columns. To consider the three-dimensional interaction, Aktan And Pecknold (1974) developed a filament model. However, their model cannot consider the effect of the local buckling, since they adopt the bilinear relation for the hysteretic model of each filament. The model we propose herein is alike the filament model but uses fewer springs which simulate the three- dimensional interaction. As the hysteretic model for each spring, we modify the Dafalias and Popov (1976) bounding-line model in order to take into account the degradation caused by the local buckling. The force-displacement relationship for each spring is determined by using curve-fitting technique, 101 102 L. Jiang and Y. Goto based on the in-plane restoring force-displacement hysteretic relation obtained by the FEM shell analysis. Liu et al (1999) also proposed an empirical hysteretic model ,but the application of this model is restricted to in-plane case. The validity of our model is examined by comparing with the results of the three-dimensional nonlinear dynamic response analysis using shell element. Fig. l:Thin-walled steel columns of elevated highways in Japan BOUNDING-LINE MODEL IN FORCE SPACE In-plane Hysteretic Behavior of Thin-Walled Circular Steel Columns From the elastic theory, Timoshenko and Gere (1961), the elastic buckling of columns with circular section is governed by two structural parameters R, and 3, . R, = R. __.ay X/3( 1-v 2) (1) t E ~ _ 2L 1 ~-~ (2) r ~ where R and t are the radius and the thickness, respectively, of the thin-walled circular column; cry is the yield stress of steel ; E is Young's modulus;v is Poisson's ratio; L is the height of column and r is the radius of gyration of cross section. In the plastic range, we assume that the hysteretic behavior of thin-walled circular steel columns is influenced by the axial load ratio P/Py (Py -Cry,, A and A is the cross-sectional area) in addition to the two structural parameters R t and X. As a result of FEM analysis, hysteretic behavior of thin-walled circular steel columns is classified into three types, depending on the value ofR t , as illustrated in Fig. 2 (a), (b) and (c).Herein, the material behavior of steel is assumed to be represented by the three-surface cyclic plasticity model proposed by Goto et al (1998). The material constants used for the three-surface model is shown in Table 1. Considering the sizes as well as the design loads of real columns, three parameters take the values as 0.1 ___ P/Py <_ 0.3,0.06 < R t < 0.12, and 0.2 < 3. < 0.5. These ranges for the three parameters indicate that the hysteretic behavior of our concern corresponds to that shown in Fig. 2 (b). This hysteretic behavior is characterized by the gradual strength degradation with the increase of cyclic plastic deformation. Three-Dimensional Hysteretic Modeling of Th&-Walled Circular Columns TABLE 1 THREE-SURFACE MODEL PARAMETERS Parameter E (Gpa) Cry (MPa) cru(MPa) v eyp" L/Cry [3 495.0 0.3 0.0183 0.581 SS400 (No.8) 206.0 289.6 103 t ~ H&i/E HmPo. 100 2 0.05 Note Note: For details see Goto et al (1998). Fig. 2: Classification of hysteretic behavior (P/Py = 0.1, A, = 0.2 ) Modified Bounding-Line Model Dafalias and Popov (1976) presented a bounding-line constitutive model to express the cyclic plasticity of metals. We modify this model to express the in-plane force-displacement relation of steel columns. As shown in Fig. 3, F and X e denote restoring force and plastic horizontal displacement at the top of the column respectively. XX and YY are bounding-lines. In order to express the strength degradation under cyclic loading, the gradient K B of the bounding-lines are assumed to be negative, being different from the original Dafalias-and-Popov model that adopts a positive gradient for the bounding-lines. The incremental force-displacement relation for the in-plane behavior of steel columns is expressed as follows, depending on whether the current state belongs to the elastic range or the plastic range. (Elastic range) AF = K e ~ AX (3) (Plastic range) AF = K e K e / (K e + K e) 9 AX (4) where K E is the elastic tangent stiffness and Kp is the plastic tangent stiffness. Based on the bounding-line model, K p is give by di Kp = K B + H ~ ~ (5) 6in 6 where K B is the slope of the bounding-line; H is the hardening shape parameter; 6 is the distance from the current force state to the corresponding bound; 6in is the value of 6 at the initiation of each loading process. In the elastic range represented by the straight lines OA and CD in Fig. 3, K e is zero; when the force reaches the bounding-line BC ,K e becomes the same as K B ;on the curves AB and DE, K p is expressed by Eq. 5. 104 L. Jiang and Y. Goto x Bounding-line Y A > "' ~in Y Fig. 3: Bounding-line model Empirical Equations to Determine Parameters Five parameters are included in our hysteretic model:Fe,Ke,6i,,K B and H. Among them, two are elastic parameters: F e is the elastic yielding force;K e is the elastic stiffness. The other three are related to bounding-line model as mentioned in the previous sub-section. From Fig. 2 (b), it is observed thatF E , K E , 6i, and K B all change with the increase of the plastic deformation. Thus, it is assumed that these four parameters are extrapolated from their initial values F e , Ke, 6i, andKB following the same rule as F e = f 9 F e (6) K e = f. K e (7) ~,. = f.o~~ (8) K B =f~ B (9) where f = 1- logo + Wp/W e 1 C )" We =-2 "Fe ~ X e is the elastic work. W e is the accumulated plastic work. C is an empirical function given by C = 37.75 - 33 ~ )~ - 25 ~ P/Py - 125. R, (10) The initial value of the four parameters: F e ,K e ,(~in ,KB and H are given by I 1o (O'y P (11) Fe -~'~ -~-) 3EI 1 KE - L 3 ~ (1+ 5.85 ~ (R/L) 2 ) (12) KB/K E = (-0.155" PIPe +0.1616)+(-0.5085" P/Py -0.1317) ~ ~, +(1.06. P/Py - 2.3) ~ R t (13) 6i,,/Fe = (2.7" P/Py + 0.48) + (-0.12" P/Py - 0.012)" X + (-22.967 ~ P/Py - 0.95)" R t (14) Three-Dimensional Hysteretic Modeling of Thin-Walled Circular Columns 105 H = -5.El0 8 9 )~ + 3.E10 8 (15) where A and I are the cross-sectional area and the second moment of inertia, respectively, of the steel columns; P is the vertical dead load. The other variables have the same meaning as in Eq. 1 . F E andK E are directly obtained from elastic theory;KB/KE, -~i,/FE and H are so determined by the least square method that our model best fits the force-displacement relationship obtained by FEM analysis using shell elements under monotonic loading. A comparison of the hysteretic loops between the present empirical hysteretic model and the FEM shell model is shown in Fig. 4 for the steel column with P/Py = 0.1,R, = 0.07 and ~, = 0.5. The present model will yield an acceptable result when applied to the practical design. Fig. 4: Comparison between FEM model and empirical hysteretic model MULTIPLE SPRING MODEL FOR THREE-DIMENSIONAL ANALYSIS Modeling of Thin-Walled Steel Columns To express the three-dimensional hyteretic behavior, the steel column is modeled into a rigid bar with multiple vertical springs at its base, as illustrated in Fig. 5. At the column base, no horizontal relative displacement is assumed to occur. Rigid body Multiple springs y -,,,, z <i- x Y os n of spring Fig. 5: Modeling of steel column Based on the three-dimensional modeling of steel columns, the following incremental force- displacement relation is obtained. 106 AFr = AFz R 2 n 79 c~ 0i) R 2 n 9 ( .~ k i 9 cos0/9 sin 0 i) 2v R " - ~ (~ ~i. cosO,) L. Jiang and Y. Goto R 2 n R " 9 i.cosO/.sinOi - .cosOi L ~ T n R2 (~~. oR (~k i~ L 2 9 k i 9 sin20i) L 9 ki L 9 sinOi) ki AX where AF x ,AFr,AF z and AX,AY, AZ are force and displacement increments, respectively, at the top of the columns; k i is the tangent stiffness for the ith spring ; 0 i is the angle that specifies the location of the ith spring; n is the total number of springs. The least number of springs that can have three-dimensional interactive effects is four. But this number of springs can not ensure the homogeneity. Fig. 6 (a) shows the non-homogeneous force-displacement relations for the column model with four springs under horizontal force directions: 0~176 ~ and 45 ~ . However, if we increase the number of springs, the column comes to exhibit homogeneity as illustrated in Fig. 6 (b). The least number of springs that is required for homogeneity is 16. Fig. 6: Homogeneity of multiple spring model Constitutive Relation for Multiple Springs The constitutive relation for the multiple springs is determined, based on the in-plane hysteretic model. From Eq.16, the in-plane force and displacement relation in the X direction is derived as R 2 n l~i'X -" "-~" (E ki " cOS20i)AX (17) ][ a By comparing Eq.17 with Eq.4, the multiple spring model parameters FE.,.pri,,g, Kespring, 6i,,.,.p,.i,,g, KBvr,,g, and H.,r, ri,,g can be related as follows to the parameters of the in-plane hysteretic model. L FEspring - F e 9 (~, g" a) (18) L 2 KEspring Ke . (._RT. g) (19) L 6i,.,prZ,g = 6i" (-R " g" a) (20) L 2 K Bspring = K. 9 ( RT " g) (21) Three-Dimensional Hysteretic Modeling of Thin-Walled Circular Columns 107 L 2 Hspring = H o (-R 5-o g~ a) (22) g = 1/'~ cos 2 0 i and a = 0.87 . / where THREE-DIMENSIONAL EARTHQUAKE RESPONSE ANALYSIS Steel Column Model In order to demonstrate the validity of the multiple spring model, a dynamic response analysis is carried out under the N-S, E-W and U-D components of the Kobe earthquake ground acceleration recorded by the Japan Meteorological Agency (JMA). Under the same ground acceleration, FEM analysis using shell elements illustrated in Fig. 7 is also conducted to examine the accuracy of our model. For the column material property, we adopts the three-surface model with the material constants summarized in Table 1. Fig. 7: FEM shell model Earthquake Response The results of the earthquake response analysis obtained by the empirical hysteretic model are shown in Figs. 8-10, in comparison with those obtained by the FEM shell model. Figure 8 illustrates the loci of the response sway displacement at the top of the column. Figure 9 shows the E-W component of the Fig. 8: Loci of response sway displacement 108 L. Jiang and Y. Goto sway displacement history, whereas Fig. 10 shows the hysteresis loops expressed in terms of the force- displacement relation. From Figs. 8-10; it is confirmed that the empirical hysteretic model can simulate the three-dimensional seismic behavior of the FEM shell model with an acceptable tolerance. Fig. 9: Sway displacement history of the column (East-West) Fig. 10: Comparison of hysteretic force-displacement relation (East-West) SUMMARY AND CONCLUDING REMARKS In view of the application to the practical design analysis, a three-dimensional hysteretic model for the thin-walled circular column is presented. This model is represented by a rigid bar with multiple vertical springs at its base. These multiple springs are used to consider both the axial force and biaxial bending interaction and the local buckling effect. The constitutive relation for each spring is determined by the curve-fitting technique, based on the in-plane hysteretic behavior of the FEM shell model. In order to examine the validity of the proposed hysteretic model, a three-dimensional earthquake response analysis is carried out for a steel column by using both the hysteretic model and the FEM shell model. As a result, it is confirmed that the proposed hysteretic model can simulate the three-dimensional seismic behavior of the FEM shell model with an acceptable tolerance. References Aktan A. E. and Pecknold A. (1974). R/C Column Earthquake Response in Two Dimensions. Journal of the Structural Division.ASCE. ST10, 1999-2015. Dafalias Y. E and Popov E. E (1976). Plastic Internal Variables Formalism of Cyclic Plasticity. Journal of Applied Mechanics. ASME. 43:12, 645-651. Goto Y. and Wang Q. Y. (1998). FEM Analysis for Hysteretic Behavior of Thin-Walled Columns. Journal of Structural Engineering. ASCE. 124:11, 1290-1301. Liu Q. Y. and Kasai A. (1999). Parameter Identification of Damage-based Hysteretic Model for Pipe- section Steel Bridge Piers. Journal of Structural Engineering. JSCE. 45A:3, 53-64. Shing-Sham L. and George T. W. (1984). Model for Inelastic Biaxial Bending of Concrete Members. Journal of Structural Engineering ASCE. 110:11, 2563-2584. Timoshenko S. P. and Gere J. M. (1961). Theory of Elastic Stability, McGraw-Hill Kogakusha, LTD. LOCAL BUCKLING OF THIN-WALLED POLYGONAL COLUMNS SUBJECTED TO AXIAL COMPRESSION OR BENDING J.G. Teng, S.T. Smith and L.Y. Ngok Department of Civil and Structural Engineering The Hong Kong Polytechnic University, Hong Kong, P.R. China ABSTRACT Thin-walled polygonal section columns are a popular form of construction due mainly to aesthetic considerations. Limited literature exists, however, on the stability of the component plate elements of these columns. A finite strip model is used in this paper to investigate the local buckling behaviour and strength of these columns subject to either axial compression or uniform bending. Cross-sections of square, pentagonal, hexagonal, heptagonal and octagonal profiles are considered. Elastic local buckling coefficients are presented for a variety of plate width-to-thickness ratios. It is shown that the dimensionless buckling stress coefficient is influenced by two parameters: the nature of the applied loading and the number of sides of the section. The buckling stress coefficient is higher for bent sections than axially compressed ones, and this difference can be quite significant. Sections with an odd number of sides have an enhanced buckling capacity over those with an even number of sides, with pentagonal sections being the strongest under either axial compression or bending. KEYWORDS Buckling, stability, columns, finite strip method, local buckling, polygonal sections. INTRODUCTION Thin-walled polygonal section columns are a popular form of construction due mainly to aesthetic considerations. Common polygonal sections include square, pentagonal, hexagonal, heptagonal and octagonal shapes. These columns are generally subjected to axial and lateral loads. Limited literature exists on the stability of the component plate elements of these columns. This paper thus considers the elastic local buckling capacity of polygonal column sections subjected to axial compression or bending. The local buckling of thin-walled columns of box sections has been quite extensively investigated. Few studies on local buckling in polygonal columns, however, are found in the literature. The local buckling of long polygonal tubes in combined bending and torsion was investigated by Wittrick and Curzon (1968) using an exact finite strip method. Bulson (1969) undertook a comprehensive test 109 . (1976) bounding-line model in order to take into account the degradation caused by the local buckling. The force-displacement relationship for each spring is determined by using curve-fitting technique,. column obtained by FEM analysis. By properly increasing the number of springs, the homogeneity of thin-walled circular columns is maintained. Finally this model is used in three-dimensional. highways in Japan BOUNDING-LINE MODEL IN FORCE SPACE In- plane Hysteretic Behavior of Thin-Walled Circular Steel Columns From the elastic theory, Timoshenko and Gere (1961), the elastic buckling