60 S.Z. Shen p/g are shown in Fig.5. It's seen that the curves sustainedly go down, do not approach a limit even as p/g=2, meanwhile the limit load has dropped to a rather low value of about 30% of the case of symmetrical loading. Based upon regression analysis the coefficient K2 of considering the effect of unsymmetrical loading can be given as Eqn.9. This formula is applicable for both net systems. Kz = 1 / [ 1 + 0.956 p/g + 0.076 (p/g)2 ] (applicable for p/g=0-2.0) ( 9 ) " ~ moveable hinge 15 N fixed hinge z lO - ., .,~ .~ .~. ~'- ~ 5 - f/L = 1/6 ~"-" ~~.~~~ f/L = 1/7 f/L= 1/8 p/g 0 0 015 i ' ' 1.5 2 Figure 5 : Limit loads of reticulated shallow shells with increase of p/g In reference to the analytical formula of linear theory for domes and based upon regression analysis, the design formula for predicting limit load of reticulated shallow shell can adopt rather simple form: x/BD For triangular system qcr = 1.29K2 ( 10 ) R1R2 4"D For orthogonal system qcr = 1"07K2 ( 11 ) R1R 2 In which R1 and R 2 are the radiuses of curvature in two directions, respectively, and the coefficient K 2 is given in Eqn.9. The effect of initial imperfections has been considered in the formulas. STABILITY OF RETICULATED SADDLE SHELLS The complete load-deflection response of the reticulated HP shells are varied with variation of geometrical and structural parameters such as the raise-span ratio, the net system, the rigidity of edge beams and etc Some load-deflection curves may rise sustainedly with no critical point emerging. Some curves may have bifurcation point appearing, but the load continues up with the rigidity matrix of the structure keeping positive definite. And in some other curves buckling of limit-point type may occur, but the load rises again after certain post-buckling path downwards. HoweVer, there exists a common character for the load-deflection curves of HP shells: the load has a general tendency to keep going up, and from practical viewpoint the load-capacity of the shells is maintained. As an example, the load-deflection curves of HP shells of L=60m with different raises (H = 6,9,12,15 and 18m) are shown in Fig.6a. It demonstrates the specific characteristics of the shells of negative Gaussion curvature. Further more, it can be supposed that the feature of monotonous rise would be revealed more obviously for load-deflection curves of the practical shells with initial imperfections. Design Formulas for Stability Analysis of Reticulated Shells 61 It seems rational to conclude that the stability problem is not significant for reticulated HP shells, and as a necessary substitutive measure the rigidity of the shell should betaken as a main structural property to be checked in practical design. The maximum deflections of the shells with different raises under service load (2kN/m 2) are shown in Fig.6b. It's seen that the rigidity of HP shells with H=9m and 6m is obviously not enough. Figure 6 : a. Load-deflection curves of reticulated HP shells with different raises b. Maximum deflections under service load of these shells CONCLUSIONS 1. Based upon the complete load-deflection analysis for more than 2800 examples of reticulated shells of prototype the varied and colorful structural behaviors developing with the loading process, the practical mechanism of structural instability and the complex effects of different factors were revealed rather thoroughly for different types of reticulated shells. 2. Based upon the regression analysis of the plentiful data obtained from the parametrical analysis as described above design formulas for predicting limit loads of reticulated domes, reticulated vaults with different supporting conditions, as well as reticulated shallow shells, rather simple for application but obtained on the basis of accurate theoretical procedure, were proposed. 3. For reticulated saddle shells it's suggested just to carry out routine rigidity check instead of the complicated stability analysis. REFERENCES Chen X. and Shen S.Z (1993). Complete Load-Deflection Response and Initial Imperfection Analysis of Single-Layer Lattice Dome. International Journal of Space Structures 8:4, 271-278 Wang N., Chen X. and Shen S.Z. (1993). Geometric and Material Non-linear Analysis of Latticed Shells of Negative Gaussion Curvature. Space Structures 4. London. 649-655 Shen S.Z. and Chen X. (1999). Stability of Reticulated Shells. The Science Publisher, Beijing, China 62 S.Z. Shen APPENDIX: Formulas for Equivalent Rigidities of Reticulated Shells The net systems used for reticulated shells can be classified into three basic types as shown in the attached figure. Attached Figure: Three typical net systems The equivalent rigidities in two main directions can be calculated as follows: 1 .For net system ( a ) and system ( b ) with single diagonal B11 EA1 EAc EI1 EIc = + sin4 a Dll = + sin4 a A 1 A c A 1 A c B22 EA 2 EA~ E12 EZc 4 = + cos4a 022 = + cos a A 2 A~ A 2 Ac 2.For net system ( b ) with double diagonals EA~ EAc E11 El c Bll = + 2 sin 4 ct D~ = + 2 sin 4 a A 1 Ac A1 Ac B22 EA 2 EA c EI 2 EI = + 2 cos4 ct D22 = + 2 cos 4 a A 2 A~ A 2 Ac 3.For net system ( c ) EA 1 EAc Bll + 2 sin4 a Dll E11 Elc = = + 2 sin 4 ct A 1 Ac A1 A c B22 2 EAc cos4 a 022 = 2 EI c = COS 4 a Ac Ac In the formulas A1,A 2 and A c are the cross-section areas of members in direction 1 and 2 and of diagonals, respectively, I l, I 2 and Ic are the corresponding moments of inertia, the intervals between members A 1 , A 2 and A c , as well as the inclination angle a are as shown as in the figure. DUCTILITY ISSUES IN THIN-WALLED STEEL STRUCTURES T. Usami 1, Y. Zheng 1, and H.B. Ge 1 1Department of Civil Engineering, Nagoya University, Nagoya, 464-8603, JAPAN ABASTRACT The ductility of thin-walled steel box stub-columns under compression and bending is studied in this paper through extensive parametric analyses, and empirical ductility equations are developed. The equations for isolated plates and pipe stub-columns proposed in the previous studies are also presented. On this basis, a simplified ductility evaluation procedure is proposed for practical steel structures with thin-walled box or pipe sections. An inelastic pushover analysis is employed and a failure criterion is introduced. The implementation of the proposed procedure is demonstrated by application to some cantilever columns and a one-story frame. Moreover, the computed results are compared with the ductility estimations through cyclic analyses reported in the literature, which leads to the validation of the proposed method. KEYWORDS Thin-walled steel structure, Ductility, Pushover analysis, Stub-column, Residual stress, Initial deflection, Box section, Pipe section, Frame, Cyclic loading. INTRODUCTION Thin-walled steel columns and frames have been widely used as substructures in urban highway bridges, suspension and cable-stayed bridge towers in Japan as well as some other countries. But the need for evaluating the seismic performance, such as the ductility capacity, of such structures has come into focus following the damage and collapse observed dr~ the 1995 Hyogoken-nanbu earthquake (Fukumoto 1997; Galambos 1998). Steel beam-column members employed in bridge structures are characterized by the use of relatively thin plates, which makes these structures vulnerable to damages caused by the local and overall interaction buckling. However, the task of accounting for such buckling can be formidable for a practical use where the balance between reliability and simplicity is required. 63 64 T. Usami et al. A simplified ductility evalUation method for steel columns and frames composed of box sections was previously proposed by the authors (Usami et al. 1995). An inelastic pushover analysis is utilized in the method and the structural ultimate state is assumed to be attained when the compressive flange strain of the most critical part reaches its failure strain. However, the method employs an empirical equation based on isolated, simply supported plate under compression (Usami et al. 1995) to calculate the failure strain, and consequently leads to somewhat conservative predictions for structures composed of moderately thin plates. This is for the reason that the interactive effects between adjacent component plates at their junctions are neglected. In this paper, aiming at proposing more refined empirical equations for failure strains, thin-walled steel box stub-columns are studied under combined action of compression and bending. Extensive parametric analyses are carried out to investigate the effects of some parameters on the behavior of stub-columns with and without longitudinal stiffeners. An elasto-plastic large deformation FEM analysis is employed. Based on the parametric analyses, empirical equations for the ductility of box stub-columns are developed. Besides, the ductility equations for isolated plates in compression and short cylinders in compression and bending proposed in the previous studies (Usami et al. 1995; Gao et al. 1998a) are also presented. By using the equations based on stub-columns, the previous ductility evaluation procedure for box-sectioned structures (Usami et al. 1995) is refined and meanwhile, is extended to both box and pipe-sectioned structures. A one-story frame with stiffened box sections and several cantilever columns with unstiffened box sections, stiffened box sections, and pipe sections are investigated as examples to demonstrate the application of the procedure. Moreover, the computed results are compared with previous results obtained through cyclic tests or numerical analyses (Usami 1996; Gao et al. 1998b; Nishikawa et al. 1999). The comparison illustrates the validity of the proposed method. DUCTILITY OF BOX STUB-COLUMNS Numerical Analytical Model Both the box stub-columns with and without longitudinal stiffeners are studied. The analytical models of such stub-columns are shown in Fig. 1, which represent a part of a long column between the diaphragms. Due to the symmetry of geometry and loading, only a half or a quarter of the stub-column is analyzed. A simply supported boundary condition is assumed along the column end plate boundaries to simulate the local buckling mode of a long column, which would deforms into several waves along the length. To Y Y P P Web I (a) Unstiffened (b) Stiffened [ s.s.: simply Supported Edge Figure 1" Analytical model of box stub-columns Ductility Issues in Th&-Walled Steel Structures 65 Figure 2: Residual stresses impose a rotation of the edge, the end sections are constrained as rigid planes by using the multi-point constraint (MPC) boundary conditions, and the rotation displacement is applied at any node on the sections. The bending moment is obtained as the reaction force of the node. The general FEM program ABAQUS (1998) and a type of four-node doubly curved shell element (S4R) included in its package are employed in the elasto-plastic large deformation analysis. An idealized rectangular form of residual stress distribution in each unstiffened panel, stiffened panel, and stiffener plate, is adopted due to the welding (see Fig. 2). The initial geometrical deflections are also considered. For unstiffened stub-columns, the shape is assumed to be sinusoidal in both flange and web plates (see Fig. 3(a)). The maximum values ofthe initial deflections in the flange and web are assumed to be B/500 and D/500 (where B and D are the breadth and depth of the box section), respectively. The directions of the initial deflections are assumed inward for flange plates while outward for web plates. The assumed initial deflection shape in the flange plate of stiffened stub-columns (Fig. 3(b)) are given by following equations: where 8=~5a+8 L (1) a ,000 sinI; 1( ) = sin ~Z y cos • Z 150 ~ ~ (3) in which cY c denotes the global initial deflections; CYL represents the local initial deflections; a is the length of the stiffened stub-columns; n is the number of the subpanels divided by the stiffeners; m is the number of half-waves of the local initial deflections in the longitudinal direction, which is assumed as an integer giving the lowest failure strain and will be further discussed below. The initial deflections in the web plates are calculated by replacing B and z in Eqs. (2) and (3) by D and x, respectively, but assumed in opposite direction (outward). A kind of steel stress-strain relation including a strain hardening part, proposed by Usami et al. (1995), is utilized in this study to define the material characteristics (see Fig. 4). Here, % and 6y denote the yield stress and strain, respectively; E is the elastic modulus (i.e., Young's modulus); 6,, is the strain at the onset of strain hardening; E~ is the initial strain hardening modulus; and E' is the strain hardening modulus assumed as E' = E,~ exp(-~ s - o%t ) (4) 6y 66 T. Usami et al. Figure 3: Initial deflections where 2j is a material coefficient. Mild steel SS400 (equivalent to ASTM A36) is utilized in the analysis of stub-columns, for which Cry = 235 MPa, E = 206 GPa, t,' = 0.3, e n = 10 e y, 2j = 0.06, and En=E/40. In this study, the ductility of the stub-column is evaluated by using the failure strain, 6u/zy, which is defined as a point corresponding to 95% of the maximum strength after the peak in the bending moment versus average compressive strain curve (Usami and Ge 1998). Parametric Study Figure 4: Material model The behavior of thin-walled box stub-columns subjected to compression and bending is considerably affected by the magnitude of axial load, P/Py (Py is the squash load), and the flange width-thickness ratio, RI, which is defined as a~ B I12(1- v2) IO'y Ry : : t 4n2x 2 E (5) in which O'cr is the elastic buckling stress; n is the number of subpanels (for unstiffened plate, n = 1). For stub-columns with stiffeners, the stiffener's slenderness ratio, 2 s , is another key parameter, given by: - 1 a 1 ~-~y A" = x/-Q r, n 3r E (6) 1 Q 2-~f [13 - ~/13z - 4Rf ] (7) 13 - 1.33Rf + 0.868 (8) in which rs is the radius of gyration of a T-shape cross section consisting of one longitudinal stiffener and the adjacent subpanel and Q is the local buckling strength of the subpanel plate (Structural Stability 1997). An alternative parameter reflecting the characteristics of the stiffener plate is the stiffener's relative flexural rigidity, y, which is interdependent on 2,. Thus, in the present study, only 2 s is considered in the ductility equations. Ductility Issues in Thin-Walled Steel Structures TABLE 1 TABLE 2 Parameters of Thin-walled Parameters of Stiffened Unstiffened Box Stub-columns Box Stub-columns D/B at = a/B Rf D/B a = a/B Rf ?" / y * 3/4, 1.0, 0.5, 0.7, 0.2, 0.4, 0.45, 0.5, 0.67, 1.00, 0.5, 0.7, 0.3, 0.35, 0.4, 0.45, 1.0, 4/3 1.0 0.55, 0.6, 0.8 1.33 1.0, 1.5 0.5, 0.55, 0.6, 0.7 3.0 67 ~s 0.180 "~ 0.751 Nevertheless, to propose ductility equations for comprehensive applications, the influence of box cross-sectional shape (say, square or rectangle) and the aspect ratio (a =a/B) has to be surveyed. And for stiffened stub-columns, the critical local initial deflection mode along the length direction giving the lowest ductility should be first determined. The thickness of the plates is assumed as 20ram and the considered axial force, P, ranges from O.OPy to 0.SPy. Other pertinent parameters are given in Tables 1 and 2, where 9" * represents the optimum value of 9" obtained from elastic buckling theory ("DIN 4114" 1953). Through parametric analyses, following conclusions are drawn for unstiffened stub-columns: (1) The effects of the cross-sectional shapes and column aspect ratios on the stub-column ductility are insignificant and the present empirical equation is based on the models with square sections and aspect ratios equal to 0.7; (2) Referring to the computed e,/ey versus R z and P/Py relations presented in Fig. 5, it is observed that the failure strain decreases as the increase of either R z or P/Py; (3) Considering the effects of axial loads, an equation of failure strain, eu/ey, versus flange width-thickness ratio, R z, is fitted as follows: ~;, 0.108(1- P / Py )1.09 ~.~ : (Rf -0.2) 3"2~ + 3.58(1- P / py)O.839 < 20.0 (9) The applicable range of this equation is R/= 0.2 0.8, D/B = 0.75 1.33, and P/Py = 0.0 0.5. It should be noted that when the failure strain, 6u/ey (which is the average strain in the compressive flange), exceeds 20.0, the local maximum strain would be very large (say, 5% or larger) and the numerical analysis results would become unreliable. Thus, the upper bound of 6,,/6y is limited as 20.0 at present time although the consequent prediction will be on the safe side for some cases. As for the stiffened stub-columns, the observations from the parametric analysis can be concluded as: (1) The critical local initial deflection mode along the length of stiffened stub-columns varies with different aspect ratios and the corresponding number of half-waves (m) is found as 2, 3, 4 and 5 for aspect ratios of 0.5, 0.7, 1.0 and 1.5, respectively; (2) The buckling mode of stub-columns has almost same shapes as the assumed initial deflection mode; (3) The influences of box cross-sectional shape and the aspect ratio on the ductility of stub-columns are not obvious and for simplification, they can be neglected in the design formulas of failure strain; (4) The effects of flange width-thickness ratio and stiffener's slenderness ~, ~0.1s ratio should be considered together and a combined parameter *-/,~s is introduced. Inversely proportional relations of the failure strains to this combined parameter and the axial load are found (see 018 Fig. 6). On this basis, an equation of eu/~y versus Rye. s " , considering the effect of axial load, are fitted as follows: ~'u _ 0"8(1- P / Py )0"94 w /,• ~- 0.18 ~'Y ~"'~f"s - 0"168) lzzs + 2.78(1 - P / P, )0.68 ~_~ 20.0 (10) Here, R/ranges from 0.3 to 0.7, ~, is in a scope from 0.18 to 0.75, and P/Py is between 0.0 and 0.5. And 68 T. Usami et al. Figure 5 Failure strains of unstiffened stub-columns Figure 6 Failure strains of stiffened stub-columns this equation is applicable to stiffened box stub-columns with a from 0.5 to 1.5 and B/D from 0.67 to 1.33. Moreover, it should be noted that this equation is fitted to give slightly smaller prediction of failure strains for the cases with smaller -" = 0.~s K:% . This is for the reason that the numerical results of present study are based on monotonically loading conditions and when applied to long columns with small 018 R/~ " , they are found to yield larger ductility predictions compared with the cyclic experimental and numerical results as presented later. DUCTILITY OF ISOLATED PLATES UNDER UNIAXIAL COMPRESSION For comparison, this paper also presents the failure strain equations based on isolated plates under uniaxial compression (Usami et al. 1995; Usami and Ge 1998). They are defined as follows: Unstiffened olates: ~" 0.07 = + 1.85 _< 20.0 (11) " % (R: -0.2y "~ Stiffened nlates: e, 0.145 : + 1.19 < 20.0 (12) 6, (x-, - 0.2) TM Equation (11) is plotted in Fig. 5 and some computed results of stiffened plates (Usami and Ge 1998) are ~ ~'- o 18 also plotted in Fig. 6, in the form of E / E y versus K/~s " 9 It is observed that the failure strains of stub-columns subjected to compression and bending are larger than those of isolated plates under pure compression. When the axial load is so large as to approach the pure compression state two procedures will give similar predictions. DUCTILITY OF SHORT CYLINDERS The ductility of thin-walled steel short cylinders in compression and bending has been also investigated in a previous study (Gao et al. 1998a). Analytical models similar to those used for box stub-columns, which have been presented above, were employed for the cylinders. Main parameters controlling their behaviors are found to be the magnitude of the axial force and the radius-thickness ratio parameter, Rt, which is in the form of Rt = 6y = ~/3(1 - v 2) oy d (13) o~, E 2t Here d and t denote the diameter and thickness of the cylinder, respectively. And an empirical equation is proposed as follows: Ductility Issues in Th&-Walled Steel Structures ~,,, O.120+4P/Py) 6 f = (R t - 0.03) 1"4s (1 + P / P,)5 + 3.6(1 - P / P,) <__ 20.0 69 (14) DUCTILITY EVALUATION PROCEDURE FOR THIN-WALLED STEEL STRUCTURES By using the empirical equations of failure strains given above, a ductility evaluation procedure is proposed for practical structures composed of thin- walled steel beam-column members. It is applicable for the design of a new structure or evaluation of a existing structure, which are in the form of cantilever-typed columns or framing structures. The procedure involves the following steps: 1. Based on the general layout and loading condition of the structure, establish the analytical model as Figure 7: Pushover analysis model shown in Fig. 7, by using beam elements, which facilitate the FEM modeling procedure but do not account for local buckling. Neither the residual stresses nor initial deflections are take into consideration. The material model defined in Fig. 4 is also utilized for the pushover analysis. 2. Carry out a planar pushover analysis. This procedure involves applying the constant vertical loads and incrementally increased lateral loads to represent the relative inertia forces which are generated at locations of sustained mass. An elastoplastic large displacement analysis is employed to account for the second-order effects. 3. The pushover analysis is terminated once the failure criterion is attained and this state is taken as the ultimate state of the structure, based on which the ductility capacity, 8u/By, of the structure can be determined. Like the previous study (Usami et al. 1995), the failure of a structure composed of the thin-walled steel box members is assumed when the average strain over an effective failure length in the compressive flange (or in the maximum compressive meridional fiber for pipe sections) reaches its failure stain (Eqs. (9), (10) and (13)). The effective failure length, I,, of a box-sectioned member is assumed as the smaller one between 0.7 times of the flange width and the distance between two adjacent diaphragms (Usami et al. 1995). For pipe-sectioned structures, based on the observations in the previous studies (Gao et al. 1999a and 1999b), an empirical equation is proposed here to define the effective failure length: 1 -1)d (15) I t = 1.2(Rt ~.0s The critical parts could be more than one place in a framing structure and all of them should be checked (see Fig. 7(b)). In a thin-walled steel structure, however, the excessive deformation tends to intensify in a local part and consequently the redistribution of the plastic stress becomes unexpected. Thus, once the failure criterion at any one of the critical parts is satisfied, the ultimate state of such a structure is though to be reached. NUMERICAL EXAMPLES To demonstrate its implementation, the proposed ductility evaluation procedure is applied to some cantilever-typed columns with box or pipe sections and a one-story rigid frame composed of box section . such structures has come into focus following the damage and collapse observed dr~ the 199 5 Hyogoken-nanbu earthquake (Fukumoto 199 7; Galambos 199 8). Steel beam-column members employed in bridge. (Usami et al. 199 5). For pipe-sectioned structures, based on the observations in the previous studies (Gao et al. 199 9a and 199 9b), an empirical equation is proposed here to define the effective. neglected. In this paper, aiming at proposing more refined empirical equations for failure strains, thin-walled steel box stub-columns are studied under combined action of compression and bending.