Analysis of the Collapse of Long-Span Reticulated Shell Structures Under Multi-Dimensional Seismic Excitations

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1 Analysis of the Collapse of Long-Span Reticulated Shell Structures Under Multi-Dimensional Seismic Excitations Ming-Fei Yang Key Laboratory of C&PC Structures of the Ministry of Education, Southeast University, Nanjing , China School of Civil Engineering, Anhui University of Science and Technology, Huainan , China Zhao-Dong Xu, Xing-Huai Huang and Han-Hu Ye Key Laboratory of C&PC Structures of the Ministry of Education, Southeast University, Nanjing , China (Received 1 August 2012; accepted 25 April 2013) The collapse processes of three typical long-span reticulated shell structures were simulated using nonlinear dynamic finite element analysis under strong seismic excitations. The plastic kinematic hardening model, which considers failure stain, was adopted for simulating steel. Both geometric and contact nonlinearities were considered in this study. The three failure states i.e., dynamic local bucking, dynamic overall buckling, and whole collapse were identified in accordance with the analysis results. Taking the Schwedler reticulated shell structure as an example, seismic waves were applied to the structure in three directions. The critical loads were obtained by the incremental dynamic analysis method (IDAM), and some critical state indices were obtained according to the dynamic responses. The results showed that all the critical indices need to be considered simultaneously in order to judge the dynamic collapse states. NOMENCLATURE M, C, K Mass matrix, damping matrix and stiffness matrix of long-span reticulated shell structures ü(t), u(t), u(t) Acceleration, velocity, and displacement array P(t) Excitation vector t, t cr Time step and critical time step ω max Largest circular frequency T min Smallest fundamental period T Period of structures 1. INTRODUCTION Long-span reticulated shell structures currently provide a widely-used structural solution to the problem of spanning large uninterrupted distances. Due to the increase in the size of such structures, it is imperative to make intensive studies about the plastic properties and the development law of plastic displacement, the buckling behaviours of long-span shell structures, and the collapse mechanism caused by strong earthquakes. 1 Nevertheless, it is difficult to investigate the collapse mechanism of long-span reticulated shell structures when the member is in the plastic state. However, finite element technology can solve the problem by considering the nonlinear material and geometric factors, and this technology has been used widely in analysis. In recent years, a great deal of research has been focused on assessing the seismic response of long-span spatial structures. 2 6 The strength of reticulated spherical domes has been generally associated with the inelastic buckling of slender members, and more and more partially restrained connections between structural members. This conclusion was verified by Battista et al. in The elastic-plastic dynamic response of single-layer reticulated shells under strong earthquake excitation, static, and dynamic loads was analysed by Li and Chen in Through the numerical example, the stability was influenced by the geometric parameters, which define the mesh of elements. In 2003, the seismic time history responses of long-span reticulated shell structures were obtained by Guo and Shen using ANSYS software, and the failure mechanism subjected to severe earthquakes was first described. 9 Taniguchi 10 quantified the relationships between the dynamic characteristics of lattice structures and those of earthquakes. In the next year, the shaking table tests of vertical and horizontal directions on long-span reticulated shell structures, with viscoelastic multi-dimensional earthquake isolation and mitigation devices, were studied by Xu And after that, a computationally effective method for evaluating the dynamic buckling and post buckling of thin composite shells was verified by Chamis in However, as of yet there is still not a unified standard to describe the dynamic collapse process and determine the collapse index of long-span reticulated shell structures. In order to judge the dynamic collapse states and obtain the dynamic collapse indices of long-span reticulated shell structures, the collapse process was numerically analysed using ANSYS/LS-DYNA software. Based on the numerical analysis results, the dynamic collapse states were described in detail. Finally, the collapse indices were proposed by the incremental dynamic analysis method (IDAM). International Journal of Acoustics and Vibration, Vol. 19, No. 1, 2014 (pp ) 21

2 2. MOTION EQUATION AND MATERIAL MODEL CONSIDERING STRAIN FAILURE ANSYS/LS-DYNA is commonly used to simulate the collapse behaviour of structures under strong seismic excitations, and it is powerful when solving motion equations. The motion equations were solved by the central difference method, in which the system damping was considered simultaneously. As for the central difference method, the velocity vector and the acceleration vector were expressed with some combination of the displacement vector in the process of solving the equations of motion. Meanwhile, the differential equations were transformed into algebraic equations. Then, the corresponding recursive formula was obtained at each time interval. Finally, the response of the whole structure was identified in this way. Due to the centralized mass matrix, motion equations were non-coupled, and the whole matrix need not be integrated, which saved storage space and computation time. In order to solve the convergence problem in the analysis, the ANSYS/LS-DYNA software used the variable time step incremental solution method at every time interval to control the current configuration stability condition Motion Equation The motion equation of the long-span reticulated shell structures can be expressed as Mü(t) + C u(t) + Ku(t) = P(t); (1) where M, C, and K represent the mass matrix, the damping matrix, and the stiffness matrix of the long-span reticulated shell structures, respectively; ü(t), u(t), and u(t) stand for acceleration, velocity, and displacement array, respectively; and P(t) is the excitation vector. As is known, when the time interval t is very small, the displacement of structures can be determined through the finite approximation difference, so the central difference method can be used for solving motion equations. First-order and secondorder central difference formulas were simultaneously substituted into Eq. (1), and then the displacement equation u(t+ t) at time t + t was obtained as: ( 1 ( t) 2 M + 1 P(t) ) 2 t C u(t + t) = ( Ku(t) 2 ) ( t) 2 Mu(t) ) u(t t). (2) ( 1 ( t) 2 M 1 2 t C The displacement, velocity, and acceleration were obtained by substituting the initial conditions of the structures into Eq. (2), and the accuracy and efficiency of the solution to displacement, velocity, and acceleration were relatively higher when the time interval t was small enough. In addition, the largest time interval was determined by the stability of the algorithm, which can be written as: { t t cr t cr = 2 ω max = Tmin T ; (3) where t and t cr are time step and critical time step, respectively; ω max is the largest circular frequency; T min is the smallest fundamental period; and T represents the period of structures Material Model The numerical model of the long-span reticulated shell structure was constructed by the ANSYS/LS-DYNA software. The peripheral joints of the structure were braced by the square column, for which the cross-section size was set as 500 mm 500 mm. The adjacent columns were connected by a beam with a cross-section size set as 300 mm 600 mm. The strength grades of concrete were both set as C40 for the column and beam. Then, the ground was assumed to be rigid. Considering the member s mechanical characteristics of long-span reticulated shell structures, the Beam161 element was selected to simulate the member element. The Hughes-Liu algorithm was adopted to calculate the structural response. In addition, the Beam161 element from the Plastic Kinematic Hardening model was used to consider failure stain. When the strain or deformation of an element increases to some extent in nonlinear dynamic analyses, the element will become an invalid element, 15 and it will be out of work. Therefore, the material failure criteria must be defined before the simulation of the collapse process of the long-span reticulated shell structures can occur. 16, 17 In this paper, the value of the material failure strain of the steel was set as 0.05, which means those elements would fail and be removed when the material strains reached In order to study the stability of the reticulated shell structure, all the bracing structures must not be damaged in the numerical simulation. 3. COLLAPSE PROCESS ANALYSIS The collapse of long-span reticulated shell structures is a dynamic, continuous process, rather than a static process, in which buckling and strength failure occur simultaneously. As for the stability of motions, the fundamental investigations and definition in the case of perturbed initial conditions dates back to Liapunov. 18 Applications and modern points of view regarding the stability of nonlinear dynamic systems were discussed by Li et al. 19 The buckling of shells very often has a local form or an overall form, and buckling does not depend essentially on the boundary conditions. This is particularly true in the case of the non-uniform geometry of a shell and the non-uniform state of stresses. In other words, the buckling can be determined by the stress state and the shape variation of a shell. 20 The collapse processes were divided into different structure states from a macroscopic perspective, through the above analysis, and there were three stages of the collapse processes. The first stage was the dynamic local buckling (i.e., local depression), which included node buckling, member buckling, and line buckling. Node buckling indicates that the combined axial forces in all of the members attached to a joint cannot balance the external load. When this happens, the node experiences a much larger displacement than the neighbouring nodes. Member buckling refers to the buckling of one member under axial pressure. Line buckling means all the nodes and members in a ring are involved in the loss of stability in a long-span reticulated shell structure, which can imply the local buckling of the structure. The second stage was dynamic overall buckling. This means that the loss of stability simultaneously appeared at some location. Meanwhile, the most typical characteristic was the rapid downward movement of the vertex of the structure when the overall buckling occurred. The last stage was whole 22 International Journal of Acoustics and Vibration, Vol. 19, No. 1, 2014

3 Figure 1. The Schwedler reticulated shell structure collapse processes: (a) peripheral joints entering plastic state, (b) local buckling, (c) overall buckling, and (d) whole collapse. Table 1. Reticulated shell structure types and components. Circumfer- The Reticulated Radial Uniform Shell Shell ential Sub- remaining shell main ribs loads span height sidiary ribs members types (mm mm) (mm mm) (mm mm) (kn/m 2 ) (m) (m) Schwedler reticulated φ150 5 φ120 5 φ shell K6 reticulated φ150 5 φ shell Lamella reticulated φ shell Note: The reticulated shell structures are braced with columns, the top of columns link on all the peripheral joints and the bottom consolidate on the ground. collapse, taking the state of one member or joint reaching the ground as the basis. The El-Centro wave in 1940 was used, and the length of 20 seconds was selected for simulation, in reference to the China Code for the Seismic Design of Buildings (GB ). Members and load distributions of three typical longspan reticulated shell structures are listed in Table 1. The collapse processes of the Schwedler reticulated shell, the K6 reticulated shell, and the Lamella reticulated shell (under the El-Centro wave) are shown in Figs. 1 3, respectively. The dynamic local buckling, the dynamic overall buckling, and the whole collapse states were both observed during the collapse process for three different reticulated shell structures. First, the peripheral joints connected to the columns entered the plastic state, as shown in Figs. 1(a), 2(a), and 3(a). The dynamic local buckling occurred after some joints on the second and third row peripheral members along radial direction also entered the plastic state, but there was no member fall failure, which can be seen in Figs. 1(b), 2(b), and 3(b). The joint stress state in this process is shown in Fig. 4. As can be seen in Fig. 4, the joint entered the yield state before the member entered the yield state. This means that the joint failed first, rather than the member. The local depressions signify continuous and simultaneous expansion along the radial and circumferential direction, and they are linked into a line or region. Therefore, some members and circumferential subsidiary ribs began to fail, and the vertex associated with structure began to move down in an obvious way. This resulted in dynamic overall buckling, as shown in Figs. 1(c), 2(c), and 3(c). With the failed numbers of subsidiary ribs increasing, some radial main ribs began to fail, which caused the structure to finally collapse, as seen in Figs. 1(d), 2(d), and 3(d). The collapse processes were similar for each of the three typical reticulated shell structures, though there were also some different points between them. Compared to the collapse processes of the Schwedler reticulated International Journal of Acoustics and Vibration, Vol. 19, No. 1,

4 Figure 2. The K6 reticulated shell structure collapse processes: (a) peripheral joints entering plastic state, (b) local buckling, (c) overall buckling, and (d) whole collapse. shell structure, the period of collapse from the dynamic local buckling to the whole collapse was shorter for the K6 reticulated shell structure. However, for the Lamella reticulated shell structure, the integrity was well-preserved after the dynamic local buckling occurred. This is because there were no ribs, and then the structure collapsed as a whole to the ground. In a word, the dynamic responses and the dynamic characteristics of long-span reticulated shell structures were affected in an obvious way by the failure of the elements, especially for the rib-less Lamella reticulated shell structure. 4. COLLAPSE INDICES ANALYSIS Incremental dynamic analysis (IDA) is a parametric method in which a structure is subjected to a series of nonlinear timehistory analyses of increasing intensity, with the objective of attaining an accurate indication of the nonlinear dynamic response of a structure under earthquake excitation In order to accurately guage the collapse indices, three kinds of seismic waves including the El-Centro wave in 1940, the Taft wave, and the artificial wave in grade II were selected for simulation; the duration of the seismic waves was set at 20 seconds. Taking the Schwedler reticulated shell structure (whose components are shown in Table 1) as an example, the peak ground acceleration (PGA) of the seismic waves was adjusted according to a certain proportion by the IDA. First, the seismic wave was adjusted based on 620 gal (the acceleration amplitude of a 9-degree, rare earthquake), which was all used to excite the structure in three directions for each seismic wave. Then, the time history analysis was carried out in the simulation process. The critical points corresponding to the structure from the dynamic local buckling state to the whole collapse state can be found on the curves of the seismic time history responses of the vertex, and the collapse indices were derived from these points value. This study analysed the structural response by using fluctuation amplitude. Taking the vertex as the reference point, the displacement fluctuation amplitude was calculated by (U 2 U 1 )/(L U 1 ), which is similar to velocity and acceleration. Where U 2 is the vertex displacement response of the structure under the analysis excitations, U 1 is the displacement response of the reference point when the PGA of the excitation is 620 gal, and L is the span of the reticulated shell structure Collapse Indices Under the El-Centro Wave In the case of El-Centro wave, the critical loads according to the three states of collapse processes were obtained by simulation, as listed in Table 2. The seismic time history responses of the vertex are shown in Fig. 5. From Fig. 5(a), it can be seen that the vertex of the Schwedler reticulated shell structure vibrated around the equilibrium position when the PGA was 24 International Journal of Acoustics and Vibration, Vol. 19, No. 1, 2014

5 Figure 3. The Lamella reticulated shell structure collapse processes: (a) peripheral joints entering plastic state, (b) local buckling, (c) overall buckling, and (d) whole collapse. Figure 4. The joint forced state. 620 gal, and the maximum vertex displacement was m at 5.60 s. When the PGA was 2590 gal, the vertex of the structure began to deviate from the position of equilibrium, and jump occurred at s. The maximum vertex displacement was m, the maximum displacement fluctuation amplitude of the vertex was 135.9%, and the structure began to buckle locally. When the PGA was 3150 gal, the vertex began to jump at 6.12 s; the maximum vertex displacement was m at s, the maximum displacement fluctuation amplitude of the vertex was 762.4%, and the structure began to buckle overall. When the PGA was 6300 gal, the vertex began to jump at 2.18 s, the maximum vertex displacement was m at s, the maximum displacement fluctuation amplitude of the vertex was %, and the whole structure collapsed. From Fig. 5(b), when the PGA was 620 gal, the maximum vertex velocity was m/s at 5.56 s. When the PGA was 2590 gal, the maximum vertex velocity was m/s at s, and the structure began to buckle locally. The maximum velocity fluctuation amplitude of the vertex was 4.8%. When the PGA was 3150 gal, the maximum vertex velocity was m/s at s, the maximum velocity fluctuation amplitude of the vertex was 17.0%, and the structure began to buckle overall. When the PGA was 6300 gal, the maximum vertex velocity was m/s at 2.42 s, the maximum velocity fluctuation amplitude of the vertex reached 68.2%, and the whole structure collapsed. As seen in Fig. 5(c), when the PGA was 620 gal, the maximum positive acceleration of the vertex was 16.4 m/s 2 at 5.60 s, and the negative was 15.5 m/s 2 at 5.54 s. When the PGA was 2590 gal, the structure began to buckle locally, the maximum positive acceleration of the vertex was 56.9 m/s 2 at 5.30 s, and the negative was 43.0 m/s 2 at 5.52 s. The maximum positive acceleration fluctuation amplitude of International Journal of Acoustics and Vibration, Vol. 19, No. 1,

6 Figure 5. Seismic time history responses of the vertex under the El-Centro wave: (a) displacement responses, (b) velocity responses, and (d) acceleration responses. Table 2. Seismic time history responses of the vertex under the EL-Centro wave. Earthquake Maximum Maximum Maximum Maximum wave displacevelocity positive negative PGA ment acceleration acceleration (gal) (m) (m/s) (m/s 2 ) (m/s 2 ) the vertex was 6.2% and the negative was 4.4%. When the PGA was 3150 gal, the structure began to buckle overall, the maximum positive acceleration of the vertex was 62.5 m/s 2 at s, and the negative was 55.7 m/s 2 at 3.44 s. The maximum positive acceleration fluctuation amplitude of the vertex was 7.0%, and the negative was 6.5%. When the PGA was 6300 gal, the maximum positive acceleration of the vertex was 78.9 m/s 2 at s, and the negative was 82.0 m/s 2 at 4.80 s. The whole structure collapsed, and the maximum positive acceleration fluctuation amplitude of the vertex reached 9.5%, while the negative reached 10.7% Collapse Indices Under the Taft Wave In the case of the Taft wave, the critical loads according to the three states of collapse processes were obtained by simulation, as listed in Table 3. The seismic time history responses of the vertex are shown in Fig. 6. From Fig. 6(a), when the PGA was 620 gal, the vertex of the Schwedler reticulated shell structure vibrated around the equilibrium position, and the maximum vertex displacement was m at 6.78 s. When the PGA was 2976 gal, the vertex of the structure began to deviate from the equilibrium position. The jump occurred at s, the maximum vertex displacement was m at s, the maximum displacement fluctuation amplitude of the vertex was 89.5%, and the structure began to buckle locally. When the PGA was 3224 gal, the vertex began to jump at 7.78 s, the maximum vertex displacement was m at s, the maximum displacement fluctuation amplitude of the vertex was 663.4%, and the structure began to buckle overall. When the PGA was 6820 gal, the vertex began to jump at 3.78 s, the maximum vertex displacement was m at s, the maximum displacement fluctuation amplitude of the vertex was %, and the whole structure collapsed. As seen in Fig. 6(b), when the PGA was 620 gal, the maximum vertex velocity was m/s at 6.74 s. When the PGA was 2976 gal and the maximum vertex velocity was m/s at s, the structure began to buckle locally, and the maximum velocity fluctuation amplitude of the vertex was 5.9%. When the PGA was 3224 gal, the maximum vertex velocity was m/s at s, and the maximum velocity fluctuation amplitude of the vertex was 18.0%, the structure began to buckle overall. When the PGA was 6820 gal, the maximum 26 International Journal of Acoustics and Vibration, Vol. 19, No. 1, 2014

7 Figure 6. Seismic time history responses of the vertex under the Taft wave: (a) displacement responses, (b) velocity responses, and (d) acceleration responses. Table 3. Seismic time history responses of the vertex under the Taft wave. Earthquake Maximum Maximum Maximum Maximum wave displacevelocity positive negative PGA ment acceleration acceleration (gal) (m) (m/s) (m/s 2 ) (m/s 2 ) vertex velocity was m/s at 7.80 s, and the maximum velocity fluctuation amplitude of the vertex reached 40.3%, the whole structure collapsed. From Fig. 6(c), when the PGA was 620 gal, the maximum positive acceleration of the vertex was 18.9 m/s 2 at 6.78 s, and the negative was 17.0 m/s 2 at 6.38 s. When the PGA was 2976 gal, the structure began to buckle locally, and the maximum positive acceleration of the vertex was 60.4 m/s 2 at s, while the negative was 64.0 m/s6 2 at s. The maximum positive acceleration fluctuation amplitude of the vertex was 5.5%, and the negative was 6.9%. When the PGA was 3224 gal, the structure began to buckle overall. The maximum positive acceleration of the vertex was 47.1 m/s 2 at 8.16 s, and the negative was 59.2 m/s 2 at s. The maximum positive acceleration fluctuation amplitude of the vertex was 3.7%, and the negative was 6.2%. When the PGA was 6820 gal, the maximum positive acceleration of the vertex was m/s 2 at 6.58 s, and the negative was m/s 2 at 5.74 s. The maximum positive acceleration fluctuation amplitude of the vertex reached 10.7%, the negative reached 13.4%, and the whole structure collapsed Collapse Indices Under the Artificial Wave In the case of the artificial wave, the critical loads according to the three states of collapse processes were obtained by simulation, as listed in Table 4. The seismic time history responses of the vertex are shown in Fig. 7. As seen in Fig. 7(a), when the PGA was 620 gal, the vertex of the Schwedler reticulated shell structure vibrated around the equilibrium position and the maximum vertex displacement was m at 6.30 s. When the PGA was 3410 gal, the vertex of the structure began to deviate from the equilibrium position. The jump occurred at s, the maximum vertex displacement was m at s, the maximum displacement fluctuation amplitude of the vertex was 41.4%, and the structure began to buckle locally. When the PGA was 3720 gal, the vertex began to jump at s. The maximum vertex displacement was m at s, the maximum displacement fluctuation amplitude of the vertex was 509.4%, and the structure began to buckle overall. When the PGA was 6510 gal, the vertex began to fluctuate at 6.80 s, the maximum vertex displacement was m at s, the maximum displacement fluctuation amplitude of the vertex was %, and the whole structure collapsed. International Journal of Acoustics and Vibration, Vol. 19, No. 1,

8 Figure 7. Seismic time history responses of the vertex under the artificial wave: (a) displacement responses, (b) velocity responses, and (d) acceleration responses. Table 4. Seismic time history responses of the vertex under the artificial wave. Earthquake Maximum Maximum Maximum Maximum wave displacevelocity positive negative PGA ment acceleration acceleration (gal) (m) (m/s) (m/s 2 ) (m/s 2 ) As seen in Fig. 7(b), when the PGA was 620 gal, the maximum vertex velocity was m/s at 6.26 s. When the PGA was 3410 gal and the maximum vertex velocity was m/s at 6.46 s, the structure began to buckle locally and the maximum velocity fluctuation amplitude of the vertex was 3.7%. When the PGA was 3720 gal, the maximum vertex velocity was m/s at s, the maximum velocity fluctuation amplitude of the vertex was 8.0%, and the structure began to buckle overall. When the PGA was 6510 gal, the maximum vertex velocity was m/s at 7.60 s, the maximum velocity fluctuation amplitude of the vertex reached 34.1%, and the whole structure collapsed. As seen in Fig. 7(c), when the PGA was 620 gal, the maximum vertex positive acceleration was 25.1 m/s 2 at 5.50 s, and the negative was 35.9 m/s 2 at 7.68 s. When the PGA was 3410 gal, the structure began to buckle locally, and the maximum positive acceleration of the vertex was 83.8 m/s 2 at 4.60 s, while the negative was 74.4 m/s 2 at 4.64 s. The maximum positive acceleration fluctuation amplitude of the vertex was 5.8%, and the negative was 2.7%. When the PGA was 3720 gal, the structure began to buckle overall, the maximum positive acceleration of the vertex was 91.1 m/s 2 at 5.50 s, and the negative was 84.1 m/s 2 at 7.88 s. The maximum positive acceleration fluctuation amplitude of the vertex was 6.6%, and the negative was 3.4%. When the PGA was 6510 gal, the maximum positive acceleration of the vertex was m/s 2 at 4.60 s, while the negative was m/s 2 at 5.28 s. The maximum positive acceleration fluctuation amplitude of the vertex reached 10.4%, the negative reached 4.5%, and the whole structure collapsed. Through the above analysis, it can be seen that the maximum vertex displacements were almost 0.02 m when the PGA was 620 gal for all three seismic waves. However, when the structure began to buckle locally, the maximum displacement fluctuation amplitudes of the vertex were 135.9%, 89.5%, and 41.4% under the EL-Centro wave, the Taft wave, and the artificial wave, respectively. When the structure began to buckle overall, the maximum displacement fluctuation amplitudes of the vertex were 762.4%, 663.4%, and 509.4% under the EL- Centro wave, the Taft wave, and the artificial wave, respectively. Obviously, the increase the maximum vertex displacement was relatively large in the local buckling or overall buckling state. Therefore, we can judge the state of the long-span reticulated shell structure by the rate of increase. However, in the 28 International Journal of Acoustics and Vibration, Vol. 19, No. 1, 2014

9 Figure 8. (a) The maximum vertex displacement and (b) number of failure member. state of whole collapse, part of the shell structure was in contact with the ground. This also meant that we did not need to judge by the displacement fluctuation amplitude of the vertex. For the fluctuation amplitudes of the velocity and acceleration, the rate of increase was relatively small when compared to the displacement; therefore, it was difficult to judge the critical states by them The Maximum Vertex Displacement and the Number of Failure Members The maximum vertex displacement and the number of failure members under three different earthquake waves are shown in Fig. 8. From Fig. 8(a), it can be seen that the slopes of the line between the point of the dynamic local buckling and the origin point were , , and under the EL-Centro wave, the Taft wave, and the artificial wave. Similarly, the slopes of the dynamic overall buckling were , , and , respectively. The slopes of the whole collapse critical load were , , and , respectively. From Fig. 8(b), there was one failure member for the dynamic local buckling critical load under the El-Centro wave, but there was no failure member under the Taft or artificial wave. When compared to total members, the ratios were 0.25%, 0%, and 0%, respectively. There were four failure members for the dynamic overall buckling critical load under the El-Centro wave, and one failure member under the Taft and artificial waves; the ratios were 1%, 0.25%, and 0.25%, respectively. There were 52 failure members for the collapse-critical load under the El-Centro wave, 66 failure members under the Taft wave, and 59 failure members under the artificial wave; the ratios were 12.75%, 16.5%, and 14.75%, respectively. Obviously, the slope value was smaller when the structure was in a state of dynamic local buckling. This was mainly caused by the small displacement of the vertex of the long-span reticulated shell structure. However, the slopes of the dynamic overall buckling and the whole collapse increased considerably when compared to local buckling. Therefore, we used the slopes to judge the state of the dynamic overall buckling and the whole collapse. In addition, the ratio of the failure members increased with the change of the state. Especially in the state of whole collapse, the number of the failure members increased when compared to the dynamic local buckling and dynamic overall buckling. 5. CONCLUSIONS The collapse model of long-span reticulated shell structures was described in detail through numerical analysis results under strong seismic excitations. For three different reticulated shell structures, the damaged positions appeared in the joints of three-row peripheral members, beginning from the bottom of the structure. The three types of instability states dynamic local buckling, dynamic overall buckling, and whole collapse all happened during the dynamic collapse processes, and this point was also confirmed by the results of the analysis. For collapse indices, the following conclusions were made through analysing the long-span reticulated shell structure. First, when the structure began to buckle locally, the maximum displacement fluctuation amplitudes of the vertex were 135.9%, 89.5%, and 41.4% under the EL-Centro wave, the Taft wave, and the artificial wave, respectively. When the structure began to buckle overall, the maximum displacement fluctuation amplitudes of the vertex were 762.4%, 663.4%, and 509.4%, under the EL-Centro wave, the Taft wave, and the artificial wave, respectively. Second, the slope values of the dynamic local buckling were , , and under the EL-Centro wave, the Taft wave, and the artificial wave. The slopes of the dynamic overall buckling were , , and , respectively. In addition, the slopes of whole collapse state were , , and , respectively. Third, the critical ratio of the failure member of the dynamic local buckling was between %, the critical ratio of dynamic overall buckling was between %, and the critical ratio was more than 12.75% when long-span reticulated shell structures collapsed. However, the collapse critical loads of long-span reticulated shell structures could not be determined from a single index from the above analysis. It is therefore suggested that the vertex displacement time history curve, the International Journal of Acoustics and Vibration, Vol. 19, No. 1,

10 curve of the PGA-maximum vertex displacement, and the critical ratio of failure member should be applied simultaneously to judge the state of the shells in analysis. ACKNOWLEDGMENTS Financial support for this research was provided by the National Natural Science Foundation Major Research Plan of China, grant ; the Doctoral Fund of the Ministry of Education of China, grant ; the Priority Academic Program Development of Jiangsu Higher Education Institutions; and the Science and Technology Project of the National Construction Ministry, grant 2012-k2-39. This support is gratefully acknowledged. REFERENCES 1 Samuel, T. Cable-Based Retrofit of Steel Building Floors to Prevent Progressive Collapse, Ph.D. Dissertation, University of California-Berkeley, USA, (2003). 2 Loh, C. and Ku, B. An efficient analysis of structural response for multiple-support seismic excitations, Engineering Structures, 17 (1), 15 26, (1995). 3 Fujita, K., Nosaka, T., and Ito, T. 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