VERIFICATION OF THE METHOD FOR IMPROVING ACCURACY OF SIMPLIFIED SEISMIC RESPONSE ANALYSIS OF STEEL RIGID FRAME VIADUCTS

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1 International Journal of Civil Engineering and Technology (IJCIET) Volume 6, Issue 10, Oct 2015, pp Article ID: IJCIET_06_10_005 Available online at ISSN Print: and ISSN Online: IAEME Publication VERIFICATION OF THE METHOD FOR IMPROVING ACCURACY OF SIMPLIFIED SEISMIC RESPONSE ANALYSIS OF STEEL RIGID FRAME VIADUCTS Tatsuo Kakiuchi JR West Japan Consultants Company (PhD Candidate), Nishinakajima, Yodogawa-Ku, Osaka, , Japan Akira Kasai Associate Profesor, Kumamoto University, Kurokami, Chuo-Ku, Kumamoto, , Japan, Shohei Okabe GSST, Kumamoto University, Kurokami, Chuo-Ku, Kumamoto, , Japan, ABSTRACT This study is aimed at verifying the usefulness of the estimation method which is able to permit several plastic hinges' occurring to the steel rigid frame viaduct which is higher statically indeterminate. For this purpose, the analytical model, named whole system model, which applies shell elements to the place where plastic hinges may form, when the structures are subjected to severe earthquakes, was constructed. Next, Pushover analysis using the whole system model was carried out for evaluating seismic performance of this structure, and simplified seismic response analysis using an equivalent single-degree-of-freedom system model and skeleton curve which resembles a bilinear model based on a result of the Pushover analysis was performed. And then, the results of simplified earthquake response analysis was compared with the result of dynamic analysis using whole system model. Finally, it was contrived to apply a trilinear model to skeleton curve on restoring forcehorizontal displacement relationship as a way to improve the predictability of the response displacement of this structure. This paper also examines how the accuracy of seismic response analysis using the equivalent single-degree-offreedom system model could be improved. Key words: Combined static-dynamic numerical method for seismic response, Plastic hinge, Pushover analysis, Seismic performance evaluation, Steel rigid frame viaducts 46 editor@iaeme.com

2 Verification of The Method For Improving Accuracy of Simplified Seismic Response Analysis of Steel Rigid Frame Viaducts Cite this Article: Tatsuo Kakiuchi, Akira Kasai and Shohei Okabe. Verification of The Method For Improving Accuracy of Simplified Seismic Response Analysis of Steel Rigid Frame Viaducts. International Journal of Civil Engineering and Technology, 6(10), 2015, pp INTRODUCTION Steel structures built in Japan have been required to have excellent seismic performance that can resist the Hyogo Earthquake in 1995, the Tohoku-Pacific Ocean Earthquake in 2011 or a major earthquake like the Tokai, Tonankai, Nankai Consolidated Type Earthquake which is predicted to occur in the near future. In the structures such as these, the steel rigid frame viaducts rigid-connected between superstructure and piers which is focused on this paper, is one of the structures which can improve seismic performance. This structure has the following characteristics; 1) the height of the viaduct part can be lowered, 2) flexible correspondence is easy for the vertical linear shape of the railroad, 3) support system between box girder and piers can be omitted, and so on. Recently, a viaduct having these advantages is adopted in a steel bridge for railway in Japan. The schematic view of the viaduct to intend for in this study is shown in Fig. 1. Vertical Longitudinal Transverse Figure 1 Conceptual diagram of bridge system in this study The bridge type in this study has the longitudinal direction of the bridge and the out-of-plane direction of the piers in the same orientation. Beams and corners of the piers of such viaducts are exposed to torsion in addition to bending during an earthquake due to horizontal inertia. For an investigation of the members subjected to complex loading, Kasai et al. [1] examined basic data from seismic safety evaluation on the beams and corners of rigid frame viaduct piers where shear would prevail. Nakai et al. [2], [3], [4] evaluated both experimentally and analytically seismic capacity of a box section subjected to torsion and bending simultaneously. However, these studies were focused on evaluating individual segments. Conventional seismic performance evaluation is based on the assumption that the ultimate state of the whole structure is reached when at least the ultimate state of a single segment. However, this bridge which is a higher statically indeterminate structure can remain sufficiently safe in terms of seismicity when a single segment 47 editor@iaeme.com

3 Tatsuo Kakiuchi, Akira Kasai and Shohei Okabe has reached the ultimate state because it does not lead to reduction in the seismic capacity of the whole structure as long as the member components have adequate plate thickness and stiffness. Seismic performance of the structure therefore can be underestimated by the conventional segment-based evaluation. Focusing on this point, the authors [5] proposed a seismic performance evaluation method using shell elements. The conventional displacement-based verification method [6] is used for a conventional equivalent single-degree-of-freedom system structure. Then, the proposed method [5] is the displacement-based verification method for a higher statically indeterminate structure. Pushover analysis using the whole system model to consider local buckling behavior was carried out, and simplified seismic response analysis using an equivalent single-degree-of-freedom system model and skeleton curve which resembles a bilinear model based on a result of the Pushover analysis was performed. In this combined static-dynamic numerical method, the most important characteristic of the proposed method is the use of shell elements for Pushover analysis which considers local buckling behavior. It is effective to handle a higher statically indeterminate structure as a whole system as proposed, not by segment as conventionally done, in seismic performance evaluation. The proposed method is carried out seismic response analysis using the equivalent SDOF model to obtain response values. However, the applicability of the equivalent SDOF model has not yet been well studied. The purpose of this study is to verify the usefulness of the seismic performance evaluation method developed for a higher statically indeterminate structure to allow for plastic hinges being set at multiple locations. 2. DESIGN CONDITIONS AND ANALYSIS MODEL Table 1 shows the design conditions of the bridge. The bridge is a four-span steel rigid frame viaduct having the pier-superstructure integrated system as shown in Fig. 1. The details are described in reference [5]. Fig. 2 shows the general view of the analysis model. The segments modeled with shell elements are the base of a viaduct pier, the beam of a viaduct pier near the rigid connection with the superstructure and a corner closest to the superstructure. The shell elements for the base and the beam were input with initial deflection. The initial deflection waveforms were created by expressing each of general and stiffener local initial deflections as sine waves of the first mode and overlapping them together. The maximum initial deflection was determined based on the guidelines for buckling design [7]. As the material constitutive rule of steel, multilinear strain hardening rule having a yield plateau was used in the Pushover analysis, and the bilinear skeleton model was used in the seismic response analysis. A universal finite element analysis software ABAQUS [8] was used for the analysis. Structural type Span length Spans Girder width Girder height Table 1 Design conditions Steel rigid frame viaduct m 39.2 m m m m 11.8 m 2.8 m 48 editor@iaeme.com

4 Verification of The Method For Improving Accuracy of Simplified Seismic Response Analysis of Steel Rigid Frame Viaducts Figure 2 Analysis model 3. COMBINED STATIC-DYNAMIC NUMERICAL METHOD 3.1 Pushover analysis Pushover analysis and seismic response analysis using the equivalent SDOF model as shown in Fig. 3 were carried out following the method proposed in reference [5] for seismic performance evaluation of the bridge. Fig. 4 shows the load-displacement relationship obtained by the Pushover analysis, where horizontal load H is the sum of reaction forces at all pier bases, and horizontal displacement δ is displacement at the center of gravity of the superstructure. Displacement at the maximum load is used as ultimate displacement δ u, with the ultimate state assumed to be reached under the maximum load. In higher statically indeterminate structures like the bridge, local buckling should have occurred at many locations when the capacity of the structure starts to decrease. Evaluation to the maximum load state in this study was an attempt to ensure adequate safety margin in the design. Figure 3 Equivalent SDOF model Figure 4 Load-displacement curve 49 editor@iaeme.com

5 Tatsuo Kakiuchi, Akira Kasai and Shohei Okabe Figure 5 Hysteresis Figure 6 Response spectra of earthquake (h=0.05) Table 2 Parameters of the restoring force characteristic K [kn/mm] ζk [kn/mm] H y [kn] δ y [mm] H u [kn] δ u [mm] Seismic response analysis using the equivalent SDOF model The Restoring force characteristic to be used for the equivalent SDOF model were obtained from the results of the Pushover analysis using the whole system model [9]. When restoring force characteristics has been made approximately, it decided to meet the following two conditions certainly. Those are, 1) that both of bilinear approximate model and whole system model are same about initial horizontal stiffness which indicates the relationship between restoring force in horizontal direction and the horizontal displacement, and 2) that the restoring force and the displacement which indicate the ultimate state in whole system model are also passed certainly in the bilinear approximate model. The first break point was determined in accordance with reference [6] so that energy absorption amount would be constant until the ultimate point. Restoring force characteristic was applied to the hysteresis of the equivalent SDOF model as shown in Fig. 5. Fig. 4 shows the load-displacement relationship in the whole system model and the restoring force characteristic used for the equivalent SDOF model. Table 2 shows the parameters of the restoring force characteristic, where K = elastic stiffness, K = elasto-plastic stiffness, H y = yield horizontal load, δ y = yield displacement, H u = ultimate horizontal load and δ u = ultimate displacement. 3.3 Time history of response displacement The analysis used epicentral ground motion specified as Level-2 earthquake in the Design Standards for Railway Structures and Commentary (Seismic Design) [10]. Analysis was performed also for the El Centro (NS) and Taft (EW) waves. Fig. 6 shows response spectra of earthquakes. The dumping ratio h = 0.05 were used in the present analysis. Figs. 7 show the seismic motion. Figs. 8 show the time history of response displacement at the top of the Pier P3 in the equivalent SDOF models. Table 3 shows maximum response displacement δ max and its ratio to ultimate displacement, δ max /δ u. According to the seismic response analysis using the equivalent 50 editor@iaeme.com

6 Verification of The Method For Improving Accuracy of Simplified Seismic Response Analysis of Steel Rigid Frame Viaducts SDOF model, maximum response displacement δ max was 270 mm for Level-2 earthquake, 90 mm for the El Centro (NS) wave, and 46 mm for the Taft (EW) wave. Figure 7 Seismic motion Figure 8 Time history of response displacement Table 3 Maximum response displacement δ max [mm] δ max /δ u Equivalent SDOF model Level-2 earthquake El Centro (NS) Taft (EW) editor@iaeme.com

7 Tatsuo Kakiuchi, Akira Kasai and Shohei Okabe 4. ELASTO-PLASTIC SEISMIC RESPONSE ANALYSIS USING THE SHELL-ELEMENT WHOLE SYSTEM MODEL 4.1 Seismic response analysis using the whole system model This section describes the results of the seismic response analysis using the whole system model and compares them with the results of the combined static-dynamic numerical method described in the previous section. Morishita et al. [11] inspected the precision of the whole system model using the shell element in a simple steel pier, according to the seismic response analysis. Figs. 9 show the time history of response displacement at the top of the Pier P3 in this structure. The response displacement was determined as relative displacement between the superstructure and the base. Figs. 10 shows the load-displacement history, where load H is the base shear. The results with the equivalent SDOF model are also shown for comparison. Table 4 shows the values of maximum response displacement δ max. 4.2 Comparison of the time history of response displacement Maximum response displacement δ max for Level-2 earthquake was 226 mm in the whole system model and 270 mm in the equivalent SDOF model, being about 19% larger in the equivalent SDOF model. The response displacement history showed that the results of the two models fitted well until around 2 seconds and started to differ at past 2 seconds. The amplitude was found to be larger in the whole system model during the period from 2 to 6 seconds and then larger in the equivalent SDOF model during the period from 6 to 10 seconds. Center of the amplitude range in the whole system model moved significantly in the positive direction in about 4 seconds, while that in the equivalent SDOF model during the same period showed no significant movement because of the small amplitude. As a result, center of the amplitude in the equivalent SDOF model stayed in the negative side, allowing for an interpretation that maximum response displacement was larger than that in the whole system model. Residual displacement also showed a tendency of being larger in the equivalent SDOF model. Maximum response displacement δ max for the El Centro (NS) wave was 85 mm in the whole system model and 90 mm in the equivalent SDOF model, being about 6% larger in the equivalent SDOF model. The response displacement history showed that the results of the two models fitted well until around 2 seconds and started to differ at past 2.5 seconds. Maximum response displacement δ max for the Taft (EW) wave was 46 mm in the whole system model and 46 mm in the equivalent SDOF model, showing no differences between the two models. The response displacement history showed that the results of the two models fitted well until around 5 seconds. Consequently, the maximum response displacement of the equivalent SDOF model that the restoring force characteristic was made the bilinear skeleton model is bigger than the maximum response displacement of the whole system model. Therefore, the seismic performance of the steel rigid frame viaduct which is higher statically indeterminate can be judged to compare the ultimate displacement δ u with the maximum response displacement δ max of the equivalent SDOF model that the restoring force characteristic was made the bilinear skeleton model editor@iaeme.com

8 Verification of The Method For Improving Accuracy of Simplified Seismic Response Analysis of Steel Rigid Frame Viaducts Figure 9 Time history of response displacement Figure 10 Load-displacement history Table 4 Maximum response displacement δ max [mm] Whole system model Equivalent SDOF model Level-2 earthquake [119%] El Centro (NS) [106%] Taft (EW) [100%] 53 editor@iaeme.com

9 Tatsuo Kakiuchi, Akira Kasai and Shohei Okabe 4.3 Comparison of the load-displacement history In Figs. 10, the whole system and equivalent SDOF models were found to show well fitting results in seismic response analysis until yield displacement. In case of Level-2 earthquake, the loops of different patterns appeared in the load-displacement curves of the two models from the point where the whole system model yielded. Load was found to be larger in the equivalent SDOF model, with the center of vibration shifted away from the origin under plastic deformation. This was likely due to some factor in the restoring force characteristic used for the equivalent SDOF model. The restoring force characteristic had been obtained by the bilinear model to skeleton curve on restoring force-horizontal displacement relationship based on a result of the Pushover analysis as described in Section 3. However, stiffness reduction after the yield displacement was slow in the Pushover analysis values. Rough approximation of such a curve with a bilinear model will result in a significant discrepancy between the analytical values and approximate values, greatly affecting the seismic response analysis results. In case of the El Centro (NS) wave, the difference of the maximum response displacement δ max between the both models has occurred a little, because the maximum response displacement δ max becomes near yield-displacement. In case of the Taft (EW) wave, the maximum response displacement δ max between the both models is almost similar, because the maximum response displacement is almost half of the yield-displacement. Consequently, both models were found to show exhibit different behaviors after yielding by using the bilinear skeleton model. To improve the accuracy of seismic response analysis using the equivalent SDOF model, for Level-2 earthquake and the El Centro (NS), the authors modified the restoring force characteristic into a trilinear model as described in the following section. 5. EXAMINATION TO IMPROVE ACCURACY OF THE EQUIVALENT SDOF MODEL 5.1 Skeleton of the equivalent SDOF model modified by the trilinear skeleton model In this chapter, case study for improving the accuracy of estimation for maximum response displacement in the typical location of this structure is discussed. The time history of response displacement for the whole system and equivalent SDOF models is compared in the previous section. And the load-displacement relationship is compared. As the result, there is a possibility that the accuracy of the maximum response displacement δ max isn't good because both the behavior isn't fitting well. This was likely due to the influence of the restoring force characteristic used for the equivalent SDOF model. The authors modified the restoring force characteristic as described in this section in an attempt to improve the accuracy of seismic response analysis. More accurate seismic response analysis would be obtained by using more accurate approximates of the Pushover analysis results for the restoring force characteristic. Accordingly, the flow chart shown in Fig. 11 was developed at first. The outline of this flow chart is as follows editor@iaeme.com

10 Verification of The Method For Improving Accuracy of Simplified Seismic Response Analysis of Steel Rigid Frame Viaducts Start Structural properties and materials Pushover analysis ( K, H u, u ) Redesign of sections Approximation for hysteresis ( K, H y, y ) n = 1 Bilinear (Reference [5]) Dynamic analysis by model ( max ) Modified approximation for hysteresis ( K, H y, y ) n 2 Trilinear (This study) Requirement of accurate max Yes No Verification of displacement max u No Yes End Figure 11 Flowchart for improvement of the equivalent SDOF model 1) Restoring force-horizontal displacement relationship is expressed by bilinear model approximately using the Pushover analysis of this structure according to Kakiuchi et al. [5]. 2) Seismic response analysis is carried out using an equivalent SDOF model and above relationship. 3) Restoring force-horizontal displacement relationship is upgraded using trilinear skeleton model, if accuracy of this numerical result is required. 4) The second break point in the modified model was set at the point of 1.1 δ max, B so that a safety margin of 10% was added to maximum response displacement δ max, B. Initial stiffness and the second point were fixed in the determination of restoring force characteristic. The first break point was set so that energy absorption amount would be constant until the second break point. As in the model before the modification, the same restoring force characteristic was applied to the hysteresis editor@iaeme.com

11 Tatsuo Kakiuchi, Akira Kasai and Shohei Okabe Figs. 12 and Table 5 shows the restoring force characteristic into a trilinear skeleton model for Level-2 earthquake and the El Centro (NS) wave. And the value of the 1st brake point and the 2nd brake point is indicated Pushover analysis Bilinear Ultimate state Trilinear-1 Trilinear Pushover analysis Bilinear Ultimate state Trilinear-1 Trilinear (a) Level-2 earthquake (b) El Centro (NS) Figure 12 Trilinear skeleton model Bilinear Trilinear-1 Trilinear-2 Table 5 Brake points of trilinear skeleton model Level-2 earthquake EL Centro 1st brake point 2nd brake point Ultimate state 1st brake point 2nd brake point Ultimate state Modification of the equivalent SDOF model Figs. 13 show the time history of response displacement in the whole system and equivalent SDOF models. Figs. 14 show the load-displacement history. Table 6 shows maximum response displacement δ max of each model. The values in square brackets in the table are percentages which are the ratios of maximum response displacement of the equivalent SDOF model to that of the whole system model. The equivalent SDOF model with the modified the restoring force characteristic using trilinear skeleton model was found to exhibit behavior closer to that of the whole system model. Maximum response displacement δ max for Level-2 earthquake was improved to an error rate of 7%, with a good fitting found in the load-displacement history between the two models. Maximum response displacement δ max for the El Centro (NS) wave was also improved to an error rate of 11%, again with a good fitting found in the loaddisplacement history between the two models editor@iaeme.com

12 Verification of The Method For Improving Accuracy of Simplified Seismic Response Analysis of Steel Rigid Frame Viaducts Figure 13 Time history of response t Displacemen Figure 14 Load- Displacement history Table 6 Maximum response displacement δ max [mm] Equivalent SDOF model Whole system model Bilinear Trilinear-1 Trilinear-2 (n=1) (n=2) (n=3) Level [119%] 242 [107%] 243 [108%] El Centro (NS) [106%] 96 [113%] 94 [111%] Moreover, ground motions which the acceleration of the design earthquake motion is set to 0.75, 1.5, 1.75 and 2.0 times uniformly are prepared, to inspect the effect of this simple method to an earthquake motion with the size of the various accelerations. Although some results when these earthquake motions are subjected to it, are gathered by an appendix specifically, results about the maximum response displacements are summarized in Table 7. The accuracy of the maximum response displacement δ max can obtain by iterating total of 3 times of the once by the bilinear skeleton model and twice by the trilinear skeleton model editor@iaeme.com

13 Tatsuo Kakiuchi, Akira Kasai and Shohei Okabe Whole system mode max,w [mm] Table 7 Approximation of the restoring force characteristic (1) max,w / u Equivalent SDOF model Bilinear (n=1) max,b [mm] (2) max,b / u (2)-(1) Trilinear-1 (n=2) max,t-1 [mm] max, T-1 / u Trilinear-2 (n=3) max,t-2 [mm] (3) max, T-2 / u (3)-(1) El Centro L L L L L u = 740 mm : The ultimate displacement max,w : The maximum response displacement of whole system model max, B : The maximum response displacement of model by the bilinear model (n=1) max, T-1 : The maximum response displacement of model by the trilinear model (n=2) max,t-2 : The maximum response displacement of model by the trilinear model (n=3) Fig. 15 shows the each maximum response displacement δ max for Level-2 earthquake, 1.5 times of Level-2 earthquake and the El Centro (NS) wave in the case of each model. In Table 5 and Figs. 12, it was found that the prediction method which bilinear skeleton model was used as a restoring force-displacement relationship of the viaduct dealt with this study is very close to seismic response displacement max, W of this structure using whole system model. Whichever earthquake motions treated by this study was used, it was found that the maximum response displacement obtained using the equivalent SDOF model was larger than the case of the whole system model, when the displacement of this structure predicted using a bilinear model as restoring force-displacement relationship. The gap of ratio of δ max /δ u was within about 6% of the ultimate displacement between the prediction displacement by bilinear approximation model and the prediction displacement by the whole system. Figure 15 Maximum response displacements-maximum earthquake motions relationship 58 editor@iaeme.com

14 Verification of The Method For Improving Accuracy of Simplified Seismic Response Analysis of Steel Rigid Frame Viaducts It was found that the accuracy of the response displacement improved in most cases, when the displacement predicted using a triilinear model as restoring forcedisplacement relationship. In this case, the gap of ratio of δ max /δ u was within about 4% of the ultimate displacement between the prediction displacement by trilinear approximation model and the prediction displacement by the whole system model as shown in Table 7. Therefore, the equivalent SDOF model using the restoring force characteristic approximated by the bilinear skeleton model is an effective method in order to verify the seismic performance of the higher statically indeterminate structure like this steel rigid frame viaduct. 6. CONCLUSION This study is aimed at verifying the usefulness of a seismic performance evaluation method developed for a higher statically indeterminate structure to allow for plastic hinges being set at multiple locations. Pushover analysis using the whole system model was carried out for evaluating seismic performance of this structure, and simplified seismic response analysis using an equivalent single-degree-of-freedom system model. The major findings were as follows: (1) The flow which applies a result of the Pushover analysis to restoring forcedisplacement relationship was proposed to carry out seismic response analysis using the equivalent SDOF model for the high-level indeterminate structures. (2) To improve the precision of the response displacement which predicted restoring force-displacement relationship by a response prediction model that resembles a bilinear model more, the method which resembles a trilinear model was developed as shown in Fig. 11. (3) It was found that the prediction method which bilinear skeleton model was used as a restoring force-displacement relationship of the viaduct dealt with this study was very close to seismic response displacement of this structure using whole system model. (4) Whichever earthquake motions treated by this study was used, it was found that the maximum response displacement obtained using the equivalent SDOF model was larger than the case of the whole system model, when the displacement of this structure predicted using a bilinear model as restoring force-displacement relationship. The gap of ratio of δ max /δ u was within about 6% of the ultimate displacement between the prediction displacement by bilinear approximation model and the prediction displacement by the whole system. (5) The gap of ratio of δ max /δ u was within about 4% of the ultimate displacement between the prediction displacement by trilinear approximation model and the prediction displacement by the whole system model. (6) The equivalent SDOF model using the restoring force characteristic approximated by the bilinear skeleton model is an effective method in order to verify the seismic performance of the higher statically indeterminate structure like this steel rigid frame viaduct. (7) The equivalent SDOF model using the bilinear skeleton model is an effective technique for practical design that redesign cross sections iteratively on verifying cross sectional force of members editor@iaeme.com

15 Tatsuo Kakiuchi, Akira Kasai and Shohei Okabe REFERENCES [1] A. Kasai, T. Watanabe, T. Usami, P. Chusilp, Strength and Ductility of Unstiffened Box Section Members Subjected to Cyclic Shear Loading, Journal of JSCE, No.703, 2002, (in Japanese). [2] H. Nakai, Y. Murayama, T. Kitada, An Experimental Study on Ultimate Strength of Thin-Walled Box Beams with Longitudinal Stiffeners Subjected to Bending and Torsion, Journal of Structural Engineering, JSCE, 38A, 1992, (in Japanese). [3] H. Nakai, T. Kitada, Y. Murayama, Ultimate Strength Analysis with Local Buckling for Horizontally Curved Box Girder Bridges, Journal of JSCE, No.513, 1995, (in Japanese). [4] H. Nakai, T. Kitada, Y. Murayama, N. Murozuka, An Analytical Study of Ultimate Strength of Box Girders Subjected to Bending and Torsion, Journal of Structural Engineering, JSCE, 42A, 1996, (in Japanese). [5] T. Kakiuchi, M. Fujita, A. Kasai, T. USAMI, S. Yajima, T. Nonaka, Seismic Performance Evaluation of Steel Continuous Bridges with Rigid Superstructure- Pier Connections, Journal of Structural Engineering, JSCE, 55A, 2009, (in Japanese). [6] T. Usami et al., Guidelines for Seismic and Damage Control Design of Steel Bridges, Gihodo Shuppan, [7] T. Usami et al., Guidelines for Stability Design of Steel Structures, 2nd Edition, JSCE, Maruzen, [8] ABAQUS, Inc. ABAQUS Standard User s Manual Ver , [9] T. Kakiuchi, A. Kasai, K. Miyazaki, T. Yamao, S. Inagaki, A Seismic Performance Evaluation of Steel Rigid Frame Viaducts Integrated Superstructure and Substructures Considering Local Buckling Behaviors, Proc. of the 6th Intl. Conference on Thin walled Structures, Romania, 2011, [10] Railway Technical Research Institute, Design Standards for Railway Structures and Commentary (Seismic Design), Maruzen, [11] K. MORISHITA, T. USAMI, T. BANNO, A. KASAI, Applicability on Dynamic Verification Method for Seismic Design of Steel Bridge Piers, Journal of JSCE, No.710, 2002, (in Japanese). [12] Hamid Afzali and Toshitaka Yamao. Seismic Behavior of Steel Rigid Frame with Imperfect Brace Members. International Journal of Civil Engineering and Technology, 6(1), 2015, pp editor@iaeme.com

16 Verification of The Method For Improving Accuracy of Simplified Seismic Response Analysis of Steel Rigid Frame Viaducts APPENDIX In this appendix, ground motions which the acceleration of the design earthquake motion is set to 0.75, 1.5, 1.75 and 2.0 times uniformly are prepared, to inspect the effect of this simple method to an earthquake motion with the size of the various accelerations model [Bilinear] model [Trilinear-1] model [Trilinear-2] Pushover analysis Bilinear Ultimate state Trilinear Trilinear Time [sec] model [Bilinear] model [Trilinear-1] model [Trilinear-2] (a) L (a) L (a) L Pushover analysis Bilinear Ultimate state Trilinear Trilinear Whole system model model [Bilinear] model [Trilinear-1] model [Trilinear-2] Time [sec] Whole system model model [Bilinear] model [Trilinear-1] model [Trilinear-2] (b) L2 1.5 (b) L2 1.5 (b) L Pushover analysis Bilinear Ultimate state Trilinear-1, model [Bilinear] model [Trilinear-1,2] Time [sec] model [Bilinear] model [Trilinear-1,2] (c) L (c) L (c) L Pushover analysis Bilinear Ultimate state Trilinear-1, model [Bilinear] model [Trilinear-1,2] Time [sec] model [Bilinear] model [Trilinear-1,2] (d) L2 2.0 (d) L2 2.0 (d) L2 2.0 Figs. A1: Load-displacement curve Figs. A2: Time history of response displacement Figs. A3: Load-displacement history 61 editor@iaeme.com