Three-Dimensional Numerical Analysis on Site Vibration around Shinkansen Viaducts. under High-Speed Running Bullet Train

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1 Journal of Structural Engineering Vol.9A (March ) JSCE Three-Diensional Nuerical Analysis on Site Viration around Shinkansen Viaducts under High-Speed Running Bullet Train Xingwen He*, Mitsuo Kawatani**, Toshiro Hayashikawa***, Takashi Matsuoto**** * Dr. of Eng., Assistant Professor, Faculty of Eng., Hokkaido University, Kita- Nishi-8, Kita-ku, Sapporo ** Dr. of Eng., Professor, Graduate School of Eng., Koe University, -, Rokkodai-cho, Nada-ku, Koe 67-8 *** Dr. of Eng., Professor, Faculty of Eng., Hokkaido University, Kita- Nishi-8, Kita-ku, Sapporo **** Ph.D., Associate Professor, Faculty of Eng., Hokkaido University, Kita- Nishi-8, Kita-ku, Sapporo This study is intended to estalish a nuerical approach to the site viration around Shinkansen viaducts under running ullet trains. In this approach, the entire train-ridge-ground interaction syste is divided into two susystes: train-ridge interaction and foundation-ground interaction. Applying the dynaic reaction forces at the pier ottos otained fro the train-ridge interaction analysis as input excitation forces, the ground viration around the viaducts is siulated and evaluated taking advantage of a general-purpose progra. The nuerical results are copared with the experiental ones to confir the validity of the analysis. Keywords: Site viration, Nuerical analysis, Train-ridge interaction, Shinkansen. INTRODUCTION Japanese high-speed railway syste, the Shinkansen, serves a vital role in the transportation network and its ain lines usually pass directly over densely populated uran areas, where the railway structure ainly coprises viaducts of reinforced concrete in the for of portal rigid frae. The ridge viration caused y running trains is propagated to the aient ground via footing and pile structures, therey causing soe environental proles related to the site viration around the viaducts. Those virations can influence precision instruents in hospitals and laoratories, or people who are studying or resting in schools, hospitals and residences. Along with the further uranization and developent of ore rapid transport facilities, there is rising pulic concern aout the environental proles in odern Japan ). Its viration regulation law legislated in 976 was the first one concerning the environental viration proles in the world. Alost concurrently, recoendations for countereasures against viration proles of the Shinkansen railway were also proposed, in which a viration acceleration level liit was specified to allay the environental ipacts of train-induced viration on iportant facilities surrounding the ain lines. Although the iportance and urgency of environental proles have een recognized, the developent and propagation echanis of site viration caused particularly y running vehicles on viaducts reains unclear ecause of its coplicated nature. In Japan, environental viration proles caused y trains running along viaducts are ainly investigated y Railway Technical Research Institute (RTRI). Without a clear grasp of the site-viration echanis through theoretical studies, these proles are traditionally evaluated and predicted ased on field test data ). The efficiency of such a process is liited to particular cases. For ore general cases, essential inforation and reliale evaluation of site virations are necessary to perfor accurate predictions and develop countereasures. For that purpose, a corresponding nuerical approach to siulate the site viration proles is anticipated. In recent years, effort has een devoted to studies of site virations induced y trains oving on the ground surface. Fujikake ) proposed a predictive ethod for viration levels of the surrounding environent. Takeiya ) conducted a siulation of track-ground virations caused y a high-speed train for predicting train-track and neary ground-orne virations. Yang et al. ) also exained train-induced wave propagation in layered soils using a.-diensional (.D)

2 finite/infinite eleent approach. Nonetheless, little is known aout the ground viration caused y trains oving over viaducts ecause of its coplicated nature: virations are transitted to the ground via piers, footings and piles. Recently, Xia et al. 6) evaluated the viration-related effects of light-rail train-viaduct syste on the surrounding environent using a two-diensional (D) interaction odel of a train-ridge syste for otaining the dynaic loads of oving trains on ridge piers and a D dynaic odel of pier-foundation-ground syste for analyzing viration responses of the ground. Wu et al. 7) attepted to estalish a sei-analytical approach to deal with ground viration induced y trains oving over viaducts. In their studies, the entire train-ridge-ground interaction syste was divided into two parts as the train-ridge interaction susyste and foundation-soil susyste. The reaction forces otained fro the train-ridge susyste at the pier ottos were used as the input to the foundation-soil susyste. However, in their analysis the train is only siplified as oving concentrated loads, which cannot accurately express the dynaic effect the train syste. In Japan, Hara et al. 8) attepted recently to clarify the site viration around Shinkansen viaducts y oth experients and analytical procedure. However in their analyses, the wheel load of the trains is also only treated as siple equivalent oving force ased on the easured results. Such approach not only cannot directly take consideration of the interaction etween the ridge and train, ut is incapale to set the wheel loads without experiental results. Based on the filed test data of ground viration around Shinkansen viaducts, Yoshida and Seki 9) indicated the iportance of considering the coupled viration of the train-ridge interaction syste when evaluating the train-induced ground viration. In this study, taking advantage of three-diensional (D) dynaic analysis, an approach to siulate the site viration around Shinkansen viaducts is estalished. In this approach, the dynaic interactions etween the train and ridge and etween the foundation and ground are considered. Currently, it is still difficult to odel the entire train-ridge-ground interaction syste as a whole, ecause of not only the extree coplexities of their interaction ut also the liit of the coputational capacity. Therefore to siplify the prole in this study, the entire interaction syste is divided into two susystes: train-ridge interaction and foundation-ground interaction. In the stage of the train-ridge interaction prole, the analytical progra to siulate the traffic-induced ridge viration is developed, with which the dynaic responses of viaducts are calculated to otain the dynaic reaction forces at the pier ottos. Then, applying those reaction forces as input excitation forces in the foundation-ground interaction prole, the site viration around the viaducts is siulated using a general-purpose progra naed SASSI ),). The analytical results are copared with experiental ones to confir the validity of the developed analytical procedure.. ANALYTICAL MODELS. Elevated Bridge Model A typical high-speed railway reinforced concrete viaduct in the for of a rigid portal frae shown in Figure is adopted in this analysis. The actual field test ) to easure the viration of Shinkensen viaduct was conducted at the Shinkansen line. Bullet trains coposed of sixteen cars were running through the viaduct with its actual operational speed of 7 k/h. The ridge viration was easured at several points of the viaduct during the ullet trains passage using acceleroeters. Then the acceleration responses of the viaduct were recorded on the data recorder fro the acceleroeters after eing processed y aplifiers. The sapling rate of the data was Hz. In this analysis, the ridge viration recorded at point- through point- of the viaducts indicated in Figure will e exained. Considering the oundary condition of the ridge, three locks (7 ) of the ridge with length of each lock are adopted as the analytical odel. Only the dynaic response of the iddle lock will e exained. Figure shows that the three-lock ridge is odeled as D ea eleents with six-dof.7.6 Moving position Rail irregularities () Point-. Side view.7.6 Fig. Bridge diensions Point- 7 Point L- L- L- L- R- R- R- R- Fig. -lock elevated ridge odel () Cross-sectional view Left Right Distance () Fig. Measured rail surface roughness

3 Tale Ground spring constant Explanation Longitudinal Transverse Vertical spring of pile top (kn/).86 6 Rotating spring of pile top (kn /rad) Horizontal spring of footing (kn/).8.7 Horizontal spring of pile top (kn/) Tale Property of rail Area ( ) Mass (t/) Moent of inertia ( ) Spring constant of track (MN/) Tale Dynaic properties of oving cars Mass (t) Spring constant k (N/) Daping coefficient c (N s/) Natural frequency (Hz) (Body) (Bogies) (Wheels) k u (Upper) k l (Lower) c u (Upper) c l (Lower) f u (Upper) f l (Lower) at each node. The luped ass syste is adopted for the ea eleents. Mass of the allast is also incorporated. Since the trains train runs etween the piers of the ridge, the viration of cantilever slas in transverse direction is not iportant. Therefore, they are not odeled with finite eleents to reduce the nuer of nodes and their asses are added to the outside nodes of the sla. On the other hand, the cantilever slas in the longitudinal direction of the ridge play an iportant role in the ridge viration and are odeled. The slas are odeled as ea eleents with the sae length as sleeper intervals, which can express the viration caused y rail fastening distance. Doule nodes defined as two independent nodes sharing the sae coordinate are adopted at the ottos of the piers to siulate the effect of ground springs. The ground springs are calculated including the effects of the footing and pile structures ). The ground spring constants are shown in Tale. Rayleigh dapin ) is adopted for the structural odel. According to the past field test results ), the daping ratio of. is assued for the first and second odes of the structure. The rail structure is also odeled as D ea eleents with six-dof at each node. Doule nodes are also defined here to siulate the elastic effect of sleepers and allast at the positions of sleepers. Properties of the rail and the spring constant of the track are shown in Tale. The vertical spring constant of the track is derived fro the ratio of the wheel load to the rail s displaceent in vertical direction. The horizontal spring constant Fig. DOF ullet train odel of the track is assued to e / of the value in the vertical direction ). Only roughness in the vertical direction of the rail is taken into account. The easured values of railway roughness are shown in Figure.. Train Model Bullet trains coposed of 6 cars, odeled as DOF syste for each car (Fig. ), are eployed for analysis. Tale shows dynaic properties of the oving trains. The suscripts u and l respectively ean the upper and lower parts of the suspension, which eans that the spring constants and and daping coefficients are the total values of the corresponding parts. As shown y the paraeters of the train odel in Tale, the natural frequency of the ogies is higher than that of the train ody, which can engender resonance in a higher-frequency field and contriute to high-frequency coponents of dynaic responses of the ridge. The train velocity is assued to e 7 k/h, referring to the actual Shinkansen operation speed.. TRAIN-BRIDGE INTERACTION ANALYSIS Bridge viration caused y running trains is propagated to the aient ground via pier and foundation structures of the viaducts. To perfor site viration analysis around the viaducts, the dynaic reaction forces at the pier ottos are deanded for use as input external excitations to the foundation-ground interaction syste. Therefore, it is first necessary to siulate the dynaic responses of the viaducts to otain those dynaic reaction forces. Dynaic responses of the high-speed railway viaducts under oving ullet trains are analyzed in consideration of the train-ridge interaction in this study. The siultaneous dynaic differential equations of the finite eleent ridge odel are derived using odal analysis. The Newark s β step-y-step nuerical integration ethod is applied to solve the dynaic differential equations. Considering the extreely high speed of the ullet train, the integral tie interval is set as. second. The rief forulas of the train-ridge interaction prole are descried as follows and ore detailed forulation process can e found in the reference ).

4 . Dynaic Differential Equations of the Bridge The dynaic differential equations of the elevated ridge can e derived as follows, ased on D Aleert s Principle. M w C w K w f () where M, C, K and w respectively denote ass, daping, stiffness atrices and the nodal displaceent vector. Herein, the daping atrix C is calculated y the linear relation etween ass and stiffness atrices, i.e. the Rayleigh dapin ). The external force vector f can e represented as Eq. (). h j l k t t f P () jlk jlk where P jlk(t) is the wheel loads of the train, and jlk (t) is the distriution vector that will distriute the wheel loads to the two nodes of a rail ea eleent when the contact point is etween the two nodes. Modal analytical technique is then applied to the aove differential equations of the ridge to siplify the analysis ).. Dynaic Differential Equations of Train Body Lateral translation of train ody: j l y ( v ( t ) () jyl Bouncing of train ody: z v ( t ) () Rolling of train ody: j l jzl I x jx yv jzl ( t ) l () v ( t ) l Pitching of train ody: z jyl l I ( v ( t ) (6) y jy l Yawing of train ody: x jzl l I z jz xv jyl ( t ) l (7) v ( t ) l y jxl Herein, the suscript j indicates the sequence nuer of the car. The suscripts relative to the otion of the train ody are descried as: l=, indicates the front and rear ogies; =, indicates the left and right sides of the train, respectively. v jxl (t), v jyl (t) and v jzl (t) denote the forces due to the expansion quantities of the upper springs of relative directions ), respectively.. Dynaic Differential Equations of Train Bogies Sway of front or rear ogie: jl jyl jylk k y ( v ( t ) v ( t ) (8) Parallel hop of front or rear ogie: jl jzl jzlk k z v ( t ) v ( t ) (9) Axle trap of front or rear ogie: I x jxl ( zv jyl( t ) ( yv jzl( t ) () v ( t ) v ( t ) k z jylk k Windup otion of front or rear ogie: I y ( t ) y jzlk k jyl zv jxlk ( t ) k () k Yawing of front or rear ogie: I z k k jz l k x v jzlk ( k k v y v x v y jxlk jylk jxl ( t) ( t) ( t) () Herein, the suscripts relative to the otion of the ogies are descried as: k=, indicates the front and rear axles of the ogie, =, indicates the left and right sides of the ogie, respectively. v jxlk (t), v jylk (t) and v jzlk (t) denote the forces due to the expansion quantities of the lower springs of relative directions ), respectively.. Siulation of Reaction Forces at Bottos of Piers Dynaic reaction forces at the ottos of piers cannot e otained accurately through odal analysis y eans of calculating the shear forces at the ends of piers ecause of the Gis phenoenon. Therefore in this study, the dynaic reaction forces are calculated with Eq. () using the influence value atrix of the reaction force: R( t ) K ( P P ) K P () R vst where R(t) and K R denote the reaction force vector and the influence value atrix of the reaction force. Vectors P vst, P vdy, and P sdy respectively denote the static and dynaic coponents of the wheel loads and the inertia forces of the ridge nodes. vdy R sdy

5 - - 6 Tie (sec) Fourier Aplitude (Gal) 6 8 Furecuency (Hz) Max:.7 Gal rs: Gal Tie (sec) Frecuency (Hz) Max:. Gal rs: 9. Gal Fourier aplitude (Gal) Fourier Aplitude (Gal) Tie (sec) Frecuency (Hz) Max:.8 Gal rs:.7 Gal Experient Tie(Sec) Tie(Sec) Tie(Sec) Max: 79. Gal rs: 6. Gal Max: 78. Gal rs: 6.78 Gal Max:.9 Gal rs:.7 Gal Analysis Point- Point- Point- Fig. Acceleration response of easured points of the ridge (vertical). DYNAMIC RESPONSES OF THE BRIDGE. Dynaic Responses of Elevated Bridge The eigenvalue analysis of the ridge odel is perfored. The predoinant frequency of the horizontal natural ode is oserved as. Hz, showing good agreeent with the value otained fro the field test, which is.9 Hz. Therefore, the ridge odel validation can e confired. The details of the eigen viration characteristics and viration odes of the ridge odel can e found in the reference ). The analytical acceleration responses and the experiental ones in vertical direction, of point- through point- of elevated ridges indicated in Figure, are shown in Figure, and those in horizontal direction of point- are shown in Figure 6. Herein, point-, point-, and point- respectively indicate the point of hanging part, the top of the first pier and the top of the third pier of the elevated ridge, with respect to the direction that the train runs towards. As shown in Figures and 6, analytical results indicate relatively good agreeent with experiental results, therey validating this analytical procedure. The hanging parts of the elevated ridge are connected with neighoring ones y rails and allast in the actual structure, ut only the rails connecting effect can e incorporated into analysis. Presualy for that reason, the virations are predoinant at lower frequencies and analytical acceleration responses display larger aplitudes than do experiental ones at point-.. Dynaic Reaction Forces at Bottos of Piers Reaction forces at the ottos of the piers in vertical and horizontal directions, as respectively shown in Figure 7, are calculated using the influence value atrix of the reaction forces. As shown in Figure, L- to L- and R- to R- respectively Tie(Sec) (a) Experient 6 7 Tie(Sec) () Analysis 6 8 indicate the piers on the left and right sides of the iddle lock of the ridge, with respect to the train s direction. The vertical reaction forces of the piers on the left side are uch stronger than those on the right side ecause the trains are assued to run along the left sides of the ridges. On the other hand, the reaction forces on the left and right sides in horizontal direction display siilar aplitudes. In particular, for oth directions in Figure 7, the aplitude at L- is soewhat larger than that of L-. Figure shows the proale reason: the axiu acceleration response that engenders a larger inertia force appears at the hanging part of the ridge. Dynaic reaction forces otained here can e used as input external excitations in further analyses of site viration proles. Max:. Gal rs: 7. Gal 6 8 Max:.7 Gal rs: 6.8 Fig.6 Acceleration of ridge (Point-, horizontal)

6 - - L- 6 Tie (Sec) Max:. Gal rs:. Gal - - L- 6 Tie (Sec) Max:.8 Gal rs:. Gal - - L- 6 Tie (Sec) Max: 8.9 Gal rs:.6 Gal - - (a) Vertical direction L- 6 Tie (Sec) Max: 7. Gal rs:. Gal () Horizontal direction R- 6 Tie (Sec) Max:. Gal rs:.6 Gal R- 6 Tie (Sec) Max:.8 Gal rs:. Gal Fig.7 Dynaic reaction forces at ottos of piers (Points as shown in Fig.). FOUNDATION-GROUND INTERACTION For the foundation-ground interaction syste in this study, eploying the previously otained reaction forces at the ottos of piers of the ridges as input dynaic forces, site virations around the viaducts of the high-speed railway are siulated using a general coputer progra naed SASSI. The soil-structure interaction (SSI) prole is analyzed conveniently using a sustructuring approach y which the linear soil-structure interaction prole is sudivided into a series of siple su-proles. Each su-prole is solved separately and the results are coined in the final step of the analysis to provide a coplete solution using the principle of superposition. Detailed inforation of the SSI syste is descried in the references ),).. Description of Soil-Structure Interaction Model Properties of the actual site around the elevated ridge used in the previous ullet train-ridge interaction syste are eployed to estalish the analytical site odel. The structural odel contains the footing and piles of the previous elevated ridges. Figure 8 depicts the surveyed points. In all, footings of the three locks of ridges used in previous ridge viration analyses are adopted to e excited here and therey siulate site viration. Black rectangles in the figure indicate the footing positions. In the figure, L and R denote the left and right sides of the ridge with respect to the oving direction. The letters a d and A D and the nuers respectively indicate the footing sequences in the three locks of the ridge. The distances etween the centers of neighoring footings on the sae side are 6. ; those etween the central lines of left and right footings are.. Surveyed points of. and lying on the line passing through the centers of footings R- and L- denote the surveyed points at which the site viration is easured in field tests. Analytical results of these points are copared with experiental results.. Site Model Tale shows surveyed values of actual site properties. The site ainly coprises three strata separated at depths of 6.8 and 7.. The velocity of an S-wave in the first stratu is 8 /s, fro which the site condition can e considered as relatively inferior. The daping constant is assued as %, deterined fro experiential values. For analysis, the site odel is divided further into thin layer eleents, whose profiles are shown in Figure 9. The axiu thickness of each layer is deterined in copliance with the criterion that it does not exceed / λs, where λs is the shortest S wavelength in that layer. Layer eleents are estalished down to the depth of 8.8, to which the structural odel is eedded. The progra then autoatically adds soe extra layer eleents and the viscous oundary at the ase to siulate the effect of half space.. Underground Structure Model One structural set consisting of one footing and seven piles is odeled as Figure. Properties of the footing and the piles are shown respectively in Tale and Tale 6. The actual footing structure is in the shape of rectangular parallelepiped at the ase and a trapezoid at the top. To siplify the analyses, the footing is approxiated as a rectangular parallelepiped divided into 6 solid eleents according to the conversion of volue. The sizes of the solid eleents also eet the criterion that they e less than / of the shortest S wavelength in the corresponding layer. The footing surface is assued to extend

7 .. Tale Properties of footing Moving position Unit ass Young s odulus Poisson s Daping L-a R-a L- R- L-c R-c L-d R-d L- R- L- R- L- R- L- R- L-A R-A L-B R-B L-C R-C L-D R-D.6. (t/ ) E (kn/ ) ratio v constant. Fig.8 Positions of surveyed points z Tale 6 Properties of piles Type Unit ass (t/ ) Layer@.6 Layer@. Layer@.6 x Cross-section area A ( ).8. Young s odulus E (kn/ ) Moent of inertia I ( ) Layer@ Poisson s ratio v.. 6 Layer@. -7. Daping constant.. Variale Depth Layer@.6 Half Space Fig.9 Profile of site odel 6@ under the ground surface. The piles are divided into two types according to their length: Type is 7 long and Type is 8. The and arks indicate the positions at which the piles are connected vertically to the footing. Herein, represents 8--long piles and represents 7--long piles. The piles are odeled as D ea eleents. The ends of the ea eleents are estalished at the soil layer interfaces. 6@.. 6. ANALYTICAL RESULTS OF SITE VIBRATION z y x Fig. Structural odel Tale Ground properties Depth of stratu () Unit ass (t/ ).6.8. Shear odulus G (kn/ ) 66 Poisson s ratio v S wave velocity Vs (/s) 8 9 Daping constant Considering the predoinant frequency coponents of the external forces that are confired within Hz, the soil daping effect, and the efficiency of the analysis, the highest frequency addressed in the analysis is deterined as Hz. By applying the dynaic reaction forces otained in the ridge viration analyses at total footings, the site response analyses using the analytical odels descried in previous sections is carried out using the SASSI coputer progra ),). Analytical results and experiental values in vertical and horizontal directions siultaneously with the axiu and rs values, of the points of. and indicated in Figure 8, are shown in Figure and Figure. For oth vertical and horizontal directions, the aplitudes of analytical results show good agreeent with the experiental ones. On the other hand, the predoinant frequency coponents, particularly in the horizontal direction, indicate soewhat disagreeent with those of the experiental ones. The reasons can e considered as follows. First, since the wheel sets of the train are not odeled, the coponents of the analytical results are inevitale to have soe differences with the actual responses especially for the horizontal direction. Furtherore, the discrepancies of the results arise fro the difference etween the actual site

8 Tie(sec) 6 8 Tie (sec) Max: 7. Gal rs:.78 Gal Experient Tie (sec) Tie (sec) Max:.6 Gal rs:.8 Gal Max: 8.9 Gal rs:.87 Gal Analysis. Fig. Acceleration of easured points of the ground (vertical) Max: 9.9 Gal rs:.8 Gal Tie (sec) Max:.6 Gal rs:.98 Gal Tie (sec) Experient Max:.8 Gal rs:. Gal Tie (sec) Max: 9.89 Gal rs:.77 Gal Tie (sec) Analysis. Fig. Acceleration of easured points of the ground (horizontal) Max:.6 Gal rs:.8 Gal properties and the idealized odel. However, considering the coplicated nature of the whole train-ridge-ground interaction syste, the analytical results otained here are considered useful to evaluate site viration around viaducts, and the nuerical approach can e elaorated to carry out ore accurate analyses of train-induced ground viration proles in the further studies. 7. CONCLUSION In this study, a DOF ullet train odel was developed and the dynaic responses of Shinkansen viaducts were siulated, considering the train-ridge interaction. Using the reaction forces at the pier ottos otained in the ridge viration analysis as input excitations, site viration around the viaducts of the high-speed railway was siulated using a general coputer progra naed SASSI. Analytical results were validated through coparison with experiental ones. Eploying the analytical procedure estalished in this study, it is possile not only to siulate and evaluate site viration around viaducts that is caused y running trains, ut also to investigate the effectiveness of presued countereasures against ridge and site viration y reinforcing the ridge structure, iproving the ground conditions, or eploying other eans. References ) Seki, M., Inoue, Y. and Naganua, Y., 997. Reduction of sugrade viration and track aintenance for Tokaido Shinkansen. WCRR 97, E. ) Yoshioka, O. and Ashiya, K., 988. Dependence of Shinkansen-induced ground viration upon their influence factors. QR of RTRI, 9( ), ) Fujikake, T.A., 986. A prediction ethod for the propagation of ground viration fro railway trains. Journal of Sound and Viration, (), ) Takeiya, H.,. Siulation of track-ground virations due to a high-speed train: the case of X- at ledsgard.

9 Journal of Sound and Viration, 6(), 6. ) Yang, Y.B., Hung, H.H. and Chang, D.W.,. Train-induced wave propagation in layered soils using finite/infinite eleent siulation. Soil Dynaics and Earthquake Engineering, (), ) Xia, H., Cao, Y.M., Zhang, N. and Qu, J.J.,. Viration effects of light-rail train-viaduct syste on surrounding environent. International Journal of Structural Staility and Dynaics, (), 7. 7) Wu, Y.S. and Yang, Y.B.,. A sei-analytical approach for analyzing ground virations caused y trains oving over elevated ridges. International Journal of Soil Dynaics and Earthquake Engineering, (), ) Hara, T., Yoshioka, O., Kanda, H., Funaashi, H., Negishi, H., Fujino, Y. and Yoshida K.,. Developent of a new ethod to reduce Shinkansen-induced wayside virations applicale to rigid frae ridges: ridge-end reinforcing ethod. Journal of Structural and Earthquake Engineering, JSCE, 68(766), -8. (in Japanese) 9) Yoshida, K. and Seiki, M.,. Influence of iproved rigidity in railway viaducts on the environental ground viration. Journal of Structural Engineering, JSCE, A, -. (in Japanese) )Lyser, J., Ostadan, F. and Chin, C.C., 999. SASSI theoretical anual A syste for analysis of soil-structure interaction. Acadeic Version, University of California, Berkeley. )Lyser, J., Ostadan, F. and Chin, C.C., 999. SASSI user s anual A syste for analysis of soil-structure interaction. Acadeic Version, University of California, Berkeley. )Kawatani, M., He, X., Shiraga, R., Masaki, S., Nishiyaa, S. and Yoshida, K., 6. Dynaic response analysis of elevated railway ridges due to Shinkansen trains. Journal of Structural and Earthquake Engineering, JSCE, 6(), 9-9. (in Japanese) )Agaein, M.E., 97. The effect of various daping assuptions on the dynaic response of structure. Bulletin of International Institute of Seisology and Earthquake Eng., 8, 7-6. )Nishiura, A., 99. A study on integrity assessent of railway rigid frae ridge. RTRI Report, (9). (in Japanese) (Received Septeer 8, )