EVALUATION OF LOCAL NONLINEAR EFFECT AROUND PILE FOUNDATION ON SEISMIC RESPONSE OF BUILDING DURING VERY LARGE EARTHQUAKES

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1 EVALUATION OF LOCAL NONLINEAR EFFECT AROUND PILE FOUNDATION ON SEISMIC RESPONSE OF BUILDING DURING VERY LARGE EARTHQUAKES Hisatoshi KASHIWA 1, Hiroshi ARAI 2 and Hiroto NAKAGAWA 3 ABSTRACT The objective of this study is to evaluate the effects of the local nonlinearity around the pile foundation on the seismic response of the building during the very large earthquakes. The Mw 7.1 main shock of the 216 Kumamoto Earthquake (16 April, 1:25 JST) was occurred nearby Mashiki Town and strong motion records on the base foundation in Mashiki Town Office and on the ground surface nearby the building were obtained. The strong nonlinear behavior such as the damage of piles or the gap between piles and ground may cause these phenomena. To perform the seismic design rationally, the strong nonlinear behavior should be accurately considered, and the effect of the strong nonlinear behavior on the seismic response should be investigated based on the observation records. Therefore, the simulation analyses about the seismic response of Mashiki Town Office during the 216 Kumamoto Earthquake were performed in order to evaluate the effects of the damage of piles and the gap between piles and ground on the seismic damage of buildings. From this study, it was confirmed that a good agreement was obtained in the tendencies of the predominant periods between the result from analysis and the observed record at the stations. It was also verified that the effects of the nonlinear behavior are remarkable on the seismic response of the building. Keywords: The 216 Kumamoto Earthquakes; Analysis for Seismic records; Nonlinear Soil-Structure- Interaction Effect 1. INTRODUCTION Very large seismic events occurred mainly in Kumamoto Prefecture in Kyusyu district in Japan. The first shock which is Mw 6.1 event occurred on 14 April at 21:26 Japan Standard Time(JST) (GMT +9h). The main shock which is Mw 7.1 event occurred on 16 April at 1:25 JST. Fig.1 shows the location view of Kumamoto Prefecture and the epicenters of the first shock and main shock. The twice large earthquakes destroyed over 7, wooden houses and residential buildings mainly in the central area of Mashiki Town. Two seismic stations are in Mashiki Town, one is KiK-net KMMH16 which is operated by the National Research Institute for Earth Science and Disaster Prevention (NIED 216) and was recorded seismic motion on the ground surface, other is Japan Meteorological Agency (JMA) seismic intensity station which is operated by local government office (JMA 216) and observed maximum value of seismic intensity which is 7 on twice seismic events. These seismic records are nearby the heavy damage area of houses as shown in Fig.2. Furthermore, JMA seismic intensity station was the records on the 1 st floor in Mashiki Town Office and this records was affected by the soil-structure interaction effects. 1 Senior Research Officer, Nat l Inst. for Land and Infrastructure Management, Japan, kashiwa-h92ta@mlit.go.jp 2 Senior Research Engineer, Building Research Institute, Japan, arai@kenken.go.jp 3 Senior Research Engineer, Building Research Institute, Japan, hiroto-n@kenken.go.jp

2 Fig.1. Location of Kumamoto Prefecture and the central area in Mashiki Town, included the epicenter of the first shock (Apr. 14) and main shock (Apr. 16), and the fault of main shock by NIED 216. This map is based on following URL: ( in Japanese.) Fig.2. Location of observation stations, KMMH16 is on the surface and in the borehole at G.L. -252m, and MTO is on the 1F in Mashiki Town Office. Furthermore, the spatial distribution of heavy damage ratio of wooden houses is illustrated in the figure (NILIM 216). The behaviors of the dynamic soil-structure interaction affect the seismic responses of buildings; especially the seismic response will be strongly affected by the nonlinear behavior between the soil and the foundation. For example, Hayashi et al (1999) investigates a slender building in heavy damaged area considering uplift behavior in the 1995 Hyogo-ken Nanbu earthquake (Kobe), and it is suggested that the nonlinear soil-structure interaction behaviors as an uplift behavior reduce structural damages. However, there are few cases that the seismic records between in the building and on the surface were obtained during very large earthquake and the nonlinear soil-structure interaction was investigated based on the observed records. To perform the seismic design rationally, the strong nonlinear behavior should be accurately considered, and the effect of the strong nonlinear behavior on the seismic response should be investigated based on the observation records. 2

3 N Value Vs (m/s) Model Soil Type [ρ:mg/m 3 ] Loam K-1 ρ=1.6 Silt with Gravel Sand with Pumise Sand w/ Gravel Sand w/ Silt Silty Sand w/ Gravel Sand w/ Gravel K-2 ρ=1.6 W.L K-3 ρ=1.9 Elastic NILIM -5 Depth (m) -5 Depth (m) (a) Site-K (KiK-net, KMMH16) Estimation NIED Analysis Model Soil Type[ρ:Mg/m 3 ] Loam Gravel Silt with Sand Sand with Pumise Sand w/ Gravel O-1 ρ=1.6 O-2 ρ=1.6 W.L O-3 ρ=1.9 Elastic Vs (m/s) Analysis =Estimation Depth (m) (b) Site-O (Mashiki Town Office) Fig.3. Soil classifications, N-value and S wave velocity profiles from the borehole investigations and microtremors estimation at Site-K (KiK-net, KMMH16) and Site-O (Mashiki town office) Therefore, the simulation analyses about the seismic response of Mashiki Town Office during the 216 Kumamoto Earthquake are performed in order to evaluate the nonlinear soil-structure interaction effects on the seismic damage of buildings based on the seismic records, which are the damage of piles and the nonlinear behavior of soil as the gap between piles and ground. During the 216 Kumamoto earthquake, many houses were collapsed in east-west (E-W) direction and the fact is implied that the seismic component on the E-W direction was dominant in the seismic response and damage of buildings. Therefore, the seismic responses on the east-west component are focused in this paper. 2. GEOLOGICAL CONDITIONS AT SEISMIC STATION Fig.3 shows the borehole dates at two sites: Site-K and Site-O. In both Site-K and Site-O the stratigraphy included about 15m of cohesive soil: loam, tuffaceous silt with gravel or sand, and tuffaceous sand with pumice or sandy soil at a depth from about 15m to engineering bedrock, whose S-wave velocity is 7m/s at the depth of about 4m in this paper. Fig.3(a) which is at Site-K has four lines of S-wave velocity: NIED indicates the result of PS logging by NIED provided from 1996, NILIM indicates the result of PS logging by NILIM conducted investigation in 216, Estimation indicates the result of estimation by microtremor observation (Aari et al. 217), and Analysis indicates the analysis model in this paper. The S-wave velocity of Estimation at the depth of -5 m and NILIM at the depth of 5- m uses in the analysis model. The S-wave velocity in Fig.3(b) which is at Site-O is the result of estimation by microtremor observation (Arai et al. 217). 3

4 Table1. List of analysis model properties about the soil nonlinearity at Site-K and Site-O based on soil tests (Dynamic deformation properties are the result of identification by modified H-D model with λ.) Strength Properties Dynamic Deformation Properties Sample Number Soil Classification Sampling Depth (m) E 5 (kpa) f cu (deg.) c cu (kpa) effective confining pressure s c ' (kpa) g 5 (x1-3 ) Model parameters [modified H-D model] h max l h min K-1 Loam -2 ~ K-2 K-3 Silt Sand -8 ~ ~ O-1 Loam -3 ~ O-2 O-3 Silt Sand -14 ~ ~ The analysis model properties about the soil nonlinearity are summarized in Table1. The consolidatedundrained triaxial compression test for strength properties and the cyclic loading triaxial test for dynamic deformation properties are conducted. The dynamic deformation properties are modelled by a composite model of the modified Hardin-Drnevich (H-D) model and the Ramberg-Osgood (R-O) model (Nakagawa et al. 211) in this paper. The shear stress-strain relationship for skeleton curve in this model is given by g G 1 g g l 5, G (1) g in which the shear stress:, the shear strain:g, the initial shear stiffness of soil: G, the reference of shear strain: g 5, the parameter: l. The hysteresis damping ratio for the hysteresis curve is given by G h hmin 1 (2) G h max hmin in which the damping ratio: h, the maximum and minimum value of damping ratio: h max, h min. The parameters in the modified H-D model are identified by using a least-square fit of the test dates to Eq. (1) and (2). When nonlinear analyses are conducted, the modified H-D model satisfied by Eq. (1) is applied to the skeleton curve and the R-O model with the Masing rule which is satisfied by Eq. (2) is applied to the hysteresis curve (Ishihara et al. 1985). The R-O model used in this analysis is given by g 1 G Gg 5 1, 1 l l g 5 g g 1 (3) g 5 g g 5 in which the maximum shear strain which was experienced: g. 4

5 3. SEISMIC RESPONSE OF GROUND USING 1D NONLINEAR ANALYSIS Fig.4 illustrates a schematic figure of the estimation methods of seismic ground response based on the seismic record on the surface by using nonlinear analysis at two sites: Site-K and Site-O. The method is implemented the basis of the six contents as follows: (1) An acceleration time history on the engineering bedrock ( A 2E ) is calculated from the observed record ( obs A K ) on the surface by deconvolution based on equivalent linear method (Yoshida 22). (2) A non-linear site response analysis is conducted at Site-K by using the program RESP-F3T (KKE inc. 217) and an acceleration time history on the surface ( k A K ) is estimated. (3) A cross spectrum between k A K and obs A K (C cal/obs ) is calculated. The amplitude and phase of the cross spectrum are estimated for a correction to the engineering bedrock motion. (4) The engineering bedrock motion is corrected in the frequency domain by the following equation: A C obs K cal / obs k 1 A2 E k A2 E (4) k AK Ccal / obs in which the corrected engineering bedrock motion: k+1 A 2E. The spectra which are and C cal/obs are smoothing with Parzen s window having a band width of.2 and 1. Hz respectively. The spectrum k+1 A 2E are filtered by a Butterworth low-pass filter (f c =1Hz). (5) The contents from (2) to (4) are iterated some times. The number of iteration (k) in this analysis is four. (6) A non-linear site response analysis is conducted at Site-O by using A 2E (= 4 A 2E ). In this analysis, the same engineering bedrock motion ( 4 A 2E ) is used at the different sites in order to focus on the investigation of the local ground response in site effects. obs A K k A K ka K : Calculated acc. on surface (3) Calculation of amplitude and phase of Fourier spectrum ratio obsa K : Observed acc. record on the surface A O (4) Correction (2) Nonlinear site response analysis at Site-K using nonlinear analysis (1) Evaluate A 2E by deconvolution using equivalent linear method ka 2E : Calculated acc. on engineering bedrock at kth time iteration A 2E : Estimated acc. on engineering bedrock (= 4 A 2E ) (5) Iteration from (2) to (4) (4 times in this analysis) Engineering bedrock (V S =7m/s) obsa B : Observed acc. record in the borehole (Not use in this analysis) (6) Nonlinear site response analysis at Site-O using nonlinear analysis Fig.4. Schematic figure of estimation of seismic ground response based on the seismic record on the surface by using nonlinear analysis at two sites: Site-K and Site-O 5

6 ps v (m/s) Depth (m) Acceleration (m/s 2 ) Acceleration (m/s 2 ) 15 : G.L. Cal. : G.L. Obs. 6 h=.5 G.L. Obs. G.L. Cal. ps v (m/s) 2E (V S =7) Time (s) (a) Acc. time history Period (s) (b) Pseudo velocity resp. spec. Fig.5. Comparison of seismic response of ground on the surface for observation and calculation for the main shock 15 : Site-O - G.L. : Site-K - G.L. Time (s) (a) Acc. time history 6 h=.5 g max (x1-2 ) Site-K - G.L. Site-O - G.L Site-O -3 Period (s) (b) Pseudo velocity resp. spec. -4 Site-K -5 (c) Max. shear strain distribution Fig.6. Comparison of seismic response of ground for difference sites: Site-K and Site-O in calculations for the main shock The 1D SH-wave propagation models of the non-linear analysis are derived as follows. The models are divided into some suitable thickness of layers, which is about 1. m at Site-K and almost the same thickness as the pile diameter at Site-O. The initial shear stiffness values are determined by the S-wave velocity and the density. The dependence of overburden pressure is considered in the non-linear parameters described in the Section 2. The initial stiffness-proportional damping which is.2% for 1 st mode is applied to the models; the natural period of 1 st mode is calculated from eigenvalue analysis of the model under fix boundary condition at the bottom. Fig.5 compares the calculated result and the observed records on the ground surface at Site-K for the main shock. It can be seen that the numerical result is able to capture the key features of the observed 6

7 26m 32m record, namely the phases of acceleration time history and the peak values and periods of pseudo velocity response spectra. Fig.6 compares the calculated result at Site-O with at Site-K. When the bedrock motions are the same, the acceleration responses on the surface between Site-O and Site-K are almost the same as shown in Fig.6(a) and (b). Otherwise, the maximum shear strain responses are slightly different as shown in Fig.6(c) due to the influence of different strata structure. The maximum shear strains over 1.% occur at a depth of nearly the boundary of layers where the S-wave velocity contrast of adjacent layers is large in both Site-O and Site-K. 4. SIMULATION OF SEISMIC BEHAVIOR OF BUILDING CONSIDERING SSI EFFECTS Fig.7 illustrates the general views of Mashiki Town Office, in which the seismic records on the 1st floor observed during the first and main shock. This building is a three-story RC building supported by pile foundation constructed in 198 and has been retrofitted with out-flame to the south side in longitudinal direction. The total number of piles is 192. The foundations under existing part consists of isolated footings. Each footings are supported by one or 3 ~ 6 autoclave piles (AC pile). which consist of pile groups. The AC pile has 4mm in diameter and 75mm in thickness and 26~32m in length. The foundations under retrofit part consists of isolated footings, each supported by two steel pile piles with wings, the pile has 318.5mm in diameter and 6.9mm in thickness and 27m in length. Postearthquake damage evaluation for the building was performed after the main shock, and the damage grade of this building was moderate. Three corner piles were investigated by visual inspection and IT test. Consequently, there were two heavy damaged AC piles nearby pile cap under the footings at the west side. The natural frequency of the building is 3.~4. Hz, where the value were measured by using microtremor (Mori 217). Existing part 1.7m G.L. 2.75m Retrofit part (by using out-frame) (a) Elevation of the south side Modelling Area Span in EW direction:6m Pile type under existing part:ac pile 24m 54m A Pile type under retrofit part:steel pile with wings A (b) Plan of the footing (c) Section of A-A in the plan:(b) Fig.7. General views of Mashiki Town Office, in which the seismic records on the first floor observed during the first shock and main shock. 7

8 Depth (m) Pilecap connection 2 types of connecting condition (Fix or Pin) Upper soil spring Polilinear skeleton + 2 types of hysteresis property (Normal or Slip) Lower soil spring Polilinear skeleton + Normal hysteresis (Normal model) Superstructure Elastic model and rigid base foundation A O-1F Pile Fiber model (PC piles) Y X A O Site-O Input ground response at Site-O to the model as boundary conditions. A K Site-K A 2E Fig.8. Schematic view of analysis model concept C D B A Cofficient.5 1 Slip characteristics h: Ratio of subgrade reaction of TAKEDA-Slip model y c : Displacement amplitude for setting slip parameter y : Unloaded displacement TAKEDA- Slip model Zero-Slip model Slip Normal h y /y c 2 Fig.9. Schematic view of slip characteristic in soil spring model and the installation depth of slip characteristic in this analysis Seismic response analyses of soil-structure interaction were conducted by using multi-lumped-massbeam-spring model as shown in Fig.8. The superstructure is assumed to maintain linear behavior and has a rigid foundation. The pile groups are modeled into one pile which has the same bending stiffness as the pile groups. The model piles are assumed to be beam elements and connected to the rigid foundation. The length of piles in this analysis model is all 26 m; the S-wave velocity of soil around the pile end is about 5 m/s. A fiber section for an estimation of nonlinear behavior is adopted to the beam element. The piles are supported on the side by interaction springs and at the end by pin-roller support. The interaction springs are Winkler-type springs and the pile group effect is assumed in the spring property. Two types of nonlinearlity of interaction springs are assumed, namely, Normal model and Slip model. Normal model has a poli-linear backbone curve and a hysteresis loop which is established by applying the Masing rule to the backbone curve. Slip model has a poli-linear backbone curve and a hysteresis loop as shown in Fig.9. The poli-linear backbone curves which are consisted by five linear lines are defined based on Japanese Railway Standard (Railway Technical Research Institute 212). The slip hysteresis loop in Slip mocel is defined by combining TAKEDA-Slip model (Edo et al. 1977) and Zero-Slip model. The ratio of subgrade reaction of TAKEDA-Slip model to Zero-Slip model is η. The hysteresis loop characteristics of Takeda-Slip model in this analysis is described as follows: 8

9 Fourier Spectral Ratio (A O-1F /A O ) Table2. Properties of superstructure model 3 Parzen :.1Hz Floor Height (m) Weight (kn) Story Mode u i Stiffness (kn/mm) T 1 (s) 2 RF F F F F F F Frequency (Hz) Fig.1. Fourier spec. ratio of analysis model during small oscillation K r p, y c c y K s p, y c c y K ' K s s y y c 2 l (5) in which unloaded displacement: y, reference displacement amplitude to define the parameter: y c, subgrade reaction at y c : p c, 2 nd break displacement: y 2, slip parameter: λ (λ=η in this analysis). Zero-Slip model cannot disipate the hysteresis energy during a cyclic loading. Therefore, the increase of the value η leads to increase the hysteresis energy disipation. The value y c is defined from the pile diameter (y c =.1B, B is pile diameter) and the value y /y c and η is considered to vary with depth up to the depth of 2/β as shown in Fig.9 in this analysis. β is defined as follows: k 4 B h (6) 4EI in which subgrade reaction coefficient when the spring deformation is 1 mm: k h, bending stiffness of pile: EI. Normal model is adopted to the characteristics of interaction springs and Slip model is only adopted up to the depth of 2/β. In addition, Kashiwa et. al. (217) have indicated that Slip model is available for the interaction spring to consider the gap behavior between the pile and the soil according to the investigation based on in-situ lateral cyclic loading tests for steel piles with wings whose diameter is 19 mm, in which the gap behavior were observed. The superstructures are modeled as a lumped mass-beam system and its properties are shown in Table 2. The time history of acceleration at bedrock (A 2E ) is input to entire model and that of velocity and displacement of the seismic response of the free ground is input to the interaction springs. Two cases are performed in the analyses, namely, Normal case and Slip+Pin case. Nomal case is the analysis by using the model which the Normal model for interaction springs and no rotational condition at the pile head are adopted. Slip case is the analysis by using the model whih the Slip model for interaction springs and rotational condition at the pile head are adopted. A rotational condition at the pile head in Slip case is an alternative model which respresents the significant degradation of bending stiffness of pile head connection. Fourier spectral ratio of acceleration at the ground surface (A O ) and on the base foundation (A O-1F ) in Normal case is shown in Fig.1. The peak frequency of Fourier spectral ratio is smaller than the natural freqency of the superstructure. 9

10 Acceleration (m/s 2 ) Acceleration (m/s 2 ) Depth (m) Exterior pile Interior pile Exterior pile Vs (m/s) Loam Gravel Silt Bending Moment (knm) M y M u Bending Moment (knm) M y M u Sand (a) Soil classification profile (b) First shock (c) Main shock Fig.11. Comparison of maximum bending moment distribution for different position of pile in Normal case 2 Observation A O-1F A K 6 Observation A O-1F h=.5 ps v (m/s) A K -2 2 Calculation A O-1F (Slip+Pin) Time (s) 6 Calculation A O-1F (Normal) Period (s) h=.5 ps v (m/s) A K A K A O-1F (Normal) Time (s) (a) Acc. time history A O-1F (Slip+Pin) Period (s) (b) Pseudo velocity spectra Fig.12. Comparison with seismic response at the deference position: on the surface at Site-K (A K ) and on the 1 st floor at Site-O (A O-1F ). These figures illustrate both observation result and calculation result. Furthermore, the figures in case of the calculation compare the different cases of analyses: Normal and Slip+Pin case. The maximum bending moment distributions of the represented exterior and interior piles in Normal case during the first shock and the main shock are shown in Fig.11. The maximum bending moments of the pile caps on the first shock are close to the ultimate state at the exterior piles. It is suggested that the stiffness of the pile caps was significant reduced before the main shock because the pile caps were damaged during the first shock. In addition, the stiffness of AC pile is significantly degraded owing to damage. Therefore, it is reasonable to assume that the rotational condition is adopted to the pile cap connections. 1

11 S a Ratio (-) S a Ratio (-) 3 h=.5 3 Obs. h=.5 2 Cal.-Normal Obs. 2 Cal.-Normal 1 Cal.-Slip+Pin Period (s) (a) First shock on 14 Apr. 1 Cal.-Slip+Pin Period (s) (b) Main shock on 16 Apr. Fig.13. Comparison of acceleration response spectra ratio for observation and calculation, the figures compare the different cases of analyses: Normal and Slip+Pin case. The acceleration time histories and pseudo velocity response spectra are shown in Fig.12 which compare the observation and the calculation. Amplitudes and phase of the acceleration history on the base foundation A O-1F is different from that of the ground surface A K in the observation. The pseudo velocity response spectrum of A O-1F is smaller than that of A K for the periods in the range of.2~.6 s, whereas larger above 1. s. The relationship between A O-1F and A K in the calculation result of Slip+Pin case is similar to the observation. In addition, the spike shape in the acceleration history in Slip+Pin case is due principally to the hardening behavior of the interaction springs. On the other hand, the feature of A O-1F in Normal case is similar to A K in the analysis in terms of the phase of the acceleration history and the peak period of the pseudo velocity response spectrum. This indicates that the local nonlinear behavior such as the slip characteristic due to gap development between pile and soil and significant reduction in the stiffness of pile cap should be considered in order to simulate the seismic behavior of the building during very large earthquake. An acceleration response spectrum ratio is defined the ratio of A O-1F to A K and the comparisons of the acceleration response spectrum ratio between observation and calculation are shown in Fig.13. The ratio in the calculation result of Slip+Pin case is similar to the observation in the both case of the first shock and main shock. Therefore, it is suggested that the observed records on the 1 st floor in Mashiki Town Office were affected by the nonlinear soil-structure interaction. However, the peak value of the ratio in the observation is more than twice as large as in the calculation. More detailed study is necessary to investigate the seismic behavior of the buildings during very large earthquakes. 5. CONCLUSIONS Simulation analyses of the RC building supported by piles which subjected very large seismic oscillation are conducted by using a multi-lumped-mass-beam-spring model in order to evaluate the nonlinear soil-structure interaction effect on seismic behavior of the superstructure and pile foundation. In this study, it is focused on the Mashiki Town Office where seismic records were observed during the Kumamoto earthquake and some recorded maximum value of seismic intensity. Furthermore, the analysis model consists based on the soil investigation and drawing and specification. The following conclusions were obtained: 1) When the bedrock motions are the same, the acceleration responses on the surface between Site-O and Site-K are almost the same, but the maximum shear strain responses are different. The maximum shear strains over 1.% occur at a depth of nearly the boundary of layers in both Site-O and Site-K. 2) The maximum bending moments of the pile caps on the first shock are close to the ultimate state in the analysis. Therefore, it is suggested that the stiffness of the pile caps was significant reduced before the main shock because pile caps were damaged during first shock. 3) When the slip characteristic due to gap development between pile and soil and significant 11

12 reduction in the stiffness of pile cap are considered, the seismic responses in this analysis are close to the observed records. Therefore, it is suggested that the observed records on the 1 st floor in Mashiki Town Office were affected by the nonlinear soil-structure interaction. 6. ACKNOWLEDGMENTS The authors would like to thank for the detail information of the office building: Mashiki Town Office and for the providing strong-motion records: K-NET and KiK-net operated by National Research Institute for Earth Science and Disaster Prevention (NIED), Japan Meteorological Agency (JMA), and the local government office. 7. REFERENCES Arai, H. and Kashiwa, H. (218). Estimation of S-wave velocity profiles from microtremor and borehole surveys in damaged area during the 216 Kumamoto Earthquakes, Japan. 16 th European Conference on Earthquake Engineering. Edo, H. and Takeda, T. (1977). Elasto-Plastic seismic response frame analyses of RC structures. AIJ Annual meeting Japan Meteorological Agency, Japan (JMA) (216). Kashiwa, H., Kobayashi, T. and Miyamoto, Y. (217). Evaluation of lateral resistance of steel pile with wing for simulation analysis based on in-situ cyclic lateral loading tests. JAEE annual meeting 217. P2-16. (in Japanese) Kozo Keikaku Engineering Inc. (217). RESP-F3T User s Manual. (in Japanese) Mashiki Town Office (1979). Report on geological investigations for town office buildings. (in Japanese) Matsu ura, M. and Hirata, N. (1982). Generalized least-squares solutions to quasi-linear inverse problems with a prior information. Journal of Physics of the Earth, 3: Mori, M. (217). Damage situation and seismic response estimation of Mashiki City Hall under the 216 Kumamoto Earthquake, Workshop about investigation for soil-structure interaction behavior and seismic design considered with SSI effects, AIJ: Nakagawa, T., Hirasawa M., Kobayashi K. and Sasaki S. (211). Modelling of non-linear hysteretic soil. J. Struct. Constr. Eng., AIJ, Vol.76, No.666: (in Japanese) National Institute for Land and Infrastructure Management, Japan (NILIM) (216). Report of committee for causal analyses on damage to buildings during the 216 Kumamoto earthquakes: (in Japanese) National Research Institute for Earth Science and Disaster Resilience, Japan (NIED) (216). Railway Technical Research Institute (212). Design Standards for Railway Structures and Commentary (Foundation Structure). (in Japanese) 12