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Experimental Study on Characteristics of Horizontal Dynamic Subgrade Reaction Using a Single-Pile Model Estudio Experimental sobre las Caracteristicas de la Reaccion Dinamica Horizontal de la Subrasante Utilizando un Modelo de Pilote Aislado Hirofumi Fukushima, Jun’ichi Nishikawa and Kouichi Tomisawa Geotechnical Division, Civil Engineering Research Institute of Hokkaido Abstract The spring constant of a pile foundation was studied to assess the seismic resistance of foundations built in sandy, clay and volcanic ash soils, the last of which is widely distributed throughout Hokkaido, Japan. Centrifuge experiments were performed to test the static horizontal load and dynamic vibration of single piles under a centrifugal acceleration of 50 G. Based on these experiments, two important factors in the seismic design of pile foundations were studied: horizontal static subgrade reaction (Kh) and horizontal dynamic subgrade reaction (Khe). The relationship between these two factors, expressed as Khe=αKh, was also studied. It was revealed that α differs according to soil type, indicating the necessity of including soil type in the assessment of seismic designs for pile foundations. Resumen La constante de resorte de la cimentacion por pilotes fue estudiada para determinar su resistencia sismica en suelos arenosos, suelos arcillosos y en cenizas volcanicas, las cuales se encuentran extensamente en el area de Hokkaido, Japon. Ademas, experimentos en centrifuga fueron llevados a cabo para evaluar la carga horizontal estatica y la vibracion dinamica en pilotes aislados bajo aceleracion centrifuga de 50 g. Con base en estos experimentos, se estudiaron los factores importantes para el diseno asismico de la cimentacion por pilotes: reaccion estatica horizontal de la subrasante (Kh) y reaccion dinamica horizontal de la subrasante (Khe). Su relacion, expresada por Khe = αKh, tambien fue estudiada. Se logro determinar que α depende del tipo de suelo, lo cual indica la necesidad de incluir el tipo de suelo en la evaluacion de un diseno asismico de cimentaciones por pilotes. rather than on ground strength, which can be 1 INTRODUCTION determined using soil survey results. Since the pile foundation is surrounded by Current seismic designs (Japan Road ground, it is generally considered to generate Association, 1996 and 2002, Ogawa and Ogata, virtually no response vibrations. However, the 1997) for pile foundations in Japan require pile foundation does react to seismic motion, analysis of the seismic behaviour of the pile using which triggers simultaneous response the traditional seismic coefficient method, displacement of the surrounding ground. During seismic horizontal load-carrying capacity method an earthquake, seismic force acts on the pile and the dynamic analysis method. These design foundation, and the force is transmitted to the methods share the same structural engineering superstructure. The response of the superstructure consideration: the provision of a certain degree of is returned to the ground as inertial force. These deformation capacity under seismic force. forces act differently from earthquake to However, in assessing the seismic resistance of earthquake because the waveforms of seismic the ground, the ground spring during an motions are different in each earthquake. For this earthquake (dynamic coefficient of horizontal reason, the seismic behavior of pile foundations is subgrade reaction) is determined largely based on very complicated, and the present seismic design the static spring of the soil, i.e., a soil constant, method does not satisfactorily address this complexity. Analysis of the seismic behavior of could accurately represent conventional steel piles the pile foundation requires a clear understanding (φ = 500 mm, t = 10 mm) (Table 1). The of spring characteristics and other complicated centrifuge generated a centrifugal acceleration of soil factors during an earthquake, such as non- 50 G. The bearing layer consisted of soil cement, linear factors. and the embedding depth was three times the To accurately assess the seismic resistance of length of the pile diameter. The soil used for the pile foundations, this research focused on the experiment was disturbed volcanic ash soil spring constant for foundations built in volcanic collected from the Lake Shikotsu area, a type of ash soil, which is widely distributed throughout soil that is considered to be unique to Hokkaido. Hokkaido. A series of centrifuge experiments was Soil was poured from a set height to ensure that conducted to test the static horizontal load and the model ground was of uniform soil quality. For dynamic vibration of single piles under a the vibration experiment, strain gauges were centrifugal acceleration of 50 G. installed on the pile. Accelerometers were placed Elements considered important to the seismic at points far enough from the strain gauges that design of pile foundations were examined by the behavior of the pile would remain unaffected comparing the characteristics of the coefficient of at the depths of the strain gauge locations. A dynamic horizontal subgrade reaction of the pile static horizontal loading test and dynamic loading (Khe) with those of the coefficient of static tests using white noise and sine waves were also horizontal subgrade reaction (Kh). conducted. In the dynamic loading tests, a 400-g weight (0.4 kg x 503 = 50 tf) was installed on the 2 OUTLINE OF THE EXPERIMENTS pile head to represent the bridge substructure. The experiments were conducted using a model 3 STATIC HORIZONTAL LOADING TEST steel pile on a scale of 1:50 (φ = 10 mm, t = 0.2 (50 G) mm) in a steel container (Figure 1) with internal dimensions of 700 x 200 x 350 mm (L x W x H). A static horizontal loading test of the pile was The scale was set at 1:50 so that the model pile conducted using the multi-cycle strain control method, which employs a horizontal Accelerome Model pile loading device. The loading rate was 70 19ch Strain gauge Weight 10 0.25 mm/min. at the model pile head. 10 30 P1 5ch Pile displacement was measured using 30 P2 4ch laser displacement gauges, and pile P3 3ch stress was determined with strain 50 gauges. The permissible displacement P4 2ch (δ) for this test was set at 0.3 mm, 300 80 which is equal to the permissible displacement of the full-size pile (15 Volcanic P5 1ch mm) divided by 50 G. This test Ash 90 displacement was used as a criterion 10 P6 17ch for pile displacement. The duration of Soil Cement 30 40 full virgin load application was approximately 15 min., in compliance 700 with the standard set by the Japanese Figure 1 Experimental container Geotechnical Society (1983). Table 1 Scale factors for the experiment To calculate the static subgrade reaction, the pile foundation was Experimental Notation Unit Scale size Real size regarded as the beam of the elastic Ground Thickness g1 H m 1/ λ 0.300 15.000 foundation of the skeletal model, and a Bearing ground thickness H m 1/λ 0.040 2.000 Embedding depth g2 L m 1/λ 0.320 16.000 Winkler spring model was constructed Outer diameter D m 1/λ 0.010 0.500 for use in the analysis of the Pile Plate thickness t 2 m 1/λ 7 0.002 7 0.010 coefficient of subgrade reaction Modulus of elasticity E Tf/m 1 2.1x10 2.1x10 Geometrical moment of inertia I4 m4 1/λ -11 7.3952x10 -11 4.6220x10 (Figure 2). The vertical distribution Cross-sectional area A2 m2 1/λ -6 6.1575x10 0.01539 and the coefficient of subgrade 3 -3 Structure Weight S M Tf 1/λ 0.4x10 50.0 reaction were used as parameters for Height H m 1/λ 0.015 0.75 Vibration acceleration S α g λ (1) (0.020) the trial calculation. This calculation Note: 1/λ = model/real = 1/50 was performed to determine the values that would produce agreement between the experimental distribution of displacement at the Y =-1.067E02 X + 2.398E-03 pile head and pile stress in the ground and those 0.00 X:Distance from estimated from the analysis. The static subgrade the pile head Upper Y:Coefficient of reaction coefficient (Kh) was defined as the area subgrade reaction layer of the coefficient of subgrade reaction (A) divided -0.05 by depth (Z), as seen in Figure 3. The coefficient -0.054 of static subgrade reaction was found to be 5,500 Distance from the pile head (m) kN/m2 from the results of the experiments and the -0.10 analysis carried out by recreating the experimental situations. In making this calculation, it was assumed that the distribution of the coefficient -0.15 was uniform (Figures 4 and 5). Fitting of bending moment curves was Spring value Lower performed using theoretical and experimental -0.20 (constant) layer values of flexural rigidity of the pile in volcanic Area (A) ash soil to analyze the vertical distribution of rigidity. In the experimental ground, rigidity -0.25 increased at a constant rate in a particular depth range. After this increase, however, rigidity became constant. The experimental ground was thought to be elastic for the following reasons: -0.30 120 0 20 40 60 80100 140 - The depths of the peak points of the bending moment were constant with respect to load value. Figure 3 Coefficient of subgrade reaction - Displacement at the pile head was Distance from the pile head (m) approximately 1 mm (10% of the pile 0.0 diameter), yet it did not affect the rigidity of the ground. - Linear representation of the strain value of the -0.1 Depth range of the spring value (upper layer) pile material was possible. Depth range of the spring value (lower layer) -0.2 2 Theoretical value 4000kN/m 2 Theoretical value 5000kN/m 2 Theoretical value 5500kN/m 2 Theoretical value 5500kN/m -0.3 Experimental value (Load: 0.0128kN) Experimental value (Load: 0.0259kN) Experimental value (Load: 0.0396kN) Experimental value (Load: 0.0526kN) -0.4 -3 -3 -3 0.0 1.0x10 2.0x10 3.0x10 δ (m) Figure 4 Distribution of vertical displacement Distance from the pile head (m) 0.0 Depth range -0.1 of the spring value (upper layer) Depth range of the spring value (lower layer) -0.2 2 Theoretical value 4000kN/m 2 Theoretical value 5000kN/m 2 Theoretical value 5500kN/m 2 Theoretical value 5500kN/m -0.3 Experimental value (Load: 0.0128kN) Experimental value (Load: 0.0259kN) Experimental value (Load: 0.0396kN) Experimental value (Load: 0.0526kN) -0.4 -3 -3 -3 -3 -3 -4.00x10 -3.00x10 -2.00x10 -1.00x10 0.00 1.00x10 M (kN m) Figure 5 Distribution of the vertical bending Figure 2 Analytical model moment 4 COEFFICIENT OF DYNAMIC SUBGRADE vibrations (i.e., they are frequency dependent). In REACTION this study, analysis was performed at the natural frequency of the pile foundation, which best 4.1 Method used for calculation of the reflects the relative displacement of the pile and coefficient of subgrade reaction the ground, to develop a new method for In the current specifications for highway calculating the coefficient of dynamic subgrade bridges, the correction coefficient, α, is used to reaction. The coefficient of dynamic subgrade assess ground rigidity and determine the reaction calculated from the interaction force of coefficient of dynamic subgrade reaction rather the pile and the ground was compared with the than the coefficient of static subgrade reaction. coefficients of dynamic and static subgrade In this study, the coefficient of subgrade reaction reaction. was calculated using two methods (p-δ and (2) Natural frequency of the pile foundation in eigenvalue analysis), and then compared with the volcanic ash soil coefficient of static subgrade reaction to A sine-wave-based experiment was conducted determine the correction coefficient. on the pile foundation by applying the vibration of white noise (a waveform with a frequency of 4.2 Analysis of the coefficient of subgrade between 10 and 350 Hz) to clarify its natural reaction using the p-δ curve (p-δ method) frequency. The analysis results for the Fourier (1) Calculation using the p-δ method transformation characteristics and the transfer To calculate the coefficient of subgrade reaction function indicated that the natural frequency of the caused by interaction between the ground and the pile foundation was between 55 and 85 Hz under pile (Khe1), the interaction force (p) was divided the conditions of white noise vibration when the by relative displacement (δ). Figure 6 shows the weight was 400 g (Figures 7, 8 and 9). Based on process used to make this calculation. The level the Fourier transformation characteristics and the of vibration was determined based on the transfer function obtained in the sine-wave- assumption that displacement is minimal, i.e., the vibration experiment (Figure 10), the natural plasticity of the pile and the ground was not frequency of the pile foundation in volcanic ash considered. soil was found to be 62.5 Hz. Acceleration and strain calculated from this frequency value were Fourier spectrum of Strain used for analysis. acceleration 150 100 Bandpass filter Bending moment P1 Strain ( µ) 50 0 -50 -100 Fourier transformation -150 for speed and displacement calculation 10 19ch Acceleration (G) 8 6 4 2 0 -2 Third-degree function Second-order Second -4 -6 using the four-point integral derivative -8 -10 method 0.5 0.4 8ch Voltage (mV) 0.3 0.2 0.1 Pile Subgrade 0.0 Ground displacement displacement reaction -0.1 -0.2 -0.3 -0.4 -0.5 21ch Acceleration (G) 10 Relative displacement 8 6 4 2 0 -2 Figure 6 Process of the p-δ method -4 -6 -8 -10 Recent research studies and analyses have 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 revealed that relative displacement and relative Time (sec) interaction force, both of which are used to Figure 7 White noise W = 400 g calculate the coefficient of dynamic subgrade Dyn. Amp. = 0.4 reaction, change with respect to the number of taken at six points, the equation for curve fitting Fourier amplitude ratio 50 400 g was approximated using a multi-term fifth-degree 40 function. Equation 1, the second derivative of this function, was used to calculate the subgrade 30 20 10 reaction. It was hypothesized that the function of 0 the subgrade reaction could be approximated 10 100 Frequency (Hz) 1000 using a curve from the third-degree function. Figure 8 19ch/17ch Transfer function d 2M Dyn. Amp. = 0.4 = − q ( x) (1) dx 2 d2y EI 2 = − M ( x) (2) Fourier amplitude ratio 500 400 400 g dx 300 200 The constants of a multi-term second-order 100 integral (Equation 2) were assessed by setting the boundary conditions of the deflection angle (θ) 0 10 100 1000 Frequency (Hz) and displacement (δ) at the pile’s lower end at Figure 9 P1/17ch Transfer function zero because that end is fixed by soil cement. Dyn. Amp. = 0.4 Due to the limitations of the experimental apparatus, direct measurement of displacement 12 was not possible in the ground. The second-order Fourier integral of the equation for ground Fourier amplitude 10 19 ch/17ch 8 acceleration was used to calculate displacement 6 (Figures 11 and 12). For calculations necessary for correcting the displacement axis, bandpassing 4 2 0 from 50 to 1,000 Hz was performed. The 40 50 60 70 80 90 100 coefficient of subgrade reaction could be Fourier amplitude 500 400 P1 ch/17ch calculated because the above procedure enables 300 the computation of displacements of the pile and 200 the ground and subgrade reaction. 100 10 8 5ch Acceleration (G) 0 6 40 50 60 70 80 90 100 4 500 2 0 Fourier amplitude 400 P2 ch/17ch -2 -4 -6 300 -8 -10 200 0.0 0.1 0.2 0.3 0.4 0.5 0.6 100 Time (sec) 0 Experimental acceleration 40 50 60 70 80 90 100 500 Figure 11 Sine wave 62.5 Hz Dyn. Amp. = 0.4 Fourier amplitude P3 ch/17ch 400 300 200 -4 100 2.0x10-4 1.5x10-4 5ch Displacement (m) 0 40 50 60 70 80 90 100 1.0x10-5 5.0x10 Nominal frequency (Hz) 0.0 -5 -5.0x10-4 Figure 10 Transfer function of the sine wave -1.0x10-4 -1.5x10-4 W = 400 g Dyn. Amp. = 0.4 -2.0x10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Time (sec) (3) Calculation of relative displacement and subgrade reaction To calculate displacement of the pile, the Figure 12 Sine wave 62.5 Hz Dyn. Amp. = 0.4 bending moment must be identified from pile strain values after which curve fitting must be performed (Figure 13). Since measurements were Volcanic ash 62.5Hz D=0.4 W=400g Volcanic ash 62.5Hz D=0.4 W=400g 0.00 0.00 7 8 10 11 12 13 14 15 9 65 4 3 1 2 17 16 -0.05 -0.05 Container depth (m) 18 15 17 Container depth (m) 16 18 1413 12 11 10 9 8 7 1 2 3 4 5 6 -0.10 -0.10 -0.15 -0.15 ∆ t=0.001 sec ∆ t=0.001 sec -0.20 1 - 0.1424 sec -0.20 1 - 0.1424 sec 5 - 0.1466 sec 5 - 0.1466 sec (Maximum shear strain) (Maximum shear strain) -0.25 9 - 0.1504 sec -0.25 9 - 0.1504 sec 18 - 0.1594 sec 18 - 0.1594 sec -0.30 -0.30 -1.0 0.0 1.0 2.0 -3 -3 -3 -3 -2.0x10 -1.0x10 0.0 1.0x10 2.0x10 Pile bending moment (kN m) Subgrade reaction of the pile (kN/m) Figure 13 Distribution of the bending moment at Figure 16 Subgrade reaction of the ground at different times different times (4) Analysis results Volcanic ash 62.5Hz D=0.4 W=400g • Dynamic coefficient of the ground 0.00 16 15 18 17 The time-history displacement (pile, ground 8 9 10 14 11 12 13 7 6 5 4 3 Container depth (m) -0.05 1 and relative), and the distribution of subgrade 2 reaction and its coefficient are shown in Figures -0.10 14 to 18. Curves were determined for ∆ t=0.001 sec approximation based on the distribution of the -0.15 coefficient of subgrade reaction. A curve that 1 - 0.1424 sec -0.20 represents the average values of all the curves was 5 - 0.1466 sec identified, and the average value of the curve -0.25 (Maximum shear strain) identified was found to be 12,400 kN/m2, from 9 - 0.1504 sec 18 - 0.1594 sec which the value of dynamic spring was estimated. -0.30 -4 -4 0.0 -4 -4 -4.0x10 -2.0x10 2.0x10 4.0x10 Volcanic ash 62.5Hz D=0.4 W=400g 0.00 Relative ground and pile displacement (m) 15 16 18 Figure 17 Relative ground and pile displacement 17 7 14 13 12 11 10 9 8 4 5 6 Container depth (m) -0.05 1 2 3 at different times -0.10 Volcanic ash 62.5Hz D=0.4 W=400g -0.15 0.00 -0.20 -0.05 Average value of all data -0.25 Container depth (m) -0.30 -4.0x10 -4 -2.0x10 -4 0.0 2.0x10 -4 4.0x10 -4 Average value of -0.10 the fitted curve Pile displacement (m) up to 0.30 m deep Figure 14 Pile displacement at different times Volcanic ash 62.5Hz D=0.4 W=400g -0.15 0.00 18 17 11 10 9 16 15 14 13 12 7 8 1 2 3 4 5 6 Container depth (m) -0.05 Fitted curve of -0.20 the average value of -0.10 all data -0.15 ∆ t=0.001 sec -0.25 -0.20 1 - 0.1424 sec 5 - 0.1466 sec (Maximum shear strain) -0.25 9 - 0.1504 sec -0.30 0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000 18 - 0.1594 sec 2 Coefficient of subgrade reaction (kN/m ) -0.30 -1.0x10 -4 -5.0x10 -5 0.0 5.0x10 -5 1.0x10 -4 Coefficient of dynamic subgrade reaction 2 Relative ground displacement (m) Estimated average value 12400 kN/m Figure 15 Relative displacement of the ground at Figure 18 Vertical distribution of the coefficient different times of subgrade reaction • Pile displacement • For analysis of vertical distribution of ground For the pile displacement mode, the primary rigidity, the distribution used in the mode is dominant because of the tremendous horizontal static subgrade reaction test is effect of one mass point (400-g weight). The utilized. distribution of bending stress for the pile • Bending rigidity of the pile is determined indicates that a large bending moment based on the bending test, and the conditions originates from a depth of approximately 40 of the pile’s lower end are constant. mm, at which the ground shows significant • The spring coefficients of the pile are from a displacement. Looking at the displacement discrete spring (Winkler) in the analysis found through curve fitting of the bending model. distribution and a second-order integral, the According to these hypotheses and the displacement of the pile indicates the first eigenvalue method, a coefficient of dynamic vibration mode. subgrade reaction of 18,000 kN/m2 was obtained. Figure 19 shows the results of the analysis. • Ground displacement It can be seen that the time-history 85 displacement of the ground found by Theoretical value based on the analysis Characteristic frequency (Hz) 80 Fitted strait line accelerometers placed in the ground does not 75 indicate the first vibration mode as is the case in Dominant frequency of the displacement of the pile. 70 volcanic ash 62.5Hz The low-frequency component is larger than 65 (experimental value) the vibrational component due to the effect of 60 Theoretical coefficient surface waves caused by the use of a fixed 55 of dynamic subgrade container. Therefore, in the dynamic reaction of volcanic ash 50 experiment, the ground displacement mode is 18000kN/m2 considered to be high because the surface part is 45 affected by both base vibrations and surface 50 60 70 80 0 90 0 1000 20 30 40 50 00 00 0 0 00 00 00 00 00 waves due to the effects of the container. 0 0 0 0 0 2 Coefficient of dynamic subgrade reaction (kN/m ) 4.3 Analysis of the coefficient of dynamic Figure 19 Computation of the coefficient of subgrade reaction using an eigenvalue dynamic subgrade reaction using the (eigenvalue analysis) dominant frequency of the ground Most research on the dynamic interaction between piles and ground employs methods such 4.4 Coefficient of dynamic subgrade reaction: as second- and third-order FEM and the Penzien comparison between dynamic reaction model for experiments and analyses. The and static reaction specifications for highway bridges, which were The coefficient of dynamic subgrade reaction revised in March 2002, however, use the normal acquired from the dynamic centrifugal experiment correction coefficient, α, to assess ground rigidity was compared with the coefficient of static and determine the coefficient of dynamic subgrade reaction. The coefficient of dynamic subgrade reaction for dynamic and other analyses. subgrade reaction is 3.3 times greater in the p-δ By focusing on this correction coefficient, this method and 2.3 times greater in the eigenvalue experiment adopted the use of an analysis model, method (Table 2). which was used for both analysis of the coefficient of static subgrade reaction and Table 2 Comparison of the coefficient of eigenvalue analysis (mode analysis by free subgrade reaction (volcanic ash) vibration) of the coefficient of dynamic subgrade reaction. The value of the spring coefficients of Khe α=Khe/Kh the pile was set as a parameter to assess the coefficient of dynamic subgrade reaction (Khe2) at p-δ method (Khe1) the natural frequency of the pile foundation 12,400 2.3 kN/m2 obtained from the experiment. The assumptions made for the eigenvalue Eigenvalue method (Khe2) 18,000 3.3 method are as follows: kN/m2 • The pile and the ground are in the linear region. The same methods were applied to assess silica dynamic analysis method are currently used in the sand and kaolin clay. The results differed design of pile foundations. These methods according to soil type (Table 3); thus, the employ a ground constant to define, in a fairly coefficient of dynamic subgrade reaction should simple manner, the dynamic subgrade reaction not be determined solely from the coefficient of (during an earthquake) of the pile foundation, Khe, static subgrade reaction. Seismic resistance which is considered to be an important factor in should be assessed with reference to soil type. assessing seismic resistance. A series of horizontal dynamic centrifuge Table 3 Comparison of the coefficient of model experiments was conducted to analyze the subgrade reaction by soil type dynamic subgrade reaction of silica sand, kaolin Silica Kaolin Volcanic clay and volcanic ash. The results differed sand clay ash according to the soil type due to dynamic interactions between the pile and the ground. Kh KN/m2 20,000 4,000 5,500 To appropriately assess the coefficient of horizontal dynamic subgrade reaction, Khe, new Khe1 KN/m2 27,400 15,500 12,400 concepts are necessary for the creation of seismic designs for pile foundations (Wang et al, 2000, Khe2 KN/m2 55,500 13,700 18,000 Meymand, 1998, Tomisawa et al, 2001). Such new seismic designs should address both the soil and response characteristics of the pile and the Khe1/Kh 1.4 3.9 2.3 ground while maintaining the advantages of the current design. 5 CONCLUSIONS In the future, dynamic characteristics according to soil type will be examined in terms of non- A series of dynamic centrifuge model linear characteristics and pile displacement. experiments were conducted in connection with seismic behavior of the pile foundation to compare the coefficient of dynamic subgrade reaction with the coefficient of static subgrade REFERENCES reaction. The following results were found: Japanese Geotechnical Society (1983): “Horizontal Loading Test Method for Piles and Instruction Manual”. (in Japanese) 1) The vibration experiment conducted using the Japan Road Association (1996): Reference centrifuge apparatus roughly clarified the Concerning Application of "Specifications dynamic characteristics of ground and piles by Concerning Restoration of Highway Bridges soil type. Damaged by the Hyogo-ken Nambu Earthquake" 2) According to the p-δ method, the coefficient of (draft). (in Japanese) Japan Road Association (2002): “Specifications for horizontal dynamic subgrade reaction, Khe1, is Highway Bridges with Instruction Manual V - 27,400 kN/m2 for silica sand, 15,500 kN/m2 Seismic Design Edition”, pp.210- 221. (in Japanese) for kaolin clay and 12,400 kN/m2 for volcanic Meymand (1998): “Shaking Table Scale Model Tests ash. The coefficient of dynamic subgrade of Nonlinear Soil-Pile-Super Structure Interaction in reaction (Khe1) divided by the coefficient of Soft Clay”. static subgrade reaction (Kh), α, is 1.4 for Ogawa and Ogata (1997): “Verification of Vibration Resistance by Dynamic Analysis”: Foundation Work, silica sand, 3.9 for kaolin clay, and 2.3 for vol. 25, No. 3. (in Japanese) volcanic ash. Tomisawa et al (2001): “Dynamic Horizontal Subgrade 3) The coefficient of dynamic horizontal subgrade Reaction of Pile by Dynamic Centrifuge Model Test”, reaction changes with respect to strain and Proceedings of the 56th Academic Lecture Meeting frequency. The coefficient of horizontal static of Japan Society of Civil Engineers. (in Japanese) Wang et al (2000): “Experimental Consideration of subgrade reaction, Khe2, determined using the Dynamic Interaction between Pile Foundation and eigenvalue method, differs depending on the Ground Using a Large Shear Soil Layer”, vibration mode and dominant frequency of the Proceedings of the Japan Society of Civil Engineers. pile (eigenvalue), both of which are affected (in Japanese) by soil type. The seismic coefficient method, the seismic horizontal load-carrying capacity method and the