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Numerical Analysis of Dynamic Soil-Box Foundation-Structure Interaction System

Xilin Lu, Peizhen Li, Bo Chen and Yueqing Chen

State Key Laboratory on Disaster Reduction in Civil Engineering, Tongji University, P.R. China

Abstract

Three-dimensional finite element analysis on soil-box foundation-structure interaction system is carried out.

General-purpose finite element program ANSYS is used in the analysis. Commonly used equivalent linearity model is chosen as constitutive relation of soil, and the changing-status nonlinearity of soil-structure interface is considered by surface-to-surface contact element. The computational model and analysis method is verified through comparison study between the calculation and the shaking table test results, and issues drawn from the computational analysis are consistent with the test results. The calculations demonstrate that sliding and separation occur between the foundation and the soil, and it shows that great error will occur when the material nonlinearity of soil and the changing-status nonlinearity of soil-structure interface are not considered in calculation of soft soil. Finally, some important findings from the calculations are concluded.

Keywords: soil-structure interaction; shaking table test; numerical analysis; ANSYS program

Introduction

The effect of Soil-Structure Interaction (SSI) on seismic response of structures has attracted an intensive interest among researchers and engineers over the world. Most of these researches focus on theoretical study and analysis, while less has been done on the experimental study, and the comparison analysis between computer simulation and test results is much less (Hadjian A et al., 1991). It is very significant that the rationality of computational model and the feasibility of model test can be verified by comparison analysis. Based on shaking table model tests and combining general-purpose finite element program ANSYS, this project plans to carry out comparison study between computer simulation and model test results, and to improve computational analysis method in forenamed process, and finally to fulfill computer simulation on practical engineering.

Shaking table model tests on SSI have been accom- plished in Chinese State Key Laboratory for Disaster Reduction in Civil Engineering. Through these tests, abundant experimental data are obtained. Three- dimensional finite element analysis on soil-box foundation-structure test is carried out in this paper.

Brief Description of Shaking Table Model Test In shaking table tests, model soil cannot be held in an infinite dimension box. Due to wave reflection on the boundary and variation of system vibration mode, certain error so called ‘box effect’ will occur in test

results. In order to reduce the box effect, a flexible container (Chen et al. 1999) and the proper construc- tional details were designed in model test, and the ratio between ground plane diameter D and structure plane size d was taken as 5 by controlling the size of the structure plane. The cylindrical container was 3000mm in diameter and its lateral rubber membrane was 5mm in thickness, and the reinforcement of Φ4@60 was used to strengthen the outside of the container. The lateral side of the cylinder was fixed with the upper ring plate and the base plate by bolt. The upper ring plate was supported by four columns fixed on the base plate.

Height adjustable screw rod was installed on the column to adjust the upper plate to horizontality and adjust the cylinder to a proper state. A universal joint was installed on the column top to enable the ring plate to displace laterally. The base plate was made of steel plate. In order to minimize relative slip between the soil and the container on the base surface, a kind of crushed rock was bonded to the base steel plate by epoxy resin to make the surface rough.

In model test, 3x3 group piles foundation and box foundation were adopted. Shanghai soft soil was used as model soil. The test was designed according to the similitude relation. The scaling factors of models were 1/10 and 1/20, respectively. Two kinds of superstruc- tures were used in the test. A single column with mass block on its top was used as one kind of superstructure.

In order to simulate different superstructures, the mass block on the column top was adjusted to change the dynamic characteristics. And in this test, the model soil was uniform sandy silt. The other kind of superstructure was a 12-story reinforced concrete frame structure with single span. And in this test, the kinds of model soil were silty clay, sandy silt and medium sand from top to bottom.

Contact Author: Xilin Lu, Professor, State Key Laboratory on Disaster Reduction in Civil Engineering, Tongji University, 1239 Siping Road, Shanghai 200092, P.R. China

Fax: +86-21-65982668 e-mail: lxlst@mail.tongji.edu.cn

(Received May 8, 2002; accepted August 8, 2002)

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Accelerometers and strain gauges were used to measure the dynamic response of the superstructure, the foundation and the soil. Soil pressure gauges were used to measure the contact pressure between foundation and soil. The measuring point arrangement model test BS10 is shown in Fig.1. Both unidirectional (X direction of shaking table) excitations and bi-directional excitations (X and Z direction of shaking table) were employed in test, including El Centro wave (1940, N-S component), Shanghai artificial wave (SHW2) and Kobe wave. The adjustment of acceleration peak value and time interval of the excitation was referred to references (Lu et al.

2002).

The emphasis of this paper is mainly put on the computer simulation of the BC20 and BS10 test. BC20 denotes the test of uniform soil-box foundation-single column system with the scaling factor of 1/20, while BS10 denotes the test of layered soil-box foundation-frame structure system with the scaling factor of 1/10.

Modeling Method

(1) Simulation of flexible soil container

The behavior of the flexible container should be reflected in modeling of shaking table model test on SSI.

The lateral rubber membrane of the container is meshed by shell element in modeling. The base plate of the container was fixed with shaking table by bolt, and proper measures were taken to make the surface of the base plate rough in test. So the relative slip between the soil and the bottom of the container can be ignored and the bottom of soil can be considered as fixed in modeling. The reinforcement loops outside the container were used to provide radial rigidity and permit soil to deform as horizontal shear layer in test. In

modeling, the reinforcement loops can be considered that the nodes along the container perimeter with same height have the same displacement in excitation direction (X-axis direction of shaking table), which can be realized by coupling of degrees of freedom in ANSYS program (Chen et al. 2002).

(2) Dynamic constitutive model of soil and simula- tion of material nonlinearity

In this paper, equivalent linearization model of soil is adopted. In calculation, assume a pair of shear modulus Gd1 and damping ratio D1 for each layer of soil, calculate corresponding effective shear strain γd1, then find out corresponding shear modulus Gd2 and damping ratio D2 in the relationship curves of Gd /G0-γd and D-γd, respectively (see Fig.2). The curve of Gd /G0

-γd denotes the relationship between the effective shear strainγd and the ratio of shear modulus Gd and the initial shear modulus G0 , while the curve of D-γd denotes the relationship between the damping ratio D and the effective shear strainγd. Repeat the above steps until the differences between twice results of the shear modulus and the damping ratio are in allowable range.

In calculation, 0.65 times of the maximal shear strain is taken as γd (Wang et al. 1997; Soil Dynamic Laboratory, 2000).

The effect of effective confining pressure of soil on initial shear modulus is considered when the initial shear modulus of each layer soil is chosen. Equation 1 (Soil Dynamic Laboratory, 2000) shows that initial shear modulus is proportional to the square root of ratio of effective confining pressure, namely

1 3

3

1 +

+

= i

i

i i

G G

s

s [1]

Where

Gi is the initial shear modulus of the i th layer soil;

Gi+1 is the initial shear modulus of the (i+1) th layer soil;

σ3 iis the effective confining pressure of the i th layer soil;

σ3 i+1 is the effective confining pressure of the (i+1) th layer soil.

The units of above four variables are all Pa.

In ANSYS program, there is a kind of parametric design language named APDL, which is a scripting

Fig.2. Typical Gd G0~gd and D~gd

Curves of Shanghai Soft Soil Fig.1. Sketch of Measuring Point Arrangement

Accelerometer in Horizontal:

A1-A7, S1-S10, SD Accelerometer in Vertical:

AZ1-AZ7, R1-R2, SZD

Shaking Direction

Pore Pressure Gauge: H1-H6 A7 AZ7 A6

A5

A4

A3

A2

A1 AZ2 AZ3 AZ4 AZ5 AZ6

R2 R1

H6

S4 S3

S2 S1

H3 H2 H1 H5

H4

S7

S6 S5

S9 S8 S10

SD SZD

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language. Users can use it to automate common tasks or even build models in terms of parameters. The equivalent linearity model is realized in ANSYS program by using the APDL, and the calculation of material nonlinearity is realized automatically.

(3) Simulation of the change-status nonlinearity on soil-structure interfaces

Due to the material characteristics difference between the soil and concrete, there are sliding and splitting phenomena on the soil-structure interface when interface stress increases to certain level, and the gappy interface could reclose under certain load condition. Earlier researchers adopted interface elements such as Goodman element, lamina element and lamina soil element to simulate this changing-status nonlinearity on the interfaces of soil and structure (foundation). In ANSYS program, contact analysis is realized by overlaying a thin layer of elements upon the contact interface of analysis model. The soil surface is taken as contact surface, while the structure (founda- tion) surface as target surface due to its greater rigidity.

Contact elements and target elements are formed on the contact surface and the target surface, respectively.

Then, the corresponding contact elements and target elements are taken as one contact pair by defining same real constant number. Rational parameters are chosen to simulate the status of sticking, sliding, splitting or reclosing on the soil-structure interface.

(4) Damping model

In SSI system, the damping ratio of soil is usually greater than that of superstructure. So the damping ratio of soil and superstructure should adopt different value.

A material-damping ratio inputting method is given in ANSYS program. Using this method, different damping ratio can be inputted according to different material.

Using the material-damping input method can take different damping ratio of soil and superstructure into account.

(5) Consideration of gravity

The initial stress produced by gravity has great influence on the status of contact. Great error will occur if the gravity is not taken into account in the dynamic calculation. The contrast of contact pressure time-history on the center of the foundation bottom between taking gravity into account and not taking is shown in Fig.3. It’s obviously that the contact pressure of taking gravity into account is greater than that of not taking gravity into account. And separation between the soil and the foundation bottom occurs when not taking gravity into account, while no separation occurs when taking gravity into account that agrees with the test results.

In this paper, gravity is applied as a dynamic load in calculation. Before the seismic wave is applied, gravity is applied on the system as a vertical acceleration field and transient analysis is carried out. After the response drove to a constant value, the seismic wave and gravity are applied to the system together and the transient analysis continued without pause. The dynamic

response can be obtained by subtracting the constant value from the total response.

Fig.4 shows meshing of BC20 and BS10 test model satisfying the above modeling principles.

Verification of the Model

(1) The effect of the material nonlinearity of soil and the changing-status nonlinearity of soil-structure interface on computational results

In the former FEM analysis on SSI, soil was gener- ally regarded as linear material, and the changing-status nonlinearity of soil-structure interface was usually

Fig.4. Meshing of Test Model (BC20 and BS10) (a) Taking Gravity into Account

Fig.3. Contact Pressure Time-history on Center of Foundation Bottom

(BC20 Test Model, Under Excitation of EL2) (b) Not Taking Gravity into Account

0 1 2 3 4

6990 7000 7010 7020 7030 7040

Contact pressure (Pa)

Time (s)

0 1 2 3 4

-40 0 40 80 120 160

Contact pressure (Pa)

Time (s)

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ignored. In order to analyze the effect of the material nonlinearity of soil and the changing-status nonlinearity, the computational analysis is carried out under following three conditions in this paper. 1) Without any nonlinearity; 2) Just taking the soil nonlinearity into account; 3) Taking both the soil nonlinearity and the changing-status nonlinearity into account.

The acceleration comparison between just taking the soil nonlinearity into account and without any nonlinearity is shown in Fig.5. Point A7 in Fig.5 is the central point on the top of the frame (as shown in Fig.1), and point S8 is on the surface of soil, which is 0.6m away from the container boundary. It’s obviously that the acceleration response remarkably decreases when considering the soil nonlinearity. The reason for this phenomenon is that the dynamic shear modulus of soil decreases and the damping ratio increases when the soil nonlinearity is considered. This conclusion can also be drawn from the comparison between other corre- sponding points in the system.

The acceleration comparison between taking the soil nonlinearity into account and taking both the soil nonlinearity and the changing-status nonlinearity into account is shown in Fig.6. It shows that the acceleration response of the point A7 changes obviously, because the status between the box foundation and the soil, which has great effect on the response of superstructure, can be simulated factually by the contact analysis. The acceleration response of point S8 hardly changes, because the changing-status nonlinearity between the soil and the structure has little influence on the point S8, which is far away from the foundation. It shows that great error will occur when the material nonlinearity of soil and the changing-status nonlinearity of the soil-structure interface are not considered in calculation of soft soil.

(2) Comparison between calculation and test results Some acceleration time-history curves of computa- tional results and test results of the BS10 test are given in Fig.7. It shows that acceleration time-history curves of corresponding points are coincident approximately.

It’s verified that the computational model is rational and appropriate for research on SSI. Furthermore, the feasibility of the test design and the reliability of the test results are certified.

Calculation Results of BC20 Test Model

(1) Separation and sliding on the bottom of the box foundation

Contact pressure time-history of the center and end points that along X axis on the foundation bottom under excitation of EL1a is shown in Fig.8. After analyzing contact pressure time-history of different points on the foundation bottom of the BC20 test model under all kinds of seismic wave excitation, it shows that zero pressure case doesn’t occur. That means separation phenomenon between the foundation bottom and the soil doesn’t occur.

Fig.9 is the sliding time-history on the foundation

bottom under excitation of EL1a. Fig.10 is the sliding contour on the foundation bottom at certain time under excitation of EL1a. Because of symmetry, only half of the foundation bottom is given in the figure. These figures show that sliding occurs on the whole bottom of foundation.

(2) Contact pressure on the bottom of the box foundation

Fig.11 shows contact pressure contour on the bottom of the box foundation at certain time under excitation of EL1a. Fig.12 shows contact pressure amplitude along Fig.5. Comparison of Computational Analysis between

Linearity and Soil Nonlinearity (BS10 Test Model, Under Excitation of EL2)

0 2 4 6 8

-0.30 -0.15 0.00 0.15 0.30

Acceleration (g)

Time (s)

A7_nonlinearity A7_linearity

0 2 4 6 8

-0.30 -0.15 0.00 0.15 0.30

Acceleration (g)

Time (s)

S8_nonlinearity S8_linearity

Fig.6. Comparison of Computational Analysis between Soil Nonlinearity and Contact Analysis

(BS10 Test Model, Under Excitation of EL2)

0 2 4 6 8

-0.12 -0.06 0.00 0.06 0.12

Acceleration (g)

Time (s)

A7_nonlinearity A7_contact

0 2 4 6 8

-0.12 -0.06 0.00 0.06 0.12

Acceleration (g)

Time (s)

S8_nonlinearity S8_contact

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the middle line on the bottom of the box foundation. It shows that the pressure amplitude in the middle is the smallest, while the pressure amplitude of the two end sides are greater than that of the middle. And the values of the pressure amplitude on a very large range of middle portion are almost constant. The pressure amplitude is related to the input seismic wave type of excitation.

(3) Separation and sliding on the side face of the box foundation

Fig.13 shows the contact pressure time-history of the

top and bottom side of the foundation’s side face under excitation of EL1a. It shows that zero pressure case occurs. That means separation phenomena between the side face and the soil occurs. Fig.14 shows separation contour on the side face at certain time under excitation of EL1a. It shows that separation phenomena between the top of side face and the soil occur at the certain time.

After analyzing contact pressure time-history of

0.0 1.5 3.0 4.5 6.0 7.5

-0.08 -0.04 0.00 0.04 0.08

Time (s)

Acceleration (g)

A7_test result A7_calculation

0.0 1.5 3.0 4.5 6.0 7.5

-0.06 -0.03 0.00 0.03 0.06

Acceleration (g)

Time (s)

S8_test result S8_calculation

Fig.7. Comparison between Calculation and Test Result (BS10 Test Model, Under Excitation of EL1)

0 1 2 3 4

6.99 7.00 7.01 7.02 7.03

Contact pressure (KPa)

Time (s)

center

0 1 2 3 4

8 10 12 14 16

Contact pressure (KPa)

Time (s)

end point

Fig.8. Contact Pressure Time-history of Center and End Points along X Axis on Foundation Bottom (BC20 Test Model, Under Excitation of EL1a)

Fig.9. Sliding Time-history of Center and End Point along X Axis on Foundation Bottom

(BC20 Test Model, Under Excitation of EL1a)

0 1 2 3 4

-0.2 0.0 0.2 0.4 0.6 0.8

Sliding (mm)

Time (s)

center

0 1 2 3 4

-0.1 0.0 0.1 0.2 0.3 0.4

Sliding (mm)

Time (s)

end point

Fig.10. Sliding Contour on Bottom of Foundation (BC20 Test Model, Under Excitation of EL1a)

Fig.11. Contact Pressure Contour on Bottom of Foundation

(BC20 Test model, Under Excitation of EL1a)

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different points on the foundation’s side face under all kinds of seismic wave excitation, it shows that separation phenomena occurs only between the soil and the top or the bottom of the side face. While the middle portion of the side face keeps contact with soil. Fig.15 shows the sliding contour on the foundation’s side face at certain time under excitation of EL1a. It is obviously that the sliding occurs on the whole side face.

Calculation Results of BS10 Test Model

(1) Separation and sliding on the bottom of the box foundation

After analyzing contact pressure time-history of different points on the foundation bottom of the BS10 test model under all kinds of seismic wave excitation, it shows that zero pressure case occur under excitation of SH4 and SH6. That means separation phenomena between the foundation bottom and the soil occurs.

Contact pressure time-history of the end point along X axis on the foundation bottom under excitation of SH6 is shown in Fig.16. Separation contour on the foundation bottom at certain time under excitation of SH6 is shown in Fig.17. Fig.16 and Fig.17 show that separation phenomena occur.

Fig.18 is the sliding contour on the foundation bottom at certain time under excitation of SH6. It shows that sliding occurs on the whole bottom of foundation.

Fig.19 is the distribution of sliding peak value on the foundation bottom. It shows that the sliding peak value

-200 -100 0 100 200

6 8 10 12 14 16 18

EL1a SH1a

Contact pressure amplitude (kPa)

Distance to bottom center (mm)

Fig.12. Contact Pressure Amplitude along Middle Line of Foundation Bottom

(BC20 Test Model)

0 1 2 3 4

-0.7 0.0 0.7 1.4 2.1 2.8

Contact pressure (KPa)

Time (s)

top

0 1 2 3 4

-1.2 0.0 1.2 2.4 3.6 4.8

Contact pressure (KPa)

Time (s)

bottom

Fig.13. Contact Pressure Time-history of Top and Bottom on Side Face of Foundation

(BC20 Test Model, Under Excitation of EL1a)

Fig.16. Contact Pressure Time-history of End Point along X Axis on Foundation Bottom

(BS10 Test Model, Under Excitation of SH6)

0 3 6 9 12

0 30 60 90

SH6

Contact pressure (kPa)

Time (s)

Fig.14. Separation Contour on Side Face of foundation (BC20 Test Model, Under Excitation of EL1a)

Fig.15. Sliding Contour on Side Face of foundation (BC20 Test Model, Under Excitation of EL1a)

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is greater at the two-end side and is smaller in the middle of foundation bottom. The sliding peak value is related to the peak value and the wave type of the seismic excitation. When the excitation wave keeps the same type, the sliding peak value increases along with the increase of the peak value of the seismic excitation. When the peak values of the excitation keep the same value, the sliding peak values change according to the change of the excitation wave type. The case under the excitation of Shanghai wave is greater than that of El Centro wave.

(2) Contact pressure on the bottom of the box founda- tion

Fig.20 shows contact pressure amplitude along the middle line on the bottom of the box foundation. It shows that the pressure amplitude in the middle is the smallest, while the pressure amplitude of the two end sides are greater than that of the middle. And the values of the pressure amplitude on a very large range of middle portion are almost constant. The pressure amplitude is related to the input wave type of excitation. And the pressure

amplitude under excitation of Shanghai wave is greater than that of El Centro wave.

(3) Separation and sliding on the side face of the box foundation

Fig.21 shows the separation contour between the side face of the box foundation and the soil at certain time under excitation of SH6. It shows that separation occurs between the side face of foundation and the soil except part of region above the line marked with ‘H’. Fig.22 shows the sliding contour between the side face of the box Fig.19. Sliding Amplitude along Middle Line of Foundation

Bottom (BS10 Test Model)

-400 -200 0 200 400

0 1 2 3 4 5

EL2 EL4 EL6

Sliding amplitude (mm)

Distance to bottom center (mm)

Fig.17. Separation Contour on Bottom of Foundation (BS10 Test Model, Under Excitation of SH6)

Fig.18. Sliding Contour on Bottom of Foundation (BS10 Test Model, Under Excitation of SH6)

-400 -200 0 200 400

0 20 40 60

EL2 EL4 EL6

Contact pressure amplitude (mm)

Distance to bottom center (mm)

Fig.20. Contact Pressure Amplitude along Middle Line of Foundation Bottom (BS10 Test Model)

Fig.21. Separation Contour on Side Face of Foundation (BS10 Test Model, Under Excitation of SH6)

Fig.22. Sliding Contour on Side Face of Foundation (BS10 Test Model, Under Excitation of SH6)

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foundation and the soil at certain time under excitation of SH6. It shows that sliding between the foundation and the soil occurs on the whole side face of the box founda- tion.

(4) Acceleration amplification factor

The acceleration amplification factors of center of the soil-structure system are shown in Fig.23. The codes of EL2, EL4, EL6 in the Fig.23 denote the excitation of El Centro wave, whose peak values of acceleration are 0.266g, 0.532g and 0.798g, respec- tively. The code EL2 denotes the unidirectional excitation of El Centro wave, while the code ELZ2 denotes the bi-directional excitation of El Centro wave.

The code KB2 denotes the excitation of Kobe wave, whose peak value of acceleration is the same as EL2’s.

The acceleration amplification factor of the medium sand layer has little change about 1.0, which demon- strates that vibration is transmitted effectively by the bottom medium sand. The intermediate layer of sandy silt filters and isolates vibration. The responses of the superstructure at different floors are different in that there are rocking and swing at the foundation. With increasing of the input acceleration, the amplification factors of the acceleration peak value are reduced due to the nonlinearity of soil. Vertical excitations have little effect on the responses of the dynamic soil-structure interaction system.

Conclusions

(1) In this paper, combing general-purpose finite element program ANSYS, research on modeling of SSI system has been carried out. By comparison analysis between the calculation and the tests, it’s verified that the modeling methods are rational and the model is feasible.

(2) Issues drawn from the calculation, which are consistent with those drawn from the tests, are as follows. 1) Soft soil can filter and isolate vibration. 2)

With increasing of the input acceleration, the amplification factors of the acceleration peak value are reduced due to the nonlinearity of soil. 3) The response of the system under the excitation of Shanghai artificial wave is obviously greater than that under the excitation of El Centro wave and Kobe wave. 4) Vertical excitations have little effect on the responses of the dynamic soil-structure interaction.

(3) Issues drawn from the calculation, which can’t be found out from the tests, are as follows. In calculation of BC20 test, the separation phenomenon occurs in the side face of foundation while it doesn’t occur in the bottom of foundation. And the sliding phenomenon occurs on both the side face of foundation and the bottom of foundation. In calculation of BS10 test, the separation and sliding phenomena occur on both the side face of foundation and the bottom of foundation.

The reason may lie in the different kind of the superstructure. And the contact pressure of the foundation bottom in both BC20 test and BS10 test are in the same distribution.

Acknowledgments

This project is carried out under the sponsorship of the key project (No.59823002 and No.50025821) of National Natural Science Foundation of China.

References

1) Hadjian A, et al. (1991) The Learning from the Large Scale Lotung Soil-Structure Interaction Experiments. Proc. 2nd Int. Conf. on Recent Advances in Geotech. Earthquake Eng. and Soil Dyn., St.

Louis, Vo1.1

2) Chen Y Q, Huang W, Lu X L, et al. (1999) Design of Shaking Table Model Test on Dynamic Soil-structure Interaction System. Structural Engineers, Supplement: 243-248

3) Lu X L, Chen Y Q, Chen B, Li P Z, et al. (2002) Shaking Table Model Test on Dynamic Soil-Structure Interaction System. Journal of Asian Architecture and Building Engineering, 1(1), 55-63

4) Chen B, Lu X L, Li P Z, et al. (2002) Modeling of Dynamic Soil-structure Interaction by ANSYS Program. 2ND Canadian Special Conference on Computer Applications in Geotechnique.

Canada, 21-26

5) Wang S T, Cao Z et al. (1997) Design Methods of Modern Aseismic Structures. China Architecture & Building Press, Beijing, China, 147-148

6) Soil Dynamic Laboratory, (2000) Test on Characteristics of Remolding soil. Research Report of Department of Geotechnical Engineering, Tongji University, Shanghai, China

0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6

0.0 0.5 1.0 1.5

Medium sand Sandy silt

Clay Ground surface

Amplitude factor

Height (m)

ELZ6 EL6 ELZ4 EL4 ELZ2 EL2

0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6

0.0 0.5 1.0 1.5

Medium sand Sandy silt

Clay Ground surface

Amplification factor

Height (m)

KBZ6 KB6 KBZ4 KB4 KBZ2 KB2

Fig.23. Distribution of the Amplification Factors of the Acceleration Amplitude

(BS10 Test Model, Contrast between Unidirectional and Bi-directional excitations)

References

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