EARTHQUAKE ENGINEERING A N D STRUCTURAL DYNAMICS, VOL. 9, 263 277 (1981)

SEPARATION AND SLIDING BETWEEN SOIL AND STRUCTURE DURING STRONG GROUND MOTION

KENZO TOKI,* TADANOBU SATOt A N D FUSANORI MIURAt

Disaster Prevention Research Institute, Kyoto University, Uji, Kyoto, Japan

SUMMARY

This paper presents an approach to the problem of separation and sliding between soil and structure in the finite element analysis of dynamic soil-structure interaction problems. Joint elements are arranged along the contact surface between soil and structure and they have a property such that tensile forces are not transmitted between the planes representing structure and soil in the finite element analysis. The dynamic properties governing the sliding are determined by the Mohr-Coulomb failure law determined from the cohesion and the friction angle between soil and structure.

The proposed method is applied to (i) a model of a reactor building resting on the free surface of layered ground and (ii) a buried foundation structure. The numerical computations reveal the following results: that the translation is dominant in the motion of the structure when sliding is taking place between soil and structure, and that the rocking is dominant in the rest of the response. The amplitude of the response during sliding is increased on any one point of the structure and decreased on any one point ofthe ground compared with that of the fixed condition at the interface. In the case of the buried structure, it is observed in the computed results that the structure and soil move in the opposite direction along the vertical contact surface and are separated from each other in the near surface region during the strong phase of the excitation.

1. INTRODUCTION

Dynamic interaction between soil and structure is defined as the phenomenon of the transmission of kinematic energy through the interface of the media. The characteristics of the soil-structure interaction are affected not only by the mechanical properties of the constituents but also by the geometrical form and condition of the interface. Therefore there is no universal method to analyse the dynamic interaction problem, while various methods such as the elastic wave propagation method, the finite element method and the springdashpot replacement of the interface effect are generally used. Moreover most studies concerned with soil-structure interaction assume a perfect bond on the contact surface between soil and structure, and the static normal stress due to the surcharge along the contact surface is not generally considered. These assumptions inevitably induce a tensile stress on the contact surface. In the actual system, the separation phenomenon may occur at the interface where the tensile stress is generated and the seismic force is transmitted through the decreased contact area resulting in an increase in the contact stress. As the stress-strain relationship of soil depends on the intensity ofconfining pressure, strong non-linear behaviour will appear in the case where the contact area on the interface varies. Moreover, sliding phenomenon may occur during strong earthquake motion, and then the response of the soil-structure system associated with sliding and separation will be greatly different from

From the above point of view, some analytical research work has been done on the separation and sliding phenomena between soil and structure in the analysis of seismic response. A number of works dealing with the partial lifting-off of the base mat of a nuclear reactor building have been published.' - 5

In Reference 1 a bilinear rocking-spring characteristic was determined for static loads using the 3- dimensional finite element representation of the elastic half-space. In References 2 and 3 the same concept as

. the response with a perfect bond assumed at the interface.

* Professor. t Associate Professor. $ Research Associate.

@ 1981 by John Wiley & Sons, Ltd. 0098-8 847/8 1 /030263 1 5 $0 1.50 Received 21 January 1980

Revised 12 September 1980 263

264 K TOKI, T. SAT0 AND E MIURA

that of Reference 1 was used, but for approximately determining the rocking-spring characteristics using the linear elastic theory of the half-space. In these works, the coupling of the vertical and rocking motion and also sliding phenomena were neglected.

Wolf4 showed a rigorous procedure to determine the non-linear impedance function of a rigid plate which is partially in contact with the elastic half-space as well as taking partial sliding into consideration. Furthermore, Wolf and Skrikerud’ examined the effect of the base mat lifting-off on the response of a nuclear reactor building by using several approximate methods. However, Wolfs method is very complicated and is difficult to apply to the response analysis of an embedded structure in the sub-soil.

Tsuboi et aL6 examined the possibility of overturning of a tall building by using the finite element method. They took into account the separation at the interface between soil and structure but did not deal with the sliding. Sasaki e t al.’ analysed the stability of sliding between two rigid masses subjected to Coulomb friction for the case where the normal and tangential components of excitation on the contact surface are simultaneously applied. In that analysis only the rigid surface slides, therefore it is difficult to apply their method to a system providing material flexibility on both sides of the interface.

On the other hand, the behaviour ofjoints and seams in rock masses has been examined experimentally and is recognized to have a great influence on the stability of rock slopes and underground excavations. Therefore. great attention has been paid to mathematical models of intact The joint element proposed by Goodman13 has provided an especially powerful tool for the finite element analysis of structures in rock.

The purpose of this study is to present an effective analysis procedure for the soil-structure system including the possibility of separation and sliding at the interface between soil and structure by use of the joint element. The method is applied to a model of a nuclear reactor building resting on the ground surface and to a buried foundation structure, and the stability of such structures with respect to sliding and overturning is discussed in detail.

2. DISCRETIZATION OF SOIL-STRUCTURE SYSTEM

Since stress and strain in the soil will attain high values in the neighbourhood of the structure during the seismic excitation, the possibility of the separation and sliding between the soil and structure is most likely along the interface. In the dynamic analysis of the soil-structure system, joint elements are set along the interface. The analytical procedures and problems in employing the joint element in the dynamic response analysis are discussed below.

2.1. Stiffness matrix of the joint element Figure 1 shows the joint element constructed from four nodal points I, J, K and L. Two pairs of nodal points

I-L and J-K are assumed to occupy the same co-ordinate at the initial state, that is, the surfaces I-J and K-L are in contact. The behaviour of the contact surface between soil and structure is expressed by the relative position of both surfaces of the joint element. The relative motion of both surfaces is classified as follows:

(i) parallel motion to the joint wall (sliding between soil and structure) (ii) perpendicular motion to the joint wall (separation and contact)

(iii) rotational motion around the centre of the joint element.

L

Figure I . Configuration of the joint element

SEPARATION AND SLIDING BETWEEN SOIL AND STRUCTURE 265

Ckl:. = 4

The deformation characteristics of the joint element are determined by the stiffness of the shear direction k,, the normal direction k, and the rotational component k,. Since k , is expressed in terms of k,, the constitutive relation of the joint element is expressed by two parameters k, and k,, as shown in Figure 2.

- - k, 0 k, 0 -ks 0 -k , 0

2k, 0 0 0 0 0 - 2 k , k, 0 -ks 0 - k , 0

2k, 0 -2k , 0 0 k, 0 k, 0

Symmetric 2k, 0 0 k, 0

- 2k” ~

-T Strain

Z: =C-O;, tan+ 61: Yield Stress C : Cohesion 9 : A@e of internal friction

Figure 2. Relationship between stress and strain at midpoint of the joint element

The stiffness matrix of the joint element [k]:, is given by

The stiffness matrix [k]:, is defined in the local co-ordinates (s, n), and should be transformed to the generalco- ordinate system in actual use.

2.2. Application of the joint element to dynamic problems Considering the general dynamic response analysis of a linear system, it is sufficient to treat only the

deformation from the static equilibrium condition. However, the normal stress in the initial state has an important role in controlling the separation and sliding in the response analysis using the joint element. The initial stress induced by gravity forces should be taken into consideration as mentioned below.

(i) The structure resting on the ground surface: The intensity of the uniformly distributed strip load q is calculated from the weight of the structure system. The ground is assumed as a homogeneous elastic half-space

266 K. TOKI, T. SAT0 AND F. MIURA

and the stress generated in the ground is calculated by Boussinesq’s equation as shown in Figure 3,

oy = - - (2e + sin 2~ . cos 2I4) - yk 4 7T

ox = - - ( 2 ~ - s i n 2 ~ . ~ 0 ~ 2 ~ ) - K , ~ ~ h 4 n

txy = - 4 sin 2 8 . sin 211/ n

where 2~ = /I2 - B l , 214 = /I2 + /I1, (/It and /j2 being shown in Figure 3), y is the unit weight of the soil mass, K , the coefficient of earth pressure a t rest and h is the depth from the surface.

I I I I I I I I I

Figure 3. Stress distribution in half-space

(ii) The structure embedded in the ground: The surficial ground is subdivided into two layers as shown in Figure 5(a). Equation 2 is used to calculate the initial stress for the second layer and the stress in the first layer is obtained from the earth pressure at rest as follows:

- \

I 2 i I 0 = -vk

ox = - K , y h

sxy = 0

While in section 2.1, the stiffness matrix of the joint element is deduced by the use of linear constitutive relations, actual joint behaviour obviously cannot be expressed by such a simple model. The shear strength of the joint is a function of normal stress and also shows work-hardening or softening phenomena. Moreover, the tensile strength is expected at the first stage of shearing deformation. Considering the modelling complexity of these phenomena and accompanying numerical diff~culties in performing the dynamic response analysis, the stress-strain characteristics of the joint element are assumed as shown in Figure 2 for the normal and shear component. The normal stress is not transmitted for E~ 2 0 (separation) and a linear constitutive relation based on the stiffness k , is valid for to. Spring coefficients k,, k, are related to the fictitious relative displacement of the interface between soil and structure, which in actual response would not occur. Therefore the spring coefficients should be as large as possible. However, numerical instability determines that their magnitude be nearly equal to the stiffness of the elements of the foundation structure. Sliding will take place when the absolute value of shear stress reaches the yield shear stress T,,, and the linear shear stress-strain relation remains below this stress level. When separation occurs, the shear stress is not transmitted through the joint wall. The yield shear stress, T,,, is determined as a function of the normal stress assuming the Mohr-Coulomb failure law.

14! zY = C -0, tan 4, for on SO (compression) T y = 0, for o, > 0 (extension) I

where C is the cohesion and 4 is the friction angle.

2.3. Numerical method for iterative solution The numerical calculation, including the separation and sliding phenomena, shows strong non-linearity,

and thus the equation of motion must be solved by the step-by-step integration method in the time domain.

SEPARATION AND SLIDING BETWEEN SOIL AND STRUCTURE 267

The required time interval of each step is on the order of 0.001s. Therefore, an excessive amount of computation time is necessary to calculate the seismic response for 10 s. The load transfer method of iterative computation is found to be most effective in this analysis, in which the stiffness matrix is kept constant during the computation and only the external force is modified so as to satisfy the equilibrium condition.

If we use the above method, systems with non-linear constitutive relations can be treated as linear systems with constant stiffness excited by quasi-external forces. The computation of the inverse stiffness matrix is required only at the first step of the response analysis and therefore the computation time is greatly reduced.

3. SEISMIC RESPONSE ANALYSIS

3.1. Models and excitations Figure 4(a) shows Model 1, which is a structure resting on the ground surface. It is a model of a typical

reactor building of a nuclear power station in Japan. The finite element mesh of Model 1 is shown in Figure 4(b). The number of elements, the number of nodal points and the degrees of freedom are also shown in the figure. Three joint elements are arranged along the interface between the structure and ground. Model 2, shown in Figure 5(a), is a pier foundation of a long span bridge in which eight joint elements are arranged along

Rigid Base WIT= rn

Figure qa). General view of soilstructure system: Model 1

Number o f nodal point 150 Number o f element 120 ( Joint element 3 ) Degree o f freedom 264 3 joint elements

Figure qb). Finite element mesh of Model 1

Figure 5(a). General view of soil-structure system: Model 2

268 K TOKJ, T. SAT0 AND F. MIURA

the interface. The finite element mesh of Model 2 is shown in Figure 5(b). The model parameters are listed in Tables I and I1 corresponding to Models 1 and 2, respectively. A value of 0.2 is used as the damping factor of the ground in order to take account of the radiation to the base ground. The Mohr-Coulomb failure law is adopted in the constitutive relation ofjoint elements. The values of the governing quantities are listed at the bottom of the tables. Non-linear behaviour is considered only in joint elements; the structure and the ground are assumed to behave as linear elastic materials.

These two models are subjected to simultaneous excitation of NS and UD components of the El Centro (1940) accelerograms shown in Figure 6, as well as sinusoidal excitations. The control point of excitation is the bottom of the finite element model, which is the rigid base beneath the surface layer. In the following sections, the acceleration and displacement time traces obtained from the non-linear analyses which consider separation and sliding are compared with linear analysis results in which the joint elements are removed. In addition, the stability of a structure against sliding and separation has been investigated and compared

Number of nodal point 138 Number o f element 112 R ( Joint element 8 ) Degree of freedom 246 J 8 joint elements

Figure 5(b). Finite element mesh of Model 2

Table I. Physical properties of model 1 ~ ~~~ ~

Shear wave Unit weight velocity Poisson's Damping

(t/m3) (m/s) ratio factor

Ground = 1.8 c , = 5 0 0 V I = 0 4 h = 0.2 Structure yz = 0.75 C, = 1720 v2 = 0.167

y3 = 1-7 c, = 1600 v j = 0.167 h = 0.05 y4 = 2.4 c, = 1600 v4 = 0.167

Joint Shear spring const. k, = 5 18200 t/m2 Normal spring const. k , = 518200 t/m2 Cohesion C = 10 t/m2 Friction angle 4 = 0" or 30"

Table 11. Physical properties of model 2

Shear wave Unit weight velocity Poisson's Damping

(W') (m/s) ratio factor

v 1 = 0.4 y 2 = 1.8 C, = 360 1 1 ~ = 0.4 / I = 0.2 Soil ;.'I = 1.8 C , = 180

Structure ;:3 = 2 4 C , = 1600 v3 = 0.167 / I = 0.05 Joint Shear spring const. k , = 200000 t/m2

Normal spring const. k, = 200000 t/m2 Cohesion C = 10 t/m' or 20 t/mZ Friction angle q5 = 0" or 30"

SEPARATION AND SLIDING BETWEEN SOIL AND STRUCTURE 269

' 0.0 2 5 5.0 75

UD

Time I sec)

Figure 6. Accelerograms of El Centro (1940) NS and UD components

with the results estimated by a provision for aseismic design of foundation structures. The time interval of computation is 0002s. It is ascertained that the numerical solution converges when the time interval is less than 0.003 s.

3.2. The response to sinusoidal excitation (a) Response analysis of model 1: Figure 7 shows the amount of sliding at the contact surface and

displacement at the centre of gravity (C.G.) for the case of 4 = 0 degrees and a horizontal sinusoidal excitation of200 gal. The displacement at the C.G. of the linear system without the joint element is denoted by the smooth

D i . at C. G.

n

0 3 4 5 6 7

':I A .-

n Sliding

6 , - 0 3 4 5 6 7

Frequency ( Hz 1 Figure 7. Frequency dependence of displacement at the centre of gravity and sliding at the interface to horizontal sinusoidal excitation

(4 = 0", excitation amplitude: 200gal)

curve. The symbols 0 and 0 correspond to the sliding and the displacement at the C.G., respectively. Both the magnitudes of sliding and displacement decrease with increasing frequency and the difference between them is kept almost constant for variation of frequency. This result implies that the sliding will occur when the structure is inclined at a specified angle, whereas the rocking motion is predominant for the linear system. Sliding disappears beyond a specified frequency for the given excitation level. To clarify the effect of excitation level on sliding, the relationship between the magnitude of sliding and acceleration amplitude is examined and shown in Figure 8. The excitation is sinusoidal with a frequency of 5Hz. It is obvious that the relative displacement at the C.G. is almost proportional to the excitation level. Sliding does not occur up to the excitation amplitude of 150 gal and the magnitude of sliding is proportional to the increment of excitation amplitude from the critical value. This figure implies that sliding will be induced when the excitation force exceeds a certain threshold level.

Table I11 lists the maximum response of acceleration and displacement at the C.G., the occurrence of sliding or lifting off and the maximum and minimum compressive stresses generated at point B in Figure 4(a) for

270 K. TOKJ, T. SAT0 AND F. MIURA

Excitation amplitude fgal) Figure 8. Displacement at the centre of gravity and sliding at the interface due to sinusoidal excitation (frequency: 5 Hz!

Table 111. Response to sinusoidal excitation (Model 1, 4 = 30”, frequency: 5 Hz)

Excitation amplitude (gal)

Maximum acceleration at the centre of gravity (gal)

Maximum displacement at the centre of gravity (mm)

Sliding at the contact surface

Lifting off Pressure Maximum

at point B Minimum (ton/m2)

300 350

358 434

4.6 5.4

No occurrence partially

No occurrence No occurrence occurred

99.1 23.2

105.6 16.9

400

61 1

6.1

partially occurred

No occurrence 11 1.8 10.9

various excitation levels with fixed frequency of 5 Hz, in the case of # = 30 degrees. Sliding of the whole structure with respect to the ground has not taken place, and local sliding has appeared only because the shear stress generated at the contact surface reaches the local yield shear stress z y which depends strongly on the position.

Lifting off and stability of overturning are also examined. Lifting off begins when the normal stress at the structure base vanishes, but does not appear, in this case, even for an input excitation of 400 gal, because the normal stress is still compressive. In addition, the same investigation carried out for an excitation level of 300 gal and a frequency of 3 Hz results in no occurrence of lifting off or sliding.

(b) Response analysis of model 2 Figure 9 compares the displacement at the top of the structure (point A in Figure 5(a)) for the case of 4 = 30 degrees and an excitation level of 300 gal. The response of the non-linear system is greater than that of the linear system for the frequency range below 6 Hz because sliding takes place between the side wall of the structure and the soil. The separation between the side wall and the soil near the

0-

!L( Linear

2 L 6 8 1 0 Frequency ( H z )

Figure 9. Displacement of structure at point A in Figure S(a) due to horizontal sinusoidal excitation (4 = 30’, cxcitation amplitude: 300 gal)

SEPARATION AND SLIDING BETWEEN SOIL AND STRUCTURE 27 1

ground surface occurs at 2 Hz, and then the displacement becomes greater than that of the linear system by 30 per cent. In the linear system, the motion of the structure is so strongly restrained by the surrounding soil that the response becomes smaller than that of the non-linear system containing joint elements, especially in the low frequency range.

The results of the non-linear response analysis are listed in Table IV. The frequency of excitation is 2 Hz and the amplitude has been varied. Numbers appearing in parentheses are the maximum acceleration or the displacement of the linear system at point A and their percentage with respect to the non-linear case. As the

Table IV. Response to sinusoidal excitation (Model 2, 4 = 30", frequency: 2 Hz)

Excitation 200 300 400

Maximum point A (gal) acceleration at 3 7 ~ ( 3 ' ~ ) 84.3 5 7 1 5 ( 4 ~ ~ ) 83.0 793(637) 80.4

amplitude (gal)

Maximum displacement at point A (cm)

Sliding at point B (mm) 1.2 2.1 5 4 Separation at point B (mm) 0 0.15 1.39

input level becomes larger, the results of the linear analysis underestimate the response of the system. The percentage of the linear case to the non-linear case is not much affected by the input level or by the amount of separation at point B. Therefore the response of the structure becomes strongly non-linear when sliding takes place between the structure wall and soil.

3.3. The response to seismic excitation (a) Seismic response analysis of model 1: Figure 10 shows the horizontal displacement at point A (Figure

4(a), where the sides of the structure and the soil share the same point before shaking). The solid line indicates the displacement of the side of the structure, and the broken line that of the soil. The relative displacement between soil and structure becomes large from 1-7 to 2.5 s, and after this time the structure vibrates with a

n - Structure

V I

0.0 2.5 5.0 7.5 94 Ti me [ sec)

Figure 10. Horizontal displacement at point A in Figure 4(a) due to El Centro accelerograms

constant residual deformation. In the figure, the displacement of the linear system is also shown by a dash-dot line. While the displacement of the structure becomes large when sliding occurs, the displacement of the soil is almost the same as that of the linear system. This figure shows that the values adopted as the spring constants of the joint element are sufficiently accurate because the non-linear and linear analyses are almost coincident before the occurrence of sliding.

Figure 11 illustrates the hysteresis loop for shear distortion at point B. The point marked by S denotes the starting time and E the ending time of the response analysis. At the contact surface, the stress-strain relation is linear until 1.71 s and sliding in the positive direction continues to 1.94 s. Sliding in the opposite direction takes

272 K. TOKI, T. SAT0 AND F. MIURA

place at 2.04 s and after this time the sliding gradually increases towards the negative direction. The residual value of the relative displacement is 11 mm.

The relationship between the input accelerogram and the times of sliding is shown in Figure 12. Sliding takes place at the times denoted by the symbol 1. It is interesting to note that the horizontal component of the accelerogram has peak values higher than 200 gal just before these marks designating the occurrence of sliding. Sliding occurs frequently in the time interval 1.7-2.3 s, in which the low frequency component predominates in the accelerogram. However the higher frequency components, such as 5-6 Hz, predominate after this period

-1.5' I -15 -10 -5 0 5 10 15

Relative displacement (mm)

Figure 11. Hysteresis loop for shear direction at point B in Figure 5(a)

00 25 50 7 5 Tlme ( sec 1

Figure 12. Occurrence time of sliding

and result in very little sliding movement. These results correspond with those of the sinusoidal excitation mentioned previously.

The calculated result for the case of 4 = 30 degrees and C = 10 ton/m2 is not shown in any figures or tables because sliding and lifting off did not occur in the response analysis. The minimum normal stress of 38.6 ton/m2 at the structure base (point B) was generated at a time 2.2 s, and the shear stress was 19.0 ton/m2. Therefore, if the internal friction angle is less than 13.1 degrees, local sliding may take place merely in the vicinity of point €3.

(b) Seismic response analysis of model 2 The hysteresis loop at point B of model 2 is shown in Figure 13 for the case of 4 = 0 degrees. Figure 13(a) shows the shear direction and Figure 13(b) the normal direction. Separation occurs at 2.19 s and the shear stress becomes zero suddenly, as shown by the vertical line in Figure 13(a). Henceforth the separation continues up to 2.37 s, under zero shear stress. During this period, the relative displacement between soil and structure in the vertical direction becomes large. The separation at point C (opposite side to point B) occurs between 1.86 and 2.09 s and between 2.70 and 2.79 s. The interface between the structure wall and soil is separated repeatedly by the rocking motion of the structure during the period 1.862.70 s. Figure 14 is the hysteresis loop at point C of model 2 for the case of 4 = 30 degrees. Separation at

SEPARATION AND SLIDING BETWEEN SOIL AND STRUCTURE 273

-1.5 -1.75 -15.0 -125 -100 -7.5 -5.0

Relative displacement (mm) Figure lYa). Hysteresis loop for shear direction at point B in Figure 5(a) (4 = 0")

'1

0- I

"i < -1 - P -

2.31

ModJ2 (9.0")

-5.1 I I 1

1.25 2.5 -25 -1.25 0 Relative displacement (mm 1

Figure lyb). Hysteresis for normal direction at point B in Figure 5(a) (4 = Oo)

this point takes place between 1-88 and 2.02s and sliding is remarkable in this period. The slope of the hysteresis curve for the shear direction varies from time to time within the period 2.14-2.24 s, because the fluctuation of normal stress with time results in the change of yield shear stress of the contact surface.

Separation and sliding between the base of the structure and the soil did not occur for the cases of 4 = 0 degrees or 4 = 30 degrees. The stability factor of safety with respect to sliding of the horizontal interface will be considered in subsection 3.4.

The horizontal displacement at the top of the structure of model 2 (point A in Figure 5(a)) is shown in Figure 15. The solid line is for the linear case, the dashdot line for the case of 4 = 0 degrees and the broken line for the case of 4 = 30 degrees. The displacement for the case of 4 = 0 degrees is greater than that of the linear case by 15-30 per cent at every peak, and the same results are obtained for the response acceleration. This means that the analysis assuming perfect bond at the interface between structure and soil does not correctly estimate the magnitude of structural response due to the overrestriction of structural movement by the sub-soil. This phenomena is especially notable in long period excitation. Consequently, in order to preclude the possibility of the separation and sliding of soilstructure systems in practical design, special consideration should be paid to seismic waves containing long-period components.

214

5-

L-

- % < 3-

3 2 - E!

b l -

0, Y - c v)

a,

5 2 O-

-1 1

K TOKI, T. SAT0 A N D F. MIURA

2.2L

198

1.88 I I 1

1 -

0 -

F -'- 0

Y 2 - 2 -

- 3

-3-

!.! c

- 4 - - 0

-5 -

8

C 9

- L -3 -2 -1 0 1

Relative displacement (mml Figure 14(b). Hysteresis for normal direction at point C in Figure 5(a) (4 = 30 )

' 0 0 2 5 50 75

Time (secl Figure 15. Horizontal displacement at the top of structure (point A in Figure 5(a))

3.4. Safety factor for sliding Figure 16 shows the safety factor for sliding of the whole structure (this factor will be referred to as total

safety factor) of model 1 for the case of 4 = 30 degrees and the sinusoidal excitation amplitude of 245 gal, corresponding to a seismic coefficient of 0.25. The safety factor for sliding is calculated as the ratio of the yield shear stress z, to the mobilized shear stress 5. The local safety factor is defined for each joint element by the

SEPARATION AND SLIDING BETWEEN SOIL AND STRUCTURE 275

I $ 3 1 1 ,

0 1 2 3 4 5 6

Frequency (Hzl Figure 16. Safety factor for sliding of entire structure of model 1 (4 = 30°, excitation amplitude: 245gal)

minimum value of this ratio through the analysed time. The safety factor of the whole structure at any instant, t = ti, is defined as the highest value of z y / t (designated as t y / z IF:;) among all the joint elements. Then the total safety factor is given as the lowest value ofzy/t l,m=8; irrespective of time. Figure 16 indicates that the total safety factor does not depend on the frequency. The broken line in the figure is obtained using the following equation obtained from typical technical guideline^'^ for aseismic design of foundation structures.

PV s, = H

where p is the coefficient of friction, V is the vertical force and H is the horizontal force. Taking I/ as the weight of structure and H = k , r/: S , becomes 2.32 for p = tan 30" and k , = 0.25. This value of S , is close to the total safety factor, and thus it may be concluded that the possibility of sliding of massive structures resting on the ground surface, such as model 1, can be ascertained by the seismic coefficient method.

Figure 17 illustrates the safety factor of sliding between the base of the structure and the soil. The excitation is a sinusoidal wave with an amplitude of 245 gal. Since the total safety factor is greater than ten, sliding of the

1 0.5 1.0 1.5 2.0

Frequency ( Hzl Figure 17. Safety factor for sliding at the contact surface between soil and structure of model 2 (#I = 30", excitation amplitude: 245 gal)

whole structure does not occur, whereas the possibility of local sliding is somewhat higher. This is due to the decrease of normal stress at the contact surface, due to the rocking motion of the structure. The broken line is the static safety factor based on the guidelines for the aseismic design of a caisson foundation with 70m width. This value is greater than the total safety factor. One reason for this is that the present guideline deals with the three-dimensional structure, considering the shear restriction of the side wall in the direction of motion. In the finite element analysis, only the two-dimensional structure is treated and the side resistance parallel to the direction of motion is not taken into account.

The safety factor defined between the structural base and the soil, which was determined from the seismic response analysis, is illustrated in Figure 18. The excitation is the El Centro accelerograms (1940, NS and UD comp.) scaled to 0.5,2 and 3 times the original amplitude. The total safety factor is greater than 1.0 even for 3 x excitation, while on the contrary the local safety factor becomes smaller than 1.0 for this case, thus indicating that sliding takes place. Local sliding is caused by the lifting off of the base of the structure in the vicinity of the

276 K. TOKI, T. SAT0 AND E MIURA

edge, due to the rocking motion. Overturning of the structure will not occur for an embedded structure such as model 2 in spite of local lifting off between the base of the structure and the ground.

2 0-

C = 20 ton/&

'#' = 30' 15

c u

Total

x\x

0 1 2 3 Magnification of El Centm acceleration

Figure 18. Safety factor for sliding at the contact surhce between soil and structure base of model 2 excited by El Centro accelerogramx

4. CONCLUSION

A general method for the seismic response analysis of soil-structure systems considering separation and sliding phenomena at the interface between soil and structure is developed in this study. The analytical procedure is based on the application of the joint element developed in static rock mechanics to the dynamic soil-structure interaction problem. The seismic response of two soil-structure models are dealt with to illustrate the applicability of the method to the actual soil-structure system and to clarify the effect of sliding and separation phenomena on the dynamic stability of soil-structure systems. From the analyses presented. the following can be concluded:

1. Dynamic separation and sliding at the interface between soil and structure become susceptible to analysis by use of the joint element in the finite element method.

2. In the analysis assuming perfect bond on the contact surface between soil and structure, the structure's motion is restricted by the surrounding sub-soil, thus possibly underestimating the actual response of the structure. However, as the response amplitude is dependent on the natural frequency of the system and on the predominant frequency of the input motion, more examination of various physical models is necessary in order to discuss the general tendency of the response amplitude when the result of non-linear analysis is compared with that of linear analysis.

3. There is a specified frequency above which separation and sliding does not take place, which depends on the excitation level. Separation and sliding become larger as the frequency becomes smaller than this specified frequency. Therefore, careful consideration should be paid, in the discussion of sliding and separation, to seismic excitations containing long-period components. 4. Sliding along the side wall of the buried structure occurs at an excitation level lower than that for

separation. The decrease of confining force associated with sliding phenomena strongly affects the structural response.

5. Safety for sliding may be examined by either the proposed analytical method or the guidelines for aseismic design of foundation structures. Both estimates of the safety factor for sliding of the whole structure show a good agreement. On the other hand, the safety for local sliding cannot be checked by the method provided in the guideline. Therefore, application of the proposed method permits analysis of the progress of sliding of the structural system.

6. Lifting off or overturning of the structure was also examined, and a high safety factor was obtained for the two examples. Although not discussed in detail, it should be noted that these phenomena will be important parameters in the aseismic design of other types of structure such as tall buildings and towers.

SEPARATION AND SLIDING BETWEEN SOIL AND STRUCTURE 277

Although the analytical examples presented herein have shown realistic behaviour for soil-structure systems including separation and sliding phenomena at the interface between soil and structure, there are many problems in performing the more rigorous response analysis. For instance, research is needed on the yielding criteria and constitutive relation of the joint, on the effect of non-linearity of soil behaviour and on the extension to three-dimensional analysis. These problems are under investigation by the authors, and the results will appear in the near future.

ACKNOWLEDGEMENTS

The authors wish to express their sincere thanks to Prof. Yuzo Ohnishi of Kyoto University and Mr. Hideaki Kishimoto of Japan Computer Consultants, Osaka, for their suggestions regarding programming of the dynamic joint element.

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