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A modified lumped parametric model for nonlinear soil-structure interaction analysis
Soil Dynamics and Earthquake Engineering 12 (1993) 273-282
A modified lumped parametric model for nonlinear soil-structure interaction analysis Xiong Jianguo, Wang Danmin & Fu Tieming Institute of Engineering Mechanics, State Seismological Bureau, Harbin, 150080, People's Republic of China (Received 19 November 1992; revised version received and accepted 30 June 1993) Most soil-structure interaction (SSI) analyses are still conducted assuming linear material behavior or simulating nonlinear effects through an equivalent linearization and the structure (foundation) being closely welded with the surrounding soil. It is recognized, however, that nonlinearities can play a significant role in the results. Two kinds of nonlinearities must be considered: those associated with inelastic soil behavior and those resulting from loss of contact between the foundation and the surrounding soil. In the present paper a modified lumped parametric model for the analysis of nonlinear SSI effects has been proposed. In the model both nonlinearities are taken into account. The results of tests of the soil-structure system model have been presented, which agree well with those obtained from analysis by using the proposed model.
vibration in the soil-structure system are neglected. With the substructure or the hybrid model for the true nonlinear analysis, on the other hand, only the near field (the structure and the near portion of the soil) is treated as though nonlinear, while the far field is assumed to be linear. The two more sophisticated models mentioned above for the analysis of the true nonlinearities associated with inelastic soil behavior are also available for the analysis of geometric nonlinearities of SSI by introducing some special elements at the interface between the foundation and the surrounding soil. 5-1° The superiority of the sophisticated models is obvious they can give a more realistic modelling of the complete structure-foundation-soil system and provide extensive information about the performance of the system from the analysis. It is important and should be used for cases involving large/heavy critical systems. However, while the trend has been toward more sophisticated modelling and analysis procedures, it is significant to note that very simple procedures can often be very effective in assessing SSI effects, 1~ in particular, in obtaining preliminary estimates of such effects for design purposes. It is imaginable that when the first kind of nonlinearities (associated with inelastic soil behavior) occur, the second (associated with geometric nonlinearities) may or may not happen, for instance, for short and stout structures with deeply buried foundations; on the other hand, as the second kind of nonlinearities take place, the first kind would be inevitable.
INTRODUCTION With soil-structure interaction analysis there are two kinds of nonlinearity. The first one, that has received most attention from researchers and practising engineers, is associated with the nonlinear behavior of the soil as a material, thus it can be called the material nonlinearity. The second kind is associated with the sliding and partial separation (uplift) of the foundation from the soil mass, resulting from the inability of the soil to resist tension. With this kind the area of the interface between the foundation and the soil is varying, thus it can be referred to as belonging to the category of geometric nonlinearities. The numerical simulation of the nonlinear behavior of soil associated with the soil-structure system may be divided into two categories: the equivalent linearization analysis 1'2 and the true nonlinear analysis. In the later case the soil-structure system is simulated in general by a complete finite element model 3 and substructure model or hybrid approach. 4 It is worth noticing that to date, in conducting a nonlinear response analysis of the soil-structure system by using an equivalent linearization technique only the nonlinearities developed in the free field are taken into account approximately, while the nonlinearities associated with the additional strain in the soil induced by the structure
Soil Dynamics and Earthquake Engineering 0267-7261/94/$07.00 © 1994 Elsevier Science Limited. 273
274
X. Jianguo, W. Danmin, F. Tieming
The lumped parametric model or the sway-rocking spring model, sometimes referred to as the S-R model, widely used in linear SSI analysis may be the most widely used in the study of the uplift effect of the structure from the supporting soil. 12'13 However, for most investigators the nonlinear S-R model differs from the linear only in the nonlinearities of the rocking stiffness and damping of the foundation soil (or the foundation impedance). In the nonlinear S-R model the stiffness and damping of the soil spring (dashpot) are in general evaluated from the frequency independent stiffness and damping adopted in the linear S-R model multiplied by some coefficients depending on the ratio of the effective contact area of the foundation base, 12'14-16 or determined by some nonlinear moment-rotation relationships established by assuming distribution of contact stresses developed after uplift occurs. 17-19 Moreover, in the nonlinear S-R model the vertical component of the response and the inelastic soil behavior are ignored, or sometimes, a primary consideration of the inelastic soil behavior is given by introduction of the yielding effect of the soil in the contact stress distribution. It should be pointed out that even in the case of only horizontal input motion the vertical component of response will result from the structure system, this is due to the fact that in linear SSI problems, as is well known, the rocking mode of the response is coupled with the horizontal, while in the nonfinear SSI associated with the uplift effect the rocking mode of the response is coupled with the vertical as well. This had been considered only by a few investigators. 13 Another insufficiency of the nonlinear S-R model is its availability being only for the surface foundation. Besides remarkable progress having been made in the numerical simulation of the SSI effects, recently, a great deal of experimental studies2°-22 have been carried out aimed at assessing the capability of predicting reliable results obtained by the analysis methods. However, few studies have been done on attempting to examine the effectiveness for evaluating the nonlinear response of the soil-structure system by comparing the numerical results with the observed ones. Shaking table tests of models for simulating a rigid foundation or SDOF structure supported on the model ground made of silicone rubber or dry sand have been conducted by Ohtomo and Iwatate 16 and Ishimura23 and the experimental data have been compared with those obtained by using the S-R model. Nevertheless, in these tests only the nonlinearities associated with the inelastic soil behavior or those resulting from uplift had been considered. Based on the previous studies24'25 performed by the first author of the present paper a modified lumped parametric model has been proposed. In which the above two kinds of nonlinearities and the vertical component of the response and so on have been considered. In simulating the effects of the surrounding soil of a rectangular embedded foundation not only the
contribution of the soil portion associated with the sidewalls of the foundation with the normal parallel to the direction of the excitation (hereafter it is called the 'parallel sidewalls'), but also those perpendicular to the direction of the excitation (the 'perpendicular sidewalls') are taken into account in a different manner. A SDOF structure-foundation-soil system model had been tested on a shaking table for the purpose of verifying the availability of the proposed analysis model. Comparison has been made between the results of model tests and those obtained by using the proposed model.
MODEL FOR ANALYSIS AND FUNDAMENTAL EQUATIONS OF MOTION Ba~cassumpfions In the establishment of the model for analysis the following basic assumptions had been made: (a) The foundation is rigid rectangular. (b) The effects of the surrounding soil are simulated by using three sets of soil springs (horizontal and vertical translational and rocking) with complex stiffness (the foundation impedance). The real part of the impedance (the stiffness of the soil spring) is divided into three parts associated with the base, the parallel sidewalls and the perpendicular sidewalls respectively, where the part associated with the foundation base is considered to be the same as that of the surface foundation. (c) Before uplift of the foundation from the supporting soil (i.e. at the linear response stage) the foundation impedance can be estimated from the existing information.27 After uplift occurs the stiffnesses of the base soil spring are given by functions of the effective width of the foundation base, these functions are the same as those before uplift. The stiffness of the spring associated with the parallel sidewalls decreases with a ratio identical to that of the base spring. The stiffness of the spring associated with the perpendicular sidewalls and the damping are constant without change during uplift. (d) The impedance-frequency dependency is approximately considered by estimating the characteristics of the springs from the initial impedance-frequency relationship by substituting the fundamental frequency of the linear soil-structure system. (e) The shear modulus-strain dependency of the soil is approximately considered by supposing the shear modulus (G) to be equal to the initial modulus (Go) multiplied by a function of uplift ratio r, 24 i.e. G = [1 - 0.9r + 0.1 sin (2"57rr)]Go where r = (B - B ) / B , here B and /~ denote the initial width and the effective width during uplift of the foundation respectively.
Nonlinear soil-structure interaction analysis From the assumptions listed above it can be seen that the main differences between the modified lumped parametric model presented and those available in the literature are as follows: (a) Not only the nonfinearities resulting from the separation of the foundation base from the supporting soil but also those induced by the separation of the sidewalls of the foundation from the surrounding soil have been considered. Numerical calculation 26 had shown that with increasing severity of uplift the sidewalls of the foundation yield more and more contribution to the total foundation impedance, and the associate nonlinearities have a great influence on the results. (b) The soil nonlinear behavior is considered through introduction of the uplift-ratio-dependent shear modulus of the soil. It is well known that in the SSI the rocking mode has a more significant effect on the results than swaying, therefore it is reasonable to relate the shear modulus of soil with the uplift ratio of the foundation, which provides a measure of the rocking of the foundation.
Impedances of rectangular embedded foundations The foundation impedance is generally written in the complex form
Kd = K~(kj + iaocj) = K/+ iKj = Kj + iwCj
(1)
in which, K] is the foundation impedance, or the complex dynamic stiffness, the subscript j denotes the jth mode, K~ is the static stiffness, ao = wB/(2Vs) is the dimensionless frequency, in which w is the circular frequency, Vs the velocity of shear wave in soil, kj and cj stand for the dynamic stiffness and damping coefficients respectively, being functions of the aspect ratio of the foundation and dimensionless frequency ao .27
Cj = (B/(2V~))K~6 represents the dashpot constant. Let us discuss the real part of the impedance at first, i.e.
Kj = Kjkj
(2)
which can be written as Kj = KjS(1 +
rlj)kj
(3)
where K)~s is the static stiffness of the surface foundation, 7/j coefficients of the embedment effect depending on the aspect and buried depth-width ratios of the foundation. 27 For the purpose of applying this to the uplift case we write the K~s as
K~ -- GB/2 Z
275
K°Rs -- G(B/2)3 R
(6) 1-v where the subscripts Z, X and R denote vertical, horizontal and rocking around the horizontal axis perpendicular to transverse axis x components respectively, v is Poisson's ratio, and
+ L
-1-4
-,q
+
/L I \-°2 L
-1.4
_ 16]
+
in which/3 = B/B, and L is the length of the foundation. Equations (4)-(9) for rectangular foundations with arbitrary aspect ratios were deduced from the formulas proposed by Pais and Kause127 for rectangular foundations (LIB <_4) and those given by Gazetas zs for strip foundations. The coefficients ~Vj and kj can be estimated by using formulas given by Pals and Kausel 27 by introducing the effective width ratio 3. According to assumptions (b) and (c), let Kj = K° + ~- + Kj = (K°s + K~ +
k~)kj
(10)
in which K°, Kj and ~j denote the contributions associated with the base, the parallel sidewalls and the perpendicular sidewalls respectively. It is easy to deduce the following expressions.
Kz=O.369(GB/2~
~-vI(L)\BI2IzIn=, ( H o ~ o's
Z
"° ]] °s XI~=I x Rx : l'386( G2B~-/2v) ( L ) (\ B/2
& = 3.2(c ( x
1
(11)
(12)
(B/2) 3
0"576~ ) R 2.4+a2 Rln= 1
(13)
(4)
1-v K~, - GB/2 x
2--v
(5)
(-o oGj
-O'369(L)\B/2]
(14)
x. Jianguo, w. Danmin, F. Tieming
276
.o/o. lj
(15)
-l386(L)(B/21
__ 3 . 2 ( L ) ( ~ 2 )
(1
0.576a~ 2.4 + a02) 1
(16)
In the above expressions (Z,X,R)]~=I represent the values of Z, X and R with/3 = I, besides B
K ° - 1G-~ - u Z.kz
(17)
B Ko :
(18)
K° = ~ R1 .- kvR
(19)
Now we turn to the imaginary part of the impedance - - the damping, i.e.
(20)
Kj = K;aocj By using cj given in Ref. 27 we have
Krz = wp[VLAb + Vs(Ax + Ay)]
(21)
for the vertical mode of vibration, and
K~ = wp[VLAx + V~(Ab + Ay)]
(22)
for the horizontal mode of vibration, and
{
K~ =cop [VL(Ib + Ix) + V~Iy]f + a~ L
3
for the rocking mode of vibration. In the above expressions p stands for the mass density of soil, Vc = 3.4Vff[Tr(1 - u)], Ao = L. B, Ax = 2(H0. B), Ay = 2 ( H 0 . L ) with H 0 the buried depth of the foundation,
foundation consists of two parts: for instance, for the translational mode of vibration, one part is the radiation damping associated with the one-dimensional (l-D) longitudinal wave propagating with Lysmer's analogy 'wave velocity' (VL) and through the foundation-soil interface with the normal parallel to the direction of excitation, and one part is that associated with the 1-D transverse wave propagating with shear wave velocity and through the foundation-soil interfaces with the normal perpendicular to the direction of excitation. For the rocking mode the situation analogous to that of the translation mode exists in the dominant part, i.e. the first term on the right side of eqn (23). The second term of the right side of eqn (23) is an addition to the radiation damping of the rocking mode modifying to the 1-D radiation damping for low and intermediate frequencies. As stated by Gazetas 29 for a rectangular foundation, the radiation damping developed in a 1-D longitudinal (transverse) wave provides a lower limit of radiation damping developed in the 3-D translation, while for the rocking mode the radiation damping developed in the 1-D wave provides an upper limit, in this case significant deviation occurs for low and intermediate frequencies. Distribution of contact stresses and estimation of effective width of foundation
For the distribution of the contact stresses under the foundation base developed after uplift three patterns, i.e. the linear, the bilinear and the quasi-static state 3° distribution have been examined. It has been found by comparison: 24 (a) With a definite vertical reaction force (N) the linear distribution results in a maximum moment (with respect to the center of the original width of the foundation base), and the bilinear distribution results in the lowest moment. (b) From the relationship between the effective width ratio (/3) and the relative moment ~¢ (I(/I=M/(NB)), as shown in Fig. 1, where Oz: O'y/[N/(B" t)], O-y being the yielding stress, the effect of a is negligible as a > 5. Therefore in the calculation the quasi-static distribution is adopted and /3 = 2-0(1 - 2)17/)
(24)
is used in determining the effective width of the foundation base.
Ib = L B 3 / 1 2
Ix = 2LH3/3 + HoLB2/2
Iy
=
.3
BI 3]
(~) [(-~) H0 qt--
The physical meaning of the above equations is evident: the radiation damping of a rectangular
Fundamental equations of motion
By dividing the soil-foundation-structure system into two, i.e. the superstructure and the foundation-soil subsystems, the equations of motion of the super
277
Nonlinear soil-structure interaction analysis
width to the original center of the foundation base, Wb is the vertical displacement of the foundation at the base center, Kx, Kz and KR denote the total complex stiffness of the horizontal, vertical and rocking soil springs respectively, K'z is the sum of the complex stiffnesses of the vertical soil spring associated with the base and the parallel sidewalls, g is the gravitational acceleration, Fx, Fz and F R are respectively the horizontal and vertical forces and moments generated by the inertia of the superstructure acting at the original center of the foundation base as follows
1.0
% N x~v..--- Or =
10
N%..
~-0.5
=
5
0¢ = 10 - - - ~
o.~5
~r~
0.50
Fx : - ~
(30)
Miii i
i=1
Bilinear distribution ---
Fz=-( ~-"~Mi)fbb
Quasi-static state distribution
(31)
i=1
Fig. 1. Contact ratio versus relative moment; effect of a ratio.
FR = - ~
Miiii(hi + Ho)
i=1
structure can be written in the well-known form:
n
[M]{//} + [C]{~} + [K]{X} = {0}
(25)
-
+/t0)
+ x,]
(32)
i=1
in which [M], [C] and [K] denote the mass, damping and the stiffness matrix of the structure, the elements in the displacement vector {u} are Ui = Ug + Ub + qOb(hi + HO) + xi
(26)
where ug is the horizontal input motion, ub and ~Ob denote the horizontal displacement of the foundation base and the rotation of the foundation respectively, xi are the elements of vector {x}, denoting the relative horizontal displacement of the ith mass point of the superstructure to the center of the top of the foundation, h/ are the distances from the ith mass point to the foundation top and H 0 is the height of the foundation. For the foundation-soil system assuming the base and parallel sidewall springs being located at the center of the effective width of the foundation base and the perpendicular sidewall spring at the original center of the foundation base the following equations of motion can be written Mb(l+'b + ZOqObqOb)+ K2 Wb + K'zelqObl = Fz(t)
(27)
gb(/J b + 20tPb ) + K x u b = - g
(28)
b iJx + Fy(t)
MbZ0(l~bqOb + Ub + 20q0b) + Ibm3+ KRqOb + K'ze(Wb sign (~0b) + e~Ob) = MbZo(g~ob -- fig) + FR(t)
Equations (27)-(29) are solved in the time domain by using Newmark's procedure.
(29)
where M b stands for the foundation mass, z0 is the height of the center of gravity of the foundation measured from its base, I b is the mass moment of inertia of the foundation about the axis through the base center, e is the eccentricity of the center of the effective
MODEL TESTS OF THE SOIL-STRUCTURE SYSTEM AND VERIFICATION OF THE ANALYSIS MODEL For verification of the proposed analysis model a model of a SDOF structure-foundation-soil system was tested on a shaking table. Content and method of the model tests The model test had to be carried out by using a ground container made from a thick wood plate and with an inner size of 120cm in length, 40cm in width and 50 cm in height. The container base was fixed on the shaking table. In order to minimize the wave reflection a piece of foam-rubber cushion was placed on each inner side of the parallel sidewall of the container. The model ground was made of prepared clay and laid in the container layer by layer and compacted. The basic characteristics of the model ground are: shear modulus 32 x 105 Pa, mass density 1"35g/cm 3 and water content 9"8%. The model structure consisted of a top steel block supported by steel plate and a rigid concrete block used to simulate the foundation. The top steel block had a mass of ll.12kg, the connected steel plate had a height of 0.32m measured from the upper surface of the concrete block to the center of the top steel block and a transverse stiffness of
X. Jianguo, I4/. Danmin, F. Tieming
278
1.85 × 104N/m, the rigid concrete block had a mass 9-30kg, and dimensions of 0"086m (height)x 0"200m (width) x 0.236 m (length). Tests of following four groups were performed:
__l Q
(1) Free field test (No. I) The free field test was aimed at: (a) estimating the characteristics of the model ground by the measured data of the response, (b) examining the availability of the model ground for simulating approximately the infinite extension of the ground in the direction of shaking, (c) observing the nonlinearities associated with the inelastic soil behavior. (2) Test of scattering field or the effective input motion
(No. I/) The test was performed by using a rigid lightweight box made of plywood buried in the ground model to simulate the massless rigid foundation, the outer size of the box was the same as that of the buried foundation described later.
(3) Test of SDOF structure model with buried foundation (No. III) The E1 Centro seismic record (1940) with reduced time scale of 1 : 6 was used as the input motion. During the test the models were subjected to increasing excitation by small increments of peak acceleration. The accelerations at different locations on the model ground surface and the structure (foundation) were measured by accelerometers, among which a pair were attached on the two opposite sides of the foundation to observe the vertical and rocking response. Also, at the base of the foundation and in the neighboring soil there were 5 pairs of switches to observe the occurrence of the uplift of the foundation from the supporting soil. The scheme of the model of the soil-structure system and the arrangement of the accelerometers are shown in Fig. 2. (4) Test of SDOF structure model with ground ~+trface supported foundation (No. IV)
I
0
9b
180
~60
Fig. 3. Amplificationfactor versus peak accelerations of input motion (No. I). Test results and discussion The variation of the amplification factor (~) with peak acceleration of the input motion at the shaking table (agmax) is shown in Figs 3 and 4 for tests No. I and No. II. The amplification factor is defined as the ratio of the peak acceleration of the response to agmax. From Figs 3 and 4 we see: (i). The free field test. The responses at all measured points on the model ground surface for a given agmaxare almost identical over a large area, this indicates the availability of the model ground for simulating the infinite extensions in the shaking direction. With increasing agrnax the response acceleration decreases, this implies the nonlinearities associated with the inelastic soil behavior. (ii). The test of scattering field. Contrary to the free field test case the amplification factor ~ of the scattering field increases with increasing a~axComparing Fig. 4 with Fig. 3 one finds that for a given input motion agraax the acceleration in the scattering field is higher than that in the free field. It is contrary to the theoretical results. This may result from the buried box used in the test of the scattering field
4-
A1
27o
agma x (gal)
=
L
x
+[j
•
3a
'~J'2 D
A6 fi
Fig. 2. Typical test model and accelerometer arrangement.
O0
~
60.*
~ 120
t~
180
•
240
agmax (gal) Fig. 4. Amplificationfactor versus peak accelerations of input motion (No. II).
Nonlinear soil-structure interaction analysis deviating from the massless foundation. Moreover, from the test results the rocking and the vertical components are obviously due to wave scattering. Figure 5 shows the horizontal acceleration time histories (AI) of the model superstructure of tests No. IV and No. III at different input motion levels of agmax. In the figure the time history of the input motion ag(t) signed with A6 is also given. From Fig. 5 we see that the horizontal response of the model superstructure with the buried foundation has the behavior of a typical periodical damped vibration, especially under a low level of input motion. Does it imply that the damping in the superstructure with the buried foundation is smaller than that with surface foundation? The relationships of the apparent fundamental frequencies of natural vibration of tests No. III and No. IV obtained from the Fourier spectra of the Al(t) with agmax are shown in Fig. 6. It shows that the frequencies of natural vibration of the system decrease with increasing agraax due to soil flexibility and the nonlinearities associated with inelastic soil behavior and resulting from loss of contact between structure and surrounding soil, especially in the case of surface foundation. From Fig. 6 and the detection of the uplift gal
r
A6 tt |
v[
_,~^ ~ - . ,,J, ,..*A. . . . . . . . . . . . . .
~l,,, 1 r-,-,~
.L,.
-,
. . . . . . . . . . . .
amax = 190.42 (agmax = 45.99)
i gal .,~ ...... ,vV~v . . . . . .
A10
,.,A^,.. ^. ,, ^ . . . . . . . . uvu---,, . . . . . . . .
^ ....... v ......
arnax = -279.40 (agmax = 123,80)
A'o Lg''"
t
.,., ...,^ ....... , VV V v., ,.wvv
amax = 426.20
gal
0 (a)
(agmax = 176.50)
1 2 No. IV
3 4 "[]me (s)
5
6
amax = -256.70 (agmax = 32.21)
gal
0[vvvvvvvvvvvvvvvvvvvw'v AIo[ ,^AAAAAA^^^^^^^,,^_.,...
...............
amax = -471.30 (agrnax = 67.51)
gal
. . . . . . . .
,.,.
.
.
.
.
.
.
.
vvmvvwvvvv-' . . . . . . . . . , amax = 652.80 (agmax = 127.00) ^^^~ • ^,./~A^. . . . . . . . ,,.^. . . . ^^^,, . . . . . . . . . . . .
[gal AAA
A1 0
J
t
"vVVV,- "" vv, .....................
jgal
amax = -711.80 (agrnax = 164.30)
,
A10 [
L
0 (b)
.,AA^^,._ , A A . . . . . ^ . =,',A . . . . . A ^ ^,,,,,V~~ •VVVv,,,-vv,- -,---, .... . . . . .
i 2 No. l l I
3 4 ] ] m e (s)
5
6
Fig. 5. Acceleration time histories; A6, input motion; AI, response of superstructure to input motion with agmax.
279
6.8
a
x No.IV
x
6.1
No. III
~ 5.4 v,,_ 4.7
4.0 0
5b
260 agmax (gal)
Fig. 6. Apparent natural frequencies versus peak acceleration of input motion. by using the switch at the foundation base it was found that the uplift of the foundation base occurs under a relatively low level of input motion. The earthquake simulating tests of the free vibration induced by striking had also been performed for the purpose of directly measuring the behavior of the natural vibration of the model structure under different foundation conditions. The results are listed in Table 1. Therefore the phenomenon of the damping generated in the structure with buffed foundation being lower than in the structure with surface foundation discovered in earthquake-simulation tests is proved by the free vibration test. However, this is in contradiction to the expectation of the elastic half space theory. The amplification factors (~) versus agmaxfor tests No. III and No. IV are shown in Figs 7 and 8. Comparison of the results of No. III with those of No. IV indicates that the response of the model structure with the buffed foundation is higher than that with the surface foundation under identical input motion. This may be due to the fact that the natural frequency of the structure with the buried foundation is more close to the predominant frequency of the input motion (6-85 Hz), with the lower damping of the structure with the buffed foundation as mentioned above. With respect to the response of the foundation, Fig. 7 along with Fig. 8 shows that the response of the foundation is independent of whether the foundation is buried or ground surface supported and differs little from that in the scattering field, with the exception of the rocking and vertical components of the response under a higher level of input motions. Table 1. Behavior of natural vibration of model structure under different foundation conditions
Foundation condition Absolutely fixed Buried in model ground Supported on the model ground
Fund. freq. (Hz) 6"75 6"37= 6.12
Damping ratio (%) 0.48 1.73 4.70
aFrom the earthquake-simulation tests this value is 6.60.
280
X. Jianguo, W. Danmin, F. Tieming
o
÷
~J'4 ~,,,,4 ¸
x
x
0
=
0 2"
0
50
1O0 agmax (gal)
150
Fig. 7. Amplification factor versus peak accelerations of input motion (No. IV). A detailed introduction to the test results can be found elsewhere. 26 Comparison between results of tests and calculations The proposed analysis model is applied to the test conditions. In the numerical analysis the response in the scattering field measured in the earthquake-simulation tests, was adopted as the input motion. The parameters of the foundation impedance used in the numerical calculation for the tested soil-structure interaction model are: The initial (before uplift occurs) spring constants of the soil (the real part of the impedance)
t
o
200
•
50
1()0 agmax (gad
150
2()0
Fig. 8. Amplification factor versus peak accelerations of input motion (No. III). agmax are given in Fig. 9. The comparisons illustrate that not only the peak values but also the whole process obtained by calculation and experiments agree well. The proposed analysis model has also been applied to an air-blast loading test, and satisfactory agreement between the results of numerical analysis and model test have also been obtained. 31 CONCLUSIONS 1. From the earthquake simulation tests it indicates: (a) The soil-structure system reaches the nonlinear state under a relatively low level of input motion.
Kz = 7"22 x 106 N / m 350 [
for vertical translation
.......
Max. 256.7 (1.64s) Max. 243.7 (1.64s)
Measured Computed
Kx = 6.56 x 106 N / m
f
for horizontal translation KR = 1.12 x 1 0 S N m / r a d
-350 ~ 0
for rocking. The initial dashpot constants:
CE = 1'24 X 104Ns/m for vertical translation
1.0
-~ tO 550 vO) tO
.......
2.0
3.0
4.0
Time (s)
5.0
6.0
7.0
Max. 471.3 (1.64s) Max. 438.2 (1.64s)
Measured Computed
Cx = 1.17 x 104Ns/m for horizontal translation
< -550
CR = 20.5 N m s/rad for rocking. These were obtained by using expressions (1)-(23) and by adopting the initial shear modulus of the soil Go = 5.05 x 106Pa and Poisson's ratio v = 0.4. The related parameters of the superstructure were given in the beginning of the present section. Typical results obtained by experiments and calculations showing the horizontal acceleration response of the model structure under input motions with different
0
1.0
850, ...... ; ! [i
2.0
3.0
4.0
Time (s)
Measured Computed
5.0
6.0
7.0
Max. 652.8 (1.45s) Max. 689.7 (1.44s) (agmax = 127.0)
I
-850
0
1.0
2.0
3.0
4.0
Tirne (s)
5.0
6.0
7.0
Fig. 9. Comparisons between acceleration responses of superstructure obtained from tests and calculations.
Nonlinear soil-structure interaction analysis
(b) Even under the nonlinear state the soil-structure system has an apparent behavior of natural vibration. With increasing input motion level the apparent natural frequencies decrease and damping ratios increase. (c) Under the action of a vertically propagating SH wave due to the nonlinear SSI effect the amplification factor of the horizontal acceleration of the structure response decreases and the vertical and rocking accelerations increase slightly. (d) Comparing with the surface foundation case the embedment of the foundation leads to an increase in the natural frequencies of the system. With increasing input motion level the apparent natural frequency of the system with the buried foundation decreases more slowly than that with the surface foundation. In certain circumstances, for instance, when the fundamental natural frequency of the system with the buried foundation is closer to the predominant frequency of the input motion, the response o f the system will be higher than that with the surface supported foundation. However, with increasing input motion level the nonlinear effect increases and the embedment effect decreases. (e) For the test case performed the nonlinearities resulting from the loss of contact between the foundation and its surrounding soil have a more significant effect on the structure response than the nonlinearities associated with inelastic soil behavior. 2. The modified lumped parametric model proposed takes into account nonlinearities associated with the inelastic soil behavior and those resulting from the loss of contact between the foundation and surrounding soil. In consideration of the latter nonlinearities the two pairs of sidewalls of a buried rectangular foundation had been treated in a different manner. Application of the proposed analysis model to the earthquake simulation tests shows that the proposed analysis model yields excellent results of structure response to low or moderate level of ground shaking and also strong. Finally, the proposed analysis model is first validated through model tests with model structure and model ground with one kind of definite characteristics under the action of definite excitation; in order to make the analysis model applicable to a wide range of soil conditions, structure behavior and earthquake loadings it is necessary to carry out additional tests. Moreover, a lot of work has to be done in future studies with respect to further modification of the proposed model, in particular, the simulation of two kinds of nonlinearities, the radiation damping, the circular foundation and so on.
ACKNOWLEDGEMENTS
This study is supported by the Joint Foundation of Seismology Science o f China (No. 90147). The authors
281
are indebted to Mrs Gong Zhenpeng, Shen Qingliang and Wang Jichun of the Institute of Engineering Mechanics, SSB for their cooperation in conducting the shaking table tests. REFERENCES
1. Schnabel, P.B., Lysmer, J. & Seed, H.B. SHAKE - - a computer program for earthquake response analysis of horizontally layered sites. Report EERC 72-12, 1972. 2. Lysmer, J., Udako, T., Seed, H.B. & Hwang, R.N. LUSH - - a computer program for complex response analysis of soil structure systems. Report EERC 74-4, 1974. 3. Ferritto, J.M. SSI and soil models, Shock and Vibration Bulletin, No. 52, part 5, 29-38, 1982. 4. Wolf, J.P. SSI analysis in time domain. Nuclear Engineering and Design, 1989, 111, 381-93. 5. Toki, K. & Miura, F. Nonlinear seismic response analysis of SSI systems. Earthquake Engineering and Structure Dynamics, 1983, 11, 77-89. 6. Miura, F. Dynamic stability analysis of rigid structures against sliding and overturning during strong earthquake motions. Proc. 8 WCEE, Vol. III. Prentice Hall, NJ, USA, 1984, 485-92. 7. Desai, C.S. & Zaman, M.M. Influence of interface behavior in dynamic SSI. Proc 8 WCEE, Vol. III. Prentice Hall, NJ, USA, 1984, 1049-56. 8. Zhao Zhendong, Cai Yongen & Xiong Jianguo, Sliding and uplift response of high-rise buildings to explosive loading. Explosion and Shock Waves, 1991, 11(4), 301-30 (in Chinese). 9. Hayashi, Y., Fakuwa, N., Nakai, S. & Kouanagi, Y. Earthquake response analysis considering soil-structure separation using contact element and dynamic flexibility of soil in time domain. Proc. 9 WCEE, Vol. VIII. 1988, 26570. 10. Aihara, S. et al. Study on the contact ratio of base mat of reactor buildings considering nonlinear SSI effects. Trans. 7 SMiRT, 1973, K9/ll. 11. Penzien, J. Closure statement special theme Session SD. Proc. 9 WCEE, Vol. VIII, 379. 12. Kennedy, R.P. et al. Effect of nonlinear SSI to base slab uplift on the seismic response of a high-temperature gascooled reactor. Nuclear Engineering and Design, 1976, 38, 323-355. 13. Wolf, J.P. SSI with separation of base mat from soil (lifting off), Nuclear Engineering and Design, 1976, 38, 357-384. 14. Sotomura, K. et aL Nonlinear response analysis of reactor building subjected to horizontal and vertical motions simultaneously. Trans. 7 SMiRT, 1983, K5/6. 15. Kobori, T. et al. Nonlinear uplift behavior of soilstructure system with frequency dependent characteristics. Proc. 8 WCEE, Vol. III. Prentice Hall, N J, USA, 1984, 897-904. 16. Ohtomo, K. & Iwatate, T. Proposal of analytical method on foundation uplift considering the damping effect. Proc. 9 WCEE, Vol. VIII. 1988, 277-82. 17. Unemori, A.L. & Whitman, R.V. Nonlinear inelastic dynamic analysis with soil flexibility in rocking. PB-256, 794. Shannon & Wilson Inc, 1976. 18. Izumi, M. et al. Rocking vibration considering uplift and yielding of supporting soil. Proc. 4th Japanese Earthquake Engineering Symposium, 1975, pp. 727-34. 19. Zinn, R. & Stangenberg, F. Base mat uplift effects on the seismic-response of a rectangular nuclear building. Trans. 7 SMiRT, 1983, K9/3.
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20. Weissman, K. & Prevost, J.H. Centrifugal modelling of dynamic SSI. National Centre for Earthquake Engineering Research Tech. Report, NCEER-89-0040, 1989. 21. Tang, H.T. et al. A large-scale SSI experiment, design and construction. Nuclear Engineering and Design, 1989, 111, 371-9. 22. Tajimi, H. State-of-the-art report: seismic observations and vibration tests for verification of SSI analysis method, Proc. 9 WCEE, 1989, Vol. III, 297-308. 23. Ishimura, K., Swaying-rocking model for simulating nonlinear response of sandy deposit with structure. Proc. 10 WCEE, Vol. III. Balkem, Rotterdam, 1992, 1897-903. 24. Xiong Jianguo, Gao Weijian et al. Overturning resistance and rotation of foundation in soil under lateral loading. Computers and Geotechnics, 1989, 8, 157-74. 25. Xiong Jianguo, Li Tong & Zhai Tong, Dynamics of liquid storage tanks considering soil flexibility and base uplift. Proc. 9th European Conf. on Earthquake Engineering, Moscow, Vol. 7B, 66-75. Kucherenko TSNIISK, Moscow, 1990.
26. Xiong Jianguo, Wang Danmin & Fu Tieming, Nonlinear soil-structure interaction analysis. Research Report, Institute of Engineering Mechanics, SSB, Harbin, 1992 (in Chinese). 27. Pais, A. & Kausel, E. Approximate formulas for dynamic stiffness of rigid foundations. Soil Dynamics and Earthquake Engineering, 1988, 7(4), 213-27. 28. Gazetas, G. Analysis of machine foundation vibrations: state of the art. Soil Dynamics and Earthquake Engineering, 1983, 2(1), 2-41. 29. Gazetas, G. Simple physical models for foundation impedances. In Dynamic Behaviour of Foundations and Buried Structures, ed. P.K. Banerjee & R. Buttertield. Elsevier, Amsterdam, pp. 45-93. 30. Beucke, K. et al. Nonlinear soil-structure interaction with base uplift. Trans. 7 SMiRT, K9/2, 1983. 31. Xiong Jianguo, Gao Weijian et al. Study on overturning of highrise building with basement under airblast loading. Research Report, Institute of Engineering Mechanics, SSB, Harbin, 1992 (in Chinese).
A modified lumped parametric model for nonlinear soil-structure interaction analysis Xiong Jianguo, Wang Danmin & Fu Tieming Institute of Engineering Mechanics, State Seismological Bureau, Harbin, 150080, People's Republic of China (Received 19 November 1992; revised version received and accepted 30 June 1993) Most soil-structure interaction (SSI) analyses are still conducted assuming linear material behavior or simulating nonlinear effects through an equivalent linearization and the structure (foundation) being closely welded with the surrounding soil. It is recognized, however, that nonlinearities can play a significant role in the results. Two kinds of nonlinearities must be considered: those associated with inelastic soil behavior and those resulting from loss of contact between the foundation and the surrounding soil. In the present paper a modified lumped parametric model for the analysis of nonlinear SSI effects has been proposed. In the model both nonlinearities are taken into account. The results of tests of the soil-structure system model have been presented, which agree well with those obtained from analysis by using the proposed model.
vibration in the soil-structure system are neglected. With the substructure or the hybrid model for the true nonlinear analysis, on the other hand, only the near field (the structure and the near portion of the soil) is treated as though nonlinear, while the far field is assumed to be linear. The two more sophisticated models mentioned above for the analysis of the true nonlinearities associated with inelastic soil behavior are also available for the analysis of geometric nonlinearities of SSI by introducing some special elements at the interface between the foundation and the surrounding soil. 5-1° The superiority of the sophisticated models is obvious they can give a more realistic modelling of the complete structure-foundation-soil system and provide extensive information about the performance of the system from the analysis. It is important and should be used for cases involving large/heavy critical systems. However, while the trend has been toward more sophisticated modelling and analysis procedures, it is significant to note that very simple procedures can often be very effective in assessing SSI effects, 1~ in particular, in obtaining preliminary estimates of such effects for design purposes. It is imaginable that when the first kind of nonlinearities (associated with inelastic soil behavior) occur, the second (associated with geometric nonlinearities) may or may not happen, for instance, for short and stout structures with deeply buried foundations; on the other hand, as the second kind of nonlinearities take place, the first kind would be inevitable.
INTRODUCTION With soil-structure interaction analysis there are two kinds of nonlinearity. The first one, that has received most attention from researchers and practising engineers, is associated with the nonlinear behavior of the soil as a material, thus it can be called the material nonlinearity. The second kind is associated with the sliding and partial separation (uplift) of the foundation from the soil mass, resulting from the inability of the soil to resist tension. With this kind the area of the interface between the foundation and the soil is varying, thus it can be referred to as belonging to the category of geometric nonlinearities. The numerical simulation of the nonlinear behavior of soil associated with the soil-structure system may be divided into two categories: the equivalent linearization analysis 1'2 and the true nonlinear analysis. In the later case the soil-structure system is simulated in general by a complete finite element model 3 and substructure model or hybrid approach. 4 It is worth noticing that to date, in conducting a nonlinear response analysis of the soil-structure system by using an equivalent linearization technique only the nonlinearities developed in the free field are taken into account approximately, while the nonlinearities associated with the additional strain in the soil induced by the structure
Soil Dynamics and Earthquake Engineering 0267-7261/94/$07.00 © 1994 Elsevier Science Limited. 273
274
X. Jianguo, W. Danmin, F. Tieming
The lumped parametric model or the sway-rocking spring model, sometimes referred to as the S-R model, widely used in linear SSI analysis may be the most widely used in the study of the uplift effect of the structure from the supporting soil. 12'13 However, for most investigators the nonlinear S-R model differs from the linear only in the nonlinearities of the rocking stiffness and damping of the foundation soil (or the foundation impedance). In the nonlinear S-R model the stiffness and damping of the soil spring (dashpot) are in general evaluated from the frequency independent stiffness and damping adopted in the linear S-R model multiplied by some coefficients depending on the ratio of the effective contact area of the foundation base, 12'14-16 or determined by some nonlinear moment-rotation relationships established by assuming distribution of contact stresses developed after uplift occurs. 17-19 Moreover, in the nonlinear S-R model the vertical component of the response and the inelastic soil behavior are ignored, or sometimes, a primary consideration of the inelastic soil behavior is given by introduction of the yielding effect of the soil in the contact stress distribution. It should be pointed out that even in the case of only horizontal input motion the vertical component of response will result from the structure system, this is due to the fact that in linear SSI problems, as is well known, the rocking mode of the response is coupled with the horizontal, while in the nonfinear SSI associated with the uplift effect the rocking mode of the response is coupled with the vertical as well. This had been considered only by a few investigators. 13 Another insufficiency of the nonlinear S-R model is its availability being only for the surface foundation. Besides remarkable progress having been made in the numerical simulation of the SSI effects, recently, a great deal of experimental studies2°-22 have been carried out aimed at assessing the capability of predicting reliable results obtained by the analysis methods. However, few studies have been done on attempting to examine the effectiveness for evaluating the nonlinear response of the soil-structure system by comparing the numerical results with the observed ones. Shaking table tests of models for simulating a rigid foundation or SDOF structure supported on the model ground made of silicone rubber or dry sand have been conducted by Ohtomo and Iwatate 16 and Ishimura23 and the experimental data have been compared with those obtained by using the S-R model. Nevertheless, in these tests only the nonlinearities associated with the inelastic soil behavior or those resulting from uplift had been considered. Based on the previous studies24'25 performed by the first author of the present paper a modified lumped parametric model has been proposed. In which the above two kinds of nonlinearities and the vertical component of the response and so on have been considered. In simulating the effects of the surrounding soil of a rectangular embedded foundation not only the
contribution of the soil portion associated with the sidewalls of the foundation with the normal parallel to the direction of the excitation (hereafter it is called the 'parallel sidewalls'), but also those perpendicular to the direction of the excitation (the 'perpendicular sidewalls') are taken into account in a different manner. A SDOF structure-foundation-soil system model had been tested on a shaking table for the purpose of verifying the availability of the proposed analysis model. Comparison has been made between the results of model tests and those obtained by using the proposed model.
MODEL FOR ANALYSIS AND FUNDAMENTAL EQUATIONS OF MOTION Ba~cassumpfions In the establishment of the model for analysis the following basic assumptions had been made: (a) The foundation is rigid rectangular. (b) The effects of the surrounding soil are simulated by using three sets of soil springs (horizontal and vertical translational and rocking) with complex stiffness (the foundation impedance). The real part of the impedance (the stiffness of the soil spring) is divided into three parts associated with the base, the parallel sidewalls and the perpendicular sidewalls respectively, where the part associated with the foundation base is considered to be the same as that of the surface foundation. (c) Before uplift of the foundation from the supporting soil (i.e. at the linear response stage) the foundation impedance can be estimated from the existing information.27 After uplift occurs the stiffnesses of the base soil spring are given by functions of the effective width of the foundation base, these functions are the same as those before uplift. The stiffness of the spring associated with the parallel sidewalls decreases with a ratio identical to that of the base spring. The stiffness of the spring associated with the perpendicular sidewalls and the damping are constant without change during uplift. (d) The impedance-frequency dependency is approximately considered by estimating the characteristics of the springs from the initial impedance-frequency relationship by substituting the fundamental frequency of the linear soil-structure system. (e) The shear modulus-strain dependency of the soil is approximately considered by supposing the shear modulus (G) to be equal to the initial modulus (Go) multiplied by a function of uplift ratio r, 24 i.e. G = [1 - 0.9r + 0.1 sin (2"57rr)]Go where r = (B - B ) / B , here B and /~ denote the initial width and the effective width during uplift of the foundation respectively.
Nonlinear soil-structure interaction analysis From the assumptions listed above it can be seen that the main differences between the modified lumped parametric model presented and those available in the literature are as follows: (a) Not only the nonfinearities resulting from the separation of the foundation base from the supporting soil but also those induced by the separation of the sidewalls of the foundation from the surrounding soil have been considered. Numerical calculation 26 had shown that with increasing severity of uplift the sidewalls of the foundation yield more and more contribution to the total foundation impedance, and the associate nonlinearities have a great influence on the results. (b) The soil nonlinear behavior is considered through introduction of the uplift-ratio-dependent shear modulus of the soil. It is well known that in the SSI the rocking mode has a more significant effect on the results than swaying, therefore it is reasonable to relate the shear modulus of soil with the uplift ratio of the foundation, which provides a measure of the rocking of the foundation.
Impedances of rectangular embedded foundations The foundation impedance is generally written in the complex form
Kd = K~(kj + iaocj) = K/+ iKj = Kj + iwCj
(1)
in which, K] is the foundation impedance, or the complex dynamic stiffness, the subscript j denotes the jth mode, K~ is the static stiffness, ao = wB/(2Vs) is the dimensionless frequency, in which w is the circular frequency, Vs the velocity of shear wave in soil, kj and cj stand for the dynamic stiffness and damping coefficients respectively, being functions of the aspect ratio of the foundation and dimensionless frequency ao .27
Cj = (B/(2V~))K~6 represents the dashpot constant. Let us discuss the real part of the impedance at first, i.e.
Kj = Kjkj
(2)
which can be written as Kj = KjS(1 +
rlj)kj
(3)
where K)~s is the static stiffness of the surface foundation, 7/j coefficients of the embedment effect depending on the aspect and buried depth-width ratios of the foundation. 27 For the purpose of applying this to the uplift case we write the K~s as
K~ -- GB/2 Z
275
K°Rs -- G(B/2)3 R
(6) 1-v where the subscripts Z, X and R denote vertical, horizontal and rocking around the horizontal axis perpendicular to transverse axis x components respectively, v is Poisson's ratio, and
+ L
-1-4
-,q
+
/L I \-°2 L
-1.4
_ 16]
+
in which/3 = B/B, and L is the length of the foundation. Equations (4)-(9) for rectangular foundations with arbitrary aspect ratios were deduced from the formulas proposed by Pais and Kause127 for rectangular foundations (LIB <_4) and those given by Gazetas zs for strip foundations. The coefficients ~Vj and kj can be estimated by using formulas given by Pals and Kausel 27 by introducing the effective width ratio 3. According to assumptions (b) and (c), let Kj = K° + ~- + Kj = (K°s + K~ +
k~)kj
(10)
in which K°, Kj and ~j denote the contributions associated with the base, the parallel sidewalls and the perpendicular sidewalls respectively. It is easy to deduce the following expressions.
Kz=O.369(GB/2~
~-vI(L)\BI2IzIn=, ( H o ~ o's
Z
"° ]] °s XI~=I x Rx : l'386( G2B~-/2v) ( L ) (\ B/2
& = 3.2(c ( x
1
(11)
(12)
(B/2) 3
0"576~ ) R 2.4+a2 Rln= 1
(13)
(4)
1-v K~, - GB/2 x
2--v
(5)
(-o oGj
-O'369(L)\B/2]
(14)
x. Jianguo, w. Danmin, F. Tieming
276
.o/o. lj
(15)
-l386(L)(B/21
__ 3 . 2 ( L ) ( ~ 2 )
(1
0.576a~ 2.4 + a02) 1
(16)
In the above expressions (Z,X,R)]~=I represent the values of Z, X and R with/3 = I, besides B
K ° - 1G-~ - u Z.kz
(17)
B Ko :
(18)
K° = ~ R1 .- kvR
(19)
Now we turn to the imaginary part of the impedance - - the damping, i.e.
(20)
Kj = K;aocj By using cj given in Ref. 27 we have
Krz = wp[VLAb + Vs(Ax + Ay)]
(21)
for the vertical mode of vibration, and
K~ = wp[VLAx + V~(Ab + Ay)]
(22)
for the horizontal mode of vibration, and
{
K~ =cop [VL(Ib + Ix) + V~Iy]f + a~ L
3
for the rocking mode of vibration. In the above expressions p stands for the mass density of soil, Vc = 3.4Vff[Tr(1 - u)], Ao = L. B, Ax = 2(H0. B), Ay = 2 ( H 0 . L ) with H 0 the buried depth of the foundation,
foundation consists of two parts: for instance, for the translational mode of vibration, one part is the radiation damping associated with the one-dimensional (l-D) longitudinal wave propagating with Lysmer's analogy 'wave velocity' (VL) and through the foundation-soil interface with the normal parallel to the direction of excitation, and one part is that associated with the 1-D transverse wave propagating with shear wave velocity and through the foundation-soil interfaces with the normal perpendicular to the direction of excitation. For the rocking mode the situation analogous to that of the translation mode exists in the dominant part, i.e. the first term on the right side of eqn (23). The second term of the right side of eqn (23) is an addition to the radiation damping of the rocking mode modifying to the 1-D radiation damping for low and intermediate frequencies. As stated by Gazetas 29 for a rectangular foundation, the radiation damping developed in a 1-D longitudinal (transverse) wave provides a lower limit of radiation damping developed in the 3-D translation, while for the rocking mode the radiation damping developed in the 1-D wave provides an upper limit, in this case significant deviation occurs for low and intermediate frequencies. Distribution of contact stresses and estimation of effective width of foundation
For the distribution of the contact stresses under the foundation base developed after uplift three patterns, i.e. the linear, the bilinear and the quasi-static state 3° distribution have been examined. It has been found by comparison: 24 (a) With a definite vertical reaction force (N) the linear distribution results in a maximum moment (with respect to the center of the original width of the foundation base), and the bilinear distribution results in the lowest moment. (b) From the relationship between the effective width ratio (/3) and the relative moment ~¢ (I(/I=M/(NB)), as shown in Fig. 1, where Oz: O'y/[N/(B" t)], O-y being the yielding stress, the effect of a is negligible as a > 5. Therefore in the calculation the quasi-static distribution is adopted and /3 = 2-0(1 - 2)17/)
(24)
is used in determining the effective width of the foundation base.
Ib = L B 3 / 1 2
Ix = 2LH3/3 + HoLB2/2
Iy
=
.3
BI 3]
(~) [(-~) H0 qt--
The physical meaning of the above equations is evident: the radiation damping of a rectangular
Fundamental equations of motion
By dividing the soil-foundation-structure system into two, i.e. the superstructure and the foundation-soil subsystems, the equations of motion of the super
277
Nonlinear soil-structure interaction analysis
width to the original center of the foundation base, Wb is the vertical displacement of the foundation at the base center, Kx, Kz and KR denote the total complex stiffness of the horizontal, vertical and rocking soil springs respectively, K'z is the sum of the complex stiffnesses of the vertical soil spring associated with the base and the parallel sidewalls, g is the gravitational acceleration, Fx, Fz and F R are respectively the horizontal and vertical forces and moments generated by the inertia of the superstructure acting at the original center of the foundation base as follows
1.0
% N x~v..--- Or =
10
N%..
~-0.5
=
5
0¢ = 10 - - - ~
o.~5
~r~
0.50
Fx : - ~
(30)
Miii i
i=1
Bilinear distribution ---
Fz=-( ~-"~Mi)fbb
Quasi-static state distribution
(31)
i=1
Fig. 1. Contact ratio versus relative moment; effect of a ratio.
FR = - ~
Miiii(hi + Ho)
i=1
structure can be written in the well-known form:
n
[M]{//} + [C]{~} + [K]{X} = {0}
(25)
-
+/t0)
+ x,]
(32)
i=1
in which [M], [C] and [K] denote the mass, damping and the stiffness matrix of the structure, the elements in the displacement vector {u} are Ui = Ug + Ub + qOb(hi + HO) + xi
(26)
where ug is the horizontal input motion, ub and ~Ob denote the horizontal displacement of the foundation base and the rotation of the foundation respectively, xi are the elements of vector {x}, denoting the relative horizontal displacement of the ith mass point of the superstructure to the center of the top of the foundation, h/ are the distances from the ith mass point to the foundation top and H 0 is the height of the foundation. For the foundation-soil system assuming the base and parallel sidewall springs being located at the center of the effective width of the foundation base and the perpendicular sidewall spring at the original center of the foundation base the following equations of motion can be written Mb(l+'b + ZOqObqOb)+ K2 Wb + K'zelqObl = Fz(t)
(27)
gb(/J b + 20tPb ) + K x u b = - g
(28)
b iJx + Fy(t)
MbZ0(l~bqOb + Ub + 20q0b) + Ibm3+ KRqOb + K'ze(Wb sign (~0b) + e~Ob) = MbZo(g~ob -- fig) + FR(t)
Equations (27)-(29) are solved in the time domain by using Newmark's procedure.
(29)
where M b stands for the foundation mass, z0 is the height of the center of gravity of the foundation measured from its base, I b is the mass moment of inertia of the foundation about the axis through the base center, e is the eccentricity of the center of the effective
MODEL TESTS OF THE SOIL-STRUCTURE SYSTEM AND VERIFICATION OF THE ANALYSIS MODEL For verification of the proposed analysis model a model of a SDOF structure-foundation-soil system was tested on a shaking table. Content and method of the model tests The model test had to be carried out by using a ground container made from a thick wood plate and with an inner size of 120cm in length, 40cm in width and 50 cm in height. The container base was fixed on the shaking table. In order to minimize the wave reflection a piece of foam-rubber cushion was placed on each inner side of the parallel sidewall of the container. The model ground was made of prepared clay and laid in the container layer by layer and compacted. The basic characteristics of the model ground are: shear modulus 32 x 105 Pa, mass density 1"35g/cm 3 and water content 9"8%. The model structure consisted of a top steel block supported by steel plate and a rigid concrete block used to simulate the foundation. The top steel block had a mass of ll.12kg, the connected steel plate had a height of 0.32m measured from the upper surface of the concrete block to the center of the top steel block and a transverse stiffness of
X. Jianguo, I4/. Danmin, F. Tieming
278
1.85 × 104N/m, the rigid concrete block had a mass 9-30kg, and dimensions of 0"086m (height)x 0"200m (width) x 0.236 m (length). Tests of following four groups were performed:
__l Q
(1) Free field test (No. I) The free field test was aimed at: (a) estimating the characteristics of the model ground by the measured data of the response, (b) examining the availability of the model ground for simulating approximately the infinite extension of the ground in the direction of shaking, (c) observing the nonlinearities associated with the inelastic soil behavior. (2) Test of scattering field or the effective input motion
(No. I/) The test was performed by using a rigid lightweight box made of plywood buried in the ground model to simulate the massless rigid foundation, the outer size of the box was the same as that of the buried foundation described later.
(3) Test of SDOF structure model with buried foundation (No. III) The E1 Centro seismic record (1940) with reduced time scale of 1 : 6 was used as the input motion. During the test the models were subjected to increasing excitation by small increments of peak acceleration. The accelerations at different locations on the model ground surface and the structure (foundation) were measured by accelerometers, among which a pair were attached on the two opposite sides of the foundation to observe the vertical and rocking response. Also, at the base of the foundation and in the neighboring soil there were 5 pairs of switches to observe the occurrence of the uplift of the foundation from the supporting soil. The scheme of the model of the soil-structure system and the arrangement of the accelerometers are shown in Fig. 2. (4) Test of SDOF structure model with ground ~+trface supported foundation (No. IV)
I
0
9b
180
~60
Fig. 3. Amplificationfactor versus peak accelerations of input motion (No. I). Test results and discussion The variation of the amplification factor (~) with peak acceleration of the input motion at the shaking table (agmax) is shown in Figs 3 and 4 for tests No. I and No. II. The amplification factor is defined as the ratio of the peak acceleration of the response to agmax. From Figs 3 and 4 we see: (i). The free field test. The responses at all measured points on the model ground surface for a given agmaxare almost identical over a large area, this indicates the availability of the model ground for simulating the infinite extensions in the shaking direction. With increasing agrnax the response acceleration decreases, this implies the nonlinearities associated with the inelastic soil behavior. (ii). The test of scattering field. Contrary to the free field test case the amplification factor ~ of the scattering field increases with increasing a~axComparing Fig. 4 with Fig. 3 one finds that for a given input motion agraax the acceleration in the scattering field is higher than that in the free field. It is contrary to the theoretical results. This may result from the buried box used in the test of the scattering field
4-
A1
27o
agma x (gal)
=
L
x
+[j
•
3a
'~J'2 D
A6 fi
Fig. 2. Typical test model and accelerometer arrangement.
O0
~
60.*
~ 120
t~
180
•
240
agmax (gal) Fig. 4. Amplificationfactor versus peak accelerations of input motion (No. II).
Nonlinear soil-structure interaction analysis deviating from the massless foundation. Moreover, from the test results the rocking and the vertical components are obviously due to wave scattering. Figure 5 shows the horizontal acceleration time histories (AI) of the model superstructure of tests No. IV and No. III at different input motion levels of agmax. In the figure the time history of the input motion ag(t) signed with A6 is also given. From Fig. 5 we see that the horizontal response of the model superstructure with the buried foundation has the behavior of a typical periodical damped vibration, especially under a low level of input motion. Does it imply that the damping in the superstructure with the buried foundation is smaller than that with surface foundation? The relationships of the apparent fundamental frequencies of natural vibration of tests No. III and No. IV obtained from the Fourier spectra of the Al(t) with agmax are shown in Fig. 6. It shows that the frequencies of natural vibration of the system decrease with increasing agraax due to soil flexibility and the nonlinearities associated with inelastic soil behavior and resulting from loss of contact between structure and surrounding soil, especially in the case of surface foundation. From Fig. 6 and the detection of the uplift gal
r
A6 tt |
v[
_,~^ ~ - . ,,J, ,..*A. . . . . . . . . . . . . .
~l,,, 1 r-,-,~
.L,.
-,
. . . . . . . . . . . .
amax = 190.42 (agmax = 45.99)
i gal .,~ ...... ,vV~v . . . . . .
A10
,.,A^,.. ^. ,, ^ . . . . . . . . uvu---,, . . . . . . . .
^ ....... v ......
arnax = -279.40 (agmax = 123,80)
A'o Lg''"
t
.,., ...,^ ....... , VV V v., ,.wvv
amax = 426.20
gal
0 (a)
(agmax = 176.50)
1 2 No. IV
3 4 "[]me (s)
5
6
amax = -256.70 (agmax = 32.21)
gal
0[vvvvvvvvvvvvvvvvvvvw'v AIo[ ,^AAAAAA^^^^^^^,,^_.,...
...............
amax = -471.30 (agrnax = 67.51)
gal
. . . . . . . .
,.,.
.
.
.
.
.
.
.
vvmvvwvvvv-' . . . . . . . . . , amax = 652.80 (agmax = 127.00) ^^^~ • ^,./~A^. . . . . . . . ,,.^. . . . ^^^,, . . . . . . . . . . . .
[gal AAA
A1 0
J
t
"vVVV,- "" vv, .....................
jgal
amax = -711.80 (agrnax = 164.30)
,
A10 [
L
0 (b)
.,AA^^,._ , A A . . . . . ^ . =,',A . . . . . A ^ ^,,,,,V~~ •VVVv,,,-vv,- -,---, .... . . . . .
i 2 No. l l I
3 4 ] ] m e (s)
5
6
Fig. 5. Acceleration time histories; A6, input motion; AI, response of superstructure to input motion with agmax.
279
6.8
a
x No.IV
x
6.1
No. III
~ 5.4 v,,_ 4.7
4.0 0
5b
260 agmax (gal)
Fig. 6. Apparent natural frequencies versus peak acceleration of input motion. by using the switch at the foundation base it was found that the uplift of the foundation base occurs under a relatively low level of input motion. The earthquake simulating tests of the free vibration induced by striking had also been performed for the purpose of directly measuring the behavior of the natural vibration of the model structure under different foundation conditions. The results are listed in Table 1. Therefore the phenomenon of the damping generated in the structure with buffed foundation being lower than in the structure with surface foundation discovered in earthquake-simulation tests is proved by the free vibration test. However, this is in contradiction to the expectation of the elastic half space theory. The amplification factors (~) versus agmaxfor tests No. III and No. IV are shown in Figs 7 and 8. Comparison of the results of No. III with those of No. IV indicates that the response of the model structure with the buffed foundation is higher than that with the surface foundation under identical input motion. This may be due to the fact that the natural frequency of the structure with the buried foundation is more close to the predominant frequency of the input motion (6-85 Hz), with the lower damping of the structure with the buffed foundation as mentioned above. With respect to the response of the foundation, Fig. 7 along with Fig. 8 shows that the response of the foundation is independent of whether the foundation is buried or ground surface supported and differs little from that in the scattering field, with the exception of the rocking and vertical components of the response under a higher level of input motions. Table 1. Behavior of natural vibration of model structure under different foundation conditions
Foundation condition Absolutely fixed Buried in model ground Supported on the model ground
Fund. freq. (Hz) 6"75 6"37= 6.12
Damping ratio (%) 0.48 1.73 4.70
aFrom the earthquake-simulation tests this value is 6.60.
280
X. Jianguo, W. Danmin, F. Tieming
o
÷
~J'4 ~,,,,4 ¸
x
x
0
=
0 2"
0
50
1O0 agmax (gal)
150
Fig. 7. Amplification factor versus peak accelerations of input motion (No. IV). A detailed introduction to the test results can be found elsewhere. 26 Comparison between results of tests and calculations The proposed analysis model is applied to the test conditions. In the numerical analysis the response in the scattering field measured in the earthquake-simulation tests, was adopted as the input motion. The parameters of the foundation impedance used in the numerical calculation for the tested soil-structure interaction model are: The initial (before uplift occurs) spring constants of the soil (the real part of the impedance)
t
o
200
•
50
1()0 agmax (gad
150
2()0
Fig. 8. Amplification factor versus peak accelerations of input motion (No. III). agmax are given in Fig. 9. The comparisons illustrate that not only the peak values but also the whole process obtained by calculation and experiments agree well. The proposed analysis model has also been applied to an air-blast loading test, and satisfactory agreement between the results of numerical analysis and model test have also been obtained. 31 CONCLUSIONS 1. From the earthquake simulation tests it indicates: (a) The soil-structure system reaches the nonlinear state under a relatively low level of input motion.
Kz = 7"22 x 106 N / m 350 [
for vertical translation
.......
Max. 256.7 (1.64s) Max. 243.7 (1.64s)
Measured Computed
Kx = 6.56 x 106 N / m
f
for horizontal translation KR = 1.12 x 1 0 S N m / r a d
-350 ~ 0
for rocking. The initial dashpot constants:
CE = 1'24 X 104Ns/m for vertical translation
1.0
-~ tO 550 vO) tO
.......
2.0
3.0
4.0
Time (s)
5.0
6.0
7.0
Max. 471.3 (1.64s) Max. 438.2 (1.64s)
Measured Computed
Cx = 1.17 x 104Ns/m for horizontal translation
< -550
CR = 20.5 N m s/rad for rocking. These were obtained by using expressions (1)-(23) and by adopting the initial shear modulus of the soil Go = 5.05 x 106Pa and Poisson's ratio v = 0.4. The related parameters of the superstructure were given in the beginning of the present section. Typical results obtained by experiments and calculations showing the horizontal acceleration response of the model structure under input motions with different
0
1.0
850, ...... ; ! [i
2.0
3.0
4.0
Time (s)
Measured Computed
5.0
6.0
7.0
Max. 652.8 (1.45s) Max. 689.7 (1.44s) (agmax = 127.0)
I
-850
0
1.0
2.0
3.0
4.0
Tirne (s)
5.0
6.0
7.0
Fig. 9. Comparisons between acceleration responses of superstructure obtained from tests and calculations.
Nonlinear soil-structure interaction analysis
(b) Even under the nonlinear state the soil-structure system has an apparent behavior of natural vibration. With increasing input motion level the apparent natural frequencies decrease and damping ratios increase. (c) Under the action of a vertically propagating SH wave due to the nonlinear SSI effect the amplification factor of the horizontal acceleration of the structure response decreases and the vertical and rocking accelerations increase slightly. (d) Comparing with the surface foundation case the embedment of the foundation leads to an increase in the natural frequencies of the system. With increasing input motion level the apparent natural frequency of the system with the buried foundation decreases more slowly than that with the surface foundation. In certain circumstances, for instance, when the fundamental natural frequency of the system with the buried foundation is closer to the predominant frequency of the input motion, the response o f the system will be higher than that with the surface supported foundation. However, with increasing input motion level the nonlinear effect increases and the embedment effect decreases. (e) For the test case performed the nonlinearities resulting from the loss of contact between the foundation and its surrounding soil have a more significant effect on the structure response than the nonlinearities associated with inelastic soil behavior. 2. The modified lumped parametric model proposed takes into account nonlinearities associated with the inelastic soil behavior and those resulting from the loss of contact between the foundation and surrounding soil. In consideration of the latter nonlinearities the two pairs of sidewalls of a buried rectangular foundation had been treated in a different manner. Application of the proposed analysis model to the earthquake simulation tests shows that the proposed analysis model yields excellent results of structure response to low or moderate level of ground shaking and also strong. Finally, the proposed analysis model is first validated through model tests with model structure and model ground with one kind of definite characteristics under the action of definite excitation; in order to make the analysis model applicable to a wide range of soil conditions, structure behavior and earthquake loadings it is necessary to carry out additional tests. Moreover, a lot of work has to be done in future studies with respect to further modification of the proposed model, in particular, the simulation of two kinds of nonlinearities, the radiation damping, the circular foundation and so on.
ACKNOWLEDGEMENTS
This study is supported by the Joint Foundation of Seismology Science o f China (No. 90147). The authors
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are indebted to Mrs Gong Zhenpeng, Shen Qingliang and Wang Jichun of the Institute of Engineering Mechanics, SSB for their cooperation in conducting the shaking table tests. REFERENCES
1. Schnabel, P.B., Lysmer, J. & Seed, H.B. SHAKE - - a computer program for earthquake response analysis of horizontally layered sites. Report EERC 72-12, 1972. 2. Lysmer, J., Udako, T., Seed, H.B. & Hwang, R.N. LUSH - - a computer program for complex response analysis of soil structure systems. Report EERC 74-4, 1974. 3. Ferritto, J.M. SSI and soil models, Shock and Vibration Bulletin, No. 52, part 5, 29-38, 1982. 4. Wolf, J.P. SSI analysis in time domain. Nuclear Engineering and Design, 1989, 111, 381-93. 5. Toki, K. & Miura, F. Nonlinear seismic response analysis of SSI systems. Earthquake Engineering and Structure Dynamics, 1983, 11, 77-89. 6. Miura, F. Dynamic stability analysis of rigid structures against sliding and overturning during strong earthquake motions. Proc. 8 WCEE, Vol. III. Prentice Hall, NJ, USA, 1984, 485-92. 7. Desai, C.S. & Zaman, M.M. Influence of interface behavior in dynamic SSI. Proc 8 WCEE, Vol. III. Prentice Hall, NJ, USA, 1984, 1049-56. 8. Zhao Zhendong, Cai Yongen & Xiong Jianguo, Sliding and uplift response of high-rise buildings to explosive loading. Explosion and Shock Waves, 1991, 11(4), 301-30 (in Chinese). 9. Hayashi, Y., Fakuwa, N., Nakai, S. & Kouanagi, Y. Earthquake response analysis considering soil-structure separation using contact element and dynamic flexibility of soil in time domain. Proc. 9 WCEE, Vol. VIII. 1988, 26570. 10. Aihara, S. et al. Study on the contact ratio of base mat of reactor buildings considering nonlinear SSI effects. Trans. 7 SMiRT, 1973, K9/ll. 11. Penzien, J. Closure statement special theme Session SD. Proc. 9 WCEE, Vol. VIII, 379. 12. Kennedy, R.P. et al. Effect of nonlinear SSI to base slab uplift on the seismic response of a high-temperature gascooled reactor. Nuclear Engineering and Design, 1976, 38, 323-355. 13. Wolf, J.P. SSI with separation of base mat from soil (lifting off), Nuclear Engineering and Design, 1976, 38, 357-384. 14. Sotomura, K. et aL Nonlinear response analysis of reactor building subjected to horizontal and vertical motions simultaneously. Trans. 7 SMiRT, 1983, K5/6. 15. Kobori, T. et al. Nonlinear uplift behavior of soilstructure system with frequency dependent characteristics. Proc. 8 WCEE, Vol. III. Prentice Hall, N J, USA, 1984, 897-904. 16. Ohtomo, K. & Iwatate, T. Proposal of analytical method on foundation uplift considering the damping effect. Proc. 9 WCEE, Vol. VIII. 1988, 277-82. 17. Unemori, A.L. & Whitman, R.V. Nonlinear inelastic dynamic analysis with soil flexibility in rocking. PB-256, 794. Shannon & Wilson Inc, 1976. 18. Izumi, M. et al. Rocking vibration considering uplift and yielding of supporting soil. Proc. 4th Japanese Earthquake Engineering Symposium, 1975, pp. 727-34. 19. Zinn, R. & Stangenberg, F. Base mat uplift effects on the seismic-response of a rectangular nuclear building. Trans. 7 SMiRT, 1983, K9/3.
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X. Jianguo, W. Danmin, F. Tieming
20. Weissman, K. & Prevost, J.H. Centrifugal modelling of dynamic SSI. National Centre for Earthquake Engineering Research Tech. Report, NCEER-89-0040, 1989. 21. Tang, H.T. et al. A large-scale SSI experiment, design and construction. Nuclear Engineering and Design, 1989, 111, 371-9. 22. Tajimi, H. State-of-the-art report: seismic observations and vibration tests for verification of SSI analysis method, Proc. 9 WCEE, 1989, Vol. III, 297-308. 23. Ishimura, K., Swaying-rocking model for simulating nonlinear response of sandy deposit with structure. Proc. 10 WCEE, Vol. III. Balkem, Rotterdam, 1992, 1897-903. 24. Xiong Jianguo, Gao Weijian et al. Overturning resistance and rotation of foundation in soil under lateral loading. Computers and Geotechnics, 1989, 8, 157-74. 25. Xiong Jianguo, Li Tong & Zhai Tong, Dynamics of liquid storage tanks considering soil flexibility and base uplift. Proc. 9th European Conf. on Earthquake Engineering, Moscow, Vol. 7B, 66-75. Kucherenko TSNIISK, Moscow, 1990.
26. Xiong Jianguo, Wang Danmin & Fu Tieming, Nonlinear soil-structure interaction analysis. Research Report, Institute of Engineering Mechanics, SSB, Harbin, 1992 (in Chinese). 27. Pais, A. & Kausel, E. Approximate formulas for dynamic stiffness of rigid foundations. Soil Dynamics and Earthquake Engineering, 1988, 7(4), 213-27. 28. Gazetas, G. Analysis of machine foundation vibrations: state of the art. Soil Dynamics and Earthquake Engineering, 1983, 2(1), 2-41. 29. Gazetas, G. Simple physical models for foundation impedances. In Dynamic Behaviour of Foundations and Buried Structures, ed. P.K. Banerjee & R. Buttertield. Elsevier, Amsterdam, pp. 45-93. 30. Beucke, K. et al. Nonlinear soil-structure interaction with base uplift. Trans. 7 SMiRT, K9/2, 1983. 31. Xiong Jianguo, Gao Weijian et al. Study on overturning of highrise building with basement under airblast loading. Research Report, Institute of Engineering Mechanics, SSB, Harbin, 1992 (in Chinese).