- Email: [email protected]
Effects of backfill soils and incident wave types on seismic response of a reinforced retaining wall
Pergamon
. _
Copyright Q 1994 Elswier BcienceLtd Printedin Great Britain.All rights reserved oc45-7949/94-s7.00 + 0.00
EFFECTS OF BACKFILL SOILS AND INCIDENT WAVE TYPES ON SEISMIC RESPONSE OF A REINFORCED RETAINING WALL Chongbin Zbao and T. P. XII School of Civil Engineering, University of New South Wales, P.O. Box 1, Kensington, N.S.W. 2033, Australia (Received 26 March 1993)
Abstract-Using the finite and infinite element coupled method, the seismic analysis of a reinforced retaining wall has been carried out in this paper. The related numerical results from this study have demonstrated that: (1) the finite and infinite element coupled method is more suitable for the seismic design of a reinforced retaining wall; (2) the stiffness of the backfill soil has a limited effect on the seismic response of the reinforced retaining wall; (3) a vertical seismic P-wave incidence may be the most dangerous condition for the seismic design of a reinforced retaining wall; (4) the types of incident seismic waves have significant effects on the seismic response of a reinforced retaining wall; (5) vertical incident seismic SV-waves and P-waves lead to different patterns of the horizontal acceleration distribution along the height of a reinforced retaining wall.
1. INTRODUCIION
The evaluation of the seismic response wall is a very important topic in the retaining project. Generally, a retaining structure and its surrounding soil
of a retaining design of the wall is a finite
is an infinite medium. Therefore, for the numerical simulation of a soil-retaining wall system, the finite element method is suitable to model the retaining wall as well as the near field of the soil and the infinite elements [l-24] or some other special techniques are useful to consider the infinite extension of the far field of the soil. Since the surrounding soil of the retaining wall is divided into the near field and far field, the nonhomogeneity of the soil in the near field can be easily modelled by finite elements. On the other hand, an earthquake wave usually contains various wave types such as Rayleigh wave, SH-wave, SV-wave and P-wave. These waves originate from the epicentre and propagate from the far field to the retaining wall. Although extensive work on these aspects has been done in the other fields [l-7, 1l-231, numerical simulation of the infinite extension of the soil and the wave propagation in the soil-retaining wall system was scarcely considered in the past for the seismic response of the soil-retaining wall system. There is no doubt that the mechanism of input earthquake waves should be modelled more appropriately in a soilretaining wall system so that realistic numerical results can be obtained. The early and conventional method for the design of a retaining structure was the so-called limit design method, in which the limiting equilibrium mechanics was used to analyse the possible collapse conditions
and suitable safety factors were required to prevent collapse. Since a soil-retaining wall system is a highly indeterminate one, the magnitudes of the forces that act upon the wall cannot be determined from statics alone and such forces are affected by various factors such as the sequence of construction and backfilling operations. To approximately calculate the earth pressures imposed on retaining structures, Coulomb and Rankine’s method was normally employed in the analysis. The procedures for determining the lateral earth pressures on the retaining walls under static conditions can be found in many geotechnical textbooks. For the evaluation of the lateral earth pressures on walls under earthquake conditions, Coulomb and Rankine’s method can still be used by simply including the horizontal and vertical initial forces to account for the earthquake loads. However, Coulomb and Rankine’s method is not exact in the sense that it does not fulfill all the necessary conditions as follows: (1) each point within the soil mass must be in equilibrium so that the pattern of stresses must satisfy the differential equations of equilibrium of the system; (2) the failure condition must not be violated at any point; (3) the strains that occur must be related to the stresses through a stress-strain relationship which is suitable for the soil; (4) the strains that occur at each point must be compatible with the strains at all surrounding points; (5) the stresses within the soil must be in equilibrium with the stresses/forces applied to the soil. Therefore, the finite and infinite element coupled method is more suitable for the seismic design of retaining structures since the above conditions can be satisfied simultaneously in the analysis. Although Zhao et al. [12, 131 have 105
106
Chongbin Zhao and T. P. Xu
H
(a)
Reinforcedretainingwall
a
I-
C
--I
account for all the effects of the original dynamic load exactly. Nevertheless, such a procedure is clearly not practical from the computational point of view. Rather, it seems more pragmatic to cut off or close the series at some frequency, above which effects of the harmonic wave components on the response of the system can be neglected. Fortunately, the cutoff frequency of an earthquake wave is not high so that F.F.T. and 1.F.F.T techniques can be used efficiently for the seismic analysis of structures. This is the reason why the frequency domain analysis technique is preferred in seismology and earthquake engineering. As expected, the current study indicates that the merit in using a finite and infinite element coupled method and related wave input procedure for infinite media lies in that not only can the structure and its natural foundation be simulated accurately but also the wave propagation mechanism can be modelled more realistically. Besides, the dynamic soil-structure interaction effects can also be considered rigorously by means of this method. Some numerical results from this study illustrate that the types of incident waves have significant effects on the seismic response of the retaining wall and hence it should be considered in the seismic design of the retaining project. Although the mechanical properties of backfill soils have considerable effects on the dynamic response of the retaining wall in the high frequency range of harmonic waves, they have limited effects on the seismic response of the retaining wall since all important harmonic wave components in an earthquake wave are usually in the low frequency range.
(b) SimpiIfIed model Fig. 1. Simplified model of a reinforced retaining wall. successfully applied the finite and infinite element coupled method to solve the static and dynamic response of gravity and arch dam systems, no work has been done on the application of this method to the seismic analysis of reinforced retaining walls. Taking the aforementioned facts into account, the finite and infinite element coupled method was developed and the dynamic response of a reinforced retaining wall during harmonic wave incidences was calculated in a previous paper [24]. Since an earthquake wave can be decomposed into several harmonic waves by a F.F.T. (Fast Fourier Transform) technique, the finite and infinite element coupled method can be extended to calculate the seismic response of retaining walls. Thus, the effect of backfill soils and incident wave types on the seismic response of a retaining wall will be investigated in this paper. It is worth noting that from a purely mathematical standpoint, the F.F.T. and I.F.F.T. (Inverse Fast Fourier Transform) techniques are only complete when infinite numbers of harmonic waves are considered in the transient analysis of structures due to dynamic loading. Consideration of all the harmonic wave components in this manner will
2. OUTLINE OF THE FREQUENCY ANALYSIS PROCEDURE
DOMAIN
Using the finite and infinite element method and the related wave input procedure [3,24], it is convenient to calculate the seismic response of a reinforced retaining wall in a frequency domain. A transient incident earthquake wave can be decomposed into a series of harmonic waves [25]. The amplitude of each harmonic wave can be expressed as +a ii,(o) =
ii,(t) e-‘“‘dt, s -z
(1)
where ii,(t) is the incident acceleration wave, w is the circular frequency of a decomposed harmonic wave, O,(W) is the Fourier spectrum amplitude of the decomposed harmonic wave. Equation (1) is the so-called Fourier transform integral and the inverse Fourier integral can be expressed as
:I
ii,(t) = i2n
ii,(w) eiw’dw.
(2)
s After the incident acceleration wave is decomposed, a unit harmonic wave with the circular frequency w can be employed to calculate the complex frequency
Seismic response of a reinforced retaining wall response function, Hj(o), for station j at the reinforced retaining wall. A detailed description about how to obtain the complex frequency response function was given in [24]. In order to obtain the transient response for station j, the harmonic response due to a given decomposed harmonic acceleration wave with amplitude o,(w) and circular frequency (0, should be obtained
iiRj(w)
=
H,(w)ii,(w),
(3)
where OR,(CD)is the acceleration response for station j at the reinforced retaining wall due to the incidence of a decomposed harmonic acceleration wave which has circular frequency o, and the amplitude of Fourier spectrum, 0, (w). By superposition of the responses for all decomposed harmonic waves, the seismic response for station j at the reinforced retaining wall can be obtained as
ii,,(t)J2n
s +0X
-*
ii, (w) eiWtdw,
(4)
where r&(i) is the acceleration response of station j at the reinforced retaining wall. In the numerical analysis, the discretized Fourier transform is used. The incident acceleration C,(l) is usually assumed to be periodic with a finite period T. In order to make use of the F.F.T. technique, the period T is divided into n equal intervals of Ar, where n is selected as a power of 2. The circular frequency o is also divided into the same number of intervals Aw as At. The lowest circular frequency and the highest circular frequency in the analysis can be expressed as
(5) where IX,,,,, is called the cut-off frequency or the Nyquist frequency which is related to the interval of time At and can be decided by the characteristics of the incident earthquake wave, Bydeliningtk =kAt =kT/n(k =0,1,2,...,n -1) and o,=qAw=q2z/T(q=O,1,2 ,..., n-l), eqns (1) and (4) can be further expressed as n-l
ul(w,) = At 2 ii,
es2”(qk/“)
k=O
(q = 0, 1,2, *. . , n - I), I&, (tk) = g
#g’(Ia,
(6)
(wq) e2rriwo
9-O
(k =0,1,2
,...,
n - I),
(7)
107
where the mean of each term is the same as above. For the purpose of a general study of the effects of backfill soils and incident wave types on the seismic response of a reinforced retaining wall, it seems that one can select an arbitrary earthquake motion. Therefore, the acceleration time history of an S25W component of the Parkfield, California Earthquake is chosen as the incident earthquake wave for the present study. This earthquake took place on June 27, 1966 and the peak value of its S25W acceleration component at station 97 is - 3.408 mjsec2. For the convenience of the analysis, this acceleration time history has been normalized by dividing the absolute value of its peak magnitude, 3.408, thus resulting in a unit acceleration wave. Of course, this unit wave has the same frequency acceleration characteristics as the original one. In order to make use of the F.F.T. technique, an equal time interval At = 0.02 set is adopted and 1024 pairs of data, which is 2 to power 9, are selected. The lowest circular frequency and highest circular frequency in this case are w,~” = 0.3068 rad/sec and w,,, = 157.08 radjsec, respectively. The predominant circular frequency is about 17 rad/sec (the corresponding predominant period is 0.37 set).
3. EFFECTS OF BACKFILL SOIL AND INCIDENT WAVE TYPE ON THE SEISMIC RESPONSE OF A RETAINING WALL It is well recognized that a reinforced retaining wall is usually very long in its longitudinal direction and the dynamic analysis of such retaining walls can be attributed to the analysis of plane strain problems. This treatment will greatly simplify the calculation so that the requirement for computer efforts will be drastically reduced. As shown in Fig. 1, a typical retaining wall with back reinforcements is made of concrete material. The portion of the system between two adjacent reinforcements is employed in the analysis and the distance between two reinforcements is defined as the analytical thickness of the wall. Supposing a real reinforced retaining wall system (Fig. l(a)) is represented by an equivalent wall with the same analytical thickness (Fig. l(b)), the equivalent wall can be divided into two subregions according to its material properties. The first one is a concrete subregion consisting of the wall and its footing and the second one is an equivalent material subregion consisting of concrete and backfill soil materials. The mechanical properties of the equivalent material can be evaluated using the weighted average method as proposed in a previous study [24]. Since the seismic analysis of a real reinforced retaining wall is simplified to the seismic anaiysis of a plane strain problem, the well-develops numerical method for wave scattering problems in infinite media can be used to solve the problem because the seismic response of the retaining wall, as a matter of fact, is a wave scattering problem due to the existence of the
108
Chongbin Zhao and T. P. Xu
@Urn b=O.3m c=6.3m H=lOm
+w. Hp3Om IV Wave input bow
Neirl
soil
Pr A
Incidentwave <>
t V
Rock
Fig. 2. Computing model of a soil-retaining wall system.
wall. It is the superposition of the incident wave field and the scattered wave field that results in the total seismic response of the retaining wall. Therefore, in order to obtain a better ~derstanding of the mechanisms of the seismic response of the retaining wall, wave motion theory should be used in the analysis. Keeping this in mind, the finite and infinite element coupled method and the related wave input procedure for infinite media are employed in this section. The main merit in using a finite and infinite element coupled method is that not only can the natural soil and the retaining wall be simulated but also the wave propagation mechanism in the system can be modelled more realistically. Since a detailed study on the finite and infinite element coupled method as well as the related wave input procedure was carried out elsewhere [l-5,21,24], it is not necessary to repeat it here, for the sake of saving space. As shown in Fig. 2, the whole soil-retaining wall system is divided into five subregions. The first and second subregions are the concrete subregion and equivalent material subregion in which the material properties can be determined by means of the method discussed in the previous study 1241.The third one is the backfill soil subregion which is generally comprised of looser materials. The fourth one is the natural soil subregion and the fifth one is the rock subregion where the epicentre of the earthquake is usually located. Moreover, in order to use the finite and infinite element coupled method more efficiently, the whole system is also divided into the near field and far field. In this case, the near field is comprised of the retaining wall (the first and second subregions), the backfill soil (the third subregion) and a part of
the natural soil; whereas the far field is comprised of the rest of the natural soil and the rock (the fifth subregion). It is noted that the near field is modelled using finite elements where the requirement for the size of elements [7] is considered to simulate the wave propagation and the far field is modelled using infinite elements where the wave propagation mechanism in the infinite medium is expressed rigorously. The following parameters are used in the analysis. For the rock and concrete materials, the elastic modulus is 28.8 x lo6 kPa, Poisson’s ratio is 0.2, the unit weight is 24 kN/m3 and the corresponding S-wave velocity is 2236m/sec. For the natural soil, the elastic modulus is 0.252 x lo6 kPa, Poisson’s ratio is 0.3, the unit weight is 20 kN/m3 and the corresponding S-wave velocity is 220 m/see. For the backfill soil, the unit weight is 20 kN/m3, Poisson’s ratio is 0.4 and two different elastic moduh, namely 0,056 x lo6 kPa (soft backfill soil) and 0.224 x lo6 kPa (stiff backfill soil) are used to investigate the effects of the backfill soil on the seismic response of the retaining wall. The analytical thickness of the wall is chosen as T = 5 m and the thickness of the reinforcement T, = 0.5 m. It is stated that in order to study how the types of incident waves and the backfill soil affect the seismic response of the retaining wall, and only for this purpose, the unit seismic acceleration wave, which is discussed in the second section, is assumed to impinge vertically onto the interface of the soil and rock as either a P-wave or an SV-wave in this study. Figures 3 and 4 show the acceleration response of the reinforced retaining wall due to the seismic P-wave vertical incidence for the stiff backfill soil and
Seismic response of a reinforced retaining wall
6
10
3
5
0
0
-3
-5 -10
-6 0
10
5
20
15
6
t(s)
t(s) (Station a,h/~> 6-
10
3-
5
o-
0
-3-
-5
4 *O
I 5
8
I 10
s
I 1s
a
-10
20
0
5
10
15
20 t(s)
t(s) (Station 69)
Fig. 3. Acceleration response of the retaining wall due to seismic P-wave incidence (stiff backfill soil).
soft backfill soil situations respectively. Figures 5 and 6 show the corresponding Fourier spectra of the wall response. In these figures, a, and a,, are the horizontal and vertical accelerations of the wall; FS, and FS,, are the Fourier spectra in the horizontal and vertical directions; Stations 1,2 and 69 represent the top, middle and bottom positions of the wall. It is observed that the overall trend of the response of the retaining wall in the case of the stiff backfill soil is very similar to that of the retaining wall with the soft backfill soil. This indicates that the stiffness of the backfill soil has very limited effect on the seismic response of the retaining wall since it has limited influence on the dynamic characteristics of the
soil-retaining wall system and all the important harmonic wave components in the input earthquake acceleration wave for this study are located in the low frequency range. If the amplitude of the total acceleration at any point of the wall is defined as A, = Jm, the amplitudes of the total accelerations at the top of the wall (Station 1) are 10.69 and 10.93 m/set* for the stiff backfill soil (Fig. 3) and soft backfill soil (Fig. 4), which occur at t = 4.8 set and t = 4.64 set respectively. Some slight differences between the Fourier spectra due to the stiff backfill soil and soft backfill soil appear at one of the resonant circular frequency of the soil-wall system, which is about 20 rad/sec in this study. It is evident
Chongbin Zhao and T. P. Xu
6 0 -6
8 4
0
4
-8
0
8,
5
10
IS
20
ax 6
0 -6
i
0
5
10
1s
20
w
0
5
10
15
t(s) (Station699)
Fig. 4. Ace&ration response of the retaining wall due to seismic P-wave in&dense (sofi backfill soil},
Seismic response of a tinforced retaining wail
111
FS@/s)
12. I
6
0 0
“,%q-Y 40
t 80
,
140
’ 1 1
o(rad/s) ‘)FS,(m/s)
6
6
6.
6
.
0,
/
40
Fig. 5. Fourier spectra for responses of the retaining wall due to shnic soil).
80
120
1643
P-wave incidence (stiff backfill
112
Chongbin Zhao and T. P. Xu
FS,h/s)
12-
I
0
1 40
80
120
0
1’60
6-
40
80
1 3
120
6-
0
40
80
120
160
0
40
80
1 D
120
o(rud,
@W/S)
:)
(Station 1
F&h/s)
F!
12
i,
40
80
120
1 3
ohd/s> (Station 69) Fig. 6. Fourier spectra for responses of the retaining wall due to seismic P-wave incidence (soft backfill soil).
Seismic response of a reinforced retaining wall
113
a&M)
5-f
2.5o-2.5-
’ ‘(Station 2)
, LI_
-2.5-
w
t(s) (Station69)
Fig. 7. Acceleration response of the retaining wall due to seismic SV-wave incidence (stiff backfill soil).
that under the seismic P-wave vertical incidence, the induced horizontal acceleration at the top of the reinforced retaining wall is much greater than that at the bottom of the reinforced retaining wall, even though the reinforced retaining wall is very stiff. The acceleration differences along the height of the wall will give rise to rotation and bending moments within the wall. Therefore, it can be concluded that the vertical seismic P-wave incidence may be the most dangerous condition for the seismic design of the reinforced retaining wall since both the induced shear force and bending moment might approach their maximum values in this situation.
Under the seismic SV-wave vertical incidence, the acceleration responses of the reinforced retaining wall for the stiff backfill soil and soft backfill soil are shown in Figs 7 and 8, while the corresponding Fourier spectra are shown in Figs 9 and 10. As can be seen from these results, the overall response of the reinforced retaining wall due to the seismic SV-wave incidence is weaker than that due to the seismic P-wave incidence so that the above conclusion is further confirmed. In the case of the seismic SV-wave incidence, nearly uniform distributions of both horizontal and vertical accelerations can be observed along the height of the wall. This fact indicates that
Chongbin Zhao and T. P. Xu
o-2.5- +
_2.5v---j
4-j
-5 : 0
I
, 5
I
I 15
Ib
I 20
t(s) (Station 2) 5 a,C+?
54a&&3
2.5-
2.5-
o-
o-
-2.5-
-2.5-
,
-5, 0
, 5
I
I 10
15
’
.
-5 20
0
tw
I 5
v
I 10
8
I 15
8 20 tw
(Station 69) Fig. 8. Acceleration response of the retaining wall due to seismic SV-wave incidence (soft backfill soil).
115
Seismic response of a reinforced retaining wall
3
1.5
0
&adls) 3
1.5
0
F&h/4
&.
(Station 2) FS,(m/s) 3
1
40
80
120
1
Fig. 9. Fourier spectra for responses of the retaining wall due to seismic SV-wave incidence (stiff backfill soil).
Chongbin Zhao and T. P. Xu
116
0
0
40
80
120
40
80
120
1’
160
w0-4s)
whds)
. (Station 2)
3
1.5
0
FSAm/s)
FWm/s)
L. 40
80
120
I
40
80
120
1
Fig. IO. Fourier spectra for responses of the retaining wall due to seismic W-wave incidence (soft backfill soil).
the effect of the induced rotation and bending moment may be unimportant to the seismic design of a reinforced retaining wall under seismic SV-wave vertical incidences. Furthermore, both the maximum magnitude and distribution of Fourier spectra under the seismic SV-wave incidence are obviously different from those of Fourier spectra under the seismic P-wave incidence. Thus, it is suggested that the type of incident seismic wave be considered carefully for the seismic design of a reinforced retaining wall since it has a significant effect on the seismic response of reinforced retaining walls.
4. CONCLUSIONS
Using the finite and infinite element coupled method and the F.F.T. technique, the seismic analysis of a reinforced retaining wall has been carried out in this paper. From the related numerical results, the following conclusions have been obtained. (1) The finite and infinite element coupled method is more suitable for the seismic design of a reinforced retaining wall since all the important factors and necessary conditions can be simultaneously satisfied in the analysis.
117
Seismic response of a reinforced retaining wall (2) The stiffness of the backfill soil has a limited effect on the seismic response of the reinforced retaining wall since it has limited influence on the dynamic characteristics of the soil-retaining wall system. (3) A vertical seismic P-wave incidence may be the most dangerous condition for the seismic design of the reinforced retaining wall since both the induced shear force and bending moment might approach their maximum values at this situation. (4) The types of incident seismic waves must be considered carefully for the seismic design of a reinforced retaining wall since they have significant effects on the seismic response of the reinforced retaining wall. (5) A vertical incident seismic SV-wave leads to the nearly uniform distribution of the horizontal and vertical accelerations along the height of the retaining wall. However, a vertical incident seismic P-wave leads to a nearly uniform distribution of the vertical acceleration along the height of the retaining wall but an obviously nonuniform distribution of the horizontal acceleration along the height of the retaining wall.
REFERENCES
1. C. Zhao, C. Zhang and G. Zhang, Analysis of 3D foundation wave problems by mapped dynamic infinite elements. Sci. Chinn (Series A) 32, 469-491 (1989). 2. C. Zhao and S. Valliappan, Vibration of threedimensional rigid raft foundation on viscoelastic medium. Earthqu. Engng struct. Dyn. 20, 1159-1177 (1991). 3. C. Zhao, A new input model of seismic waves for infinite foundations of arch dams and other large structures. J. Rock Mech. Engng 7, 248-256 (1988). 4. S. Valliappan and C. Zhao, Dynamic response of concrete gravity dams including dam-water-foundation interaction. Int. J. Numer. Anal. Geomech. 16, 19-99 (1992). 5. C. Zhao and S. Valhappan, A numerical model for wave scattering problems in infinite media due to P and SV wave incidences. Int. J. Numer. Meth. Engng 33, 1661-1682 (1992). 6. C. Zhang and C.’ Zhao, Coupling method of finite and infinite elements for strip foundation wave problems. Earthqu. Engng struct. Dyn. 15, 839-851 (1987). 7. C. Zhao, C. Zhang and G. Wang, A modelling of wave propagation in semi-infinite domain using dynamic infinite elements. Science Report of Tsinghua University, TH88003, Beijing (1988).
8. P. Bettess, Infinite elements. Int. J. Nurner. Meth. Engng 11, 53-64 (1977). 9. Y. K. Chow and I. M. Smith, Static and periodic infinite solid elements. ht. J. Numer. Meth. Engng. 17, 503-526 (1981).
10. F. Medina and R. L. Taylor, Finite element techniques for problems of unbounded domains. Int. J. Numer. Meth. Engng 19, 699709 (1983). 11. C. Zhao, C. Zhang and G. Zhang, Study on the characteristics of mapping dynamic infinite elements. Earthqu. Engng Engng Vibr. 7(3), l-5 (1987).
12. C. Zhao, C. Zhang and G. Zhang, Simulation of semi-infinite plane elastic foundation using infinite elements. J. Tsinghua Univ. 26, 51-64 (1986): 13. C. Zhao, C. Zhang and G. Zhang, Simulation of arch dam foundation using infinite elements. J. Hydruul. Engng No. 2, 27-36 (1987).
14. C. Zhao. C. Zhane and G. Zhane. Analvsis of free field motions on arch dam canyons and the effects due to existence of weathered rock strata. J. Hydraul. Engng No. 11, 29-40 (1988). 15. C. Zhao, C. Zhang, G. Wang and G. Zhang, Effects of the characteristics of soil layers on field seismic response using infinite elements. J. Scientif. Res. Hydraul. Engng I,
.
Transport. No. 4, 31-45 (1988).
16. C. Zhao, C. Zhang and G. Zhang, Infinite element and its development. J. Shaanxi Inst. Mech. Engng No. 1, 89-97 (1989).
17. C. Zhao and C. Zhang, Analysis of the differences between finite and infinite model for infinite media. J. Scientif. Res. Hydraul. Engng Transport. No. 2, 143-153 (1990). 18. C. Zhao, C. Zhang and G. Zhang, Dynamic simulation for infinite foundation. J. Hydraul. Engng No. 6, 20-27 (1990). 19. C. Zhao and S. Valliappan, An approximate method for simulating infinite media. Comput. Struct. 41, 104-1049 (1991). 20. C. Zhao, C. Zhang and G. Zhang, A dynamic model for elastic and v&o-elastic infinite foundations. J. Geotechn. Engng 13, 1l-23 (1991).
21. C. Zhao and S. Valliappan, An efficient wave input procedure for infinite media, Cornman. appl. Numer. Meth. 9, 407-415
(1993).
22. C. Zhao and S. Valliappan, Incident P and SV wave scattering effects under different canyon topographic and geological conditions. Int. J. Numer. Anal.-Geomech. 17. 73-94 (1993).
23. CZhao and S. ‘Valliappan, Transient infinite elements for seepage problems in infinite media. Int. J. Numer. Anal. Geomech. 17. 323-341 (1993). 24. C. Zhao and S. Valliappan: Dynamic analysis of a reinforced retaining wall using finite and infinite element coupled method. Comput. Struct. 47, 239-244 (1993). 25. J. P. Wolf, Dynamic Soil-Structure Interaction in Time Domain, Prentice-Hall, Englewood Cliffs, New Jersey (1988).
. _
Copyright Q 1994 Elswier BcienceLtd Printedin Great Britain.All rights reserved oc45-7949/94-s7.00 + 0.00
EFFECTS OF BACKFILL SOILS AND INCIDENT WAVE TYPES ON SEISMIC RESPONSE OF A REINFORCED RETAINING WALL Chongbin Zbao and T. P. XII School of Civil Engineering, University of New South Wales, P.O. Box 1, Kensington, N.S.W. 2033, Australia (Received 26 March 1993)
Abstract-Using the finite and infinite element coupled method, the seismic analysis of a reinforced retaining wall has been carried out in this paper. The related numerical results from this study have demonstrated that: (1) the finite and infinite element coupled method is more suitable for the seismic design of a reinforced retaining wall; (2) the stiffness of the backfill soil has a limited effect on the seismic response of the reinforced retaining wall; (3) a vertical seismic P-wave incidence may be the most dangerous condition for the seismic design of a reinforced retaining wall; (4) the types of incident seismic waves have significant effects on the seismic response of a reinforced retaining wall; (5) vertical incident seismic SV-waves and P-waves lead to different patterns of the horizontal acceleration distribution along the height of a reinforced retaining wall.
1. INTRODUCIION
The evaluation of the seismic response wall is a very important topic in the retaining project. Generally, a retaining structure and its surrounding soil
of a retaining design of the wall is a finite
is an infinite medium. Therefore, for the numerical simulation of a soil-retaining wall system, the finite element method is suitable to model the retaining wall as well as the near field of the soil and the infinite elements [l-24] or some other special techniques are useful to consider the infinite extension of the far field of the soil. Since the surrounding soil of the retaining wall is divided into the near field and far field, the nonhomogeneity of the soil in the near field can be easily modelled by finite elements. On the other hand, an earthquake wave usually contains various wave types such as Rayleigh wave, SH-wave, SV-wave and P-wave. These waves originate from the epicentre and propagate from the far field to the retaining wall. Although extensive work on these aspects has been done in the other fields [l-7, 1l-231, numerical simulation of the infinite extension of the soil and the wave propagation in the soil-retaining wall system was scarcely considered in the past for the seismic response of the soil-retaining wall system. There is no doubt that the mechanism of input earthquake waves should be modelled more appropriately in a soilretaining wall system so that realistic numerical results can be obtained. The early and conventional method for the design of a retaining structure was the so-called limit design method, in which the limiting equilibrium mechanics was used to analyse the possible collapse conditions
and suitable safety factors were required to prevent collapse. Since a soil-retaining wall system is a highly indeterminate one, the magnitudes of the forces that act upon the wall cannot be determined from statics alone and such forces are affected by various factors such as the sequence of construction and backfilling operations. To approximately calculate the earth pressures imposed on retaining structures, Coulomb and Rankine’s method was normally employed in the analysis. The procedures for determining the lateral earth pressures on the retaining walls under static conditions can be found in many geotechnical textbooks. For the evaluation of the lateral earth pressures on walls under earthquake conditions, Coulomb and Rankine’s method can still be used by simply including the horizontal and vertical initial forces to account for the earthquake loads. However, Coulomb and Rankine’s method is not exact in the sense that it does not fulfill all the necessary conditions as follows: (1) each point within the soil mass must be in equilibrium so that the pattern of stresses must satisfy the differential equations of equilibrium of the system; (2) the failure condition must not be violated at any point; (3) the strains that occur must be related to the stresses through a stress-strain relationship which is suitable for the soil; (4) the strains that occur at each point must be compatible with the strains at all surrounding points; (5) the stresses within the soil must be in equilibrium with the stresses/forces applied to the soil. Therefore, the finite and infinite element coupled method is more suitable for the seismic design of retaining structures since the above conditions can be satisfied simultaneously in the analysis. Although Zhao et al. [12, 131 have 105
106
Chongbin Zhao and T. P. Xu
H
(a)
Reinforcedretainingwall
a
I-
C
--I
account for all the effects of the original dynamic load exactly. Nevertheless, such a procedure is clearly not practical from the computational point of view. Rather, it seems more pragmatic to cut off or close the series at some frequency, above which effects of the harmonic wave components on the response of the system can be neglected. Fortunately, the cutoff frequency of an earthquake wave is not high so that F.F.T. and 1.F.F.T techniques can be used efficiently for the seismic analysis of structures. This is the reason why the frequency domain analysis technique is preferred in seismology and earthquake engineering. As expected, the current study indicates that the merit in using a finite and infinite element coupled method and related wave input procedure for infinite media lies in that not only can the structure and its natural foundation be simulated accurately but also the wave propagation mechanism can be modelled more realistically. Besides, the dynamic soil-structure interaction effects can also be considered rigorously by means of this method. Some numerical results from this study illustrate that the types of incident waves have significant effects on the seismic response of the retaining wall and hence it should be considered in the seismic design of the retaining project. Although the mechanical properties of backfill soils have considerable effects on the dynamic response of the retaining wall in the high frequency range of harmonic waves, they have limited effects on the seismic response of the retaining wall since all important harmonic wave components in an earthquake wave are usually in the low frequency range.
(b) SimpiIfIed model Fig. 1. Simplified model of a reinforced retaining wall. successfully applied the finite and infinite element coupled method to solve the static and dynamic response of gravity and arch dam systems, no work has been done on the application of this method to the seismic analysis of reinforced retaining walls. Taking the aforementioned facts into account, the finite and infinite element coupled method was developed and the dynamic response of a reinforced retaining wall during harmonic wave incidences was calculated in a previous paper [24]. Since an earthquake wave can be decomposed into several harmonic waves by a F.F.T. (Fast Fourier Transform) technique, the finite and infinite element coupled method can be extended to calculate the seismic response of retaining walls. Thus, the effect of backfill soils and incident wave types on the seismic response of a retaining wall will be investigated in this paper. It is worth noting that from a purely mathematical standpoint, the F.F.T. and I.F.F.T. (Inverse Fast Fourier Transform) techniques are only complete when infinite numbers of harmonic waves are considered in the transient analysis of structures due to dynamic loading. Consideration of all the harmonic wave components in this manner will
2. OUTLINE OF THE FREQUENCY ANALYSIS PROCEDURE
DOMAIN
Using the finite and infinite element method and the related wave input procedure [3,24], it is convenient to calculate the seismic response of a reinforced retaining wall in a frequency domain. A transient incident earthquake wave can be decomposed into a series of harmonic waves [25]. The amplitude of each harmonic wave can be expressed as +a ii,(o) =
ii,(t) e-‘“‘dt, s -z
(1)
where ii,(t) is the incident acceleration wave, w is the circular frequency of a decomposed harmonic wave, O,(W) is the Fourier spectrum amplitude of the decomposed harmonic wave. Equation (1) is the so-called Fourier transform integral and the inverse Fourier integral can be expressed as
:I
ii,(t) = i2n
ii,(w) eiw’dw.
(2)
s After the incident acceleration wave is decomposed, a unit harmonic wave with the circular frequency w can be employed to calculate the complex frequency
Seismic response of a reinforced retaining wall response function, Hj(o), for station j at the reinforced retaining wall. A detailed description about how to obtain the complex frequency response function was given in [24]. In order to obtain the transient response for station j, the harmonic response due to a given decomposed harmonic acceleration wave with amplitude o,(w) and circular frequency (0, should be obtained
iiRj(w)
=
H,(w)ii,(w),
(3)
where OR,(CD)is the acceleration response for station j at the reinforced retaining wall due to the incidence of a decomposed harmonic acceleration wave which has circular frequency o, and the amplitude of Fourier spectrum, 0, (w). By superposition of the responses for all decomposed harmonic waves, the seismic response for station j at the reinforced retaining wall can be obtained as
ii,,(t)J2n
s +0X
-*
ii, (w) eiWtdw,
(4)
where r&(i) is the acceleration response of station j at the reinforced retaining wall. In the numerical analysis, the discretized Fourier transform is used. The incident acceleration C,(l) is usually assumed to be periodic with a finite period T. In order to make use of the F.F.T. technique, the period T is divided into n equal intervals of Ar, where n is selected as a power of 2. The circular frequency o is also divided into the same number of intervals Aw as At. The lowest circular frequency and the highest circular frequency in the analysis can be expressed as
(5) where IX,,,,, is called the cut-off frequency or the Nyquist frequency which is related to the interval of time At and can be decided by the characteristics of the incident earthquake wave, Bydeliningtk =kAt =kT/n(k =0,1,2,...,n -1) and o,=qAw=q2z/T(q=O,1,2 ,..., n-l), eqns (1) and (4) can be further expressed as n-l
ul(w,) = At 2 ii,
es2”(qk/“)
k=O
(q = 0, 1,2, *. . , n - I), I&, (tk) = g
#g’(Ia,
(6)
(wq) e2rriwo
9-O
(k =0,1,2
,...,
n - I),
(7)
107
where the mean of each term is the same as above. For the purpose of a general study of the effects of backfill soils and incident wave types on the seismic response of a reinforced retaining wall, it seems that one can select an arbitrary earthquake motion. Therefore, the acceleration time history of an S25W component of the Parkfield, California Earthquake is chosen as the incident earthquake wave for the present study. This earthquake took place on June 27, 1966 and the peak value of its S25W acceleration component at station 97 is - 3.408 mjsec2. For the convenience of the analysis, this acceleration time history has been normalized by dividing the absolute value of its peak magnitude, 3.408, thus resulting in a unit acceleration wave. Of course, this unit wave has the same frequency acceleration characteristics as the original one. In order to make use of the F.F.T. technique, an equal time interval At = 0.02 set is adopted and 1024 pairs of data, which is 2 to power 9, are selected. The lowest circular frequency and highest circular frequency in this case are w,~” = 0.3068 rad/sec and w,,, = 157.08 radjsec, respectively. The predominant circular frequency is about 17 rad/sec (the corresponding predominant period is 0.37 set).
3. EFFECTS OF BACKFILL SOIL AND INCIDENT WAVE TYPE ON THE SEISMIC RESPONSE OF A RETAINING WALL It is well recognized that a reinforced retaining wall is usually very long in its longitudinal direction and the dynamic analysis of such retaining walls can be attributed to the analysis of plane strain problems. This treatment will greatly simplify the calculation so that the requirement for computer efforts will be drastically reduced. As shown in Fig. 1, a typical retaining wall with back reinforcements is made of concrete material. The portion of the system between two adjacent reinforcements is employed in the analysis and the distance between two reinforcements is defined as the analytical thickness of the wall. Supposing a real reinforced retaining wall system (Fig. l(a)) is represented by an equivalent wall with the same analytical thickness (Fig. l(b)), the equivalent wall can be divided into two subregions according to its material properties. The first one is a concrete subregion consisting of the wall and its footing and the second one is an equivalent material subregion consisting of concrete and backfill soil materials. The mechanical properties of the equivalent material can be evaluated using the weighted average method as proposed in a previous study [24]. Since the seismic analysis of a real reinforced retaining wall is simplified to the seismic anaiysis of a plane strain problem, the well-develops numerical method for wave scattering problems in infinite media can be used to solve the problem because the seismic response of the retaining wall, as a matter of fact, is a wave scattering problem due to the existence of the
108
Chongbin Zhao and T. P. Xu
@Urn b=O.3m c=6.3m H=lOm
+w. Hp3Om IV Wave input bow
Neirl
soil
Pr A
Incidentwave <>
t V
Rock
Fig. 2. Computing model of a soil-retaining wall system.
wall. It is the superposition of the incident wave field and the scattered wave field that results in the total seismic response of the retaining wall. Therefore, in order to obtain a better ~derstanding of the mechanisms of the seismic response of the retaining wall, wave motion theory should be used in the analysis. Keeping this in mind, the finite and infinite element coupled method and the related wave input procedure for infinite media are employed in this section. The main merit in using a finite and infinite element coupled method is that not only can the natural soil and the retaining wall be simulated but also the wave propagation mechanism in the system can be modelled more realistically. Since a detailed study on the finite and infinite element coupled method as well as the related wave input procedure was carried out elsewhere [l-5,21,24], it is not necessary to repeat it here, for the sake of saving space. As shown in Fig. 2, the whole soil-retaining wall system is divided into five subregions. The first and second subregions are the concrete subregion and equivalent material subregion in which the material properties can be determined by means of the method discussed in the previous study 1241.The third one is the backfill soil subregion which is generally comprised of looser materials. The fourth one is the natural soil subregion and the fifth one is the rock subregion where the epicentre of the earthquake is usually located. Moreover, in order to use the finite and infinite element coupled method more efficiently, the whole system is also divided into the near field and far field. In this case, the near field is comprised of the retaining wall (the first and second subregions), the backfill soil (the third subregion) and a part of
the natural soil; whereas the far field is comprised of the rest of the natural soil and the rock (the fifth subregion). It is noted that the near field is modelled using finite elements where the requirement for the size of elements [7] is considered to simulate the wave propagation and the far field is modelled using infinite elements where the wave propagation mechanism in the infinite medium is expressed rigorously. The following parameters are used in the analysis. For the rock and concrete materials, the elastic modulus is 28.8 x lo6 kPa, Poisson’s ratio is 0.2, the unit weight is 24 kN/m3 and the corresponding S-wave velocity is 2236m/sec. For the natural soil, the elastic modulus is 0.252 x lo6 kPa, Poisson’s ratio is 0.3, the unit weight is 20 kN/m3 and the corresponding S-wave velocity is 220 m/see. For the backfill soil, the unit weight is 20 kN/m3, Poisson’s ratio is 0.4 and two different elastic moduh, namely 0,056 x lo6 kPa (soft backfill soil) and 0.224 x lo6 kPa (stiff backfill soil) are used to investigate the effects of the backfill soil on the seismic response of the retaining wall. The analytical thickness of the wall is chosen as T = 5 m and the thickness of the reinforcement T, = 0.5 m. It is stated that in order to study how the types of incident waves and the backfill soil affect the seismic response of the retaining wall, and only for this purpose, the unit seismic acceleration wave, which is discussed in the second section, is assumed to impinge vertically onto the interface of the soil and rock as either a P-wave or an SV-wave in this study. Figures 3 and 4 show the acceleration response of the reinforced retaining wall due to the seismic P-wave vertical incidence for the stiff backfill soil and
Seismic response of a reinforced retaining wall
6
10
3
5
0
0
-3
-5 -10
-6 0
10
5
20
15
6
t(s)
t(s) (Station a,h/~> 6-
10
3-
5
o-
0
-3-
-5
4 *O
I 5
8
I 10
s
I 1s
a
-10
20
0
5
10
15
20 t(s)
t(s) (Station 69)
Fig. 3. Acceleration response of the retaining wall due to seismic P-wave incidence (stiff backfill soil).
soft backfill soil situations respectively. Figures 5 and 6 show the corresponding Fourier spectra of the wall response. In these figures, a, and a,, are the horizontal and vertical accelerations of the wall; FS, and FS,, are the Fourier spectra in the horizontal and vertical directions; Stations 1,2 and 69 represent the top, middle and bottom positions of the wall. It is observed that the overall trend of the response of the retaining wall in the case of the stiff backfill soil is very similar to that of the retaining wall with the soft backfill soil. This indicates that the stiffness of the backfill soil has very limited effect on the seismic response of the retaining wall since it has limited influence on the dynamic characteristics of the
soil-retaining wall system and all the important harmonic wave components in the input earthquake acceleration wave for this study are located in the low frequency range. If the amplitude of the total acceleration at any point of the wall is defined as A, = Jm, the amplitudes of the total accelerations at the top of the wall (Station 1) are 10.69 and 10.93 m/set* for the stiff backfill soil (Fig. 3) and soft backfill soil (Fig. 4), which occur at t = 4.8 set and t = 4.64 set respectively. Some slight differences between the Fourier spectra due to the stiff backfill soil and soft backfill soil appear at one of the resonant circular frequency of the soil-wall system, which is about 20 rad/sec in this study. It is evident
Chongbin Zhao and T. P. Xu
6 0 -6
8 4
0
4
-8
0
8,
5
10
IS
20
ax 6
0 -6
i
0
5
10
1s
20
w
0
5
10
15
t(s) (Station699)
Fig. 4. Ace&ration response of the retaining wall due to seismic P-wave in&dense (sofi backfill soil},
Seismic response of a tinforced retaining wail
111
FS@/s)
12. I
6
0 0
“,%q-Y 40
t 80
,
140
’ 1 1
o(rad/s) ‘)FS,(m/s)
6
6
6.
6
.
0,
/
40
Fig. 5. Fourier spectra for responses of the retaining wall due to shnic soil).
80
120
1643
P-wave incidence (stiff backfill
112
Chongbin Zhao and T. P. Xu
FS,h/s)
12-
I
0
1 40
80
120
0
1’60
6-
40
80
1 3
120
6-
0
40
80
120
160
0
40
80
1 D
120
o(rud,
@W/S)
:)
(Station 1
F&h/s)
F!
12
i,
40
80
120
1 3
ohd/s> (Station 69) Fig. 6. Fourier spectra for responses of the retaining wall due to seismic P-wave incidence (soft backfill soil).
Seismic response of a reinforced retaining wall
113
a&M)
5-f
2.5o-2.5-
’ ‘(Station 2)
, LI_
-2.5-
w
t(s) (Station69)
Fig. 7. Acceleration response of the retaining wall due to seismic SV-wave incidence (stiff backfill soil).
that under the seismic P-wave vertical incidence, the induced horizontal acceleration at the top of the reinforced retaining wall is much greater than that at the bottom of the reinforced retaining wall, even though the reinforced retaining wall is very stiff. The acceleration differences along the height of the wall will give rise to rotation and bending moments within the wall. Therefore, it can be concluded that the vertical seismic P-wave incidence may be the most dangerous condition for the seismic design of the reinforced retaining wall since both the induced shear force and bending moment might approach their maximum values in this situation.
Under the seismic SV-wave vertical incidence, the acceleration responses of the reinforced retaining wall for the stiff backfill soil and soft backfill soil are shown in Figs 7 and 8, while the corresponding Fourier spectra are shown in Figs 9 and 10. As can be seen from these results, the overall response of the reinforced retaining wall due to the seismic SV-wave incidence is weaker than that due to the seismic P-wave incidence so that the above conclusion is further confirmed. In the case of the seismic SV-wave incidence, nearly uniform distributions of both horizontal and vertical accelerations can be observed along the height of the wall. This fact indicates that
Chongbin Zhao and T. P. Xu
o-2.5- +
_2.5v---j
4-j
-5 : 0
I
, 5
I
I 15
Ib
I 20
t(s) (Station 2) 5 a,C+?
54a&&3
2.5-
2.5-
o-
o-
-2.5-
-2.5-
,
-5, 0
, 5
I
I 10
15
’
.
-5 20
0
tw
I 5
v
I 10
8
I 15
8 20 tw
(Station 69) Fig. 8. Acceleration response of the retaining wall due to seismic SV-wave incidence (soft backfill soil).
115
Seismic response of a reinforced retaining wall
3
1.5
0
&adls) 3
1.5
0
F&h/4
&.
(Station 2) FS,(m/s) 3
1
40
80
120
1
Fig. 9. Fourier spectra for responses of the retaining wall due to seismic SV-wave incidence (stiff backfill soil).
Chongbin Zhao and T. P. Xu
116
0
0
40
80
120
40
80
120
1’
160
w0-4s)
whds)
. (Station 2)
3
1.5
0
FSAm/s)
FWm/s)
L. 40
80
120
I
40
80
120
1
Fig. IO. Fourier spectra for responses of the retaining wall due to seismic W-wave incidence (soft backfill soil).
the effect of the induced rotation and bending moment may be unimportant to the seismic design of a reinforced retaining wall under seismic SV-wave vertical incidences. Furthermore, both the maximum magnitude and distribution of Fourier spectra under the seismic SV-wave incidence are obviously different from those of Fourier spectra under the seismic P-wave incidence. Thus, it is suggested that the type of incident seismic wave be considered carefully for the seismic design of a reinforced retaining wall since it has a significant effect on the seismic response of reinforced retaining walls.
4. CONCLUSIONS
Using the finite and infinite element coupled method and the F.F.T. technique, the seismic analysis of a reinforced retaining wall has been carried out in this paper. From the related numerical results, the following conclusions have been obtained. (1) The finite and infinite element coupled method is more suitable for the seismic design of a reinforced retaining wall since all the important factors and necessary conditions can be simultaneously satisfied in the analysis.
117
Seismic response of a reinforced retaining wall (2) The stiffness of the backfill soil has a limited effect on the seismic response of the reinforced retaining wall since it has limited influence on the dynamic characteristics of the soil-retaining wall system. (3) A vertical seismic P-wave incidence may be the most dangerous condition for the seismic design of the reinforced retaining wall since both the induced shear force and bending moment might approach their maximum values at this situation. (4) The types of incident seismic waves must be considered carefully for the seismic design of a reinforced retaining wall since they have significant effects on the seismic response of the reinforced retaining wall. (5) A vertical incident seismic SV-wave leads to the nearly uniform distribution of the horizontal and vertical accelerations along the height of the retaining wall. However, a vertical incident seismic P-wave leads to a nearly uniform distribution of the vertical acceleration along the height of the retaining wall but an obviously nonuniform distribution of the horizontal acceleration along the height of the retaining wall.
REFERENCES
1. C. Zhao, C. Zhang and G. Zhang, Analysis of 3D foundation wave problems by mapped dynamic infinite elements. Sci. Chinn (Series A) 32, 469-491 (1989). 2. C. Zhao and S. Valliappan, Vibration of threedimensional rigid raft foundation on viscoelastic medium. Earthqu. Engng struct. Dyn. 20, 1159-1177 (1991). 3. C. Zhao, A new input model of seismic waves for infinite foundations of arch dams and other large structures. J. Rock Mech. Engng 7, 248-256 (1988). 4. S. Valliappan and C. Zhao, Dynamic response of concrete gravity dams including dam-water-foundation interaction. Int. J. Numer. Anal. Geomech. 16, 19-99 (1992). 5. C. Zhao and S. Valhappan, A numerical model for wave scattering problems in infinite media due to P and SV wave incidences. Int. J. Numer. Meth. Engng 33, 1661-1682 (1992). 6. C. Zhang and C.’ Zhao, Coupling method of finite and infinite elements for strip foundation wave problems. Earthqu. Engng struct. Dyn. 15, 839-851 (1987). 7. C. Zhao, C. Zhang and G. Wang, A modelling of wave propagation in semi-infinite domain using dynamic infinite elements. Science Report of Tsinghua University, TH88003, Beijing (1988).
8. P. Bettess, Infinite elements. Int. J. Nurner. Meth. Engng 11, 53-64 (1977). 9. Y. K. Chow and I. M. Smith, Static and periodic infinite solid elements. ht. J. Numer. Meth. Engng. 17, 503-526 (1981).
10. F. Medina and R. L. Taylor, Finite element techniques for problems of unbounded domains. Int. J. Numer. Meth. Engng 19, 699709 (1983). 11. C. Zhao, C. Zhang and G. Zhang, Study on the characteristics of mapping dynamic infinite elements. Earthqu. Engng Engng Vibr. 7(3), l-5 (1987).
12. C. Zhao, C. Zhang and G. Zhang, Simulation of semi-infinite plane elastic foundation using infinite elements. J. Tsinghua Univ. 26, 51-64 (1986): 13. C. Zhao, C. Zhang and G. Zhang, Simulation of arch dam foundation using infinite elements. J. Hydruul. Engng No. 2, 27-36 (1987).
14. C. Zhao. C. Zhane and G. Zhane. Analvsis of free field motions on arch dam canyons and the effects due to existence of weathered rock strata. J. Hydraul. Engng No. 11, 29-40 (1988). 15. C. Zhao, C. Zhang, G. Wang and G. Zhang, Effects of the characteristics of soil layers on field seismic response using infinite elements. J. Scientif. Res. Hydraul. Engng I,
.
Transport. No. 4, 31-45 (1988).
16. C. Zhao, C. Zhang and G. Zhang, Infinite element and its development. J. Shaanxi Inst. Mech. Engng No. 1, 89-97 (1989).
17. C. Zhao and C. Zhang, Analysis of the differences between finite and infinite model for infinite media. J. Scientif. Res. Hydraul. Engng Transport. No. 2, 143-153 (1990). 18. C. Zhao, C. Zhang and G. Zhang, Dynamic simulation for infinite foundation. J. Hydraul. Engng No. 6, 20-27 (1990). 19. C. Zhao and S. Valliappan, An approximate method for simulating infinite media. Comput. Struct. 41, 104-1049 (1991). 20. C. Zhao, C. Zhang and G. Zhang, A dynamic model for elastic and v&o-elastic infinite foundations. J. Geotechn. Engng 13, 1l-23 (1991).
21. C. Zhao and S. Valliappan, An efficient wave input procedure for infinite media, Cornman. appl. Numer. Meth. 9, 407-415
(1993).
22. C. Zhao and S. Valliappan, Incident P and SV wave scattering effects under different canyon topographic and geological conditions. Int. J. Numer. Anal.-Geomech. 17. 73-94 (1993).
23. CZhao and S. ‘Valliappan, Transient infinite elements for seepage problems in infinite media. Int. J. Numer. Anal. Geomech. 17. 323-341 (1993). 24. C. Zhao and S. Valliappan: Dynamic analysis of a reinforced retaining wall using finite and infinite element coupled method. Comput. Struct. 47, 239-244 (1993). 25. J. P. Wolf, Dynamic Soil-Structure Interaction in Time Domain, Prentice-Hall, Englewood Cliffs, New Jersey (1988).