Woodward Análisis seudo estático y dinámico muros gravitacionales

Geotechnical and Geological Engineering, 1996, 14 269-290

Comparison of the pseudo-static and dynamic behaviour of gravity retaining walls P.K. WOODWARD" and D.V. GRIFFITHS t

"Department of Civil & Offshore Engineering, Hermt-Watt Untversity, Edinburgh EH14 4AS, UK t

and Geomechamcs Research Center, Colorado School of Mines, Golden, Colorado, 80401, USA

Received 12 May 1995 Accepted 22 August 1996

Summary

Pseudo-static and dynamic non-linear finite element analyses have been performed to assess the dynamic behaviour of gravity retaining walls subjected to horizontal earthquake loading. In the pseudo-static analysis, the peak ground acceleration is converted into a pseudo-static inertia force and applied as a horizontal incremental gravity load. In the dynamic analysis, an actual measured earthquake acceleration time history has been scaled to provide peak ground acceleration values of 0.1 g and 0.3 g. Good agreement is obtained between the pseudo-static analysis and analytical methods for the calculation of the active coefficient of earth pressure. However, the results from the dynamic analysis require careful interpretation. In the pseudo-static analysis, the increase in the point of application of the resultant active force with the horizontal earthquake coefficient kh from the one-third point to the mid-height of the wall is clearly observed. In the dynamic analysis, the variation in the point of application is shown to be a fimction of the type of wall deformation. Both finite element analyses indicate the importance of determining the magnitude of the predicted displacements when assessing the behaviour of the wall to seismic loading.

Keywords: pseudo-static analysis, gravity retaining wall, earthquake, dynamic finite element analysis

Introduction

A review of the increase in the lateral earth pressure on retaining walls during earthquakes was conducted by Nazarian and Hadjian (1979). They reported that the increase in lateral earth pressure during seismic loading induces sliding and/or tilting to retaining structures, causing three types of structural displacements: rigid body translation, rigid body rotation, and flexure. They also reported that the response of the wall is a function of the relative soil-structure displacements, structural rigidity, backfill properties, foundation stability and the characteristics of the applied input motion.

Ortiz et al. (1983) conducted dynamic centrifuge testing of a cantilever retaining wall and showed that the earth pressure distribution behind the wall was non-linear. If the wall does not fail, then deflections of the wall during the earthquake may be greater than the

0960-3182 �9 1996 Chapman & Hall

270 Woodward and Griffiths

final value. They also confirmed the work of Seed and Whitman (1970), in that a residual pressure acts on the wall after the earthquake has subsided, which can be substantially greater than the static pressure before the earthquake and can be a significant proportion of the maximum pressure developed during shaking.

Neelakantan et al. (1990) used shake-table tests to look at tied-back walls subjected to earthquakes and found that the passive resistance due to the embedment of a flexible wall greatly enhanced its stability during shaking. Siller et aL (1991) also looked at tied-back walls and found that the stiffness of the wall does not affect the peak displacements. The wall tended to follow the soil motion and was not capable of modifying the free-field displacements. The most popular method used to estimate the increase in lateral earth pressure during earthquakes is the Mononobe-Okabe active earth pressure theory (Okabe, 1926; Mononobe and Matsuo 1929). The method is based on Coulomb's theory for the active earth pressure on retaining walls due to a dry cohesionless backfill and modified to take into account vertical and horizontal accelerations. These accelerations produce an additional inertia force which causes an increase in the lateral earth pressure on the retaining wall. The horizontal and vertical inertia forces can be described in a non- dimensional form as

kh = (horizontal component of earthquake acceleration) (1)

kv = (vertical component of earthqake accelerator) (2)

g = 9.81 m/s 2 (3)

It can be shown that the dynamic active earth pressure, with earthquake effects can then be given as

1 7//2( 1 _ kv)KA~ (4) PAE =

where 3' = unit weight of the backfill, H is the height of the wall and KAe represents the active coefficient of earth pressure (Fig. 1), with earthquake effects,

KAE = C082((fi - - 1~ - - O) (5 )

I f sin(a+~)sin(~-i-O) ]~l/212 eosOeos2/3cos(~5+/3+O) 1 + / . ~ ~ J

The main disadvantages of the Mononobe-Okabe solution can be summarized as follows:

(1) the inertia of the wall is neglected and the dynamic amplification of the backfill is not considered:

Pseudo-static and dynamic behaviour of retaining walls 271

(2) the method is based on rigid body motions and so does not predict deformations; (3) reversal of the ground motion is not considered; (4) the method does not predict the increase in the point of application of the resultant

active force. Despite these deficiencies, centrifuge and shake-table studies have shown that the Mono- nobe-Okabe solution does predict the correct earth pressure coefficients for a dry cohesionless backfill (e.g. Ortiz et al. 1983). Steedman and Zeng (1990) have also found good agreement with the Mononobe-Okabe solution and showed the influence of phase on the calculation of the resultant active force.

Richard and Elms (1979) proposed a design procedure for gravity retaining walls. They showed that when designing for earthquake loading, the inertia of the wall is an important factor and the weight of the wall should be increased over the static value in order to prevent excessive displacements.

Finite element analyses of retaining walls during seismic shaking have been performed by Siller and Bielak (1986). They found that permanent relative displacements between the retaining wall and its base were lower for the rotating wall than for the non-rotating wall. Other finite elements analyses of the dynamic behaviour of retaining walls have been reported by Yogendrakumar et al. (1992) and Finn et al. (1992).

Centrifuge and shake-table studies have shown that the point of application of the resultant active force (Ha) increases from the one-third point (H/3) towards a dynamic

C

T H

A

khW

B

? W

/ S

ot

L . . . .

O N

F

Fig. 1. Derivation of the Mononobe-Okabe solution


value of Ha = HI2 during seismic loading. Prakash and Basavanna (1969) showed that Ha = HI2 when kh = 0.3 and found an average value of Ha = 0.42H has also been suggsted.

Description of the problem

The mesh used in both the pseudo-static and dynamic finite element analysis is shown in Fig. 2. The gravity retaining wall was assumed to be 5 m in height with a Young's modulus of E = 250.0 x 105 kN/m 2 and a Poisson's ratio of u = 0.15. The wall was assumed to behave elastically with a unit weight of 2 /= 25 kN/m 3. The backfill was assumed to be dry, cohesionless and obey a Mohr-Coulomb failure criterion with an angle of friction r = 30 and a dilation angle of ~ = 0 ~ A Young's modulus of E = 1.0 x 105 kN/m 2, Poisson's ratio of u = 0.3 and a unit weight of 7 = 17 kN/m 3 were also assumed. Seed and Whitman (1970) showed that the influence of the wall/ backfill friction angle on Kae, using the Mononobe-Okabe theory, was small. The wall is obviously more likely to fail through the development of active detbrmation than passive deformation. For simplicity, the interface between the back of the wall and the granular backfill was therefore assumed to be smooth. To achieve this, the same degree- of-freedom number is assigned to both the wall and the backfill in the horizontal direction, but separate degrees of freedom are assigned in the vertical direction. This creates an interface which will allow the soil to slip down the back of the wall (i.e. 6w/b = 0), but still allow horizontal interaction. In a very simple way, a discontinuity in the vertical direction has been introduced to model the soil/structure interaction. It should be noted however, that Richard and Elms (1979) commented that the wall/ backfill friction angle is an important factor when assessing the inertia of the wall, given by the inertia factor Cte.

In the pseudo-static analysis, the interface between the bottom of the wall and the foundation was assumed to be rough (i.e. the same freedom numbers for the wall and soil elements at the interface). In the dynamic analysis, two different boundary conditions were considered for the foundation.

In the first case, called the 'smooth' analysis, the horizontal degrees of freedom between the wall and foundation interface were disconnected (i.e. ~Sw/f = 0), but the vertical ones were not. This creates a smooth interface at the base in the horizontal direction and again introduces a very simple discontinuity. The wall will not displace indefinitely as it is still connected horizontally to the backfill and vertically to the

Fig. 2. Mesh used in the analysis

wall = 5m

Pseudo-static and dynamic behaviour of retaining walls

Table 1. First four natural frequencies

273

Mode Natural frequency (Hz)

1 3.601 2 4.939 3 6.569 4 6.816

foundation. It should also be noted that the acceleration time history is constantly changing direction, forcing the wall to displace either into or away from the backfill. This type of boundary condition, although unrealistic, gives a failure which is almost entirely translational in nature. The earth pressure coefficient K can be monitored and compared directly to the pseudo-static finite element value, in which a purely translational failure was induced, and to the Mononobe-Okabe solution.

In the second case, called the 'rough' analysis, both the horizontal and vertical degrees of freedom were connected across the interface (i.e. ~w/f = qS). This condition is obviously the more realistic case and allows for both translational and rotational behaviour of the wall (note: behaviour is also dependent on the geometry of the wall). If a particular gravity retaining wall were to be analysed, then better interface modelling could be achieved through the use of interface or slip elements.

The horizontal initial stresses were assumed to be a function of Ko (i.e. Ko ,.~ 1 - sin0), although they could be set to simulate the particular method used to construct the wall.

Natural frequency analysis

Before the pseudo-static and dynamic analyses were performed, a natural frequency analysis was performed in order to assess the basic dynamic characteristics of the wall. As expected, the first mode shape was essentially horizontal motion of the backfill. Table 1 shows the first four natural frequencies.

In both the pseudo-static and dynamic analysis the coefficient of earth pressure (K) was calculated by summing the horizontal forces behind the wall (PToT) and re-arranging Equation 4 to give

2PToT K -- 7H2( 1 _ kv) (7)

The point of application of the resultant force (Hk) can be found through the following equation

Hk = ~ H Pile PTOT (8)

where Pe and le are respectively the elemental horizontal force component and point of application of the elemental force component above the base of the wall. In all the analyses performed a vertical coefficient of kv = 0.0 was assumed.


Pseudo-static finite element analysis

In the pseudo-static analysis, the excitation was in the form of a suddenly applied constant horizontal acceleration, which was converted to an incremental gravity load in the horizontal direction. Purely translational displacement increments were then applied to the whole wall in order to mobilize active stresses.

Figure 3a shows typical results from the pseudo-static analyses, for horizontal earthquake coefficients of kh = 0.0, 0.1 and 0.3. The lateral stresses (expressed in the Figure as the earth pressure coefficient K), initially increase sharply as the acceleration is applied, hence the sharp increase in K, and then gradually fall as the wall is displaced, until active failure of the wall is achieved and the value of K becomes constant and equal to KAE. Figure 3b shows the increase in the point of application of the resultant active force for kh = 0.0, 0.1 and 0.3 and clearly shows this point moving towards the mid-point of the wall.

Figure 4a shows the computed active earth pressure coefficients KA~ compared to the Mononobe-Okabe solution. Although the computed results predict slightly higher values of Kae at larger peak accelerations, they are generally in excellent agreement. The authors conducted similar analyses for friction angles of r = 25 ~ and 35 ~ and found similar results. Figure 4b shows the increase in the point of application of the resultant active force Ha with increasing peak acceleration. The tendency for this parameter to move towards the mid-point of the wall (Ha = 0.5H) with increasing kh is clearly observed.

Dynamic finite element analysis

The pseudo-static analysis does not assume a pre-defined failure plane and so active failure of the wall occurs along the actual induced failure plane. The pseudo-static analysis can predict earth pressure distributions, points of application of the resultant active force, changes in the shear strength due to the inertia forces and an estimate of predicted displacements. This is a distinct advantage over the Mononobe-Okabe solution. However, the pseudo-static analysis still cannot predict amplification of the ground motion or inertia effects of the waU, but some consideration can be given to reversal of the ground motion. These deficiencies in the pseudo-static approach can be dealt with in the dynamic environment. As mentioned previously, to examine translational and rotational behaviour of the wall the wall/foundation interface was assumed to be both 'smooth' (translational behaviour) and 'rough' (translational and rotational behaviour).

Input motion and dynamic algorithm

The input (ground) acceleration time history (Fig. 5a) was scaled to peak acceleration values of 0.1 g and 0.3 g. The earthquake was actually measured at bedrock level near to the Long Valley Dam in California (Griffiths and Prevost, 1988). Although this paper only considers the behaviour of the retaining wall to one earthquake (the purpose of this paper is to examine and compare a pseudo-static and dynamic finite element approach) in the actual design of the retaining wall several acceleration time histories would have to be applied to assess its overall performance. The decision on which acceleration time history


1.8

"~ 1.6

1.4 o o 1.2

o 0.8 r .~ 0.6

0.4

0.2 kh O0

r J i i i i i i i I . . . . ~ . . . . i . . . . i ~ i i

0 0.001 0 . 0 0 2 0 . 0 0 3 0 . 0 0 4 0.005

Displacement (m)

(a)

i 1 , , i

0 . ~

kh = 0 . 3

, I r ,

0.007 , J

0.008

o

,C

o

o

0.5

0.4

0.3

0.2

0.1

kh = 0 . 1

J " kh = 0.0

I

kh = 0.3

T T r I 1 r T t T 1 T T I I I I T T I l T I T T l I T T I 1 I I I 1 I I 1 I I

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008

Displacement (m)

(b)

Fig. 3. Typical results from the pseudo-static analysis. (a) Earth pressure coefficient K. (b) Normalized point of application of the resultant earth pressure coefficient (Hk/H)

to apply, to represent the next earthquake, poses one of the biggest uncertainties in the field of earthquake engineering.

The acceleration time history used in this paper was applied as an inertia force (i.e. the product of mass times ground acceleration) in the following equation of motion,


0 0 r,9 0

0 b . q

0.8

0.6

0.4

0.2

P s e u d o - S t a t i c ~

, , , i , , , , I i ~ i i I i i r [ [ i . . , i . . . . I i t i i I i i i i I , i ~ i i . . . .

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Horizontal Earthquake Coefficient kh

0.54 Z:

O

r

< ~s

O

0.5

0.46

0.42

0.38

0.34

. 3 . . . . ~ , , , , i , , , , i i , i i i , r i i ~ . . . . i . . . . i . . . . i . . . . i . . . .

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Horizontal Earthquake Coefficient kh

(b)

Fig. 4. Results from the pseudo-static analysis for 0.0 < kh <_ 0.5. (a) Earth pressure coefficient KAE. (b) Normalized point of application of the resultant earth pressure coefficient (H~/m

[M] i, o tz j + [c] ~-~ + [K] {r} = -[34] I ~ J (9)

where [K], [M] and [C] are the consistent stiffness, mass and damping matrices respec-

tively; {r}, {2} and { ~ } are the vectors of relative displacement, velocity and accel-

eration respectively; and {-~}g is the grotmd acceleration time history. To examine the


0.1=

277

0.08

0.06 0.o4

0.o2

o

~ -0.02 ~ 4).04

-0.06 -0.08

-0.1 2 4 6 8 10 12

Time (s) (a)

0.12

0.1

,~ 0.08

~ 0.06

~ 0.04

0.02

0 0 4 8 12 16 20

Frequency (Hz)

(b)

Fig. 5. Scaled horizontal ground motion used in the dynamic analysis. (a) Acceleration time history. (b) Fourier transform of the acceleration time history

influence of phase on the retaining wall (Steedman and Zeng, 1990) a more sophisticated method of applying the acceleration time history to the base of the mesh should be used. The Fourier transform of the earthquake is shown in Fig. 5b. The main frequency content of the earthquake was around 2.0 Hz, which is lower than the natural frequency of the wall. However, a large spike is observed at frequencies around 3.5 Hz and some dynamic amplification of the backfill is to be expected.

The dynamic implicit algorithm used in the analysis was the Newmark %/3 method (Newmark, 1959) in which the equation of motion (Equation 9) is solved at time t + At. In this method the new displacements, velocities and accelerations are calculated from the following recurrence relationships:

3 f O2r ~ g [ K ] / { r ) , + A , - [M] [ ~ t 2 [M] + ~ - ~ [C] +

3


1 1 or 1 [M][~--~-~{r}t+~tt{-~}t+(~-1) (02 r ] ] +

Or 1 - 7 02r [C] [~-7~t { r } t - ( 1 - ~ ) ( ~ } t - ( 2 - f l ) A t { ~ } t ] (lO)

{_~}t+At='y Or 1 _ 7 /3kt {r}t+At- (-~tkt {r}t- (1- -~) {-Ot }t- ( -~) At l ~ ~. t ~1 "~ (11)

{ 02r'~ _ 1 1 1 Or 1 ot2jt+~t ~3At 2 {r}t+~t-(fl~-~{r}t+~t {~} t+(-~- l) ~O2r'~ j (12)

The results presented are for "y = 0.55 and/3 = 0.28, where,

/3-- ('Y -}- 1) 2 (13) 4

The value of these parameters represent a small amount of numerical damping, which was considered justified as the discretization process and the elastic-perfectly plastic assump- tion would generate some spurious high frequency noise. The elastic-perfectly plastic soil model underestimates the level of hysterietic damping and so for each peak ground acceleration Rayleigh damping values of { = 5 and 10% were used for the backfill. The damping matrix [C] was therefore calculated based on the following relationship,

[c] = <a//l +/~[K] (14)

where,

Oe = 2coacob{ (cOb -- COa) (co coN) (15)

/3 = 2~ (cob - coa) (16) (co coa

where coa and cob are the frequencies used to define the damping curve and were set at the fundamental frequency of 3.6 Hz and 9.0 Hz respectively. A constant value of ~ = 5% was assumed throughout the analyses for the wall. Variable damping techniques (Idriss et aL, 1974) could also be used to specify the viscous damping eoefficient based on the shear strain level in each element.


To prevent stress waves being reflected at the mesh boundaries, viscous boundaries (Lysmer and Kuhlemeyer, 1969) were applied. Effectively a 'dash-pot' is positioned at each node where the viscous boundary is to be applied, thus imposing a non-zero force at the node. The elemental boundary consistent viscous damping matrix [C]~ can be written as

[C]~ = f [N]T[C]*[Nlds (17) J

where [N] are the element shape functions and

Into (18)

where p is the mass density, Vp and V, are the compression and shear velocities, and a and b are constants. The element viscous damping matrices at the boundaries of the mesh are added to form a global viscous damping matrix which is subsequently added to the consistent material damping matrix.

In all of the results presented, the following convention was used: positive displacements indicate active movement of the wall, and negative displacements indicate passive movement of the wall.

Peak ground acceleration = O. 1 g

Figure 6a and b show the horizontal displacement time histories of the top and bottom, and the bottom vertical displacement time history, of the retaining wall subject to 0.1 g peak ground acceleration, for the smooth and rough analyses respectively, assuming Rayleigh damping at ~ = 5%. Both analyses indicate some translation and rotation of the wall (rocking) with the horizontal displacements being more significant in the smooth analysis. The Figures also show that provided the wall has not 'failed' (i.e. large plastic deformation of the wall), then displacements induced during the earthquake can be greater than after the shaking has stopped, indicating elastic behaviour. Figure 6c and d show the effect of increasing the Rayleigh damping coefficient to ~ = 10%. Although the peak displacement has not been significantly reduced the displacements after the peak have.

Figure 7a and b show the change in the earth pressure coefficient K with time, at damping ratios of ~ = 5 and 10% respectively. The smooth boundary analysis predicts a larger variation in K, but both analyses show the difficulty in predicting a value of K which could be compared to an analytical solution, since K continuously varies with time as the wall is displaced away from the soil and then back into it. Both analyses predict an increase in the earth pressure coefficient after the earthquake has stopped, although a higher increase in the residual value of K is observed in the rough boundary analysis. Figure 8a and b show the change in the point of application of the resultant force with time. The rough boundary analysis predicts a larger variation in the point of application of K, which suggests a more significant change in the non-linear stress distribution behind the wall during shaking.


0.007 0.006 ~ ~ Top Horizontal 0.005 - - - - Bottom Horizontal 0.004 - . . . . . . Bottom Vertical 0.003 ~

o.o01

0

-0.001 -O.OO2 - -0.003 -

-O.004 - 0 2 4 6 8 10

Time (s)

(a)

12

0.007

g

o

0.006

0.005

0.004

0.003

0.002 0.001

0

-0.001 -0.002

-- - - Top Horizontal - - - - -- Bottom Horizontal

:=-- . . . . . . Bottom Vertical

L

12

-0 .003 -O.OO4 i

0 2 4 6 8 10

Time (s)

(b)

Fig. 6. Displacement time histories of the wall for 0.1 g peak ground acceleration, from (a) 'smooth' and (b) 'rough' analyses, at a damping ratio of ~ = 5%, and from (c) 'smooth' and (d) 'rough' analyses, at a damping ratio of ( = 10%

Peak ground acceleration = 0. 3 g

Figure 9a and b show the horizontal displacement time histories of the top and bottom and the bottom vertical displacement time history, of the retaining wall subject to 0.3 g peak ground acceleration for the smooth and rough analyses respectively, assuming Rayleigh damping at ( = 5%. Figure 9a shows that the wall has 'failed' since the displacements after the earthquake has subsided are approximately the same as those at the peak ground acceleration. In Fig. 9b a significant reduction in the peak displacements is observed for the rough boundary analysis, although failure of the wall has probably still occurred. There is also a large difference between the displacements of the top of the wall and the bottom, indicating a significant rotation of the wall during shaking. It is interesting to note that the vertical displacements at the bottom of the wall are higher than the horizontal displacements at the bottom of the wall. Figure 9c and d show the effect of increasing the Rayleigh damping coefficient to ( = 10%. The peak displacements are significantly


0.007

t~ o

0.006

0.005

0.004

0.003

0.002

0.001

0 -0.001

-0.002

-0.003

-0.004

0.007

- - Top H o r i z o n t a l

- - - - - - B o t t o m H o r i z o n t a l

- . . . . . . B o t t o m V e r t i c a l

2 4 6 8 10 12

Time (s)

(c)

v

o

?5

0.006

0.005

0.004

0.003

0.002

0.001

0 -0.001

-0.002

-0.003

~ 0

- - Top H o r i z o n t a l

- - - - B o t t o m H o r i z o n t a l

. . . . . . B o t t o m V e r t i c a l

2 4 6 8 10

Time (s) 12

(d)

Fig . 6. Con td .

reduced, especially for the smooth analysis. This demonstrates the importance of knowing the correct level of material damping if the wall is approaching failure.

Figures 10a and b show the change in the earth pressure coefficient K with time, at damping ratios of ~ = 5 and 10% respectively. The smooth and rough analyses predict large variations in K, but again both analyses show the difficulty in predicting a value of K which could directly be compared to an analytical solution. Both analyses predict an increase in the earth pressure coefficient after the earthquake had subsided. However, in the rough analysis the residual earth pressure is substantially greater than the initial earth pressure and is equal to the peak earth pressure experienced during shaking.

Figure 11 a and b show the change in the point of application of the resultant force with time. The rough boundary analysis again predicts a larger variation in the point of application of K, which again suggests a more significant change in the non-linear stress distribution behind the wall during shaking. After the earthquake had subsided, it is interesting to note that the point of application of the resultant force remains high in the


0.7

g

.g

0.65

0.6

0.55

0.5

0.45

Smooth

~ - - - Rough ]1~

0 . 4

0 h , ' ' ' r , , , , I . . . . I . . . . I , , , ,

2 4 6 8 10 12

Time (s)

0.7

~'~ 0.65

0.6

0.55

0.5

0 . 4 5

0 . 4 _ _ a . _ . . ~

0 2

Smooth

. . . . Rough

4 6 8 1 0 1 2

Time (s)

(b)

Fig. 7. Variation in the earth pressure coefficient (K) for 0.1 g peak ground acceleration from the 'smooth' and 'rough' analyses at damping ratios of (a) ~ = 5%, and (b) ( = 10%

smooth analysis, but falls close to the initial point of application in the rough analysis. This is to be expected in the rough analysis as the wall experiences a significant rotation lowering the point of application.

Figure 12a and b show the displaced mesh at T = 5.5 s, corresponding to the peak ground acceleration. For the smooth analysis, translational deformation of the wall is clearly observed, whereas the rough analysis shows translation with rotation.

Discussion

In this paper, the dynamic active earth pressure problem has been addressed using a non- linear pseudo-static and dynamic finite element approach. The pseudo-static analysis gave good agreement with an existing analytical solution for the determination of the earth pressure coefficient KAE and indicated that the point of application of the resultant force increases with increasing peak horizontal acceleration.

Figure 3a showed that as the coefficient of earthquake acceleration kh increases, the


0.44

283

0.42

= 0.4 o

0.38

.~ 0.36

0.34

~u 0.32

0.3

Smooth I

-- -- Kougn 11

II

bll I i I i I i~l / t ~ I | I I l l I l l i I . h . I i i ^ e . . . .

'~ I I~ I I l ~ J | ' l " ' N t a ' ~ ' " ~ ~ " " / -

1

2 4 6 8 10

T i m e (s)

(~)

12

0.44

lal 0.42

,~ 0.4

o.38

0.36

O.3,*

0.32

0.3

Smooth ---- Rough

q

' , , , , , . . . . . . r . . . . i , , , I , , ,

0 2 4 6 8 10 12

Time (s)

(b)

Fig. 8. Variation in the normalized point of application (Hk/H) of the resultant earth pressure coefficient (K) for 0.1 g peak ground acceleration from the 'smooth' and 'rough' analyses at damping ratios of (a) ~ = 5% and (b) ~ = 10%

magnitude o f the displacements required to cause a constant horizontal stress distribution behind the wall (i.e. K = KAe) also increases. Typically, over seven times more displacement is required when kh = 0.3, than when kh = 0.0. The displacements calculated from a pseudo-static analysis therefore suggest an initial estimate of the displacements at the onset of active failure. Although a homogeneous backfill was assumed in this paper, the pseudo-static analysis can be used for backfills in which the material properties vary.

The results from the dynamic analysis show that there is a significant variation in the earth pressure coefficient K during earthquakes, but due to the time varying input it is difficult to find an exact value of K which would represent "failure' o f the wall, as expressed in pseudo-static and analytical methods. This is due to the soil-structure interaction effects with the value of K decreasing during active movement and increasing during passive movement of the wall. An indication to whether the wall has failed can he obtained by computing the increase in the weight of the wall due to inertia, as proposed by Richard and Elms (1979). In their paper, the weight of the wall including inertia effects


0.06

g

C2

0.05

O.O4

0.03

0.02

0.01

0

-0.01 0

- - - Top H o r i z o n t a l

_ ot om Horizontal

_ �9 ..... B o t t o m V e r t i c a l J I " l ~ " . r / " r ' / z ' " ~ . . . . . .

I

I , i , , i , , i , I , l , , I , , , , l , b, , , ,, , , �9 ,

2 4 6 8 10 12

Time (9

(a) .

0.06

0.05

0.O4

0.03

0.02

0.01

0

-0.01 0

I - - Top H o r i z o n t a l

- - - - - B o t t o m H o r i z o n t a l

. . . . . . B o t t o m V e r t i c a l

2 4 6 8 10 12

Time (S)

(u)

Fig. 9. Displacement time histories of the wall for 0.3 g peak ground acceleration at a damping ratio of = 5% for (a) 'smooth' and (b) 'rough' analyses, and at a damping ratio of~ = 10% for (c) 'smooth' and (d) 'rough' analyses

Ww is given by:

where

Ww = GeEA~ = CIE-127HZ(1 - k v ) K A e (19)

CtE = cos (6 + fl) - s in (6 + fl) t a n ~bb (20) (1 -- kv) (tan Cb -- ta i l 0)

For the rough wall, fi = 6 = 0, ~bb = ~b = 30 ~ and k~ = 0. Table 2 shows the results o f Equations 19 and 20 for kh = 0.1 and kh = 0.3. The wall used in this paper has a mass o f Wm= 21.0 x 103 kg/m and so a factor o f safety F,.e,eh~ can be considered on the wall

Pseudo-static and dynamic behaviour o f retaining walls 285

0.06

v

ff05

0.04

0.03

0.02

0.01

0

-0.01 0

0.06

Top Horizontal

- - - - - B o t t o m Horizonta l

. . . . . . B o t t o m Vertical

"V

2 4 6 8 10 12

Time (s)

(~)

v

0.05

0.04

0.03

0.02

0.01

0

4).01 f 0

Top Horizonta l

m _ _ ~ B o t t o m Horizontal

. . . . . . B o t t o m Vertical

2 4 6 8 10 12

Time (s)

Fig. 9. Contd.

mass as

W m Fwezght - Ww (21)

Table 2 indicates that when kh = 0.1 the wall has not failed, but when kh----0.3, Fwe~ght < 1 indicating failure. By observing the displacement trace of the wall in the rough analysis, where kh = 0.3 and ~ = 5%, it can be seen that the residual displacements of the wall are the same as the peak values, indicating significant plastic deformation during shaking, i.e. failure. It should be noted however that Equations 19 and 20 relate to a wall which has failed by sliding failure. The dynamic analysis therefore suggests that the magnitude of the peak and residual displacements should also be used as a guide to the suitability of a retaining wall to withstand seismic loading when performing finite element simulations. The pseudo-static analysis indicated that failure of this wall, in terms of a


1.1

g

o

.o

m

- ~ Smooth

1 - - - - Rough

0.9 /l{

0.8 _-- , . ,^1 .Ix I 0.7 ~ ' ' ' \ l ~ ' ]~ ']

o.6 - - ~

0.5

0.4 0 2 4

1.1

, ~ / : / l l ( / , / . . , I

6 8 10 12

Time (s)

v 1

�9 ~ 0.9

o 0.8 o

0.7

0.6

0.5

0.4

I ~ Smooth

- - - - Rough

- t lU I \/~,

2 4 6 8 10 12

Time (s)

(b)

Fig. 10. Variat ion in the earth pressure coeff icient (K) for 0.3 g peak ground accelerat ion from the

' smooth ' and ' rough ' analyses at damping ratios o f (a) ~ = 5%, and (b) ~ = 10%

constant value of KAE, will occur at displacements of 0.008-0.01 m when kh = 0.3 (Fig. 3a). Both the dynamic smooth and rough analyses predict permanent displacements greater than this for ~ = 5% and ~ = 10%.

The simple elastic-perfectly plastic soil model underestimates the level of hysteretic damping and so Rayleigh damping was introduced. This highlights the problem of using simple constitutive soil models in a dynamic finite-element analysis when simulating the cyclic behaviour of the soil. To simulate accurately the dynamic behaviour of the backfill, more realistic cyclic soil models should be used in future analyses which can reproduce the correct cyclic behaviour of the soil. Cyclic constitutive soil models are often kinematic in nature and often use multiple yield surfaces to simulate phenomenon such as cyclic mobility. It is desirable to use these types of models when simulating the build-up of excess pore water pressures. The first author is currently implementing the kinematic elasto-plastic soil model ALTERNAT (Molenkamp, 1982, 1990) which can successfully model this type of phenomenon (Woodward, 1993).

Comparisons between the smooth and rough analyses show the importance of the wall/


0.44

287

0.42

0.4

"r-. 0.38

0.36

0.34

*~ 032

~u 0.3

0.28

O

O g~

I Smooth , I I

�9 Rough : ' / : l l ; It tk 1 ~ , ~ . ~ . A~ .

i | l l ( l l l l l V ~

,,,, ,i'i;k,s k<...,.,dv,/'t:';?'""" i?i}, k XJ'~.JJ �9 ' l l I ' ' I ~ l~ ' l CL-IX.X._C~FfkF /X,.~-

~ YV ~1 ii I I I I| t~l ' t" l \ ~ I k / J " \ \ "

! , r r , ~ , , , r ~ . . . . I , , , r I , , , r r , , . . . .

0 2 4 6 8 10 12 Time (s)

(a)

0.44 F Smooth , I~ II

0.42 E-- n �9 7k II ' II I I I-- l~ougn t ~.. I ~ i I / I I i i n II I I . ~ .,', .

0.3 ~- ' ;' ~ ~t

028 ~ 4

0 2 4 6 8 10 12 Time (s)

(b)

Fig. 11. Variation in the normalized point of apphcation (Hk/H) of the resultant earth pressure coefficient (K) for 0.3 g peak ground acceleration from the 'smooth' and 'rough' analyses at damping ratios of (a) ~ = 5%, and (b) ~ = 10%

foundation resistance. If only small resistance is provided, the wall experiences large translational deformation leading to failure. However, a large resistance to movement causes both translational and rotational deformation; although the translational displacements were significantly lower, the rotational ones were higher than for the smooth base. Richards and Elms (1979) commented on the use of Franklin and Chang's (1977) curves for calculation of the total relative displacement of the wall. Unfortunately these curves depend only on the earthquake acceleration time history (in fact, maximum values of acceleration and velocity) and not upon any retaining wall parameters, and do not account for the mode of deformation of the wall during shaking. Displacement comparisons of the wall using a simple model combined with viscous damping is therefore unrealistic as the peak displacements were influenced by the damping coefficient ~ when kh = 0.3.

To compare the earth pressure coefficient KA~ between the pseudo-static and dynamic analysis, the smooth analysis must be used, as this corresponds to the same mode of deformation. KAe must also be compared for kh = 0.3 (Table 2) as this value of earth pressure coefficient produced excessive plastic deformation (failure). Figure 10a and b

2 8 8 Woodward and Griffiths

t

__ t_ E t

- - i - i

i

. . . . . . . I -

. . . . . . . . . . . I . . . . . . . . . . t

i i i E i

i i r E i

. . . . i . . . . Z__ : __2 . . . . . . . 1

(a)

_ _ 1 _ _ _ I

l d

i

= _ , . . . . . . . . i i i

- - , . . . . . . . . . . . j . . . . . . . . . . . . ~ . . . . . . .

i

i

J

(b)

Fig. 12. Displaced mesh at time t = 5.5 s for a peak ground acceleration of 0.3 g from (a) 'smooth' and (b) 'rough' analysis. Magnification factor = 50 x

indicate that the final earth pressure coefficient of the wall (i.e. residual) is approximately 0.62, which is close to the pseudo-static analysis.

The rough analysis predicts significantly higher residual values of K for a peak ground acceleration of 0.3 g than the smooth analysis, due to rotation of the wall. In reality the residual earth pressure would probably lie somewhere in between the smooth and rough residual values. The pseudo-static analysis does not directly give residual values of the earth pressure coefficient. The point of application of the resultant active force in the dynamic analysis seemed to occur around Hk = Ha ~ 0.42 --+ 0.44H (slightly lower than

Table 2. Factors of safety of wall with inertia effects

kh W ~ ( k g / m ) Fw~wht

0.1 19.0 x 103 1.1 0.3 46.9 x 103 0.45


in the pseudo-static analysis), whereas the resultant passive force seemed to remain close to the one-third point (Hk = Hp = HI3).

A pseudo-static finite element analysis will give more information than the Mononobe- Okabe solution, in terms of the distribution of earth pressures, points of application of resultant earth pressures and estimates of relative displacements. The method can also be used for non-homogeneous soils. The method cannot however examine the dynamic behaviour of the wall and backfill (dynamic amplification, inertia effects of the wall, etc.) and so a dynamic finite element analysis is therefore necessary.

Conclusions

Both a pseudo-static and dynamic finite element analysis can be used to estimate the response of a gravity retaining wall subjected to seismic loading in terms of earth pressures and displacements. When equal modes of deformation between the dynamic and pseudo-static analyses were compared, similarities between earth pressure coefficients and expected initiation of failure (start of excessive plastic displacements) were observed.

The conclusions of the paper can be summarized as: (1) The pseudo-static finite element approach gave good agreement to analytical

methods for predicting the increase in lateral earth pressure during earthquakes. This observation is especially beneficial if the properties of the backfill vary with depth;

(2) The pseudo-static finite element approach can also be used to estimate the increase in the point of application of the resultant active force;

(3) The dynamic analysis showed that if failure of the wall (in terms of excessive plastic displacements) has not occurred, deflections during excitation are greater than the final values;

(4) The dynamic analysis also showed that the earth pressure coefficient K and the point of application of K can vary considerably during the earthquake, as the direction of wall movement is constantly changing with the input motion. If failure of the wall has occurred, then the active earth pressure coefficient with earthquake effects KAe can be compared directly to the pseudo-static finite element analysis once the mode of deformation is taken into account;

(5) The earth pressure coefficient after shaking may be greater than the earth pressure coefficient before the event and seems to be dependent on the magnitude of the wall rotation, the peak acceleration and the level of material damping.

Acknowledgements

This research was supported by the SERC and W.S. Atkins North-West in the form of a CASE studentship to the first author.

References

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