(\ Proceedings: Second International Conference on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics, ~ March 11-15, 1991, St. Louis, Missouri, Paper No. 4.8

Seismic Earth Pressures Against Structures with Restrained Displacements P. Ortigosa Professor of Civil Engineering, IDIEM, University of Chile

H. Musante Professor of Civil Engineering, IDIEM, University of Chile

SYNOPSIS A simplified kinematic method to compute seismic earth pressures against structures with restrained horizontal displacements is described. The retaining structure has a linear elastic behaviour and the retained soil is considered linear as well as a nonlinear material. Seismic pressures for different maximum free field accelerations were obtained by applying free field horizontal displacements at the base of the interaction springs conecting the soil and the retain­ing structure. For perfectly rigid structures without horizontal displacements and linear retained materials, seismic pressures obtained with the simplified kinematic method are compared with those obtained using elastic models; for nonlinear retained soils comparisons are established with results obtained using the finite element method. Finally, a parametric analysis was performed using the kinematic method with nonlinear soils for both perfectly rigid and flexible retaining structures with restrained horizontal displacements.

INTRODUCTION

The analysis of seismic earth pressures against retaining structures is commonly divided into two main groups:

- Retaining structures with enough flexibi­lity or with foundations able to undergo some amount of horizontal sliding or tilting during an earthquake.

- Retaining structures with large flexural rigidity or with rigid lateral supports, founded on rigid soils, where lateral displacements are restrained to small levels.

First Group

For the first group, displacement of the retain­ing wall is assumed to be sufficient to produce a state of plastic equilibrium in the retained soil, resulting in minimum active pressures. Probably the earliest design method to carry out with this approach is the classical Mononobe and Okabe (M-0) formulation, which is an extension of Coulomb theory. In the M-0 method the failure plane in the soil is assumed to pass through the toe and be inclined at some angle, so the soil wedge acts as a rigid body with uniform vertical and horizontal accelerations.

By using the earth thrust given by M-0 formula and applying the Newmark's approach, Richards and Elms (1979) established an expression to compute permanent horizontal displacements when the earthquake acceleration pulses at the wall base are larger than a threshold horizontal acceleration. This threshold acceleration is defined as the minimum value to trigger horizon­tal slide at the wall base (or other critical section) including inertia forces on the wall itself. This definition assumes a lower threshold acceleration to trigger sliding than

621

the threshold acceleration to trigger tilting. At this extend, to compute bending moments and shear forces in the retaining structure, the maximum horizontal inertia force to be applied at the M-0 wedge and the retaining structure seems to be the threshold value. For flexible retaining structures where permanent displace­ments at the wall base are constrained to small levels, it seems that the maximum horizontal acceleration to be applied is the maximum earthquake acceleration.

More recently, Nadim and Whitman (1982) used the finite element method (F.E.M.) to compute seismic pressures and permanent displacements on rigid retaining walls. To model the failure plane in the soil and soil-wall interfaces they use contact elements with elasto-plastic beha­viour. For the retained soi 1 nonlinear elements were introduced, with shear modulus and damping factors depending on the maximum shear strain induced by an earthquake record applied at the wall base (actually they use an equivalent linear behaviour following the procedure outlined by Seed and Idriss, 1970).

Many cases of damage reported in the literature can be attributed to increased lateral pres­sures during earthquakes inducing sliding or tilting on the retaining structures, or both. A recent example of this behaviour was observed at Valparaiso Harbour during the 1985 Chilean earthquake. Figure 1 illustrates typical profiles of the gravity retaining walls at berths 1 to 5, built during 1913-1924 using concrete blocks without shear keys. Due to the foundation soil characteristics at berths 1 through 4, failure was triggered as a permanent sliding between concrete blocks at 12m depth; due to poorest soil conditions below the retaining wall at berth 5, a permanent rotation at the wall base was measured. Figure 2 shows the horizontal displacements at the top of the walls.

12

16 E

.c 20 -0.

"' 0 24

28

32

36

Fig. 1

140

120

E 100 ~

c: 80 "' E

"' u 60 ~

0. VI

0 40

Fig. 2

0 20

Berth

1' 2, 3 4, 5

Rock Depth

26m 31m

1,0 60 80 N (blows/foot)

Gravity Retaining Walls at Valparaiso Harbour.

Horizontal Length at Berth 5, (m)

Seismic Horizontal Displacement at the Top of the Retaining Walls­Valparaiso Harbour.

Second Group

For retaining structures with restrained lateral displacements, most of the methods assume a linear elastic behavior (i.e. Arias, 1982 and wood, 1973). Nadim and Whitman (1982) applied the F.E.M. to compute seismic pressures on perfectly rigid walls without horizontal displa­cement and nonlinear retained soils, but computer time and efforts to produce the input data can preclude solutions using this approach, especially for parametric analysis. That is why a simplified kinematic method was developed to compute seismic pressures against linear elastic structures with both linear and nonlinear retained soils. The method was used to perform a parametric analysis on perfectly rigid and flexible retaining structures with restrained horizontal displacements.

622

a max L

Retained soi 1: Gcz y '(

Base

T z

Fig. 3 Earth Retaining Structure used in the Kinematic Method.

KINEMATIC METHOD

Figure 3 shows a retaining structure with linear elastic behaviour. The retained soil can be considered as a linear or nonlinear material, with a length L ~ 4.0H in order to avoid border effects (Wood, 1973). Other parameters are as follows:

H height of the retaining structure.

h span between horizontal supports (struts, slabs, etc).

t

E

I

wall thickness.

Young modulus of the retaining wall.

moment of inertia of the wall cross section per unit length.

maximum free field horizontal accelera­tion expressed as a fraction of g.

a = horizontal acceleration at the base expressed as a fraction of g.

Gcz shear modulus of the retained soil at a depth, z.

'Ycz

maximum free field horizontal displace­ment at depth, z, relative to the base.

maximum free field shear strain at depth z.

v Poisson modulus of the retained soil.

'Y unit weight of the retained soil.

KH horizontal spring representing soil­structure interaction.

Ke rocking spring representing soil­structure interaction.

kz modulus of horizontal subgrade reaction at depth, z, between the structure and the retained soil.

For simplicity, in the analysis and results that follow values of h, t, E, v and 'Y are considered to be constant with depth, KH is assumed to be very large and no dashpots are introduced to take into account soil structure radiational damping.

The simplified kinematic method computes the seismic pressures, as, by applying the maximum free field horizontal displacements at the base of the subgrade springs. Hence, values of as can be expressed as:

as ( l)

where Osz = horizontal displacement of the structure at depth z. Extreme condition for a perfectly rigid structure without horizontal displacement (6sz = 0) can be reached by setting very large E and Ke values. On the other hand, two independent conditions must be satisfied when solving the problem:

( 2)

( 3)

First condition was introduced when dealing with nonlinear retained soils. It means that seismic plus initial static horizontal pressure, oh, at a given depth can reach a limit value given by the passive soil pressure at that depth, op. Passive resistance was computed using conven­tional static expressions becouse maximum seismic pressures act when inertia forces go from the soil towards the retaining structure; static horizontal pressures were computed as oh = K0 ·ov, where K0 is the at rest coeffi­cient and ov the vertical pressure expressed as y. z. Values of K0 = 1 -sencj> were used, where cj> represents the angle of internal friction of the retained soil. For compacted granular fills the initial static horizontal pressure is computed using the procedure outlined by Broms (1971).

Second condition means no relative horizontal displacement at any depth between the structure and the soil immediately close to the wall. This condition was introduced to maintain approxima­tely the at rest free field condition used to compute 6z values.

Finally, it is necessary to point out that the kinematic method was set for small inertia forces coming from the structure (i.e. mostly buried structures, underground structures with light weight aerial members, etc).

Free Field Displacements

Maximum free field horizontal displacements were computed using a simplified approach given by the following equations:

Ycz

z f Ycz · dz

H

Gcz · g

( 4)

( 5)

( 6)

623

( 7) 3

(1- O.Ol67z) ( 8)

Equation (5) gives the maximum shear strain using a reduction coefficient, rd. According to Seed and Idriss (1971) and others analyses, a simplified expression to obtain rd is given by equation (8).

Equation (6J applies for cohesionless granular soils and gives the shear modulus in ton/m2 if the mean initial confining pressure, oc, is expressed in ton/m2. The shear coefficient, K2, used in that equation is strain independent when dealing with a linear elastic behavior. However, for nonlinear granular soils, K2 depends on the soil characteristics as well as on the shear strain, Ycz· Figure 4 gives the strain dependent variation of K2/K2max for granular soils, where K2max is the shear coefficient obtained for small shear strains (i.e. Yc<l0-4%). For cohesive soils equations (6) and (7) are replaced by direct correlations between Gc/Su and Yc• where Su is the undrained strength of the soil (Seed and Idriss, 1970). Accordingly, if the soil is characterized by K2max or Su, it is possible to obtain the shear modulus as a function of the shear strain level (actually a secant equivalent linear modulus).

By combining equations ( 4) through ( 8), an iterative computation process was used to obtain free field horizontal displacements.

Modulus of Subgrade Reaction

The horizontal modulus of subgrade reaction was computed using the equation given by Gonzalez and Ortigosa (1977), which applies for walls rotating about the top, about the base or subjected to horizontal translation:

N 0.4 :,::

kz 2 Gcz

1 - v

Range for 0.2 sandy gravels

(Ortigosa 1 1987)

1 ( 9)

0~------~--------~--------~------~ 10- 4 10-3 10-2 10-1

Shear strain I 'lc ("f.)

Fig. 4 Strain dependent Variation of the Shear Coefficient for Granular Soils.

N

• ..c ~ 3~---~~----~-------~-------,--­Q.o

0

Fig. 5 Horizontal Modulus of Subgrade Reaction Tunnel Section within the Santiago Sandy Gravel.

Figure 5 shows measured kz values along the wall of a tunnel section at the Santiago subway, located within a very dense sandy gravel deposit. Predicted kz values were obtained by means of equation ( 9 l using a Poisson ratio from triaxial tests on large size undisturbed samples. The shear modulus vs depth used into equation (9) for predicting kz values exhibited a well defined profile either from triaxial tests, building settlement measurements or horizontal plate tests (Ortigosa, 1987 l. Predicted values compare pretty well with those measured, so the kinematic method used equation ( 9 l for computing kz values. Final Gcz values given by the iterative process when computing maximum free field horizontal displacements were introduced into that equation.

Water Table

For simplicity, water table in granular soils was handled assuming no excess pore pressures (i.e. dense or high permeability soils behind the retaining structure). Saturated unit weight was introduced into eq. (5) when dealing with soil layers below the water table and bouyant unit weight was used for Oc computations. For saturated cohesive soils a similar approach was used when using eq. (5), but shear modulus was obtained using direct correlations between Gc/Su VS Yc·

RESULTS

Linear Behaviour

The kinematic method was applied for retained materials with elastic linear behavior and perfectly rigid structures without horizontal displacements (6sz = 0). Figure 6 compares the dimensionless seismic pressure, cr, obtained using a constant shear modulus with depth,

624

~ H

0 r----,-----r---....----rrO--e.--, -- Wave equation

0.2

0.4

0.6

0.8

lO 0 0.8

Cr =Cis JL d'H a

l2

Arias (1982)

F E.M. rough wall, Wood (1973)

FE.M. smooth wall, Wood (1973)

---+-- Kinematic method

-o-- FE.M. smooth wall, Kern County earthquake Nadim-Whitman (1982)

Arias, Wood and Kinematic method use constant acceleration

1.6

Fig. 6 Dimensionless Seismic Earth Pressures against Rigid Walls without Horizontal Displacement: Constant Elastic Shear Modulus with Depth.

z H

0.4

FE. M. smooth wall Q6 Kern County quake1959

Nadim-Whitman (1982)

Cr = Cis _g_ tH a

Fig. 7 Dimensionless Seismic Earth Pressures against Rigid Walls without Horizontal Displacement: Variable Elastic Shear Modulus with Depth.

v=0.30, L= 5H and a constant horizontal accele­ration, a, acting simultaneously in all points of the retained material. Figure 6 also includes maximum seismic pressures, Osr reported by Nadim and Whitman for a rigid wall with H= 10m, subjected at its base to the S69E acceleration record of the Kern County 1952 earthquake, scaled to represent a peak accele­ration of 0.4g. To set Nadim and Whitman results into a dimensionless factor, Os values were divided by the peak acceleration of the earthquake record. Results under similar conditions are given in Fig. 7 using a shear modulus Gcz = K ~. where K is a constant.

According to these results, for linear elastic materials the simplified kinematic method predicts seismic pressures with enough engi­neering accuracy when compared with more sophisticated solutions.

0

2

4 E

N 6

8

10 0

--.;;:::, r-=-::.: ------.:: ~ ~ r--

...... -v ' -- Kinematic Method ' ' --- F.E.M. smooth wall r-. / Kern County quake 1959 l( Nodim-Whitmon ( 1982)

/ I v I

v / /

/

/ ..---- ......

~ ~ 0.2 0.4 0.6 0.8 1.0 1.2

Cr = ~ _g_ ~H Omax.

1.4 1.6

Fig. 8 Dimensionless Seismic Earth Pressures against a Rigid Wall without Horizontal Displacement: Comparison for a Dry Sand with Nonlinear behavior.

Nonlinear Behaviour: Rigid Structures

As a first step, the kinematic method was used on perfectly rigid structures without horizontal displacement (Osz = 0). The retained soil was a dry sand with ~=30°, y=2ton/m3 , v=0.3, K2max=50, average K2/K2max vs Yc given in Fig.4 and a maximum free field acceleration amax=0.4g. Fig. 8 compares the dimensionless seismic pressures with those reported by Nadim and Whitman using F.E.M. for a rigid wall with H=lOm, the same sand and the S69E Kern County acceleration record scaled to a peak accelera­tion of 0.4g applied at the wall base. To set Nadim-Whitman's results into a dimensionless factor, maximum Os values given by these authors were divided by the peak acceleration of the earthquake record. Similar comparisons are in progress using wall heights ranging 4 to 13m and other acceleration records such as El Centro 1940 and Chilean earthquake records on rock and deep sand deposits, scaled to represent a peak acceleration ranging O.lg to 0.4g.

Comparisons were also established with measured seismic pressures on the basement walls of a building in Yokohama (Ikuta et al, 1979). The Yokohama Tenri Building has two basement floors and 27 stories above ground level. The founda­tions comprise cast-in place piles supporting the central core and basement walls extended to 26-28m from ground level, forming a continuous piling wall supporting the perimeter. Maximum seismic pressure records were taken on the

1.0

0.8

N" E 0.6 ...... c 0

0.4

0.2

0

• _,

0

/ ~eosured

• • 0--. .~ r--...

['--.--' I ~. ~ Kinematic \ Method, omax=0.013g

10 15 20 25 Depth z(m)

Fig. 9 Comparison between Predicted and Measured Seismic Earth Pressures.

basement walls during the earthquake of 12 June 1978 (magnitude 7.4 and epicentral distance 380km) which caused a maximum acceleration at the 2nd basement of about 12.5 gals (0.013g). The soil profile comprised a thick alluvial deposit of soft saturated silt with y = 1.52 ton/m3, K0 =0.58 obtained from measured static pressures and Su ranging 1.25 to 4.7 ton/m2, which reached the hard support layer of a diluvial deposit at a depth of 22 to 28m. The kinematic method was applied using the average Gc/Su vs Yc correlation for saturated clays proposed by Seed and Idriss, a Poisson modulus v=0.5, a perfectly rigid retaining wall going down 25m and a maximum free field acceleration of O.Ol3g. Figure 9 shows a more or less good comparison in spite of ignoring inertia forces coming from the building and uncertainties due to unknown free field accelerations.

A parametric study followed using the kinematic method on perfectly rigid structures with Osz=O and granular retained soils. Table I summa­rizes parameters used in such analysis, which are referred as Basic Cases, and Fig. 10 shows typical results.

To see the influence of the K2/K2max vs Yc curve on seismic pressures, some Basic Cases with H=4m and H=lOm were selected using K2max=50 and 150, and the upper and lower bounds for sands given by Fig. 4~ or K2max=310, the average curve for sands and the lower bound for sandy gravels. Typical results are illustrated in Fig. 11 showing a maximum variation of the average seismic pressure along the wall height of ± 7%. This percentage increases up to ± 12% when considering the whole set of selected cases. The influence of the angle of friction was also analyzed using ~=38° and 45° on some selected Basic Cases, proving to be less than 10%.

TABLE I. Parameters for Basic Cases: Rigid Walls with Nonlinear Granular Retained Soils

Soil Characterization H amax K2 v y --- vs Yc

K2max ~ (m) K2max (tonjm3)

50 38° 4 O.lOg Average 0.30 2.0 (dry) 150 42° 7 0.15g curve 2.25 (water 310 45 ° 10 0.20g for sands table at the

13 0.30g retained soil 0.45g surface)

625

0 0

~ '\ " a max = 0.45 g /

2

3 E

N 4

5

6

2

~ 3 E

N 4

6

7

,\ \ '\ 0.1g I o.39

0.2g

1

I I v

IM v

2

4

E

N 6

Dry Soil K2max = 150

H =10m K2max = 150

H = 7m 8

--- Upper bound for sands.

-- Lower bound

10 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 0 4 8 12 16 20

Cis (ton/ m2) <Ts(ton/m2) G's(ton/m2)

Fig. 10 Seismic Earth Pressures against a Rigid Wall without Horizontal Displacement: Basic Ca­ses for a Dry Nonlinear Granular Soil.

Fig. 11 Rigid Wall: Influence of the Shear Modulus Strain depen­dent Variation.

Values of the dimensionless average seismic pressure along the wall height, cr, obtained from the Basic Cases analysis are plotted in Fig. 12. These values can vary up to ± 12% due to changes on the K2/K2max vs Yc curves as pointed out before. Dimensionless seismic pressures in Fig. 12 were computed using averaged results for H=4m to 13m, so additional variations on Cr must be considered due to a non perfect normalization with the wall height, ending with a maximum final percentage of variation of the order of ±15%. As a reference, Fig. 12 includes the dimensionless average seismic pressures along the wall height given by the M-0 formula. The M-0 average pressures were plotted as a function of K2max• using K2max vs ~ correlation established for the Basic Cases as shown in Table I.

~li j'I ~""

" L. lu

Fig. 12

1.4 -- Dry ( t = Yl ---- Submerged ( Y = J1 sat l

1.2

1.0

0.8

\ _Q.45 g

~~

~ I RIGID WALL Jsz=Of '

0.30g .... ' ~ ~--

::::.::::...:!.;), -- --- - -- -0.6

0.4

amax =0.1g1

.... -- ~-r 0.45g

0.3 g\ lM-0 I ---~- - -- -4· l:;.z:::~

0.2 amax =0.1g

- -

I 100 200 300

K 2max

Dimensionless Average Seismic Pressure for Granular Soils from Basic Cases. ·

626

Results in Fig. 12 show a significant increase of seismic pressures when going from yielding retaining structures, where M-0 formula ap­plies, to rigid walls without horizontal displacement. In spite of assuming no pore pressure increments, soil bouyancy also produce an increase of seismic pressures for both yielding and rigid structures. However, increase due to wall rigidity is significantly greater than the increase due to soil bouyancy.

Nonlinear Behaviour: Flexible Walls

Retaining structures similar to the basement wall shown on Fig. 3 were analyzed using the kinematic method. For simplicity, the analysis was performed assuming a constant wall thick­ness, t, and a constant span lenth, h, between rigid horizontal supports. Soil-structure interaction at the wall base was modeled using a very large horizontal stiffness (KH=~l and a rocking spring representing a hinge (Ke=O) or a fixed support (Ke=~J. The retained soil was a dry or a submerged nonlinear granular material. Accordingly, a parametric analysis was perfor­med using the soil and wall characteristics given in Table II. Typical results are illustrated by Fig. 13 for one and three span walls with hinged support at the wall base. This figure show a larger seismic pressure attenuation as long as the span flexural flexibility, Fs, increases. This parameter was defined as:

1 Fs = I ( 10)

EI 12

The whole set of results from the parametric analysis were plotted as As vs Fs, where As is the seismic pressure attenuation coefficient defined as Cf/Cr· Values of cr represent the dimensionless average seismic pressure along the wall height as defined previously for a rigid retaining wall (Fs=O) without horizontal displacement (Osz=O), and Cf is the correspon­ding dimensionless average seismic pressure for

TABLE II. Parametric Analysis: Flexible Walls with Nonlinear Granular Retained Soils

Soil Characterization Number of h t Ka E a max K2 v r --- vs "Yc

K2max q, spans (m) (m) (ton/m2) K2max ( ton/m3)

50 38 ° 1 2 0.15 0 2.sx1o6 O.lg Average 0.30 2.0 (dry) 150 42° 2 3 0.20 "' 0.2g curve 2.25 (water 310 45 ° 3 4 0.25 0.3g for sands table at

0.30 0.4g the retai-0.45 ned soil 0.60 surface)

-~ Rigid

ll' ~ J \

fll

1-- !ve I .... c::,· I lj / 1 Span

_j_ ~ v h =I. m K2max =150

0

2

3

I.

0 I .,._._

~igid

"" ~ r-- 3 Span ~'\

f-- h = l.m ~ K2max = 150 -~

./// \ I 'V

t = 0.1.5m .# l ..... V/

.... ::;. ~ rc- 0.60 ~ ~

2

" E 6 N

8

10

12 0 2 3 " 0 8

<Is (tonfm2)

Fig. 13 Seismic Earth Pressures against Flexible Walls with Hinged Support: Dry Nonlinear Granu­lar Soils with amax = 0.3g.

the same wall with some degree of flexural flexibility (Fs>O), KH="' and Ka=O or "'· Figures 14 and 15 summarize the As vs Fs correlation for Ka=O and h=2 and 4m, respective­ly. Quite similar results were obtained for submerged granular soils. Some important features related to Figs. 14 and 15 must be pointed:

Each curve has been plotted until reaching a limit value of the span flexural flexibility, Fsmax• to satisfy the condition given by equation(3); for Fs>Fsmax at least one point on the retaining wall exhibits an horizontal displacement greater than the free field seismic displacement ( 6sz>6z). Greater attenuation coefficients can be obtained if condition given by equation ( 3) is not satisfied at some points along the wall.

Each curve represents an average correlation between As and Fs, with a maximum variation of ±12% for K2max=50 and ±8% for K2max ~ 150. These variations arise from a non perfect normalization with the wall height and the maximum free field acceleration.

As a reference, it is shown the seismic pressure attenuation coefficient using Cf values from the M-0 formula.

Attenuations for the limit span flexural flexibility are lower than those obtained with the M-0 formula. This is in accordance with the maximum wall displacement associated with that limit (i.e. for one span wall, displacements are of the order of O.OlH,

627

0.002H and O.OOlH for K2max=50, 150 and 310, respectively). These displacements seem to be lower than those required to reach a full state of plastic equilibrium in the retained soil, which is an implicit condition when using the M-0 formula.

For a given span flexural flexibility, seis­mic pressure attenuation increases as long as the soil stiffness, typified by K2maxr increases.

FINAL REMARKS

A simplified kinematic method was developed to compute seismic pressures against retaining structures with restrained horizontal displa­cements. The method can be applied using linear and nonlinear behavior for granular and cohesive soils under dry and submerged condi­tions. Submerged granular soils are assumed with no pore pressure increments. The method is suitable for parametric analysis using a microcomputer, but it is constrained to solve problems with small inertia forces coming from the structure. More empirical evidences are needed to validate results given by the kinema­tic method or other analytical approach.

ACKNOWLEDGEMENT

The writers wish to acknowledge their apprecia­tion for the finantial support offered by the National Commission for Science and Technology of Chile (Research Project FONDECYT 89-964)

2 SPAN

1. 0.----,,---r-r--rrn-n-.,----r..-rTTTTl

0.9 +-----""1..::----+-r~r++-H---~

0. 8 +---f-t-+-+++++1---+-+-+--m+H

3 SPAN 0. 6+---f-+-t--t+Htt--+--.-,--,rrrrrl

0. 5+---f-+-t--t+Htt--+--+-Hr-t+lti 0.5+--1--~~H+H--+-+4-H++H

Fs max (Typ)

0.4 ~M-0 ~ = 38~-45~

0. 3 3 2 1 10- 10- 10- 10-2

Fs ( m3 I ton) Fs (m3tton)

10-2 Fs(m3/ton)

Fig. 14 Seismic Pressure Attenuation Coefficient for Flexible Walls with Hinged Support and h=2m Dry Nonlinear Granular Soils with amax = O.lg to 0.4g.

1 SPAN 0.9 ~ r--,b, 0.9+--T-t""'k-t+-?rrt--+-+4'~

O.A 1\._\~ ~ ~ 1'\ ~

0.7+----1~'\-~'-o-+"<+~-'--&.uo'++--4.:\~

0. 8+----1i--t-t-+tt++f----t--t~-++1f-H-i

0. 7 +----11--t-+-++++++----'---'-'---.L..L..If-H-i

0. 8+--t-t-+-t++-1+1--+-++-+++++1

0. 7 +---lt--t-t-++++++--'---.L....I.--'-'-+f-H

2 SPAN 3 SPAN

0.6 1\ IN \ 0.:>1;.-t-----lr--t--t--i 1 ~

~

0. 6 +----11--t-+-++++++-----,----,,---.--rif-H-i

0. 5+---t--f-+-++++++--+--+-+++++-H

0.4 ~\~' ~-0

0.3 3 10-

Fs max CTyp)

10-2 10-2 Fs(m3/ton) Fs ( m3 /ton)

Fig. 15 Seismic Pressure Attenuation Coefficient for Flexible Walls with Hinged Support and h=4m: Dry Nonlinear Granular Soils with amax = O.lg to 0.4g.

REFERENCES

Arias, A. (1982), "Seismic Analysis of Retaining Walls", Revista Ing. Sismica, N° 26, Sociedad Mexicana de Ing. Sismica.

Brems , B. B. ( 1 9 71 ) , due to Compaction Proc. 4th. Conf. pp 373-384.

"Lateral Earth Pressures of Cohesionless Soils", on Soil Mech., Budapest,

Ikuta, Y., M. Maruoka, T. Mitoma and M. Naganou (1979), "Record of Lateral Pressure taken during Earthquake", Soils and Foundations, Vol. 19, No 4, pp 85-92.

Nadim, F. and R. V. Whitman (1982), "A Numerical Model for Evaluation of Seismic Behavior of Gravity Retaining Walls", MIT Research Report R82-33.

Gonzalez, R. and P. Ortigosa (1977), "Lateral Soil Restriction on Piers", Revista del !DIEM, Vol. 16, N° 2, pp 73-102.

628

Ortigosa, P. (1987), "Horizontal Plate Tests on Gravels", Proc. VI I I Panamerican Conf. on Soil Mech. and Foundation Eng., Cartagena, Colombia, pp 597-606; see also pp 607-619.

Richards, R. Jr. and D. G. Elms (1979), "Seismic Behavior of Gravity Retaining Walls", Journal Geotech. Eng. Div., ASCE, Vol. 105, N° GT4, pp 449-464.

Seed, H. B. and I. M. Idriss (1970), "Soil Moduli and Damping Factors for Dynamic Response Analysis", Report N° EERC 70-10, University of California, Berkeley.

Seed, H. B. and I. M. Idriss (1971), "Simpli­fied Procedure for Evaluating Soil Liquefac­tion Potential", Journal Soil Mech. and Found. Div., ASCE, Vol. 97, N° SM9, pp 1249-127 3.

Wood, J. H. (1973), "Earthquake - Induced Soil Pressures on Structures", Report N° EERL 73-05, CALTECH, Pasadena, California.