1987 Permanent Slope Displacements, Houston, Houston, Padilla

Houston, S. L.; Houston, W. N. and Padilla, J. M. (1987). Microcomputer-Aided Evaluation of Earthquake-Induced Permanent Slope Displacements. Microcomputers in Civil Engineering, p.p. 207-222.

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MICROCOMPUTER-AIDED EVALUATION OF EARTHQUAKE-INDUCED

PERMANENT SLOPE DISPLACEMENTS

Sandra L. Houston, William N. Houston, and Manuel Padilla.

Arizona State University

A Geotechnical engineering microcomputer program has been developed for the

determination of permanent slope displacements resulting from earthquake shaking. The

Newmark procedure, in which accelerations in excess of yield accelerations are double

integrated to obtain displacements, has been incorporated into the program DISPLMT.

Several user options are available for describing the yield acceleration function, including

variation with time and variation with displacement. Screen graphics are available which

allow the user to observe the down slope movements of the Newmark sliding block as they

increase with time during the simulated earthquake. DISPLMT has been used to calculate

the permanent movements of an earth dam using acceleration-time histories and soil shear

stresses determined from a separate analysis. Modification to the conventional Newmark

procedure, by introduction or a “slip layer” has been made in the analysis of a dam.

Permanent slope displacements can be analyzed using the recommended procedure, which

involves fairly simple computations that can be performed in a reasonable period of time

with the exclusive use of microcomputers.

INTRODUCTION

Approximately four methods of assessing seismic slope stability are currently being

used be or are under study. The most ideal of these analyses is a two or three dimensional

finite-element analysis which employs a complex general constitutive model which is

capable of quantifying both elastic recoverable and plastic permanent strains. The output

from such an analysis includes stresses and strains as a function of time, displacement

versus time, and a description of the final deformed shape of the slope. Although recent

research activities have been devoted to development of such an analysis [ ]2,6 , it requires

considerable run time on a mainframe computer, and it is probably beyond the state-of-the-

practice at the present time.


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The most simplified approach would be a pseudo static analysis, in which an

“equivalent” static force is applied to the slope, and a factor of safety is determined. This

type of analysis provides no information on permanent movements, which is one of the

most important pieces of information in assessing the safety of an embankment or earth

dam against earthquake shaking. Another technique which has been used recently is to

perform a static analysis “after the earthquake” using post-earthquake shear strengths,

which are less than static strength values [ ]1 . This method also provides no information on

the permanent slope displacements due to ground shaking.

Probably the most frequently used technique for computing earthquake-induced

slope displacements is the Newmark method of double integration of the “excess”

accelerations [ ]3 . The assumption is made that only the permanent strains are of interest or

that all strains which occur prior to the point when the dynamic stress equals the dynamic

strength of the soil are negligible.

First, the slope is modeled using the sliding block analogy as shown in Figure 1.

When the acceleration of the block is such that the frictional sliding resistance is just

overcome, the factor of safety against sliding equals one, and the acceleration is said to

equal the yield acceleration, ya . They corresponding point for the real slope is when the

acceleration is just large enough to drive the factor of safety to one. Using a slope stability

program which employs seismic coefficients, the coefficient which brings the factor of

safety to one can be determined. This coefficient is the yield acceleration. A dynamic

response analysis is then performed, and the acceleration-time histories at points along the

“critical slip surface” are determined. A representative time history is typically scaled so

that the maximum acceleration is equal to the average value of several points along the

potential slip surface. The accelerations in excess of the yield acceleration are then double

integrated to get displacements.


3

Figure 1. The sliding block analogy.

Microcomputer slope stability programs are available which can be used to obtain

the seismic coefficient required to bring the slope to a factor of safety of one. Acceleration-

time histories which are representative for the slope can be obtained using a two-

dimensional finite-element analysis, or approximated using a series of one-dimensional site

response analyses. Microcomputer programs for one-dimensional site response analyses are

also available. Of course, if the site response analysis is sufficiently complex and complete

so as to include permanent strains as output, the Newmark method would not be required.

Probably the most commonly used program for the site response analysis is a wave

propagation program called SHAKE [ ]7 , which does not provide permanent strains as

output.

DISPLMT, developed as a part of this study, can be used to perform the double

integration of the accelerations in excess of the yield acceleration. Thus, the evaluation of

the permanent slope displacement resulting from earthquake shaking can be made using

only microcomputer computations and the relatively simple procedure described below.

RECOMMENDED PROCEDURE

When the “sliding block on an inclined plane” analogy is applied to a potential

landslide, as depicted in Figure 2, a number of difficulties arise. First, the slip surface

(shear zone) may be curved rather than planar. However, the problems to which this

analogy are usually applied typically involve slip surfaces without severe curvature and

with relatively gentle slopes, which may be represented by a noncircular slip surface as

shown in Figure 2.


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Secondly, the block and the base are both rigid, whereas neither the sliding wedge

nor the material below it in Figure 2 are rigid. The lack of rigidity in the real soil profile

comes into play when the dynamic response of the soil column to an earthquake is

calculated [ ]3 . This calculation will be discussed shortly. First however, the computation of

the static factor of safety, staticFS , and the yield acceleration, ya , for the sliding wedge in

Figure 2 will be addressed.

Figure 2. Typical slope failure to which block analogy is applied.

The static factor of safety is defined as the ratio of the maximum available shearing

resistance between base and block to the shearing resistance between base and block

required for equilibrium when the base is not shaking. If this static factor of safety is

greater than one, then the block does not slide for “static” conditions.

The yield acceleration is computed by performing a pseudo-static slope stability

analysis in which the entire sliding wedge is indeed assumed to be rigid - or at least the

“slices” are assumed rigid. Each and every slice is assumed to experience the same

acceleration at the same time. This inertial force is added vectorially to the gravitational

forces and the factor of safety is computed. The acceleration applied to all slices is varied

by trial, and error until the computed factor of safety is one. The corresponding acceleration

is ya . In general, ya would not be constant, but would vary with time or displacement. This

is because the shearing resistance of the soil, which relates directly to ya , may decrease


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with time due to the damaging effects of the shaking. The yield acceleration is likely to

begin at the value of ya corresponding to the static soil strength, and end with the value of

ya corresponding to the post-earthquakes strength. Alternatively, it would be reasonable to

allow variation of the yield acceleration as a function of displacement. Both of these

options, as well as constant yield acceleration, can be selected in DISPLMT.

Usually, only the downslope value of ya is computed. However, the upslope

component is also required for this analysis of permanent slope movements. The upslope

component of ya can be computed using the static factor of safety and the downslope

component of ya , i.e.,

staticMRFSSR

=

( )y downslope

MR SRam−

=

( )y upnslope

MR SRam+

=

Where,

MR = Maximum available shearing resistance

SR = Shearing Resistance needed for static equilibrium

m = Mass of the block

Upon rearranging and combining these three equations, it can be found that the ratio

of the downslope value to the upslope value of ya is equal to:

( )( )

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static

static

FSFS

−+

The dynamic response of the soil slope to an earthquake motion imposed at

“bedrock” (point R in Figure 2) can best be computed by a two-dimensional finite-element


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analysis. A number of comparisons have shown, however, that a satisfactory approximation

can be obtained by using one-dimensional wave propagation programs [ ]7,9 .

A series of three profiles are analyzed, one near the crest, one near the center, and

one near the toe of the potential sliding wedge. Examples are shown as profiles 1, 2, and 3

in Figure 2. The results are then averaged in order to obtain a response which is quite close

to the average response that would have been obtained from a two-dimensional finite-

element in term of acceleration-time history. The input motion at point R propagate

upward toward points A , 3B , 2B , and 1B . Trey may be ether amplified or attenuated, but

they are, in general, modified as they propagate upward.

For typical earthquakes and typical soil profiles, the elastic strains are moderately

small and the computed maximum accelerations at point 1B , 2B , and 3B in Figure 2 will

often exceed ya . In fact, if these acceleration did not exceed ya , then no problem with

seismic stability would exist. As cited earlier, a problem does arise when the calculated

accelerations exceed ya . In the “sliding block on an inclined plane” model that is being

utilized, the acceleration in the block or the sliding wedge can never exceed ya because

movements would occur. Therefore, computed accelerations above ya in the sliding wedge

are inconsistent with the model being employed.

In order to avoid this inconsistency, it is necessary to introduce a “slip layer” with

“softened” properties at the location of the shear zone. The properties of the “slip layer” are

varied by trial and error until the average maximum acceleration at points 1B , 2B , and 3B is

approximately equal to ya . When this convergence is achieved, then the corresponding

acceleration-time history at point A (just below the “slip layer”) can be compared to ya

and the “excess” accelerations double integrated to obtain a permanent movement estimate

for profile 2, in Figure 2. Without the introduction of such a slip layer, dynamic response

programs, such as SHAKE, would predict accelerations within the sliding mass which are

in excess of the yield acceleration. Perhaps more importantly, the acceleration-time history

User

Resaltado

User

Resaltado


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at point A may depend somewhat on whether or not a “slip layer” that prevents the

acceleration of the block from exceeding ya is present. This is because the acceleration-

time history at any point will depend on how much energy is being dissipated in the

vicinity of the point. Therefore, in order to get the best estimate of the acceleration-time

history at point A (when maxa within the wedge is essentially equal to ya ) the “slip layer”

should be present.

Similar movement estimates can be made for profiles 1 and 3 and then an average

movement can be calculated. Although the average is probably the “best estimate”, the

average as well as the maximum should be reported. These movement computations can

readily be made with a computer program such as DISPLMT.

It should be noted that the sliding block model does not apply perfectly to the

landslide problem. In fact, in view of the various approximations required, it probably

should be viewed as a tool to assist the engineer in deciding whether the probable slope

movement is: (l) a fraction of an inch, or (2) a few inches, or (3) a few feet. This level of

distinction is usually adequate to enable an engineering or management decision. The

computational method cannot be used to realistically distinguish between 0.33 feet and 0.57

feet, for example. However, such a distinction is often made during the analysis stage in

order to evaluate parameter sensitivity.

PROGRAM DISPLMT

DISPLMT is a FORTRAN microcomputer program which calculates permanent

slope displacements resulting from earthquake shaking. The Newmark method, in which

accelerations in excess of yield accelerations are double integrated to obtain displacements,

has been incorporated into DISPLMT. The inputs to the program are the static factor of

safety, the yield acceleration, and the acceleration-time history at the point along the slip

surface which is being evaluated. Three options for specifying the yield acceleration

function are available to the user:


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1. Constant yield acceleration throughout the earthquake.

2. A yield acceleration function which varies as a function of time. The variation is

prescribed by the user.

3. A yield acceleration function which varies as a function of the computed permanent

displacement. The variation is prescribed by the user.

Program outputs consist of a table of the maximum velocity and final permanent

displacement or a table containing the entire velocity and displacement history. In addition,

screen graphics are available which allow the user to observe the movement of the sliding

block as it progresses throughout the computation. Also plotted are the acceleration-time

history and the yield acceleration versus time. Screen graphics have been accomplished

using a graphics kernel software package, which is a set of subroutines written in assembly

language that can be called from FORTRAN. A listing of the program, which includes

general input instructions, has been included in Appendix II to this paper.

The “excess” accelerations are double integrated in DISPLMT to obtain the

permanent displacement of the slope. Figure 3 shows a sketch of a yield function which

varies linearly with time from the computed yield acceleration. using pre-earthquake soil

strength parameters, to the yield acceleration computed using post-earthquake soil strength

parameters. The trapezoidal method, a simple numerical integration scheme, is used to

perform the integration. The interval from the initial time of zero to the desired final time,

ft , is subdivided and the areas of the trapezoids (shown in Figure 3) are summed to obtain

an estimate of the area under the “excess” acceleration curve. Only the contributions to the

integral which fall above the yield acceleration line are included in the summation. The

corresponding velocity curve is then integrated using the trapezoidal rule to obtain an

estimate of the permanent slope displacements. The error in the computation described

above is, of course, reduced by decreasing the time step used in describing the acceleration-

time record.


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Figure 3. Numerical integration of the acceleration-time history.

Figure 4. Flowchart for program DISPLMT.

The time interval for computation of the area under the acceleration curve is

normally taken as the time step between the given data points of the acceleration-time

history. However where there are consecutive acceleration values above and below the

yield acceleration line, that particular time interval is subdivided so that two time intervals

are generated in which the acceleration record is either entirely above the yield

acceleration, or entirely below the yield acceleration line. If this subdivision is not made


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then the computed displacement would be over estimated. Thus, not all time intervals used

in the computation are the same, as is the usual practice in implementing the trapezoidal

rule.

To specify an arbitrary yield acceleration as a function of time or displacement, the

user must input a series of values of yield acceleration and time or yield acceleration and

displacement. The yield accelerations between specified values are obtained by linear

interpolation.

Figure 4 shows the DISPLMT program flowchart. The input data is read first. Since

only the downslope yield acceleration is given, the program computes the ratio of ( )y upslopea

to ( )y downslopea using the static factor of safety, so that the upslope yield acceleration can be

computed at any later time.

After setting the variables to their initial values and setting up the graphical display,

the main loop for computing permanent slope displacements begins. First, time

incremented by t∆ . Then ( )y downslopea is determined using the actual time or actual

displacement, depending on the given function as prescribed in the input data. The

( )y upslopea value is also computed. Negative accelerations, velocities and displacements are

upslope.

Next, the program checks for any downslope movements. If the ground response

acceleration, ( )a t , is greater than ( )y downslopea or the initial velocity for this interval is

greater than zero, then there is downslope movement. If there is movement, then the

program calls a subroutine to compute the values or velocity and displacement. In this

subroutine, the actual double integration is performed. Before double integrating the

“excess" accelerations, the program checks for a need for subdividing the original time

interval, t∆ .


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If there is no downslope movement, then a check is made for upslope movement. If

the ground response acceleration, ( )a t , exceeds or ( )y upslopea or the initial velocity for this

interval is less than zero then there is an upslope movement. Again, if there is movement,

the program calls the subroutine to compute velocity and displacement.

At the end of this main loop, the graphic screen is updated with the new values or

displacement and accelerations. The analog block is moved to the new displaced value.

This is done by first erasing the block at its original position and then plotting it at the new

computed position. The original position or the block is shown in all cases by a dashed

outline. Time, t, is incremented by t∆ and the main loop is entered again.

When the final time is reached, the acceleration-time history is reversed by

changing the sign to each one or its values and the process previously described is

performed again for the reverse earthquake. This reversal is done in case the input

acceleration-time history is significantly and unsymmetrical.

It is generally assumed that both the first run and the reversal run are of equal

validity. The two runs are used to indicate the probable range in results.

IMPLEMENTATION OF RECOMMENDED PROCEDURE TO AN EARTH DAM

An existing earth dam was recently enlarged for the purposes of improving flood

control. The original earth dam was approximately 650 feet long at the crest and about 79

feet above the original ground surface. The enlargement raised the dam 8 feet in height.

The dam is located in a small valley with a 45 ft. layer of alluvium overlying bedrock. It is

in a region of seismic activity, in which several major earthquakes have occurred within

about 50 miles of the dam, ranging from a Richter magnitude of 8.3-5.8 over the past 80

years or so. Several major faults are located from 10 to 40 miles of the site, and several

small inactive faults and shear zones are present in the area near the dam

The recommended procedure, described in the previous sections, was used in

conjunction with the newly developed microcomputer program, DISPLMT, to estimate the


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permanent slope displacements of the earth dam using the Newmark method of double

integrating the excess accelerations. The critical noncircular slip surfaces which were

analyzed for the upstream and downstream slopes of the dam are shown in Figures 5 and 6

respectively.

The design earthquake for the seismic response computations was an 8.25 Richter

magnitude. Because there are no available measured records for earthquakes of this

magnitude, the synthetic “Seed-Idriss” record was used as the input motion.

DOWNSTREAM SLOPE ANALYSIS

The critical slip surface for the downstream slope and static loading conditions, as

shown in Figure 5, was noncircular. The static factor of safety of 1.37 was determined

using the Morgenstern and Price method and the microcomputer program TSLOPE [ ]10 .

The yield acceleration, using pre-earthquake soil parameters, was also determined using the

pseudo-static method in TSLOPE. The yield acceleration for the critical slip surface was

0.14 g, which was assumed to remain constant throughout the earthquake shaking. The

yield acceleration is computed as the acceleration which brings the slope to a factor of

safety of 1.0.

The procedure used in the computation of permanent slope displacement utilizes

one-dimensional seismic response analyses to obtain the acceleration-time histories which

are input into DISPLMT. The “average” acceleration-time history from the one-

dimensional analysis, which can be obtained using microcomputer versions of programs

such as SHAKE, is typically very close to that which is obtained using the more complex

two-dimensional site response analyses performed using finite-element programs on a

mainframe computer. This has been verified by the authors for several different soil profiles

and slope geometry.


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Figure 5. Critical downstream slip surface.

Figure 6. Critical upstream slip surface.


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In the analysis of the downstream slope of the earth dam, acceleration-time histories

were computed for several soil profiles within the sliding mass. Acceleration-time histories

were obtained for vertical soil profiles corresponding to slices 1, 5, and 10, shown in Figure

5. According to the recommended procedure, discussed previously, a “softened layer” was

introduced at the elevation corresponding to the critical slip surface in performing the

seismic response computations. The acceleration-time histories for points immediately

below the shear zone were then input into DISPLMT for the purposes of computing

permanent displacements. Acceleration-time histories were obtained at the three sections

along the slope using SHAKE, which incorporated an equivalent linear soil modulus.

Using the constant yield acceleration assumption, the permanent slope

displacements were estimated using DISPLMT. For the soil profile corresponding to slice 1

of the downstream slope, the maximum permanent deformation was 0.07 ft. downslope.

The maximum acceleration in the acceleration-time history for slice 1 was 0.253 g. The

middle of the slope, slice number 5, resulted in a maximum downslope movement of 0.27

ft. for a maximum acceleration of 0.328 g. The profile at the bottom of the slope, slice 10,

resulted in a maximum downslope movement of 0.85 ft. for a maximum acceleration of

0.437 g.

The average value of the permanent downslope displacement for the three profiles is

probably the best estimate for the slope, because the rigid block assumption require that the

displacements be equal for the entire slope. Therefore, the average permanent downslope

displacement for the downstream slope of the earth dam was estimated to be about 0.3 ft.

However, it is considered good practice to report the maximum calculated value as well.

The graphical output from program is shown in Figure 7 for the slope downstream

slope computation for the profile corresponding to slice 5 in the center of the critical sliding

mass. An interval of the acceleration-time history and the yield acceleration as a function of

time are shown along with the “moving block” in the output.


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Figure 7. Graphical output from DISPLMT showing downstream slope movement.

UPSTREAM SLOPE ANALYSES

A static slope stability analysis was performed on the upstream slope of the dam

using the microcomputer program TSLOPE. The static factor of safety was found to be

2.17 to the critical noncircular slip surface, and the yield acceleration, using pre-earthquake

strength values, was found to be 0.25 g.

Acceleration-time histories were computed to several soil profiles within the sliding

mass. In the upstream slope analysis, profiles corresponding to slice number 1, 8, and 14,

shown in Figure 6, were used in the one-dimensional site response computation to

determine the range of acceleration-time histories for the slope. In performing the one-

dimensional seismic analyses, a softened layer was introduced in the vicinity of the

“critical” slip surface. Using the recommended procedure, the acceleration-time histories

for points immediately below the slip surface were used to obtain “representative”

acceleration-time histories for input into DISPLMT.

A constant yield acceleration was assumed for the DISPLMT computations for the

each dam. For the soil profile corresponding to slice 8, the maximum downslope

deformation was 0.035 ft. For slices 1 and 14, the maximum accelerations computed from

the one-dimensional seismic analysis did not exceed the yield acceleration (0.246 g.), and

therefore no displacements were computed.


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The graphic output form DISPLMT for the slice 8 profile is shown in Figure 8. The

graphic output includes the “moving” block, as well as the plot of an interval of the

acceleration-time history and yield acceleration.

Figure 8. Graphical output from DISPLMT showing upstream slope movement.

The effect of introducing the slip layer with softened properties is summarized in

Table 1 for both the up-stream and downstream slopes.

Table 1. Effects of slip layer. Downstream Upstream

Maximum acceleration at base of wedge

Maximum acceleration at base of wedge

Profile a

( )g With slip

layer ( )g

Without slip layer ( )g

Profile

a

( )g With slip

layer ( )g

Without slip layer ( )g

Slice 1 0.253 0.202 Slice 1 - 0.253 Slice 5 0.14 0.328 0.256 Slice 8 0.25 0.372 0.273 Slice 10 0.437 0.273 Slice 14 - 0.252

For this particular analysis, the average maximum acceleration at the base of the

sliding mass is greater by about 35% when the layer with softened properties is introduced.

This will lead to larger permanent slope displacements than for the case where no slip layer

was incorporated in the seismic response analysis. In general, it would be expected that the

acceleration at the base of the sliding mass would be a function or the slip layer properties


17

because the dissipation of energy and the amount of strain depend on the slip layer

properties. The 35% increase in maximum acceleration observed is for this particular earth

dam, and the difference in acceleration with and without the slip layer would be dependent

upon the particular soil profile under study.

SUMMARY

Modifications to the conventional Newmark sliding block procedure for the

determination of permanent slope displacements resulting from earthquake shaking were

presented. Following the recommended procedure for estimating permanent slope

displacements, the entire analysis can be performed with microcomputer facilities alone.

The proposed method introduces a layer with softened properties at the location of the shear

zone for the computation of the seismic ground response. This “slip layer” prevents the

calculated accelerations within the slope from exceeding the yield acceleration, a condition

which is inconsistent with the sliding block analogy. The computed seismic ground

acceleration-time response just below the slip layer acceleration is then double Integrated to

obtain the permanent displacements of the slope.

A microcomputer program has been developed to double integrate the seismic

ground response for determining the permanent slope displacements. The program allows

the yield acceleration to be a function of time or displacement. Graphical output from the

computer program DISPLMT allows the user to observe the downslope movements of the

sliding block as they increase each time the yield acceleration is exceeded.

The recommended procedure was used to estimate the permanent slope

displacements of an earth dam. A one-dimensional microcomputer analysis using an

equivalent linear soil model was used for computing the acceleration-time history at three

representative profiles along the slope. DISPLMT was then used to calculate the permanent

downslope movements.


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REFERENCES

[ ]1 Castro, G., Poulos, S. J., and Leathers, F. D. A Re-examination of the Slide of the

Lower San Fernando Dam. Journal of Geotechnical Engineering, 3(9): 1093-1107

(1985).

[ ]2 Daddazio, R. P., Ettouney, M. M., and Sandler, I. S. Nonlinear Dynamic Slope

Stability Analysis. Journal of Geotechnical Engineering, 113(SM3): 285-298

(1987).

[ ]3 Lin, J. S. and Whitman, R. V. Earthquake Induced Displacement of Sliding blocks.

Journal of Geotechnical Engineering, 112(1): 44-59 (1986).

[ ]4 Makdisi, F. I. and Seed, H. B. Simplified Procedure for Estimating Dam and

Embankment Earthquake-Induced Deformations. Journal of Geotechnical

Engineering, 104(GT7): 849-867 (1978).

[ ]5 Newmark. N. M. Effects of Earthquakes on Dams and Embankments.

Geotechnique, 15(2): 139-160 (1965).

[ ]6 Prevost , J. H., Abdel-Ghaffar, A. M. and Lacy, S. J. Nonlinear Dynamic Analyses

of an Earth Dam. Journal of Geotechnical Engineering, 111(2): 882-897 (1985).

[ ]7 Schnabel, P. B., Lysmer, J., and Seed, H. B. Shake - A Computer Program For

Earthquake Response Analysis Of Horizontally Layered Soils. Report No. EERC

72-12. University of California, Berkeley, December 1972.

[ ]8 Seed, H. B. and Martin, G. R. The Seismic Coefficient in Earth Dam Design.

Journal of the Soil Mechanics and Foundations Division, 92(SM3): 25-58 (1966).

[ ]9 TESS1 – A Computer program for nonlinear ground response analyses. TAGA

Engineering Software Services. Berkeley, California, 1985.

[ ]10 TSLOPE – Computer program for limit equilibrium slope stability analysis. TAGA

Engineering Software Services, Berkeley, California, 1984.


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APPENDIX I – NOTATION

a = Acceleration

ya = Yield acceleration

( )a t = Acceleration-time history

FS = Factor of safety

p′ = Effective vertical stress

uS = Undrained shear strength

t∆ = Time incremental

φ = Friction angle

Documents

1987 Permanent Slope Displacements, Houston, Houston, Padilla