Analisis Sismico de ad de Taludes

Kyrou: Seismic Slope Stability Analysis

Seismic Slope Stability Analysis

Dr. Kyriacos Kyrou

1 Introduction Failures occur in both natural and manmade slopes. In our natural environment landslides occur

frequently and they are part of the ongoing evolution of landscape. More rarely failures occur in manmade earth slopes which are designed specifically to resist the forces of nature.

Landslides have caused great amounts of damage and loss of life throughout history. Examples of catastrophic landslides in the 20th century include the earthquake induced landslides in the Ningxia province of China in 1920 (> 100000 deaths), the earthquake induced landslides in Alaska in 1964 (tremendous damage) and the reservoir induced landslide at Vaiont dam, Italy in 1963 (2300 deaths).

Earthquake induced ground shaking is one of the most frequent causes of landslides. This is happening in marginally or moderately stable slopes where earthquake inertial forces may be sufficient to trigger a failure. In the case of weak foundations or embankment fill materials, repeated ground shaking may cause loss of strength of the soil materials (e.g. liquefaction) and subsequent slope failure.

2. Earthquake induced landslides The possibility of the occurrence of a landslide where a slope is subject to earthquake loading, depends on

numerous factors which include the geometry, the geology of the slope, the soil engineering properties, the ground water regime, the presence of pre-existing shear zones, the weather etc. It is not uncommon for a slope to survive stronger earthquake shaking and fail under lower earthquake shaking because some of the above factors were more favourable for the second case.

Landslides induced by earthquakes may be classified into three broad categories (Varnes, 1978). (i) Disrupted slides and falls (ii) Coherent slides (iii) Lateral spreads and flows

In the case of disrupted slides and falls, the soil or rock material in the slide is sheared and distorted in a nearly random manner. The slopes involved are usually steep and failures take place very suddenly. The damages and loss of life from such slides in developed areas may be devastating. Disrupted slides and falls include disrupted soil/rock slides, soil/rock falls and soil/rock avalanches.

Coherent slides generally occur at deeper failure surfaces in moderate to steeply sloping ground and they involve rotational and translational failures of coherent soil and/or rock blocks. These failures include rock/soil slumps, rock/soil block slides and slow earth flows. They develop at slow to rapid velocities.

Keefer (1984) studied the effect of earthquake Magnitude and epicentral distance on the occurrence of earthquake induced landslides. The study was based on 300 U.S. earthquakes and it showed that for local magnitudes less than 4.0, landslides rarely occur. He produced a relationship between maximum epicential distance and Magnitude for different types of landslides (see Fig. 1) and presented data that relate the area affected by earthquake induced landslides, with earthquake magnitude. His data show that the area over which earthquake induced landslides can be expected increases rapidly as earthquake Magnitude increases.

3. Evaluation of slope stability The evaluation of slope stability is a process that requires the collection of information on the geology,

topography and hydrology of the site, the prevailing weather conditions, the material engineering properties etc.

Slope stability analysis can yield sufficiently accurate results, only if the above factors are evaluated carefully and the appropriate input parameters are used in a slope stability calculation.


Considerable information on the factors that affect the stability of a site may be obtained through a desk study that involves a review of available published information such as geological/geotechnical maps and reports, topographic maps, hydrological records, aerial photographs etc.

A field reconnaissance is always necessary in order to detect site characteristics that may relate specifically to potential slope instability.

Monitoring of surface and deep seated movements and pore water pressures is always desirable and in some cases necessary prior to the performance of a slope stability calculation.

Subsurface investigation and laboratory testing form an integral part of a slope stability assessment as these processes will yield the engineering parameters such as strength, stress-strain relationships etc, which are necessary for a slope stability calculation.

4. Static slope stability Slopes became unstable when the shear stresses on a potential failure surface exceed the shearing

resistance of the soil. In the case of slopes where stresses on the potential failure surface are high the additional earthquake induced stresses needed to trigger failure are low. In this sense the seismic slope stability is dependent on the static slope stability. The most commonly used methods of slope stability analysis are the limit equilibrium methods. Stress-deformation analysis, using the finite element method, are performed more rarely, especially in the case of major projects.

4.1 Limit equilibrium methods The limit equilibrium methods have been used extensively for many years for the analysis of natural and

manmade slopes. They have been calibrated against actual slope failures and with careful selection of appropriate input parameters these methods can yield sufficiently accurate results. With these methods the force or moment equilibrium of a mass of soil above a potential failure surface is considered. Shearing is assumed to take place on the potential failure surface with the soil above assumed to be rigid. The soil on the potential failure surface is assumed to be rigid-perfectly plastic and its shear strength is mobilized concurrently on all points on the failure surface. Since the states of stress and mobilized strength are the same for all elements of soil on the failure surface, the factor of safety is constant over the entire failure surface. In real slopes however the factor of safety for each element of soil on the failure surface is not constant.

A number of limit equilibrium procedures have been developed and applied in practice. Some of these procedures are listed below together with the type of slope failure these are most applicable

for:- 1. Culman method (plane failures) 2. Wedge methods (failure on two or three planes) 3. Fellenius method (circular and log spiral failures- homogeneous soils) 4. Bishop´s simplified and Bishops modified (circular and log spiral failures – homogeneous soils) 5. Morgestern and Price (1965) ) Non circular failures - non 6. Spencer (1967) ) homogeneous anisotropic soils 7. Janbu (1968) )

In practice a slope is considered stable if the factor of safety produced from the stability analysis is significantly greater than 1. Factors of safety are introduced in order to allow for certain uncertainties which relate to the accuracy of the method (how accurate is the failure mechanism), the accuracy of input parameters and also other important factors such as the potential consequences of a slope failure. For permanent slopes the minimum factor of safety is usually 1.5 and for temporary slopes 1.3. In cases of permanent structures such as motorway embankments where after completion the factor of safety increase with time due to dissipation of pore water pressures, a minimum factor of safety of 1.3 may be acceptable at the end of the construction period.

An important aspect, which needs to be given proper attention in a slope stability calculation is the possibility of a progressive failure mechanism. The various limit equilibrium methods treat the soil as a rigid – perfectly plastic material but in reality many soils exhibit brittle, strain-softening stress-strain behaviour.


This means that the peak shear strength of the soil may not be mobilized simultaneously at all points on the failure surface and with this kind of mechanism it is possible to have a failure even if the factor of safety based on peak shear strength is above 1.0. Kramer (1996) suggests that the stability of strain softening materials is analysed reliably only by using residual shear strengths.

4.2 Stress – deformation analyses Stress-deformation analyses can be performed mainly with finite element models which allow the

simulation of the complicated stress-strain behaviour of soils. The finite element method is powerful tool which can cope with irregular geometries, complex boundary conditions and pore water pressure regimes and can simulate complicated construction operations.

The method can predict stresses, movements and pore water pressures due to construction procedures and also predict the most critically stressed zones within a slope. In this way the most likely mode of failure can be identified and deformations up to and sometimes beyond the point of failure can be calculated.

5. Seismic slope stability Seismic slope stability analyses are further complicated by two additional factors

(i) the dynamic stresses induced by earthquake shaking and (ii) the effect of dynamic stresses on the stress stain behaviour and strength of slope

materials. Depending on the behaviour of the soil during seismic shaking, seismic instabilities may be grouped into

two categories: (i) inertial instabilities and (ii) weakening instabilities

In the case of inertial instabilities the strength of the soil remains relatively unaffected by the earthquake shaking and any permanent deformations are produced when the strength of the soil is exceeded during small intervals of time by the dynamic stresses. In the case of weakening instabilities the earthquake shaking produces a substantial loss of strength which gives rise to very large displacements and instability. The most common causes of weakening instability are flow liquefaction and cyclic mobility. There are numerous analytical techniques that deal with the above two categories and these are either based on limit equilibrium or stress-deformation analyses.

5.1 Analysis of inertial instability When the dynamic normal and shear stresses on a potential failure surface are superimposed upon the

corresponding static stresses, these may produce inertial instability of the slope if the shear stresses exceed the shear strength of the soil.

The problem is approached either by performing a pseudostatic analysis that produces a factor of safety against slope failure or by attempting to calculate permanent slope displacements produced by earthquake shaking.

5.1.1 Pseudostatic Analysis The pseudostatic approach has been used by engineers to analyse the seismic stability of earth structures

since the 1920´s. This method of analysis involves the computation of the minimum factor of safety against sliding by including in the analysis static horizontal and vertical forces of some magnitude.

These horizontal and vertical forces are usually expressed as a product of horizontal or vertical seismic coefficients and the weight of the potential sliding mass. The horizontal pseudostatic force decreases the factor of safety by reducing the resisting force and increasing the driving force. The vertical pseudostatic force typically has less influence on the factor of safety since it affects positively (or negatively) both the driving and resisting forces and for this reason this is ignored by many engineers.

The factor of safety of a slope critically depends on the value of seismic coefficient Kh. In the mid 60´s when pseudostatic analyses were widely used, one of the biggest problems facing the engineer was that of selecting a value of the seismic coefficient to be used for design purposes. At the time the selection of values


for Kh was mostly empirical and in typical U.S. practice the values used varied between 0.10 and 0.15. In Japan the earth dam code specified values between 0.15 and 0.25. Ambraseys (1960) was the first to make specific suggestions regarding a rational selection of seismic coefficients. He recommended the use of seismic coefficients based on maximum and root-mean square values as determined by elastic response analyses for 20% critical dumping. A comparison of the methods for determining seismic coefficients is shown in Fig. 2 (Seed, 1966), which is based on a numerical example of a long dam, with a homogeneous section 300 ft high, constructed of compacted soil having a shear wave velocity 1000 ft/second subjected to the N-S component of the El Centro earthquake of 1940, for which the maximum ground acceleration was about 0.3 g. The figure includes the seismic coefficients from U.S. practice and the Japanese Codes, a design coefficient equal to the maximum acceleration corresponding to an assumption of rigid body response and Ambraseys´ recommended values determined from elastic response analysis, it is apparent from the figure that an engineer had a wide range of choice.

Typical seismic coefficients and factors of safety used in practice today are given in table 1.

Table 1 - Typical Seismic Coefficients & Factors of Safety used in Practice

Seismic Coefficient

Remarks

0.10

0.15

0.15-0.25 0.05-0.15

0.15 ⅓-½ PGA

½ PGA

Major Earthquake, FOS>1.0 Corps of Engineers Manual EM-111-2-1902 (1982) Great Earthquake, FOS>1.0 Corps of Engineers Manual EM-111-2-1902(1982) Japan, FOS>1.0 State of California Seed (1979), with FOS>1.15 and a 20 percent strength reduction Marcuson and Franklin (1983), FOS>1.0 Hynes and Franklin (1984)>1.0 and a 20 percent strength reduction

The recommendation by Seed (1979) was based on a study of earth dams constructed of ductile soils (those that do not generate pore water pressures and show no more than 15% loss of strength upon cyclic loading) with crest accelerations less than 0.75g. He indicated that in these cases deformations would be acceptably small if the earthquake coefficients are 0.10-0.15 with factors of safety greater than 1.0.

The recommendation by Hynes-Griffith and Franklin (1984) were based on deformation calculations using 354 accelerograms (see Fig. 3) which showed that the use of horizontal earthquake coefficients equal to 50% of peak ground acceleration and factors of safely greater than 1.0 would not developed dangerously large deformations.

Although the pseudostatic approach to stability analysis is simple and straight forward producing an index of stability (F.S.) which engineers are used to appreciating, it suffers from many limitations as it can not really simulate the complex dynamic effects of earthquake shaking through a constant unidirectional pseudostatic acceleration.

These limitations were recognized by many researchers including Terzaghi (1950), Seed (1966), Seed et al. (1969), Marcuson and Hynes (1980) etc. More specifically in case of soils that build up large pore water pressures or have a degradation in strength of more than say 15% due to the earthquake shaking the analysis can be unreliable. As shown by Seed (1979) a number of dams such as the upper & lower San Fernando Dams, Sheffield Dam etc have in fact failed due to earthquake shaking although the calculated factors of safety were well above 1.

In the last couple of decades methods based on the assessment of the permanent slope deformations induced by seismic shaking find increasing applications. These methods are particularly suited to the case of earth dams where the magnitude of induced deformations is not only a measure of the stability of the embankment but also a measure of the effectiveness of the filter protection system.


5.1.2 Permanent deformation analyses Newmark (1963) first proposed the important concept that the effects of earthquakes on embankment

stability should be assessed in terms of the deformations they produced rather than the minimum factor of safety. He presented a method of analysis based on this concept in his Rankine Lecture in 1965 (Newmark, 1965). The method assumes rigid-plastic materials and presumes a knowledge of the time history of the acceleration acting on the embankment during the earthquake. Newmark´s method is partly presented herebelow using the notation adopted by Kramer (1996).

If the inertial forces acting on a potential failure mass (static plus dynamic) exceed the available resisting forces the factor of safety reduces to below 1 and the failure mass is no longer in equilibrium. Newmark (1965) used the analogy of a block resting on an inclined plane to develop a method of prediction of the permanent slope displacements (see Fig. 4).

Consider the case of block of weight W in static equilibrium with the block’s resistance to sliding being purely frictional. Under conditions of static equilibrium (see fig. 5a) the factor of safety is given by

1

(Φ = angle of friction) Under dynamic conditions (during earthquake shaking), at a particular instant of time the horizontal

acceleration on the inclined plane will be equal to

(Kh – horizontal earthquake coefficient) inducing an inertial force on the block equal to KhW. The effects of vertical acceleration in this analysis are ignored. By resolving forces with respect to the inclined plane (see Fig. 5b) the dynamic factor of safety at a given point in time is given by

2

By increasing the value of Kh the factor of safety reduces and there will be appoint when this will reach the value of 1 (see Fig. 6). The horizontal earthquake coefficent producing a factor of safety of one is termed the yield coefficient (Ky) and the corresponding acceleration the critical yield acceleration (ay). From equation 2 the yield coefficient required to produce sliding in the downslope direction Ky = tan (Φ-β).

If during seismic shaking the inclined plane is subject to pulses of acceleration that exceed the yield acceleration then there will be piecewise movements of the block in a downslope direction. Consider the case of a rectangular acceleration pulse of amplitude A and duration ∆t acting on the inclined plane (Fig. 7). During the interval between to and to + ∆t when A is greater than ay the acceleration of the block relative to the plane will be

3

The relative velocity of the block relative to the plane may be obtained by intergrading eq. 3 once and the

relative displacement by equation eq. 3 twice as shown below: 4

W cosβ tanΦ tanΦ FS = -------------------- = -------- W sinβ tanβ

ah (t) = Kh (t) g

(cosβ - Kh(t) sinβ) tanΦ FSd (t) = ----------------------------------- sinβ + Kh (t) cosβ

arel = A – ay

vrel(t) = ∫ t arel (t) dt = [A - ay] (t - to) to


to ≤ t ≤ to + ∆t 5

to ≤ t ≤ to + ∆t

At time to + ∆t when the relative velocity reaches its maximum value

vrel = [A - ay] ∆t drel = ½ [A - ay] ∆t2 After this point in time the block will continue to slide on the plane but at a decreasing velocity until it

stops. Between time (to+∆t) and time (t1) when the velocity becomes zero the relative acceleration is equal to 0-ay = -ay. During this time interval the relative velocity is given by

6 when the relative velocity drops to zero t = t1 = to + A∆t/αy

The relative displacement of the block during the same time internal is given by

7 The total relative displacement at time t1 is given by

8 For a single acceleration pulse, displacement is related to the amplitude and duration of the pulse. Since

in the case of an actual earthquake there will be several occurrences where the earthquake induced acceleration exceeds the yield acceleration producing a number of increments of displacement see fig. 8. It is logical to expect that the total displacement will depend on

(i) strong motion duration (ii) amplitude and (iii) frequency content

Yegian et al. (1991) have shown that by applying the sliding block method to rectangular, sinusoidal and triangular periodic base motions, the permanent displacement calculated is proportional to the square of the period of the base motion.

Using the rectangular pulse solution, Newmark (1965) related the slope displacement due to single pulse, to peak base velocity as follows

9 By analyzing several earthquake motions Newmark (1965) suggested that the effective number of pulses

during an earthquake is approximately equal to A/ay and a reasonable upper bound of permanent slope displacements, for values of ay /amax ≥ 0.17 may be given by

drel(t) = ∫ t [A - ay] [t - to] dt = ½ [A - ay] [t - to] 2 to

vrel = [A - ay] ∆t + ∫ t arel dt = A∆t – ay (t – to) to+∆t

drel(t) = [ A∆t - ay (t - to) ] dt

= A∆t (t - to – ∆t ) - ½ [ t2 - (to + ∆t )2 ]

drel(t1) = ½ ( A - ay ) ∆t2 A/ay

v2 max 1 – ay drel = ------------- ( ---------- )

2ay Α


10 Studies of permanent displacements predicted by the sliding block method for actual earthquakes by

Ambraseys and Menu (1988) have shown that those are similar to the displacements produced by sinusoidal and triangular waves for ay/amax greater than 0.5. They suggested that permanent slope displacements (u) are calculated using the formula

11

By studying an extensive database of records prepared by Franklin & Chang (1977), Yegian et al (1991) proposed a formula for calculating permanent slope displacements, which takes account of the frequency content and duration of shaking. The formula calculates the median permanent normalized slope displacement u* as follows:

12

where Neq is an equivalent number of cycles and T is the predominant period of the input motion. To account for any sources of uncertainty relating to this formula Yegian et al. (1991) produced contours of equal probability of exceedance of normalised permanent displacement (see Fig. 9).

In the above formulae, peak acceleration is the sole descriptor of ground motion and Jibson (1994)

proposed an alternative formula for calculating sliding block displacements which is based on the Arias intensity as follows: 13

where u is in cm, Ia in m/sec and ay is g´ s.

In the sliding block method of calculating permanent slope displacements the potential failure mass is assumed to be rigid perfectly plastic. As shown by Tika-Vasilikos et el. (1993) the residual shear strength of many soils depends on displacement and rate of shearing (see fig. 10). They suggested procedures for incorporating the rate dependent strengths into the calculation of permanent slope displacements using the sliding block method. Application of these procedures using accelerograms from three earthquakes have shown that consideration of fast loading effects (as in the case of earthquakes) may reduce significantly the displacements calculated with the constant strength soil model.

One of the limitation of the sliding block model is the assumption that the potential failure mass and the embankment are rigid. Although this assumption may be partly satisfied in the case of embankments composed of very stiff or hard soils or slopes subject to low frequency motion, this may not be true for softer soils. In the latter case, lateral displacements throughout the potential failure mass may be out of phase, with inertial forces at different points in the potential failure mass acting in opposition directions. This effectively means that the resultant inertial force and the resulting permanent displacements calculated with the rigid sliding block method are significantly overestimated. This effect was studied by Chopra (1966) who employed dynamic stress- deformation finite element analyses to produce the time varying horizontal

v2max amax

dmax = --------- ------- 2ay ay

ay ay logu = 0.90 + log [ (1 – ------- ) 2.53 ( ------- )-1.09 ]

amax amax

u ay ay ay log u* = log (---------------- ) = 0.22 – 10.12 ------ + 16.38 (--------)2 – 11.48 (------)3

amax Neq T2 amax amax amax

log u = 1.460 log Ia - 6.642 ay + 1.546


resultant force acting on the potential failure surface, by integrating the horizontal components of dynamic stresses on the potential failure surface (Fig. 11). The average acceleration of the potential failure mass is then produced by dividing the horizontal resultant force by the potential failure mass. With this method the average acceleration time history may be of higher of lower magnitude than the base acceleration time history and as Kramer (1996) suggests this is the most realistic input motion for sliding block analyses.

5.1.3 Makdisi –Seed analysis The method proposed by Makdisi and Seed (1978) for calculating permanent slope deformation of earth

dams produced by earthquake shaking is based on the sliding block method but uses average accelerations computed with the procedure of Chopra (1966) and the shear beam method (Fig. 12). The method uses a plot that relates the average maximum acceleration with the depth of the potential failure surface (see Fig. 13) and a plot of normalised permanent displacement with yield acceleration for different earthquake magnitudes (see Fig. 14). The latter was produced by subjecting several real and hypothetical dams to several actual and synthetic ground motions, scaled to represent different earthquake magnitudes.

The following procedure is used in order to evaluate the permanent slope deformation of a potential failure surface. (i) The maximum acceleration (at crest level) is calculated with an iterative method. With this method the

values of the shear modulus, G, dumping ratio, D and shearwave velocity, Vs are selected and the three dominant modal periods calculated. The spectral accelerations Sa1, Sa2 and Sa3 are then estimated from tabulated spectral data and thus the maximum acceleration dmax at crest level calculated.

(ii) The critical yield acceleration for the particular failure surface is calculated using the dynamic yield strength (on the failure surface).

(iii) Using the average value on the graph of Fig. 13, the value of amax αν/amax z=0 is calculated, and the value amax aν of the potential sliding mass determined.

(iv) By calculating the ratio ay/amax αv, the value of the normalized permanent displacement can be estimated from Fig. 14.

5.1.4 Stress – deformation analysis Stress deformation analyses are usually carried out using dynamic finite element programs. The

permanent deformation of a slope are produced by integrating the seismically induced permanent strains in each finite element. Various have been used for calculating permanent strain within individual finite elements namely: (i) the strain potential approach (Seed et al., 1973) (ii) Stiffness reduction approach (Lee, 1974; Serf et at., 1976) (iii) Non linear analysis approach (Finn et al., 1986)

The stain potential and stiffness reduction approaches are very approximate. Most accurate results may be obtained with the non-linear analysis approach, which employ non linear soil models (stress-strain relationships). The biggest difficulty in employing these models is to obtain soil stress-strain models that are representative of the soil in situ behaviour.

5.2 Analysis of weakening instability Weakening instability occurs when the earthquake induced stresses and strains result in a substantial

reduction of shear strength. They are usually associated with the phenomenon of liquefaction and they fall into two main categories.

(i) flow failures and (ii) deformation failures. Flow failures occur when due to earthquake shaking the shear strength of the soil drops below the soil

strength required to maintain static equilibrium. These failures may occur under static conditions after the earthquake shaking was ceased and they usually produce very large deformations which take place very suddenly and without warning. They can cause tremendous amount of damage and loss of life.


Deformation failures are usually smaller in magnitude than flow failures and they occur when the shear strength of the soil is reduced by earthquake shaking to such an extent that this is temporarily exceeded by pulses of earthquake induced shear stresses.

A number of empirical, semi-empirical and analytical methods have been used for analyzing weakening instabilities.

5.2.1 Flow failure analysis Flow failure analysis is performed in two steps. 1. Evaluation of potential flow slide instability and 2. Analysis of deformations A procedure for evaluating potential flow slide instability has been proposed by Marcuson et al. (1990).

Using dynamic finite element analyses and assessment of liquefaction resistance using insitu CPT and SPT tests, factors of safety against liquefaction are calculated for all soil elements on the potential failure surface. For those locations on the failure surface where the factor of safety is less than 1, residual strength values are assigned from Seed´s SPT correlations for liquefied soils. For those zones where the factor of safety is greater than one, strength values are based on knowledge of soil deformation characteristics and pore water pressures generated during the earthquake. The overall factor of safety is subsequently calculated using conventional, static limit equilibrium, slope stability analyses. If the factor of safety is less than one, flow sliding is expected to occur. Theoretically if the factor of safety is greater than one the slope will remain stable. In the latter case instability due to a progressive failure mechanism must be examined.

In the cases where flow sliding is expected, a crude estimate of the deformations involved can be obtained using a simplified procedure proposed by Lucia et al. (1981). This procedure is based on simple plane strain limit equilibrium procedures and involves several simplifying assumptions and requires an estimate of the strength of the liquefied soil. The procedure estimates the failure distance and deformations involved in a flow slide over a gentle slope slightly greater than 3o.

A powerful analytical technique for performing flow failure analysis has developed by Finn at el. (1977). This model is based on the work presented by Martin et al (1975) which demonstrated that pore water pressures induced by earthquake shaking can be predicted from simple shear tests on dry sand.

A finite element program for non-linear dynamic stress analysis (TARA-3FL) was developed by Finn and Yogendrakumar (1989) which allows reduction of the strength of each element of soil to the residual strength at the time of liquefaction. The program can cope with large deformations through a finite element mesh regeneration procedure. Finn´s approach has been used at Sardis dam in Mississippi to estimate the deformed shape of the unremediated dam after the occurrence of an earthquake (see Fig. 15) and to aid in the design of remedial treatment (Finn et. al., 1991).

5.2.2 Deformation failure analysis

Deformation failures usually produce smaller ground deformation but they can cause considerable damage. The most common type of deformation failure is lateral spreading. Procedures for estimating ground movements for this kind of failure mechanism are usually more empirical.

By studying actual permanent earthquake induced ground displacements in uniform sands of medium grain size Hamuda et. al. (1986) proposed an empirical relationship for calculating permanent horizontal downslope ground displacement.

D = 0.75 H½ Θ1/3 where D is ground displacement in metres H is thickness of liquefied layer in metres and Θ is the largest of the ground slope and the slope of the lower boundary of the liquefied layer (%)

Youd and Perkins (1987) proposed the Liquefaction Severity Index (LSI) for a conservative estimate of ground displacements related to lateral spreads on wide flood plains, deltas or other gently sloping fluvial deposits.


Using a model of a slope consisting of a crust of intact soil resting on a layer of liquefied soil and by employing work energy principles Byrne (1991) proposed expressions for calculating permanent slope displacements. Byrne et. al. (1992) extended his approach to calculate stiffness reduction factors for use in finite element analyses. The failure of the upper San Fernando dam was modeled using this approach and the deformation predicted were in good agreement with the actual deformations.

Other methods used for calculating permanent lateral displacements include the Baziar et. al. (1992) approach which was based on the sliding block analysis and the Bartlett and Youd (1992) approach which employs empirical expressions based on the study of lateral spreading cases.

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Figure 1: Maximum epicentral distance for different types of landslides (after Keefer, 1984)

Figure 2: Seismic coefficients suggested in 1966 for use in pseudostatic analysis (after Seed,1966)


Figure 3: Normalized permanent displacements from 354 horizontal accelerograms (after Hynes-Griffith & Franklin,1984)

Figure 4: Analogy between (a) potential landslide and (b) block resting on inclined plane

Figure5: Forces acting on a block resting on an inclined plane (a) static conditions (b) dynamic conditions.


Figure 6: Variation of the factor of safety as a function of the horizontal seismic coefficient

Figure 7: Variation of relative velocity and relative displacement between sliding block and plane due to rectangular acceleration pulse (from Kramer,1996)


Figure 8: Development of permanent slope displacements for actual earthquake ground motion (after Wilson and Keefer,1985)

Figure 9: Contours of equal probability of exceedance of normalized permanent displacement.(After Yegian et al.1991.)


Figure 10: Typical behaviour of pre-existing shear surfaces during fast shearing, as identified from ring shear test (after Tika – Vasilikos et al.,1993)

Figure 11: Evaluation of average acceleration for slope in embankment


Figure 12: Calculation of average acceleration from finite element response analysis (after Makdisi & Seed, 1978)

Figure 13: Variation of “maximum acceleration ratio” with depth of sliding mass (after Makdisi and Seed,1978)


Figure14: Variation of average normalized displacement with yield acceleration (after Makdisi and Seed, 1978)

Figure15: Pre-earthquake geometry (dashed lines) and post-earthquake deformed shape (solid lines) of Sardis Dam estimated using TARA-3 (after Finn et al., 1991.

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