Sea to Sky Geotechnique 2006

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1 UNDERSTANDING SLOPE DEFORMATIONS TRIGGERED BY EARTHQUAKES Muhsin Élie RAHHAL, CLERC, École Supérieure d Ingénieurs de Beyrouth, Université Saint Joseph, PO BOX 54, BEIRUT, LEBANON, s: muhsin.rahal@fi.usj.edu.lb or clerc@fi.usj.edu.lb Mazen ABI AZAR, École Supérieure d Ingénieurs de Beyrouth, Université Saint Joseph, LEBANON ABSTRACT Earthquake induced ground shaking is one of the most recurrent causes of slope instabilities that might be leading to landslides. In the present research, a typical clayey slope is analysed. Empirical methods such as Newmark method, and Makdisi Seed method, are considered for the determination of the maximum deformation in a slope under dynamic loading. Stress deformation methods using finite element techniques are also used to estimate slope deformations under same conditions. A relation is established between the yield acceleration, the cohesion of soil and the slope angle. Then, the evolution of the dynamic factor of safety is analysed in terms of the yield acceleration of the slope. A procedure is developed to establish a relation between slope deformation and dynamic factor of safety. Curves linking the factor of safety to yield acceleration are established for different values of cohesion and friction angle of soil. RÉSUMÉ Les mouvements du sol induits par les séismes sont une cause récurrente des instabilités d une pente, pouvant mener à des glissements de terrains. Dans ce travail, une pente argileuse typique est considérée. Des méthodes empiriques, comme les méthodes de Newmark et Makdisi Seed, sont utilisées pour la détermination de la déformation maximale de la pente sous sollicitations sismiques. Les méthodes contraintes- déformations basées sur les éléments finis sont utilisées pour estimer les déformations de la pente dans des conditions similaires. Une relation est établie entre l accélération de ruine, la cohésion du sol et l angle de la pente. L évolution du facteur de sécurité dynamique est aussi analysée en termes de l accélération de ruine. Une procédure est développée pour établir une relation entre les déformations calculées et le facteur de sécurité dynamique. Des courbes reliant le facteur de sécurité à l accélération de ruine sont tracées pour différentes valeurs des paramètres mécaniques.. INTRODUCTION When an earthquake occurs, the effect of earthquake induced ground shaking is often sufficient to cause failure of slopes that were to some extent stable before the earthquake. The resulting damage may be catastrophic depending on the geometric and material characteristics of the slope. When compared to static slope stability analysis, seismic slope stability is complicated by the need to consider the effects of dynamic stresses induced by earthquake shaking and the effects of those on the strength and stress-strain behaviour of slope materials. The stability of slopes is influenced by many factors, and a complete slope stability evaluation must consider the effects of each. Geological, hydrological, topographical, geometrical, and material characteristics all influence the stability of a particular slope. Field reconnaissance, field monitoring, subsurface investigation and material testing could help obtaining information on these characteristics. Earthquake induced slope failures and landslides are secondary effects, which are defined as non-tectonic surface processes that are directly related to earthquake shaking. Different examples of earthquake induced slope movements exist (Yeats et al. 997). There would appear to be a shaking threshold that is needed to produce earthquake induced slope movements. Furthermore, it is logical to expect that the extent of earthquake induced landslide activity should increase with increasing earthquake magnitude, and that there would be a minimum magnitude below which earthquake induced landslide would rarely take place. Three main types of earthquake induced slope movements may exist in soils (Keefer 984); soil falls (typical in the case of granular slightly cemented soils), soil slides (typical in the case of loose, partly to completely saturated sands; minimum slope angles between 5º and 3º), and flow slides and lateral spreading (typical in the case of sensitive clays, or uncompacted loose sands and silts partly to completely saturated; slope angles between º and º). The seismic evaluation can be grouped into two general categories: inertia slope stability analysis and weakening slope stability analysis. Inertia slope stability analysis is preferred for those materials that retain their shear strength during the earthquake. Inertial instabilities are most commonly determined by pseudostatic, sliding block (Newmark 965), or stress-deformation analyses (Karray et al. 2). The Makdisi-Seed method (Makdisi and Seed 978) based on the results of the sliding block analyses is also frequently used. The weakening slope stability analysis is preferred for those soils that will experience a significant reduction in shear strength during the earthquake. Two cases of weakening slope stability analysis are: flow slides and lateral spreading. In the geotechnical literature, many simplified and interesting methods for seismic slope stability evaluation are found (Rathje and Bray 999; Ashford and Sitar 22). In the present paper, a typical clayey slope is analysed. Empirical traditional methods, as well as stress 382

2 deformation methods using finite elements, are used to evaluate slope deformation. Special attention is given to the yield acceleration estimation. A comparison between the results is established. Also, a procedure is developed to set up a relation between slope deformation and dynamic factor of safety. A parametric study is carried out to investigate the effects of some geotechnical parameters on the evaluation of seismic slope stability and the induced deformations. 2. METHODOLOGY OF CALCULATION The stress-deformation approach used in this research is presented. The relation between the factor of safety and the measured deformations in the slope is established. 2. Evaluation of the stress-deformation method The stress deformation method is frequently used to perform static calculations of slope stability. It is based on the finite elements method, in which soil is considered to be generally elastic. The precision of stress-deformation analysis is mainly influenced by the accuracy with which the stress strain model represents actual material behaviour. The accuracy of simple models is usually limited to certain ranges of strain. Extending the stress deformation method to a dynamic case is based on dividing the earthquake signal into many time intervals covering the biggest range of acceleration possible. The first step consists in considering static soil properties in the calculations, and then in every following time interval, the method uses modified soil properties. Modified soil properties depend on how the dynamic soil properties vary with shear strain. So iterative calculations are to be carried out. At the end of each step, a stress-deformation profile is obtained and is used as calculation input for the next step. And so on, the next acceleration corresponding to the following time interval is used and the slope is submitted to a system of forces corresponding to the mass of the element multiplied by the acceleration. A new system of stress deformation is calculated according to the following equation of motion of such a system. [M]{ä} + [D]{å} + [K]{a} = {F} () Where [M] is the mass matrix, [D] the damping matrix, [K] the rigidity matrix; {ä} the nodal accelerations vector, {å} the nodal speed vector, {a} the nodal deformations vector, and {F} the forces vector. The vector of forces could be made up by different forces: {F} = {F b} + {F s} + {F n} (2) Where {F b} is the body force, {F s} is the force due to surface boundary pressures, and {F n} is the concentrated nodal force. This method is applied till the end of the earthquake input motion. At the end of each cycle, calculations are launched again using modified dynamic properties. Iterations continue until obtained deformations in two successive cycles do not differ by more than 5%. It is important to remind that dynamic properties G (shear modulus) and D (damping) depend on the shear strain. As far as soil constitutive models are concerned, linear elastic conditions are considered in static loading. In the case of dynamic loading, equivalent linear conditions are considered to account for properties changing with shear strain. An equivalent linear model can approximate nonlinear behaviour of soil. The equivalent linear model is actually non linear, but it is equivalent to a linear model because it transforms the irregular earthquake shaking into equivalent uniform cycles. It is non linear in that new calculated values of G and D at the end of each cycle are used in the calculations of the next cycle, and the process is repeated until differences between two successive cycles become very small. 2.2 Relation between deformation and factor of safety By analysing the results of the stress deformation method and considering the modification of parameters due to dynamic loading, the factor of safety can be calculated at each step. A curve representing the factor of safety as a function of time could be drawn, time being the duration of the earthquake loading. The method allows the determination of instantaneous nodal deformations as a function of duration of earthquake motion. For a specific node, the maximum deformation can be determined and hence the corresponding acceleration is found. Then using this same acceleration in the pseudo-static slope stability evaluation, a factor of safety for the slope is obtained. This factor of safety is a dynamic factor of safety since it depends on time; a curve representing the factor of safety as a function of time may be drawn. By comparing the two curves (Displacement-time; Dynamic factor of safety-time), the relation between the factor of safety and the deformation can be illustrated. Of course, curves obtained depend on the earthquake acceleration and the response point in the slope, which is usually the crest. Analysis is based on the acceleration signal used. The choice of time interval when subdividing the earthquake acceleration signal should be treated with care and great precision in order not to omit any peak value. Frequency content influences the choice of time interval; with high frequency content earthquake acceleration signals, smaller time intervals are to be considered. 3. CASE STUDY A typical clayey slope, representing many existing cases, is considered. The slope angle is 26.6º and the total height of the slope is 2m (from toe to crest). The mechanical static properties are cohesion c = 2 kpa and friction angle φ = 25º. The Poisson s ratio is considered equal to.35. The dynamic properties at small shear strain are: maximum shear modulus (G max) equal to MPa and damping ratio D equal to 2%. As for the variation of dynamic soil properties with increasing shear strain due to earthquake, values are given in Table. 383

3 Bedrock is located at a depth of 2m below the slope crest. Table. Variations of G/G max and D with Shear Strain γ Shear Strain (%) G/G max (%) D (%) Yield acceleration Yield acceleration is calculated using the pseudostatic analysis in which the effects of an earthquake are represented by constant horizontal and vertical accelerations. The magnitude of the pseudostatic accelerations should be related to the severity of the anticipated ground motion. The pseudostatic acceleration required to bring a slope to the point of imminent failure is called the yield acceleration (Kim and Sitar 24). For a given slope, by imposing a factor of safety equal to in the pseudostatic approach, the yield acceleration is obtained. In the case of the slope studied in this paper (slope angle = 26.6º), the pseudostatic slices method was used in the calculations to define the yield acceleration. A large number of surface failures as well as rotation centres were used to determine the critical failure surface. The yield acceleration was found to be.52g. A detailed analysis was carried out to understand the relation between the yield acceleration and the static geotechnical parameters of the slope, namely the cohesion c. Figure shows the variation of yield acceleration in the case of the analysed slope. The friction angle φ is maintained constant = 25º. It can be noted that the yield acceleration, which increases with increasing cohesion in the soil, is a linear function of cohesion. Figure 2 shows the effect of changing the slope angle on the yield acceleration knowing that geotechnical parameters are maintained constant equal to the original values. Higher yield accelerations are needed for smaller slope angles. It can be seen that the yield acceleration is a linear function of the slope angle. Yield Acceleration (g),2,,6,4,2, Slope Angle (degree) Figure 2. Yield acceleration as a function of slope angle. Finally, Figure 3 relates the yield acceleration to the dynamic factor of safety obtained by the pseudostatic approach. Naturally, higher yield accelerations are required for slopes showing higher safety factors. This means that knowing the factor of safety of a homogeneous clayey slope, and since the factor of safety depends on the properties of the soil, Figure 3 can help giving an approximate value of the yield acceleration for that slope. Once again, the yield acceleration proves to be almost a linear function of the factor of safety.,2,7,6 Yield Acceleration (g),6,4,2 Yield Acceleration (g),5,4,3,2, ,5 2 2,5 3 Cohesion C (kpa) Factor of Safety Figure. Yield acceleration as a function of soil cohesion. Figure 3. Yield acceleration as a function of safety factor. 384

4 3.2 Calculating slope deformations by different methods. Since the serviceability of a slope after an earthquake is controlled by deformation, analyses that estimate slope displacements give a more useful mean of measuring slope stability. Three methods have been used to calculate the deformation induced by an earthquake in the studied slope. These are respectively: The Newmark method, the Makdisi-Seed method and the stress deformation method. Since earthquake induced accelerations vary with time, the pseudo-static factor of safety will vary throughout an earthquake as indicated earlier. When the total inertial forces acting on a potential mass become large enough that driving forces exceed the available resisting forces, the factor of safety drops below. The first method (Newmark 965) considered behaviour of a slope under such conditions. When the factor of safety drops below unity, the potential failure surface becomes unstable and the unbalanced force will accelerate it. The situation is analogous to that of a block resting on an inclined plane. Newmark (965) developed this analogy to predict permanent displacement of a slope subjected to any ground motion. Permanent slope displacements depend on the relationship between the yield acceleration and the maximum induced earthquake acceleration. As it could be observed in Table 2, if the yield acceleration of a slope is greater than the maximum acceleration of a particular ground motion, no displacements will occur. Since the yield acceleration was determined to be equal to.52g, slope displacements increase quickly when earthquake accelerations became higher than the yield acceleration. The second method used in the calculations consists of determining permanent slope deformations of earth dams and embankments produced by earthquake loading and based on the sliding block method (Makdisi and Seed 978). By simplifying assumptions about the results of dynamic finite elements and shear beam analyses of such structures, a simplified procedure for prediction of permanent displacements was developed. Knowing the fundamental period of vibration of the dam (or embankment) and the yield acceleration of the slope, simple charts are used to estimate earthquake induced permanent displacements. The method uses a plot that relates the average maximum acceleration with the depth of the potential failure surface and a plot of normalised permanent displacement with yield acceleration for different earthquake magnitudes. The latter was obtained by subjecting many real and hypothetical dams to several actual and synthetic ground motions scaled to represent different earthquake magnitudes. The application of this method was extended to the studied slope, and results are shown in Table 2. Deformations shown in Table 2 are calculated on the crest of the slope and are nil for earthquake acceleration values below the yield acceleration of the slope. Deformation values increase drastically for high acceleration values. Because this procedure is based on the dynamic response characteristics of embankments and dams, its results should be watchfully used when applied to slopes. The third method for determining slope deformations is based on stress deformation analyses carried out using a dynamic finite elements program. Integrating the seismically induced permanent strains in each finite element produced the permanent deformations of a slope. The finite elements mesh considered square elements having a 2m-side length, and a few triangular elements directly on the slope. As far as the boundary conditions are concerned, the left and right vertical boundaries are supposed to move freely in the horizontal directions. Although vertical boundaries might move up and down during an earthquake, they are not allowed to because they are connected to the ground beyond the problem. This assumption is reasonable when the vertical boundaries are far from the main area of interest. Also, a fixed boundary is considered at the roc interface, and a totally free motion is considered at the surface of the slope. Figure 4 shows part of the slope mesh. Table 2. Crest Deformations calculated by Traditional Methods (Newmark and Makdisi-Seed) a max (g) Newmark Makdisi-Seed Figure 4. Part of the finite elements mesh for the slope as well as the deflected shape after seismic loading. Figure 4 shows the original mesh as well as the deflected shape of the slope. Obviously, maximum deflection is 385

5 obtained on the crest. Different elements were observed inside and on the surface of the slope. A special attention was given to nodes 66 (crest), 85 (Midpoint on slope), and 9 (toe of slope). The observation of the deformation at the three nodes allows the understanding of the slope behaviour. Dynamic properties of the slope are given earlier. Earthquakes are subdivided in small time intervals of.2s. Equivalent linear soil model explained in a previous paragraph is used in the analysis. frequency content. Finally the third record (Figure 7) comes from Mexico City (February 25 th 2) with a max =.8g and is rich in low frequencies. During calculations the three accelerograms were scaled in order to get different a max values and at the same time analyse the effect of frequency content. Three different earthquake records have been used as shown in Figure 5, Figure 6 and Figure 7. The three earthquake loadings are different as far as maximum acceleration a max and frequency contents are concerned. The first accelerogram (Figure 5) comes from the well known El Centro (May 8 th 94) with a max =.34g obtained after 2.34s of the beginning. Figure 7. Earthquake record from Mexico-City (25/2/). Figure 5. Earthquake record from El Centro (8/5/4). Figure 6. Earthquake record from Alaska (27/2/4). The second record (Figure 6) comes From Alaska (December 27 th 24) with a max =.2g and is rich in high Table 3 gives the results of the calculations for the three earthquakes with different scaled a max values. In the total, 39 loading cases were studied. It is clear that deformations are measured for all acceleration values even for values below the yield acceleration. Stress deformation analyses are more precise than traditional empirical methods presented in Table 2. Table 3. Crest deformations calculated by stress deformation method using three input different input earthquake motions a max (g) Deformation (cm) El Centro Deformation (cm) Alaska Deformation (cm) Mexico City When analysing further the results of Table 3, it can be seen that for the same a max, higher deformations are measured in the case of El Centro earthquake, than in the 386

6 case of the Alaska and Mexico City earthquakes. The observed differences are due to the frequency content of the input motions. Finally, when comparing results of Tables 2 and 3, the Newmark method gives values closer to the ones measured in the stress deformation analysis, and seems to be less approximate than the Makdisi-Seed method. The higher values of deformations in the Makdisi-Seed method could be due to the reduced soil strength parameters used in the method. 3.3 Relation between slope deformations and factor of safety Factor of Safety 3 2,5 2,5 This paragraph applies the methodology presented earlier in the paper concerning the establishment of a relation between the deformation in the slope and the dynamic factor of safety. The equivalent linear approach is used in the dynamic calculations. A huge number of calculations have been carried out. The measured deformation at the crest of the slope is represented as a function of the factor of safety of the slope. The El Centro earthquake has been used as a reference signal in the analysis. Figure 8 represents the safety factor as a function of maximum deformation in the case of a slope having a friction angle φ equal to 2º and for different values of the cohesion. In Figure 9, the same principles are presented but with a friction angle φ equal to 25º. 2,5,5,34,4 c=3,fi=25 c=2,fi=25 c=,fi=25 c=,fi=25 3,3 5,3 7,8,7 c=25,fi=25 c=5,fi=25 c=5,fi=25 Figure 9. Factor of safety as a function of deformation for different cohesion values and φ = 25º. Factor of Safety 2,5,5 The selected values for cohesion and friction angle are intended to cover as many existing cases of clayey slopes as possible. Static calculations determined the stress conditions in the slope according to the finite elements mesh used. Mohr circles representing the stress state at each node could be obtained. Soil elastic model was used at this stage. The finite element stress analysis in this case is straightforward. Besides being simple, using elastic models always ensures a solution and there is no difficulty in obtaining convergence. The analysis is enough adequate to obtain a reasonable picture of the stress conditions within the slope.,34,4 3,3 c=3,fi=2 c=2,fi=2 c=,fi=2 c=,fi=2 5,3 7,8,7 c=25,fi=2 c=5,fi=2 c=5,fi=2 Figure 8. Factor of safety as a function of deformation for different cohesion values and φ = 2º. As for the dynamic calculations, the equivalent linear soil model was used. The strain dependence of shear modulus and damping is accounted for by this equivalent linear approach. Iterative calculations were done up to reaching a convergence with less than % difference between two successive steps. The deformations at different nodes could be illustrated as a function of time (seismic loading time). Maximum deformations were measured at the crest. The new modified strength properties of the slope after the shaking are used to recalculate the factor of safety using the pseudostatic slices approach. A factor of safety varying with time is obtained and the one corresponding to maximum deformation is selected to be represented in Figures 8 and 9. Analysing the results of both figures proves that the absence of cohesion puts the slope in a state of failure or impending failure. It can also be observed how the factor of safety is related to crest deformation in a homogeneous slope of known cohesion and friction angle 387

7 values. On the other hand, it is interesting to watch how a same crest deformation might be measured, for different cohesion values of a slope and the corresponding safety factors. procedures for earth structures. Canadian Geotechnical Journal, 36 (): Yeats, R.S., Sieh, K. and Allen C.R. (997) The geology of earthquakes, Oxford University Press, New York, USA. 4. CONCLUDING REMARKS The endeavour of this research is to contribute to the understanding of deformations in a homogeneous clayey slope as well as the relation between these deformations and the measured factor of safety. The study proved the importance of determining the yield acceleration and clarified the relation between the yield acceleration and the cohesion of the soil. A curve representing the yield acceleration as a function of the factor of safety of a slope has been drawn. As far as the measured crest deformations are concerned, a comparison between traditional empirical methods and finite elements dynamic stress methods has been established for different values of acceleration a max and different earthquake records frequency contents. The use of dynamic stress deformations methods gives a better view of soil behaviour under earthquake loading. Furthermore, the relation between the crest deformation and the global dynamic factor of safety has been thoroughly analysed for different values of cohesion and friction angle of the soil. Analysis is still underway to comprehend the effect of dynamic properties, mainly the decrease of shear modulus, as well as the frequency content of the earthquake input record on the seismic stability of a slope. More results are forthcoming. REFERENCES Ashford, S.A. and Sitar, N. (22) Simplified method for evaluating seismic stability of steep slopes, ASCE Journal of Geotechnical and Geoenvironmental Engineering, 28 (2): Karray, M., Lefebvre, G. and Touileb, B. (2) A procedure to compare the results of dynamic and pseudostatic slope stability analyses, Proceedings 54 th Canadian Geotechnical Conference, Calgary, Canada: Keefer, D.K. (984) Landslides caused by earthquakes, Geologic Society of America Bulletin, 95 (2): Kim, J. and Sitar, N. (24) Direct estimation of yield acceleration in slope stability analyses, ASCE Journal of Geotechnical and Geoenvironmental Engineering, 3 (): -5. Makdisi, F.I. and Seed, H.B. (978) Simplified procedure for estimating dam and embankment earthquake induced deformations ASCE, Journal of Geotechnical Engineering Division, 4 (7): Newmark, N. (965) Effects of earthquakes on dams and embankments, Géotechnique, 5 (2): Rathje, E.M. and Bray, J.D. (999) An examination of simplified earthquake induced displacement 388