Beyond Critical Acceleration Pseudo-Static Analysis

Common geotechnical earthquake engineering design approaches utilize acceleration levels to evaluate slope failure triggering and resulting permanent displacements, using pseudo-static stability analyses together with the Newmark method. These approaches assume a single sliding mechanism and do not take into account the possibility of multiple (increased in size) mechanisms due to higher accelerations than the critical one. This paper presents a “quasi dynamic” analysis framework which allows evaluation of the effect of higher acceleration (beyond the critical one) on the mechanism and the volume of the slide. The approach combines concepts from pseudostatic analysis together with plastic flow, such that the developed mechanisms are restricted from transferring greater stresses than their yield value (and by that preventing factor of safety lower than 1.0). The approach is applied to slopes characterized by ideal elasto-plastic material to demonstrate that in certain conditions an increase in the acceleration level, by itself, may alter the failure mechanism. Introduction The pseudo-static analysis is one of the most commonly used techniques for evaluation of the seismic risk of manmade and natural slopes. It may be used as part of factor safety (FS) evaluation under the input of a given acceleration (e.g. Seed, 1979), or for determination of the slope yield acceleration (e.g, Sarma and Bhave, 1974; Sarma 1975; Baker et al. 2006). The latter may be combined with Newmark (1965) analysis for determination of an earthquake induced permanent displacement (e.g. Makadisi and Seed, 1978). The approach assumes that once a mechanism develops, its size and shape remain constant, and the failing slope accelerates with the relative acceleration of a(t)-ay (e.g. Newmark, 1965), where ay is the yield acceleration (i.e. the acceleration that leads to a factor of safety of 1.0 in a pseudo-static analysis). Pseudo-static analyses, by their nature, cannot deal with “beyond critical accelerations” (i.e. accelerations higher than those which lead to sliding), as this will only produce a factor of safety smaller than 1.0 and will not reveal information regarding the influence on the mechanism. In fact, sliding mechanisms resulting from pseudo-static analyses with FS<1 are meaningless, as they are associated with artificial, unrealistic, strength properties greater than the real ones Associate Professor, Department of Structural Engineering and Construction Management, Faculty of Civil and Environmental Engineering, Technion – Israel Institute of Technology, Haifa, Israel, klar@technion.ac. il Graduate Student, Department of Structural Engineering and Construction Management, Faculty of Civil and Environmental Engineering, Technion – Israel Institute of Technology, Haifa, Israel. Associate Professor, The Institute of Earth Sciences, The Hebrew University, Givat Ram, Jerusalem 91904, Israel. Senior researcher, Geological Survey of Israel, 30 Malkhe Israel St., Jerusalem 95501, Israel. (φmob>φ, and cmob>c). The Newmark (1965) assumption that the slope accelerates by a(t)-ay is correct only if the failure mechanism does not change and the kinematics allows deformation in the direction of the applied acceleration a(t)-ay. This assumption, however, may not always hold. In order to capture the possible impact of greater acceleration on the developed mechanism and displacements, the current paper suggests a “quasi-dynamic” analysis framework which can deal with beyond critical acceleration levels. The paper is composed of 3 main sections. Firstly the concept of “quasi-dynamic” analysis is presented. Secondly, numerical implementation and an example are provided. Finally, discussion and conclusions are presented. The “Quasi-Dynamic” Analysis Framework The Achilles’ heel of the pseudo-static analysis is that it cannot limit the mobilized stresses to their yield value. Consequently, analyses with beyond critical accelerations results in mobilized stresses higher than the yield value (hence the FS=τy/τmob<1). In order to examine the slope behavior under accelerations higher than the critical value, one must allow the unbalanced stresses (or loads) to be translated into motion (i.e. acceleration). Let us further discuss this concept using a simple condition of a failure along a predefined, fixed, planar slip surface. Figure 1a shows a given slope subjected to a horizontal acceleration and its free body diagram. Note that due to introduction of kinematics into the current analysis, the dilation angle, ψ, is of importance (unlike conventional pseudo-static limit equilibrium analysis). Figure 1b shows the force polygon associate with the limit state (i.e. FS=1), from which the critical (or yield) acceleration, ky (=ay/g), may be defined. Note that the contribution of cohesion is equivalent to an inclined reduced weight at an angle of δ = arccot[W/(cLcosα)-tanα]. Solution of the force polygon shown in Figure 1b, results in a critical acceleration coefficient, ky, of: ) cos( ) sin( ) cos( cos φ α α δ φ δ α α − − + − = y k (1) where, α is the inclination of the slip surface, φ is the soil friction angle, and δ angle capturing the effect of cohesion as described earlier. When c = 0 (or δ = 0), kRyR degenerates into the wellknown solution of a sliding frictional block (with friction φ) over an inclined slope of α, kRyR = tan(φ α) (e.g. Kramer, 1996). Once acceleration values are higher than kRyR,R Ra D’Alembert’s force must be introduced into the force polygon in order to consider “static equilibrium” as shown in Figure 1c. Note that the total acceleration body force, kW, is explicitly divided into the yield component, kRyRW, and the post yield component, (k-kRyR)W, to emphasize the contribution of the beyond critical acceleration component. As can be seen, the normal force and frictional forces depend on the acceleration value, and do not remain constant as acceleration increases beyond critical value. The unbalanced force, or the D’Alembert’s force, shown as ma (mass times the acceleration), is directed opposite to the movement direction, and is influenced by the dilation angle, ψ. Only when k = kRyR the dilation angle does not affect the result, as the force polygon degenerates into that of Figure 1b. An expression for the failed mass acceleration may be obtained from the force polygon (introducing m=W/g): ) cos( ) cos( ) ( ψ φ α φ − − − = y k k g a (2) Note that the failed mass accelerates at a value different than (k-ky)g (the value suggested by Newmark, 1965), and in fact depends on the dilation angle besides the relative acceleration and sliding directions. For the case of horizontal acceleration input, as discussed here, the mass will accelerate by (k-ky)g only if α and ψ are equal to zero.