Loukidis, D., Bandini, P. & Salgado, R. (2003). Ge ´otechnique 53, No. 5, 463–479 463 Stability of seismically loaded slopes using limit analysis D. LOUKIDIS  , P. BANDINI  and R. SALGADO  Numerical limit analysis is used to assess the stability of slopes subjected to seismic loading. The soil is assumed to follow the Mohr–Coulomb failure criterion. The lower and upper bound theorems are formulated as linear problems to be solved using linear programming techni- ques. Based on finite element discretisation of the slope, the velocity field is optimised to find the lowest upper bound, and the stress field is optimised to obtain the highest lower bound. Limit equilibrium computations and log-spiral upper bound solutions were also per- formed for comparison purposes. Additionally, finite ele- ment analyses were done for selected cases. Results from the limit equilibrium and finite element methods are in excellent agreement with the rigorous lower and upper bounds for all cases studied. The slip surfaces obtained from both the limit equilibrium and log-spiral upper bound methods lie within the plastic zones obtained for the slopes from both finite element and numerical limit analysis. Plots are presented of the horizontal pseudo- static acceleration ratio k c a c /g required to cause col- lapse of simple homogeneous slopes as a function of the slope inclination and shear strength parameters. KEYWORDS: earthquakes; limit state design/analysis; numer- ical modelling and analysis; plasticity; slopes Nous utilisons des analyses de limite nume ´rique pour e ´valuer la stabilite ´ de talus soumis a ` des charges sismi- ques. Nous pre ´supposons que le sol suit le crite `re de rupture de Mohr-Coulomb. Nous formulons des the ´o- re `mes de limite infe ´rieure et supe ´rieure sous forme de proble `mes line ´aires a ` re ´soudre en utilisant des techniques de programmation line ´aire. En nous basant sur la ‘dis- cre ´tisation’ des e ´le ´ments finis du talus, nous optimisons le champ de ve ´locite ´ afin de trouver la limite supe ´rieure la plus basse et nous optimisons le champ de contrainte pour obtenir la limite infe ´rieure la plus haute. Nous avons e ´galement effectue ´ des calculs d’e ´quilibre limite et formule ´s des solutions de limite supe ´rieure log-spirale a ` des fins de comparaison. De plus, nous avons fait des analyses d’e ´le ´ment fini dans certains cas choisis. Les re ´sultats provenant des me ´thodes d’e ´quilibre limite et d’e ´le ´ment fini correspondent parfaitement a ` des limites infe ´rieures et supe ´rieures rigoureuses pour tous les cas e ´tudie ´s. Les surfaces de glissement obtenues avec les me ´thodes d’e ´quilibre limite et de limite supe ´rieure log- spirale se trouvent a ` l’inte ´rieur de zones plastiques obtenues pour les talus d’apre `s les analyses d’e ´le ´ment fini et les analyses de limite nume ´rique. Nous pre ´sentons les trace ´s des taux d’acce ´le ´ration pseudo-statique horizontale k c a c /g, ne ´cessaires pour causer l’effondrement de talus homoge `nes simples, comme fonction des parame `tres d’in- clinaison des pentes et des forces de cisaillement. INTRODUCTION Recent advances in numerical simulation allow the perform- ance of elaborate analyses of geotechnical stability problems, including the seismic stability of earth slopes. However, the conventional pseudo-static approach is still widely used in engineering design. The concept of the pseudo-static ap- proach relies on the representation of the earthquake-induced loading by statically applied inertial forces. In a pseudo- static analysis, the effect of the actual time history of the response of the slope to the ground motion and other time- dependent or deformation-dependent phenomena (such as the excess pore pressure generation) are neglected. More- over, the seismic stability of the slope is expressed in terms of a single parameter, the critical (or yield) seismic coeffi- cient, k c . The critical seismic coefficient, k c , is the ratio of the seismic acceleration, a c , yielding a factor of safety equal to unity, to the acceleration of gravity g (¼ 9·81 m/s 2 ). The seismic acceleration is usually assumed to be horizontal and uniform with respect to depth. The pseudo-static approach is generally not applicable to saturated soils with high liquefaction potential, or to soils that will soften considerably when cycled. Accurate analysis of problems involving these soils requires elaborate dynamic finite element modelling with advanced constitutive relations capable of simulating the pore pressure generation during the earthquake. The critical seismic coefficient is usually computed using limit equilibrium methods. In limit equilibrium, neither static nor kinematic admissibility is necessarily satisfied. The slope is assumed to be in a state of incipient failure along a slip surface, and the shear strength of the soil is assumed to be mobilised simultaneously along the slip surface. Although limit equilibrium is a crude approach to the real problem, it manages to provide a fair estimation of the collapse load, with minimal computational cost compared with stress– deformation analyses. The pseudo-static approach has been implemented in various limit equilibrium methods to deter- mine the factor of safety. Sarma (1973) proposed a method that determines the critical seismic coefficient k c directly, eliminating the need for iterations. Sarma’s method is a so- called ‘rigorous’ limit equilibrium method, as it satisfies both force and moment equilibrium. Sarma (1973) showed that the determination of the critical seismic coefficient is easier than the determination of the factor of safety. Spencer (1978) reformulated his original method (Spencer, 1967, 1973) in a way that allowed direct assessment of k c . Spencer’s method, which also satisfies both force and mo- ment equilibrium for all slices, assumes a distribution of the inclination of the interslice forces across the sliding mass. Sarma (1979) proposed a method that can be used for direct calculation of k c , where the slice interfaces are allowed to be non-vertical. Leshchinsky & San (1994) modified the variational limit equilibrium formulation of Baker & Garber (1978) to account for inertia forces, and produced a number Manuscript received 28 February 2002; revised manuscript accepted 8 January 2003. Discussion on this paper closes 1 December 2003; for further details see p. ii.  School of Civil Engineering, Purdue University, West Lafayette, USA.