Soil Dynamics and Earthquake Engineering 27 (2007) 957–969 An alternative to the Mononobe–Okabe equations for seismic earth pressures George Mylonakis à , Panos Kloukinas, Costas Papantonopoulos Department of Civil Engineering, University of Patras, Rio 26500, Greece Received 23 July 2006; received in revised form 23 January 2007; accepted 25 January 2007 Abstract A closed-form stress plasticity solution is presented for gravitational and earthquake-induced earth pressures on retaining walls. The proposed solution is essentially an approximate yield-line approach, based on the theory of discontinuous stress fields, and takes into account the following parameters: (1) weight and friction angle of the soil material, (2) wall inclination, (3) backfill inclination, (4) wall roughness, (5) surcharge at soil surface, and (6) horizontal and vertical seismic acceleration. Both active and passive conditions are considered by means of different inclinations of the stress characteristics in the backfill. Results are presented in the form of dimensionless graphs and charts that elucidate the salient features of the problem. Comparisons with established numerical solutions, such as those of Chen and Sokolovskii, show satisfactory agreement (maximum error for active pressures about 10%). It is shown that the solution does not perfectly satisfy equilibrium at certain points in the medium, and hence cannot be classified in the context of limit analysis theorems. Nevertheless, extensive comparisons with rigorous numerical results indicate that the solution consistently overestimates active pressures and under-predicts the passive. Accordingly, it can be viewed as an approximate lower-bound solution, than a mere predictor of soil thrust. Compared to the Coulomb and Mononobe–Okabe equations, the proposed solution is simpler, more accurate (especially for passive pressures) and safe, as it overestimates active pressures and underestimates the passive. Contrary to the aforementioned solutions, the proposed solution is symmetric, as it can be expressed by a single equation—describing both active and passive pressures—using appropriate signs for friction angle and wall roughness. r 2007 Elsevier Ltd. All rights reserved. Keywords: Retaining wall; Seismic earth pressure; Limit analysis; Lower bound; Stress plasticity; Mononobe–Okabe; Numerical analysis 1. Introduction The classical equations of Coulomb [1–4,10] and Mononobe–Okabe [5–11] are being widely used for determining earth pressures due to gravitational and earthquake loads, respectively. The Mononobe–Okabe solution treats earthquake loads as pseudo-dynamic, generated by uniform acceleration in the backfill. The retained soil is considered a perfectly plastic material, which fails along a planar surface, thereby exerting a limit thrust on the wall. The theoretical limitations of such an approach are well known and need not be repeated herein [11–13,16–18]. Given their practical nature and reasonable predictions of actual dynamic pressures (e.g. Refs. [9,14,16–18]), solutions of this type are expected to continue being used by engineers for a long time to come. This expectation does not seem to diminish by the advent of displacement-based design approaches, as the limit thrusts provided by the classical methods can be used to predict the threshold (‘‘yield’’) acceleration beyond which permanent dynamic displacements start to accumulate [11,15,19–21,43]. Owing to the translational and statically determined failure mechanisms employed, the limit-equilibrium Mono- nobe–Okabe solutions can be interpreted as kinematic solutions of limit analysis [22]. The latter solutions are based on kinematically admissible failure mechanisms in conjunction with a yield criterion and a flow rule for the soil material, both of which are enforced along pre- specified failure surfaces [10,19,23,24,40,42]. Stresses out- side the failure surfaces are not examined and, thereby, ARTICLE IN PRESS www.elsevier.com/locate/soildyn 0267-7261/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.soildyn.2007.01.004 à Corresponding author. Tel.: +30 2610 996542; fax: +30 2610 996576. E-mail address: mylo@upatras.gr (G. Mylonakis).