Generalized Rankine Solutions For Seismic Earth Pressures: Validity, Limitations & Refinements

A sustained interest towards developing simple analytical solutions for assessing gravitational and seismic earth pressures on retaining walls is evident in literature. An important family of such solutions are based on stress limit analysis formulations - a branch of plasticity theory developed following Rankine’s pioneering work for establishing ultimate limit states in the soil, by directly solving the relevant equilibrium differential equations - without resorting to ad-hoc assumptions such as the failure mechanisms assumed in Coulomb’s seminal method. Further, the prospect of establishing solutions by means of a uniform stress field in the backfill makes the Rankine approach particularly attractive for routine use in design. In this paper, the conditions that can generate a Rankine stress field under generalized geometry and loading conditions involving combined gravitational and seismic action in the realm of pseudo-dynamic analysis, are examined. Exact closed-form solutions are derived for the critical values of the governing problem parameters required to generate a Rankine stress field (i.e., critical wall inclination and roughness, backfill inclination and friction angle, amplitude of seismic body force), as well as for the corresponding earth pressure coefficients. It is shown that for the active limit state, when the critical value of a problem parameter (“Rankine value”) is higher than the actual (“physical”) value, the Rankine solution is conservative that is, it overestimates the active earth pressures. The opposite is true for the passive state. A parametric investigation of these solutions is provided, with emphasis on practical situations and numerical examples, shedding light into the physics of the problem.