A mixed smoothed finite element limit analysis formulation for static and seismic collapse loads

Abstract

This paper introduces a new formulation of limit analysis based on nodal integrations for calculating static and seismic collapse loads in geotechnical engineering. Unlike the classical kinematic limit analysis, our newly proposed formulation of upper-bound limit analysis using mixed elements is expressed in terms of the stress fields rather than displacement fields. The numerical framework approximates stress and velocity fields using low-order triangular elements with a strain smoothing technique. Subsequently, the weak form of the equilibrium conditions and flow rule are imposed over nodal smoothing cells rather than elements. The final form of stress mixed formulation is established on nodal smoothing cells and is cast as a set of conic constraints, allowing the stress fields to be directly determined using conic programming algorithms. Additionally, the determination of kinematically admissible displacement fields is achieved through duality theory. We demonstrate the robustness and accuracy of our numerical scheme through benchmark examples involving static and seismic collapse loads, such as bearing capacity and tunnel stability, showcasing its practical application. Although the proposed scheme outperforms other traditional numerical schemes and smoothed limit analysis in terms of accuracy and efficiency, the gain in performance is offset by a loss of rigour. Furthermore, we incorporate a simple non-associated plasticity scheme into the analyses to assess dilation-dependent collapse loads. The newly proposed numerical scheme of the stress-based upper-bound limit analysis is then utilised to assess the influence of the dilation on the static and seismic collapse loads and their failure mechanism, giving some valuable insights into the dilation-dependent collapse loads under seismic conditions.