Effective Young's modulus of a spatially variable soil mass under a footing

This study investigates the possibility of representing the effective Young's modulus (E-eff) for a footing problem supported on a spatially variable medium - the Young's modulus actually felt by the footing - using a spatial average. The E-eff is simulated by a homogenization procedure that matches the responses between a random finite element analysis (RFEA) and a homogeneous finite element analysis. Emphasis is placed on whether the spatial average can well represent the numerical value of E-eff in each spatially varying realization, not just the statistics of E-eff within an ensemble (a weaker requirement). It is found that the conventional spatial averaging model that treats all soil regions equally important cannot satisfactorily represent E-eff. Extensive numerical results show that the concept of mobilization is essential: highly mobilized soil regions close to the footing should be given larger weights than non-mobilized remote regions. Moreover, the non-uniform weights can be prescribed prior to RFEA, that is, they do not depend on the specific response corresponding to a specific random field realization. The prescribed mobilization for the spatially variable Young's modulus can be contrasted with the emergent mobilized shear strength in a spatially variable medium that results from the emergent nature of the critical failure path - it cannot be predicted prior to random finite element analysis. A key contribution of this paper is the development of a simple method based on the pseudo incremental energy to estimate the non-uniform weights for the spatial averaging using a single run of a homogeneous finite element analysis.