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 (Eeff) for a footing problem supported on a spatially variable medium - the Young’s modulus actually “felt” by the footing - using a spatial average. The Eeff 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 Eeff in each spatially varying realization, not just the statistics of Eeff 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 Eeff. 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.