Reply to the discussion on “effect of soil spatial variability on failure mechanisms and undrained capacities of strip foundations under uniaxial loading” by Zhe Luo

Abstract

The authors recently published work on failure mechanisms and undrained capacities of strip foundations on spatial variable soils (Shen et al., 2021, called the original paper hereafter). In order to improve the computational efficiency of random finite element analysis (RFEA), a non-uniform mesh was employed in the original paper. Luo (2021) stated that such non-uniform mesh resulted in a different degree of variance reduction over the random field (RF) and thus disobeyed the stationarity of the RF based on viewpoints of spatial averaging (SA) discretization method. However, as introduced in Section 3.2 in the original paper, the midpoint (MP) discretization method was used to discretize the RF, rather than the SA discretization method. Note that MP and SA are two very different methods for the discretization of RF in the RFEA. The material value for an element is represented by the RF value calculated at its centroid for the MP discretization method, whereas the material value for an element is represented by the spatial average of RF over the element for the SA discretization method. There is spatial averaging for SA, which results in significant variance reduction for a very coarse mesh. In contrast, there is no spatial averaging for MP, and thus no variance reduction (Tabarroki and Ching, 2019). Therefore, the discusser’s judgement based on viewpoints of the SA discretization method to a RF discretized by the MP method is not relevant.