The bearing capacity of square and/or rectangular footings in geotechnical foundation designs traditionally is determined based on experimental observations and/or deterministic analysis assuming uniform soil profiles. However, soils are spatially varying, and this spatial variability can significantly affect the bearing capacity of the foundation soils. Probability-based design methods can address this problem explicitly. However, a full three-dimensional (3D) probabilistic simulation, such as that involving the random finite-element method, generally is prohibitive, because it involves numerous Monte Carlo runs of a complicated nonlinear elastoplastic algorithm. This paper developed and validated an approximate analytical method based on local averaging theory and geometric averages of soil properties directly under the footing. It was found that the theoretical prediction of the first two moments of a square footing bearing capacity agrees very well with crude Monte Carlo simulation. The analytical prediction of the probability of a design failure was validated through simulation and can be used directly in reliability-based designs against bearing failure.

Shape factors traditionally are used to relate the bearing capacity of square, rectangular, and/or circular footings to that of strip footings in geotechnical foundation designs, in order to take account of additional resistances developed in front of and behind strip footings. They traditionally are determined by experimental observations or deterministic analysis assuming uniform soil profiles. However, soils are spatially varying, and this variability can have a substantial influence on the bearing capacity of foundations resting on them. Therefore, the empirical shape factors used in foundation designs are quite uncertain, depending on the in situ soil variability. This paper investigated how soil spatial variability may affect the shape factors for the bearing capacity of square footings and the reliability of the traditionally determined shape factors. By quantifying the uncertainties involved, shape factors can be chosen based on site conditions and a target reliability. It was found that the traditional shape factor used in design has the lowest reliability for sites with intermediate soil spatial variability, and thus lower shape factors may be warranted to attain a required reliability in such cases.

The random finite element method has been increasingly used in the geotechnical community to investigate the influence of soil spatial variability and to bridge the gap between a traditional design and a reliability-based design. There are two approaches to calculate the reliability curves as a function of the traditional/global factor of safety in the literature. However, it is not clear how these two approaches may be related and why. This paper is devoted to answering this question, through the aid of an implemented auto-search algorithm within the strength reduction method and the quantification of the potential sliding volumes in the various possible Monte Carlo realisations of the soil spatial variability. The equivalences and differences between the two approaches, and thereby their respective merits and disadvantages, are explained and discussed for the most commonly used distribution types of soil strength properties, that is, normal and lognormal distribution. Computational efficiency has also been addressed in the form of pseudocodes, which can be readily implemented.