New Formulations of Boussinesq Solution for Vertical and Lateral Stresses in Soil

Abstract

Calculation of stresses within a soil body due to surface loading is a required step when designing buried commodities and subgrade wall structures. The Boussinesq equation is commonly used for determining stresses in soil due to surface loading, with examples for use found industry-wide. However, the available formulations have limitations, or they only cover simple cases. The purpose of this work is to review the derivation of the Boussinesq equation for vertical and lateral stresses in a soil body and to present several new, closed-form solutions for various surface load cases, including finite line and finite area loads. The formulations are presented as functions of Cartesian coordinates such that the stress at any point in the subsurface plane of interest can be found, not just the peak stress or the stress contour at a specific line. This is particularly useful when considering load distribution at a lateral extent from a finite loading, which may be significantly lower than the peak loading.

Background

For a while at my workplace, I was responsible for performing calculations that evaluated buried utilities and subgrade structures, which often includes checking for loads superimposed by vehicles or other types of surface surcharge. One of my co-workers developed methodology for determining the subgrade loading using the Boussinesq method, which is an elastic solution for loading on the surface of a semi-infinite half-space. Essentially, his solution performed numeric integration of Boussinesq’s equation over the bounds of the loading in Mathcad.

While this solution worked, it’s very computationally expensive and can take a while to execute depending on the complexity of the problem. Additionally, not everyone has access to Mathcad (or other computer-aided calculation packages capable of numeric integration). Not wanting to leave well-enough alone, I spent the next few years on-and-off trying to figure out if there was a better way to go about this. The ideal solution would be a closed-form equation representing the subsurface stress for different load patterns.

Solutions such as these already exist for simpler load patterns, such as infinite line and infinite strip loads (see the AASHTO LRFD Bridge Design Specification, Section 3.11.6). While these can usually be sufficient approximations, sometimes you need more refinement in the analysis, so the use of finite loadings becomes necessary. For example, as far as I had been able to find, formulations for finite line and finite area loadings (see figure below) did not exist in the literature.

Some time later, I had developed several new, closed-form solutions for determining the vertical and lateral stresses on subgrade structures, which I figured was worth writing a paper on. For those interested, you can download the final draft of this paper directly. This is not the final, typset and copyedited version (available on the ASCE website, although behind a paywall), but the information in it is the same as the final version.