Applicability of the Mindlin Solution from the Perspective of Saturated Soil

Abstract

The Mindlin solution is extensively utilized in the field of geotechnical engineering, including settlement computations and assessing the geoenvironmental impacts of shield tunneling and foundation pit excavation. Despite its widespread use, there is a lack of in-depth discussion regarding the solution’s essence and applicability in various engineering contexts. This paper for the first time examines and understands the essence of the Mindlin solution from the perspective of saturated soil by delving into the derivation and discussion of the Mindlin solution based on the displacement function method, which is very different from the traditional viewpoint of the single-phase method. Utilizing the Biot consolidation equations in cylindrical coordinate system, the steady-state solution problem for both the axisymmetric and nonaxisymmetric Biot equations in half-space can be simplified to a set of decoupled harmonic equations by employing the McNamee and Schiffman displacement function. Then the partial differential harmonic equations can be converted into ordinary differential equations through the application of the Hankel integral transform technique. According to the boundary and continuous conditions, the steady-state solutions for stresses and displacements within the transform domain can be meticulously derived. The solutions in the original time domain can be subsequently obtained by the Hankel inverse integral transform method. The results theoretically proof that the Mindlin solution is essentially a steady-state solution at the end of soil consolidation, which is suitable for long-term conditions such as settlement problems and short-term drained conditions in sandy soil. The solution may underestimate the response caused by construction in short-term undrained conditions, potentially posing risks to engineering projects. Therefore, it is advisable to employ elastic parameters under undrained conditions, expressed in total stress form, when addressing such short-undrained cases.