Elastic–Plastic Analysis of Rigid Passive Piles in Two-Layered Soils

The paper analyses the behavior of a rigid passive pile embedded in a soil profile consisting of a stable layer underlying an unstable layer subjected to a uniform soil displacement. Pile-soil interaction is considered by modeling the soil by a series of elastic-plastic springs along the pile shaft. The modulus of horizontal subgrade reaction is assumed to linearly increase with depth in the unstable layer and constant in the stable one. The ultimate soil resistance is assumed increasing with depth in both layers. The results of analysis are presented in dimensionless form in terms of shear force developed at the slip surface as a function of the pile embedment into the stable layer and the distribution of soil characteristics over depth. The method allows capturing pile response not only at the soil ultimate state but also at the intermediate states. Specifically, the governing equations for the elastic, elastic-plastic and plastic cases are discussed and, whenever possible, a set of closed-form expressions is provided to estimate the maximum bending moment along the shaft and the pile head deflection, so that for an assigned value of the required stabilizing force both ultimate and serviceability limit state of the pile can be checked. A numerical example is given to illustrate the application of the proposed procedure.