Winkler model for axially loaded piles in inhomogeneous soil

J J Crispin,C P Leahy,G Mylonakis

doi:10.1680/jgele.18.00062

Abstract

Analytical closed-form solutions are developed for the elastic and elasto-plastic settlement of axially loaded piles in inhomogeneous soil. The soil is modelled by way of a bed of Winkler (‘t–z’) springs with stiffness varying as a power function of depth, described by two dimensionless inhomogeneity parameters. The associated governing differential equation is solved in an exact manner using Bessel functions, which reproduce the solution for homogeneous soil. Additional limiting cases are explored including: (a) infinitely long piles, (b) short piles, (c) perfectly floating piles and (d) perfectly end-bearing piles. The solution is extended to the non-linear range by employing elastic–perfectly plastic Winkler springs. A systematic approach for predicting the full load–settlement curve is presented and applied to tests from a site in London. Dimensionless charts are provided for routine design.

Highlights

The Winkler model and related load-transfer analyses have been used extensively to model piles under axial load (e.g. Randolph & Wroth, 1978; Poulos & Davis, 1980; Scott, 1981; Mylonakis & Gazetas, 1998; Salgado, 2008; Guo, 2012)
For stages 3 and 4, the shaft resistance is exhausted, the interface depth is at the pile base, Lp = L, and the load and displacement at this depth are given by the base load–settlement curve, w(Lp) = wb and P(Lp) = Pb
Note that as the base spring is modelled as linear elastic, the pile head load, P, varies linearly with pile head settlement, w0, in stage 3

Summary

INTRODUCTION

The Winkler model and related load-transfer analyses have been used extensively to model piles under axial load (e.g. Randolph & Wroth, 1978; Poulos & Davis, 1980; Scott, 1981; Mylonakis & Gazetas, 1998; Salgado, 2008; Guo, 2012). Scott (1981) and Guo (2012) extended the Winkler model to inhomogeneous soil and provided closed-form solutions for pile head stiffness when soil stiffness varies as a power function of depth. For the simple case of homogeneous soil, the Winkler modulus is constant with depth and the head stiffness, K0, of a pile of length, L, is given by the familiar solution (Mylonakis, 1995; Mylonakis & Gazetas, 1998; Randolph, 2003; Fleming et al, 2008; Salgado, 2008). The derivation of these solutions and alternative representations are provided in sections A2 and A3 of the Supplementary Appendix. IÀν Iþν an=2 qn=2S3 þ ΩRS4 qn=2S5 þ ΩRS6 an=2 S3 S5 an=2 S4 S6 an=2 IνÀ1ðχ0Þ À I1Àνðχ0Þ IÀνðχ0Þ À Iþνðχ0Þ

Ω sinhðλLÞ þ coshðλLÞ

Findings

CONCLUSIONS