Theoretical t-z Curves for Axially Loaded Piles

Abigail H Bateman, Jamie J Crispin, Paul J Vardanega, George Mylonakis*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

2 Citations (Scopus)
636 Downloads (Pure)


Estimation of nonlinear pile settlement can be simplified using one-dimensional “t-z” curves that conveniently divide the soil into multiple horizontal “slices.” This simplification reduces the continuum analysis to a two-point boundary-value problem of the Winkler type, which can be tackled by standard numerical procedures. Theoretical “t-z” curves can be established using the “shearing-of-concentric-cylinders” theory of Cooke and Randolph-Wroth, which involves two main elements: (1) a constitutive model cast in flexibility form, γ=γ(τ); and (2) an attenuation function of shear stress with radial distance from the pile, τ=τ(r). Soil settlement can then be determined by integrating shear strains over the radial coordinate, which often leads to closed-form solutions. Despite the simplicity and physical appeal of the method, only a few theoretical “t-z” curves are available in the literature. This paper introduces three novel attenuation functions for shear stresses, inspired by continuum solutions, which are employed in conjunction with eight soil constitutive models leading to a set of 32 “t-z” curves. Illustrative examples of pile settlement calculation in two soil types are presented to demonstrate application of the method.

Original languageEnglish
Article number04022052
Number of pages20
JournalJournal of Geotechnical and Geoenvironmental Engineering
Issue number7
Early online date6 May 2022
Publication statusPublished - 1 Jul 2022

Bibliographical note

Funding Information:
The first and second authors would like to thank the Engineering and Physical Sciences Research Council for their support (Grant Nos. EP/T517872/1 and EP/N509619/1, respectively). Partial funding was received by EU/H2020 under grant agreement number 730900 (SERA) with George Mylonakis as Principal Investigator for University of Bristol.

Publisher Copyright:
© 2022 This work is made available under the terms of the Creative Commons Attribution 4.0 International license,.


  • piles
  • settlement
  • non-linear analysis
  • "t-z" curves
  • soil/structure interaction


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