Fatigue Theory for Cyclic Loading of Soils
Authors: van Eekelen H.A.M. 
Published: December 2007

American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc.
This paper was prepared for the SPE-European Spring Meeting 1976 of the Society of Petroleum Engineers of AIME, held in Amsterdam, The Netherlands, April 8–9, 1976. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of where and by whom the paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor of the appropriate journal, provided agreement to give proper credit is made. Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers office. Such discussion may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines.

Abstract
For a prediction of the foundation behaviour of gravity structures or for earthquake foundation design, a model for the behaviour of soil under cyclic loading is needed. It is common practice to use only a limited set of laboratory tests, for instance at constant stress or constant strain amplitude, and to extrapolate from these tests to random wave loading conditions, under e.g. a prolonged ocean storm. Most extrapolation procedures used to date are based on an implicit assumption that the influence of cyclic loading can be represented by one single fatigue parameter.
A mathematical model of soil fatigue is given, from which several possible (or impossible) choices for such a fatigue parameter emerge. It is shown how these choices can be verified or falsified, either on theoretical ground or from the results of simple experiments. A practical extrapolation procedure is proposed, for which only constant stress and constant strain amplitude tests are needed.

Introduction
One of the basic ingredients needed for the prediction of the behaviour of foundations under earthquake or storm loading, is a model for the behaviour of soil under time-dependent random cyclic stressing. As it is not practicable to perform laboratory tests for practicable to perform laboratory tests for every possible cyclic stress history, we have to work with a limited set of experimental data, and the problem arises how to extrapolate from, say, cyclic triaxial compression tests at constant stress and/or strain amplitude, to triaxial compression cycling at variable stress amplitude.
In this paper we discuss one type of loading, be it triaxial compression or simple shear, one-way or two-way, drained or undrained. We assume that the entire cyclic loading process is defined by successive values of one process is defined by successive values of one single stress amplitude: pi = pin for cycle number n. The reaction or response of the sample to this loading process may be expressed in such observables as cyclic and cumulative strains, pore pressures, or volume changes. pore pressures, or volume changes. Our aim is to give a simple theory for the response of a material to cyclic loading, in terms of a single fatigue parameter.