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* Correspondence to: CeH cile Cremer, GeHodynamique et Structure, Bagneux, France

Received 9 May 2000Copyright 2001 John Wiley & Sons, Ltd. Revised 20 March 2001

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech., 2001; 25:1257}1284 (DOI: 10.1002/nag.175)

Cyclic macro-element for soil}structure interaction: material andgeometrical non-linearities

CeHcile Cremer*, Alain Pecker and Luc Davenne

GeHodynamique et Structure, Bagneux, France Laboratoire de MeHcanique et de Technologie-ENS Cachan, Cachan, France

SUMMARY

This paper presents a non-linear soil}structure interaction (SSI) macro-element for shallow foundation oncohesive soil. The element describes the behaviour in the near "eld of the foundation under cyclic loading,

reproducing the material non-linearities of the soil under the foundation (yielding) as well as the geometricalnon-linearities (uplift) at the soil}structure interface. The overall behaviour in the soil and at the interface isreduced to its action on the foundation. The macro-element consists of a non-linear joint element, expressedin generalised variables, i.e. in forces applied to the foundation and in the corresponding displacements.Failure is described by the interaction diagram of the ultimate bearing capacity of the foundation undercombined loads. Mechanisms of yielding and uplift are modelled through a global, coupled plasticity}upliftmodel.

The cyclic model is dedicated to modelling the dynamic response of structures subjected to seismic action.Thus, it is especially suited to combined loading developed during this kind of motion. Comparisons ofcyclic results obtained from the macro-element and from a FE modelization are shown in order todemonstrate the relevance of the proposed model and its predictive ability. Copyright 2001 John Wiley& Sons, Ltd.

KEY WORDS: macro-element; soil}structure interaction; shallow foundation; plasticity; uplift

1. INTRODUCTION

Numerous studies have been performed on the bearing capacity of a shallow foundation under

inclined eccentric loading. Guided by experimental results, Meyerhof [1], Vesic [2], Butter"eldand Gottardi [3] have proposed a solution for a shallow foundation lying on a sand layer. Later,

Salenion and Pecker [4,5], Paolucci and Pecker [6], Ukritchon et al. [7] and Houlsby and

Puzrin [8], have elaborated solutions for frictional and/or cohesive medium. The proposed

bounding surfaces allow the determination of the ultimate forces supported by the foundation,

but do not allow prediction of the amplitude of permanent displacements, which may in certain

cases become excessive and lead to instability of the structure.

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M

H

V

B(1- )B

ex

e z

x

z

Figure 1. System de"nition.

The concept of macro-element has been applied to the soil}structure interaction by di!erentauthors. They have especially studied the case of a foundation on sand. Among them, Tan [9],

Nova and Montrasio [10], Gottardi et al. [11] have performed a lot of experimental tests, for

di!erent monotonic loading paths, that guided them in the elaboration of a macro-element.Martin [12] has applied the same concept, but for o!shore foundations on cohesive soil. These

models lead to a good prediction of plastic displacements, especially settlements, but only undermonotonic loading. Recently, Pedretti [13] has extended the Nova and Montrasio model to

cyclic loading using the hypoplasticity theory.

In this paper, we propose a new cyclic soil}structure interaction macro-element for a shallowfoundation on cohesive soil. Besides the plastic behaviour of the soil, the model takes into account

the non-linearities at the soil}foundation interface. Uplift has the e!ect of signi"cantly reducingthe forces in the structure. Di!erent studies have shown that it is a predominant factor at the baseof slender structures during seismic action [14]. This paper presents the cyclic plasticity-uplift

coupled model, and proposes a macro-element, which is rather easy to use and which represents

a very e$cient tool in designing a structure}foundation system.

2. BEHAVIOUR CHARACTERIZATION

2.1. System dexnition

Assuming that the foundation is a rigid body, its movement can be described with global

variables expressed at the foundation centre (Figure 1). The behaviour of the soil}foundationsystem will thus be modelled through the forces applied at the base of the foundation (vertical

force

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Figure 2. 2D "nite element mesh (parameters of the clay constitutive law: c"c#g Z with c

"30 kPa

and g"3 kN/m; G/c"1300 (G"39 MPa); "0.45; "1.9 t/m;

"0.016).

characterize the foundation behaviour and to identify the model parameters. These simulations

have been performed with the "nite element code Dyna-ow (PreHvost, version 1998), whichprovides adequate constitutive laws for the description of the soil behaviour under cyclic loading.

The 2D numerical model (Figure 2) consists of a foundation lying on a largely discretized soil

medium, presenting a constant rate of increase of cohesion with depth. The contact elements at

the soil}foundation interface are governed by a no-tension criterion that allows separationbetween the soil and the foundation.

For the soil modelling, the PreHvost [15] multi-yield constitutive law for cohesive soil is used.This is an analytical model which describes the anisotropic, elastoplastic, path-dependent

stress}strain}strength properties of saturated clays under undrained loading conditions. Thefailure is de"ned by the Von Mises criterion. The hardening is purely kinematic and is describedby successive yield surfaces translating inside the failure criterion. A plastic modulus is associated

with each of the yield surfaces, and an associative #ow rule is used to compute the plastic strains.The cyclic behaviour is presented in Figure 3 for a triaxial compression and extension simulation.

The constitutive parameters have been identi"ed from experimental tests on undrained normallyconsolidated saturated clay (plasticity index PI"20 per cent).

On the basis of these numerical simulations, carried out for a large number of di!erent loadingpaths, the foundation behaviour has been interpreted.

2.3. Description of behaviour

Di!erent response diagrams M} (moment}rotation), H}x (horizontal force}horizontal displace-ment), M}z (moment}vertical displacement) and M} (moment}uplift ratio) are presented inFigures 4}7 for a foundation with an ultimate vertical force equal to