cyclic macro-element for soil–structure interaction: material and geometrical non-linearities

*Correspondence to: CeH cile Cremer, GeH odynamique 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

CeH cile 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 underinclined 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 andPuzrin [8], have elaborated solutions for frictional and/or cohesive medium. The proposedbounding 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 certaincases become excessive and lead to instability of the structure.

M

H

V

δB

(1- δ)B

ex

e z

xz θ

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, fordi!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. Thesemodels lead to a good prediction of plastic displacements, especially settlements, but only undermonotonic loading. Recently, Pedretti [13] has extended the Nova and Montrasio model tocyclic loading using the hypoplasticity theory.In this paper, we propose a new cyclic soil}structure interaction macro-element for a shallow

foundation on cohesive soil. Besides the plastic behaviour of the soil, the model takes into accountthe 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-upliftcoupled model, and proposes a macro-element, which is rather easy to use and which representsa 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 globalvariables 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 (verticalforce<}horizontal forceH}momentM) and through the corresponding kinematic displacementsmeasured at the centre (vertical displacement z}horizontal displacement x}rotation �). Theseparation ratio of the foundation of width B is noted as �; it is de"ned as the ratio of the length ofthe foundation not in contact with the soil divided by the foundation width.

2.2. Numerical data base

Due to the di$culty and cost of carrying out experimental tests on a foundation lying ona cohesive medium (mainly due to the di$culties of the model preparation due to clay consolida-tion) for di!erent loading paths, the elaboration of a numerical data base allowed us to

1258 CED CILE CREMER, ALAIN PECKER AND LUC DAVENNE

Copyright � 2001 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech., 2001; 25:1257}1284

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 simulationshave been performed with the "nite element code Dyna-ow (PreH vost, 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 atthe soil}foundation interface are governed by a no-tension criterion that allows separationbetween the soil and the foundation.For the soil modelling, the PreH vost [15] multi-yield constitutive law for cohesive soil is used.

This is an analytical model which describes the anisotropic, elastoplastic, path-dependentstress}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 associatedwith 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 loading

paths, the foundation behaviour has been interpreted.

2.3. Description of behaviour

Di!erent response diagramsM}� (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 <

��"2.4 MN. It is

subjected to a constant vertical force < (<"0.6 MN; </<��

"0.25) and a cyclic radial loadingpath in the M}H plane (M/H"10 m). The behaviour is strongly non-linear and highly

CYCLIC MACRO-ELEMENT FOR SOIL}STRUCTURE INTERACTION 1259


Figure 3. PreH vost's multi-yield model for anisotropic undrained clays (1978): Triaxial compres-sion and extension stress}strain curves.

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-0.002 -0.0015 -0.001 -0.0005 0 0.0005 0.001 0.0015 0.002

θ (rad)

M (M

Nm

)

cyclicmonotonic

V=0.62MNM/H=10m

Figure 4. Overturning moment}rotation.

dissipative. Non-linearities come from soil yielding but also from uplift at the interface. The non-linearities due to uplift are partially recoverable (Figure 7), and thus clearly noticed in theunloading part of the M}� and M}z curves. Indeed, we observe in the unloading M}� curve(Figure 4), the increase in sti!ness due to restored contact of the foundation with the soil. In theM}z diagram (Figure 6), which is dominated by the foundation settlement (increasing withcycles), we note the e!ect of the lift-up of the centre of gravity during uplift and its lowering duringunloading.

2.4. Model design

From the observation of this kind of behaviour, it has been chosen to build the macro-elementaround two di!erent models, one in plasticity and one in uplift, separated but coupled. Coupling



-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-0.01 -0.005 0 0.005 0.01

x (m)

H (M

N)

cyclicmonotonic

V=0.62MNM/H=10m

Figure 5. Horizontal Force}horizontal displacement.

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.002 0.004 0.006 0.008 0.01 0.012

z (m)

M (M

Nm

)

cyclicmonotonic

V=0.62MNM/H=10m

Figure 6. Overturning moment}vertical displacement.

between both submodels accounts for the in#uence of plastic yielding on uplift and vice versa, ofuplift on plastic yielding. Indeed, the reduction of the foundation width in contact during upliftinduces an increase of the stresses under the foundation. This leads to a larger soil yielding, whichitself modi"es the uplift behaviour of the foundation. Following this philosophy, the (z, x, �)displacements calculated for a given (<,H,M) forces vector are obtained by summing thedi!erent components, i.e. the elastic and plastic displacements issued from the plasticity model,and the uplift displacement issued from the uplift model:

utot"uel#upl#uup

The assumption that the global behaviour of the foundation behaves purely elastically duringunloading, allows one to assign all the non-linear e!ects visible on the unloading part of the



-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-60 -40 -20 0 20 40 60

Uplift δ (%)

M (M

Nm

)cyclicmonotonic

Uplift left side

Upliftright side

virgin loading

unloading-reloading

V=0.62MNM/H=10m

Figure 7. Overturning moment}uplift.

curves to geometrical non-linearities. They are all gathered into the uplift model. These areassociated with the rotation and the vertical displacement (lift-up and lowering of the gravitycentre) induced exclusively by uplift. These expressions are "rst deduced from an uplift model foran elastic soil. They are further modi"ed to take into account the coupling with plasticity. Oneconsequence of that coupling is that uplift becomes partially irreversible and that the momentM for which it is initiated becomes a function of the foundation bearing capacity.The plasticity model includes all the other non-linearities. These come from soil yielding under

the dead weight, under the increase in loads and also under the increase in stresses in the soilduring uplift. The e!ect of uplift is introduced by a failure criterion which corresponds toa soil-failure mechanism for a foundation with uplift. The hardening law is also deduced from theobservation of the uplift behaviour. Finally, all the plastic parameters are identi"ed fromthe plastic displacements from which the non-linear uplift components have been previouslysubtracted.This substructure approach decomposes a highly non-linear, coupled problem in order to

study each nonlinear e!ect separately, but without dismissing any coupling.

3. MODELLING

3.1. Plasticity model

3.1.1. Constitutive laws. The constitutive law is written as

F� "K : (u� !u� pl) (1)

where u� "u� el#u� pl with u� el being the increment of elastic reversible displacement, u� pl theincrement of plastic permanent displacement.



Figure 8. Failure overturning mechanism with uplift.

The global variables (forces and displacements) have been rendered dimensionless to work witha system independent of the foundation width B and of the plastic properties of the soil (cohesion):

F,�<

H

M�" 1

<��

<

H

M/B�, u,�z

x

��,1

B �z

x

B�� (2)

where K is the dimensionless elastic sti!ness matrix (see Section 3.1.2), <��

"q��

B, withq��being the foundation ultimate pressure under vertical centred load [16] or [17].

The plastic displacement increment is de"ned by the tensor

u� pl"�Q �P (3)

where P de"nes the #ow direction of plastic displacements in the forces' space, �Q is the plasticmultiplier; �Q �"�Q if �Q *0, else �Q �"0.

3.1.2. Elastic stiwness matrix. The dimensionless elastic sti!ness matrix is written as

K,�K

��0 0

0 K��

0

0 0 K�� with K��

"

K��

q��

, K��

"

K��

q��

, K��"K��

B�q��

(4)

This is a diagonal matrix where the diagonal termsK��,K

��,K�� correspond to the real part of the

static impedances of a shallow strip foundation, de"ned for instance by Gazetas [18]. They arefunctions of the geometrical properties of the foundation (width B) and of the elastic properties ofthe soil (shear modulus G, Poisson's ratio �).Following the common practice for surface foundations, the o!-diagonal terms have been

neglected because they are very low with respect to the diagonal terms and do not signi"cantlyin#uence the foundation response.

3.1.3. Failure criterion. Bearing-capacity solutions, under any loading (combination of<,H,M),for a strip foundation lying on a homogeneous cohesive half-space, have been proposed bySalenion and Pecker [4,5] and Pecker [19] for a soil obeying the Tresca criterion with andwithout tensile strength. These solutions have been obtained within the framework of the yielddesign theory with static and kinematic approaches (Figure 8).



Figure 9. Bounding surface for cohesive soil.

For a homogeneous cohesive soil without tensile strength, the equation of the bounding surfaceis (Figure 9)

f"�H

a<�(1!<)��#�

Mb<�(1!<)��

�!1"0 (5)

with the coe$cients (values given in Table I):

� a and b that de"ne the size of the bounding surface of elliptic shape in a (H,M) plane;� c, d and e, f that de"ne the parabolic shape of the bounding surface in the (<,H) and (<,M)planes, respectively.

These coe$cients have been determined from the curves proposed by Ukritchon et al. [7] whohave derived solutions for a heterogeneous cohesive soil pro"le, exhibiting a constant gradient ofcohesion with depth [20]. The coe$cients are thus functions of q

��/q

��, where:

� q��is the foundation ultimate bearing capacity for a soil pro"le with cohesion c

�at the surface,

exhibiting a gradient of cohesion, constant with depth;� q

��is the ultimate bearing capacity of the same foundation for a homogenous soil pro"le with

cohesion c�.

3.1.4. Loading surfaces. Knowing the failure criterion, one has now to determine the evolution ofthe loading surface, dragged by the forces point, towards the failure surface when the forcesincrease. The choice has been guided by the need for reproducing the behaviour of the soil andfoundation, initially submitted to the weight of the structure, and then solicited along any loadingpath. Since the macro-element is built with the ultimate aim of modelling soil}structure interac-tion under seismic action, loading paths followed by the structure under seismic conditions havebeen favoured. If it is assumed that the dynamic gravity centre of the structure varies only slightly(one predominant mode in the response), the paths are mainly radial in the (H,M) plane, witha vertical force being almost constant (Figure 10). The model will be particularly well adapted forthese paths (parameters identi"cation).



Table I. Proposed relationships for the parameters determination without any test.

Parameter Relationship

ElasticStatic sti+ness for a shallow strip foundation on a heterogeneous soil medium with a gradientof shear modulus constant with depth [18]:

K��, K

��,K

��K

��"

0.73

1!�G

�(1#2 ); K

��"

2

2!�G

��1#2

3 �; K��"

�2(1!�)

G��

B

2��

�1#1

3 �

where de"nes the G shear modulus gradient following:

G"G�(1# �) and �"

2z

B

with z, the depth, G�, the shear modulus at depth z"0, G, the shear modulus at depth z,

�, the Poisson ratio, B, the foundation width

Plastic;ltimate bearing capacity of a shallow strip foundation for a vertical centred loading ona cohesive soil [17]):

q��

Homogeneous soil with constant cohesion c�:

q��

"5.14c�

Heterogeneous soil with a gradient of cohesion constant with depth:

q��

q��

"��c��5.14#

�gB4c

��

where �g de"nes the c cohesion gradient following c"c�#�gz

with z, the depth, c�, the cohesion at depth z"0, c, the cohesion at depth z, B, the width

of the foundation, ��, coe$cient depending on �gB/c

�and on B/h, with h, the height of

the soil layer. Diagrams of ��are given in Matar and Salenion [17]

a, b, c, d, e, f Coe.cients of the failure criterion [20];a"0.32/�; b"0.37/��; c"0.25; d"0.55; e"0.8; f"0.8;where �"q

��/q

��

�, � Parameters of the plastic potential:

�+0.23; �+0.18

;pliftMoment for which the uplift is initiated, as a function of < :

M�

M�"0.25< exp��

Magnifying unloading factor to apply to the loading slope of the (M/<, �) relationship:� �"4!3 exp(!4<)

During the initialization phase of the gravity loads (Figure 11), the loading surface is reduced toa straight line segment along the <-axis (M"H"0):

f"< with <3[0, �] and �3[0, 1]. (6)

At the end of the initialization, <"�"N/<��, where N is the weight of the structure.



CG

h

m γ

B

M

H

h1 h2

h3

h4

for a constant V

Figure 10. Type of loading paths preferentially developed during seismic action for a structurewith one predominant mode.

0 χ 1

V

Figure 11. Initialization under vertical load.

Figure 12. Hardening rule: evolution of the loading surface.

For any loading, the surface, initially reduced to a straight line segment, is dragged along by theforces point F. It simultaneously undergoes an isotropic growth and a kinematic translation ofthe ellipse centres in the (H,M) plane, with a movement of the extreme point P along the<-axis(Figure 12). This point moves in such a way that, when point F reaches the failure surface, pointP simultaneously reaches the extreme failure point <"1.To describe this evolution, the loading surfaces are written as

f"�H! ��

��

��#�

M!��

��

��!1"0 with ��,�

0

��

��,�

0

a<�(�!<)�

a<�(�!< )�� (7)

and�"�#(1!�) (�#�)



Figure 13. Relationship between isotropic and kinematic parameters.

� �,�0

� � is the kinematic hardening vector, with �"(� : �)��"� �#��;

� � is the isotropic hardening parameter.

The condition �"�#(1!�)(�#�) ensures that point Pmoves from<"� to 1 as a function ofthe � and � hardening parameters. Under the assumption of a radial loading path in the (H,M)plane, it may be veri"ed that point F reaches the failure surface for �#�"1. At that moment, theextreme point P of the loading surface reaches point <"1.

3.1.5. Hardening laws. The hardening laws describe the evolution of the � and � parameters, sothat the model correctly reproduces the foundation behaviour under cyclic loading as well asunder monotonic loading. These laws are deduced from three relationships: (a) the relationshipthat will link the isotropic parameter to the kinematic which is determined from the observationof the foundation behaviour during uplift; (b) the criterion of non-interpenetration of the failuresurface by the loading surface which will allow us to de"ne the direction of the kinematichardening vector, a criterion that is commonly used in multi-surface plasticity models (see forinstance Prevost, Mroz); (c) the consistency rule which will determine the amplitude of thekinematic hardening vector.For that, it will be posed that

�� "�R��

(8)

where �R that de"nes the amplitude of the kinematic increment and �/�� its direction.

(a) Relationship between �R }�R : It is observed for the M}� relationship that loading of thefoundation on one side almost does not in#uence its behaviour on the other side. This is easilyexplained by considering the role of uplift. The soil area located under the edge of the foundation,which has been strongly solicited and has yielded during loading in one direction, does notcontribute, if uplift occurs, to the foundation response during loading in the opposite direction.The behaviour for the evolution of the loading surfaces (Figure 13) is expressed by imposing the

condition that point F, representative of the forces state reached during loading in one direction,becomes a "xed point of the loading surface during loading in the opposite direction.



Figure 14. Non-interpenetration criterion.

This condition is written as (for a radial path)

�R "�R (9)

(b) Criterion of non-interpenetration of the failure surface by the loading surface: Following thephilosophy of Prevost's tangent rule [15], this criterion ensures an evolution of the loadingsurface such that, at failure, the representative point of the forces state coincides with the meetingpoint of the loading surface with the failure surface.The vector direction representing the increment of the kinematic hardening vector �/�� is

calculated by imposing the condition that the next forces point F��

(Figure 14) lies along thesegment de"ned by F

F�, with F

�belonging to the failure surface, such that O

F

//OFf .

The following equations are obtained:

�,�0

��cos�!H

��

��sin�!M

��

�, �"a tan��

��

M!��

H! �� (10)

where ��, ��

are equivalent to ��

�, ��

but de"ned for the failure surface, i.e. for �"1

��,�0

��

��,�

0

a<�(1!<)�

a<� (1!<)��(c) Consistency rule: The consistency rule

fQ"0 (11)



determines the amplitude �R of the kinematic hardening incremental vector. By introducingEquations (8) and (9) into the consistency rule, and noting that

�Q "1

hQ : dF (12)

�R "!�Q �h

�f/�� : �/�� #�f/��(13)

Determination of the plastic modulus h from the M}� relationship: The plastic modulus hdescribes the evolution of the force}plastic displacement relationship for monotonic loading. Aswe are mainly interested in the rotational foundation behaviour, h is identi"ed with the M}�relationship.By introducing relationships (12) and (18) into (3) and then (3) into (1), and by isolating the h

parameter in the rotational component, it is given by

h"K�� (dM/d�)

K��!(dM/d�)2M��

�f�<

d<dM

#

�f�<

d<dM

#

�f�M� (14)

By identifying the M}� relationship, it is proposed for dM/d� that

dMd�

"K�� exp(!K��/M�) with M

�: moment corresponding to failure. (15)

Determination of the initial settlement of the foundation under the weight of the structure(initialization phase under gravity): For the calculation of the foundation settlement undera vertical centred load, the relationship proposed by Nova and Montrasio [10] has been chosen:

<"1!exp�� (16)

3.1.6. Flow rule. The choice of a plastic potential g di!erent from the loading surface f (non-associated model) is, in this particular case, required to correctly model the evolution of theplastic displacements.The plastic potential g has been chosen as

g"�H� �

�#�

M� �

�#<�!1"0 and P"

�g�F

(17)

with

�g�<

"2<,�g�H

"

2H��,

�g�M

"

2M��

(18)

It describes an ellipsoid centred at the axes origin (<, H,M).The choice of a non-associated model is justi"ed because the outer normal to the yield surface

is not convenient for de"ning the direction of the plastic displacements u� pl. This is particularly



Figure 15. Non-associated model. External normals of the loading surface and of the plastic potential.

-0.0045

-0.004

-0.0035

-0.003

-0.0025

-0.002

-0.0015

-0.001

-0.0005

0

0 0.0005 0.001 0.0015 0.002 0.0025

θpl

FE resultsmodel

M/H=2

M/H=20

M/H=5

M/H=10

x ’pl

Figure 16. Plastic #ow: plastic rotation}horizontal displacement.

obvious for loading with a small < (Figure 15). In that zone (i.e. for < smaller than the valuecorresponding to the summit of the parabola in the (H, <) or (M,<) plane), the componentalong< of the outer normal to the yield surface n

�is always negative whatever the loading is (and

particularly even if < increases). The choice of an ellipsoid centred at the axes origin as plasticpotential allows for increasing plastic vertical displacements, even in this zone.This potential linearizes the relationship between the plastic displacements for a radial loading

path of the corresponding forces. Figure 16 shows that this choice, when compared to FEsimulations, is acceptable.For this example (radial loading path in the (H,M) plane for a constant < ), we have (from

Equations (3) and (18))

�Q ��xR ��

"

��

��MH

(19)



M

M0=VB/4

θ0

MC=VB/2

θ

K θθ

Figure 17. Moment}rotation (elastic soil).

3.1.7. Elasto-plastic tangent stiwness matrix. In the previous chapters, all the components re-quired to de"ne a plasticity model have been set. The terms of the elastic sti!ness matrixcorrespond to the static sti!nesses of a shallow strip foundation. The interaction diagram of thefoundation bearing capacity (limit analysis for an overturning failure mechanism with uplift) hasbeen chosen for the failure criterion. The evolution of the loading surface and the hardening rulehas been expressed to describe the cyclic behaviour of the foundation specially under seismicloading. Finally, the choice of the plastic potential leads to a linearization of the relationshipsbetween plastic displacements for a radial path of the corresponding forces.Knowing all that, the expression of the elasto-plastic tangent sti!ness matrix K elpl can now be

derived:

F� "�K!

1

h#h�

(K :P)�(Q :K )� : u� with h�"Q :K :P (20)

3.2. Uplift model

The uplift model for a foundation lying on an elastic soil is "rst presented. The expression of theseparation ratio �, as a function of the M/< ratio, is given, as well as the evolution of therotational and vertical components of displacement due to uplift. Then the in#uence of soilyielding on the uplift behaviour is studied. Indeed, yielding modi"es the stress distribution underthe foundation and a!ects the value of the soil resisting forces for a given uplift. An uplift surface,moving in the forces space with the loading history, is proposed. The expressions of the rotationand vertical displacement of the elastic model are adapted to take into account the coupling withplasticity.

3.2.1. Uplift model for an elastic soil [21]Determination of the percentage of uplift width � for a (M, <) forces couple: For a shallow (perfectlyrigid) foundation lying on an elastic, homogeneous soil medium, under the assumption of anactual, vertical stress distribution under the foundation (vertical stresses tending towards in"nityat the edges of the foundation), CreHmer [21] proposed the following relationships (Figure 17):

Before uplift M(M�, M"K��

Uplift onset M"M�, M

�"K��



MC=VB/2

M0=VB/4

M

δ0 10.5

M01

Figure 18. Moment}uplift (elastic soil).

During uplift M'M�,

M

M�

"2!��

(21)

�"

��

1!�(22)

with M�"<B/4, where M

�is the moment for which uplift is initiated, �

�is the rotation

corresponding to M�.

M�"<B/2, whereM

�is the critical moment for which the structure overturns (o! centre vertical

force of a half foundation width).From these relationships (Equations (21) and (22)), it results that the length B� along which

separation occurs is a linear function of the ratio M/< (Figure 18) and is written as

�"

M

M�

!1 with M�"<B/4 (23)

Note: It should be noticed that the vertical stress distribution at the soil}foundation interface isvery slightly in#uenced by the soil pro"le (layer of limited depth, constant gradient of the elasticproperties with depth). Figure 19 compares the vertical stress distribution under vertical loadingfor an homogeneous soil pro"le (Poulos and Davis [22] analytical solution) and for a pro"le witha linearly increasing elastic sti!ness, corresponding to the reference case of Figure 2 (Dyna#owFE simulations). However, for a layer of very limited depth or for strong gradient, the in#uence ofuplift on the horizontal degree of freedom may no longer be neglected.

Calculation of the components in rotation �� and vertical displacement z�� induced by uplift: Setting�"��#�� and M"K�� during uplift, the expression of ��, the rotational componentcoming exclusively from uplift, is computed as

��"��

��

1!�(24)

By writing the purely kinematic relationship existing between rotation and vertical displacementof the foundation centre, the expression of the vertical displacement due to uplift can also be



0

0.5

1

1.5

2

2.5

3

3.5

4

σ zz/

σ zzm

ean

linearly increasing stiffness (numericalsolution)homogenous soil (analytical solution)

-B/2 0 B/2

Figure 19. Vertical stress distribution at the soil}foundation interface for a vertical loading.

computed. This is found by calculating "rst the in"nitesimal increment dz at the foundationcentre produced by an increment d�, under the assumption of a vertically "xed point located atthe centre of the foundation of reduced width (1!�)B. By integrating dz from �

�to �, and by

replacing � by its � expression (Equation (22)), it "nally gives

z��"B�

�2 �

�1!�

#ln (1!�)� with �'0 (25)

Note: All the equations discussed above will be, for the elasto-plastic model, expressed inthe dimensionless variables ( ) of our system in order to include the e!ect of the bearingcapacity <

��.

3.2.2. Uplift surface for an elasto-plastic soil. In the plasticity model, a loading surface growinginside the failure criterion and referring to a certain yielding level has been proposed. Followingthe same idea, an uplift surface moving inside an uplift domain de"ning the magnitude ofsoil}foundation separation is built. This uplift surface is expressed in the forces space.As for an elastic soil, it is observed that � is only a function of theM and < variables, and it

may be assumed that it is independent of theH force. For this reason, the following developmentswill be made in the (<,M) plane, for a givenH. The surfaces of &iso'-separation (same separationratio) are presented in Figure 20, superimposed with the failure criterion. Their shapes have beendeduced fromDyna-ow numerical simulations. Those surfaces de"ne, in the forces plane, an upliftdomain limited between the 0 and the 100 per cent separation surfaces.The slopes at the origin of the surface corresponding to the onset of uplift (�"0 per cent) and

that corresponding to maximum separation (�"100 per cent) are

�dMd< �

��

"� and �

dMd< �

��

"��

(26)

which is consistent with the uplift behaviour of the foundation on elastic soil.



Figure 20. &Iso'-uplift surfaces and failure criterion in the (M, <) plane.

M /V

M 0/V

δ

1/4

V <

V >

elastic

1

Figure 21. Evolution of the uplift for di!erent values of <.

The only values of separation, which may be reached during loading, are de"ned by the area ofthe uplift domain located inside the failure criterion. Thus it is obvious that, with plasticity, soilfailure occurs before uplift has led to 100 per cent separation between the soil and the foundation.The greater the < force, the smaller the maximum separation reached at failure.

3.2.3. Evolution of the uplift surfaceMonotonic loading: The momentM

�for which uplift is initiated is not (contrary to the elastic case)

a linear function of < (see M�curve in Figure 20). Its variation may be approximated by

a function of form

M�"

<4exp� � where A is a parameter to identify (27)

Moreover, it is observed that separation is still a linear function of the M/< ratio (Figure 21)with the same coe$cient, whatever the yielding level. By applying the uplift results in elasticity(Equation (23)), a coe$cient of �

is found for the slope.

The separation evolution is then de"ned by

�"

4

<(M!M

�) (28)



M /V

M 0(0)

δp(1)

η/41

11/4

M 0(1)

M 0(2)/V

M p(1)

M p(2)

δp(2) δ

/V

/V

/V/V

Figure 22. Uplift irreversibility (cyclic loading).

Knowing the expression of the surface at the onset of uplift and knowing also that separation isa linear function of the M/< ratio, the evolution of the uplift surface is, in the (M, <) plane,entirely de"ned for a monotonic loading.

Cyclic loading: For an elasto-plastic soil, uplift is no longer totally reversible (Figure 7). Duringcyclic tests, it is observed that, after some cycles, soil yielding, located around the edges of thefoundation, leads to a bumping of the interface, and then to the appearance of holes at each edge.This additional permanent non-linearity is visible on the (M/<!�) curve where the slopeduring unloading and reloading (till the previously applied loading has been reached) is signi"-cantly increased with respect to the original loading one (Figure 22).This behaviour leads to a division of the uplift domain into an elastic}uplift and a plastic}uplift

domain. The elastic domain de"nes the set of forces that have already been applied to thefoundation. The plastic domain covers the set of forces that have not yet been reached during theloading history. The surface, which separates the two domains, is nothing but the loadingsurface of the plasticity model. Each domain is characterized by a di!erent slope in the(M/<}�) diagram. The plastic domain has a slope of �

. The elastic domain has a larger slope

�/4, where � is a magnifying factor (ranging from 1 for an elastic soil to about 4 for a highlyplastic soil) depending on the yielding and thus on the < force. The � parameter needs to beidenti"ed.To summarize (Figure 23), for a given H and <, uplift occurs if M'M��

�and:

� if M'M��: �"

4

<(M!M��

�) with M��

�"

<4exp� � ,

� if M)M��: �"

4

�<(M!M��

�) M��

�"M��

�!

�<4

��,

whereM��is the moment belonging to the loading surface. It is calculated from Equation (10) for

the given H and <, ��is the separation corresponding to M��

�through the equation

��

"(4/<) (M��

!M��).



surface

M M

δ=100%

V δ10

Failure

δp(i) δ max

M c

M max

M 0(i)

M 0(0)

M p(i)Plastic

Elastic

Elastic Plastic

δ=0%(0)

δ=0%(i)

Figure 23. Coupling between plasticity and uplift.

3.2.4. Rotation �� and vertical displacement z�� induced by uplift. As noted just above, thepercentage of separation reached at failure is smaller than 100 per cent, depending on the applied<. Thus in order to ensure that �� tends towards in"nity when the soil fails under a purelyoverturning mechanism (i.e. for HP0), � is replaced, in Equations (24) and (25), by the ratio�/�

��. �

��is the maximum percentage of separation width reached at failure for a given < and

for H"0 (Figure 20). Following Equation (28), we write

��

"

4

<(M

��!M

�) (29)

where M��is the maximum moment reached at failure for a given < and H"0, i.e. (from

Equation (5))

M��

"b<� (1!<) (30)

The expressions of �� and z�� are now written as

��"(1!< ) ��

(�/��)�

1!�/��

(31)

z��"(1!< )��2 �

�/��

1!�/��

#ln (1!�/��)� (32)

In both of them, the (1!<) coupling coe$cient has been introduced because it is noted that, fora given value of �/�

��, the greater < (the larger yielding), the smaller the �� and z�� uplift

components. This expression is consistent because for <P0 (elastic soil), the coe$cient is equalto 1 (no yielding) and for <P1 (vertical force tending towards the ultimate bearing capacity ofthe foundationPmaximum yielding and failure), the coe$cient is equal to 0 (no uplift).

3.2.5. Uplift tangent stiwness matrix. To summarize, knowing the current state of yielding (loca-tion of the loading surface), the percentage of separation corresponding to the forces <, H, Mcan be calculated. Then, from the � value, the �� and z�� displacement components are obtained.



dF, duel

dFdutot = duel + dupl + duup

Elasticity

Plasticity

Uplift

dF, dupl

dF, duup

ModŁle de plasticitØdF=Kelpl (duel + dupl)

Plasticity modeldF=Kelpl (duel + dupl)

Uplift modeldF=Kup duup

dF=Kelplup dutot

Figure 24. Structure of the global model.

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-0.0008 -0.0006 -0.0004 -0.0002 0 0.0002 0.0004 0.0006 0.0008

θ (rad)

M (

MN

m)

Figure 25. Plastic model (el. #pl. comp.).

By writing the "rst order Taylor development of �� and z�� ,

d��"��<

d<#��M

dM and dz��"�z��<

d<#�z��M

dM (33)

the linearized terms of the compliance matrix can be computed. After inversion, the uplift tangentsti!ness matrix Kup is calculated.

3.3. Global model (plasticity#uplift)

The total displacement increment u� �� experienced by a foundation submitted to a F� forces vectorincrement is simply obtained by summing the elastic, plastic and uplift components (Figure 24).

u� tot"u� el#u� pl#u� up (34)

Figures 25}27 show an example for the rotational component.



-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-0.0003 -0.0002 -0.0001 0 0.0001 0.0002 0.0003

θ (rad)

M (

MN

m)

Figure 26. Uplift model (up. comp.).

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-0.001 -0.0005 0 0.0005 0.001

θ (rad)

M (

MN

m)

Figure 27. Global model (el. #pl. #up. comp.).

4. PARAMETER IDENTIFICATION

4.1. Tests required for the identixcation

Contrary to the parameter identi"cation of a material constitutive model, the identi"cation ofa macro-element may not be completed with one or two classical tests. The di$culty comes herefrom the concept of the macro-element itself. As it describes the behaviour of a global system (andnot of a material), the identi"cation has to be carried out on the entire system (foundation#soil).The complete identi"cation of all model parameters requires one con"guration of foundation

and soil (one material setting-up for experimental test or one mesh discretization for numerical



simulation) for which it is necessary to carry out only one test consisting of two di!erent loadingphases.

� First phase: Monotonic increase of the vertical centred force until the force reaches the weight ofthe studied structure (P<}z).

� Second phase: For the constant vertical force applied at phase 1, increase followed by decreaseof the moment and horizontal force in a constant ratio equal to the height of the gravity centreof the studied structure (PH}x, M}�, M}z, M}�).

4.2. Identixcation procedure

Besides the parameters that de"ne the studied system, and which are considered to be known asgeometrical parameters of the foundation (width B) and material properties of the soil (shearmodulus G, the Poisson ratio �, cohesion c and their gradient with depth), the parameters toidentify are as follows

� elastic parameters: elastic sti!nesses, K��, K

��, K�� (Equation (4));

� plastic parameters: parameters de"ning the plastic potential, � and � (Equation (17));� uplift parameters: plastic moment for which the uplift is initiated,M

�; magnifying factor �, to

apply to the loading slope of the relationship (M/<, �).

Note: The parameters de"ning the shape of the failure criterion are considered to be constant(c, d, e, f ) or a function of the q

��/q

��ratio (a, b). They are given in Table I.

In agreement with the model development (Chapter 2.4), the identi"cation has to follow thisparticular procedure. First the elastic parameters, then the uplift ones and "nally the plastic onesshould be identi"ed. Identi"cation of the uplift parameters has to precede that of the plasticparameters and the latter ones have to be identi"ed on the plastic displacements from which theuplift components are subtracted.TheK

��elastic sti!ness is "tted to the initial sti!ness of the curve (<, z) issued from the "rst test

phase. The K��and K�� ones are similarly identi"ed on the H}x andM}� curves of the second

phase. These sti!nesses are not purely elastic as the soil has already (depending on the appliedweight) yielded during the "rst phase. The purely elastic ones may be captured on the unloadingpart when the uplift is zero.The uplift M

�and � parameters are identi"ed from the (M}�) curve, which has been trans-

formed into the dimensionless variables (M/<, �). For �, it is "rst veri"ed that the slope of theloading part is in agreement with the elastic case (slope of about �

) and the � magnifying factor,

which should be applied to the loading slope, is derived on the unloading part.The � and � plastic parameters are derived from the relationships existing between the plastic

displacements (as shown in Chapter 3.1.6 for the (��, z��) relationship). These are obtained fromthe total displacements u�� from which the elastic u�� and the uplift u�� components aresubtracted. The elastic displacements are calculated with the elastic sti!nesses and the uplift onesare computed following Equations (31) and (32) with � issued from the uplift identi"cation.

4.3. Proposed relationships for the parameters determination without test

As is known, carrying out experimental tests on a foundation lying on a cohesive soil is a tricky,di$cult and costly task. Also, running numerical analyses on a largely discretized soil medium



-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-0.0015 -0.001 -0.0005 0 0.0005 0.001 0.0015

θ (rad)

M (

MN

m)

FE modelmacro-element

B=10mVmax=2.4MNV=0.6MNM/H=10m

Figure 28. Overturning moment}rotation (<"0.6 MN).

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-0.005 -0.004 -0.003 -0.002 -0.001 0 0.001 0.002 0.003 0.004

x (m)

H (

MN

)



Figure 29. Horizontal force}horizontal displacement (<"0.6 MN).

with a suitable soil constitutive law and contact elements requires special capabilities and is timeconsuming.For these reasons, a set of relationships is proposed for the determination of the model

parameters allowing one to dismiss, if necessary, any identi"cation test. The relationships, forsome of them, are known expressions, coming from literature, and are regularly used in practicalapplications. For the others, they have been "tted here from numerous FE simulations. Fora preliminary study, the use of these relationships (without any identi"cation test) may besu$cient, depending on the required accuracy.These relationships are given in Table I.



-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-60 -40 -20 0 20 40 60

δ (%)

M (

MN

m)



Figure 31. Overturning moment}uplift (<"0.6 MN).

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.002 0.004 0.006 0.008 0.01

M (

MN

m)



z (m)

Figure 30. Overturning moment}vertical displacement (<"0.6 MN).

5. COMPARISON MACRO-ELEMENT/FE MODEL

The comparison between the results obtained from the FE modelization (Dyna-ow) and from themacro-element is presented in Figures 28}35. The same geometry and the same characteristics forthe soil as those presented in Figure 2 have been used, as well as the same loading path. Theparameters identi"cation has been carried out as described in Chapter 4.1.2. Results are presentedfor three di!erent vertical loads (<"0.3, 0.6, 0.9 MN). From Figures 28}31 (<"0.6 MN), it maybe concluded that the cyclic behaviour of the foundation obtained from the macro-elementreproduces very well all the trends observed with the FE model. From Figures 32}35 (<"0.3



-1.5

-1

-0.5

0

0.5

1

1.5

-0.0015 -0.001 -0.0005 0 0.0005 0.001 0.0015θ (rad)

M (

MN

m)



Figure 32. Overturning moment}rotation (<"0.3 MN).

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004

z (m)

M (

MN

m)




and 0.9 MN), it is noticed that the model is better suited for describing moderate plasticitybehaviour, but with strong uplift.

6. CONCLUSION

A non-linear soil}structure interaction macro-element for a shallow strip foundation lying ona cohesive medium has been proposed. Expressed in global variables, the macro-element



-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

-0.0015 -0.001 -0.0005 0 0.0005 0.001 0.0015

θ (rad)

M (

MN

m)



Figure 34. Overturning moment}rotation (<"0.9MN).

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 0.005 0.01 0.015 0.02

z (m)

M (

MN

m)




reproduces the cyclic behaviour of the foundation, including the e!ects of non-linearities occur-ring in the near "eld. Yielding of the soil under the foundation is modelled through a globalplasticity model. The contact non-linearities induced by the uplift of the foundation are integratedin an uplift model. Although developed separately, these two models remain strongly dependentin order to take into account the coupling e!ect. This substructure approach allows thedecomposition of a highly non-linear, coupled problem, but without dismissing any couplings.The comparison of the results obtained from the macro-element with those from an FEmodellinghas enhanced the relevance of the proposed model.It is well known that an alternative model of the foundation behaviour obtained by the "nite

element method with suitable non-linear soil constitutive laws and special contact elements,



requires a high degree of modelling competence and is time consuming. The macro-elementprovides a practical and e$cient tool, which may replace e$ciently, in a "rst approach, acostly FE soil model, and which ensures the accurate integration of the e!ect of soil}structureinteraction.Specially developed for seismic loading, the macro-element will be adapted to a dynamic

behaviour. The radiation of waves in an in"nite half-space, as well as energy dissipation, will beintroduced by the concept of dynamic impedances.

REFERENCES

1. Meyerhof GG. The bearing capacity of foundations under eccentric and inclined loads. Proceedings of the ¹hirdInternational Conference on Soil Mechanics Foundations Engineering, Zurich, vol. 1, 1953; 440}445.

2. Vesic AS. Bearing capacity of shallow foundations. In Foundation Engineering Handbook, Winterkorn HF, and FangH-Y (eds). van Nostrand Reinhold: New York, 1975; 121}147.

3. Butter"eld R, Gottardi G. A complete three-dimensional failure envelope for shallow footings on sand. Ge&otechnique1994; 44:181}184.

4. Salenion J, Pecker A. Ultimate bearing capacity of shallow foundations under inclined and eccentric loads. Part I:purely cohesive soil. European Journal of Mechanics, A/Solids 1995; 14(3):349}375.

5. Salenion J, Pecker A. Ultimate bearing capacity of shallow foundations under inclined and eccentric loads. Part II:purely cohesive soil without tensile strength. European Journal of Mechanics, A/Solids, 1995; 14(3):377}396.

6. Paolucci R, Pecker A. Seismic bearing capacity of shallow strip foundations on dry soils. Soils and Foundations,Japanese Geotechnical Society 1997; 37(3):95}105.

7. Ukritchon B, Whittle AJ, Sloan SW. Undrained limit analysis for combined loading of strip footings on clay. Journalof Geotechnical and Geoenvironmental Engineering 1998; 124(3):265}276.

8. Houlsby GT, Puzrin AM. The bearing capacity of a strip footing on clay under combined loading. Proceedings of theRoyal Society of ¸ondon A, 1999; 455:893}916.

9. Tan FSC. Centrifuge and theoretical modelling of conical footings on sand. PhD thesis, University of Cambridge,1990.

10. Nova R, Montrasio L. Settlements of shallow foundations on sand. Ge&otechnique 1991; 41(2):243}256.11. Gottardi G, Houlsby GT, Butter"eld R. Plastic response of circular footings on sand under general planar loading.

Ge&otechnique 1999; 49(4):453}469.12. Martin CM. Physical and numerical modelling of o!shore foundations under combined loads. DPhil ¹hesis,

University of Oxford, 1994.13. Pedretti S. Nonlinear seismic soil-foundation interaction: analysis and modelling method. PhD thesis, Dpt Ing

Strutturale, Politecnico di Milano, 1998.14. Psycharis IN. Dynamic behavior of rocking structures allowed to uplift. Journal of Earth Engineering and Structural

Dynamics 1983; 11:57}76 and 501}521.15. PreH vost JH. Anisotropic undrained stress}strain behaviour of clays. Journal of the Geotechnical Engineering Division

1978; GT8:1075}1090.16. Davis EH, Booker JR. The e!ect of increasing strength with depth on the bearing capacity of clays. Ge&otechnique

1973; 23(4):551}563.17. Matar M, Salenion J. CapaciteH portante des semelles "lantes. Revue franmaise de Ge&otechnique 1979; 9:51}76.18. Gazetas G. Foundations vibrations. In Foundation Engineering Handbook, Fang H-Y (ed.), Chapter 15. van Nostrand

Reinhold: New York 1991.19. Pecker A. Analytical formulae for the seismic bearing capacity of shallow strip foundations. In Seismic Behaviour of

Ground and Geotechnical Structures, Seco e Pinto (ed.), Balkema: Rotterdam, 1997; 261}268.20. CreHmer C, Pecker A, Davenne L. Elaboration d'un macro-eH leHment d'interaction sol-structure avec prise en compte du

deH collement de la fondation. Proceedings of the <�& �� Colloque National Afps, Cachan, vol. 1, 1999; 197}206.21. CreHmer C. Elaboration of a soil}structure macro-element for foundation uplift. Me&moire de stage de DEA, ENS-

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cyclic macro-element for soil–structure interaction: material and geometrical non-linearities

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