International audienceIn earthquake engineering, several approaches exist to take into account Soil–Structure Interaction (SSI): the following work is based on the “macro-element” concept. The particularity of the macro-element lies in the fact that the movement of a foundation is entirely described by a system of generalised variables (forces and displacements) defined at the foundation centre. The non-linear behaviour of the soil is reproduced using the classical theory of plasticity. This Note improves an already existing macro-element by adding the uplift behaviour of the foundation according to the plasticity theory. Comparisons with experimental results of a foundation submitted to cyclic loadings show the performance of the approach.Un macro-élément d'Interaction Sol-Structure pour une fondation superficielle prenant en compte un mécanisme de décollement. Dans le domaine du génie parasismique, plusieurs méthodes existent pour considérer l'Interaction Sol-Structure : ce travail est basé sur le concept de « macro-élément ». La particularité du macro-élément réside dans le fait qu'il est formulé en variables globales (forces et déplacements) décrites au centre de la fondation. Le comportement non-linéaire du sol est reproduit selon la théorie de plasticité. Cette Note propose une amélioration d'un macro-élément existant dans la littérature par la prise en compte du décollement de la fondation selon la théorie de la plasticité. Des comparaisons avec des résultats expérimentaux d'une fondation soumise à des chargements cycliques montrent la performance de cette approche

Grange, Stéphane

Kotronis, Panagiotis

Mazars, J.

English

HAL-Univ-Nantes

A macro-element for a shallow foundation to simulate Soil-Structure Interaction considering uplift

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A macro-element for a shallow foundation to simulate
Soil-Structure Interaction considering uplift
Stéphane Grange, Panagiotis Kotronis, J. Mazars
To cite this version:
Stéphane Grange, Panagiotis Kotronis, J. Mazars. A macro-element for a shallow foundation to
simulate Soil-Structure Interaction considering uplift. Comptes Rendus Mécanique, Elsevier Masson,
2008, 336 (11 12), pp.856-862. ￿10.1016/j.crme.2008.10.002￿. ￿hal-01004841￿
A macro-element for a shallow foundation to simulate
Soil–Structure Interaction considering uplift
Stéphane Grange ∗, Panagiotis Kotronis, Jacky Mazars
Laboratoire 3S-R (Sols, Solides, Structures-Risques), INPG/UJF/CNRS, domaine universitaire, BP 53, 38041 Grenoble cedex 9, France 
Abstract
In earthquake engineering, several approaches exist to take into account Soil–Structure Interaction (SSI): the following work is 
based on the “macro-element” concept. The particularity of the macro-element lies in the fact that the movement of a foundation is 
entirely described by a system of generalised variables (forces and displacements) defined at the foundation centre. The non-linear 
behaviour of the soil is reproduced using the classical theory of plasticity. This Note improves an already existing macro-element 
by adding the uplift behaviour of the foundation according to the plasticity theory. Comparisons with experimental results of 
a foundation submitted to cyclic loadings show the performance of the approach. 
Résumé
Un macro-élément d’Interaction Sol–Structure pour une fondation superficielle prenant en compte un mécanisme de 
décollement. Dans le domaine du génie parasismique, plusieurs méthodes existent pour considérer l’Interaction Sol–Structure : 
ce travail est basé sur le concept de « macro-élément ». La particularité du macro-élément réside dans le fait qu’il est formulé en 
variables globales (forces et déplacements) décrites au centre de la fondation. Le comportement non-linéaire du sol est reproduit 
selon la théorie de plasticité. Cette Note propose une amélioration d’un macro-élément existant dans la littérature par la prise en 
compte du décollement de la fondation selon la théorie de la plasticité. Des comparaisons avec des résultats expérimentaux d’une 
fondation soumise à des chargements cycliques montrent la performance de cette approche. 
Keywords: Soils; Soil–structure interaction; Uplift; Macro-element; Foundation; Plasticity
Mots-clés : Sols ; Interaction sol–structure ; Décollement ; Macro-element ; Fondation ; Plasticité
* Corresponding author.
E-mail addresses: Stephane.Grange@hmg.inpg.fr (S. Grange), Panagiotis.Kotronis@inpg.fr (P. Kotronis), Jacky.Mazars@inpg.fr (J. Mazars).1
Version française abrégée
Dans le domaine du génie parasismique, l’Interaction du Sol avec la Structure (ISS) est un phénomène important à
considérer pour espérer rendre compte du comportement réel d’une structure et donc évaluer sa vulnérabilité. Plusieurs
méthodes existent pour prendre en compte cette interaction. Basée sur le concept de « macro-élément », la formulation
presentée dans nos anciens travaux permettait de modéliser, moyennant un adimensionnement adéquat des variables,
une fondation superficielle de forme circulaire reposant sur un massif de sol semi infini en considérant la plasticité du
sol [1]. Dans cet article, une amélioration du macro-élément initial est proposée par la prise en compte du décollement
de la fondation formulé aussi suivant la théorie de plasticité. La particularité de ce décollement est qu’il est considéré
avoir un comportement non-linéaire et non-réversible (Fig. 3(a)) et que les deux sens de chargements (moment positif
ou négatif) sont découplés. En d’autres termes, deux surfaces de charge évoluant en chargement et en déchargement
peuvent être actives (l’Éq. (6)). L’évolution des variables d’écrouissage est différente en chargement et en décharge-
ment (l’Éq. (5)). Du fait de l’apparition de décollements résiduels dus à la plastification du sol (Fig. 3(b)), le domaine
élastique peut totalement disparaitre ce qui conduit à l’activation simultanée des deux mécanismes (Fig. 4).
Le nouveau macro-élément est implémenté dans FEDEASLab, un code élément finis développé dans Matlab [2].
La comparaison avec des résultats expérimentaux d’une fondation soumise à des chargements cycliques [3] montre
l’importance de la composante de décollement notamment dans le calcul du soulèvement du centre de la fondation
(Fig. 5).
1. Introduction
In the field of earthquake engineering, Soil–Structure Interaction (SSI) is a phenomenon that has to be taken into
account in order to reproduce correctly the non-linear behaviour of a structure and thus to be able to predict its relative
displacements at the top. Several methods exist: the macro-element approach consists in condensing all non-linearities
into a finite domain. It works with generalised variables (forces and displacements) at the centre of the foundation,
thus allowing decreasing the necessary degrees of freedom of the numerical model.
Several macro-elements can be found in the literature [4–6]. The macro-element presented in [1] reproduces the
behaviour of a shallow circular foundation considering the plasticity of the soil. An extension is introduced in this
Note considering the uplift of the foundation according to the plasticity theory. The macro-element is implemented
into FEDEASLab, a finite element MATLAB toolbox [2]. After the mathematical description of the macro-element,
numerical results compared with experimental tests under cyclic loadings [3] are provided to show the performance
of this new numerical tool.
2. Associated generalised variables
As usual is the case for a macro-element, the associated generalised variables (displacement and force vectors) are
dimensionless [1]. The are defined hereafter: vertical force V ′, horizontal forces H ′x , H ′y and moments M ′x , M ′y , but
also the corresponding displacements, vertical settlement u′z, horizontal displacements u′x , u′y and rotations θ ′x , θ ′y .
Torque moment (M ′z) is not taken into account (Fig. 1).
3. Decomposition of three mechanisms: elasticity plasticity and uplift
The new SSI macro-element takes into account three different mechanisms: elasticity, plasticity of the soil and
uplift of the foundation. The total displacement must thus be decomposed as a sum of an elastic, plastic and uplift
part:
u = uel + upl + uup (1)
Uplift is defined as the negative vertical displacement of the centre of the foundation. It is the result of rocking, i.e.
the fact that the foundation rotates around θx or θy (part of the foundation looses contact with the soil), see Fig. 2.
In order to compute uplift, the simple plasticity of the soil is not sufficient and a new non-linear mechanism must be
introduced in the macro-element. The reason is that the actual plasticity mechanism of the macro-element can take
into account only positive values of the vertical displacement (u′z > 0). Plasticity and uplift are strongly coupled [4].2
Fig. 1. Generalised variables: (a) forces and (b) displacements for a circular foundation.
Fig. 1. Variables généralisées : (a) forces et (b) déplacements pour une fondation circulaire.
Fig. 2. Kinematics of a foundation for the uplift mechanism.
Fig. 2. Cinématique de la fondation pour un mécanisme de décollement.
4. Mathematical description of the macro-element
4.1. Elastic behaviour and plastic mechanism
The elastic and plastic mechanisms are very briefly described hereafter. For more information the reader is invited
to look in [7,1].
The elastic part of the constitutive law is defined as F = Keluel, where the displacement uel and force vectors
F are dimensionless. The elastic stiffness matrix Kel is calculated using the real part of the static impedances. It is
considered diagonal, i.e. there is not coupling between the different directions of the loading.
For the plastic mechanism, a failure criterion, a loading surface and a flow rule are needed. The failure criterion
is defined for an overturning mechanism with uplift. It comes from [8]. The adaptation for a 3D loading is presented
in [7]. It relies on the fact that failure criteria are similar for different shapes of foundation when described into the
appropriate space of dimensionless variables. Thus, the extension to the case of a circular foundation is straightfor-
ward due to the axial symmetry of the problem. For a rectangular foundation however, we assume continuity of the
behaviour between the two principal axes of the foundation. This is of course a more questionable assumption that
needs improvement in the future. However, due to the nature of the macro-element (being a simplified numerical tool),
we consider this hypothesis acceptable. The loading surface follows the same philosophy and one obtains 5D surfaces.
A flow rule is finally needed. It is associated in the (H ′x,M ′y,H ′y,M ′x) hyperplane and non-associated in the (H ′x,V ′),
(M ′y,V ′), (H ′y,V ′), (M ′x,V ′) planes [7].
4.2. Uplift behaviour
The mechanism presented hereafter describes in a phenomenological way uplift using a unique state variable δ.
This variable represents the percentage of the surface of the uplifted footing [4], Fig. 2 (for any a, a˙ defines the
derivative with respect to time). Indeed, this is the only way to introduce the influence of the change of the geometry,
the macro-element being just a point. We assume that uplift is not influenced by horizontal forces.
For 2D loading, the behaviour of the foundation during uplift is presented in Fig. 3. This figure presents the relation
between M ′ −δ (M ′ being M ′x or M ′y ), see also [4]. One can clearly see that the behaviour is not reversible and that the3
Fig. 3. Moment-δ relationship for a plastic soil.
Fig. 3. Evolution moment-décollement sur sol plastique.
unloading slope (ηq2/q1) is increased with respect to the original loading one (q2/q1). The two directions of loading
(M ′ > 0 or M ′ < 0) are uncoupled. Furthermore, this behaviour is physically translated into a residual uplift at each
edge of the foundation, see [1] and Fig. 3(b). The symbols ⊕ and  are used to discriminate the directions of loading.
For a circular foundation q1 = 6, q2 = 2, for a rectangular foundation q1 = 4, q2 = 1 and the evolution of η is
provided by [4]:
η = 4 − 3e−4V ′ (2)
4.2.1. Failure criterion and loading surfaces
For the uplift mechanism, failure occurs when the foundation is completely detached from the soil, i.e. when
δ = 1. A simple analytical analysis for a circular foundation lying on elastic soil allows finding the relation M ′ =
V ′/2 between the overturning moment and the given vertical force. This equation can be actually considered as a
failure criterion. On a plastic soil, the relationship between the overturning moment and the vertical force is more
complicated [4]. For a loading in two different directions (for M ′ > 0 and M ′ < 0) we obtain [1]:
f∞ ≡ M ′2 −
(
V ′
q1
(
e−AV ′ + q2
))2 = 0 (3)
where A = 2.5 is a dimensionless parameter of the constitutive law.
The uplift behaviour being a non-linear, non-reversible mechanism with the unloading slope increased with respect
to the original loading one, the evolutions of the loading surfaces are activated even during unloading. Nevertheless,
an initial elastic domain often exists (except if the loading is important leading to high non-linearities in the soil and
thus to a total suppression of the elastic domain).
In order to activate the loading surfaces in loading but also in unloading, their mathematical expression is chosen
to be always positive whatever the sign of the loading. For the case of the direction ⊕, the corresponding hardening
variable β⊕ evolves between δ and δmax (maximal percent of uplift reached during the loading). It is defined as:
β⊕ = δmax(1 − η) + ηδ (4)
The evolution of the hardening variable β⊕ is also given by Eq. (5) as a function of the rotation due to uplift θ ′up
and the rotation θ ′0 related to the initial overturning moment, see also Fig. 2 (β⊕max being the maximal value of β⊕
reached during the loading).⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
˙β⊕ = θ˙
′up
θ ′0
(1 − β⊕)2
β⊕(2 − β⊕) if β
⊕ = β⊕max
˙β⊕ = θ˙
′up
θ ′0
η
(
1 − (β⊕−(1−η)β⊕max
η
))2
β⊕−(1−η)β⊕max (2 − (β⊕−(1−η)β⊕max )) if β⊕  β⊕max
(5)η η
4
Fig. 4. Loading surfaces, failure criterion and elastic limits with their signs.
Fig. 4. Surfaces de charge, critère de rupture et limites élastiques avec leurs signes.
The mathematical expression of the loading surface for the direction ⊕ is finally provided by Eq. (6).
f ⊕ ≡
∣∣∣∣M ′ − V ′q1
(
e−AV ′ + q2β⊕
)∣∣∣∣= 0 (6)
It is also necessary to introduce the surface defining the elastic limit zone. The loading surfaces being always
positive, a test on the elastic surface allows knowing in which mechanism (uplift or elastic) is the model. The elastic
limit is a function of the maximum percentage δmax. Its mathematical expression is given in Eq. (7):
f ⊕el ≡ M ′ −
V ′
q1
q2β
⊕
max(1 − η) −
V ′
q1
e−AV ′ = 0 (7)
If residual uplift occurs on the ⊕ side of the foundation, the elastic domain entirely disappears. Conditions for
choosing the activated mechanisms are given by the equations:{
f ⊕el (M ′,V ′) 0 or f ⊕(M ′,V ′, β⊕) = 0 ⇒ elasticity
f ⊕el (M ′,V ′) > 0 and f ⊕(M ′,V ′, β⊕) > 0 ⇒ uplift
(8)
The same equations can be written for the other direction by replacing ⊕ with . The graphical representation of
all the surfaces is finally given in Fig. 4.
As for the plasticity mechanism, the adaptation of the uplift mechanism in 3D is done assuming continuity of the
behaviour between the two principal axes of the foundation (the two directions are coupled considering a projection
of the moment in the principal base [1]). Coupling of the plasticity and the uplift mechanisms is done following the
classical theory of the multi-mechanisms [1]. The choice and the computation of the normal vectors on the different
plasticity mechanisms follow the algorithm presented in [1] and in [9].
4.2.2. Flow rule
The flow rule for the uplift mechanism is found through geometrical considerations, assuming that the centre
of rotation of the foundation stays always at the middle of the non-uplifted segment (Fig. 2, [1]). Its mathematical
expression takes the following form (β is equal to β⊕ or β):⎧⎪⎪⎨
⎪⎪⎩
∂g
∂M ′
= fg|fg|
∂g
∂V ′
= − δ
2
fg
|fg|
with fg ≡ M ′ − V
′
q1
(
e−AV ′ + q2β
)= 0 (9)
5. Numerical simulations and comparisons with experimental results for a cyclic loading
Within the European program TRISEE, several experimental tests are performed on a shallow 1 m×1 m rectangular
foundation lying on sand [3]. Two tests are presented hereafter concerning different types of sand (High density sand5
Fig. 5. Tests TRISEE, moment–rotations and time history of the vertical settlements for the High density (HD) and the Low density sand (LD) for
the phase III.
Fig. 5. Tests TRISEE, moment–rotations et déplacement vertical en fonction du temps pour le sable haute densité (HD) et le sable de faible densité
(LD) pour la phase III.
(HD) and Low density sand (LD)). Phase III (where the loading was the most severe) of the experimental program
is simulated using the macro-element. Fig. 5 shows the comparison of the experimental with the numerical results.
One can identify the influence of the uplift mechanism looking at the S-shaped moment–rotation curve. Rotations
due to the plasticity of the soil are found almost equal to the ones coming from uplift (phase III with HD sand). In
other words, uplift and plasticity of the soil have similar contributions on the moment–rocking angle curve. The waves
present in the settlement curve are also not reproduced if uplift is not taken into account.
6. Conclusions
The 3D macro-element developed within this work gives satisfactory results for simulating the non-linear be-
haviour of shallow rigid foundations lying on an infinite space submitted to a cyclic loading. It takes into account the
plasticity of the soil and the uplift of the foundation. Using global variables it presents the advantage of inducing low
computational costs (couple of minutes for each simulation).
All the results presented in this paper are for loadings in a plane (2D). The 3D behaviour of the element has not
been validated due to the difficulty to find experimental results with loadings in 2 horizontal directions. This point
should constitute a future validation for the element.
References
[1] S. Grange, Modélisation simplifiée 3D de l’interaction sol–structure: application au génie parasismique, PhD Thesis, laboratoire 3S-R, INP
Grenoble, France, 2008.
[2] F.C. Filippou, M. Constandines, FedeasLab getting started guide and simulations examples, Department of Civil and Environmental Engineer-
ing, UC, Berkeley, 2004.
[3] TRISEE, 3D Site Effects and Soil-Foundation Interaction in Earthquake and Vibration Risk Evaluation, Part 4, Large-scale geotechnical exper-
iments on soil–foundation interaction. European Commission, Directorate General XII for Science, Research and Development, 1998.
[4] C. Crémer, A. Pecker, L. Davenne, Cyclic macro-element for soil–structure interaction: material and geometrical non-linearities, International
Journal for Numerical and Analytical Methods in Geomechanics 25 (13) (2001) 1257–1284.
[5] C. Crémer, A. Pecker, L. Davenne, Modelling of non-linear dynamic behaviour of a shallow strip foundation with macro-element, Journal of
Earthquake Engineering 6 (2) (2002) 175–211.6
[6] R. Nova, L. Montrasio, Settlements of shallow foundations on sand, Géotechnique 41 (2) (1991) 243–256.
[7] S. Grange, P. Kotronis, J. Mazars, A macro-element for a circular foundation to simulate 3D soil–structure interaction, International Journal for
Numerical and Analytical Methods in Geomechanics 32 (10) (2008) 1205–1227.
[8] A. Pecker, Analytical formulae for the seismic bearing capacity of shallow strip foundations, in: Seco e Pinto (Ed.), Seismic Behavior of Ground
and Geotechnical Structures, Rotterdam, Balkema, 1997, pp. 261–268.
[9] F. Collin, Couplages Thermo-hydro-mécaniques dans les sols et les roches tendres partiellement saturés, PhD Thesis, Université de Liège, 2003.7


Stéphane Grange

Panagiotis Kotronis

Jacky Mazars

Comptes Rendus Mécanique (CRM)

C. R. Mecanique 336 (2008) 856–862http://france.elsevier.com/direct/CRAS2B/A macro-element for a shallow foundation to simulateSoil–Structure Interaction considering upliftStéphane Grange ∗, Panagiotis Kotronis, Jacky MazarsLaboratoire 3S-R (Sols, Solides, Structures-Risques), INPG/UJF/CNRS, domaine universitaire, BP 53, 38041 Grenoble cedex 9, FranceReceived 21 July 2008; accepted after revision 6 October 2008Available online 29 October 2008Presented by Jean SalençonAbstractIn earthquake engineering, several approaches exist to take into account Soil–Structure Interaction (SSI): the following work isbased on the “macro-element” concept. The particularity of the macro-element lies in the fact that the movement of a foundation isentirely described by a system of generalised variables (forces and displacements) defined at the foundation centre. The non-linearbehaviour of the soil is reproduced using the classical theory of plasticity. This Note improves an already existing macro-elementby adding the uplift behaviour of the foundation according to the plasticity theory. Comparisons with experimental results ofa foundation submitted to cyclic loadings show the performance of the approach. To cite this article: S. Grange et al., C. R.Mecanique 336 (2008).© 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméUn macro-élément d’Interaction Sol–Structure pour une fondation superficielle prenant en compte un mécanisme dedécollement. Dans le domaine du génie parasismique, plusieurs méthodes existent pour considérer l’Interaction Sol–Structure :ce travail est basé sur le concept de « macro-élément ». La particularité du macro-élément réside dans le fait qu’il est formulé envariables globales (forces et déplacements) décrites au centre de la fondation. Le comportement non-linéaire du sol est reproduitselon la théorie de plasticité. Cette Note propose une amélioration d’un macro-élément existant dans la littérature par la prise encompte du décollement de la fondation selon la théorie de la plasticité. Des comparaisons avec des résultats expérimentaux d’unefondation soumise à des chargements cycliques montrent la performance de cette approche. Pour citer cet article : S. Grange etal., C. R. Mecanique 336 (2008).© 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Keywords: Soils; Soil–structure interaction; Uplift; Macro-element; Foundation; PlasticityMots-clés : Sols ; Interaction sol–structure ; Décollement ; Macro-element ; Fondation ; Plasticité* Corresponding author.E-mail addresses: Stephane.Grange@hmg.inpg.fr (S. Grange), Panagiotis.Kotronis@inpg.fr (P. Kotronis), Jacky.Mazars@inpg.fr (J. Mazars).1631-0721/$ – see front matter © 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crme.2008.10.002S. Grange et al. / C. R. Mecanique 336 (2008) 856–862 857Version française abrégéeDans le domaine du génie parasismique, l’Interaction du Sol avec la Structure (ISS) est un phénomène important àconsidérer pour espérer rendre compte du comportement réel d’une structure et donc évaluer sa vulnérabilité. Plusieursméthodes existent pour prendre en compte cette interaction. Basée sur le concept de « macro-élément », la formulationpresentée dans nos anciens travaux permettait de modéliser, moyennant un adimensionnement adéquat des variables,une fondation superficielle de forme circulaire reposant sur un massif de sol semi infini en considérant la plasticité dusol [1]. Dans cet article, une amélioration du macro-élément initial est proposée par la prise en compte du décollementde la fondation formulé aussi suivant la théorie de plasticité. La particularité de ce décollement est qu’il est considéréavoir un comportement non-linéaire et non-réversible (Fig. 3(a)) et que les deux sens de chargements (moment positifou négatif) sont découplés. En d’autres termes, deux surfaces de charge évoluant en chargement et en déchargementpeuvent être actives (l’Éq. (6)). L’évolution des variables d’écrouissage est différente en chargement et en décharge-ment (l’Éq. (5)). Du fait de l’apparition de décollements résiduels dus à la plastification du sol (Fig. 3(b)), le domaineélastique peut totalement disparaitre ce qui conduit à l’activation simultanée des deux mécanismes (Fig. 4).Le nouveau macro-élément est implémenté dans FEDEASLab, un code élément finis développé dans Matlab [2].La comparaison avec des résultats expérimentaux d’une fondation soumise à des chargements cycliques [3] montrel’importance de la composante de décollement notamment dans le calcul du soulèvement du centre de la fondation(Fig. 5).1. IntroductionIn the field of earthquake engineering, Soil–Structure Interaction (SSI) is a phenomenon that has to be taken intoaccount in order to reproduce correctly the non-linear behaviour of a structure and thus to be able to predict its relativedisplacements at the top. Several methods exist: the macro-element approach consists in condensing all non-linearitiesinto a finite domain. It works with generalised variables (forces and displacements) at the centre of the foundation,thus allowing decreasing the necessary degrees of freedom of the numerical model.Several macro-elements can be found in the literature [4–6]. The macro-element presented in [1] reproduces thebehaviour of a shallow circular foundation considering the plasticity of the soil. An extension is introduced in thisNote considering the uplift of the foundation according to the plasticity theory. The macro-element is implementedinto FEDEASLab, a finite element MATLAB toolbox [2]. After the mathematical description of the macro-element,numerical results compared with experimental tests under cyclic loadings [3] are provided to show the performanceof this new numerical tool.2. Associated generalised variablesAs usual is the case for a macro-element, the associated generalised variables (displacement and force vectors) aredimensionless [1]. The are defined hereafter: vertical force V ′, horizontal forces H ′x , H ′y and moments M ′x , M ′y , butalso the corresponding displacements, vertical settlement u′z, horizontal displacements u′x , u′y and rotations θ ′x , θ ′y .Torque moment (M ′z) is not taken into account (Fig. 1).3. Decomposition of three mechanisms: elasticity plasticity and upliftThe new SSI macro-element takes into account three different mechanisms: elasticity, plasticity of the soil anduplift of the foundation. The total displacement must thus be decomposed as a sum of an elastic, plastic and upliftpart:u = uel + upl + uup (1)Uplift is defined as the negative vertical displacement of the centre of the foundation. It is the result of rocking, i.e.the fact that the foundation rotates around θx or θy (part of the foundation looses contact with the soil), see Fig. 2.In order to compute uplift, the simple plasticity of the soil is not sufficient and a new non-linear mechanism must beintroduced in the macro-element. The reason is that the actual plasticity mechanism of the macro-element can takeinto account only positive values of the vertical displacement (u′z > 0). Plasticity and uplift are strongly coupled [4].858 S. Grange et al. / C. R. Mecanique 336 (2008) 856–862Fig. 1. Generalised variables: (a) forces and (b) displacements for a circular foundation.Fig. 1. Variables généralisées : (a) forces et (b) déplacements pour une fondation circulaire.Fig. 2. Kinematics of a foundation for the uplift mechanism.Fig. 2. Cinématique de la fondation pour un mécanisme de décollement.4. Mathematical description of the macro-element4.1. Elastic behaviour and plastic mechanismThe elastic and plastic mechanisms are very briefly described hereafter. For more information the reader is invitedto look in [7,1].The elastic part of the constitutive law is defined as F = Keluel, where the displacement uel and force vectorsF are dimensionless. The elastic stiffness matrix Kel is calculated using the real part of the static impedances. It isconsidered diagonal, i.e. there is not coupling between the different directions of the loading.For the plastic mechanism, a failure criterion, a loading surface and a flow rule are needed. The failure criterionis defined for an overturning mechanism with uplift. It comes from [8]. The adaptation for a 3D loading is presentedin [7]. It relies on the fact that failure criteria are similar for different shapes of foundation when described into theappropriate space of dimensionless variables. Thus, the extension to the case of a circular foundation is straightfor-ward due to the axial symmetry of the problem. For a rectangular foundation however, we assume continuity of thebehaviour between the two principal axes of the foundation. This is of course a more questionable assumption thatneeds improvement in the future. However, due to the nature of the macro-element (being a simplified numerical tool),we consider this hypothesis acceptable. The loading surface follows the same philosophy and one obtains 5D surfaces.A flow rule is finally needed. It is associated in the (H ′x,M ′y,H ′y,M ′x) hyperplane and non-associated in the (H ′x,V ′),(M ′y,V ′), (H ′y,V ′), (M ′x,V ′) planes [7].4.2. Uplift behaviourThe mechanism presented hereafter describes in a phenomenological way uplift using a unique state variable δ.This variable represents the percentage of the surface of the uplifted footing [4], Fig. 2 (for any a, ȧ defines thederivative with respect to time). Indeed, this is the only way to introduce the influence of the change of the geometry,the macro-element being just a point. We assume that uplift is not influenced by horizontal forces.For 2D loading, the behaviour of the foundation during uplift is presented in Fig. 3. This figure presents the relationbetween M ′ −δ (M ′ being M ′x or M ′y ), see also [4]. One can clearly see that the behaviour is not reversible and that theS. Grange et al. / C. R. Mecanique 336 (2008) 856–862 859Fig. 3. Moment-δ relationship for a plastic soil.Fig. 3. Evolution moment-décollement sur sol plastique.unloading slope (ηq2/q1) is increased with respect to the original loading one (q2/q1). The two directions of loading(M ′ > 0 or M ′ < 0) are uncoupled. Furthermore, this behaviour is physically translated into a residual uplift at eachedge of the foundation, see [1] and Fig. 3(b). The symbols ⊕ and  are used to discriminate the directions of loading.For a circular foundation q1 = 6, q2 = 2, for a rectangular foundation q1 = 4, q2 = 1 and the evolution of η isprovided by [4]:η = 4 − 3e−4V ′ (2)4.2.1. Failure criterion and loading surfacesFor the uplift mechanism, failure occurs when the foundation is completely detached from the soil, i.e. whenδ = 1. A simple analytical analysis for a circular foundation lying on elastic soil allows finding the relation M ′ =V ′/2 between the overturning moment and the given vertical force. This equation can be actually considered as afailure criterion. On a plastic soil, the relationship between the overturning moment and the vertical force is morecomplicated [4]. For a loading in two different directions (for M ′ > 0 and M ′ < 0) we obtain [1]:f∞ ≡ M ′2 −(V ′q1(e−AV ′ + q2))2 = 0 (3)where A = 2.5 is a dimensionless parameter of the constitutive law.The uplift behaviour being a non-linear, non-reversible mechanism with the unloading slope increased with respectto the original loading one, the evolutions of the loading surfaces are activated even during unloading. Nevertheless,an initial elastic domain often exists (except if the loading is important leading to high non-linearities in the soil andthus to a total suppression of the elastic domain).In order to activate the loading surfaces in loading but also in unloading, their mathematical expression is chosento be always positive whatever the sign of the loading. For the case of the direction ⊕, the corresponding hardeningvariable β⊕ evolves between δ and δmax (maximal percent of uplift reached during the loading). It is defined as:β⊕ = δmax(1 − η) + ηδ (4)The evolution of the hardening variable β⊕ is also given by Eq. (5) as a function of the rotation due to uplift θ ′upand the rotation θ ′0 related to the initial overturning moment, see also Fig. 2 (β⊕max being the maximal value of β⊕reached during the loading).⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩˙β⊕ = θ̇′upθ ′0(1 − β⊕)2β⊕(2 − β⊕) if β⊕ = β⊕max˙β⊕ = θ̇′upθ ′0η(1 − (β⊕−(1−η)β⊕maxη))2β⊕−(1−η)β⊕max (2 − (β⊕−(1−η)β⊕max )) if β⊕  β⊕max(5)η η860 S. Grange et al. / C. R. Mecanique 336 (2008) 856–862Fig. 4. Loading surfaces, failure criterion and elastic limits with their signs.Fig. 4. Surfaces de charge, critère de rupture et limites élastiques avec leurs signes.The mathematical expression of the loading surface for the direction ⊕ is finally provided by Eq. (6).f ⊕ ≡∣∣∣∣M ′ − V ′q1(e−AV ′ + q2β⊕)∣∣∣∣ = 0 (6)It is also necessary to introduce the surface defining the elastic limit zone. The loading surfaces being alwayspositive, a test on the elastic surface allows knowing in which mechanism (uplift or elastic) is the model. The elasticlimit is a function of the maximum percentage δmax. Its mathematical expression is given in Eq. (7):f ⊕el ≡ M ′ −V ′q1q2β⊕max(1 − η) −V ′q1e−AV ′ = 0 (7)If residual uplift occurs on the ⊕ side of the foundation, the elastic domain entirely disappears. Conditions forchoosing the activated mechanisms are given by the equations:{f ⊕el (M ′,V ′)  0 or f ⊕(M ′,V ′, β⊕) = 0 ⇒ elasticityf ⊕el (M ′,V ′) > 0 and f ⊕(M ′,V ′, β⊕) > 0 ⇒ uplift(8)The same equations can be written for the other direction by replacing ⊕ with . The graphical representation ofall the surfaces is finally given in Fig. 4.As for the plasticity mechanism, the adaptation of the uplift mechanism in 3D is done assuming continuity of thebehaviour between the two principal axes of the foundation (the two directions are coupled considering a projectionof the moment in the principal base [1]). Coupling of the plasticity and the uplift mechanisms is done following theclassical theory of the multi-mechanisms [1]. The choice and the computation of the normal vectors on the differentplasticity mechanisms follow the algorithm presented in [1] and in [9].4.2.2. Flow ruleThe flow rule for the uplift mechanism is found through geometrical considerations, assuming that the centreof rotation of the foundation stays always at the middle of the non-uplifted segment (Fig. 2, [1]). Its mathematicalexpression takes the following form (β is equal to β⊕ or β):⎧⎪⎪⎨⎪⎪⎩∂g∂M ′= fg|fg|∂g∂V ′= − δ2fg|fg|with fg ≡ M ′ − V′q1(e−AV ′ + q2β) = 0 (9)5. Numerical simulations and comparisons with experimental results for a cyclic loadingWithin the European program TRISEE, several experimental tests are performed on a shallow 1 m×1 m rectangularfoundation lying on sand [3]. Two tests are presented hereafter concerning different types of sand (High density sandS. Grange et al. / C. R. Mecanique 336 (2008) 856–862 861Fig. 5. Tests TRISEE, moment–rotations and time history of the vertical settlements for the High density (HD) and the Low density sand (LD) forthe phase III.Fig. 5. Tests TRISEE, moment–rotations et déplacement vertical en fonction du temps pour le sable haute densité (HD) et le sable de faible densité(LD) pour la phase III.(HD) and Low density sand (LD)). Phase III (where the loading was the most severe) of the experimental programis simulated using the macro-element. Fig. 5 shows the comparison of the experimental with the numerical results.One can identify the influence of the uplift mechanism looking at the S-shaped moment–rotation curve. Rotationsdue to the plasticity of the soil are found almost equal to the ones coming from uplift (phase III with HD sand). Inother words, uplift and plasticity of the soil have similar contributions on the moment–rocking angle curve. The wavespresent in the settlement curve are also not reproduced if uplift is not taken into account.6. ConclusionsThe 3D macro-element developed within this work gives satisfactory results for simulating the non-linear be-haviour of shallow rigid foundations lying on an infinite space submitted to a cyclic loading. It takes into account theplasticity of the soil and the uplift of the foundation. Using global variables it presents the advantage of inducing lowcomputational costs (couple of minutes for each simulation).All the results presented in this paper are for loadings in a plane (2D). The 3D behaviour of the element has notbeen validated due to the difficulty to find experimental results with loadings in 2 horizontal directions. This pointshould constitute a future validation for the element.References[1] S. Grange, Modélisation simplifiée 3D de l’interaction sol–structure: application au génie parasismique, PhD Thesis, laboratoire 3S-R, INPGrenoble, France, 2008.[2] F.C. Filippou, M. Constandines, FedeasLab getting started guide and simulations examples, Department of Civil and Environmental Engineer-ing, UC, Berkeley, 2004.[3] TRISEE, 3D Site Effects and Soil-Foundation Interaction in Earthquake and Vibration Risk Evaluation, Part 4, Large-scale geotechnical exper-iments on soil–foundation interaction. European Commission, Directorate General XII for Science, Research and Development, 1998.[4] C. Crémer, A. Pecker, L. Davenne, Cyclic macro-element for soil–structure interaction: material and geometrical non-linearities, InternationalJournal for Numerical and Analytical Methods in Geomechanics 25 (13) (2001) 1257–1284.[5] C. Crémer, A. Pecker, L. Davenne, Modelling of non-linear dynamic behaviour of a shallow strip foundation with macro-element, Journal ofEarthquake Engineering 6 (2) (2002) 175–211.862 S. Grange et al. / C. R. Mecanique 336 (2008) 856–862[6] R. Nova, L. Montrasio, Settlements of shallow foundations on sand, Géotechnique 41 (2) (1991) 243–256.[7] S. Grange, P. Kotronis, J. Mazars, A macro-element for a circular foundation to simulate 3D soil–structure interaction, International Journal forNumerical and Analytical Methods in Geomechanics 32 (10) (2008) 1205–1227.[8] A. Pecker, Analytical formulae for the seismic bearing capacity of shallow strip foundations, in: Seco e Pinto (Ed.), Seismic Behavior of Groundand Geotechnical Structures, Rotterdam, Balkema, 1997, pp. 261–268.[9] F. Collin, Couplages Thermo-hydro-mécaniques dans les sols et les roches tendres partiellement saturés, PhD Thesis, Université de Liège, 2003.

A macro-element for a shallow foundation to simulate Soil–Structure Interaction considering uplift

Hal - Université Grenoble Alpes

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A macro-element for a shallow foundation to simulate Soil-Structure Interaction considering uplift

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Hal - Université Grenoble Alpes