A MACRO-ELEMENT FOR DYNAMIC SOIL-STRUCTURE INTERACTION ANALYSES OF SHALLOW FOUNDATIONS

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1 4 th International Conferene on Earthquake Geotehnial Engineering June 25-28, 2007 Paper No A MACRO-ELEMENT FOR DYNAMIC SOIL-STRUCTURE INTERACTION ANALYSES OF SHALLOW FOUNDATIONS Charisis T. CHATZIGOGOS 1, Alain PECKER 2, Jean SALENÇON 3 ABSTRACT The sope of the paper is to present some aspets of the development of a maro-element for dynami soil-struture interation analyses of shallow foundations. Initially the onept of maroelement is introdued and is illustrated with the aid of a very simple example originating from strutural engineering. Then the link is made with the modeling of the dynami response of shallow foundations and the objetives and struture of suh a tool are desribed with referene to the speifi onfiguration of a irular footing resting on the surfae of a heterogeneous purely ohesive soil. The prinipal features of the maro-element are then presented; the soil-struture interation domain is redued to a point that oinides with the enter of the footing and all the (material and geometri) non-linearities are lumped at this point. A disussion on the most appropriate way to treat these nonlinearities is undertaken based on experiene gained with earlier works. It is suggested that the nonlinearities be inorporated in the model within a unified formalism making use of the theory of multimehanism plastiity. Initial results onerning the definition of the ultimate surfae for suh a plastiity model, orresponding to the seismi bearing apaity of the foundation, are finally presented. Keywords: Soil-struture interation, maro-element, shallow foundations, performane-based design. INTRODUCTION The onept of maro-element In previous works Cremer et al. (2001, 2002) made use of the onept of maro-element as a onvenient tool for a fast but onise and aurate predition of the response of shallow foundations during time history analyses of strutures. The whole issue was atually motivated by the fat that, although an extraordinary progress has taken plae during reent years in terms of omputational apaity and effiieny available to the engineering ommunity, the dynami soil-struture interation (hereafter SSI) analyses still remain exeedingly time onsuming. Among other reasons, this is also due to the fat that, in priniple, non linear SSI models need to be implemented at the sale of the onstituent materials of the foundation and the soil, for whih appropriate onstitutive laws and strength riteria are required. These models offer a detailed desription of the dynami response of the soil, the foundation and the superstruture, but they inrease the size of the addressed problem onsiderably. The onept of maro-element offers an alternative simplifying approah whih redues the size of the problem signifiantly while preserving the essential features of the dynami response of the system. The onept of maro-element an be understood by introduing the following sales desribing the examined soil-foundation-superstruture system: 1 PhD Candidate, Laboratoire de Méanique des Solides UMR CNRS 7649, Éole Polytehnique, Frane, harisis@lms.polytehnique.fr 2 Professor and Managing Diretor of Geodynamique et Struture. 3 Professor Emeritus.

2 1. The loal sale, whih is the sale of the onstituent materials of the soil, the foundation and the superstruture. Elements at this sale are desribed by onventional onstitutive laws for soil, onrete et. 2. The global sale, whih is the sale of the system in its entirety. 3. The meso-sale, whih an be viewed as an intermediate sale between the loal and the global sales. This is, for instane, the sale of some strutural elements, parts of the superstruture, suh as beams, olumns, footings et. Within this ontext, the onept of maro-element an be viewed as a hange in sale within the global model, where one passes from the loal sale of the onstituent materials of a speifi part of the global model, say a partiular strutural element, to the meso-sale of this strutural element viewed in its entirety. In making suh a hange in sale, what was originally desribed by a large number of elements in the loal sale, now onstitutes a single maro-element in the meso-sale, thus reduing the size of the global model signifiantly, whih an then be treated in a muh more inexpensive and effiient way. The maro-element, viewed simply as a part of the global model, must be desribed by a onstitutive law ompatible with the rest of the global model elements. This onstitutive law must be seleted in suh a way so as to ensure that the response of the system, examined at the meso-sale (i.e. with the maro-element) orretly reprodues the features of the atual response of the model (i.e. at the loal sale) that were retained in making the passage from the loal to the meso-sale. This is an essential remark, sine the passage from the loal sale to the meso-sale wipes out all the harateristis of the loal sale (e.g. stresses and displaements at any point in the soil domain near the footing et.) but for those that are deemed essential for the overall behavior of the global model. The features of the system to be retained at the meso-sale model are usually defined by means of a number of generalized stress variables and by the orresponding generalized strain variables aording to the type of the examined problem. An illustrative example To illustrate these ideas, a simple example from strutural engineering will be presented: it onerns a steel I-beam as in Figure 1a subjet to bending moment from the ation of a onentrated load at midspan. The loal sale here refers to the onstituent material of the beam, i.e. the steel, whih is assumed to be desribed by an elasti-perfetly plasti onstitutive relationship. The solution of this problem in the loal sale reveals the reation of a zone of plasti deformations around the entral setion of the beam. As the load inreases, the zone of plasti deformations expands until the whole entral setion is plastified. On the inner and outer fibers of the beam the zone of plasti deformations has a finite width b. The beam annot support any further load inrease; it has reahed the state of plasti ollapse. The passage to the meso-sale is done by onsidering the generalized urvilinear ontinuous medium as in Figure 1b, whih oinides with the lous of the neutral axis of the I-beam. The load inreases up to its ultimate value P u ; the bending moment at the enter of the beam equals the moment of plasti ollapse Mu of the entral beam setion and a plasti hinge is reated at that point; a mehanism of plasti ollapse is reated and the beam an support no further load inrease. It is obvious that the plasti hinge an be viewed as the maro-element, whih atually represents the zone of plasti deformations in the loal sale. In passing to the meso-sale, knowledge about fibers other than the neutral axis is ignored and annot be retrieved. Moreover, all the non-linearity is lumped at one single point, namely the plasti hinge.

3 P u P u M u = Pa u /2 (a) (b) Figure 1. An elasti perfetly plasti I-beam subjet to pure bending: a) Modeling at the loal sale, b) Modeling at the meso-sale with a plasti hinge as a maro-element For this system, the bending moment M is the generalized stress variable (the effet of shear fore is ignored), while its orresponding generalized strain variable is the urvature of the beam χ. Figure 2a represents the moment urvature diagram in the loal sale; the response of the beam is linear up to a value My whih orresponds to the initiation of plasti deformations in the beam, namely on the inner and outer fibers of the beam. The beam passes to a phase of elasto-plasti response until the moment reahes its ultimate value M u. Figure 2b represents the response of the system at the meso-sale. All the non-linear part has been shrunk to one single point whih orresponds to an elasti-perfetly plasti response of the beam. This fat reflets the introdution of the plasti hinge as a maroelement, whih replaes the entire zone of plasti deformations in the beam. Figure 2. a) The urvature-moment diagram in the loal sale and b) the idealized elastiperfetly plasti urvature-moment diagram in the meso-sale The solution of the problem of the elasto-plasti response of the beam has thus been simplified by the introdution of the generalized urvilinear ontinuum and the plasti hinge, as a maro-element, replaing the zone of plasti deformations in the beam. The elasto-plasti onstitutive relations for the generalized urvilinear ontinuum an be written with respet to the generalized stress and strain variables as follows: el pl χ = χ + χ (elasti and plasti omponents of total rate of urvature).

4 el K If M < Mu : χ = χ =, with M K denoting the slope of the elasti branh of the urvaturemoment diagram. el K If M = Mu, M < 0: χ = χ = M pl χ = χ = λ, λ 0, ifm = Mu If M = Mu, M = 0: pl χ = χ = λ, λ 0, ifm = Mu The above set of equations an be integrated along a presribed loading path and all through the generalized urvilinear ontinuum representing the beam and allow for the determination of quantities suh as the vertial defletions or the rotations of the beam. The alulation of suh quantities is evidently muh simpler and quiker in the examined meso-sale model where a maro-element (i.e. the plasti hinge) was introdued, than in any loal sale model that would ontain the geometry of the setions and would be supplied with a set of onstitutive elasto-plasti equations for steel. The maro-element in the literature Although, the onept of maro-element, as desribed above, has been extensively used in strutural engineering (by the onstrution of generalized media for beams, membranes, plates, shells et.), its use in geotehnial engineering is up to now rather restrited. The term maro-element was initially introdued by Nova & Montrasio (1991) in their study of the settlements of shallow foundations on sand; they onsidered the foundation and the soil as a maro-element for whih the loading is expressed by a number of generalized stress variables and the displaement and rotation of the footing by the orresponding generalized strain variables. A set of inremental plastiity-type onstitutive equations were introdued to link the generalized stress and strain variables. The initial model by Nova & Montrasio was developed for quasi-stati monotoni loading. Pedretti (1998) extended the model so as to desribe quasi-stati loading unloading yles more effiiently. In parallel, Paolui (1997) proposed a numerial tool based on the model by Nova & Montrasio permitting the study of the response of simple strutures subjet to dynami (seismi) loading and taking into aount the oupling between the non-linear response of the soil-foundation system and the response of the superstruture. The maro-element was further extended by Cremer et al. (2001, 2002) by the onsideration of all the material and geometri non-linearities at the soil-footing interfae (to be disussed later on), the oupling between them and their oupling with the response of the superstruture. The maro-element of Cremer et al. (2001, 2002) was developed for strip footings. Similar appliations of the onept of maro-element have been developed for foundations of offshore platforms subjet to quasi-stati yles of loading, as the model by Houlsby & Cassidy (2002). Nova & di Priso (2003) presented further appliations of the maro-element in problems of rok impat on the ground, soil-pipeline interation problems et. In parallel, Muir Wood & Kalasin (2004) presented a maro-element model for the dynami response of gravity walls. PRESENTATION OF THE MACRO-ELEMENT FOR SHALLOW FOUNDATIONS The examined onfiguration The maro-element desribed in this study refers to a generi onfiguration presented in Figure 3. The onfiguration onsists of a very simple superstruture of mass m, whih is lumped at a single point. The foundation of the superstruture is omposed of a irular perfetly rigid footing of diameter D and mass m 0, whih is resting on the surfae of the soil. In terms of its strength riterion, the soil is onsidered to be purely ohesive with a ohesion inreasing linearly with depth. The struture is assumed to be exited by the propagation of a seismi wave induing the development of inertial fores in the horizontal diretion and ausing the exitation of the whole soil-foundation-

5 superstruture system. The footing is thus subjet to an inlined and eentri fore originating from the dynami response of the superstruture. Figure 3. Examined onfiguration for the development of the maro-element Figure 4 presents the results from a typial finite element analysis at the loal sale with the footing being subjet to an inlined and eentri fore. The results highlight the loalization of plasti deformation in a zone around the footing orresponding to a bearing apaity failure with uplift of the footing. Outside this zone, the deformations remain elasti. By making the link with the example of the elasto-plasti beam, the envisaged maro-element will replae the footing and the soil in a similar way as the plasti hinge replaes the zone of plasti deformations in the beam. Figure 4. Results of a footing soil system in the loal sale. Creation of a bearing apaity failure (unonfined plasti flow) with uplift of the footing. The generalized variables The proposed maro-element will replae the footing and the foundation soil in the global model; the entire soil domain and the footing will be redued to a single point to whih the maro-element will be attahed. The seleted point is the enter of the footing. Sine the footing has been onsidered perfetly rigid, its motion an be desribed by the motion of its enter. Introdution of the maroelement implies that information at a loal sale within the soil is lost and annot be retrieved. Dimensionless variables The desription of the motion of the footing enter is aomplished by the introdution of a system of generalized variables. To simplify the presentation, we onsider motion only in one horizontal diretion (e.g. only within the xz -plane) and we introdue the resultant vertial fore N, the horizontal fore V x and the moment My ating at the enter of the footing, and the orresponding

6 displaements: the vertial displaement u z, the horizontal displaement u x and the rotation around the y -axis, θ y. The variables of the system are rendered dimensionless as follows: N N uz uz 1 1 F = V x Vx, u u x u = = = x N max D M y My / D θ y D θ y (1) In (1), the fores are normalized with respet to the maximum vertial fore Nmax supported by the foundation. In other words, Nmax represents the bearing apaity of the irular footing for a entered vertial load. The generalized variables of fores and displaements are presented in Figure 5. Figure 5. Generalized fores and displaements in the maro-element Sope Having defined the generalized fores and displaements variables that desribe the maro-element, the main objetive is to derive the relationship that establishes the link between them. The relationship between the generalized fores and the generalized displaements is path dependent and not invertible. However, it an be inverted if it is written in inremental form, as in the following: ( ) d u = f F, u,df (2) () t (-1) t (-1) t () t ( ) d F = f F, u,du (3) 1 () t ( t 1) ( t 1) () t The above equations represent two possibilities of onstruting the solution algorithm for the maroelement: (2) provides the solution for given fore inrements (i.e. at a given time step t, given u( t 1), F ( t 1),dF ( t) find du () t ) and (3) for given displaement inrements. In the impending developments, the seletion of a solution sheme with respet to the displaements (as in (3)) will be followed that will allow the implementation of the maro-element within onventional FEM odes for dynami strutural analysis. Loading modes For the types of ontemplated appliations, the established relationship should allow the reprodution of the foundation response under the following loading modes:

7 Quasi-stati monotoni loading Quasi-stati loading - unloading yles Dynami loading (taking into aount inertial and damping effets) Struture of the maro-element The struture of the envisaged maro-element will follow the sheme presented in Figure 6. The soil domain is divided in two parts: the far field, whih oneptually desribes the area where the response of the system remains linear, and the near field where all material and geometri non-linearities are lumped. The two fields are oneptually separated by a boundary. Aordingly, the response of the far field will be desribed by the linear part of the onstitutive relationship in the maro-element while the response of the near field will orrespond to the non-linear part of the established onstitutive relationship. In the following the near and the far field properties are desribed. Figure 6. Struture of the maro-element Far field The linear response of the far field is desribed by a system of dashpots and elasti springs that ouple all the degrees of freedom along the boundary that separates the near and the far field. For the development of the maro-element, it will be onsidered that the interation horizon is redued to a single point as in Figure 6. This point oinides with the enter of the footing. The spring and dashpot attahed to this point desribe the far-field response of the system and they are supplied with the dynami impedane (stiffness and damping) oeffiients of the foundation. Sine the soil response remains linear (no material damping due to soil hystereti behavior is onsidered in the far field), only radiation damping will be onsidered. Moreover, it will be assumed that the impedanes related to the onsidered three degrees of freedom are unoupled. They an thus be assembled in the following stiffness and radiation damping matries: Kzz 0 0 Czz 0 0 K = 0 Kxx 0, C 0 Cxx 0 = 0 0 Kθθ 0 0 C yy θθ yy (4) Both the stiffness and the radiation dashpot oeffiients in (4) are in general frequeny-dependent. Values of the oeffiients in (4) for a large variety of foundation onfigurations and soil profiles are summarized in Mylonakis et al. (2006). Near field As it was mentioned, the near field response of the system is assoiated with all the non-linearities generated by the soil-struture interation phenomena at the soil-footing interfae. Two types of nonlinearities are onsidered; the material non-linearities that refer to the development of permanent

8 irreversible displaements of the footing due to non linear soil or interfae behavior and the geometri non-linearities mainly referring to the phenomenon of roking and uplift of the footing aompanied by the reation of a detahment area at the soil-footing interfae. Within the maro-element, the aforementioned soures of non-linear behavior will be addressed by means of a unified formalism based on the theory of multi-mehanism plastiity as developed by Koiter (1960) and Mandel (1965). Within the framework of this approah, one first mehanism of plasti behavior will desribe the development of permanent displaements due to the soil non-linear behavior and a seond one will desribe the uplift of the footing. It is noted that when uplift ours it is aompanied by a vertial displaement and a rotation of the footing enter that may eventually be aumulated during a seismi exitation. Consequently, these two distint mehanisms are oupled. The Koiter - Mandel theory makes it possible to treat these two oupled mehanisms simultaneously by writing them in similar forms. Using the adjetive plasti to refer to the interation phenomena at the soil-footing interfae, the inrement of plasti deformation will be expressed in both mehanisms using a plasti multiplier. It is noted that the implementation of suh an approah shall onstitute a signifiant improvement with respet to the maro-element of Cremer et al. (2001, 2002) where the two mehanisms where treated separately within the maro-element solution algorithm. As Cremer (2001) indiates, the adoption of the Koiter Mandel formalism would onstitute the most straightforward development for the maro-element model. In order for the plasti models of uplift and of soil yielding to be implemented, three elements need to be established: a) the surfae of ultimate loading or ultimate surfae, b) the shape of the yield surfae with the hardening laws and ) the flow rules. The yield surfae for both mehanisms defines the loading ombinations below whih the system response is elasti whereas the ultimate surfae expresses the maximum ombinations of loading that an be supported by the footing, i.e. the bearing apaity of the footing. For loading ombinations beyond the ultimate surfae, it is assumed that an unonfined plasti flow ours in the soil. Note that suh a situation, although unaeptable for quasi-stati loading, an be aepted for dynami loading as long as the aumulated displaements and rotations remain below a ertain limit. Figure 7. Ultimate and yield surfaes for the mehanisms of uplift and the development of permanent displaements (soil yielding) with regions where eah mehanism is ative The ultimate and the yield surfaes are presented shematially in Figure 7, in the spae of the loading parameters ( N, M ), together with the region of purely elasti behavior (EL), the region where only the mehanism of uplift (UP) or only the mehanism of permanent soil displaements (soil yielding) (PD) is ative and finally the region where both mehanisms are ative (UP-PD). The shape and the position of the yield surfaes are hanging as the loading evolves aording to the hardening law. The inrements of plasti deformation are given aording to the flow rule and to whether the mehanisms are ativated/deativated at eah load inrement. The flow rule for both mehanisms is in general non-assoiative. On the ontrary, the shape and position of the ultimate surfae remains invariant.

9 Ultimate Surfae Up to the present stage of the development of the maro-element model, the ultimate surfae has been thoroughly investigated whereas the shape and evolution of the yield surfae and the flow rule will be addressed in the future. In partiular, Chatzigogos et al. (2006) studied the maximum ombinations of seismi loads of a irular footing resting on the surfae of a purely ohesive soil with a vertial ohesion gradient. This was made possible using the kinemati approah of the Yield Design theory (Salençon, 2002) in whih, a series of three-dimensional virtual veloity fields were examined to provide optimal upper bounds for the maximum ombinations of seismi loads supported by the footing. An essential element of the study is that, besides the generalized fores ( NV, x, My ) ating on the footing, the effet of the inertial fores ating within the soil volume was taken into aount through the onsideration of a dimensionless parameter, funtion of the maximum horizontal seismi aeleration. The results have been presented in the form of surfaes in the spae of the loads ( NV, x, M y ) for a range of values of the inertial seismi fores in the soil or in the form of interation diagrams between two loading parameters. Conerning the examined virtual veloity fields, eah one of them orresponds to a partiular virtual mehanism of bearing failure. Three large lasses of virtual failure mehanisms have been onsidered. It is noted that the onsidered virtual failure mehanisms ontain the ase of uplift of the footing and the ase of sliding along the soil-footing interfae. Consequently, the ultimate surfae thus established is unique for both plastiity-type mehanisms of uplift and soil yielding ontained in the maro-element. A detailed desription of the solution proedure, the onsidered virtual mehanisms of failure and the results an be found in Chatzigogos et al. (2005, 2006). The most interesting onlusion of the study is that the analytial expression proposed by Peker (1997) and adopted in the Euroode 8 for the desription of the seismi bearing apaity of strip footings on homogeneous purely ohesive soils or in saturated ohesionless soils an also be used with minor modifiations for irular footings in both homogeneous ohesive soils and in soils with a vertial ohesion gradient. This expression is written as follows for irular footings on homogeneous soils: T ( ) ( β ) a N ( ) ( h ) M ( ) ( γ ) T M 1 ef V 1 ff M h h b d N N N N 1 mf N ( N ) ( 1 mf ) N h (5) subjet to the onstraints: 0 ( 1 ) N N < N mf, V 1/6.06 h. In (6), we define N, V, M as in (1) and Fh = ρad h /2, where D is the footing diameter, the soil ohesion, ρ is the soil mass density, and a h a harateristi horizontal seismi aeleration. The rest of the parameters in (5) are onstants and are given in Table 1. Table 1. Values of numerial parameters in equation (5) A = 0,70 d = 1,81 m = 0,21 = 2,00 β 2,57 T B = 1,29 e = 0,21 N = 1,22 = 2,00 γ = 1,85 M C = 2,14 f = 0,44 = 1,00 = 1,00 N M CONCLUSIONS The onept of maro-element stands as a onvenient alternative approah for fast but onise and aurate non-linear SSI analyses for shallow foundations provided it takes the different soures of non-linearity into aount. It appears that the multi-mehanism plastiity formalism would allow the simultaneous treatment of the two basi mehanisms of non-linearity (soil yielding and uplift) enountered in the problem. Initial results from the maro-element development proess, whih

10 onern the definition of the ultimate surfae of the plastiity-type model for the footing are already available from the determination of the seismi bearing apaity of the footing where it was onluded that the Euroode 8 equation for strip footings on homogeneous soils an be extended with minor modifiations to the ase of irular footings on soils with a vertial ohesion gradient. ACKNOWLEDGEMENTS The first author wishes to thank the Éole Polytehnique (Laboratoire de Méanique des Solides) and the Publi Benefit Foundation Alexandros S. Onassis for the finanial support during the exeution of this study. REFERENCES Chatzigogos, C., T. "Comportement sismique des fondations superfiielles : Vers la prise en ompte d un ritère de performane dans la oneption", Thèse de Dotorat, Éole Polytehnique, Palaiseau, 2007 (in preparation). Chatzigogos, C., T., Peker, A., Salençon, J. "Seismi bearing apaity of irular foundations", Proeedings, 1st Greee-Japan Workshop: Seismi design, observation and retrofit of foundations, Gazetas, Goto and Takashi (eds.), Athens, , Chatzigogos, C., T., Peker, A., Salençon, J. "Seismi bearing apaity of a irular footing on a heterogeneous ohesive soil", Soils and Foundations, submitted for publiation, Cremer, C. "Modélisation du omportement non linéaire des fondations superfiielles sous séisme", Thèse de Dotorat, Éole Normale Supérieure de Cahan, Cremer, C., Peker A., Davenne L. "Cyli maro - element of soil - struture interation: material and geometrial non-linearities", International Journal of Numerial and Analytial Methods in Geomehanis, 25, , Cremer, C., Peker A., Davenne L. "Modelling of non linear dynami behaviour of a shallow foundation with maro - element", Journal of Earthquake Engineering, 6, Issue 2, , Houlsby, G., T., Cassidy, M., J. "A plastiity model for the behavior of footings on sand under ombined loading", Géotehnique, 52, Issue 2, , Koiter, W., T. "General theorems for Elasti-Plasti solids", Progress in Solid Mehanis (Sneddon, I., N., Hill, R., eds.), Vol. I, North Holland, Amsterdam, , Mandel, J. "Généralisation de la théorie de plastiité de W. T. Koiter", International Journal of Solids and Strutures, 1, , Muir Wood, D., Kalasin, T. "Maroelement for study of dynami response of gravity retaining walls" in "Cyli behaviour of soils and liquefation phenomena", Triantafyllidis, T., (ed.), Balkema Publishers, Leiden, , Mylonakis, G., Nikolaou, S., Gazetas, G. "Footings under seismi loading: Analysis and design issues with emphasis on bridge foundations", Soil Dynamis and Earthquake Engineering, 26, Issue 9, , Nova, R, Montrasio, L. "Settlements of shallow foundations on sand", Géotehnique, 41, Issue 2, , Nova, R., di Priso, C., "The maro-element onept and its appliation in geotehnial engineering", Fondations Superfiielles, Magnan et Droniu (ed.), Presses de l ENPC/LCPC, Paris, , Paolui, R. "Simplified evaluation of earthquake indued permanent displaements of shallow foundations", Journal of Earthquake Engineering, 1, Issue 3, , Peker, A. "Seismi bearing apaity of shallow foundations", State-of-the-art: 11 th World Conferene on Earthquake Engineering, Aapulo, Mexio, June 23-28, Peker, A. "Analytial formulae for the seismi bearing apaity of shallow strip foundations", "Seismi behavior of ground and geotehnial strutures", Seo & Pinto (eds.), Balkema, , Pedretti, S. "Nonlinear seismi soil-foundation interation: analysis and modeling method", PhD thesis, Dpt Ing Strutturale, Politenio di Milano, Salençon, J. "de l Élastoplastiité au Calul à la Rupture", Presses de l Éole Polytehnique, Palaiseau, 2002.