Seismic response of rigid shallow footings

Seismic response of rigid shallow footings Claudio di Prisco — Federico Pisanò Department of Structural Engineering Politecnico di Milano Piazza Leonardo da Vinci, 32 20133 Milano, Italy {Cdiprisc, pisano}@stru.polimi.it This chapter concerns the seismic design of shallow footings and, more specifically, a critical analysis of the soil-structure interaction problem. To highlight the non-linear mechanical response of shallow footings, several experimental test results are reviewed and the so-called “macro-element theory” is introduced as a promising interpretative and simulation tool. Various macro-element approaches are discussed and their employment for seismic design is exemplified by means of some engineering applications. ABSTRACT.

RÉSUMÉ. Cet article concerne la conception parasismique des fondations superficielles et, en particulier, une analyse critique du problème d’interaction sol-structure. Pour mettre en évidence la réponse mécanique non linéaire des fondations superficielles, plusieurs résultats expérimentaux ont été pris en compte et la théorie du « macro-élément » a été introduite comme instrument pour interpréter et simuler les résultats. Différentes approches « macroélément » sont discutées et leur utilisation pour la conception parasismique est mise en exemple à travers des applications. KEYWORDS: shallow foundations, seismic design, cyclic loading, macro-element, experimental tests, numerical analyses. MOTS-CLÉS : fondations superficielles, conception parasismique, chargement cyclique, macroélément, essais expérimentaux, analyses numériques.

DOI:10.3166/EJECE.15SI.185-221 © 2011 Lavoisier, Paris

EJECE. Special Issue 2011, pages 185 to 221

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EJECE. Special Issue 2011

1. Introduction In the context of seismic design, according to capacity principles, it is generally recognized that any damage to foundations is to be avoided. This implies that the non-linear capacity of the system is exclusively exploited at the superstructure level, typically allowing energy dissipation at ad hoc selected points through either the formation of plastic hinges or the insertion of isolation/dissipation devices. This choice is partially motivated by budget considerations, but it is also justified by the lack of well-established methods to analyse the post-yielding behaviour of soilfoundation systems under strong seismic loading. Conversely, when the seismic performance of already existing buildings is to be assessed, this approach cannot obviously be adopted and performance-based approaches are needed. Indeed, the interest towards performance-based approaches for seismic design and seismic adequacy assessment is rapidly growing, spreading an increasing awareness about the effects of the interaction between foundation and superstructure. However, while it is widely accepted that the role of foundation on the overall seismic capacity of structures (see e.g. ATC-40 1996; Martin and Lam, 2000; Pecker, 2006) cannot be neglected, on the other side a lack of reliable methods for the seismic analysis of foundations is still apparent. For this purpose, non-linear dynamic finite element (FE) simulations of large numerical models, including the superstructure, the foundation and the surrounding soil, are likely not to be particularly suitable, because of their excessive computational costs when sophisticated soil constitutive laws are adopted. To overcome this shortcoming preserving a satisfactory description of the dynamic soilstructure interaction, the macro-element concept can be fruitfully employed (Nova et al., 1991; Paolucci, 1997; Cremer et al., 2001 and 2002; Le Pape et al., 2001). This basically consists in modelling the soil-foundation system as a unique non-linear macro-element with a limited number of degrees of freedom (DOF). However, although the macro-element approach seems to be very promising, it has not been supported so far by adequate experimental evidences, at least for seismic applications. Indeed, few experimental results are available on the non-linear soilfoundation dynamic interaction (Negro et al., 2000; Faccioli et al., 2001; PWRI, 2005; Zeng et al., 1998; Gajan et al., 2005; Maugeri et al., 2000). This chapter consists of five parts: i) a general introduction to the problem is followed by a discussion on the main approaches available in literature; ii) the mechanical response of shallow foundations under monotonic/cyclic loading as it results from experimental tests is then outlined; iii) the macro-element theory in its different versions (elasto-perfectly plastic, elasto-strain-hardening plastic, bounding surface plastic and hypo-plastic) is described; iv) the hypotheses both of the Ultimate Limit State (ULS) approach and the Direct Displacement-Based Design (DDBD) in the light of the macro-element concept are discussed; v) three application examples on the use of the macro-element for solving practical problems are reported.

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2. General theoretical aspects The soil-structure interaction problem is usually approached by subdividing the whole spatial domain into two sub-structures: i) the infinite half-space where seismic waves travel and ii) the superstructure, whose dynamic response is analysed to prevent damages/collapses. From a dynamic point of view, the responses of the two substructures are apparently coupled, nevertheless uncoupling strategies are usually followed for the sake of simplicity. In the case of either towers or chimneys, the superstructure can be schematised as an isostatic structure, so that the actions transmitted to the soil are not affected by the cyclic accumulation of settlements (as long as second-order effects are neglected). In contrast, non-negligible second-order effects arise in redundant tall structures and the aforementioned coupling becomes therefore more pronounced. To highlight the mechanisms governing the dynamic response of the soilfooting-superstructure system, the seismic event can be artificially partitioned into distinct phases, not necessarily subsequent.

Figure 1. Sketch of the seismic soil-structure interaction

As a first phase, the input motion (arrow 1 in Figure 1) coming from the bedrock propagates throughout the soil stratum. Its propagation is mainly influenced by the stratigraphic profile and the local topography. Factors of site amplifications are usually employed in national and European standards to account these effects for. Sometimes, when seismic waves cross saturated loose sand strata, the phenomenon of soil liquefaction can locally take place, with dramatic consequences for buildings and lifelines. In some other cases, seismic actions may cause catastrophic landslides, with even more disastrous effects on civil structures. In both cases, the triggering

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mechanisms of such events are assumed to be totally independent both of the structure and its foundation; on the contrary, the inception both of soil liquefaction and slope failures is essentially driven by the local initial state of stress, that is in turn dominated by the presence of the superstructure. In this case, the soil-structure dynamic interaction markedly transforms the input motion (arrow 2 in Figure 1). A possible uncoupling strategy consists in considering uniquely the soil domain and applying on its boundaries the stresses transmitted by the superstructure due to permanent/static loads. In some critical situations, most energy is dissipated within the soil domain and the majority of the damage eventually suffered by the superstructure is basically a consequence of soil failure. More frequently, the incoming dynamic signal can be assumed to get the foundation without any marked change neither in amplitude nor in frequency, so that the signal transformation takes place at the superstructure level: there, owing to the mass and the stiffness of the superstructure, the input motion is indeed modified in phase, amplitude and frequency; then, the superstructure transmits to the footing, and this to the soil, a transformed dynamic signal (arrow 4 in Figure 1): this process is commonly referred to as inertial interaction. If the input signal has a “sufficient” duration and the natural period of the superstructure is not too large, the input (signal 2) and the output (signal 4) experience a constructive interference, this making the analysis and the interpretation of the interaction phenomenon further complex. The most simple and standard way to uncouple and solve the soil-structure interaction problem implies the assumptions of rigid foundation and rigidly constrained superstructure. Accordingly, the input motion (arrow 2 in Figure 1) is recognised to coincide with the free field motion (arrow 1). As is well known, this is also the approach commonly recommended by national standards in the context of pseudo-static seismic analyses. According to this strategy, the evaluation of actions on the soil (arrow 4) is therefore rather easy, since these do not depend on the type of foundation and underlying soil. According to the ULS method, all the actions are known and do not have to activate any failure mechanism within the foundations soil. In essence, the soil-structure interaction is interpreted as rigid-plastic and a “safe” distance from the ultimate plastic limit is recommended. Analogously, as far as the input motion is concerned (arrow 2 in Figure 1), the same pseudo-static ULS approach is usually suggested. In this case, the limit conditions are evaluated by introducing pseudo-static horizontal equivalent forces Fi (Figure 2), corresponding with the inertial forces in the soil during the seismic excitation. Such an approach is based on the hypothesis of a synchronous motion for the soil underneath the footing, hypothesis that is acceptable only in the case of small footing widths B) and large values of the soil stiffness. In fact, the variable governing the phenomenon is the  B ratio, where l = VS f , f is the frequency


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of the input motion (arrow 1) and VS is the shear wave velocity depending on the shear stiffness and the density of the soil. If input and output signals are assumed to be simultaneous (arrow 4 in Figure 1), the pseudo-static actions coming from the superstructure and those within the soil have thus to be superimposed. However, this hypothesis is usually considered to be too prudent, since the case of in-phase input and output is quite unlikely to occur. To overcome the evident drawbacks of standard pseudo-static-ULS methods, the macro-element approach is hereafter employed. This theoretical framework is very useful to clarify many aspects of the problem, since it can conjugate the description of limit conditions and pre-failure response. The macro-element theory has, however, two remarkable shortcomings: i) the shallow footing is still assumed to be rigid and ii) inertial effects in the underlying soil mass are neglected.

Figure 2. Schematic representation of pseudo-static seismic forces

Under seismic conditions, the use of the macro-element theory implies the subdivision of the whole spatial domain into three sub-domains: i) the far field, ii) the near field and (iii) the superstructure. The far field is the part of soil domain unaffected by the presence of the superstructure, i.e. where the displacement field can be assumed to be known. Conversely, in the near field irreversible mechanisms due to the soil-structure interaction become dominant. From this perspective, the identification of the zone where significant plastic/irreversible strains develop is essential: for instance, in the case of shallow strip footings, the size of the domain is governed both by the foundation width and its embedment. Sometimes the definition of the boundaries of this domain is somewhat ambiguous and the geometry of this latter evolves with time. On this point, the case of shallow footings on loose sand strata will be briefly discussed in the following. As for plastic hinges in steel frames, the mechanical response of the entire substructure is thus interpreted as the sum of two contributions: the former associated with the elastic response of the entire infinite half-space, the latter with the inelasticities developing in the plastic hinge.

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3. Experimental evidences All the results here below are based on the assumption of perfectly rigid footing, this allowing a significant reduction in the DOF number. For instance, under plane strain conditions, the mechanical interaction can be described in terms of three generalised stresses (the vertical load V , the horizontal load H and the overturning moment M ), components of vector Q , and three generalised strains (the vertical displacement v , the horizontal displacement u and the rotation  ), components of vector q (Figure 3). Under seismic conditions, particularly crucial is to consider the response of the footing both to eccentric and inclined loads, associated with the inertial horizontal forces in the superstructure and, for this reason, some hints inferred from quasi-static experimental results are here below briefly summarised: – under monotonic loading, the response of the footing is non-linear from the very beginning; – the coupling between the different generalised stress/strain variables is evident from very low generalised stress levels and gets dominant at failure. For example, when a monotonic horizontal load is applied, vertical displacements develop even for constant vertical load and nil overturning moment;

V, v M, θ H, u

z

B

Figure 3. Generalised stresses and strains for a shallow strip footing

– bearing capacity is severely affected both by the inclination and the eccentricity of the loads imposed; the so-call interaction domain describes this dependence. The interaction domain is a function of the nature of the foundation soil (the relative density severely influences its size and shape), as well as of the roughness, shape and embedment of the footing itself. As is suggested by Figure 4, if the penetration mechanism was uncoupled with respect to either the sliding or the toppling mechanism, the interaction domains in the two planes H  V and M  V , respectively, would be uncoupled, too. Indeed, sliding would always concern uniquely the interface zone and toppling would consist solely in the result of the detachment between the footing and the underlying soil. Conversely, the compliance


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and the limited strength of the soil necessarily introduce the aforementioned coupling, so that interaction domains cannot be described by the four straight lines in Figure 4; – the geometry of failure mechanisms severely depends on the combination of generalised stresses: each point of the interaction domain corresponds with a unique failure mechanism; – the experimental results for strip footings can be easily interpolated in a threedimensional space by employing expressions, quite common in literature (Butterfield et al., 1979; Georgiadis et al., 1988; Nova et al., 1991; Butterfield et al., 1994; Montrasio et al., 1997), such as that by Nova et al. (1991): 2

2

 M  H V  2 F        V 1   B      VM 

2

0

[1]

where  , m ,  and VM are parameters describing shape and size of the failure locus. A geometrical representation of the expression [1] is reported in Figure 5 in a suitable dimensionless fashion. M

H penetration mechanism

penetration mechanism rigid toppling mechanism

rigid sliding mechanism

V

a)

V

b)

Figure 4. Uncoupled interaction domains for a shallow rigid footing in the a) H-V and b) M-V planes

h

m ξ

Figure 5. Dimensionless interaction domain (M = M/BVM, h = H/µVM,  = V/VM)

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– when overturning moments are applied (or better, under generalised strain controlled conditions, tilting angles are imposed), local measures testify a process of progressive concentration of the vertical stresses under the footing. As is suggested by Figure 6a, the uplift of the foundation dominates the response of the system and the rB zone progressively detaches from the underlying soil. In Figure 6b the vertical stress distributions are reported as a function of the tilting angle  imposed during a cyclic test performed on a dense sand stratum (PWRI, 2005); – as is well known, standard general shear failures develop for stiff soils, while, as the soil stiffness reduces (loose sand strata), a punching mechanism is more likely to take place and the corresponding bearing capacity becomes hard to be evaluated. Indeed, owing to second-order effects (the foundation sinking requires large displacements to be accounted for), the corresponding generalised stress-strain curve is characterised by a limitless increase in stress (i.e. no peaks and/or plateaus). A plateau can be envisaged solely if the foundation level is artificially maintained coincident with the ground level, as it can be done in the laboratory; – the shape of both failure mechanisms and the interaction domain is abruptly affected by soil inhomogeneity. Moreover, the symmetry of the interaction domain with respect to the V axis is lost in the case of either inclined strata or anisotropic soil;

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s rB v

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200

(1-r)B 0 100

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200

a)

0 100 200

b)

Figure 6. Laboratory small scale experimental results after Shirato et al. (2008): a) uplift mechanisms, b) vertical stress distribution during a cyclic test at constant vertical load and variable tilting angle

– under cyclic loading, the system dissipates energy and, in general, accumulates irreversible generalised strains; – when the soil is sufficiently rigid (dense sands), during rotation-controlled tests with constant vertical load, the cycles in the M   plane assume for large  the typical backbone shape in Figure 7a and the uplift phenomenon is apparent. In the case of loose sands, the uplift is less pronounced and the reduction in the rotational stiffness during unloading seems to disappear (Figure 7b);

Seismic response of rigid shallow footings 2

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Figure 7. Experimental data concerning a square shallow foundation cyclically tilted: a) dense and b) loose sand subgrade (after PWRI, 2005)

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Figure 8. Experimental data concerning a rectangular shallow foundation under load-controlled cyclic tests (after Pedretti, 1998): a) rocking moment vs rotation angle (dense sand stratum), b) rocking moment vs rotation angle (loose sand stratum), c) loading history vs time, d) vertical settlements vs time both for dense (black line) and loose sand stratum (grey line) – if at each cycle the ultimate load is not attained (Figure 8), in case of both loose and dense sand strata, the mechanical response at each unloading phase is characterised by a monotonic decrease in the rotational stiffness (Figure 8a-b); – if the experimental response of rigid footings to symmetric cycles is interpreted in terms of the well-known concepts of secant stiffness K and damping factor   D 4W (Figure 9d), a clear decay in K and an increase in  are evident at increasing values of either rocking or horizontal displacements. An example from the data by Pedretti (1998) and PWRI (2005) is illustrated in Figure 9a-b with reference to dense sands. In the same plots numerical simulations obtained by

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employing the macro-element concept are also reported, but these numerical data will be commented in the following. It is worth noting that even for relatively small foundation rocking, e.g. 1 mrad, the reduction in foundation stiffness, depending on soil relative density, ranges from about 40% to 60%. The equivalent damping ratio h in the rocking mode, computed as the ratio between the dissipated energy D (area of the hysteresis loop) and the stored elastic energy W , is also plotted in the same figures. For rocking values up to 1 mrad, the  ranges from 5% to 10%, while it significantly increases for larger rocking angles, up to 20% for dense sands and 30% for medium dense sands. When the rocking angle is sufficiently large, the uplift dominates the response of the system and  stops evolving; – the coupling between tilting/horizontal and vertical displacements is particularly severe for loose sands, but yet evident for dense sands as well (Figure 8d); 1

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Figure 9. a) Normalized rocking, b) translational stiffness, c) damping factor for Dr = 90% (after Paolucci et al., 2007) at increasing values of rocking angle and horizontal displacement, respectively and d) definition of rocking/translational stiffness and damping factor


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– for a very large number of cycles, the accumulation rate of irreversible settlements progressively decreases: usually a sort of stabilisation takes place, at least in the absence of any damage phenomenon. When the generalised stress path is symmetric, only vertical displacements accumulate with the number of cycles, but when either the initial generalised stress state or the generalised stress path is asymmetric, the ratcheting phenomenon involves not only vertical settlements but also either horizontal displacements or rocking angles. To exemplify this aspect, in Figures 10-11 some experimental results after di Prisco et al. (2003a), concerning small scale tests on a footing placed on a loose sand stratum, are illustrated. In particular, in Figure 10a a schematic view of the generalised stress path (nil tilting moment) is reported, whereas in Figure 10b the loops in the H  u plane are plotted. The accumulation in generalised strains is observed to be essentially a function of i) the cyclic stress path, ii) the amplitude of the cycles and iii) the image point in the generalised stress space which the loading path is imposed around and in particular of its position with respect to the failure locus. In Figure 11 all the results concern the same amplitude, while only the position of cycles in the H -V plane is varied: when the image point approaches the limit locus the accumulation rate markedly increases (this is particularly true for curves 1 and 4, characterised by a large obliquity H V of the fixed image point).

H [kg]

H

V

10 9 8 7 6 5 4 3 2 1 8.64

8.66

8.68

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u [mm]

a)

8.72

8.74

b)

Figure 10. Experimental test results (after di Prisco et al., 2002) obtained by performing V-constant tests with cyclically varying H: a) generalized stress path, b) horizontal load vs horizontal displacement

a)

b)

Figure 11. Experimental test results (after di Prisco et al., 2003b) obtained by performing V-constant tests with cyclically varying H: a) vertical and b) horizontal displacements vs number of cycles n

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4. Numerical analyses In the case of large interaction problems, like those here considered, numerical analyses can be performed to better understand the mechanical processes governing the problem. These latter can be employed as heuristic tools but even to calibrate the constitutive parameters governing the macro-element constitutive relationships discussed in the following section. For the sake of completeness, four different approaches are mentioned here below: i) a very simple contact spring model will be employed to describe the uplift phenomenon, ii) limit analysis to determine the interaction domain, iii) finite and iv) discrete element numerical approaches to describe the dependence of the interaction domain shape on both the relative depth of the foundation plane and topographic conditions. Contact springs models are usually employed to show the interaction problem to be mainly dominated by the unilaterality of the local contact. Indeed, if bilateral elastic contact springs are employed, and a V -constant q -controlled test is performed, a linear M   curve is obtained (straight line in Figure 12a). In contrast, if elastic unilateral springs are used, the response is still linear until M  M 0 (that is until all the springs are under compression), while he subsequent linearity is lost and an ultimate overturning moment is asymptotically approached. A different ultimate load M 2 is achieved if the springs are also assumed to be elasto-perfectly plastic (Figure 12a). By numerically performing different tests at different V values, even the M  V interaction domain can be obtained: this satisfactorily matches the experimental envelope as well as that provided by Equation [1]. It is worth noting again that in case of a rigid substratum the boundary of the interaction domain of Figure 12b simply reduces to a straight line passing through the origin. Since the late 50’s, the definition of both the H  V and M -V interaction domains has been a challenging issue for engineers and geomechanicians, so that a large number of solutions according to classical theories (e.g. Limit Equilibrium Method, Limit Analysis, Characteristic Line Method) has been proposed. Many of these earliest results are summarized in Vesic (1975), while it is worth citing the later contributions by Salençon et al. (1995a-b), who proposed improved Limit Analysis solutions for strip footings under inclined/eccentric loads on purely cohesive soils; further advances in the field of Limit Analysis approaches have been also recently developed by Randolph et al. (2003). On the other side, in the last decade modern numerical techniques have been exploited to the same purposes. In particular, with special reference to Offshore Engineering applications, many authors (Bransby et al., 1998; Bransby et al., 1999; Gouvernec et al., 2003; Bransby et al., 2007; Gouvernec, 2008) have been performing undrained FE numerical analyses to determine the interaction domain for


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shallow footings on clay strata under combined loading, by employing suitable both soil constitutive models and soil-footing contact laws.

M [kNm/m]

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Nova-Montrasio Model Elasto-Plastic Model

0

100

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Figure 12. Simplified interpretation of the footing response to an overturning moment: a) linear elastic, non-linear elastic and elasto-plastic response, b) interaction domain in the M−V plane obtained by using a generalised spring interaction model and comparison with the locus by Nova et al. (1991)

In the last decade, several authors have also been investigating how the V  H  M failure locus can be affected by different mechanical/geometrical factors, such as the spatial inhomogeneity of soil properties (Gouvernec et al., 2003) or the embedment of the foundation (Bransby et al., 1999; Bransby et al., 2007; Gouvernec, 2008). For instance, an interesting – and less-studied – issue concerns the shape of the M  H cross section, whose relevance both for offshore and seismic applications is self-evident. On this point, Gouvernec (2008) confirmed the intrinsically asymmetric shape of the M  H envelope, highlighting its marked dependence both on the embedment ratio B / D and the normalised vertical load   V VM (Figure 13).

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Figure 13. Normalised failure M-H envelopes for   0, 0.5, 0.75,1 : a) D B  0 (surface foundations); b) D B  0.25 ; c) D B  0.5 ; d) D B  1 (after Gouvernec (2008)) Alternatively, the Discrete Element Method (Cundall et al., 1979) has been also recognized as a suitable tool for studying soil-structure interaction problems (Calvetti et al., 2004). For instance, Gabrieli et al. (2009) numerically studied the influence of the local topography (sloping ground) on the vertical bearing capacity, focussing on the influence of the i) footing-slope distance, ii) kinematical constraints (free vs fixed horizontal/rotational displacements) and iii) base roughness (Figure 14). a)

b)

Figure 14. Force-displacement curves for a foundation close to a sloping ground. Effect of a) roughness, position and b) constraint configuration of the footing (after Gabrieli et al., 2009)


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5. The macro-element approach Any incremental time-independent constitutive relationship between the generalised stress and strain rate variables can be written as:





  D Q, Q  , ψ  q p  q Q

[2]

where D stands for an incremental constitutive matrix depending on the current  and on a set of hidden variables ψ , taking track of the stress Q , the stress rate Q previous history through the generalised plastic strain q p . The macro-element theory was initially conceived for rigid strip foundations on homogeneous dry sand strata under monotonic inclined/eccentric loads; more recently it has been extended to describe the response of shallow footings under cyclic loading (di Prisco et al., 1998; Cremer et al., 2001; di Prisco et al., 2003a; di Prisco et al., 2003b). The theory has been also generalised to deal with rectangular footings (Grange et al., 2008; Grange et al., 2009) and in this case both vectors Q and q become six-dimensional. If an isotropic elastic soil is assumed, the stiffness matrix in Equation [2] diagonalises and its diagonal terms can be evaluated by employing well-known either empirical or analytical solutions (Sieffert et al., 1991). In contrast, when irreversible strains progressively develop within the soil up to a local failure mechanism, the process concentrates and a sort of plastic hinge – as for metal beams (Figure 15) – develops. Thus the stiffness matrix in Equation [2] is intended to describe the response of the system during the entire process, from the very beginning up to the final collapse. To better understand the macro-element concept, hereafter analogies and differences of this approach with respect the classical limit design theory for steel beams are briefly discussed. As is well known, if the steel beam schematically shown in Figure 15a, is considered and a rigid-plastic constitutive relationship for the plastic hinge is assumed, the structural response of Figure 15d, is obtained, where PU is the beam ultimate load. In the macro-element theory, this corresponds with the bearing capacity of the shallow footing. Analogously, the inclination of the straight line passing through the origin in Figure 15d, related to the elastic response of the beam, will correspond with the elastic response of the soil stratum. If the mechanical local response of the cross section, where the plastic hinge develops, was more precisely described and the progressive plasticization of the cross section accounted for, the rigid-plastic scheme suggested in Figure 15c would have to be abandoned and a sort of generalised strain hardening relationship to be chosen. Analogously, in the case of shallow footings, irreversible strains locally develop much earlier than the failure mechanism activation and more sophisticated relationships have to be introduced. The previously defined analogy between the beam and the considered interaction problem becomes a bit more obscure when the local problem is tackled. Indeed, as is well-known, in the case of the beam, the generalised stresses are, for instance, the

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bending moment, the axial force, the torsional moment; by contrast, in the case of shallow footings, the generalised stresses, as are indicated in Equation [2], coincide with the external loads themselves. This lack of symmetry between the two mechanical systems is essentially due to the types of processes here compared: the beam response is governed by its flexural behaviour (both constraints and length play a crucial role in affecting the M value), in contrast, for the footing, the local process of failure dominates the system response. Local Scale

Pu

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Mu=Pu a/2

b

Zone of platic deformations

Platic hinge

a)

M

b)

Pu

Mp

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θp

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δ

d)

Figure 15. Structural analogy: a) steel beam under point loading, b) development of the plastic hinge, c) plastic hinge constitutive relationship (rigid-plastic assumption) and d) structural response 5.1. Elasto-plastic strain hardening macro-element models Once the generalised stresses and strains are defined, the simplest way to describe the previous experimental evidences consists in setting up an homogenized constitutive relationship of elasto-perfectly plastic type. This requires the definition of a suitable failure locus F  Q, α F  and a plastic potential G  Q, α G  , where vectors α F and α G are two sets of shape parameters. Within the failure/yield locus the mechanical behaviour is assumed to be elastic and uncoupled, while coupling exclusively characterises the ultimate conditions. Either DEM or FEM codes can be employed to numerically evaluate the direction of the irreversible generalised strain vector at failure. For instance, in case of shallow footings, if an associated flow rule was accepted when sliding mechanisms are activated, that is when in the plane M  0 the straight line H / V   is approached, a meaningless negative (i.e. upwards directed) unlimited vertical displacement would occur. An example of elasto-perfectly plastic macro-element for rigid footings under seismic actions is


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given in Paolucci (1997); a practical application of Paolucci’s model will be illustrated in a subsequent session of this chapter. A more sophisticated modelling approach consists in conceiving strainhardening elasto-plastic macro-element models. Accordingly, both the loading function f and the plastic potential g can be conveniently defined by assuming their shape to coincide with that of F and G : the failure locus is, therefore, the special yield locus for which the variable Vc coincides with VM (Nova et al., 1991). The development of plastic strains causes an increment in Vc according to a suitable hardening rule. For instance, Nova et al. (1991) proposed for a strip shallow footing:  V dVc   1  c  VM

 R0 dv p   du p   B d p  V  M





[3]

where B stands for the footing width, R0 is a constitutive parameter governing the stiffness of the system under pure vertical loads, whereas  and g are constitutive parameters influencing the system response when either inclined or eccentric loads are applied. In a dimensionless representation (Figure 5) and defining  c  Vc VM , such a hardening can be illustrated as in Figure 16. This simple approach allows to satisfactorily reproduce the mechanical response of shallow footings under any monotonic loading path, this statement being supported by the comparisons with both FE analyses and experimental test results. Equation [3] implies an increase in size of the yield locus and, as a consequence, always a positive hardening. In contrast, some authors have also proposed more complex hardening rules to account for even a reduction in the yield function size and, consequently, to simulate a softening regime (Gottardi et al., 1999; Martin, 1994; Cassidy et al., 2002; Nova et al., 2008). Furthermore, very recently Hodder et al. (2010) with regard to normally consolidated clays, whereas di Prisco et al. (2010) for loose sands, modified the aforementioned hardening rules to reproduce the mechanical interaction between rigid bodies and largely deformable soils, that is to model large displacements as well. As undrained tests can be employed to infer the yield locus shape of soils, in a similar fashion the so-called swipe-tests have been conceived for shallow foundations (Gottardi et al., 1999; Butterfield et al., 2003). These consist in applying to rigid shallow footings, horizontal/rotational displacements by inhibiting the vertical displacement at rotational/horizontal constant loads. An extension of this theory to visco-plasticity according to Perzyna’s approach (Perzyna, 1963) was also recently suggested by di Prisco et al. (2006) to numerically reproduce the impact of rigid boulders on granular materials. In this case, the dynamic interaction problem is approached by accounting for both the boulder mass and the inertial actions arising within the deformable sand stratum. This is obtained by means of: i) a suitable definition for the viscous nucleus, ii) the definition,

202


according to a radial mapping, of a sort of overstress. According to this theory, the generalised stress state can get external to both the yield function and the interaction domain. This implies that, during the evolution of time, the load applied by the boulder to the soil can be much larger than that can be statically reached. Thus, the interest of this approach, even for future applications under seismic conditions, lies in the idea of employing delayed plasticity to handle with local inertial effects. 0.3 ρc

0.25

h

0.2 0.15 0.1 0.05 0 0

0.2

0.4

ξ

0.6

0.8

1

Figure 16. Representation of the isotropic hardening in the dimensionless plane h−ξ 5.2. An application example: the case of the medieval Ghirlandina bell tower (Modena, Italy) As is well known, tall historical buildings very often suffers stability problems against tilting, these being obviously amplified in the presence of significant horizontal loads (seismic, aeolian, etc) and compliant foundation soils. From this viewpoint, an interesting example is represented by the leaning Ghirlandina tower in Modena (northern Italy, Figure 17a), recently under restoration works.

a)

b)

Figure 17. a) The cathedral and the Ghirlandina bell tower in Modena (Italy), b) scheme of the pseudo-static seismic analysis


203

This example aims at stressing the relevance of soil-structure interaction in seismic analyses, even when standard pseudo-static approaches are adopted. From the rotational equilibrium of the tower under the action of the self-weight FV and of the horizontal seismic force FH (Figure 17b), it results that: FH h  FV h sin   k   

[4]

where  is the rotation angle and k the rotational stiffness of the soil-foundation system. In Equation [4] the dynamic nature of the interaction problem is neglected, however two essential aspects are accounted for: i) the influence of the vertical weight FV as a second-order effect due to large displacements; ii) the non-linear dependence of k on the unknown rotation  . Moreover, in the light of modern seismic standards, the k   relationship also affects the expected seismic action FH : indeed, FH can be derived from the so-called “response spectrum” of the tower, as long as its “ k -dependent” natural period is known. Apparently, the definition of a realistic k   relationship is crucial. The use of a macro-element model seems to be very appropriate to this purpose, since the required stiffness k can be extracted from the full elasto-plastic stiffness matrix (see previous sections). Hereafter, the key points for the formulation/calibration of a suitable macroelement model are discussed, this requiring a reliable geometrical/mechanical characterization of the soil-footing system (Figure 18). Then, as seismic excitations are usually fast and the tower subgrade is mainly formed by clayey saturated materials, undrained loading conditions can be reasonably assumed. As was previously mentioned, the fundamental “ingredients” in the formulation of a strain-hardening macro-element are: i) the elastic stiffness matrix Del , ii) the failure locus F V , H , M   0 , iii) the yield function f and the plastic potential g and iv) the hardening rule and hereafter, with reference to Ghirlandina bell tower, their determination is briefly discussed.

a)

b)

Figure 18. a) Vertical section of the tower foundation and b) its schematic representation

204


Elastic stiffness matrix If the pre-yielding soil behaviour is assumed to be isotropic linear-elastic, the elastic stiffness matrix is diagonal, D el  diag kvvel khhel kel  . Nevertheless, the three elastic stiffnesses cannot be evaluated through classical solutions for shallow strip foundations, because of its square-shaped base (Figure 18) and its nonnegligible embedment. This difficulty can be overcome by employing the results by Gazetas et al. (1985) and Gazetas et al. (1988), who numerically evaluated ad hoc correction factors to deal with arbitrarily-shaped embedded foundations. In this case, “averaged” undrained elastic parameters are used for the soil (Poisson’s ratio  u  0.5 and Young modulus Euel , to be estimated from in situ and/or laboratory tests). Failure locus The definition of the general failure locus F V , H , M   0 for “non-standard” foundations is a non-trivial engineering matter. As the foundation embedment is not negligible, the modified expression by di Prisco et al. 2004 can be adopted: F = h

d1

+ m

d2

2g

- ( x - xc )

1 Q = éê x h m ùú = ë û V -V M1 M2

2b

( 1 - x + xc ) é êV ëê

H m

=0 ù VM 2 M ú; x = yB ûú c VM 1 - VM 2

[5]

where Q is the vector of generalised dimensionless stresses and  ,  , 1 ,  2 ,  ,  are six constitutive parameters, Thus, the bearing capacity under pure compression ( VM 1 ), tension ( VM 2 ), shear ( H 0 ) and moment ( M 0 ) are to be evaluated. While expression [5] always guarantees F  , h  m  0   V  VM 1  V  VM 2 ,

 ,  and a unique relationship    , c  are set to ensure the V  H and V  M cross sections to attain the maximum at   0 (Bransby et al., 1999; Gouvernec 2008). Finally, the free parameters 1 ,  2 and  can be set to match the shape of V  H , V  M and M  H failure loci, as derived through standard formulas for inclined/eccentric loads (Bowles, 1996) and/or from FE analyses (swipe tests). The analytical expression [5] fails in reproducing the asymmetry of the M  H cross section, however it can be employed as a first approximation.

Yield function and plastic potential As was proposed by di Prisco et al. (2004), the yield function can be simply defined by slightly modifying expression [5]: 1

f h m

2

     c c 

2

   1   c   c 

2

[6]


205

 c being the unique isotropic hardening parameter, so that f  F as  c  1 . As far as the plastic potential g is concerned, an associated plastic flow ( g  f ) can be conveniently assumed for undrained analyses, this reducing the amount of constitutive parameters. The mechanical soundness of this assumption – not only at local soil level but for the overall foundation as well – has been numerically confirmed by Bransby et al. (1998). Hardening rule The evolution of the yield locus toward the failure envelope can be described through the following hardening rule (di Prisco et al., 2004): a1 d e p a2 d J p R æ d rc = ( 1 - rc ) 0 2 ççç d h p + + m y VM 1 çè p

q = éê h e J ùú = (VM 1 - VM 2 ) éê v mu ë û ë p

yB q ùú û

ö÷ ÷÷÷ ø÷

[7]

p

where q p is the vector of generalised plastic strains, while R0 , a1 , a2 are further constitutive parameters. The identification of hardening parameters is as important as uncertain: indeed, although they determine the plastic stiffness of the macroelement (and thus k ), no standard calibration procedures are yet available. However, it could be shown that, under pure vertical load, R0 is directly linked to the plastic stiffness of the foundation dV dv p , provided VM 1 , VM 2 and the current load V (here the weight of the tower): R0 =

dV dv p

éæV -V öæ V ö÷÷ùú ÷çç M2 ÷ êçç M 1 1 ÷ êçç V ÷øèçç V ø÷÷ú êëè M1 M 1 úû

[8]

As a first estimate, dV dv p can be evaluated through the elastic formula by Gazetas et al. (1985), as long as a proper value for the undrained plastic modulus Eup is used for the soil. Eup can be for instance derived from oedometric tests in the light of a suitable constitutive clay model. Conversely, while no simple calibration seems to be possible for 1 and  2 , the use of elasto-plastic FE analyses is recommended to this purpose. Provided the full set of constitutive parameters, the macro-element model can be easily employed to simulate ad hoc loading path and thus to derive k   . For the present pseudo-static seismic analysis, V -constant radial M  H paths are likely to be the most meaningful, whence the importance of the M  H envelope shape in seismic applications results.

206


5.3. Cyclic macro-element models A way for overcoming the shortcomings of the previously defined class of models, as is done for soil REV constitutive relationships, is to conceive either generalised plastic or hypo-plastic constitutive models. For instance, the above macro-element model by Nova et al. (1991) was modified by di Prisco et al. (1998), by introducing within the yield locus a subloading surface (Figure 19), such that any intersection with the outer “bounding surface” is avoided. A convenient mapping rule allows to relate any point within the yield locus to an appropriate point on it. The plastic multiplier is evaluated for the bounding surface and suitably scaled according to the distance between the current point Pi and the corresponding image point belonging to the bounding surface I i (Figure 19). When such distance reduces to zero, the scaling function converges towards unity. The inner locus is therefore exclusively employed to define the elastic domain and to determine the image point I i . In this way, both the occurrence of permanent generalised strains – even when the stress point is within what is usually considered a purely elastic region – and the accumulation of plastic distortions during cyclic or transient loading can therefore be simulated (di Prisco et al., 2003a; di Prisco et al., 2003b). The model becomes inevitably more complex and a larger number of constitutive parameters is necessary. A validation example of the model concerning the behaviour of a plinth, 1 m wide, founded on a dense sand stratum and subjected both to cyclic horizontal load and overturning moment (experimental data after Pedretti (1998)) is given in Figure 20. Loads are applied at low frequencies, so that dynamic effects can be neglected; both the overturning moment and the horizontal force are varied at constant ratio. It is evident that this constitutive approach is suitable for capturing in a satisfactory way the essential features of the experimental evidences and this conclusion regards both dense and loose sand strata (at least in the case of symmetric cycles applied on symmetric systems). H

Failure locus F=0 bounding surface f=0

Ii Inner yield locus

A Pi B

V

Figure 19. Sub-loading function and bounding surface in the model by di Prisco et al., (2003b)


207

b)

a)

v [mm]

0

2

4 0

40

t [sec]

80

c)

Figure 20. Comparison of measured (dotted lines) and calculated (solid lines) displacements of a real scale foundation after Pedretti (1998); a) horizontal load vs horizontal displacements; b) overturning moment vs rotation; c) accumulated vertical settlement vs time

This class of models can be also employed to rapidly derive interesting heuristic information, that obviously have to be experimentally confirmed. For instance, as already discussed, the experimental data in Figure 9a have been obtained by imposing generalised stress paths similar to those plotted in Figure 21a ( M H constant loading paths with increasing cyclic amplitude, Figure 21b). The large scatter in the experimental results can be theoretically explained in the light of the aforementioned constitutive model (di Prisco et al., 1998). H/V

H

V

Vfond

step

VMAX

a)

b)

Figure 21. a) constant load path and b) loading history with symmetric cycles of increasing amplitude

208


In Figure 22, the different K K 0 numerical curves concern two different relative densities and different vertical loads V . It is evident that at decreasing values of V VMAX the decay in K K 0 becomes more rapid, while the increase in damping is less rapid. These numerical curves can be numerically interpolated by means of analytical expressions of the following type: Kf Kf0



1 ;  f   f ,min   f ,max   f ,min  1  exp  b  1  a m

[9]

1

1

0.8

0.8

Kθ / Kθ0 (-)

Kθ / Kθ0 (-)

The values of dimensionless parameters a , m and b allowing the best fitting of numerical results are listed in Table 1. These expressions will be fruitfully employed in the practical application briefly outlined in Section 5.4.

0.6

0.4

0.6

0.4

dense sand 0.2

medium density sand

VMAX/V = 2 VMAX/V = 4 VMAX/V = 6 VMAX/V = 8 VMAX/V = 10

0.2

0 1E-005

0 0.0001

0.001

0.01

rocking angle θ (rad)

0.1

1E-005

a)

0.4

0.3

damping (-)

damping (-)

0.001

0.01


0.1

b)

medium density sand


0.2

0.1


0.2

0.1

0 1E-005

0.0001

0.4

dense sand 0.3


0 0.0001

0.001


0.01

c)

1E-005

0.0001

0.001


0.01

d)

Figure 22. Influence of the V/VMAX ratio on the dependence of a, b) the secant rotational stiffness K and of c, d) the damping factor for dense and medium dense sand on the rocking angle, respectively

Unfortunately, the previously mentioned class of models fails in reproducing three very important aspects of the cyclic response of rigid shallow foundations: i) the large settlements induced by the first unloading, ii) the reduction in stiffness during the first phase of the unloading when the footing is previously largely tilted, iii) the ratcheting when asymmetric loading paths are imposed.


209

To capture at least the first item, as already suggested by Cremer et al. (2001), the hardening of the plastic surface should be anisotropic. This can be easily inferred from FE numerical results. Indeed, the soil domain interested by the development of severe plastic strains, at increasing M and constant V , is very small and close to one of the corners. When the overturning moment is reversed, a new plastic zone develops close to the opposite corner and the two zones interact only marginally. For this reason, a previous loading characterised by large positive values of M cannot cause a hardening involving the yield locus in the zone where M is negative. This necessarily implies that elasto-plastic constitutive models with an isotropic hardening rule cannot satisfactorily reproduce this peculiar experimental aspect.

Table 1. Values of parameters a, b and m in Equation [9] (from Paolucci et al., 2010) Dr = 90%

Dr = 60%

VM V

a

m

b

a

m

b

2

458.36

1.30

37.73

686.26

1.30

39.39

3

281.95

1.11

32.76

386.24

1.11

47.61

4.5

262.90

1.00

43.93

339.87

0.98

67.79

6

292.81

0.94

62.25

352.13

0.92

90.64

7.5

324.76

0.91

66.96

398.44

0.89

104.49

9

378.05

0.89

85.08

433.12

0.86

119.20

10

415.50

0.88

95.60

452.44

0.84

130.85

15

575.36

0.83

164.42

653.02

0.79

210.42

20

1010.99

0.86

233.70

1219.47

0.83

285.15

25

2461.06

0.95

305.97

2461.06

0.89

367.70

30

5192.13

1.02

382.51

5192.13

0.96

442.47

Following the theoretical approach conceived for cohesive strata by Salençon et al. (1995a-b), Cremer et al. (2001) suggested the two different expressions for the failure locus F and for the yield locus f , respectively, reported here below:

210


 H F  c  a V  1  V d 

2

  M   e   bV  1  V  f  

2

2

  1  0  

[10]

2

 H    H   M    M  f     1  0      H     M 

where V  , H  and M  are normalised dimensionless variables as those introduced by Nova et al. (1991) (  , h and m ), whereas a  , b , c , d  , e and f  are constitutive parameters;   ,   and   are hardening variables defining the isotropic hardening the former and the kinematic hardening of the yield function the latter two, whereas: 0   0    d     c Γ    H    aV    V       f   M   aV e   V   

    1       

[11]

  2   2 For          1 , f coincides with the failure locus F . When the yield function isotropically hardens and a centred vertical load is applied,       and    0 . These quite complex analytical definitions provide a convenient formulation for the anisotropic hardening as is schematised in Figure 23, both in the H   V  and M   V  planes. An interesting application of this class of models has been recently published by Grange et al. (2007a); (2007b), (2008) and (2009). These authors employed a suitable evolution rule for ’, a linear constraint between   and  

(    k   ) and, to satisfactorily match the experimental results, parameter k was abruptly reduced. Unfortunately this choice necessarily implies the influence of previous loading histories to be lost. The second aforementioned aspect, concerning the unloading rotational stiffness after getting large tilting angles, is instead more important and can be clarified by commenting the experimental results obtained on large scale models at Public Works Research Institute (PWRI, 2005; Shirato et al., 2007) and already shown in Figure 7. These experimental data have been obtained by imposing to a prototype rigid structure, placed on a caisson filled with sand, a horizontal cyclically varying displacement. During the cyclic phase, the footing is therefore loaded by a constant vertical load due to the steel frame weight and by cyclically varying overturning moment and horizontal force. As the test is performed under displacementcontrolled conditions, even the reduction in generalised loads can be observed. As already commented, when the soil density is sufficiently high, during the unloading


211

a typical S-shaped trend is observed. As was already mentioned, this mechanical response during unloading in the case of dense sands is essentially due to the uplift of the foundation: the reduction in the contact surface between the footing and the soil due to the detachment between the two materials generates a sort of damage of the system that could be described coherently for instance by introducing an elastoplastic coupling. Recent efforts along this direction have been spent by Cremer et al. (2002), Shirato et al. (2007), Paolucci et al. (2008), Grange et al. (2008) and (2009). In particular, Paolucci et al. (2008) suggested a sort of pseudo-empirical damage rule, whereas Chatzigogos et al. (2011) a non-linear damaging elasticity accounting for the detachment. Finally, as far as the third issue listed above is concerned, it is worth noting that both standard bounding-surface-plasticity and anisotropic strain hardening constitutive models severely overestimate the phenomenon of ratcheting, when asymmetric loading paths are imposed. In the case of elastic isotropic strainhardening, unloading/reloading cycles do not cause any accumulation in irreversible displacements, whereas the standard bounding surface approaches cannot generate any loop, so that ratcheting is too marked. To reduce the accumulation of the irreversible generalised strains and to reproduce, both qualitatively and quantitatively, the experimental results, the previous di Prisco bounding surface model was further modified by the author by introducing a sort of artificial memory surface rotating in the   h  m space and “re-modulating” the I i  Pi distance in Figure 19. An alternative approach based on the hypo-plastic theory was instead very recently proposed by Salciarini et al. (2010). H’

0

H’

Pi

Pi+1

1

V’ 0 Fi

M’ Fi+1 Ff

a)

b)

Figure 23. Failure locus and evolving yield function according to the approach of Salençon and Pecker (1995a, b) a) in the H’−V’ and b) in the H’−M’ plane

212


5.4. A DDBD exemplifying procedure

The aforementioned model by di Prisco et al. (1998) can be employed to numerically define the degradation curves K / K 0   , with the aim of accounting for the non-linear soil-structure interaction. In this perspective it is possible to employ the so-called Direct Displacement-Based Design (DDBD) recently proposed by Priestley et al. (2007). This method is particularly suitable for taking the effects of the soil-structure interaction into account, since, according to this method, the structure is schematized as a 1DOF system characterized by an equivalent damper. In fact, by disregarding the footing mass and by assuming the interaction to be essentially dominated by the rotational degree of freedom, it is possible to demonstrate (Wolf 1985) that any ideal 2DOF system can be substituted by an equivalent 1 DOF system (Figure 24) for which natural period, damper and input acceleration are defined as it follows: T  T  T ;  eq  2 eq

2 s

2 f

K eq Ks

s 

K eq H 2 Kf

 s ; ageq 

K eq Ks

ag

[12]

where Ts , T f and Teq are the natural periods of the structure, footing and equivalent system, respectively; K s , K f , K eq and  s ,  f , eq are the stiffness and the damping ratio of the structure, of the macro-element and 1DOF equivalent system, respectively; H is the elevation of the centroid, a g and a geq are the soil acceleration and the equivalent seismic input, respectively. Therefore, it becomes possible adapting the DDBD method to the equivalent system.

Figure 24. a) 2DOF and b) equivalent 1DOF systems

Paolucci et al. (2010), to account for the macro-element non-linearity, proposed an iterative algorithm, based on the use of the interpolating functions given in [9]. For a detailed description of DBDD methods, the aforecited volume by Priestley et al. (2007) is suggested.


213

Taking the non-linear rotational behaviour of the soil-structure system into account, it is possible to show, for instance in the case of bridge piers, that the contribution to the total displacement due to the footing rotation is larger than the one due to the structural distortion. As a consequence, the superstructure does not suffer any damage, as it mostly remains in the linear elastic range. In contrast, the footing behaves as a damper and governs the total energy dissipation of the overall system. The shear loading at the toe markedly decreases with respect both to the case of fixed and elastic constraints. The choice of considering the non-linear soilstructure interaction allows to reduce the width of the footing, as in this case the calculated overturning moment applied to the foundation are much lower. From a practical point of view, it is possible to state that the non-linear soilstructure interaction is particularly relevant in the case of tall and flexible structures, such as bridge piers. The flexibility implies that the shear force at the pier toe due to the seismic input is rather small, whereas a large overturning moment arises. The failure mechanism mainly develops within the dissipating soil, while the superstructure essentially remains in the elastic domain. These conclusions are in essence opposite to what would be derived from the analysis of a system with both elastic macro-element and superstructure. 6. The dynamic response

In the previous engineering applications, the macro-element approach has been used only to provide a tool to be implemented into either pseudo-static approaches (Section 5.2) or pseudo-dynamic methods (Section 5.4). In this session, a more effective use of the macro-element is proposed and a real, though simplified, dynamic analysis is described. 6.1. The shaking table experimental setup and test series

As an example of dynamic tests on shallow foundations, it is worth citing the tests recently performed at the Public Works Research Institute (PWRI), Tsukuba (Japan). There, a large scale shaking table equipment was set up and a laminar box 2.1 m high and 4 m x 4 m in plan was placed on a shaking table. The box was filled with dry Toyoura sand, compacted in layers, so that nearly homogeneous soil conditions were obtained (relative density Dr = 80%, mass density  = 1 600 kg/m3). By means of standard drained triaxial tests, an internal friction angle   = 42.1° was evaluated. A footing prototype (Figure 25a) was located at the centre of the box on the ground level (Figure 25b). The footing model consisted of three main structural components: a steel rack at the top, 5 227 N heavy, a 0.5 m sided square foundation block at the bottom and a short steel beam with I cross-section connecting the two massive blocks. The total height of the model is 0.753 m, while the height of the centroid is 0.420 m from the base of the

214


foundation. The stiffness of the I-beam connection is much higher than the one of the soil-foundation system, so that it is the latter one to drive the dynamic response of the whole model. The recording system consists of 70 accelerometer sensors.

Figure 25. The experimental shaking table setup, including the footing prototype

The shaking table was excited by means of four different seismic inputs (Figure 26, Paolucci et al., 2007). The first, referred to as Case 1-2, was recorded at Schichihou Bridge, Hokkaido, Japan, during the 1993 Hokkaido Nansei Oki Earthquake (MW = 7.8), and consists of a very long duration, high-frequency excitation. The further three excitations (Cases 1-4, 1-5, 2-2) consisted of the NS component recorded during the 1995 Kobe Earthquake (MW = 6.9) at Japan Meteorological Agency.

Figure 26. Earthquake records used as base excitations for shaking table tests: a) 1993 Hokkaido Nansei Oki Earthquake Schichihou Bridge record, b) and c) 1995 Kobe Earthquake JMA record, d) 80% scaled 1995 Kobe JMA record


215

6.2. Numerical simulations

A computational procedure to simulate the above shaking table experiments was proposed by Paolucci (1997) and consists in the use of a two-dimensional 4DOF soil-structure interaction model (Figure 27). One DOF is for the horizontal motion of the elastic superstructure, while the soil-foundation system is described by means of a 3DOF (horizontal, rocking and vertical motion) elasto-perfectly plastic macroelement. This is characterised by a yielding function defined according to Nova– Montrasio’s model (1991) and a plastic flow rule derived from Cremer et al. (2002). An important features of the numerical model consists in considering that during the strongly non-linear phases of the excitation the instantaneous foundation-soil contact area decreases, because of cyclic foundation rotations. This aspect has been implemented by introducing a simple degradation rule for the foundation stiffness parameters ( K 0 , K v and K r , see Figure 27), described by the following equation: B   B 1  D 

[13]

where B is the foundation width and D is a degradation parameter defined as: D  p  

D1 1  1 D2 p

[14]

D1 and D2 being model parameters related to the ultimate D value and to the

degradation rate, respectively, while  p is the cumulated plastic foundation rotation.

Figure 27. The 4DOF model used for dynamic non-linear soil-structure interaction analyses

A sample of results for Case 2-2 is shown in Figure 28 in terms of the normalized overturning moment, foundation rocking and vertical settlement, during the most severe 10 s of the excitation. The foundation behaviour is highly nonlinear and frequently reaches a threshold eccentricity ratio that in the experiments turns out to be slightly higher than 0.4 (Figure 28a). The results with and without stiffness

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degradation are compared to show that, although the simplest approach without degradation is already rather satisfactory, the improvement obtained through the stiffness degradation rule [13]-[14] leads to capture some important details of the nonlinear response, including the period elongation following the major yielding phases of the model response. Results are quite satisfactory both in terms of eccentricity ratio (Figure 28a) and of foundation rocking (Figure 28b), taking also into account that a single parameter set was used for all the experiments.

Figure 28. From top to bottom: time history of the eccentricity ratio e/B = M/VB (B: foundation width; M: overturning moment; V: vertical load – thick line: observed; thin line: simulated), for loading case 2-2 (scaled Kobe JMA), obtained with and without the degradation rule 3); corresponding foundation rocking time histories (thin line: with degradation rule; dotted line: no degradation rule); the foundation vertical settlement (thick line: observed) 7. Concluding remarks

This chapter has concerned the analysis of the seismic response of shallow footings. The problem has been tackled in terms of experimental test results (almost exclusively for rigid shallow footings on sand strata under quasi-static conditions) and macro-element theories. Three practical applications of the macro-element concept for design purposes have been discussed.


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The fully uncoupled strategy to solve the seismic soil-structure interaction problem, i.e. by evaluating the actions transmitted by the soil to the foundation, and vice versa, has been outlined in the case of a footing rigidly constrained to the soil. Three alternative strategies have been briefly described. The case of a tall historical bell tower has been approached within the framework of a standard ULS pseudostatic analysis, with the aim of stressing both the relevance of second-order effects and the difficulty in calibrating constitutive parameters. Then, the case of slender bridge piers has been considered to highlight, according to the Priestley DDBD method, the isolating effect of non-linearities developing in the so-called near field (especially in the case of strong seismic motions). Finally, the case of a laboratory prototype dynamically excited has been employed to validate the most basic version of macro-element (the elastic-perfectly plastic version) implemented in a four degrees of freedom dynamic system. This example has allowed to underline potential/shortcomings of the approach when implemented into standard structural finite element codes. As is evident from the contents of this chapter, a large amount of work has been carried out in the last decades to investigate the seismic soil-structure interaction problem, however many open questions are still waiting for satisfactory answers. These mainly concern i) the role of soil inertiae in the “near field”, ii) the effect of the interstitial pore water, iii) the definition of macro-element constitutive relationships for complex soil profiles and irregular topographies. 8. References ATC-40, Seismic evaluation and retrofit of concrete buildings, Chapter 10: Foundations effects, Technical Rep. SSC 96-01, Seismic Safety Commission, State of California, 1996. Bowles J.E., Foundation analysis and design (5th edition), New York, McGraw Hill, 1996. Bransby M.F., Randolph M.F., “Combined loading of skirted foundations”, Géotechnique, Vol. 48, No. 5, 1998, p. 637-655. Bransby M.F., Randolph M.F., “The effect of embedment depth on the undrained response of skirted foundations to combined loading”, Soils & Foundations, Vol. 39, No. 4, 1999, p. 19-33. Bransby M.F., Yun G., “The horizontal-moment capacity of embedded foundations in undrained soil”, Canadian Geotechnical Journal, Vol. 44, 2007, p. 409-424. Butterfield R., Gottardi G., “Determination of yield curves for shallow foundations by swipe testing”, In Fondations superficielles, Magnan and Droniuc (Eds), Paris, Presses de l’ENPC/LCPC, 2003. Butterfield R., Gottardi G., “A complete three dimensional failure envelope for shallow footings on sand”, Géotechique, Vol. 44, No. 1, 1994, p. 181-184.

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