Advanced soil dynamics and earthquake engineering pdf

New York: Dover Publications. Newmark, N. Procedures and criteria for earthquake resistant design. Building practices for disaster mitigation.

Building Research Series, 46, — National Bureau of Standards, US department of commerce. Nigam, N. Calculation of response spectra from strong motion earthquake records. Bulletin of the Seismological Society of America , 59 2 , — Prakash, S. Shallow foundations for seismic loads.

Ramberg, W. Description of stress strain curves by three parameters. Technical note Rausch, A. An expression for cyclic resistance ratio. Austin: University of Texas. Richart, Jr. Vibrations of soils and foundations. Robertson, P. Schnabel, P. SHAKE a computer program for earthquake response analysis of horizontally layered site. Research Report 72— Seed, H.

Ground motion and soil liquefaction during earthquake. Influence of SPT procedures in soil liquefaction resistance evaluations. Research Report 84— Srbulov, M. Practical soil dynamics. In Case studies in earthquake and geotechnical engineering. Springer Publications. Swaran, S. Soil dynamics and machine foundation. New Delhi, India: Galgotia Publication. Suzuki, Y. An in-depth study of dynamic soil properties and the methods of their determination provide the basics to tackle the dynamic soil—structure interaction problems.

Practical problems of dynamics of beam—foundation systems, dynamics of retaining walls, dynamic earth pressure theory, wave propagation and liquefaction of soil are treated in detail with illustrative examples. Soil Dynamics and Liquefaction Author : A. Soil Dynamics Author : T. The accuracy is therefore higher than in a test where the modulus is computed from force and displacement measurements.

The resonant column test measures the soil characteristics in a strain range included between to approximately 5 for torsional tests and at smaller strains in axial tests. With hollow cylinders, strains up to can be reached Anderson, These latter tests have the advantage of creating a uniform strain field within the sample, but render the sample set-up extremely diffi- cult.

For strain amplitudes smaller than , most soils behave elastically and the test is non destructive; it is then possible to perform several tests on the same sample by varying the ambient conditions stresses, temperature,….

To conclude, it is worth noting that resonant column tests are accurate and reproducible. At the present stage, the test has been standardized Drnevich et al, In these tests, a known cyclic stress or strain is applied to the soil sample and the induced strain or stress is measured.

Typically, tests are performed at frequencies of the order of 1Hz 0. The hysteresis loop is determined and, according to the stress path, the relevant parameters for the characterization of the soil behaviour are computed: for instance, in a simple shear test, the secant shear modulus and the equivalent damping ratio for the visco-elastic linear equivalent soil constitutive model.

Nowadays, with the increasing development of more and more sophisticated constitutive relationships, numerous and complex parameters are required. Measurement of these parameters has become possible with the significant technological advances in the apparatus, test driving and servo-control equipment, data acquisition systems provided by micro computers.

Cyclic triaxial tests or torsional cyclic tests on hollow cylinders have the capability of applying widely different complex stress paths, which makes possible the determination of other parameters than a modulus of deformation or a hysteresis loop. More generally, forced vibration tests are well-suited for measuring characteristics in a strain range extending to 5 up to failure.

It is no possible with commercial apparatus to reach smaller strains, associated with displacements of a few microns for standard size laboratory samples diameter of the order to 70mm. Only specially designed devices give access to smaller deformations. As a general rule, forced vibration tests are a good complement to resonant column tests for the evaluation of soil characteristics in a higher strain range. Figure 9 adapted from Woods, summarizes the domain of validity of each test.

Shear strain amplitude Resonant column Cyclic triaxial Cyclic simple shear Torsional hollow cylinders Machine Seismic Nuclear Vibration Motions Explosions Figure 9 : Domains of application of laboratory tests Cyclic triaxial test The cyclic triaxial test has first been used by Seed and Lee and is presently the most widely used laboratory apparatus, especially for the evaluation of the cyclic strength.

Pecker the apparatus is similar to the triaxial apparatus used in Soil Mechanics for monotonic tests; some adaptations are however required to increase the accuracy of the measurements, especially at small strains. One can mention: a stiffer frame, the absolute requirement of having the load gage located inside the triaxial cell to get rid of piston friction,… With these modifications, the cyclic triaxial apparatus presents all the needed versatility and most of the advantages one can expect from a laboratory equipment: good definition of an otherwise homogeneous stress field, possibility of saturation of samples, capability of applying isotropic or anisotropic consolidation stresses, pore pressure measurement devices,…To improve the accuracy of the strain measurements, El Hosri introduced proximity transducers located in the central third of the sample; the displacement measurement is done without any contact with an induction coil and a metallic target placed in the magnetic field of the coil.

The accuracy on displacements is of the order of 0. Cyclic triaxial tests can be performed under strain, force or stress controlled condi- tions. The shear modulus secant modulus and the shear strain J are computed from these parameters.

The equivalent damping ratio E is obtained either directly from the area of the hysteresis loop, or from the phase shift between the force and the displacement eq. Data processing techniques de- scribed for field tests cross correlation function can be used to evaluate the phase shift. The cyclic triaxial test is also used for the evaluation of the cyclic undrained strength of sands.

In that case, the test is performed under force, or better stress, controlled conditions. The test is pursued until failure of the sample by liquefaction occurs; during the test, the applied stress, in- duced strain and pore pressure are continuously recorded.

Cyclic triaxial tests are reliable and reproducible; cyclic undrained strengths measured on the same material with similar testing pro- cedures in eight different laboratories using different equipment proved to be similar Silver et al, Cyclic simple shear test The cyclic simple shear-test has been developed to study the stress-strain behaviour of soils under pure shear stress fields.

Samples can be tested under plane strain conditions with possible rotation of the principal stresses during the test. It must not be confused with the direct shear test, developed by Casagrande, which is not suited for the study of the stress-strain behaviour.

This test has long been considered as the test which most closely duplicates the field stress conditions in a soil element subjected to the vertical propagation of shear waves. The first tests under cyclic loads have been reported by Peacock and Seed and Silver and Seed Presently, the most commonly used apparatus are derived from the original Norwegian Geo- technical Institute NGI device; the cylindrical sample is wrapped in a reinforced rubber membrane. The cyclic shear stress is applied at the top horizontal plane of the sample with a hydraulic or pneumatic device; the membrane stiffness enforces a near simple shear deformation of the sample.

Undrained tests are performed by keeping the volume constant, assuming that the vertical stress variation required to maintain the sample height constant is equal to the pore pressure variation that would be measured in truly undrained tests De Groot et al, Therefore, the distribution of shear and normal stresses is no longer uniform.

However, experimental studies Finn et al, , Vucetic — Lacasse, ; De Groot et al, have shown that, except for the behaviour at large strains, beyond the peak shear stress, tests results do not seem to be affected by this lack of homogeneity.

The use of the simple shear device is therefore justified. However, the versatility of the test is less than for the triaxial test difficulty to control the volume change, no control on the radial stress, only one possible stress path,….

This test is however used either for the measurement of stress strain properties shear modulus or for the evaluation of the cyclic undrained shear strength.

Torsional cyclic shear tests To ensure more homogeneous stress fields within the sample and to have control on the radial stress, Hardin et Drnevich have developed a torsional cyclic shear device to test hollow cylinders. Obviously, this apparatus cannot be used for testing undisturbed cohesionless samples and poses many difficulties for the set-up of cohesive samples.

It is therefore not used in everyday practice. With the capability of executing on the same sample resonant column tests and forced vibration tests, this device gives access to the cyclic stress strain behaviour over the whole strain range of interest. It can also be used for the evaluation of the undrained cyclic shear strength.

References Anderson D. Dynamic modulus of cohesive soils — Ph. University of Michigan, Aubry D. Identification of elastic coefficients through resonant tests of soils samples, Proceedings 2nd Soils Dynamics and Earthquake Engineering Conference, Southampton, Ballard R. Seismic field methods for in situ moduli. Campanella R.

Seismic cone analysis using digital signal processing for dynamic site characterization, Canadian Geotechnical Journal, 29 3. De Groot D. An automated electropneumatic control system for direct simple shear testing, Geotechnical Testing Journal, 14 4 , Drenvich V. Resonant column testing. Problems and solutions. Duncan J. Pecker Hardin B. Haskell N. The dispersion of surface waves on multilayered media, Bulletin of Seismological Society of America, vol.

Hicher P. Hvorslev M. Nigbor R. Woods Editor, Balkema. Prevost J. Reanalysis of simple shear testing, Canadian Geotechnical Journal, vol. Roscoe K. An apparatus for the application of simple shear to soil samples. Silver M. Skoglund G. Stewart W. In situ damping measurement with the seismic cone penetration tests, Geophysics.

Stokoe K. Wright S. Sutterer K. Thomson W. Transmission of elastic waves through a stratified soil medium, Journal of Applied Physics, vol. Vucetic M. Whiteley R. Woods R. Standards Drnevich V. Standard Method for deep quasi-static, cone and friction cone penetration tests of soils. Usually in the seismic design of ordinary building, soil structure interaction is neglected and the dynamic response of the structure is evaluated under the assumption of a fixed based response.

However during seismic loading the soil undergoes deformations which are imposed to the foundation; the question naturally arises of knowing if the motion in the vicinity of the structure is altered by the presence of the structure and how the structure response is modified by the compliance of the supporting soil.

This interaction between the structure and the soil is named soil-structure interaction SSI. The purpose of this chapter is to illustrate whether and under which conditions SSI is important and what are its conse- quences on the dynamic response of the structure. These two phenomena are referred in the technical literature as inertial and ki- nematic loading. The relative importance of each factor depends on the foundation characteristics and nature of the incoming wave field.

However, more often, design engineers refer to inertial loading as SSI, ignoring the kinematic compo- nent. This situation stems from the fact that: x kinematic interaction may in some situations be neglected; x aseismic building codes, except for very few exceptions like Eurocode 8, do not even mention it; x kinematic interaction effects are far more difficult to evaluate rigorously than inertial in- teraction effects.

Figure 1 illustrates the key features of the problem under study Gazetas-Mylonakis, It is presented in the general situation of an embedded foundation supported on piles but all the conclusions are valid for any foundation type.

The soil layers away from the structure are subjected to seismic excitation consisting of numerous incident waves: shear waves S waves , dilatational waves P wave , surface waves R or L waves. The nature of the incoming waves is dictated by seismological conditions but the geometry, stiffness and damping characteristics of the soil deposit modify this motion; this modified motion is the free field motion at the site of the founda- tion.

Pecker Lysmer , the design motion is usually specified at only one location, the ground surface, and the complete wave field cannot be back-calculated from this incomplete information; that is the problem is mathematically ill posed. Assumptions have to be made regarding the exact composi- tion of the free field motion and it can be stated that no satisfactory solution is available to date. Even without the superstructure, the motion of the foundation will be different from the free field motion because of the differences in rigidity between the soil on the one hand, and the piles and foundations on the other hand; the incident waves are reflected and scattered by the foundation and piles which in turn are stressed developing curvatures and bending mo- ments.

This is the phenomenon of kinematic interaction. The motion induced at the foundation level generates oscillations in the superstructure which develop inertia forces and overturning moments at its base. Thus the foundation, the piles, and eventually the surrounding soil experience additional dynamic forces and displacements. This is the phenomenon of inertial interaction. Obviously the foundation, in a broad sense, must be checked for the combined inertial and kinematic loading.

For the evaluation of SSI effects of linear systems the most appropriate constitutive model for the soil is either the linear elastic or, more commonly, the equivalent viscoelastic linear model.

The connection between the structure and the foundation is ensured by a rigid beam. The foundation rests on the soil deposit and its interaction with the soil is, for the time being, modeled by springs and dashpots, called foundation impedances that will be defined later. The spring represents the stiffness of the supporting medium and the dashpot reflects the dissipation of energy arising from the soil itself material damping and from the radiation of the seismic waves away from the foundation.

For the sake of simplicity, material damping is neglected with respect to radiation damping, which is a valid approximation for homogeneous soil deposits at moderate strain amplitudes. The system depicted in Figure 2 possesses 3 degrees of freedom: x The horizontal displacement of the mass um x The horizontal displacement of the foundation u0 x The rotation T of the foundation It is subjected to a horizontal harmonic support displacement with circular frequency Z and amplitude ug.

Figure 3 clearly shows that soil structure interaction is more pronounced for soft soil conditions increasing s and for heavy structures increasing m.

That formulation is presented within the framework of the finite element method. In fact the complexity of the problem to solve is beyond the capability of closed form solutions and numerical solutions are required. However, other numerical techniques can be used, such as boundary element techniques; nevertheless, the concepts that are presented below are general and not restricted to finite element solutions; as a matter of fact the results could also have been obtained from the Principle of Virtual Rate of Work.

The dynamic equilibrium equations are obtained with reference to Figure 4, which is a schematic representation of a SSI problem. This boundary is assumed to be sufficiently remote from the structure in order that the motion at the boundary is not influenced by the presence of the structure.

The total displacement for the SSI problem is then given by equation To simplify the demonstration let us leave out the damping term in equation 15 and restrict our problem to that of a structure at the surface of a horizontally layered soil profile subjected to the vertical propagation of body waves. Consequently, interaction is only generated by inertial forces in the structure; this phenomenon is named inertial interaction.

Let us consider now an embedded structure, the mass of which is zero above the ground and equal and equally distributed to the soil mass for the embedded part.

Even with the same mass, there is interaction; this kind of interaction is named kinematic interaction. It arises from the stiffness of the foundation that prevents it from following the displacements imposed by the soil.

That kind of interaction may be equal to zero as shown previously for surficial foundations, or negligible under certain circumstances, like very flexible piled foundations.

However, for stiff embedded structures it may be very significant. In the most general situation, soil structure interaction arises from both phenomena: inertial and kinematic interaction. Figure 4 and the previous developments illustrate the two broad approaches for evaluating SSI; Figure 4 a corresponds to the direct methods the solution of which is obtained by a direct solution of equation This approach does not involve any superposition and is therefore well suited for non linear systems.

Alternatively, substructure methods take advantage of the decomposition of Figure 4 b and c , or of similar decompositions, to solve the global problem in successive steps. These methods are obviously only applicable to linear problems. The validity of these methods relies on the superposition theorem established by Kausel and Roesset Equivalence between equation 18 and equations 20 - 21 is established by simple addition taking into account equation 19 and the previous definition of the mass matrices.

Its solution provides the kinematic interaction motions that are used as input motions for the solution of equa- tion In the solution of equation 21 the soil can be modeled with finite elements or equivalently by a stiffness matrix representing the condensation of all the foundation-soil degrees of freedom at the interface; this condensation is only possible in the frequency domain. In that framework the stiffness matrix is formed with the complex valued moduli taking into account material damping.

The terms in the matrix are frequency dependent. For a rigid foundation, it is legitimate to replace the N x N stiffness matrix N being the number of degrees of freedom at the interface by a 6 x 6 matrix providing the rigid body motions of the foundation.

This matrix is called the impedance matrix and can be conceptually viewed as an assemblage of springs and dashpots. It follows that the kinematic interaction motions are the rigid body motions of the massless structure. Therefore, under the assumption of a rigid foundation, it is pertinent to split the global problem into there sub-problems: x determination of the motion of the massless rigid foundation subjected to the seismic design motion; this steps represents the solution of equation 20 ; x determination of the foundation impedance matrix; this matrix is composed of a real and an imaginary component, both being frequency dependent; x calculation of the dynamic response of the structure connected to the foundation imped- ances and subjected, at its support, to the kinematic interaction motions.

As long as the foundation is perfectly rigid, this three steps approach is strictly equivalent to the resolution of the global problem equation The advantage of this decomposition is obvious if one of the successive steps can be simplified or ignored: the first step always exists except for a surficial foundation resting at the surface of a horizontal layered soil profile subjected to the up- ward propagation of body waves; in the latter situation solution to step 1 is identical to solving the free field site response since kinematic interaction is nil.

Solution to the second step can be sim- plified for common geometries by using published results in the literature. The third step is always required; however it is simpler and more common to structural engineers since it resorts to classical dynamic analyses. References Gazetas G. Kausel E. Lysmer J. Roesset J. Wolf J. Earthquake foundation design is a challenging task that requires analytical capa- bilities and extensive understanding of soil behaviour and soil structure interaction.

The classical approach involves the determination of the forces applied to the foundation, the seismic demand, and the verification of the bearing capacity, the seismic capacity. However not all situations can be tackled with analyses. Seismic building codes and in particular their chapters on foundation detailing are fundamental to achieve a safe design. Leaving aside the seismic retrofit of existing foundations, which is an even more difficult issue, the design of new foundations raises issues which are far from being totally resolved.

One of the main reasons stems from the complexity of the problem which requires skills in soil mechanics, foundation engineering, and soil-structure interaction along with, at least, some knowledge of structural dynamics. A parallel between static design and seismic design reveals some similarity but also very marked differences. In the early days, static design of foundations put much emphasis on the so-called bearing capacity problem failure behavior ; with the introduction of an appropriate safety factor, close to 3, the short term settlements were deemed to be acceptable for the structure.

It is only with the increase in the understanding of soil behavior and the development of reliable constitutive models that sound predictions of settlements could be achieved. Not surprisingly, earthquake geotechnical engineers have focused their attention on the cyclic non linear behavior of soils and on the evaluation of the cyclic deformations of foundations.

This was clearly dictated by the need for an accurate evaluation of the soil-structure interaction forces which govern the struc- tural response. It is only during the last decade that seismic bearing capacity problems and evaluation of permanent displacements have been tackled. These studies have clearly been moti- vated by the foundation failures observed in the Mexico City and Kobe earthquakes.

These two aspects of foundation design have reached a state of development where they can be incorporated in seismic building codes; Eurocode 8 - Part 5 is certainly a pioneering code in that respect. Nevertheless, a new trend is emerging in earthquake engineering, known as "Performance Based Design" PBD , which definitely needs to be accounted for in earthquake foundation engi- neering.

In this lecture we will focus only on the evaluation of the seismic demand and seismic capacity and review the code approach, and foundation detailing, the earthquake resistant desin of foundations.

Pecker 2 Aseismic Design Process The aseismic design process for foundations is a "very broad activity requiring the synthesis of insight, creativity, technical knowledge and experience" Pender, Information is required and decisions have to be made at various stages including Pecker and Pender, : 1. Obviously the process described above is not a linear progression. Several iterations may be required, at least from step 2 to step 7 , before arriving at a feasible, reliable and economic design.

In the following we will focus on steps 3 and 5. We will assume that all the required information related to the soil characterization and structural performance is available. This in no way means that these two items are of secondary importance; the data listed under these items are probably the most difficult to assess and considerable experience is required as well as the exercise of judgment.

The belief is that SSI always plays a favorable role in decreasing the inertia forces; this is clearly related to the standard shape of code spectra which almost invariably possess a gently descending branch beyond a constant spectral acceleration plateau. Lengthening of the period, due to SSI, moves the response to a region of smaller spectral accelerations Figure 1.

However there is evidence that some structures founded on unusual soils are vulnerable to SSI. Examples are given by Gazetas and Mylonakis for instance. This has been recognized in some codes. The effects of soil-structure interaction on piles shall be assessed With the tremendous development of computer facilities, there does not seem to be any rational reason for neglecting soil-structure interaction.

Most building codes now require that the structural response be evaluated using a multimodal analysis, as opposed to a former monomodal analysis and this can be performed with most computer codes available on the market.

Pecker x the system remains linear; x kinematic interaction can be neglected; x dynamic impedance functions are readily available. Although the superposition theorem is exact for linear soil, pile and structure, it can neverthe- less be applied to moderately non linear systems. This can be achieved by choosing reduced soil characteristics which are compatible with the free field strains induced by the propagating seismic waves: this is the basis of the equivalent linear method, pioneered by Idriss and Seed This engineering approximation implies that all the soil non linearities arise from the passage of the seismic waves and that additional non linearities, developed around the edges of a mat foundation or along the piles shafts, are negligible.

Experience shows that it is a valid approximation in many situations where large soil instabilities do not occur. For some situations, kinematic interaction can be neglected and the second step of the multistep approach can be bypassed.

It must be realized however that, if kinematic interaction is thought to be significant, there is no simple means for evaluating it; as a matter of fact, evaluation of kinematic interaction is almost as difficult as solving the complete SSI problem. Obviously kinematic in- teraction is exactly zero for shallow foundations in a seismic environment consisting exclusively of vertically propagating shear waves or dilatational waves.

Gazetas has demonstrated that when the piles are flexible with respect to the surrounding soil, kinematic interaction is significant for small to medium frequencies. During the last decade, numerous solutions for the dynamic impedances of any shape foundations and of piles have been published Gazetas, They are available for homogeneous soil deposits but also for moderately heterogeneous ones. In addition, simplified methods are available in the case of pile foundations to account for the group effect Dobry and Gazetas, However to be fully efficient, and to allow for the use of conventional dynamic computer codes, the impedance functions which are frequency dependent Figure 2 must be represented by frequency independent values.

From the published results, it appears that only under very restrictive soil conditions homogeneous halfspace, regular foundations can these dynamic impedances be represented by constant springs and dashpots. Nevertheless, structural engineers still proceed using these values which, more than often, are evaluated as the static component zero frequency of the impedance functions.

However, fairly simple rheological models can be used to properly account for the frequency dependence of the impedance functions. These models can be developed using curve fitting techniques, or with physical insight, such as the series of cone models developed by Wolf Figure 3 shows examples of such models: Figure 3a is the model proposed by De Barros and Luco based on a curve fitting technique; Figure 3b is a class of cone models proposed by Wolf.

With such models, which are most conveniently used in time history analyses, the actual dynamic action of the soil can be properly accounted for; even "negative stiffnesses", which are frequently encountered in layered soil profiles, can be apprehended with those models.

As an illustrative example, Figure 2 presents the application of model 3a to an actual bridge pier foun- dation; the foundation is a large circular caisson, 90 m in diameter, resting on a highly heterogeneous soft soil profile. The "exact" impedances were computed using a frequency domain finite element analysis. Note the very good fit achieved by the model square symbols even for the negative stiffness of the rocking component.

Clearly, implementation of such simple rheological models does not impose a heavy burden to the analyst and represents a significant improvement upon the lengthy and tedious iteration process in which springs and dashpots are updated to become compatible with the SSI frequencies. Table 1 taken from Eurocode 8 acknowledges that with increasing ground acceleration the soil adjacent to a shallow foundation will experience increasing shear strains and consequently the stiffness will decrease and the material damping increase.

Table 1 suggests how the apparent average shear modulus and material damping of the soil adjacent will change with increasing peak ground acceleration and envisages that an elastic SSI calculation would be done with the modified values for the soil stiffness and damping. Pecker plification there is, of course, no frequency dependence on the stiffness and damping parameters for the foundation. As noted previously, some non linearities, such as those related to the propagation of the seismic waves, can be introduced but the non linearities specifi- cally arising from soil-structure interaction are ignored.

The generic term "non linearities" covers geometrical non linearities, such as foundation uplift, and material non linearities, such as soil yielding around the edges of shallow foundations, along the shafts of piles, and the formation of gaps adjacent to pile shafts. Those non linearities may be beneficial and tend to reduce the forces transmitted by the foundation to the soil and therefore decrease the seismic demand.

This has long been recognized for foundation uplift for instance see ATC Giving up the mathematical rigor of the superposition theorem, an engineering approximation to these aspects can be reached by substructuring the supporting medium into two sub-domains Figure 4.

The exact boundary between both domains is not precisely known but its location is irrelevant for practical purposes. This concept of far field and near field domains can be easily implemented if one assumes that the degrees of freedom of the foundation are uncoupled: the far field domain is modeled with the linear or equivalent linear impedance functions whereas the near field domain is lumped into a non-linear macro-element.

A simplified rheological representation of this sub-structuring is shown in Figure 5 Pecker, : the macro-element is composed of a finite number of springs and Coulomb sliders which are determined from curve fitting to the non-linear force-displacement or moment-rotation backbone curve, computed for instance with a static finite element analysis push over analysis.

Calibration of this simplified rheological model against a rigorous 2D dynamic finite element analysis, including all the non linearities mentioned previously, shows very promising results. This model can be extended in a more rig- orous way to account for the coupling between the various degrees of freedom of the foundation, especially between the vertical and rotational ones when uplift occurs Cremer et al, At this point a major difference appears between static, permanently acting loads, and seismic loads.

In the first instance excessive loads generate a general foundation failure whereas seismic loads, which by nature vary in time, may induce only permanent irrecoverable displacements. Therefore, failure can no longer be defined as a situation in which the safety factor becomes less than unity; it must rather be defined with ref- erence to excessive permanent displacements which impede the proper functioning of the structure.

Pecker applied to dam engineering Newmark, , its implementation in a code format is far from being an easy task. One of the difficulties is to define acceptable displacements of the structure in relation to the required performance. Another difficulty obviously lies in the uncertainty linked to the estimation of permanent displacements. The rise of pore water pressure under cyclic loading should be considered either in the form of undrained strength or as pore pressure in effective stress analysis.

For important structures, non linear soil behavior should be considered in determining possible permanent deformation during earthquakes. These two terms are explained below. The design action represents the set of forces acting on the foundations.

For the bearing ca- pacity problem, they are composed of the normal force Nsd, shear force Vsd, overturning moment Msd and soil inertia forces F developed in the soil. The actions Nsd, Vsd, and Msd arise from the inertial soil-structure interaction. The term design action is used to reflect that these forces must take into account the actual forces transmitted to the foundation i.

The design resistance represents the bearing capacity of the foundation; it is a function of the soil strength, soil-foundation interface strength and system geometry for instance foundation width and length. Obviously, inequality 1 must include some safety factors.

One way is to introduce partial factors, as in Eurocode 8. The Eurocode approach is preferred because it gives more insight in the philosophy of safety; on the other hand it requires more experimental data and nu- merical analyses to calibrate the partial factors. In Eurocode 8, the following values are used: 1.

It acts like the inverse of a strength reduction factor applied to the resistance in an LRFD code. This factor reflects the fact, that to evaluate the system re- sistance some approximations must be made: a theoretical framework must be developed to compute the resistance and like any model it involves simplifications, and assumptions which deviate from reality.

It will be seen later on that the model factor is essential and can be used with benefit to differentiate a static problem from a seismic one.

It is interesting to note that the results have been later completed by additional lower bound solutions which confirm the merit of the upper bound solutions and help to narrow the gap between upper and lower bound solutions Ukritchon et al, Finally the results, mainly based on the upper bound solutions are cast in the general format Pecker, : I.

Figure 6 : Ultimate loads surface for cohesive soils Inequality 3 expresses the fact that any combination of the loading parameters lying outside the surface corresponds to an unstable situation; any combination lying inside the bounding surface corresponds to a potentially stable situation.

The word potentially is used to point out that no assurance can be given since the solutions were derived from upper bound solutions. Indications on the merit of the solutions are obtained by comparison with the lower bound solutions and the model factor of Eq. The uncertainty is twofold: the solution is obtained from an upper bound approach and, although various kinematic mechanisms were investigated, their number remains necessarily limited when a comprehensive implementa- tion of the upper bound theorem would require that all the conceivable mechanisms be investigated.

These results are put in a simple mathematical expression and implemented in the current ver- sion of Eurocode 8 Annex F and are applicable to cohesive and purely frictional materials. The other parameters entering equation 4 are numerical parameters derived by curve fitting to the "exact" bearing capacity, the values of which can be found in Pecker, However such an evaluation is anything but an easy task.

Probably the most rigorous approach would be to use a global model finite element model including both the soil and the structure.