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Based on the above discussion, it can be claimed that at least from a static point ofview, the use of lateral soil resistance-deflection curves (even linear) is a convenientapproach for the estimation of the dynamic characteristics of the bridge. Nevertheless,despite the wide application of the P-y approach for the assessment of the structuralresponse in the design practice, there are certain limitations that have to be stressed:(a) the uncertainty of estimating the parameters involved when load tests are notavailable (especially of defining pu and k), is disproportionally high comparedto the simplicity of the approach. It is notable that although Eq. 2.19 is adoptedby both the Multidisciplinary Centre for Earthquake Engineering Research andATC (MCEER/ATC 2003) and the California Department of Transportation(CALTRANS) guidelines, the proposed sets of the required subgrade modulidiffer on average by a factor of 4 (Finn 2005).(b) the relationships available that relate the 1D (expressed in terms of modulus ofsubgrade reaction k) to 2D and 3D soil stiffness (Fig. 2.46), the latter expressed in terms of modulus of elasticity Es, and Poisson’s ratio n, are not directlyrelated nor verified by 2D and 3D FE analyses. As a result, a set of calibrationassumptions is required for establishing a correspondence between theWinklerand plane-strain FE approaches (Kappos and Sextos 2001) based on the initialformulations proposed by Vesic (1961):k ¼ Dkh ¼ 0 65Esð1 v2Þ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEsD4ð1 v2ÞEpIp12s(2.20)where kh is the modulus of subgrade reaction and D the pile diameter. More-over, the transformation from one soil parameter to the other is straightforwardonly in the case that they are assumed constant with the depth while a set ofadditional and rather case-dependent calibrations is required to obtain agree-ment in the inelastic range.(c) the pile group effect is essentially neglected. Even if the piles are staticallyconnected using appropriate single valued springs to represent the increasedflexibility of a pile group compared to the summation of the stiffness of allinpidual piles (a practice that is acceptable for static analysis), the actualdynamic impedance and the subsequent damping are completely neglected.(d) the extension of the above P-y relationships or other curves calibrated fromstatic analysis or loading testing for use in the framework of dynamic analysis issubjective.(e) estimating the effect of soil-structure interaction solely on the basis of theincreased foundation flexibility, is an oversimplification that may lead tounconservative response estimates under certain circumstances (Mylonakisand Gazetas 2000).(f) the convenience of the (particular statically based) P-ymethod, often leads to theextension of its application for the case of inelastic dynamic analysis in the timedomain.
Such an extension, although tempting for special cases of structuraldesign (i.e. performance based design of new or retrofit of existing importantstructures) leads to the misleading perception of modelling refinement withoutproper understanding and consideration of the complex dynamic nature of SSIphenomena. As a result, important aspects of the soil-foundation-superstructuresystem response are hidden under allegedly ‘all-purpose’ 3D linear/nonlinear,static/dynamic stick models.It can be claimed therefore that as soil-foundation-structure interaction is amulti-parametric and strongly frequency-dependent phenomenon, it inevitablyhas to be seen from a dynamic point of view, through a very careful selection ofFE models, associated parameters and modelling assumptions. As a result, thepseudo-static Winkler approach is deemed appropriate only for cases wherein:(a) what is of interest is the identification of the dynamic characteristics of theoverall soil-foundation-superstructure system and not the actual seismicresponse of the system in the time domain. (b) a preliminary (standard or modal) pushover analysis is performed in order toquickly assess the inelastic mechanisms that are expected to be developed in thebridge under earthquake excitation.(c) a response spectrum analysis is performed or a linear response history analysisis run for a relatively low level of seismic forces.(d) an inelastic dynamic analysis is conducted but the energy absorption isexpected to be mainly concentrated on the superstructure while the materialand radiation damping at the soil-foundation interface is a-priori judged ofsecondary importance (i.e. in cases that the underlying soil formations are stiffand uniform with depth).2.6.2 Linear Soil-Foundation-Bridge Interaction Analysisin the Time DomainCurrently, despite the lack of specific design guidelines on how to model, computeand consider Soil-Structure Interaction effects in the framework of the seismicexcitation of bridges in the time domain, it is common knowledge that foundation isflexible, dissipates energy and interacts with the surrounding soil and the super-structure, in such a way that it both filters seismic motion (kinematic interaction orwave scattering effect) while it is subjected to inertial forces generated by thevibration of the superstructure (inertial interaction). This phenomenon is complexand its beneficial or detrimental effect on the dynamic response of the structure isdependent on a series of parameters such as (Pender 1993;Wolf 1994) the intensityof ground motion, the dominant wavelengths, the angle of incidence of the seismicwaves, the stromatography, the stiffness and damping of soil, as well as the size,geometry, stiffness, slenderness and dynamic characteristics of the foundation andthe structure.The basic methods to be used in the analysis of Soil-Structure Interaction effectsthat implement Finite Element Discretization are the Complete Finite Elementapproach and the Substructure Method. Due to the substantial computational timerequired, the Complete Finite Element approach is primarily used when the overall2D or 3D geometry of the problem is of interest and the response of the soil-foundation-superstructure is mainly linear elastic. It has to be noted though, that ingeneral, the use of Finite Element representation as a means of numericallypredicting the seismic wave propagation, is relatively inferior to more specializedapproaches such as the Finite Difference method. In order to enhance the accuracyof the FE analysis therefore, it is deemed critical to ensure realistic boundaryconditions and optimal mesh dimensions. The first is commonly tackled with theincorporation of Kelvin elements (separately for the horizontal andvertical directions) which is easier to be implemented compared to other artificialboundary conditions such as superposition, paraxial or extrapolation boundaries.It is noted that with proper selection of model parameters, the boundary inter-ference may indeed be eliminated, as has been shown by comparing numerical with experimental results (i.e. Pitilakis et al. 2004). The frequency dependent constantsof the Kelvin elements can be calculated using the solution developed by Novakand Mitwally (1988), with coefficients proposed for the two horizontal and thevertical direction respectively. Additionally, it is important to achieve adequate(albeit not exhaustive) mesh refinement in order to minimize the error related tofinite wave propagation. A very practical limit for the characteristic length lc of theelements has been proposed by Lysmer and Kulemeyer (1969) and is still exten-sively used in FE discretization.According to the Substructure Method on the other hand, the soil-foundation-structure system is pided into substructures, typically the superstructure, the nearfield soil domain inclusive of the foundation and the far field domain. Due toconsiderable computational economy the substructure method has been moreextensively used in the past either in the form of:(a) Coupled FEM/BEM approaches (Renault and Meskouris 2004; Savidis et al.2000 among others): the advantage of this approach is that the soil can bediscretizised only in the interaction horizon, while the boundary conditions areconsistent and hence, the wave propagation in the free-field can be accuratelycalculated considering non-relaxed boundary conditions.(b) Uncoupling and superposition of kinematic and inertial interaction (Kausel andRoesset 1994;