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This curve is widely used in current practice, especially in the U.S., and it isapplied also for dynamic or non-linear problems. The latter is achieved with asimple transformation of Eq. 2.1 to a bi-linear relationship by assuming a specificdeformation Dy to enter the inelastic range (typically equal to 2.5 cm forSpringer, Dordrecht, ISBN: 9789400739437 The efficiency of the numerical model was demonstrated by simulating theexperimental tests performed at the LESE on reduced scale bridge pier modelstested under cyclic loads. Important characteristics of the test models were ade-quately captured using the analyticalmodel. The shear capacity of each pier was alsoaccurately captured by the model developed by Priestley et al. (1996), with shearfailure in PO1-N4, and limited flexural ductility in PO1-N6, which has twice asmuch shear reinforcement. The ability of the detailed analytical model to captureshear deformations is due to the refined element modelling it incorporated.Distributed shear degradation and failure is generally hard to be replicated by simplemacromodels.2.6 Modelling of Dynamic Interaction Between Piers,Foundation and Soil2.6.1 Pseudo-static Winkler ApproachThe most commonly adopted engineering method for calculating the pseudo-staticinteraction between the piles of a bridge foundation and the soil is theWinklermodelin which the soil reaction to pile movement is represented by independent (linear ornon-linear) unidirectional translational spring elements distributed along the pileshaft to account for the soil response in the elastic and inelastic range respectively.Although approximate,Winkler formulations are widely used not only because theirpredictions are in good agreement with results from more rigorous solutions but alsobecause the variation of soil properties along the pile length can be relatively easilyincorporated. Moreover, they are efficient in terms of computational time required,thus allowing for easier numerical handling of the structural inelastic response.The corresponding mechanical parameters for the springs are frequentlyobtained from experimental results (leading to P-y curves for lateral and T-z curvesfor axial loading) as well as from very simplified models. A commonly used P-ycurve is the lateral soil resistance vs. deflection relationship proposed by theAmerican Petroleum Institute (1993):P ¼ 0 9pu tanhkH0 9puy (2.19)where pu is the ultimate bearing capacity at depth H, y is the lateral deflection and kis the initial modulus of subgrade reaction which is both depth and diameter-dependent despite the fact that in many cases (i.e. NAVFAC 1982) the modulusof the subgrade reaction is assumed to be independent of diameter.This curve is widely used in current practice, especially in the U.S., and it isapplied also for dynamic or non-linear problems. The latter is achieved with asimple transformation of Eq. 2.1 to a bi-linear relationship by assuming a specificdeformation Dy to enter the inelastic range (typically equal to 2.5 cm for cohesionless soils) and a second branch stiffness reduced to 1/4 of the initial soilstiffness (Kappos and Sextos 2001). Alternatively to the above procedure, the staticstiffness detached from the complex dynamic stiffness matrix (as discussed in thefollowing section) is also used in practice.To complete the foundationmodelling, a horizontal inelastic soil spring can be usedat the top of the pile to represent the strength and stiffness provided by passive soilresistance against the pile cap while a vertical inelastic spring is commonly used at thepile tip to account for downward and upward capacity of the supporting soil.For the case that particular soil layers are considered to be susceptible toliquefaction, recent studies suggest that both the lateral subgrade reaction of pilesand the maximum reaction force of the laterally-spreading soils have to be reducedat the corresponding locations along the pile length. This reduction factor lies in therange of 0.1–0.2 (Finn 2005) or 0.05–0.2 (Tokimatsu 1999). An average 10%therefore of the lateral stiffness provided along the liquefied soil layers is deemedreasonable, however, the designer has to bear in mind that the particular assumptionis strongly earthquake magnitude dependent.