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2.44e, f, it can be observed that the strainin the transverse and longitudinal reinforcement bars under 1.43% drift ratioexceeded the yield strain as evidenced by ductilities higher than 1. The transversereinforcement strain pattern shows the significant influence of the shear forces,reflected by stirrup yielding along the entire pier height suggesting development ofa strut-and-tie shear mechanism. The longitudinal bar strains indicate yielding inthe outer bars above the foundation, but with a maximum ductility of slightly above1 suggesting very small flexural plastic deformations. These results are in generalagreement with those observed in the experimental studies.The calculated tensile strain ratios for the initial cycles (Fig. 2.45a) showsignificant cracking at webs and flanges, mainly concentrated at the pier base.The compressive damage pattern for 1.43% drift ratio (Fig. 2.45b and c) indicatesmaximumdamage near the pier base, which is consistent with observed damage thatconsisted of minor concrete spalling at the corners. The deformed mesh for 1.43%drift ratio at the top of pier PO1-N6 shows significantly less shear deformationcompared with PO1-N4 (Fig. 2.45d) because of the higher transverse steel ratio. Compared to transverse bar strains in PO1-N4, the strains shown in Fig. 2.45eare considerably lower due to the higher amount of transverse steel in PO1-N6. Finally,the longitudinal rebar strain distribution shows that extent of yielding is significantlyhigher than that of PO1-N4. This because the higher amount of transverse steelreduced shear degradation and allowed for higher flexural strains to develop.2.5.7.2 Concluding RemarksAn application of refined constitutive models for estimating the nonlinear seismicbehaviour of hollow box reinforced concrete bridge piers was described. A consti-tutive model based on the Continuum Damage Mechanics was used for the con-crete, incorporating two independent scalar damage variables to reproduce thedegradation under tensile or compressive stress conditions. Steel reinforcementwas discretized via 2-noded truss elements, and the corresponding behaviour wassimulated using the Menegotto-Pinto cyclic model. 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.