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Figure 2.41 shows the results for the medium pier. It can be seen that there is a goodagreement in terms of both the amplitude and the frequency content of the response.The agreement is representative of the correlation for the other two piers. Table 2.8lists the ratios of the maximum absolute response obtained from numerical calcula-tion to that from the tests. It is noted that the force response of the squat pier is notreproduced with full accuracy, whereas displacements are very well predicted. Thenumerical overestimation of the action at the top of the short pier can be explainedby the fact that the fibre-based element formulation did not account for sheardeformation.2.5.6.5 Closing RemarksStructural behaviour is inherently nonlinear, particularly in the presence of largedisplacements or material nonlinearities, the structural response can be accuratelycaught only by means of nonlinear dynamic analyses. 57198
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The fibre modelling approachemployed in the current work is shown to be capable of associating simplicity of use,even for not highly experienced users. Moreover, its ability to simulate the nonlineardynamic response of reinforced concrete bridges to seismic loads has been proven bysimulating large-scale experimental pseudo-dynamic tests.Results of the dynamic andmodal analyses performed reveal a good agreement with the pseudo-dynamic tests,both in terms of displacements and forces at the top of the tall and medium-heightpiers. At present, shear strains across the element cross-section are not included in thefibre-element formulation adopted, i.e. the strain state of a section is fully representedby the curvature at centroidal axial strains alone: this approach is not accurate enoughfor representing the squat pier deformation state, where shear deformations are ofrelevance. In this case, despite the relevance of the shear response, the prediction of thedeformation of the squat member was still fairly good.Table 2.8 Ratios of theabsolute maximum responseobtained from numericalcalculation to that from testsTall pier (%) Med pier (%) Short pier (%)Displacement 88 94 102Top shear 90 95 188This section has thus illustrated how the use of simple-to-calibrate fibre structuralmodels can be employed to reproduce with good level of accuracy the nonlinearstructural response of continuous span bridge structures. In other words, it is believedthat such an advanced analytical tool can be readily handled within a commonengineering practice framework, provided a basic level of awareness on the decisionsthat the designer has to face, discussed herein is available.2.5.7 Analytical Modelling of Hollow Box ColumnsIn Sects. 2.5.4–2.5.6 several modelling methods with different levels of sophistica-tion were described and demonstrated. Another example of the development andapplication of microscopic models is presented in this section.A series of quarter scale hollow columns were tested under cyclic loading atPorto University (Delgado et al. 2006, 2007, 2009).
These columns were analyzedusing CAST3M computer code (CEA 2003), which is a general purpose finiteelement analysis program. A wide variety of non-linear elements are included inCAST3M, particularly, a damage model developed at the Faculty of Engineering ofPorto University (FEUP) (Faria et al. 1998; Costa et al. 2005). Studies have shownthat the damage model is suitable for seismic behaviour analysis of RC bridgepiers (Faria et al. 2004). The damage model is Continuum Damage Mechanicsbased constitutive model for the concrete zone discretized into 3D finite elementsincorporating two independent scalar damage variables that account for the degra-dation due to tensile or compressive stress conditions. The Giuffre ´-Menegotto-Pinto model (Giuffre ` and Pinto 1970) for the cyclic behaviour simulation of thesteel reinforcement discretized via truss elements is used.In analytical studies of the piers, the second and third repeated cycles of eachdisplacement level were removed to facilitate comparison with experimental data.Only one-half of the cross section was modelled due to symmetry. In addition tocomparing the measured and calculated hysteresis curves, the strains in variouscomponents were calculated and pided by the yield strain and are discussed in thefollowing sections.2.5.7.1 Comparison of Analytical and Experimental ResultsFigures 2.42 and 2.43 compare the calculated and measured force-displacementrelationships for two of the columns under moderate amplitude loading. It can beseen that generally good correlation was observed between the measured andcalculated results.The result of tensile damage pattern in PO1-N4 is illustrated in Fig. 2.43 forthe initial cycles in which the first shear and flexural cracks were observed alongnearly the entire pier height on the webs and concentrated at the flange bases.The compressive strain pattern is shown in Fig. 2.44b, c for 1.43% drift ratio whensome damage was observed at the pier base during the tests. The deformed mesh for2 Modelling of Bridges for Inelastic Analysis 57Springer, Dordrecht, ISBN: 9789400739437This section has thus illustrated how the use of simple-to-calibrate fibre structuralmodels can be employed to reproduce with good level of accuracy the nonlinearstructural response of continuous span bridge structures. In other words, it is believedthat such an advanced analytical tool can be readily handled within a commonengineering practice framework, provided a basic level of awareness on the decisionsthat the designer has to face, discussed herein is available.2.5.7 Analytical Modelling of Hollow Box ColumnsIn Sects. 2.5.4–2.5.6 several modelling methods with different levels of sophistica-tion were described and demonstrated. Another example of the development andapplication of microscopic models is presented in this section.A series of quarter scale hollow columns were tested under cyclic loading atPorto University (Delgado et al. 2006, 2007, 2009). These columns were analyzedusing CAST3M computer code (CEA 2003), which is a general purpose finiteelement analysis program. A wide variety of non-linear elements are included inCAST3M, particularly, a damage model developed at the Faculty of Engineering ofPorto University (FEUP) (Faria et al. 1998; Costa et al. 2005). Studies have shownthat the damage model is suitable for seismic behaviour analysis of RC bridgepiers (Faria et al. 2004). The damage model is Continuum Damage Mechanicsbased constitutive model for the concrete zone discretized into 3D finite elementsincorporating two independent scalar damage variables that account for the degra-dation due to tensile or compressive stress conditions. The Giuffre ´-Menegotto-Pinto model (Giuffre ` and Pinto 1970) for the cyclic behaviour simulation of thesteel reinforcement discretized via truss elements is used.In analytical studies of the piers, the second and third repeated cycles of eachdisplacement level were removed to facilitate comparison with experimental data.Only one-half of the cross section was modelled due to symmetry. In addition tocomparing the measured and calculated hysteresis curves, the strains in variouscomponents were calculated and pided by the yield strain and are discussed in thefollowing sections.2.5.7.1 Comparison of Analytical and Experimental ResultsFigures 2.42 and 2.43 compare the calculated and measured force-displacementrelationships for two of the columns under moderate amplitude loading. It can beseen that generally good correlation was observed between the measured andcalculated results.The result of tensile damage pattern in PO1-N4 is illustrated in Fig. 2.43 forthe initial cycles in which the first shear and flexural cracks were observed alongnearly the entire pier height on the webs and concentrated at the flange bases.The compressive strain pattern is shown in Fig. 2.44b, c for 1.43% drift ratio whensome damage was observed at the pier base during the tests. The deformed mesh for the drift ratio of 1.43% is shown in Fig. 2.44d, in which significant shear deforma-tion is evident. In the results shown in Fig.