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32 Figure 4-1. Finite element model of the bridge used for model verification. Figure 4-2. Grillage model of the bridge used The typical cross sections are shown in Figures 4-3, 4-5, 4-7, for model verification. and 4-9. The section geometry remains the same for various bridge span lengths, with the exception of the overall section pier stiffness. The stiffer columns were 6 ft by 8 ft which can depth. Also, in order to simplify the analysis cases and automate also be considered similar in behavior to a multi-column pier. the model generation, without jeopardizing the global behavior The softer columns are 6 ft by 6 ft, 7 ft by 7 ft, and 8 ft by 8 ft of the bridge, the cross section was idealized as shown in Fig- for the short, medium, and long bridges respectively. All ures 4-4, 4-6, 4-8, and 4-10. The cross-section properties used column heights were assumed to be 50 ft to the point of fixity for the analytical models are shown in Tables 4-2 through 4-5 at the base. It was assumed that the pier and abutment For each cross-section type, five typical bridge models were diaphragms were relatively stiff, i.e., they had a moment of in- considered: ertia of 5000 ft4. Single Span 200 ft long; Each of the above 32 bridges was configured as a straight Single Span 100 ft long; bridge and as curved bridges with radii of 200, 400, 600, 800, Three Span 200, 300, and 200 ft long; and 1000 feet, resulting in 192 bridge configurations. Three Span 150, 200, and 150 ft long; and Three Span 75, 100, and 75 ft long. Structural Analysis Each of the three-span bridges was analyzed with two types Each bridge configuration was modeled as a spine model of piers to identify the effect of softer versus stiffer transverse (in which one line of elements was used for superstructure, Table 4-1. Comparison of results, grillage vs. FEM two cell box. Action Location DL LL2C Grillage FEM Grillage/FEM Grillage FEM Grillage/FEM Midspan Span 2 Deflections Left -4.40 -4.31 1.02 -0.23 -0.26 0.91 (inches) Right -3.87 -3.80 1.02 -0.25 -0.23 1.10 Gross 52103 53523 0.97 3791 3829 0.99 Midspan Span 2 Moments Left Girder 14016 16290 0.86 1197 1190 1.01 (ft. kips) Center Girder 21523 21735 0.99 1566 1504 1.04 Right Girder 16565 15499 1.07 1028 1135 0.91 Gross -91261 -94623 0.96 -3095 -3235 0.96 Negative Bent 3 Moments Left Girder -27469 -27309 1.01 -1007 -845 1.19 (ft. kips) Center Girder -38265 -39494 0.97 -1342 -1526 0.88 Right Girder -25528 -27820 0.92 -746 -864 0.86

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33 Figure 4-3. Typical single-cell cast-in-place cross section. located along the centerline of the bridge), and also as a The section properties for the grillage model included a set grillage analogy (grid) model (where each web line was of properties for the exterior girder, a set of properties for the modeled as a line of elements along its centerline and trans- interior girder, and a set of properties for the transverse ele- verse elements along radial lines were used to connect them ment. Furthermore, the transverse element shear area was in the transverse direction). Typical spine and grillage calculated via a special formula to account for the warping on modeling techniques are shown in Figures 4-11 and 4-12. the cells. An example of these properties for the two-cell sec- Each model uses several elements per span in the longitu- tion is shown in Table 4-3. The transverse element properties dinal direction of the bridge. Each span element is limited may be different for each element based on its width and to a central angle of 3.5 degrees as recommended in therefore is shown here for a unit width. The members on the Appendix C. outside of the curve were longer than the ones on the inside. The section properties for the spine model were based on Likewise, the widths of transverse members along the outside the entire section. An example of these for the two-cell cross of the curve were larger than those of the members along the section is shown in Table 4-2. inside. The actual member property used in analysis was Figure 4-4. Idealized single-cell cast-in-place cross section.