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39 Figure 42. HL-93 truck configurations of the multiple-span continuous cases (right bridges, interior girder, negative moment). Full width was typically acting at cross sections where the purpose of this study, special case bridges were divided it was most needed, i.e., where moments and hence per- into Cable-Stayed Bridges and Validation Cases. formance ratios would be highest. Where the effective width was less than full width at 2.5.1 Cable-Stayed Bridge Investigation such cross sections, those cross sections had consider- able excess flexural capacity. This section summarizes the investigation of effective slab width in cable-stayed bridges. Further detail on the cable- stayed investigation is provided in Appendix I. 2.5 SPECIAL CASE BRIDGES Special cases such as cable-stayed and prestressed girder 2.5.1.1 Cable-Stayed Bridges Investigated bridges typically confirmed the trends observed in the para- metric study reported above, although girder spacings wider Five cable-stayed bridges were investigated, four of them than 4.8 m were beyond the realm of the parametric study. For having been analyzed previously by Byers (1999). The fifth

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40 Figure 43. HL-93 truck configurations of the multiple-span continuous cases (skewed bridges, interior girder, negative moment). bridge analyzed was the Cooper River Bridge. All had two 2.5.1.2 Two-Level Modeling Scheme edge girders on a cross section, two pylons, and a semi-harp cable configuration with two planes of cables. Table 10 and Each structure was modeled on a "global" and a "sub- Figure 55 summarize principal dimensional differences among structure" or "local" level. The former takes into account the the bridges investigated. behavior of the bridge as a whole, while the latter focuses on The first number in the bridge designation (e.g., "8" in parts of the structure with a more detailed model and assesses "8_15") indicates the number of cables on each side of the how the bridge performs under the loads considered. tower. The second number indicates the distance from the There were four global models, one for each bridge. The centerline (CL) of the slab to the centerline of the edge girder solution obtained for the global model was used as input to in meters. the local model of the structure where a part of the bridge

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41 Figure 44. HL-93 truck configurations of the multiple-span continuous cases (right bridges, exterior girder, negative moment). between two cables was modeled in greater detail. In this were discretized into 3-D beam elements along their centroidal way the stresses in the composite cross-section could be axes. There were "rigid" beam link elements connecting the obtained, and the effective width could be computed. Only deck to the floor beams; two edge girders and no middle dead load was applied. girder existed in each model. The cable areas were such that they all provided approx- Global Models. Figure 55 shows the element types used in the imately the same vertical stiffness. The towers were consid- global models. All materials were considered linear-elastic. ered to be fixed at their bases. Each linear element (e.g., The structural elements were modeled as follows. The deck beam or tower) was located at the equivalent member's cen- was modeled as a thick plate (each element had width and troidal axis. The concept of rigid linear elements was used length not significantly higher than the thickness) at the level to ensure that members that were connected shared common of its mid-surface. The cables were modeled as truss ele- displacements. Part of a global bridge model (8_8) is shown ments. The beams (floor beams and girders) and the towers in Figure 56.

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42 Figure 45. HL-93 truck configurations of the multiple-span continuous cases (skewed bridges, exterior girder, negative moment). Local Models. Each local model represented the part of in ABAQUS notation). Concrete was used for the slab. Steel the structure lying between two adjacent cables. The floor was used for the beams. beams and the girders shared common nodes at the points where they met given that those points were connected and 2.5.1.3 Cable-Stayed Bridge Results should have had the same displacements. The cable-stayed local models had the same level of detail as the models used Results were categorized on the basis of which of three in the parametric study, except that deck rebar was neglected. regions along the bridge they were in: The slab, for example, was divided into four layers. The material properties and beam dimensions were the Type I (positive moment and low axial force regions same as those given in the description of the global models. close to the center of the main span), All the elements were 3-D eight-noded solid elements (C3D8 (text continued on p. 47)

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43 Figure 46. beff /b and bending moment versus x/L for the multiple-span continuous cases (right bridges, interior girder, positive moment, Service II).

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44 Figure 47. beff /b and bending moment versus x/L for the multiple-span continuous cases (skewed bridges, interior girder, positive moment, Service II).

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45 Figure 48. beff /b and bending moment versus x/L for the multiple-span continuous cases (right bridges, exterior girder, positive moment, Service II).

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46 Figure 49. beff /b and bending moment versus x/L for the multiple-span continuous cases (skewed bridges, exterior girder, positive moment, Service II).

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47 TABLE 8 Effective slab width ratio ( beff /b) for multiple-span continuous bridges (positive moment) Bridge Interior Girder Exterior Girder ID 0.3L1 0.35 L1 0.4 L1 0.45 L1 0.5 L1 0.3L1 0.35 L1 0.4 L1 0.45 L1 0.5 L1 CS-01 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 CS-03 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 CS-07 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 CS-09 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 CS-19 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 CS-21 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 CS-25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 CS-27 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 CS-41 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 CS-55 0.95 0.95 0.96 0.96 0.96 1.00 1.00 1.00 1.00 1.00 CS-57 0.96 0.95 0.94 0.94 0.94 1.00 1.00 1.00 1.00 1.00 CS-61 0.85 0.92 0.98 0.95 0.92 1.00 1.00 1.00 1.00 1.00 CS-63 0.99 0.99 0.99 0.99 1.00 1.00 1.00 1.00 1.00 1.00 CS-73 0.86 0.89 0.92 0.95 0.97 1.00 1.00 1.00 1.00 1.00 CS-75 0.87 0.81 0.74 0.72 0.70 1.00 1.00 1.00 1.00 1.00 CS-79 0.94 0.95 0.96 0.95 0.98 1.00 1.00 1.00 1.00 1.00 CS-81 0.97 0.99 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 Type II (positive moment and high axial force regions width ratio along the length of the main span in the Cooper close to the support of the main span), and River Bridge. Results from Cooper River were similar to those Type III (negative moment and high axial force regions obtained from the other four bridge models. very close to the tower). Figure 57 shows the transverse distribution of normal slab 2.5.2 Validation Cases stresses in a Type I region, while Figures 58 and 59 show slab stresses in Type II and Type III regions, respectively. These The remaining bridges analyzed included the following in were for the 8_15 and 12_15 bridges. Analogous distributions addition to steel multi-girder bridges with geometric parame- were also observed for the Cooper River Bridge, as shown in ters beyond those of the parametric study presented earlier: Figures 60 and 61. Some shear lag was evident in these figures, but substantial effective widths were realized anyway-- Two-girder continuous steel girder bridges with both considerably beyond the 4.8-m girder spacing maximum cast-in-place and prestressed deck slabs and very wide investigated in the parametric study presented earlier. (7.68-m) girder spacing, Table 11 summarizes the values of effective width extracted A continuous hybrid steel girder bridge, from all three regions of all five cable-stayed bridges ana- Simply-supported and continuous tub-girder bridges, and lyzed. As in the main parametric study, the short wide bridge Simply-supported and continuous prestressed bulb-tee (Bridge 8_15) had the smallest effective width. girder bridges. Figures 62 through 72 show the variation of effective width ratio beff /b along the length in representative regions of the Full effective slab width was obtained for all these cases. first four bridges. Figure 73 shows the variation of effective Further details on the validation cases appear in Appendix J.

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48 Figure 50. beff /b and bending moment versus x/L for the multiple-span continuous cases (Span 2 loading, interior girder, positive moment, Service II).

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49 Figure 51. beff /b and bending moment versus x/L for the multiple-span continuous cases (right bridges, interior girder, negative moment, Service II).

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50 Figure 52. beff /b and bending moment versus x/L for the multiple-span continuous cases (skewed bridges, interior girder, negative moment, Service II).

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51 Figure 53. beff /b and bending moment versus x/L for the multiple-span continuous cases (right bridges, exterior girder, negative moment, Service II).

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52 Figure 54. beff /b and bending moment versus x/L for the multiple-span continuous cases (skewed bridges, exterior girder, negative moment, Service II).

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53 TABLE 9 Effective slab width ratio ( beff /b) for multiple-span continuous bridges (negative moment) Bridge Interior Girder Exterior Girder ID 0.9L1 0.95 L1 1.0 L1 0.05 L2 0.1 L2 0.9L1 0.95 L1 1.0 L1 0.05 L2 0.1 L2 CS-01 0.90 1.00 1.00 1.00 0.80 1.00 1.00 1.00 1.00 1.00 CS-03 0.92 0.92 0.92 0.92 0.92 0.87 0.87 0.87 0.87 0.87 CS-07 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 CS-09 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 CS-19 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 CS-21 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 CS-25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 CS-27 1.00 1.00 1.00 1.00 0.95 1.00 1.00 1.00 1.00 0.95 CS-41 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 CS-55 0.80 0.90 1.00 0.80 0.75 0.90 0.95 1.00 0.95 0.90 CS-57 0.75 0.75 0.75 0.75 0.75 0.93 0.97 1.00 1.00 0.90 CS-61 0.80 0.90 1.00 1.00 0.91 0.96 0.98 1.00 1.00 0.90 CS-63 0.70 0.73 0.70 0.73 0.73 0.80 0.90 1.00 1.00 1.00 CS-73 0.80 0.90 1.00 0.90 0.82 1.00 1.00 1.00 1.00 1.00 CS-75 0.85 0.85 0.85 0.85 0.85 1.00 1.00 1.00 0.90 0.80 CS-79 0.94 0.95 0.96 0.95 0.98 1.00 1.00 1.00 1.00 1.00 CS-81 0.97 0.99 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 TABLE 10 Cable-stayed bridges investigated Bridge ID Overall Main Edge Girder Deck Slab Length Span Spacing Thickness (m) (m) (m) (mm) 8_8 495 255 16 250 8_15 735 375 30 250 12_8 495 255 16 250 12_15 735 375 30 250 Cooper River 867 471 38.4 240 Figure 55. Side views of Bridges 12_8 & 12_15 (total length = 735 m) and Bridges 8_8 & 8_15 (total length = 495 m).

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54 Figure 56. Elements composing the global model; close-up of the area near the cable. Figure 57. Transverse distribution of normal stresses in the middle of Bridge 12_15 (3-D plot).

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55 Figure 58. Transverse distribution of normal stresses in the positive moment region close to the support of Bridge 8_15 (3-D plot). Figure 59. Transverse distribution of normal stresses in the negative moment region close to the support of Bridge 12_15 (3-D plot).

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56 Figure 60. Transverse distribution of normal stresses in the positive moment region close to the support of the Cooper River Bridge (3-D plot). Figure 61. Transverse distribution of normal stresses in the negative moment region close to the support of the Cooper River Bridge (3-D plot). TABLE 11 Cable-stayed effective width FEM results Bridge ID Girder Spacing beff/b S (m) Type I Region Type II Region Type III Region 8_8 16 1.0 0.95 0.82 8_15 30 1.0 0.95 0.68 12_8 16 1.0 0.99 0.90 12_15 30 1.0 1.0 0.80 Cooper River 38.4 0.99 0.99 0.83

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57 Figure 62. Longitudinal distribution of the normalized Figure 66. Longitudinal distribution of the normalized effective width for Bridge 8_8 (Region I). effective width for Bridge 12_8 (Region II). Figure 63. Longitudinal distribution of the normalized Figure 67. Longitudinal distribution of the normalized effective width for Bridge 8_8 (Region II). effective width for Bridge 12_8 (Region III). Figure 64. Longitudinal distribution of the normalized Figure 68. Longitudinal distribution of the normalized effective width for Bridge 8_8 (Region III). effective width for Bridge 8_15 (Region I). Figure 65. Longitudinal distribution of the normalized Figure 69. Longitudinal distribution of the normalized effective width for Bridge 12_8 (Region I). effective width for Bridge 8_15 (Region II).

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58 Figure 70. Longitudinal distribution of the normalized Figure 71. Longitudinal distribution of the normalized effective width for Bridge 8_15 (Region III). effective width for Bridge 12_15 (Region I). Figure 72. Longitudinal distribution of the normalized effective width for Bridge 12_15 (Region II). Figure 73. Longitudinal distribution of the normalized effective width for the main span of the Cooper River Bridge.