Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges (2008)

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31 This chapter summarizes the work performed for global response analysis and the results and conclusions obtained from this study. Additional detailed results are presented in Appendix E. Objective The global behavior of a curved bridge can be distinctly different from a straight bridge. The curvature results in off-center placement of loads and, subsequently, such loads induce torsion into the superstructure. The torsion, in turn, causes the shear stresses on the outside of the curve to increase. Also, the curved geometry of the bridge will result in development of transverse moments, which can increase the normal stresses on the outside edges of the bridge and can result in higher tension and/or compression stresses. Post-tensioned bridges also have an additional equivalent transverse load, which can result in significant tension on the inside of the curve and compression on the outside edge. The magnitudes of such effects depend on the radius of curvature, span configuration, cross-sectional geome- try, and load patterns among other parameters. The struc- tural analysis required to capture such effects, in most cases, is beyond the scope of day-to-day normal bridge de- sign activities. The objective of the global analysis study in this research was to quantify the effect of increased shear force and nor- mal stresses in the cross section, identify common trends, and find approximate modeling methods to obtain accurate results for design purposes. In particular, shear forces and normal stresses due to dead loads, live loads, and post- tensioning were studied to obtain analysis modeling limitations and develop empirical adjustment factors for simplified analysis. A set of special studies was also per- formed to review the effect of diaphragms, pier connection (bearing versus monolithic), and skewed abutments on curved bridges. Model Verification 3-D finite element analysis using plate and shell elements is accepted as the most accurate level of analysis available for box- girder bridges. However, the magnitude of analysis cases desir- able for parametric studies in this project was such that a more simplified model was desirable. Given that the parametric stud- ies are based on bridges with radial supports, available guide- lines for grillage analysis were used. In order to make sure that the grillage and finite element models produced similar results, a detailed model of a three-span bridge on a tight curve (400-ft radius) was created and results were compared for various load effects. The models are shown in Figures 4-1 and 4-2. The re- sults for superstructure dead load and a concentrated midspan load obtained from the two models (grillage and finite element) for a two-cell box are compared in Table 4-1. These results in- dicate a very close comparison. As a result, grillage models were used throughout the rest of the study as the basis of comparisons and such results are deemed accurate for all practical purposes. Guidelines for performing an analysis with a grillage analogy are included in Appendix C. These guidelines may be used for de- sign purposes when the bridge configuration requires it. Parameter Studies Analysis Cases To study the effect of various bridge parameters on the re- sponse of curved bridges, a parametric study was performed which focused on the variation of span configuration and length, bridge cross-section geometry, and loading. Four bridge cross-section shapes were considered: • Single-cell box based on a typical cast-in-place cross section, • Single-cell box based on a typical precast cross section, • Two-cell box based on a typical cast-in-place cross section, and • Five-cell box based on a typical cast-in-place cross section. C H A P T E R 4 Global Response Analysis Studies

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

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

calculated by multiplying the values shown here by the average width of each element, except that “Izz” is multiplied by L3. The bent cap and abutment diaphragms were assumed to be relatively rigid. Cap and column element properties are shown in Table 4-4. Loads Each bridge configuration was subjected to dead load (self weight), live load, and post-tensioning loads. A con- centrated load of 100 kips was used to simulate the live load. 34 Figure 4-5. Typical single-cell precast cross section. Figure 4-6. Idealized single-cell precast cross section. Figure 4-7. Typical two-cell cast-in-place cross section.

35 Figure 4-8. Idealized two-cell cast-in-place cross section. Figure 4-9. Typical five-cell cast-in-place cross section. Figure 4-10. Idealized five-cell cast-in-place cross section. This loading captures the effect of concentrated axle loads and may magnify its effect on curved bridges to some extent; therefore, it is justified as a simplification for a conservative upper bound. This load was applied at the middle of the bridge and transversely was located at various positions: (1) on outside web, (2) on inside web, (3) on center of bridge, and (4) on all webs, i.e., equally distributed over the bridge width. Maximum stresses occur when the entire bridge width is loaded, therefore, the results from this case were studied in more detail when developing guidelines for design. Post-tensioning was also applied along all webs of the section. The prestress effects are modeled as equivalent loads at nodes, i.e., the axial forces along the prestress path are applied at each end of each element which in effect is the same as modeling the prestress tendon as a series of straight lines. In case of grillage models, additional load cases were studied by loading only the inside and only the outside webs. Results Review The following results were obtained for each load case and compared from spine and grillage models: • Midspan Deflection at middle of center span; • Midspan Rotation at middle of center span; • Midspan Longitudinal Bending Moment at middle of center span;

• Midspan Transverse Bending Moment at middle of center span; • Midspan End Shear at first abutment in single span and at start of middle span in 3-span case; • Midspan Normal Stress at bottom outside corner of cross section, at middle of center span; and • Midspan Normal Stress at bottom inside corner of cross section, at middle of center span. The graphical review of results included scatter-grams of each response quantity from spine and grillage models. An example of such graphs is shown in Figure 4-13. The ratio of stresses and shear forces from the spine model to those of the grillage model were also obtained to review the effect of radius of curvature and modeling technique. An example of such graphs is shown in Fig- ures 4-14a and 4-14b. Summary of Results Numerous scattergrams were plotted for each of the re- sults listed in the previous section and bridge types shown in Figures 4-4 through 4-10. Figure 4-14 shows the plots with the most divergent results (i.e., with values furthest away from 1.00) between the spine and grillage models. As a result, the outside web shear force and longitudinal stress are the main effects that need attention in design. To better quantify these results and the effects of curvature on various bridge types, ratios of results for various bridge models were combined and plotted as shown in Figures 4-15 through 4-18. Average 36 200 ft. Single Span 100 ft. Single Span 200/300/200 ft. Multi-Span 150/200/150 ft. Multi-Span 75/100/75 ft. Multi-Span CG (ft) 4.53 2.24 5.46 4.06 2.01 Area (ft2) 81.73 66.73 87.73 78.73 65.23 Avy (ft2) 56.60 56.60 56.60 56.60 56.60 Avz (ft2) 30.00 15.00 36.00 27.00 13.50 Iyy (ft4) 1,326.25 256.23 2,028.29 1,037.00 197.80 Izz (ft4) 9,544.20 7,488.61 10,366.44 9,133.08 7283.05 Jxx (ft4) 1,559.82 365.08 2,213.30 1,267.24 286.74 Depth (ft) 10.0 5.0 12.0 9.0 4.5 Table 4-2. Section properties for two-cell curved line (spine) bridge model. EXTERIOR GIRDER 200 ft. Single Span 100 ft. Single Span 200/300/200 ft. Multi-Span 150/200/150 ft. Multi-Span 75/100/75 ft. Multi-Span CG (ft) 4.23 2.06 5.13 3.78 1.85 Area (ft2) 25.03 20.03 27.03 24.03 19.53 Avy (ft2) 18.87 18.87 18.87 18.87 18.87 Avz (ft2) 10.00 5.00 12.00 9.00 4.50 Iyy (ft4) 386.50 73.28 594.75 301.19 56.45 Izz (ft4) 216.76 208.19 219.34 215.31 207.11 Jxx (ft4) 519.94 121.69 737.77 422.41 95.58 INTERIOR GIRDER 200 ft. Single Span 100 ft. Single Span 200/300/200 ft. Multi-Span 150/200/150 ft. Multi-Span 75/100/75 ft. Multi-Span CG (ft) 5 2.50 6.00 4.50 2.25 Area (ft2) 31.67 26.67 33.67 30.67 26.17 Avy (ft2) 18.87 18.87 18.87 18.87 18.87 Avz (ft2) 10.00 5.00 12.00 9.00 4.50 Iyy (ft4) 541.74 106.59 823.13 425.04 82.44 Izz (ft4) 399.43 399.02 399.60 399.35 398.97 Jxx (ft4) 519.94 121.69 737.77 422.41 95.58 Transverse Element 200 ft. Single Span 100 ft. Single Span 200/300/200 ft. Multi-Span 150/200/150 ft. Multi-Span 75/100/75 ft. Multi-Span CG (ft) 5 2.50 6.00 4.50 2.25 Area (ft2) 1.63 1.63 1.63 1.63 1.63 Avy (ft2) 0.01 0.01 0.01 0.01 0.01 Avz (ft2) 1.63 1.63 1.63 1.63 1.63 Iyy (ft4) 34.38 7.21 50.94 27.32 5.61 Izz (ft4) 0.14 0.14 0.14 0.14 0.14 Jxx (ft4) 68.58 14.25 101.69 54.47 11.05 Table 4-3. Section properties for two-cell grillage bridge model.

ratios and standard deviations were also calculated for each group of bridges with same cross-section type and loading. These results are the primary source of the recommenda- tions for design at the end of this study. Figures 4-15 and 4-17 show the ratios of Curve to Straight Bridge spine models. These figures reveal the limit of radius of curvature beyond which the curvature effects may be ignored altogether and the bridge may be analyzed as if it were straight. The ratios of “Spine to Grillage Model” shown in Figures 4-16 and 4-18 reveal if a curve spine model can be used in cases of tighter curvatures and the limit of this type of model to obtain accurate results for design purposes. In the case of the spine models, shear forces were obtained by considering the shear flow from torsion in addition to the vertical shear. The stresses were also obtained from the combined effect of axial force and longitudinal and transverse moments. Figures 4-15 through 4-19 are for the four bridge cross sec- tions (cast-in-place single-cell (CIP1), precast single-cell (PC1), cast-in-place two-cell (CIP2), and cast-in-place five- cell (CIP5)) with different pier configurations and span lengths. Therefore, points designated as PC1_6x6_sp3m_dl are for a single cell precast bridge (PC1) with a 6 x 6 pier (6x6), 3-span configuration of medium length spans (sp3m) and dead load response (dl). Detailed results from the parameter studies are presented in Appendix E. Conclusions of the Parametric Study Study of the above results led to the following conclusions: • Bridges with L/R less than 0.2 may be designed as if they were straight. Figures 4-15 and 4-17 reveal results are within 4% of a curved spine model when this is done. • Bridges with L/R less than 0.8 may be modeled with a single-girder spine model using a curved (spine) model and lateral effects shall be included in the analysis. Figure 4-18 shows that longitudinal stresses will be 37 Figure 4-11. Typical curved line (spine beam) bridge model (showing 3-span unit). Cap Section 6' x 18' Column 8' x 8' Column 7' x 7' Column 6' x 6' Column Area (ft2) 108 108 64 49 36 Avy (ft2) 108 108 64 49 36 Avz (ft2) 108 108 64 49 36 Iyy (ft4) 5000 324 341.33 200.08 108 Izz( ft4) 5000 2916 341.33 200.08 108 Jxx( ft4) 5000 1296 1365.33 800.33 432 Table 4-4. Section properties for cap and column elements. Obtuse (Element 1) Acute (Element 21) Straight Value Ratio Value Ratio Radial -389.35 1 -387.84 1 Skew-Left -605 1.55387184 -190.65 0.49156869 Skew-Both -392.94 1.0092205 -342.23 0.88239996 Obtuse (Element 1) Acute 400’ Radius Value Ratio Value Ratio Radial -577.89 1 -189.78 1 Skew-Left -782.91 1.3547734 -58.279 0.30708715 Skew-Both -614.14 1.0627282 -183.66 0.96775213 Table 4-5. Dead load shear results for 200 ft/single span skew.

• Bearing forces (i.e., shear at the abutments) obtained from a spine model must consider the effect of torsion. Bearings must be designed considering the curvature effect. Special Studies In addition to the above parametric study, special studies were performed to get a better understanding of the effect of diaphragms, bearing conditions, skewed abutments, and long-term creep when combined with curved geometry. Diaphragm All 5-cell grillage bridge models used in the parametric study were also modified to have a stiff diaphragm in the center of each span. The results from each model with and without diaphragms were compared for dead load, live load, and post-tensioning. These results were compared using scatter-grams and ratios (line diagrams) similar to the results shown in Figures 4-19 and 4-20. The overall conclusion from these results is that interior diaphragms have minimal effect on the shear and stress responses and therefore may be elim- inated altogether. To verify the conclusions relative to interior diaphragms, the two-cell finite element model used in the model verifica- tion studies was modified to include interior diaphragms. These diaphragms were placed at the center of one of the end spans and at one of the third points in the center span. These diaphragms were located on one side of the midpoint of the bridge, which was otherwise symmetrical. Results were compared on both sides of the bridge and found to be nearly 38 Two-Cell CIP Dead Load: Midspan Moment -70000 -60000 -50000 -40000 -30000 -20000 -10000 0 -70000 -60000 -50000 -40000 -30000 -20000 -10000 0 Grid Model Si ng le L in e M od el Figure 4-13. Scatter-gram comparison of results from spine and grillage models. Figure 4-12. Typical curved grillage bridge model (showing 3-span unit with 3 webs). unconservative by less than 4% up to this limit unless the bridge has a low span length-to-width ratio (i.e., short- span 5-cell sections) • Bridges with L/R larger than 0.8 shall be analyzed with more detailed analysis models such as grillage or finite ele- ment models. Figures 4-16 and 4-18 reveal that spine beam analysis will generally become increasingly unconservative beyond the 0.8 L/R limit. • Curved bridges with length-to-width ratios of less than 0.2 and an L/R larger than 0.2 also require detailed analysis as revealed by the unconservative results for tightly curved short-span 5-cell bridges in Figures 4-16 and 4-18.

39 All 3-Span Vr/Vs Dead Load 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 0 0.5 1 1.5 2 Length/Radius (L/R) Vr /V s CIP1_6x18_sp3l_dl CIP1_6x18_sp3m_dl CIP1_6x18_sp3s_dl CIP1_6x6_sp3l_dl CIP1_6x6_sp3m_dl CIP1_6x6_sp3s_dl PC1_6x18_sp3l_dl PC1_6x18_sp3m_dl PC1_6x18_sp3s_dl PC1_6x6_sp3l_dl PC1_6x6_sp3m_dl PC1_6x6_sp3s_dl CIP2_6x18_sp3l_dl CIP2_6x18_sp3m_dl CIP2_6x18_sp3s_dl CIP2_7x7_sp3l_dl CIP2_7x7_sp3m_dl CIP2_7x7_sp3s_dl CIP5_6x18_sp3l_dl CIP5_6x18_sp3m_dl CIP5_6x18_sp3s_dl CIP5_8x8_sp3l_dl CIP5_8x8_sp3m_dl CIP5_8x8_sp3s_dl Figure 4-15. Ratio of outside web dead load shear forces in curved (Vr) to straight (Vs) bridges where length equals middle span length. 0.9 0.905 0.91 0.915 0.92 0.925 0.93 0.935 0.94 0.945 Cu rv e/ G rid 1.02 1.04 1.06 1.08 1.1 1.12 1.14 Cu rv e/ G rid Radius (ft) Midspan Outside Corner Stress (ksi) 200 400 600 800 1000 Straight Radius (ft) 200 400 600 800 1000 Straight Two-Cell CIP Dead Load: Midspan Outside Shear (kips) a) Midspan Outside Girder Shear Ratio b) Longitudinal Stress Ratio Figure 4-14. Line graphs of shear and stress ratios of results from spine and grillage models. symmetrical. Therefore, the diaphragms were shown to have very little effect on the global response of the bridge. Bearings at the Bents The purpose of this study was to determine if bridges with integral and non-integral bents respond differently to curve effects. All three-span five-cell spine and grillage bridge mod- els used in the parametric study were also modified to have a point bearing support at the piers; i.e., free to move in trans- verse or longitudinal directions. The results were studied using scatter-grams and ratio (line) diagrams comparing the results from spine to grillage models. It was found that, in general, magnitudes of curve to straight results are in the same order as the integral bridges. Therefore, the final con- clusion is that, as long as the support conditions are modeled correctly, the same guidelines for modeling limitations are equally applicable to integral and non-integral conditions. Skewed Abutments A two-cell single-span (200 ft long) and a two-cell three-span bridge (200ft-300ft-200ft) were modified to have 30-degree skew support at the left support and another case with 30-degree

40 All 3-Span fr/fsDead Load 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 0 0.2 0.4 0.6 0.8 1.61.41.21 Length/Radius (L/R) fr/ fs CIP1_6x18_sp3l_dl CIP1_6x18_sp3m_dl CIP1_6x18_sp3s_dl CIP1_6x6_sp3l_dl CIP1_6x6_sp3m_dl CIP1_6x6_sp3s_dl PC1_6x18_sp3l_dl PC1_6x18_sp3m_dl PC1_6x18_sp3s_dl PC1_6x6_sp3l_dl PC1_6x6_sp3m_dl PC1_6x6_sp3s_dl CIP2_6x18_sp3l_dl CIP2_6x18_sp3m_dl CIP2_6x18_sp3s_dl CIP2_7x7_sp3l_dl CIP2_7x7_sp3m_dl CIP2_7x7_sp3s_dl CIP5_6x18_sp3l_dl CIP5_6x18_sp3m_dl CIP5_6x18_sp3s_dl CIP5_8x8_sp3l_dl CIP5_8x8_sp3m_dl CIP5_8x8_sp3s_dl Figure 4-17. Ratio of outside corner dead load longitudinal stress in curved to straight bridge. 0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 Vl in e/ Vg rid CIP1_6x18_sp3l_dl CIP1_6x18_sp3m_dl CIP1_6x18_sp3s_dl CIP1_6x6_sp3l_dl CIP1_6x6_sp3m_dl CIP1_6x6_sp3s_dl PC1_6x18_sp3l_dl PC1_6x18_sp3m_dl PC1_6x18_sp3s_dl PC1_6x6_sp3l_dl PC1_6x6_sp3m_dl PC1_6x6_sp3s_dl CIP2_6x18_sp3l_dl CIP2_6x18_sp3m_dl CIP2_6x18_sp3s_dl CIP2_7x7_sp3l_dl CIP2_7x7_sp3m_dl CIP2_7x7_sp3s_dl CIP5_6x18_sp3l_dl CIP5_6x18_sp3m_dl CIP5_6x18_sp3s_dl CIP5_8x8_sp3l_dl CIP5_8x8_sp3m_dl CIP5_8x8_sp3s_dl All 3-Span Vline/Vgrid Dead Load Total 0 0.5 1 1.5 2 Length/Radius (L/R) Figure 4-16. Ratio of outside web dead load shear forces from spine to grillage model.

41 All 3-Span fline/fgrid Dead Load 0.94 0.95 0.96 0.97 0.98 0.99 1 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 Length/Radius (L/R) fli ne /fg rid CIP1_6x18_sp3l_dl CIP1_6x18_sp3m_dl CIP1_6x18_sp3s_dl CIP1_6x6_sp3l_dl CIP1_6x6_sp3m_dl CIP1_6x6_sp3s_dl PC1_6x18_sp3l_dl PC1_6x18_sp3m_dl PC1_6x18_sp3s_dl PC1_6x6_sp3l_dl PC1_6x6_sp3m_dl PC1_6x6_sp3s_dl CIP2_6x18_sp3l_dl CIP2_6x18_sp3m_dl CIP2_6x18_sp3s_dl CIP2_7x7_sp3l_dl CIP2_7x7_sp3m_dl CIP2_7x7_sp3s_dl CIP5_6x18_sp3l_dl CIP5_6x18_sp3m_dl CIP5_6x18_sp3s_dl CIP5_8x8_sp3l_dl CIP5_8x8_sp3m_dl CIP5_8x8_sp3s_dl Figure 4-18. Ratio of outside corner dead load longitudinal stress from spine to grillage model. 0.997 0.999 0.9985 0.998 V_Out Sp3_L Sp1_M Sp3_M Sp1_S Sp3_S 0.9975 0.9995 1 1.0005 W ith /W ith ou t D ia ph ra gm Radius (ft) 200 400 600 800 1000 Straight Midspan Outside Shear (kips) Figure 4-19. Scatter-gram comparison of live load results from grillage models with and without interior diaphragm. 0.994 0.998 0.997 0.996 St_O Sp3_L Sp1_M Sp3_M Sp1_S Sp3_S 0.995 0.999 1 1.003 1.002 1.001 W ith /W ith ou t D ia ph ra gm Radius (ft) 200 400 600 800 1000 Straight Midspan Outside Corner Stress (ksi) Figure 4-20. Line graph of ratio of results from grillage models with and without interior diaphragm. skew support at both ends, see Figure 4-21. The abutments were supported on rollers. The obtuse and acute abutment shear values were compared in straight and skewed bridges with and without curved alignment. The simple span bridge results were compared for dead load and live load and the three span results were compared for dead load responses. The results are shown in Tables 4-5 through 4-7. The final outcome of these results is that the curved alignment does not aggravate the effect of skewed abutments and therefore, any consideration taken for straight bridges can be equally valid for curved bridges. These corrections will be necessary when the bridge is designed using a spine beam analysis. Skew can be automatically accounted for in a grillage analysis approach as shown in Figure 4-21. Long-Term Creep The effect of the time-dependant properties of concrete (principally creep) on the response of curved bridges was investigated using the LARSA 4D computer program, which

42 Obtuse (Element 1) Acute Straight Value Ratio Value Ratio Radial -297.35 1 -296.59 1 Skew-Left -378.79 1.27388599 -197.35 0.66539668 Skew-Both -378.02 1.27129645 -197.66 0.66644189 Obtuse (Element 1) Acute 400’ Radius Value Ratio Value Ratio Radial -370 1 -202.2 1 Skew-Left -452.84 1.22389189 -135.81 0.67166172 Skew-Both -453.89 1.22672973 -135.81 0.67166172 Table 4-7. Dead load shear results for 400 ft radius, 200/300/200 ft multi-span skew. Skew at One Abutment Only Skew at Both Abutments Figure 4-21. Skew configurations studied. Obtuse (Element 1) Acute Straight Value Ratio Value Ratio Radial -577.89 1 -189.78 1 Skew-Left -782.91 1.3547734 -58.279 0.30708715 Skew-Both -614.14 1.0627282 -183.66 0.96775213 Obtuse (Element 1) Acute 400’ Radius Value Ratio Value Ratio Radial -81.618 1 -12.007 1 Skew-Left -116.15 1.42309295 12.526 -1.0432248 Skew-Both -85.079 1.04240486 -9.1316 0.76052303 Table 4-6. Live load shear results for 200 ft/single span skew. can consider both the 3-D geometry of the bridge plus the time-dependant behavior of concrete. The same structure used in the comprehensive example problem was analyzed over a period of 10 years. The bridge was modeled as a 3-D spine beam. In addition, the model was modified by chang- ing the curve radius in two cases and changing the length of the end span in another case. Therefore, four models were evaluated. In all cases the abutments were fixed against tor- sion. The end span and radii of these bridges are shown in Table 4-8 The long-term deflection of Models 1 through 3 did not appear to be affected by the radius of the bridge. Therefore, methods used for adjusting cambers for straight bridges would appear to be applicable to curved bridges analyzed as three-dimensional spine beams. The major concern, however, was abutment bearing re- actions. These will change over time. This is manifested by the change in the torsion reaction at the abutment, which in turn will affect the bearing reactions. Table 4-9 summa- rizes the dead load and prestress results from the four models investigated. The following conclusions can be drawn from this limited study: • The torsion reaction and thus the relative bearing forces in continuous curved concrete box-girder bridges vary over time and depend on both the radius of the curve and the relative length of the end span with respect to the cen- ter span. The forces in outside bearings will increase and inside bearings will decrease. Given the many possible bridge-framing configurations, it is difficult to make an accurate assessment of time-dependent bearing forces. • The above study assumed bridges constructed on false- work. Segmentally constructed bridges are expected to behave differently. • The LARSA 4D program does not consider torsion creep, as is the case with most other commercially available software. Torsion creep is expected to mitigate long-term changes in bearing forces, as would modeling the flexibil- ity of bearing systems subjected to torsion loads from the superstructure. Given these conclusions, it would appear to be safe to an- alyze curved concrete box-girder bridges using commercially available software, particularly for segmentally constructed bridges. It is recommended that the vertical flexibility of bear- ings be considered in these analyses.

In the absence of such an analysis, it is recommended that dead load torsion reactions at the abutment from a 3-D spine beam analysis be increased by approximately 20% for the final condition and bearings or bearing systems be designed to envelope both the initial and final condi- tions. This is a crude recommendation, but given the miti- gating factors not considered in this limited study, it should provide a reasonable hedge against bearing failure. In the case of a grillage analysis, the same adjustment can be made by resolving bearing reactions into a torsional moment, in- creasing that moment by 20%, and recalculating the new bearing forces. 43 Model Number Radii (ft) End Span Length (ft) 1 400 200 2 800 200 3 1600 200 4 400 140 Table 4-8. Models of two-cell bridge used to study the effect of creep. Model Number Torsion Moment 7 Days (ft-kips) Torsion Moment 3600 Days (ft-kips) % Difference 1 -8826 -10034 13.7 2 -7357 -9262 25.9 3 -6395 -8500 32.9 4 -4012 -5842 45.6 Table 4-9. Time dependence of abutment torsion moment.