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From page 39... ... 39 4.1 Redundancy of Bridge Systems under Lateral Load Traditionally, structural design codes have defined a structure's capacity in terms of the ability of its individual members to sustain the applied loads using a linear-elastic analysis. Given that the failures of individual members do not necessarily lead to the collapse of the system, structural redundancy is defined as the ability of a structural system to continue to carry load after one critical member reaches its load carrying capacity. Read the entire page →
From page 40... ... 40 the confinement ratio of the column being evaluated to the average confinement ratios observed in typical confined and unconfined columns. This chapter summarizes the results of the analyses conducted during this project and those extracted from NCHRP Report 458. Read the entire page →
From page 41... ... 41 Figure 4.1. The 3-D, profile, and elevation views of the twin box-girder bridge. Read the entire page →
From page 42... ... 42 Bridge Columns The bridge columns are the most important contributors for the resistance of the bridge to lateral load. The lateral load is applied on the top of the bents, in particular on the middle point of the cap beams. Read the entire page →
From page 43... ... 43 Figure 4.5. Column M-phi curve for different values of axial load. Read the entire page →
From page 44... ... 44 bearings are assumed to allow for longitudinal expansion and for transverse displacement. The same stiffness values are assumed for the bearings in the longitudinal and lateral directions. Read the entire page →
From page 45... ... 45 linear system represented by the force Pu and the maximum displacement at failure du. The displacement d1 is the maximum displacement if the pushover analysis is performed on a single column. Read the entire page →
From page 46... ... 46 Figure 4.7. Lateral capacity by Equation 4.2 vs. Read the entire page →
From page 47... ... 47 Table 4.6. Results summary of multi-cell box-girder bridge with two-column bent. Read the entire page →
From page 48... ... 48 The validation of Equation 4.2 for the ultimate capacity of the system is verified in Figure 4.10, which plots Pu from Equation 4.2 versus the one obtained from the SAP2000 analysis. Figure 4.10 shows that the data points are almost aligned along the equal force line, indicating that Equation 4.2 is reasonably accurate with a regression coefficient R2 = 0.86. Read the entire page →
From page 49... ... 49 tudinal girders to the center of the cap beams through the bearings. Figure 4.13 shows cross sections of the column, cap beam, and girder. Read the entire page →
From page 50... ... 50 properties affect the response of the bridge system. Specifically, the sensitivity analysis performed in this section describes the effect of changes in the following parameters: (1) Read the entire page →
From page 51... ... Base Case 36 331 2.41 761 1060 0.00096 7.5 7.8 32.5-ft 36 391 2.41 719 1044 0.00096 9.6 9.8 37.5-ft 36 451 2.41 706 1047 0.00096 11.5 11.9 Pile Foundation 36 331 2.41 784 1095 0.00096 8.3 8.1 Spread Foundation 36 331 2.41 794 1071 0.00096 7.7 8.1 Base Pinned 36 331 2.41 569 839 0.00096 10.6 10.5 Longitudinal Load 36 331 2.41 619 922 0.00096 12.6 7.8 Integral Top/Fixed Base 36 331 2.41 768 1063 0.00096 7.5 7.8 Integral Top/Pinned Base 36 331 2.41 564 834 0.00096 10.4 10.4 Integral Longitudinal 36 331 2.41 752 1096 0.00096 7.8 7.8 Column Top Pinned 36 331 2.41 579 831 0.00096 10.4 10.4 50% Phi 36 331 2.41 761 929 0.00048 5.0 5.1 75% Phi 36 331 2.41 761 1011 0.00072 6.4 6.4 Category B (column lateral confinement ratio=0.3%) New data using M-phi curve from multi-girder bridge in Category B, Category B below is different from the above one Category B Diameter (in.) Read the entire page →
From page 52... ... 52 Figure 4.14 shows that the data points are clearly aligned along the equal force line indicating that Equation 4.2 is reasonably accurate with a regression coefficient R2 = 0.99. The COV of the ratio between the value from Equation 4.2 and the SAP2000 is less than 9%. Read the entire page →
From page 53... ... 53 Table 4.8. Parameters for two-column bent. Read the entire page →
From page 54... ... 54 Table 4.12. Results summary of two-column bent in NCHRP Report 458. Read the entire page →
From page 55... ... 55 bent, the results show the ultimate capacity Pu assuming that the columns are confined and also assuming that the columns are unconfined. These results are compared to those obtained when the first column reaches its limiting capacity, assuming linear behavior. Read the entire page →
From page 56... ... 56 Table 4.13. Results summary of four-column bent in NCHRP Report 458. Read the entire page →
From page 57... ... 57 Piles_Stiff 19.7 ×19.7 255.9 1.85 535 597 597 3.96E-04 1.692E-03 Spread_Normal 59 ×59 255.9 1.85 10340 12261 15674 2.86E-04 1.211E-03 Spread_Stiff 59 ×59 255.9 1.85 10710 13034 15850 2.86E-04 1.211E-03 Extension_Soft 59 ×59 255.9 1.85 8166 8735 10110 2.86E-04 1.211E-03 Extension_Normal 59 ×59 255.9 1.85 8307 9452 12110 2.86E-04 1.211E-03 Extension_Stiff 59 ×59 255.9 1.85 8813 10505 14496 2.86E-04 1.211E-03 Piles_Soft 59 ×59 255.9 1.85 9402 10662 14219 2.86E-04 1.211E-03 Piles_Normal 59 ×59 255.9 1.85 10793 13248 16508 2.86E-04 1.211E-03 Piles_Stiff 59 ×59 255.9 1.85 12242 15286 16541 2.86E-04 1.211E-03 Spread_Normal 39.4 ×39.4 255.9 0.60 1739 2484 2529 4.46E-04 1.981E-03 Spread_Stiff 39.4 ×39.4 255.9 0.60 1795 2541 2542 4.46E-04 1.981E-03 Extension_Soft 39.4 ×39.4 255.9 0.60 1189 1527 1846 4.46E-04 1.981E-03 Extension_Normal 39.4 ×39.4 255.9 0.60 1305 1797 2347 4.46E-04 1.981E-03 Extension_Stiff 39.4 ×39.4 255.9 0.60 1417 2118 2463 4.46E-04 1.981E-03 Piles_Soft 39.4 ×39.4 255.9 0.60 1277 1727 2285 4.46E-04 1.981E-03 Piles_Normal 39.4 ×39.4 255.9 0.60 1476 2280 2488 4.46E-04 1.981E-03 Piles_Stiff 39.4 ×39.4 255.9 0.60 1628 2527 2536 4.46E-04 1.981E-03 Spread_Normal 39.4 ×39.4 255.9 3.10 5791 6595 7107 3.19E-04 1.344E-03 Spread_Stiff 39.4 ×39.4 255.9 3.10 5988 6749 7131 3.19E-04 1.344E-03 Extension_Soft 39.4 ×39.4 255.9 3.10 3652 3864 4190 3.19E-04 1.344E-03 Extension_Normal 39.4 ×39.4 255.9 3.10 3792 4206 4827 3.19E-04 1.344E-03 Extension_Stiff 39.4 ×39.4 255.9 3.10 4023 4600 5575 3.19E-04 1.344E-03 Piles_Soft 39.4 ×39.4 255.9 3.10 3809 4126 4667 3.19E-04 1.344E-03 Piles_Normal 39.4 ×39.4 255.9 3.10 4241 4837 5980 3.19E-04 1.344E-03 Piles_Stiff 39.4 ×39.4 255.9 3.10 4650 5459 7042 3.19E-04 1.344E-03 (Column lateral confinement ratio=0.6%) Confined core concrete ultimate strain is 0.015 First Member Unconfined Confined Unconfined Confined Foundation Type b×d (in.) Read the entire page →
From page 58... ... 58 force versus lateral deformation curve. This has caused some difficulty in defining the exact failure point in the NCHRP Report 458 model. Read the entire page →
From page 59... ... 59 calculated from the ultimate plastic analysis of the column's cross section. Values for Fmc, Cj, jtunc, and jtconf have been extracted from the analysis of a large number of bridges with two-column, three-column and four-column bents. Read the entire page →
From page 60... ... 60 subjected to lateral load. Another parameter that identifies the variation between Equation 4.2 and the analysis results is the COV of the ratio between the equation and the analysis results. Read the entire page →
From page 61... ... 61 difference between the target reliability margin and the reliability margin that a system designed using current methods provides. Following the same rationale, it can be assumed that bridges under lateral loads should produce a target reliability margin Dbu target to be classified as sufficiently redundant. Read the entire page →
From page 62... ... 62 capacity under lateral load can be represented by an equation of the form: = + ϕ − ϕ ϕ − ϕ    φ (4.2) 1P P F Cu p mc u tunc tconf tunc where Pp1 gives the capacity of a bridge system under lateral load assuming linear-elastic behavior as typically done when using a force-based analysis, Fmc is a multi-column factor, Cj is a curvature factor, ju is the ultimate curvature of the weakest column in the bent, jtunc is the average curvature for a typical unconfined column, jtconf is the average curvature for a typical confined column. Read the entire page →
From page 63... ... 63 rs = 0.003. The analysis of the section determines that the moment capacity of the column meets the requirement and is equal to Mp = 3.85 × 10-4 kip-in. Read the entire page →
From page 64... ... 64 the engineer calculates the ultimate load capacity of the sys tem to be 714 1.10 0.24 0.974 10 3.64 10 1.55 10 3.64 10 874 1 3 4 3 4 P P F C kips u p mc u tunc tconf tunc = + ϕ − ϕ ϕ − ϕ     = + × − × × − ×     = φ − − − − The redundancy ratio is obtained as 874 7141 R P Pu u p = = = 1.22 . This indicates that this bridge with twocolumn bents does provide some level of redundancy. Read the entire page →
From page 65... ... 65 Typically, P-d only becomes significant at unreasonably large displacement values, or in especially slender columns. • P-D effect, or P-"big-delta," is associated with displacements relative to member ends. Read the entire page →
From page 66... ... 66 typical substructure design cases. To investigate situations where the foundations may be overdesigned or underdesigned, this section analyzes the multi-girder bridge with three-column bents assuming different foundation stiffnesses. Read the entire page →
From page 67... ... Table 4.17. Summary of results for three-column bridge bents with various foundations. Read the entire page →
From page 68... ... 68 Effect of Inadequate Cap Beams In a properly designed bridge system, the cap beam should be at least as strong as the columns. Thus, no plastic hinges are expected to form in the cap beam during a pushover analysis. Read the entire page →
From page 69... ... 69 lower than that of the column but higher than the column's plastic moment. In Case C, the cap beam's moment capacity is lower than the column's plastic moment. Read the entire page →
From page 70... ... 70 Case A Diameter Column Height System Capacity Plastic Capacity Ultimate Column Curvature C/D Ratio Moment C/D Ratio Curvature System Capacity by Eq.4.22 Error D (in.) Read the entire page →
From page 71... ... 71 the proposed approach when evaluating the ultimate load carrying capacity of a bridge system subjected to distributed lateral load. Implementation Example: Weak Cap Beam Case A Case A is the situation when C/Dcurvature of the cap beam leads to a reduction in the curvature capacity of the beamcolumn connection, which can be modeled by an equivalent reduction in the column's curvature. Read the entire page →
From page 72... ... 72 This example is for a three-span continuous bridge with two three-column bents where the lateral confinement reinforcement ratio of each column is rs = 0.3% (detail category B) Read the entire page →
From page 73... ... 73 4. Estimate the ultimate system capacity. Read the entire page →
From page 74... ... 74 reaches its bending moment capacity. This ratio will be represented by the term C/Dshear. Read the entire page →
From page 75... ... 75 such that gVju ≈ effective curvature for columns weak in shear. Case C: If the final shear resistance of the column is sufficient to withstand the maximum shear force due to plastic hinging, [Vf(c) Read the entire page →
From page 76... ... 76 which is assumed to be reached when the displacement is at 5 Dy. A linear interpolation is used to find the shear capacity when the displacement is between 2 Dy and 5 Dy. Read the entire page →
From page 77... ... 77 hinging, [Vi(c) Read the entire page →
From page 78... ... 78 hinge during the pushover analysis. Therefore, according to Equation 4.27, the lateral load capacity of the entire bridge system is 6,355 1,769 1,974 5,6951P P V c V d kipu p i d ( ) Read the entire page →
From page 79... ... 79 capacity when the load is Pp = 6355 kip. At that load, the shear force in the column is assumed to be equal to the demand shear force Vu(d) Read the entire page →
From page 80... ... 80 This bridge system will fail in ductile bending and the columns should be evaluated using a system factor equal to exp exp 6,330 5,263.8 0.890.6 0.5 F Cs mc u tunc tconf tunc u targetφ = + ϕ − ϕ ϕ − ϕ     =   = −ξ×∆β ϕ − × The system factor is calculated to be, fs = 0.89 which is higher than that needed when the failure is due to shear. 4.5 Conclusions This chapter presents the proposed model for estimating the system capacity of bridge systems subjected to uniform lateral load at the superstructure level and how the model can be used to define system factors that can be applied during the safety evaluation of new and existing bridge systems. Read the entire page →

From page 39...

... 39 4.1 Redundancy of Bridge Systems under Lateral Load Traditionally, structural design codes have defined a structure's capacity in terms of the ability of its individual members to sustain the applied loads using a linear-elastic analysis. Given that the failures of individual members do not necessarily lead to the collapse of the system, structural redundancy is defined as the ability of a structural system to continue to carry load after one critical member reaches its load carrying capacity.