Bridge System Safety and Redundancy (2014) / Chapter Skim
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From page 81... ...
81 5.1 Measures of System Safety and Redundancy Figure 5.1 gives a conceptual representation of the performance of a structure under increasing loads and the different levels that should be considered when evaluating member safety, system safety, and system redundancy. For example, the green line labeled "Intact system" may represent the applied load versus maximum vertical displacement of a ductile multi-girder bridge superstructure.
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From page 82... ...
82 the incremental analysis by R′ = LF × HS20. The applied live load P = LL × HS20 is the expected maximum live load that will be applied on the superstructure within the appropriate service period.
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From page 83... ...
83 The load multipliers, LFi, and the system reserve ratios in Equation 5.2 provide deterministic estimates of system safety and redundancy while Equation 5.1 can be used to determine the reliability index, b, for any member or system limit state. The reliability indices corresponding to the load multipliers LF1, LFu, or LFd of Figure 5.1 may be expressed respectively as bmember, bultimate, and bdamaged.
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From page 84... ...
84 bridges varying in length between 45-ft and 150-ft with a composite concrete deck supported by 4, 6, 8, and 10 beams spaced at 4-ft, 6-ft, 8-ft, 10-ft, and 12-ft are analyzed. Also, over 50 prestressed concrete I-girder bridges with two continuous spans varying in length between 100-ft and 150-ft supported by 4, 6, 8 and 10 beams spaced at 4-ft, 6-ft, 8-ft, 10-ft, and 12-ft are investigated.
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From page 85... ...
85 strength and the dead load. The plot shows that the points lie close to a 45° line.
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From page 86... ...
86 and in red for continuous span bridges. These results are for bridges loaded by two side-by-side trucks in the middle of one span.
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From page 87... ...
87 box sections, number of boxes, and box spacing. During the analyses of the bridges, wide bridges were loaded by two sideby-side trucks; narrow bridges were loaded by only one truck.
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From page 88... ...
88 line having an equation of the form y = 1.0 x with a coefficient of regression R2 = 1.0. The COV of the error between the predicted values and the analytical results is 4.11%.
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From page 89... ...
89 Figure 5.6. Verification of model for three-span continuous steel box-girder bridges.
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From page 90... ...
90 Figure 5.8. Verification of model for three-span continuous prestressed concrete box-girder bridges.
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From page 91... ...
91 did not sufficiently improve the redundancy of four beams at 4-ft which remained at a lower level of redundancy. Only four narrow continuous steel I-girder bridges were analyzed having 4 beams at 60-ft and six beams at 4-ft spacing with compact and noncompact sections in negative bending.
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From page 92... ...
92 wide bridges while those loaded in two lanes show practically no redundancy with LFu = 1.03 LF1. Continuous Span I-Girder Bridges with Noncompact Sections When the section in negative bending is noncompact, the dominant failure mechanism may be due to trucks in different contiguous spans of the bridge rather than the two sideby-side trucks in the same span.
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From page 93... ...
93 Continuous-Span Steel Box-Girder Bridges with Noncompact Negative Sections The AASHTO LRFD Bridge Design Specifications indicate that steel box-girder sections in negative bending should be considered to be noncompact. For this reason, several analyses were performed assuming that the boxes in the negative bending regions have no ductility and that they will fail as soon as their moment capacity is first reached.
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From page 94... ...
94 LF1-<1.75LF1+. The transition between the two curves describing the two failure modes is not sudden and can be expressed using the parameter g defined in Equation 5.9.
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From page 95... ...
95 mate load carrying capacities of originally intact typical bridge configurations and compared these to the load capacity of critical members. Consistent patterns for the relationship between these parameters describing the performance of bridges are possible for evaluating the redundancy of bridge superstructures using a few characteristic parameters that, as described in Equations 5.7, 5.8, and 5.9, consist of the member moment resistance in positive and negative bending regions, the applied dead load moment, the moment from live load applied on the most critical member and, for continuous bridges, the parameter g which gives the relative stiffness of the beam near the support to the stiffness of the slab.
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From page 96... ...
96 The modification factor g takes into account the stiffness of continuous box-girder bridges relative to slab stiffness as well as the negative bending strength capacity of the box. EIbox is the stiffness for the cracked section of the box girder in negative bending, which ignores the portion of the concrete in tension.
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From page 97... ...
97 other hand, fs should serve to lower the reliability index for the system when the available Dbu and Dbd are higher than the target values. The amount by which bultimate and bdamage should be increased should be equal to the deficit in the available Dbu and Dbd when compared to the target values while the amount by which bultimate and bdamage should be decreased should be equal to the surplus in the available Dbu and Dbd when compared to the target values.
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From page 98... ...
98 within the range of typical new designs. Further review of the NCHRP Report 406 data augmented by the results of the analyses performed as part of this project show that a better approximation for the relationship between LFu and LF1 for all simple-span and continuous I-girder bridges is obtained from an equation of the form = × +1.16 0.75 (5.22)
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From page 99... ...
99 design life of the structure LL75 for the 80-ft span loaded in two lanes is 1.81. Assuming a lognormal model, the reliability index bmember for the failure of the first member can be expressed as ln LF LL ln 1.13 6.96 1.81 0.19 0.135 6.31 1 75 2 2 2 2V V member LL LF β = + = × + = b.
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From page 100... ...
100 systems under vertical load. The relationships in Table 5.1 can be presented by equations of the form = + (5.27)
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101 Figure 5.19. Plot of system factor vs.
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102 depending on the moment capacity in each region (R) , the dead load moment in each region (D)
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103 to be R = 7200 kip-ft. The dead load effect is found to be 3500 kip-ft.
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104 bridges. For the box-girder bridges, the main damage scenario also consisted of removing an entire web.
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105 Figure 5.21. Effect of span length on LFd/LF1 for steel I-girder bridges.
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From page 106... ...
106 The plot of LFd versus LF1 for a set of simple-span bridges is presented in Figure 5.24. The results show a strong linear relationship between LFd and LF1 with an equation of the form = 0.60 (5.30)
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From page 107... ...
107 torsional rigidity of the box is not affected by the damage. The results of these analyses are summarized in Figure 5.26.
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From page 108... ...
108 fatigue fracture. This is done by removing about a 1-ft segment in each of the webs and the bottom flange at the midpoint along the length of one box.
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From page 109... ...
109 how LFd decreases for the same LF1 as the beam spacing increases. The trend with beam spacing is similar to that depicted in Figure 5.20 with LFd = 0.48 LF1 for the systems analyzed in this case with beams at 8-ft, LFd = 0.43 LF1 for bridges with beams at 6-ft, and LFd = 0.37 LF1 for bridges with beams at 12-ft.
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From page 110... ...
110 Effect of Dead Weight of Damaged Beam The results of simple-span bridges provided in NCHRP Report 406 also demonstrate that bridge redundancy for the damaged limit state depends on the weight of the damaged beam that must be transferred to the adjacent undamaged beams as the bridge is loaded. To better establish this relationship, additional bridge models having a load factor LF1 ranging from 3.0 and up to 8.0 were analyzed.
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From page 111... ...
111 the redundancy ratio. The weight correction factor is defined as gweight 1.23 0.23 (5.38)
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From page 112... ...
112 verification of the effect of diaphragm capacity with different spacing is shown in Figure 5.36. The base line is the deck with a transverse bending moment capacity equal to Mtransverse = 13.5 kip-ft/ft (or 1615 kip-in.
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From page 113... ...
113 adjacent main bridge girders; Fbr = bracing chord force determined from the applicable limit state for the bolts (see AISC Steel Construction Manual, 2011, Part 7) , welds (see AISC, 2011, Part 8)
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From page 114... ...
114 listed in Table 5.3, the redundancy equation can be represented in the form =LF C LF (5.43) d red1 1 where the parameter Cred1 is equal to the redundancy ratio Rd.
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From page 115... ...
115 bridges, as well as bridges that have suffered major damage to the most critical bridge member. The results were fitted into equations that best described the redundancy based on a set of simple parameters that describe the bridge geometry and main load carrying characteristics.
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