Bridge System Safety and Redundancy (2014)

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1 S U M M A R Y Bridge System Safety and Redundancy This report develops a method to calibrate system factors that can be applied during the design and load capacity evaluation of highway bridges to account for bridge redun- dancy and system safety. The proposed system factors can be used during the design and safety assessment of bridges subjected to distributed lateral load being evaluated using the displacement-based approach specified in the AASHTO Guide Specifications for LRFD Seismic Bridge Design or the traditional force-based approach. Also, the report presents system factors calibrated for application with bridge systems subjected to vertical vehicular loads. The proposed system factor tables are presented for each load case as described below. More details are provided in the body of this report. The proposed system factors are used to modify the design/safety-check equation so that the required member capacity is evaluated using the following equation: ∑φ φ = γ (S1)R Qs nN i i where RNn is the required member capacity accounting for bridge redundancy, fs is the system factor specified in this Summary, f is the member resistance factor as specified in the current AASHTO LRFD Bridge Design Specifications, gi is the load factor for load i, Qi is the load effect of load i. The recommended values for the system factors are provided in the tables and equations below. Bridge Systems under Distributed Lateral Load Evaluated Using the Displacement-Based Method The displacement-based approach was found to explicitly consider the system effects of the entire bridge system. Accordingly, there is no measurable reserve system capacity that could be used to take advantage of bridge system redundancy. Therefore, the recommended system factor should serve to improve the reliability of the system and it takes the form φ = ( )− ∆βexp (S2)0.60 u targets where the 0.60 value in the exponential equation is used to account for the uncertainties associated with estimating the system capacity and the seismic demand. Dbu target is the target reliability index margin that should be specified by the code writers. Dbu target is the additional reliability that a system should provide beyond the reliability index that is used for the design of individual members. The AASHTO LRFD Bridge Design Specifications were calibrated so that bridge members under vertical gravity load generally produce a reliability index bmember = 3.5. However, it is generally expected that bridge systems provide additional reserve strengths so that system collapse does not take place should one member reach its limit capacity.

2A Dbu target provides a reliability measure of that reserve strength. The reliability index for mem- bers of bridge systems subjected to seismic load or other lateral loads has not been determined. Nevertheless, it is herein proposed that a Dbu target on the order of 0.50 be used for bridges evalu- ated when subjected to lateral load. Because the displacement-based approach does not provide any additional system reserve strength, Equation S2 is proposed in order to obtain a reliability index for the system similar to that observed when using the traditional force-based method. Bridge Systems under Distributed Lateral Load Evaluated Using the Force-Based Method When the evaluation of bridge systems under lateral load is undertaken using the force- based method, the proposed system factor takes the form φ = + γ ϕ − ϕ ϕ − ϕ     −ξ×∆β φ ϕ exp (S3)u target F Cs mc u tunc tconf tunc where x in the exponential equation is the dispersion coefficient used to account for the uncertainties associated with estimating the system capacity and the hazard demand. Dbu target is the target reliability index margin that should be specified by code writers. It is herein pro- posed that a Dbu target equal to 0.50 be used in order to obtain a reliability index of the bridge system similar to that observed in NCHRP Report 458 for unconfined multi-column bridge bents when subjected to other than seismic load. Fmc is a multi-column factor, Cj is a curva- ture factor, ju is the ultimate curvature of the weakest column in the bent, gj is the curvature correction factor for cases with weak connecting elements and weak details, jtunc is the aver- age curvature for a typical unconfined column, jtconf is the average curvature for a typical confined column. The values recommended for each of the parameters in Equation S3 are provided in Table S1. The value for the ultimate curvature at failure ju is calculated from the ultimate plastic analysis of the column’s cross section. The values for Fmc, Cj, jtunc and jtconf provided in Table S1 were extracted from the analysis of a large number of bridges with two-, three-, and four-column piers and bents. The piers and bents covered a range of column sizes, vertical reinforcement ratios, and confinement ratios. The analyses also considered the effect of different foundation stiffnesses. The values for jtunc and jtconf are the average curvatures obtained from the analysis of the column sizes used in NCHRP Report 458. The columns analyzed in NCHRP Report 458 represent typical column sizes and reinforcement ratios collected from a national survey conducted as part of that study. The values for jtunc and jtconf are used in Equation S3 to compare the confinement ratio of the column being evaluated to the average confinement ratios observed in typical confined and unconfined columns. A correction factor is applied in Equation S3 to reduce the ultimate column curvature for the cases where the shear capacity or the detailing of the columns or the capacity of the cap beams and pile caps are not sufficient to allow the columns to reach their full ultimate capacities, but the bridge columns do reach their plastic moment capacities. The correction factor is given as γ = − − ≥ ≥ γ = ϕ ϕ ≥ ϕ < ϕ γ = ≥ ϕ ≥ ϕ < ϕ ϕ ϕ if if and (S4) 1.0 if and system is non-redundant (see Table S1) if M M M M M M M M M M M M M available p column u column p column u column available p column u connection u available u column u connection u available u column u connection u available p column

3 where Mavailable = moment capacity of the connecting elements such as cap beams and pile caps or the reduced moment that can be supported by the column based on the available shear reinforcement, development length, splice, or connection detailing. Details on how to calculate the available moment capacity for a member with weak detail- ing are available in the FHWA Seismic Retrofitting Manual for Highway Structures, Part 1, Bridges, as Mp column = plastic moment capacity of column, Mu column = ultimate overstrength moment capacity of column calculated using nonlinear sectional capacity analysis programs or conservatively estimated to be 1.15 Mp column, ju = ultimate curvature of the weakest col- umn in the bent, and ju connection = minimum ultimate curvature of the connecting elements. Bridge Systems under Concentrated Lateral Load The analysis of systems subjected to statically applied concentrated lateral loads to a girder or column have demonstrated that they primarily cause local effects with little contribution from the remaining members of the system. Therefore, the system factor for evaluating the effect of concentrated lateral forces is φ = ( )− ∆βexp (S5)0.35 u targets where the 0.35 value in the exponential equation is used to account for the uncertainties associated with estimating the capacity of the member and the force applied. Variable Applicability Recommended Value u target, target reliability index margin All systems 0.50 , dispersion coefficient Seismic loads 0.60 All other lateral loads 0.35 s, system factor for One-column bents Longitudinal loading of systems with bearing connections between superstructures and substructures Systems where failure is controlled by shear or where failure is in the connections or where the detailing is not sufficient to allow plastic moment capacity of the members to be reached Systems evaluated using the displacement-based approach targetexp us mcF , multi-column factor based on number of columns in each bent when the bridge is loaded laterally for both integral and bearing superstructure-substructure connections; also mcF is a multi-bent factor based on the number of bents between expansion joints when a bridge with integral column/superstructure connections is loaded longitudinally Two-column subsystems 1.10 Three-column subsystems 1.16 Multi-column subsystems 1.18 C , curvature factor All systems 0.24 tunc , typical unconfined column ultimate curvature All systems 3.64 x 10-4 (1/in) tconf , typical confined column ultimate curvature All systems 1.55 x 10-3 (1/in) Table S1. Recommended values for redundancy parameters for straight bridges with one-column and multi-column bents of equal height under horizontal load.

4Bridge Systems under Vertical Vehicular Load System factors for multi-beam I-girder and box-girder bridges are proposed for evaluat- ing the redundancy of originally intact systems subjected to vehicular overloads and for damaged bridges that have been previously exposed to local member damage. Redundancy of Originally Intact Systems under Overloads The system factors for I-girder bridges and spread box-girder bridges are provided in Tables S2 and S3 in function of the capacity of the bridge to resist first member failure represented by the variable, LF1, the dead load to resistance ratio of the members and based on the material and geometric properties of the bridge. The proposed systems factors were calibrated so that the system provides a reliability index margin Dbu = 0.85 as recommended in NCHRP Report 406. This Dbu target value was obtained based on the reliability evaluation of bridge systems that have traditionally shown good system performance. The reliability index margin Dbu reflects the required reliability of the system beyond the reliability of the first member to fail. D/R is the dead load to resistance ratio for the bridge beams. LF1 is a factor related to the capacity of the system to resist the failure of its most critical bridge member calculated from a linear structural analysis of the bridge up until the first member fails. LF1 gives the number of HS-20 trucks that the bridge member can carry in addition to the dead load. It can be expressed as = = − ≥ = = − < + + + + − + − − − − − + when 1.0 when 1.0 (S6) 1 1 1 1 1 1 1 1 1 1 LF LF R D L LF LF LF LF R D L LF LF That is, LF1 in Tables S2 and S3 represents the load carrying capacity of the weakest section of the beam that can be either the positive bending section or the negative bending section depending on the moment capacity in each region (R), the dead load moment in each region (D), and the effect of the applied live load moment on the most critical beam (L1) where the live load represents two side-by-side HS-20 trucks applied at the middle of the span or two trucks in one lane applied in each of two contiguous spans. The positive superscript in LF1, R, D, and L1 is for the positive bending region, the negative superscript is for the negative bending regions. Table S2. System factors for overloads on I-girder bridges. Bridge Cross-Section Type System Factor Simple-span 4 I-beams at 4 ft 0.80 0.16s D R Simple-span 4 I-beams at 6 ft 0.90 0.09s D R Simple-span 6 I-beams at 4 ft 0.95 0.05s D R Continuous span 4 I-beams at 4 ft with compact members 0.93 0.07s D R Continuous steel I-girder bridges with noncompact negative bending sections and 1 11.16 0.75LF LF 0.80 0.16s D R All other simple-span and continuous I-beam bridges 2 2 1 1 1.5 / 1 1s D R LF

5 The parameter L1 gives the live load applied on the most critical member, which is defined as the member that fails first. It can be calculated as = ×. . (S7)1L D F LL where D.F. is the load distribution factor and LL is the effect of the HL-93 truck load with no impact factor and no lane load. Redundancy of Damaged Systems under Vertical Loads The system factors for damaged I-girder and spread box-girder bridges are provided in Tables S4 and S5 as a function of the redundancy ratio = 1 R LF LF d d which gives the capacity of a damaged bridge system that has previously lost the load carrying capacity of a main member given as LFd and the ability of the originally intact bridge to resist first member failure which Table S3. System factors for overloads on spread box-girder bridges. Bridge Cross-Section Type System Factor Narrow simple-span box-girder bridges less than 24-ft wide 0.83 0.14s D R All other simple-span box-girder bridges 2 2 1 1 1.5 / 1 1s D R LF Narrow continuous box-girder bridges less than 24-ft wide 2 2 1 1 1.5 / 1 1s D R LF Continuous steel box-girder bridges with noncompact negative bending sections and 1 11.75LF LF 2 2 1 1 1.5 / 1 1s D R LF All other continuous box-girder bridges 2 2 1 1 1.5 / 1 4 1s D R LF Bridge Cross-Section Type Redundancy Ratio 1 d d LFR LF System Factor Simple-span and continuous prestressed concrete I-girder bridges with four beams at 4 ft 0.56d transverse weightR 0.47 (0.47 ) d s d R DR R Simple-span and continuous compact steel I- girder bridges with four beams at 4 ft 0.64d transverse weightR All other simple-span I- girder bridges 1 0.056d transverse weightR S Continuous noncompact steel I-girder bridges with four beams at 4 ft 0.58d transverseR All other continuous noncompact steel I-girder bridges 1.00 0.08d transverseR S All other continuous compact steel and prestressed concrete I- girder bridges 1.35 0.08d transverseR S where S = beam spacing in feet. Table S4. System factors for damaged I-girder bridges under vertical loads.

6is represented by the variable, LF1, defined in Equation S6. For the spread box-girder bridges, three different damage scenarios are considered. In the first scenario, one box is assumed to have been exposed to a fatigue-type fracture that sliced through the entire bottom flange and two webs. The second scenario assumed major damage to one web while maintaining the torsional capacity of the box. The third scenario considered that the failure of the web also led to the loss of the torsional rigidity of the box. Tables S4 and S5 list the expressions for Rd as a function of beam spacing, slab strength, and the dead weight applied on the damaged member for the bridge types analyzed in this study and the corresponding system factor. The proposed systems factors were calibrated so that a damaged system provides a reliability index margin Dbd = -2.70 as recommended in NCHRP Report 406. This Dbd target value was obtained based on the reliability evaluation of damaged bridge systems that have traditionally shown good system performance. The reli- ability index margin Dbd reflects the required reliability of the damaged system compared to the reliability of the first member to fail in an originally intact system. The effect of the weight of the damaged beam that must be carried by the remaining sys- tem is considered using 1.23 0.23 (S8) total dead weight on the damaged beam in kip per unit length. beam kip ftweight beam ( )γ = − ω ω = The effect of the slab, bracings, and diaphragms is considered using γ = +0.50 13.5 . 0.50 (S9) M kip ft ft transverse transverse (S10)M M Mtransverse slab br L= + where Mtransverse = combined moment capacity for lateral load transverse expressed in kip-ft per unit slab width, Mslab = moment capacity of slab per unit width, and Mbr/L = contribution Table S5. System factors for damaged spread box-girder bridges under vertical loads. Bridge Cross-Section Type Redundancy Ratio 1 d d LFR LF System Factor Fractured simple-span steel box-girder bridges less than 24-ft wide Non-redundant s=0.80 Narrow simple-span steel box-girder bridges less than 24 ft with no torsional rigidity 0.46d transverseR 0.47 (0.47 ) d s d R DR R All other simple-span box- girder bridges 0.72d transverseR Continuous steel box-girder bridges with noncompact negative bending sections and 1 11.75LF LF 0.72d transverseR All other continuous box- girder bridges 1 4.500.59d transverseR LF

7 of the bracing and diaphragms to transverse moment capacity calculated using Equa- tions S11 or S12. The equivalent transverse moment capacity for cross bracing as defined in the FHWA Steel Bridge Design Handbook: Bracing System Design (2012) can be obtained as = (S11)M F h L br L br b b The equivalent transverse moment capacity for diaphragms is given by = (S12)M M L br L br b where Mbr = moment capacity of diaphragms contributing to lateral transverse distribu- tion of vertical load between adjacent main bridge girders; Fbr = bracing chord force deter- mined from the applicable limit state for the bolts (see AISC Steel Construction Manual, Part 7), welds (see AISC, Part 8), and connecting elements (see AISC, Part 9); Lb = spacing of the cross frames or diaphragms; and hb = distance between the bracing top and bottom chords. The range of applicability of gtransverse has been verified for I-girder bridges and showed an upper limit value in the range of gtransverse = 1.10 to 1.20 with 1.10 being a conservative value. Multi-cell box-girder bridges have been found to be sufficiently redundant for the ulti- mate system reserve strength condition and the ability of the originally intact system to resist collapse if member strength is exceeded with a recommended system factor fs = 1.0. Multi-cell box-girder bridges are highly redundant for system strength of damaged bridge condition with a recommended system factor fs = 1.2 for systems that may have sustained damage to one of the webs. Single-cell box-girder bridges are not redundant and the rec- ommended system factor for both the ultimate condition and the damaged state condi- tion is fs = 0.80. Recommended system factors for typical straight superstructures are specified in Table S6 for single-cell and multi-cell boxes for resistance to collapse of the originally intact system and Table S7 for single-cell and multi-cell boxes in damaged state condition. Bridge Cross-Section Type System Factor Single-cell box-girder bridges 0.80s Multi-cell box-girder bridges 1.00s Table S6. System factors for single-cell and multi-cell box-girder superstructures for resistance to collapse conditions under vertical loading. Bridge Cross-Section Type System Factor Single-cell box-girder bridges 0.80s Multi-cell box-girder bridges 1.20s Table S7. System factors for single-cell and multi-cell box-girder superstructures in damaged state condition under vertical loads.

8General Comment The analyses in NCHRP Report 406 and NCHRP Report 458 concentrated on bridges that closely met the member strength design requirements. The analyses performed in this study considered the redundancy of deficient bridges as well as overdesigned bridges that expand the applicability of the proposed system factors. The proposed system factors are expressed in terms of a limited number of parameters related to the relevant geometric properties of the system and the strength and material properties of the primary bridge members.