Fracture-Critical System Analysis for Steel Bridges (2018)

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13 This chapter summarizes the findings after the research methodology described in Section 2.2 was carried out. First, the development of an FEA methodology for the redundancy evaluation of typical steel bridges is discussed, as follows: • Description of the analysis procedures, techniques, and inputs required to construct finite element models to evaluate steel bridges for redundancy, in Section 3.1; • Discussion of reliability-based load models that character- ize bridge loading conditions in the faulted condition, in Section 3.2; • Establishment of minimum performance requirements for steel bridges in the faulted condition, in Section 3.3; and • Calculation of the dynamic amplification of load caused by sudden failure of a steel tension member, in Section 3.4. As discussed in Section 2.2, the FEA procedures, tech- niques, and inputs were benchmarked by comparison with available field data for five bridges. These structures were also evaluated in accordance with the developed FEA methodol- ogy. Dynamic analyses of the five bridges—and an additional sixth bridge—were carried out for the calculation of dynamic amplification of load after sudden failure of a steel member. A summary of the results obtained from FEA for the six steel bridges for benchmarking and for the calculation of dynamic amplification factor is shown in Table 1. Further detail with regard to these bridges can be found in Appendix B, which summarizes the analyses conducted for these structures. A set of requirements for the design and fabrication of new steel bridges is presented in Section 3.5. The main objective of these requirements is to provide adequate strength, as well as resistance, to fatigue and fracture. Section 3.6 presents screening criteria to assess whether a structure with mem- bers initially designated as FCM is an acceptable candidate to undergo system analysis with the objective of re-designating such members as SRM. Section 3.7 explains the procedures needed for adoption of the research findings described in this chapter on bridge engineering practice; particularly, imple- mentation of the proposed guide specifications—accompanied by complementary background information and application examples—is discussed. 3.1 Finite Element Analysis Procedures, Techniques, and Inputs As stated, detailed three-dimensional finite element models of several structures traditionally classified as nonredundant with FCMs were developed in the current study. These models were built to develop a general-purpose methodology suitable to perform such system analyses for various bridge configura- tions. The models that were developed are capable of predict- ing the dynamic behavior of a steel structure after sudden failure of a tension member. Abaqus, a general-purpose com- mercial finite element software package, was used to develop the methodology. During the research, several essential behavioral and mechanics-based characteristics were deemed important and needed to be accurately modeled. They are described below. While Abaqus was the software package used, any software package that is capable of modeling these characteristics would have been acceptable. Since many ana- lysts will not have access to Abaqus, detailed requirements to check the adequacy of a particular FEA software have been developed and are included in the proposed guide specifica- tions in Appendix E and in Appendix D, as well. Appendix D is included in the background material as part of the stand- alone proposed guide specification. In summary, the basic capabilities of any FEA software must include the following: • Ability to perform three-dimensional dynamic analysis in which nonlinear geometry (large deformation theory with finite strains and finite rotations) is considered. The analy- sis procedures employed are described in Section 3.1.1. C H A P T E R 3 Findings and Application

14 • Ability to include material models that can be used to sim- ulate nonlinear behavior of steel and reinforced concrete. Additionally, the analyst must be able to specify the density, material damping, and field-variable–dependent material properties. The material models used—with the required inputs described in Section 3.1.2—are as follows: – Combination of linear elasticity, classical metal plastic- ity, and a failure criteria for modeling steel; and – Combination of linear elasticity and concrete damage plasticity for modeling concrete. • Solver that either contains an extensive element library or includes the capability for the user to add specific ele- ments. This includes special-purpose techniques to model the following: – Shear stud behavior (Section 3.1.3.2), – Substructure flexibility (Section 3.1.3.3), and – Connection failure (Section 3.1.3.4). • Automated meshing procedures in which the required ele- ments are included, essential for efficiency. The elements and meshes used to model the bridge are described in Sec- tion 3.1.3.1. • Ability to model kinematic constraints, which include kinematic coupling, mesh tie, and embedment. The con- straints used are described in Section 3.1.3.1. • Ability to model contact and friction accurately. A contact algorithm, in which frictional tangential behavior can be specified and is applicable to solid-to-solid, shell-to-solid, and shell-to-shell interaction must be included in the solver. The contact algorithm and the necessary inputs are described in Section 3.1.3.5. • Solver that must include tools to define a variety of bound- ary conditions; particularly, prescribed displacements, surface tractions, and body forces. The applied boundary conditions are described in Section 3.1.3.6. 3.1.1 Analysis Procedures Two analysis procedures have been developed in this study. The first is intended for calculating the dynamic amplification factor following the sudden failure of a tension member. Thus, since the objective is to predict real behavior, no load factors were applied for this specific phase of the analysis. Applying load factors would effectively skew the results to be either conservative or nonconservative. Thus, the nominal dead and live loads are appropriate. The second analysis procedure was developed to establish whether the system possesses satisfactory redundancy in the faulted con- dition. For this analysis, load factors for both dead and live load were developed as described in Section 3.2. These load factors were developed to achieve specific target reliability indices in the faulted state. Both analysis procedures were composed of an initial implicit static analysis and a final explicit dynamic analysis in which the results from the initial implicit static analysis are imported. Static implicit analysis is used when inertial effects can be neglected and arrives at a result by solving a system of equations involving the state of the system at both the beginning and at the end of the time increment. It uses an iterative procedure (Newton’s method) that requires inversion of the stiffness matrix, and it is unconditionally stable (the magnitude of the time increment does not affect the solution) for linear problems. The implicit static analysis can arrive at the solution with a small number of increments (Simulia 2017). Bridge Type of Structure Type ofFailure Summary of Results Successfully Benchmarked Calculated DAR Performance Criteria Satisfied Neville Island 3-span continuous 2-plate girder (350 ft) Full-depth girder fracture Yes 0.39 Strength: Yes Service: Yes Hoan Bridge 3-span continuous 3-plate girder (217 ft) Multiple full- depth girder fractures Yes 0.21 Strength: Yes Service: Yes University of Texas twin- tub girder Simple-span twin-tub girder (120 ft) Simulated full-depth fracture Yes 0.30 Strength: No Service: No Milton–Madison Truss Simple-span truss (147 ft) Lower chord partial and full fracture Yes 0.36 Strength: Yes Service: Yes White River 2-span continuous 2-plate girder (155 ft) Girder fracture Yes 0.32 Redundancy evaluation not performed Dan Ryan Expressway Cross-girder (40 ft) Partial-depth fracture Limited data available 0.19 Not applicable a Note: main span length in parentheses. aPerformance criteria do not apply since it is a light rail commuter bridge. Table 1. List of bridges selected for benchmarking and calculation of dynamic amplification factor (DAR).

15 Explicit dynamic analysis is used when inertial effects need to be considered or when the problem becomes non- linear and prevents convergence of an implicit analysis. It uses a central-difference time integration rule, in which the solution at the end of a time increment is computed based on the state of the system at the beginning of the time incre- ment. The stability of the solution is constrained to a small stable time increment that depends on the mass, stiffness, and size of the finite elements used. The explicit dynamic analysis does not require inversion of the stiffness matrix; hence, it can compute a large number of increments in a short time (Simulia 2017). Thus, the analysis procedure is a sequence of analyses developed to optimize the computa- tional time required. 3.1.1.1 Initial Implicit Static Analysis Implicit static analysis was used to calculate the state of the structure prior to hardening of the concrete in the slab. An implicit static analysis was used because—although non- linearity is implemented in the analysis—the bridge behavior is linear, and inertial effects can be neglected as the bridge is in the undamaged condition. As the slab does not carry any load and does not contribute to the stiffness of the system before the concrete hardens, during the initial implicit static analysis the following were specified in the FEA: • Very low stiffness is specified for the elements composing the slab; that is, the elements modeling concrete and rebar. A stiffness of 1/1000 times the respective modulus of elas- ticity of each material was used. This is done so that the load carried by the reinforced slab and the contribution to the stiffness of the system is negligible. For the steelwork (girders, floor beams, bracing, stringers, and so on), no modifications to the stiffness needed to be applied. • Instead of defining contact interaction between the slab and the steelwork, a mesh tie was specified. The nodal dis- placements of the concrete slab elements are tied to the displacements of the top flanges of girders, floor beams, and stringers caused by dead load. As a result, the slab deforms with the steelwork. It does not sag between the girders, floor beams, and stringers, as would be observed in a real bridge. The remainder of the finite element modeling is identi- cal between the initial implicit static analysis and the final explicit dynamic analysis. The specific steps in the initial implicit static analysis are described as follows: 1. Apply load caused by self-weight of the structural steel components as a body force. This load is not factored for calculating dynamic amplification factors, since the objec- tive is to predict real behavior. However, when a redun- dancy analysis is performed, these loads are factored according to the load combination to achieve a desired reliability index. 2. Apply load caused by self-weight of the wet slab com- ponents as a body force. This load is not factored for the calculation of dynamic amplification factors, since the objective is to predict real behavior. However, when a redundancy analysis is performed, these loads are factored according to the load combination to achieve a desired reliability index. 3. Fix the system with regard to position; that is, the displace- ment degrees of freedom are not allowed to change. 4. Deactivate the elements that compose the slab. 5. Reactivate the elements that compose the slab. During this reactivation, the strain in the elements that compose the slab is reset to zero. 3.1.1.2 Final Explicit Dynamic Analysis As contact algorithms, softening material behavior, and nonlinear geometry are required to be part of the FEA, implicit solution procedures present unavoidable conver- gence problems in most FEA solvers. In addition, to calcu- late the dynamic response of the bridge or its capacity after sudden failure of a tension component, a dynamic explicit analysis needs to be carried out. The results obtained from the initial implicit static analysis are imported into the final explicit dynamic analysis. In other words, the state of the sys- tem (stresses, strains, displacements, and forces) at the begin- ning of the final explicit dynamic analysis is defined by the state of the system computed at the end of the initial implicit static analysis. In the initial implicit static analysis, the slab was modeled with softened stiffness to reflect that it is not hardened, and a mesh tie constraint was used to assure that the slab deformed with the steelwork and to prevent slump between girders and stringers. After the state of the system is imported, the follow- ing changes are made to capture the response of the structure after the concrete has hardened: • The modulus of elasticity of the concrete and rebar ele- ments in the slab is changed to their final real values. For the steelwork (girders, floor beams, bracing, stringers, and so on), no modifications need to be applied. • The mesh tie constraint between the slab concrete elements and the top flanges of girders and stringers is replaced by a frictional contact interaction. All of the body forces applied during the initial implicit static analysis (i.e., the dead load of the structure) are main- tained throughout the final explicit dynamic analysis.

16 For calculating the dynamic amplification factors caused by the fracture event, the following step for modeling sudden fracture—continued from Steps 1 to 5 in Section 3.1.1.1—is carried out in the final explicit dynamic analysis: 6. Instantly delete the elements in the member under con- sideration. The system is then allowed to oscillate and dampen until negligible kinetic energy is present. Only the nominal dead load of the structure is applied to the model. If live loads were to be applied in the calcula- tion of dynamic amplification, the resulting inertial effects would be lower because of load stiffening of the structure. Further, if live loads were to be applied, the masses associated with those loads and the interaction between those masses and the structure need to be considered, as well. Because of the complexity of the aforementioned phenomena, the authors consider that inclusion of live loads in the calcu- lation of dynamic amplification could yield inconsistent or nonconservative results. Once a dynamic amplification caused by the fracture is cal- culated, this factor can be used to greatly simplify the analysis of future bridges since the results from a static analysis can simply be amplified by this factor. This approach is analo- gous to how truck load effects obtained from a static analysis are amplified to account for the dynamic effects associated with impact. To evaluate the capacity of the structure in the faulted state, the following steps—continued from Steps 1 to 5 in Section 3.1.1.1—were carried out in the final explicit dynamic analysis: 6. Slowly reduce the stiffness of the elements located at the fracture location under consideration. The stiffness is reduced slowly to minimize any dynamic effects. If dynamic effects are significant, the system is allowed to oscillate until these effects are dampened. 7. Apply factored live loads caused by traffic. These loads are also very slowly applied to minimize any dynamic effects. If dynamic effects are significant, the system is allowed to oscillate until these effects are dampened. 3.1.1.3 Effect of Pouring Sequence Inclusion As previously stated, the analysis procedure sequence described in Section 3.1.1.1 and Section 3.1.1.2 is intended to capture the effects of concrete pouring and hardening. When concrete is placed, the slab does not contribute to stiff- ness of the system, and the weight of the slab is carried by the steelwork. Once the concrete hardens, the slab is able to carry loads and contribute to the stiffness of the structure. In the analysis procedure sequence, it is assumed that all of the concrete is poured and that it hardens at the same time. However, in general, concrete is typically poured in stages, with the objective of reducing dead load deflections. In this section, three scenarios are compared: • No-pouring sequence scenario, in which it is assumed that the concrete is hardened from the beginning. This situa- tion is similar to shored construction, in which the struc- ture is continuously supported and is then released. • Simplified pouring sequence scenario, in which it is assumed that all concrete is placed and hardens at the same time. This scenario is described in the analysis procedure sequence of Section 3.1.1.1 and Section 3.1.1.2. • Detailed pouring sequence scenario, in which the place- ment sequence described in the construction plans is modeled. In this case, it is assumed that each slab section hardens before the next section is placed. These three scenarios were modeled for the Neville Island Bridge and the Hoan Bridge. For further information with regard to the geometry of these structures, refer to Appendix B. For the comparison of the three scenarios, only implicit static analysis procedures are used. In the no-pouring sequence, the moduli of elasticity of concrete and reinforcement are 3,600 ksi and 29,000 ksi, respectively. The interaction between the slab and the steelwork is modeled by a contact interaction, as described in Section 3.1.2.2, and two analysis steps are carried out: 1. Apply load caused by self-weight of the structural steel components as a body force. This is a nominal (unfactored) load. 2. Apply load caused by self-weight of the slab components as a body force. This is a nominal (unfactored) load. In the simplified pouring sequence, the modulus of elas- ticity of concrete and reinforcement initially are 1/1,000 of their respective values, and the interaction between the slab and the steelwork is modeled by a tie constraint, as described in Section 3.1.3.5. The analysis procedure has the following six steps: 1. Apply load caused by self-weight of the structural steel components as a body force. This is a nominal (unfac- tored) load. 2. Apply load caused by self-weight of the slab components as a body force. This is a nominal (unfactored) load. 3. Fix the system with regard to position; that is, the displace- ment degrees of freedom are not allowed to change. 4. Deactivate the elements that compose the slab. 5. Reactivate the elements that compose the slab. During this reactivation, the strain in the elements that compose the slab is reset to zero.

17 6. Change the moduli of elasticity of concrete and reinforce- ment to 3,600 ksi and 29,000 ksi, respectively. The tie con- straint between the slab and the steelwork is replaced with a contact interaction, as described in Section 3.1.3.5. For the detailed pouring sequence scenario, the analysis procedure is similar to the simplified pouring sequence sce- nario, except that Steps 2 through 6 are carried out sequen- tially for each segment of the slab described in the concrete placement sequence in the construction plans of each struc- ture until the entire slab is placed and hardened, as shown in Figure 2 and Figure 3 for the Neville Island Bridge and the Hoan Bridge, respectively. The finite element models were constructed in accordance with the techniques and inputs described throughout Sec- tion 3.1.2 and Section 3.1.3, except for the tension behavior in the concrete material model. Cured concrete was modeled using the concrete damage plasticity model (Lubliner et al. 1989). A modulus of elasticity of 3,600 ksi and a Poisson’s ratio of 0.2 were used for initial linear elastic response. The compression inelastic behavior was based on the experimen- tal model proposed by Popovics (1973), and tension inelastic behavior was modeled as perfectly plastic with a yield stress in tension of 0.4 ksi. Ideally, a softening behavior could be employed for the tension inelastic behavior of concrete, but it presented convergence difficulties in a static implicit analy- sis. However, for the comparative purposes of this study, suc- cessful inclusion of that softening behavior would not have had a significant effect on the results. Figure 4 shows the uni- axial behavior of the material modeled employed for hard concrete. After completion of the analysis for the Neville Island Bridge, focus was paid to the forces carried by the primary member—in this case, the girders—and the concrete slab. Figure 5 and Figure 6 show the longitudinal normal stress along the center of the bottom flange of Girder G3 and Girder G4, respectively, for the three scenarios. According to these results, the stress in the bottom flange is not significantly affected by taking the concrete hardening into account. How- ever, the longitudinal normal stress along the top flange of the girders differ significantly, depending on the modeling approach. Figure 7 and Figure 8 show the longitudinal nor- mal stress along the center of the top flange of Girder G3 and Girder G4, respectively. The stress on the top flange is severely underestimated in the no-pouring sequence scenario. This is because concrete Figure 2. Pouring sequence of the Neville Island Bridge, segments labeled 1 through 3. Figure 3. Pouring sequence of the Hoan Bridge, segments labeled 16 through 18.

18 –5 –4 –3 –2 –1 0 1 –0.015 –0.01 –0.005 0 0.005 0.01 0.015 St re ss (k si) Strain (–) Note: – = magnitude has no unit. Figure 4. Uniaxial behavior of concrete damage plasticity model employed in the analysis of the effects of pouring sequence inclusion. –30 –20 –10 0 10 20 30 –3,000 –2,000 –1,000 0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 Lo ng itu di na l N or m al S tr es s (k si ) Distance from Pier 9 (Fixed Pier) (in.) Detailed Sequence Simplified Sequence No Sequence Figure 5. Comparison of longitudinal normal stress along the bottom flange of Girder G3 in the Neville Island Bridge. –30 –20 –10 0 10 20 30 –3,000 –2,000 –1,000 0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 Lo ng itu di na l N or m al S tr es s (k si ) Distance from Pier 9 (Fixed Pier) (in.) Detailed Sequence Simplified Sequence No Sequence Figure 6. Comparison of longitudinal normal stress along the bottom flange of Girder G4 in the Neville Island Bridge.

19 of tensile inelastic straining takes place at shared edges of slab segments poured and hardened at different stages. The no-sequence scenario shows large concentrations of tensile inelastic behavior in the concrete slab at locations over inte- rior supports (negative-moment regions), which will not take place in the real structure since the concrete in the slab hardens in the deformed shape. The results of the application of the three pouring sequence scenarios to the Hoan Bridge were similar to those previously reported for the Neville Island Bridge. Figure 11, Figure 12, and Figure 13 show the longitudinal normal stress along the center of the bottom flange of Girder D, Girder E, and Girder F, respectively, for the three scenarios. The stress in the bottom flange is similar for the three pouring sequence scenarios, with the no-sequence scenario resulting in slightly smaller stress. –30 –20 –10 0 10 20 30 –3,000 –2,000 –1,000 0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 Lo ng itu di na l N or m al S tr es s (k si ) Distance from Pier 9 (Fixed Pier) (in.) Detailed Sequence Simplified Sequence No Sequence Figure 7. Comparison of longitudinal normal stress along the top flange of Girder G3 in the Neville Island Bridge. –30 –20 –10 0 10 20 30 Lo ng itu di na l N or m al S tr es s (k si ) –3,000 –2,000 –1,000 0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 Distance from Pier 9 (in.) Detailed Sequence Simplified Sequence No Sequence Figure 8. Comparison of longitudinal normal stress along the top flange of Girder G4 in the Neville Island Bridge. was assumed to have hardened and to be composite with the steelwork at a stage when it is wet and unable to carry any self-weight. On the other hand, when any of the two pouring– hardening sequence approaches is taken (simplified sequence scenario or detailed sequence model), the results are similar. The simplified sequence scenario resulted in stress values up to 2.5 ksi larger at the middle span and up to 7.5 ksi smaller at the end spans than in the detailed sequence scenario. The no-sequence scenario underestimated stress in the middle and end spans—by up to 12 ksi—in comparison with the simpli- fied and the detailed sequence scenarios. Figure 9 and Figure 10 show the extent of tension crack- ing, based on inelastic tensile strain in the deck. There are no inelastic tensile strains in the simplified sequence sce- nario and, in the detailed sequence scenario, a small amount

20 No-Sequence Model One-Sequence Model Detailed Sequence Model Pier 8 Pier 9 Pier 10 Pier 11 Figure 9. Comparison of tension cracking in concrete slab of the Neville Island Bridge, viewed from top. No-Sequence Model Simplified Sequence Model Detailed Sequence Model Pier 8 Pier 9 Pier 10 Pier 11 Figure 10. Comparison of tension cracking in concrete slab of the Neville Island Bridge, viewed from bottom. –20 –10 0 10 20 –1,000 0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 Lo ng itu di na l N or m al S tr es s (k si ) Distance from Pier 5S (in.) Detailed Sequence Simplified Sequence No Sequence Figure 11. Comparison of longitudinal normal stress along the bottom flange of Girder D in the Hoan Bridge.

21 However, larger differences in the longitudinal stress in the top flange was computed, as shown in Figure 14, Figure 15, and Figure 16 for Girder D, Girder E, and Girder F, respectively. In the case of the Hoan Bridge, the simplified pouring sequence scenario resulted in larger stress values in all spans, up to 2 ksi larger than the detailed sequence scenario. The no-pouring sequence scenario resulted in underestimated longitudinal stress in all three spans, up to 8 ksi smaller than in the simpli- fied and detailed pouring sequence scenarios. Figure 17 and Figure 18 show the extent of inelastic tensile strains, which indicates occurrence of tensile cracking in the concrete slab. While no tensile inelasticity was calculated for the detailed and simplified pouring sequence scenarios, the no-pouring sequence scenario resulted in the slab cracking in tension over the piers of the middle span. This cracking would not occur in the real structure, since the concrete in the slab hardens in the deformed shape. The effect of concrete pouring and hardening must be explicitly considered in the analytical recommendations produced through this research, especially when analyz- ing long-span structures since their dead load demands are comparable to their live load demands. It is evident that not considering the self-weight of wet concrete (no-pouring sequence model) will result in large underestimation of the load carried by the steelwork. Further, if there are continu- ous spans, the concrete slab could undergo significant tensile cracking at the negative-moment locations if the no-pouring sequence scenario modeling approach is used. Detailed simulation of the concrete pouring–hardening sequence (detailed pouring sequence scenario model) revealed that a small amount of localized tensile cracking may develop at the interfaces between pouring segments in the concrete slab. The stress distribution along the bridge girders is similar to that obtained from the model in which all of the concrete slab –20 –10 0 10 20 –1,000 0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 Lo ng itu di na l N or m al S tr es s (k si ) Distance from Pier 5S (in.) Detailed Sequence Simplified Sequence No Sequence Figure 12. Comparison of longitudinal normal stress along the bottom flange of Girder E in the Hoan Bridge. –20 –10 0 10 20 –1,000 0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 Lo ng itu di na l N or m al S tr es s (k si ) Distance from Pier 5S (in.) Detailed Sequence Simplified Sequence No Sequence Figure 13. Comparison of longitudinal normal stress along the bottom flange of Girder F in the Hoan Bridge.

22 –20 –10 0 10 20 –1,000 0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 Lo ng itu di na l N or m al S tr es s (k si ) Distance from Pier 5S (in.) Detailed Sequence Simplified Sequence No Sequence Figure 14. Comparison of longitudinal normal stress along the top flange of Girder D in the Hoan Bridge. –20 –10 0 10 20 –1,000 0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 Lo ng itu di na l N or m al S tr es s (k si ) Distance from Pier 5S (in.) Detailed Sequence Simplified Sequence No Sequence Figure 15. Comparison of longitudinal normal stress along the top flange of Girder E in the Hoan Bridge. –20 –10 0 10 20 –1,000 0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 Lo ng itu di na l N or m al S tr es s (k si ) Distance from Pier 5S (in.) Detailed Sequence Simplified Sequence No Sequence Figure 16. Comparison of longitudinal normal stress along the top flange of Girder F in the Hoan Bridge.

23 is assumed to be poured and hardened at once (simplified pouring sequence scenario). Depending on the concrete placement sequence described in the construction plans, the stress values at the top flanges of members in contact with the slab calculated by the simplified pouring sequence scenario modeling approach may be under- estimated or overestimated in comparison with those calcu- lated with the detailed pouring sequence scenario modeling approach. This difference may be greater in bridges with larger spans or with very complex concrete placement sequences. Taking into account (1) that the concrete placement sequence is rarely known in detail by a structural analyst evaluating an existing structure, (2) the similarity between the results cal- culated in the simplified and the detailed pouring sequence scenarios for both structures, and (3) the complexity and additional time cost of comprehensively modeling the con- crete placement and hardening effects, the modeling approach of the simplified pouring sequence scenario is recommended for most steel bridges. The analysis procedures used in the finite element models discussed in the current dissertation— and described in Section 3.1.1.1 and Section 3.1.1.2—are based on the simplified pouring sequence scenario. No-Sequence Model Detailed Sequence Model Simplified Sequence Model Pier 5S Pier 4S Pier 3S Pier 2S Figure 17. Comparison of tension cracking in concrete slab of the Hoan Bridge, viewed from top. No-Sequence Model Detailed Sequence Model Simplified Sequence Model Pier 5S Pier 4S Pier 3S Pier 2S Figure 18. Comparison of tension cracking in concrete slab of the Hoan Bridge, viewed from bottom.

24 3.1.2 Material Models The majority of the primary load carrying components in a bridge superstructure are made either of steel or of con- crete. During this research, a linear elastic–Mises plastic rela- tion with linear kinematic hardening and a specified failure strain for modeling steel components was used and is recom- mended. For the modeling of concrete, linear elastic–concrete damage plastic relation was used and is recommended. Unless otherwise specified, nominal values are used to construct con- crete and steel material models. However, measured material properties may be used if they are available. 3.1.2.1 Steel Material Model The stiffness of the steel elements was defined by an initial linear elastic behavior, followed by Mises plasticity with linear kinematic hardening and a specified failure strain. The linear elastic behavior was defined by the specification of the modu- lus of elasticity and Poisson’s ratio, which were 29,000 ksi and 0.3, respectively. Onset of plasticity takes place at nominal yield strength, and the material hardens until nominal ulti- mate strength is reached. The hardening is linear and kine- matic. Instead of specifying failure strains related to uniaxial tension tests, the failure strain was conservatively assumed to be 0.05 (to recognize possible biaxial stress states and the presence of welds that limit ductility). It is possible that cer- tain steel types will have a nominal elongation at failure that is less than 0.05. In those cases, the nominal elongation at failure should be taken as the failure strain. Once the failure strain is attained, the element fails. The stress–strain relation for steel in the finite element models is shown in Figure 19. 3.1.2.2 Concrete Material Model A combination of linear elasticity and concrete damage plasticity models was used to define the material model for concrete (Lubliner et al. 1989). For the initial linear elas- tic behavior, a modulus of elasticity calculated in accor- dance with American Concrete Institute (ACI) ACI 318-14 ( )= ′ ≤ ′33 570001.5E w f fc c c c —where Ec is the modulus of elasticity of concrete in poundforce per square inch (psi), wc is the density of concrete in pounds per cubic foot, and f c′ is the nominal compressive strength of concrete in psi—and a Poisson’s ratio of 0.2 were used (ACI 2014). Concrete dam- age plasticity assumes that the main two failure mecha- nisms are tensile cracking and compressive crushing of the concrete material. The evolution of the yield (or failure) surface is controlled by two hardening variables—tensile and compressive equivalent plastic strains—linked to fail- ure mechanisms under tension and compression loading, respectively. Under uniaxial tension, the stress–strain response follows a linear–elastic relation until the value of the failure stress is reached. The failure stress corresponds to the onset of micro- cracking in the concrete material. Beyond the failure stress, the formation of microcracks is represented macroscopically with a softening stress–strain response, which induces strain localization in the concrete structure. A tensile strength ft, in accordance with the 2010 FIB Model Code for Concrete Struc- tures, is used (Fédération Internationale du Béton 2010). In the following expression, f c′ and ft are in psi: [ ] ( ) ( ) = ′ ′ ≤ ′ + −     1.58 7,250 307 ln 2,610 2,240 2 3 f f for f psi f otherwise t c c c Tabular data for the tension-softening behavior is developed using the fracture energy model contained in the 2010 FIB Model Code for Concrete Structures. This fracture energy is based on Wittmann et al. (1988), which used compact ten- sion (CT) specimens to determine fracture energy and strain softening of concrete and developed a bilinear softening rela- tion that—once applied to finite element models—resulted in good agreement with experimental data. In general, the fracture energy of concrete Gt can be assumed to be ( )= ′ +0.17 1,160 0.18G ft c where f c′ is in psi, and Gt is in psi • in. The uniaxial tensile stress–crack opening relation used in the finite element models is shown in Figure 20. Under uniaxial compression, the response is linear until the value of initial yield. In the plastic regime, the response is typically characterized by stress hardening followed by strain softening beyond the ultimate stress. Nominal compressive strength of concrete is used as peak yield stress, and tabu- lar data for the compressive hardening–softening behavior is developed using Popovics’ experimental relation. Popovics’ stress–strain relation is entirely defined by the compressive Note: σ = uniaxial stress in ksi, ε = uniaxial strain, σu = nominal ultimate strength in ksi, and σy = nominal yield strength in ksi. Figure 19. Steel stress–strain relation input in FEA.

25 strength and two constants, one related to the cementitious material type (concrete, mortar, or paste) and another related to the type of aggregate and test method used. This relation was compared against a comprehensive set of experimental tests resulting in good correlation. According to Popovics (1973), the compression stress for a given total strain f(e) can be calculated using the following equation: ( )ε = ′ ε ε     − + ε ε                1 f f n n c c c n where f c′ is the compressive strength of concrete, ec is the total strain at compressive strength, and n is a parameter calcu- lated from experimental data (Popovics 1973). The concrete uniaxial compressive stress–strain relation used in the finite element models is shown in Figure 21. Note that e is the total strain (e = eelastic + eplastic). The total strain at compressive strength ec , the experimental param- eter n, and the plastic strain eplastic, can be calculated as: ( ) = ′+ ε = ′ ε = ε − ε 0.0004 1.0 0.000224 n f f f E c c c plastic c 3.1.3 Analysis Techniques and Inputs For the six bridges analyzed in the development of the FEA methodology, the geometry provided in the original design plans was used to develop the finite element models. An effort was made to produce finite element models as computation- ally inexpensive as possible, while maintaining a high level of detail. Hence, a variety of elements and modeling techniques were used. The types of elements used to predict the behavior of each of the structural members and their application are described in this section. The special procedures used and developed to model shear studs, substructure flexibility, con- nection failure, contact interactions, and load application are also described. 3.1.3.1 General Elements and Constraints Used The reinforced concrete slab may be modeled with either a combination of solid elements and embedded truss ele- ments or shell elements in which the effect of the layers of reinforcements is implicitly included. When using a com- bination of solid elements and truss elements, the solid ele- ments are used to model concrete and the truss elements to model steel reinforcement. The literature review and experience have shown that concrete can be modeled best with solid elements. Specifically, eight-node linear bricks with reduced integration and hourglass control were used (C3D8R per Abaqus designation). It was found that at least eight solid concrete elements should be used through the thickness of the slab to achieve accurate results. Further, a maximum aspect ratio (length of longest edge divided by length of shortest edge) of 5 and corner angles (angle at which two element edges meet) between 40° and 140° should be maintained. At the locations in contact with steel- work (e.g., bottom slab haunches), the mesh density should be higher than the mesh density of the steelwork to ensure proper enforcement of the contact interaction. The reinforc- ing steel was modeled using two-node truss elements with linear displacement (T3D2 per Abaqus designation). The Note: δ = crack opening displacement. Figure 20. Concrete uniaxial tensile stress– displacement relation input in finite element models (f c , ft, and ft1 are in psi; Gt is in psi • in.; ct and ct1 are in in.). Figure 21. Concrete uniaxial compressive stress–strain relation input in finite element models.

26 length of the truss elements was approximately equal to the length of the longest edge of the solid concrete ele- ments. These truss elements were also embedded within the solid concrete elements. At the nodes of the embedded truss elements, the translational degrees of freedom were eliminated and the nodal translations were constrained to interpolated values of the nodal translations of the host solid concrete element. When using shell elements to model the behavior of rein- forced concrete slabs, the elements modeling the reinforced concrete slab were four-node linear shells with reduced inte- gration, finite membrane strains, and a minimum of five Simpson-thickness integration points (S4R, per Abaqus des- ignation). Other shell-thickness integration schemes were not tested in the current work; however, it is likely that an alternative thickness integration scheme, such as Gauss integration, would be adequate though that would require additional verification. The effect of the reinforcement was accounted for in the integration of the shell section. Each layer of reinforcement was assumed to act uniaxially, treated as a smeared layer with a constant thickness equal to the area of each reinforcing bar divided by the reinforcing bar spacing. In general, the mesh density shall be similar to the one used for the steel elements. At the locations in contact with steel- work (e.g., bottom slab haunches), the mesh density should be higher than the mesh density of the steelwork to ensure proper enforcement of the contact interaction. The reinforced shell element was found to be accurate in flexure-dominated applications; however, because of the material model used, shear damage will not be adequately modeled. In the concrete damage plasticity material model, the two failure modes are assumed to be tensile rupture and compressive crushing. While these assumptions approximate the shear inelastic behavior of concrete when employed in solid elements with relatively fine meshes, when used in shell element theory they fail to properly model inelastic shear behavior of concrete. Therefore, when employing shells to model the reinforced concrete slab, it was necessary to verify by hand that the nominal shear capacity of the slab was not exceeded (e.g., obtaining the shear stress values at the integra- tion points of the shell element and comparing then with the nominal concrete shear strength). Girders, stringers, and fabricated plate floor beams were modeled with four-node shells with reduced integration, hourglass control, and finite membrane strains (S4R elements per Abaqus designation). These elements are highly robust and have been found to provide accurate prediction of behav- ior of steel members, even in the highly inelastic range by many other researchers. A minimum of four elements along the flange width and the web height were found to give satis- factory results. Further, the maximum aspect ratio should be kept to about 5 and corner angles kept between 60° and 120°. It was also found that truss-type floor beams, bracings, and cross frames can be modeled with shell elements (as described for girders, stringers, and fabricated plate floor beams) or with beam elements by using at least three two-node linear shear–flexible (Timoshenko) beam elements (B31 elements per Abaqus designation) per component. When specifying the section properties, it is necessary to account for the eccentric- ity in the cross section of the member. The transfer of load taking place at the connections between steel members is modeled through the use of mesh ties. In a mesh tie, the motion of a slave surface or node group is linearly interpolated from the motion of a master surface or node group. When the nominal capacity of a connection was calculated to be lower than the nominal capacity of the connected members, additional procedures as described in Section 3.1.3.4 were applied. Vertical stiffeners were either explicitly modeled with four-node shells with reduced inte- gration, hourglass control, and finite membrane strains (S4R elements per Abaqus designation) or with kinematic coupling constraints. In a kinematic coupling constraint, the motion of a surface or a node set is constrained to the motion of a master node. When modeling stiffeners, kinematic couplings were used to prevent cross-sectional distortion. 3.1.3.2 Shear Stud Modeling The tensile and shear behavior of shear studs is critical in the load transfer between the steel members and the concrete slab in composite steel bridges, as they help provide addi- tional load paths after the failure of a primary steel member. The superior ability of composite steel bridges to transfer load was shown by Neuman (2009), who performed full- scale experiments in a simple-span twin-tub girder bridge that underwent failure of the bottom flange and web of one of the tub girders. Therefore, given their essential role in com- posite action, the behavior of shear studs needs to be prop- erly modeled to capture the transfer of load from a faulted composite member to the rest of the structure. To this end, a methodology was developed in NCHRP 12-87A to imple- ment shear, tensile, and combined shear and tensile behavior of shear studs in finite element models of steel bridges. The suggested methodology is valid for up to three transversely grouped shear studs. The primary focus of the shear stud behavior study was the calculation of tensile stiffness, strength, and inelastic behav- ior of transversely grouped shear studs. Shear studs under high tensile load may fail because of one of three modes: steel rupture of the shear studs’ shaft, pullout of the shear studs from the concrete slab (and/or haunch), or breakout of a sec- tion of the concrete slab (and/or haunch). The tensile force– displacement relations for shear stud groups is dependent upon these failure modes and requires different definitions

27 of the inelastic response of the shear stud group, as well as different expressions for the calculation of the initial stiffness, nominal tensile strength, and maximum cumulative tensile displacement. In general, concrete breakout strength is lower than the steel rupture strength or concrete pullout strength, hence becoming the governing failure mode. The concrete capac- ity design (CCD) approach in ACI 318-14 (ACI 2014) pro- vides the best approximation to calculate concrete breakout strength; however, this formulation does not consider the effects of the haunch. Mouras et al. (2008) developed a new modification factor for the CCD approach that considers the slab haunch effect. The existing methodology for the slab haunch effect is presented in the AASHTO LRFD BDS Sec- tion 6.16.4.3 Shear Connectors (AASHTO 2014). Nonethe- less, Mouras et al. (2008) performed a very limited number of experiments, which may not be enough to develop an accu- rate modification factor. Moreover, neither ACI 318-14 nor AASHTO LRFD BDS include any information about shear stud tensile stiffness and load-displacement behavior. During NCHRP 12-87A, it was concluded that the CCD approach needed to be enhanced and made suitable for implementation in finite element analysis procedures for steel bridges developed herein. With that goal, a finite ele- ment analysis methodology was developed to estimate the effect of several parameters on the concrete tensile breakout strength, stiffness, and ductility of several shear stud configu- rations. First, detailed finite element models were calibrated and benchmarked to the full-scale subassembly testing of shear studs noted in the ACI 355 database (ACI n.d.) and performed by Mouras et al. (2008). The finite element analy- sis procedures developed during the benchmark process were then used to conduct a parametric study in which the effects of several parameters on the tensile behavior of transversely grouped shear studs were assessed. In the parametric study, the tensile behavior was influenced by the following parame- ters: (1) concrete compressive strength, (2) shear stud height, (3) stud spacing in longitudinal direction, (4) stud spacing in transverse direction, (5) top flange width, (6) top flange thickness, (7) haunch thickness, and (8) number of shear studs in a group. A total of 80 finite element models were analyzed in the parametric study to develop load–displacement relations. Based on the results of the parametric study, tensile force– displacement relations dependent upon the dominant fail- ure mode were developed. When the failure mode is tensile rupture of the shear stud shafts, the behavior is initially linear elastic until the tensile yield strength of the shear stud shaft is reached, followed by plasticity with linear hardening. As yielding continues, failure is assumed to occur when the tensile rupture strength of the shear stud shaft is reached at a maximum axial displacement equal to 5% of the effec- tive stud height (i.e., height of shaft). A triangular load– displacement curve is characteristic of concrete breakout and shear stud pullout failure modes; the behavior is ini- tially linear elastic until the concrete breakout strength or the shear stud pullout strength is reached, followed by linear softening until the axial ductility of the shear stud group is exhausted. NCHRP 12-87A developed the following provisions to model the behavior of shear connectors: • Equations to calculate the initial stiffness of transversely grouped shear studs that account for the combined effect of the flexibility of the shear stud shaft, the concrete sec- tion under the head of the shear stud, and the bending stiffness of the flange. • Modification factors for the calculation of concrete break- out strength that account for the haunch effect—and other geometrical features—to be applied to the CCD expressions (ACI n.d.). • Modification factors applied to the expressions in ACI 318-14 to calculate steel rupture strength and con- crete pullout strength. These factors incorporate the effect of unequal load distribution among transversely grouped shear studs. • Maximum tensile displacement values dependent upon the governing failure mode and the number of shear studs in the group. In addition to the tensile load–displacement relations studied, it is necessary to define the shear behavior of trans- versely grouped shear studs to completely capture the behav- ior of the shear stud group. The shear force–displacement relations developed by Ollgaard et al. (1971) were employed in the current research and recommended. The nominal shear strength, nonlinear shear force–displacement behavior, and maximum shear displacement at failure are determined according to Ollgaard’s model, which is also prescribed in AASHTO LRFD BDS to calculate the shear resistance of shear studs. Ollgaard’s shear force–displacement relations are com- bined with the tensile force–displacement relations develop in NCHRP 12-87A through the shear–tension interaction equation in AASHTO LRFD BDS 6.16.4.3 (AASHTO 2014). To implement the shear stud behavior in finite element models of composite steel bridges, it is necessary to use connector elements. These elements are multidimensional springs for which coupled force–displacement curves can be assigned, which allow the engineer to characterize the stiff- ness, capacity, and ductility of the shear stud group at discrete locations. Linear or nonlinear force–displacement curves need to be assigned for each relative motion component. The implementation of the shear stud modeling methodology in finite element models of composite steel bridges was also

28 benchmarked against the full-scale experiments conducted by Neuman (2009). Further detail on the research conducted in NCHRP 12-87A with regard to shear stud behavior, as well as comprehensive explanations of the application procedures to implement coupled shear and tensile load–displacement relations in finite element models of composite steel bridges can be found in Appendix A. 3.1.3.3 Substructure Flexibility Modeling Fracture of main tension members can result in significant changes in the torsional stiffness of the structure. This may result in very large horizontal reaction forces at fixed sup- ports when rigid boundary conditions (prescribed displace- ments) are specified. Alternatively, there may be significant displacements at expansion bearings, with a concern being the member slips off the support. In either case, the support con- ditions must be accurately modeled. Since no forces should be generated at expansion bearings, the greater concern is at fixed bearings to which significant load may be transmitted. Assuming full fixity (i.e., zero translation) in a model may be unrealistic since (1) the support or bearing may fail under the extreme load, and (2) even small amounts of flexibility within the substructure or bearing will greatly reduce the forces attracted to the support. Hence, the reaction forces calculated without the inclusion of substructure flexibility may be greatly overestimated. Connector elements were used to account for substructure longitudinal and transverse flexibility. These elements allow for the definition of coupled force–deformation relations. Connector elements can be thought of as multidimensional springs. A Cartesian connector was determined to capture the intended behavior best. These elements provide a con- nection between two nodes, where the change in position is measured in three directions local to the connection. One of the nodes is fixed (or connected to ground), and the other node is the support point in the superstructure. The connec- tor element is rigid in the vertical direction and has a cou- pled linear–elastic relation in the two horizontal directions (longitudinal and transverse). Depending on the complex- ity of the supporting element (geometry, skewed, material homogeneity, and so on), the horizontal force–deformation relation can be obtained by hand calculations or through a simple FEA. Since elastic connector elements were used to characterize substructure flexibility, its application was limited to static and quasi-static phenomena. To include the effect of sub- structure flexibility in dynamic phenomena, the damping characteristics of the substructure would have to be included. Further, the interaction with the foundations themselves would have to be modeled. This is a complex task in itself and one that the authors are confident will not have appre- ciable effect on the overall dynamic response of the system. Therefore, during the calculation of the dynamic amplifica- tion factor used to account for sudden fracture, substructure flexibility was neglected. 3.1.3.4 Connection Failure Modeling When a connection is likely to fail before yielding of the member, an additional step—beyond the use of mesh ties to attach the components—may be necessary to capture connec- tion failure. Although it is possible to develop force/moment– displacement/rotation relations, which can be applied to a connector element, a simpler approach was developed and used in all models. If the members are directly connected to each other (i.e., there are no connection plates or other components that significantly affect the flexibility of the connection), a sec- tion at the end of the members can be treated as a fuse. That end fuse section will be assigned a failure criterion, based on the stress or load level that will produce failure of the connection. This level of stress or load will be different from the rest of the member. If connection plates exist—and they increase the flexibility of the connection—they should be included in addition to the previously described procedure. This approach was used in all models. As the load factors used in the analysis consider the statistical variance of both load and resistance, the strength of the fuse element input in the finite element model shall be the nominal (unfactored) strength. In general, for a bolted connection weaker than the con- nected members, the behavior can be modeled by a linear- elastic–perfectly plastic relation. The initial linear elastic stage is typically dictated by the combined bearing stiffness of the plate and the shear stiffness of the bolts. Henriques et al. (2014) provided expressions to evaluate the stiffness of basic components of a connection, which were adopted in the Eurocode 3 (Comité Européen de Normalisation 2007). The stiffness of a plate in tension kplate can be calculated as follows: =k EA P plate b where E is the modulus of elasticity of steel, A is the gross cross-sectional area of the plate, and Pb is the pitch distance parallel to the load transfer. The shear stiffness of a bolt group kbolts can be calculated as follows: = 12.7 2k n d fbolts b ub where nb is the number of bolts in the group, d is the diameter of the bolt, and fub is the nominal ultimate strength of the

29 bolt. The bearing stiffness of the plate can be calculated as follows: ( ) ( ) = = + + = 8 0.630 min 0.25 050, 0.25 0.375, 1.25 min 12 0.630 , 250 k in n k k df k e d p d k in t bearing b b t u b b b t where nb is the number of bolts in the group, d is the diameter of the bolt, fu is the nominal ultimate strength of the plate, eb is the edge distance parallel to the load transfer, pb is the pitch distance parallel to the load transfer, and t is the thick- ness of the plate. Once kplate, kbolts, and kbearing are calculated, the individual stiffness values can be combined through a series sum as follows: = + +    − 1 1 1 1 k k k k total plate bolts bearing where ktotal is the total stiffness of the connection, and k1, k2, . . . , kn are the individual stiffness contributions. The onset of the perfectly plasticity takes place at the capacity of the connection that can be calculated per the provisions in the AASHTO LRFD BDS (AASHTO 2014). The maximum displacement at failure may be calculated as the larger of 2.5 times the capacity to stiffness ratio and 0.18 times the diameter of the bolt as shown in the following expression (Sarraj 2007): =   max 2.5 , 0.18u R k dmax conn total b where umax is the displacement at failure, Rconn is the nominal (unfactored) connection capacity, ktotal is the stiffness of the connection, and db is the diameter of the bolt. 3.1.3.5 Contact Modeling Contact between the bottom of a concrete slab and the top of the steelwork (top flanges of girders and stringers) was modeled using two approaches. The first approach was a mesh tie between all exterior nodes of the slab and the top flanges of girders and slabs, and it was used before the slab hardened. In a mesh tie, the motion of a slave surface or node group is linearly interpolated from the motion of a master surface or node group. In this case, the exterior nodes of the slab were slaved to the top flanges of girders and stringers. In this way, the deformation of the slab conformed to the defor- mation of the steelwork while preventing unrealistic sagging between supporting elements. This approach also prevented the wet concrete barrier from tipping over. Once concrete hardens, it is necessary to allow separation between the steel and the concrete and to consider the fric- tional behavior. This was achieved by substituting the previ- ously applied mesh tie with a contact interaction. The normal behavior of the contact interaction is modeled through a penalty stiffness. The penalty stiffness is several orders of magnitude larger than the normal stiffness of the underlying contacting elements and allows an infinitesimal penetration so a pressure can be calculated. The tangential behavior of the contact interaction is modeled through an algorithm based on Coulomb friction with a limit on the allowable shear. A fric- tion coefficient (µ) of 0.55—as suggested by Lai et al. (2014) and Lai and Varma (2015)—and an interfacial shear strength (tLIMIT) of 0.06 ksi, based on the lower bound provided in the commentary of the American Institute of Steel Construction (AISC) AISC 360-10, were specified (AISC 2010). 3.1.3.6 Loads and Boundary Conditions Two types of loads were applied in the finite element models: body forces and surface tractions. Body forces were applied for component dead loads (DC per AASHTO desig- nations). These are simply the product of mass, gravity, and applicable load factors. Surface tractions were applied for traffic live loads (LL per AASHTO designation). These are based on the HL-93 load model described in the AASHTO LRFD BDS (AASHTO 2014). None of the bridges modeled included any bituminous pavement. However, if the load effect of the pavement must be included, it may be modeled by specifying a layer of relatively soft solid elements (analyst should refer to SHRP-A-388, as asphalt stiffness varies signifi- cantly with temperature [Tayebali et al. 1994]) and applying the correspondent body force. Component dead loads were linearly applied in the initial implicit static analysis. Traffic live loads and dynamic ampli- fications were applied in the final explicit dynamic analysis. Their dynamic effects were minimized by applying them slowly through the use of smooth step, per the following equation as implemented in Abaqus: ( ) ( ) ( )( ) = − +6 15 105 4 3LLR t tT tT tT where LLR(t) is the fraction of live load at a load application time t, and T is the duration of the load application (Simulia 2017). The duration of the load application was larger than the fundamental period of the structure to avoid vibration of the structure effects. It must be noted that the dynamic amplification effects are captured through the use of dynamic

30 amplification factors, which are discussed in Section 3.2 and Section 3.4. Although analyses focused on the dynamic behavior of the structure were carried out to determine the dynamic amplification factor, the FEA methodology was developed so that its application did not require such com- plex analysis. With regard to prescribed boundary conditions, at all sup- port points vertical translations were assumed to be zero. For horizontal (longitudinal and transverse) support conditions at fixed support points, two approaches were taken. If the modeled phenomenon was static or quasi-static, substruc- ture flexibility was included, as described in Section 3.1.3.3. For dynamic phenomena, the horizontal translations at fixed supports were assumed to be zero (i.e., substructure flexibil- ity is ignored). 3.2 Proposed Load Model The load model developed herein was specifically crafted to reasonably represent the loading conditions that take place during the event for which the structure is evaluated. In the current case, such an event is the sudden failure of a primary steel tension member and the subsequent extended service period in the faulted state. Based on the research, which has included input from a wide array of stakeholders, these are two scenarios that need to be considered separately. As docu- mented in Section 2.1.2, bridges that have undergone failure of a member designated as an FCM and do not collapse will likely provide service after the failure has occurred; therefore, it is not sufficient to only establish that the bridge will survive the sudden failure. In addition, the post-failure strength of the system needs to be checked, since the bridge remains in service in the faulted state for a finite period of time. It may be argued that in current design and evaluation procedures the strength of the system is not checked after the effects of some event, such as an earthquake or a barge impact. This is reasonable, as the bridge under these considerations is in an unfaulted condition (i.e., a member is not specifically assumed to have failed)—as in the case of system analysis described herein—nor is such member failure permitted. For example (based on the procedures in the AASHTO LRFD BDS), when an engineer has designed a bridge for Strength I and is checking Extreme Event II, there is no need to re-check Strength I with possible damage incurred because of the earthquake included. If the bridge does not have ade- quate capacity to meet Extreme Event II, the design needs to be revised and reevaluated for all load combinations, but in no case is the bridge designed to operate in a faulted condition. Another possible argument against the consideration of two loading scenarios was that they could and should be com- bined into a single load combination. While that would be convenient in a redundancy evaluation, unless one of the two loading scenarios always controls, the combination of both scenarios will result in an overly conservative load combina- tion. Redundancy as a single-limit state does not necessarily mean that one load combination suffices. Strength is a single- limit state, as well. Yet, it requires several load combinations representing different scenarios, as currently stipulated in the AASHTO LRFD BDS. For example, the current AASHTO LRFD BDS does not use an artificially low strength-reduction factor (phi) for net section fracture to avoid checking block shear. Though one could take this approach to save time in design, it is not economical and actually masks true behav- ior. Accordingly, since redundancy has two loading scenarios, two loading combinations were developed in which the fail- ure event and the post-failure service period were not consid- ered separately. These loading combinations are referred to as Redundancy I and Redundancy II. The Redundancy I load combination is intended to cap- ture the dead and live load that the bridge is subjected to at the instant in time at which the failure of a primary steel ten- sion member occurs. The Redundancy I load combination is somewhat analogous to the Extreme Event II load combi- nation, as the recurrence interval of the extreme event (i.e., member failure plus dynamic amplification) is assumed to exceed the design life of the bridge, and the probability of the simultaneous concurrence of the maximum vehicular live load and the extreme event (or member failure plus dynamic amplification) is very low. With regard to this last statement, the likelihood of fracture heavily depends on material tough- ness, the degree of constraint, magnitude of residual stresses, temperature, and detailing, all of which have as large if not larger influence on the likelihood of fracture as the level stress produced by vehicular live load. The Redundancy II load combination is intended to cap- ture dead and live load relating to the normal vehicular use of the bridge in the faulted condition without wind load, similar to the Strength I load combination. While in the Redundancy I load combination, the dynamic amplification of load caused by sudden failure of a primary member may occur, with levels of live load representing average or point- in-time conditions. The Redundancy II load combination requires larger return periods on the live load, as it is associ- ated with an extended service period. Since the loading com- binations calculated in the current research need to be valid independent of the inspection strategy, statistical parameters for a return period of 75 years were used, as in the initial calibration of the AASHTO LRFD BDS. Live load return periods lower than 75 years can only be jus- tified when coupled to a particular inspection strategy. Stipu- lation of inspection strategies or procedures that differ from ruling mandates is not within the scope of the work reported here. Hence, the assumption throughout the load study for redundancy evaluations assumes that a member redesignated

31 as SRM would not be subjected to FCM hands-on inspec- tions again. Additionally, since the calculated load factors are rounded to 1⁄20 (i.e., 0.05), and the statistical parameters for live loads with return periods between 5 and 75 years are very similar, prescribing a lower return period for live load based on an inspection rationale will not have a significant effect on the calculated load factors. 3.2.1 Target Reliability Level in the Faulted State Despite the similarities between Extreme Event II and Redundancy I, as well as Strength I and Redundancy II, there is a fundamental difference: the state of the structure. While the load combinations in the AASHTO LRFD BDS apply to unfaulted bridges, Redundancy I and Redundancy II apply to the bridge in the faulted condition, which changes what constitutes a reasonable target reliability level for the faulted structure. In other words, a bridge in which a primary member has failed cannot be expected to operate at the same reliability level of an intact bridge. The authors of the current research hosted an expert consensus meeting with owners, fabricators, designers, and other researchers to gain insight into establish- ing the appropriate target reliability levels for Redundancy I and Redundancy II load combinations. In other words, the appropriate reliability level for safe bridge operation after complete failure of a member is traditionally classified as an FCM. After much discussion, the consensus was that a target reliability index (β) equal to 2.5 was thought to be reasonable for a bridge in which a primary member has failed. During the meeting, attendees overwhelmingly felt that the implementation of the FCP was successful in lowering the probability of fracture in steel tension members, as sup- ported by the lack of cases of failure of FCMs fabricated to the FCP in the literature. Unfortunately, a meaningful, formal, and accurate estimation of the probability of fail- ure of an FCM—or a quantitative analysis of the benefits of FCP fabrication procedures—is not possible, as there is not sufficient data available. While one could make up data, it would imply a level of sophistication and accuracy that is simply not justified. Rather than perform such an academic exercise, the authors took a risk-based approach to incorpo- rating the benefits of the FCP. In such an approach, conse- quence and likelihood are treated separately when analyzing risk (risk = consequence • likelihood). In the AASHTO LRFD BDS, the strength-limit state load combinations are calibrated to a target reliability index (β) equal to 3.5, which equates to a failure rate of 1/4,300. In other words, the accepted probability of the loads associated normal vehicular use of the bridge exceeding the capacity of the bridge (Strength I) is equal to 1/4,300 (consequence is 1/4,300). Further, since normal vehicular use is always going to take place (i.e., likelihood is 1/1), the total risk of bridge failure caused by normal vehicular use is 1/4,300 (again, risk = consequence • likelihood = 1/1 • 1/4,300 = 1/4,300). With regard to Redundancy II, which can be described as normal vehicular use of the bridge in the faulted stated, the target reliability index (β) is equal to 2.5, which equates to a failure rate of 1/160. Therefore, given that the failure of a primary member has occurred, the accepted probability of normal vehicular use resulting in bridge failure is 1/160 (con- sequence is 1/160). Since a total risk of 1/4,300 is deemed acceptable for Strength I, it is reasonable to adopt the same total risk of bridge failure caused by normal vehicular use in the faulted state (risk is 1/4,300), which requires the maxi- mum probability of a primary member failing to be at a rate of 1/27 (likelihood = risk / consequence = 1/4,300 / 1/160 = 1/27). After consulting with owners, engineers, and fabrica- tors, a primary member failure probability of 1/27 was con- sidered adequately conservative, based on the observed cases of primary member failure. In other words, for bridges not fabricated to Section 12 of the AWS D1.5, the overall engi- neering opinion of the consensus meeting participants was that member failures do not occur at such a high rate. Hence, the use of a target reliability index (β) equal to 2.5 is conser- vative for this family of structures, as the overall failure rate is no greater than associated with Strength I. As stated during the consensus meeting, participants agreed that when a bridge is constructed in accordance with Section 12 of the AWS D1.5 the likelihood of member failure caused by fatigue or fracture would be expected to be much lower than 1/27. Thus, one could effectively use a lower target reliability in the faulted state (i.e., a lower β) while still achiev- ing the same overall failure rate if the likelihood of member fracture was lower. The authors then explored using a target reliability index of the Redundancy II load combination (β) equal to 1.5, which corresponds to a failure rate of 1/15. If one maintains that the accepted bridge failure risk is 1/4,300 (risk = 1/4,300), and the probability of load exceeding the resistance in the faulted stated is 1/15 (consequence = 1/15), then the required maximum failure probability for a primary member constructed to Section 12 of the AWS D1.5 is 1/290 (likelihood = risk / consequence = 1/4,300 / 1/15 = 1/290). In other words, members would have to fail at a rate of 1/290 or more to result in a lower overall reliability in the faulted state than in Strength I. During the consensus meeting, attendees unanimously agreed that the failure rate of members built to Section 12 of the AWS D1.5 Bridge Welding Code was less than 1/290. (Considering that no such failures have ever been observed in members fabricated to the FCP, the rate is clearly much less than 1/290.) In summary, those bridges in which primary steel tension members have been fabricated to Sec- tion 12 in the AASHTO/AWS D1.5M/D1.5—or equivalent— may be evaluated for a target reliability index (β) equal to 1.5.

32 3.2.2 Load Factors for Redundancy II Load Combination Once the required reliability level is established for the load combinations to be used in the analysis, the load factors can be calculated using the same procedures used in the develop- ment of those already included in the AASHTO LRFD BDS. For ease of understanding, the calculation of load factors for the Redundancy II load combination are explained first. As previously mentioned, the Redundancy II load combina- tion is used to evaluate the ability of the bridge to provide an extended period of service after the failure of a primary mem- ber has occurred; that is, strength against normal vehicular use of the bridge (Strength I load combination in AASHTO LRFD BDS) in the faulted state. Therefore, the load factor calculation procedure is a recalculation of Strength I for lower reliability levels following the methodology in NCHRP Report 368 (Nowak 1999). The work summarized in NCHRP Report 368 is focused on the calculation of load and resistance factors for a design equation that is as follows: ∑φ ≥ γ ,R Qn i n i i where φ is the resistance factor; Rn is the nominal resistance; and for a particular load effect i, γi is the load factor; and Qn,i is the nominal load effect. These load and resistance factors are calculated through structural reliability methods to take into account the variability of load and resistance. In NCHRP Report 368, a modification of the algorithm developed by Rackwitz and Fiessler (1977) was used to calculate reliability indices for a number of representative load components and resistance for bridges. A similar procedure, also based on the algorithm of Rackwitz and Fiessler (1977), was developed to calculate load factors for lower reliability levels in the current research. To verify the validity of the procedures developed, the authors recalculated the reliability indices for representa- tive load components and resistances for steel girder bridges (composite moment, noncomposite moment, and shear) that were calculated in NCHRP Report 368. The reliability indices (β) calculated for the load factor design (LFD) load combi- nation (1.3(DC + DW) + 2.17(LL + IM)) and the LRFD load combination proposed in NCHRP Report 368 (1.25DC + 1.50DW + 1.75(LL + IM)) are shown in Table 2. (The data used as input in the calculations and comparison can be found in Appendices E and F in NCHRP Report 368). The average percentage difference in the calculation of reliability indices is 0.99%, and the maximum error is 5.68%. These are very slight discrepancies attributed to the rounding of the data input into the calculation procedure. Therefore, the calculation proce- dure used by the authors was concluded to be accurate and consistent with the methods employed in the calibration of strength-limit states in the AASHTO LRFD BDS. After completing the benchmarking of the calculation pro- cedure, load factors were calculated for different reliability indices. To do so, the definition of load factors with regard to the bias and coefficient of variation of each load component used in NCHRP Report 368 is used. ( )γ = λ +1 kVi i i where k is a scalar associated with a particular reliability index; and for a particular load effect i, γi is the load factor; li is the bias; and Vi is the coefficient of variation. For each of the rep- resentative load components and resistances for steel girder bridges in NCHRP Report 368 (Tables E-3, E-4, and E-7 in Appendix E of NCHRP 368), reliability indices (β) are calcu- lated for incremental values of k with the purpose of obtaining load factors that guarantee minimum reliability indices. The results of this calculation are summarized in Table 3. In the calculation of the load factors presented in Table 3, a resistance factor φ, equal to 1.0 was selected. When attempting to accurately model nonlinear behavior, simply reducing the capacity of a material or component through resistance fac- tors in FEA is not appropriate, as there is a direct impact on the predicted behavior. For example, the finite element models in the current research require nonlinear material constitutive models and nonlinear element formulations. The application of resistance factors when the response is not linear elastic is questionable at best. Further, some resistance factors are tied to design assumptions used to calculate member capacities, such as the effective width of composite girders. Some resistance factors have also been established to address the consequences associated with a failure mode. As an overall example, resistance factors for net section fracture are different than those for gross section yielding. Including these factors in the FEA methodol- ogy will result in the predicted response being markedly differ- ent, and in some cases incorrect, than if the nominal resistance values had been used. Hence, the calculated load factors are to be employed in analyses in which the modeled capacity is not factored (φ = 1.0), acknowledging that the uncertainty in resis- tance is accounted for in the load factor calculation procedure. The resulting load factors for reliability indices (β) equal to 3.5, 2.5, and 1.5 are summarized in Table 4. In the AASHTO LRFD BDS and the AASHTO MBE it is customary to round the calculated load factor to 1/20 (0.05). The rounding pro- cedure and combination that was performed for the Strength I load combination was approximately followed for the Redun- dancy II load combination, as shown in Table 4, for bridges not built in accordance with Section 12 of the AWS D1.5 (β = 2.5) and for bridges built in accordance with Section 12 of the AWS D1.5 (β = 1.5). Reliability indices (β) were recal- culated for each of the representative load components and

33 Component Span(ft) Space (ft) (LFD) (LRFD) NCHRP 368 Calculated Difference (%) NCHRP 368 Calculated Difference (%) Noncomposite moment 30 4 2.00 2.02 0.97 3.77 3.78 0.26 6 2.66 2.64 0.91 3.78 3.73 1.19 8 3.10 3.08 0.69 3.78 3.75 0.76 10 3.43 3.39 1.26 3.78 3.73 1.23 12 3.69 3.67 0.66 3.78 3.74 1.14 60 4 2.90 2.87 1.02 3.75 3.71 1.19 6 3.54 3.52 0.58 3.75 3.74 0.27 8 3.96 3.95 0.14 3.76 3.72 0.93 10 4.25 4.27 0.46 3.75 3.72 0.86 12 4.47 4.51 0.91 3.75 3.70 1.24 90 4 2.85 2.80 1.70 3.71 3.68 0.71 6 3.39 3.36 1.01 3.71 3.66 1.29 8 3.76 3.75 0.31 3.71 3.67 1.08 10 4.03 4.03 0.10 3.71 3.71 0.13 12 4.22 4.24 0.54 3.69 3.67 0.56 120 4 2.75 2.70 1.70 3.68 3.68 0.08 6 3.24 3.20 1.16 3.68 3.67 0.25 8 3.57 3.54 0.75 3.68 3.62 1.54 10 3.81 3.80 0.32 3.67 3.64 0.95 12 3.99 3.99 0.04 3.66 3.66 0.06 200 4 3.19 3.15 1.15 3.57 3.57 0.03 6 3.56 3.55 0.35 3.58 3.55 0.91 8 3.82 3.83 0.26 3.59 3.57 0.57 10 4.00 4.02 0.58 3.58 3.58 0.12 12 4.12 4.17 1.10 3.56 3.54 0.44 Composite moment 30 4 2.00 2.01 0.74 3.77 3.77 0.07 6 2.66 2.65 0.53 3.78 3.74 0.96 8 3.10 3.08 0.74 3.78 3.75 0.82 10 3.43 3.39 1.30 3.78 3.73 1.26 12 3.69 3.67 0.49 3.78 3.74 0.97 60 4 2.90 2.87 1.10 3.75 3.70 1.27 6 3.54 3.52 0.68 3.75 3.74 0.37 8 3.96 3.95 0.20 3.76 3.72 0.98 10 4.25 4.27 0.42 3.75 3.72 0.88 12 4.47 4.51 0.89 3.75 3.70 1.23 90 4 2.84 2.80 1.46 3.71 3.68 0.80 6 3.39 3.36 0.99 3.71 3.66 1.28 8 3.76 3.75 0.39 3.71 3.67 1.16 10 4.03 4.03 0.11 3.71 3.71 0.13 12 4.23 4.24 0.32 3.69 3.67 0.52 120 4 2.74 2.69 1.79 3.68 3.67 0.35 6 3.24 3.20 1.25 3.68 3.67 0.35 8 3.57 3.54 0.83 3.68 3.62 1.62 10 3.81 3.80 0.37 3.67 3.63 1.00 12 3.99 3.99 0.06 3.66 3.66 0.07 200 4 3.18 3.15 1.01 3.58 3.56 0.43 6 3.56 3.55 0.30 3.59 3.55 1.13 8 3.83 3.83 0.11 3.60 3.57 0.71 10 4.01 4.03 0.52 3.58 3.59 0.37 12 4.13 4.17 0.93 3.56 3.55 0.34 Table 2. Comparison of reliability indices (a) for representative load components and resistance for steel girder bridges. (continued on next page)

34 Shear 3.30 3.94 1.57 3.86 3.95 0.49 4.27 3.95 2.57 4.53 3.95 0.12 4.83 3.95 1.50 2.53 4.01 2.27 3.15 4.01 1.52 3.73 4.01 1.68 3.84 4.01 0.72 4.15 4.00 1.98 2.16 3.81 3.88 2.48 3.81 0.45 2.86 3.82 0.36 3.02 3.81 2.51 3.32 3.81 0.22 1.87 3.98 0.76 2.33 3.98 0.56 2.71 3.98 1.08 2.83 3.98 0.85 3.05 3.96 0.57 30 60 90 120 200 2.30 4.00 2.19 2.72 4.01 0.90 2.99 4.02 0.43 3.21 4.01 0.81 4 6 8 10 12 4 6 8 10 12 4 6 8 10 12 4 6 8 10 12 4 6 8 10 12 3.36 3.90 4.36 4.49 4.69 2.66 3.23 3.66 3.85 4.06 2.04 2.53 2.92 3.11 3.32 1.92 2.37 2.71 2.89 3.07 2.32 2.74 3.02 3.22 3.37 3.33 1.88 0.95 2.02 0.99 3.07 5.00 2.41 1.80 0.29 2.28 5.68 1.89 2.14 2.93 0.01 2.73 1.66 0.07 2.04 0.53 0.88 0.57 1.14 0.45 1.16 3.99 3.88 3.93 3.85 3.95 4.01 3.92 3.95 4.08 3.98 4.08 3.96 3.79 3.81 3.71 3.82 3.95 4.00 4.02 3.95 3.98 4.09 4.05 4.04 4.04 4.02 0.80 Component Span(ft) Space (ft) (LFD) (LRFD) NCHRP 368 Calculated Difference (%) NCHRP 368 Calculated Difference (%) Table 2. (Continued). kMAX DC,factory made DC,cast in place DW LL 0.5 -0.75 0.97 0.97 0.81 1.04 1 -0.33 1.00 1.02 0.92 1.13 1.5 0.12 1.04 1.06 1.03 1.23 2 0.59 1.08 1.11 1.15 1.33 2.5 1.10 1.12 1.17 1.28 1.44 3 1.63 1.16 1.22 1.41 1.55 3.5 2.19 1.21 1.28 1.55 1.67 4 2.78 1.26 1.34 1.70 1.80 Note: kMAX = maximum scalar computed for a reliability index, DC = load factor for dead load of structural components and nonstructural attachments, DW = load factor for dead load of wearing surfaces and utilities, and LL = load factor for vehicular live load. Table 3. Calculation of steel girder bridge load factors for target reli- ability levels (a). DC DW LL 3.5 1.21–1.29 1.55 1.67 3.5 Rounded 1.25 1.50 1.75 2.5 1.12–1.17 1.28 1.44 2.5 Rounded 1.15 1.25 1.50 1.5 1.04–1.06 1.03 1.23 1.5 Rounded 1.05 1.05 1.30 Table 4. Load factors for Strength I (a = 3.5), Redundancy II (a = 2.5), and Redundancy II for bridges built to AWS D1.5, Section 12 (a = 1.5). resistances for steel girder bridges with the rounded load fac- tors to ensure that they met the target reliability levels. The resulting Redundancy II load combination is as follows: [ ] [ ] ( ) ( ) + + + + + +       : 1.15 1.25 1.50 1.5, .12 1.05 1.05 1.30 1.5, .12 Redundancy II DC DW LL IM NOT built to AWS D Sec DC DW LL IM built to AWS D Sec

35 where DC = dead load of structural components and nonstruc- tural attachments, DW = dead load of wearing surfaces and utilities, LL = vehicular live load, and IM = dynamic load allowance. 3.2.3 Load Factors for Redundancy I Load Combination After the load factors of the Redundancy II load combina- tion have been calculated, the load factors for the Redundancy I load combination can more easily be defined. As previously stated, the Redundancy II load combination is used to address the strength of the bridge in the faulted state after the failure of a primary member has occurred, while the Redundancy I load combination is intended to characterize the loading conditions that take place during sudden failure of a primary member. Because of the similarities between the events char- acterized by the Extreme Event II load combination in the AASHTO LRFD BDS and the occurrence of primary mem- ber failure with regard to recurrence interval, the selection of load factors for the Redundancy I load combination mirrors the extreme event rationale. When considering an event for which the recurrence inter- val exceeds the design life of the structure and that is not fully correlated with the level of live load on the bridge, a calcula- tion of load factors based on the structural reliability principles used in NCHRP Report 368 is not possible because there is a lack of data to produce a statistically meaningful analysis. As a result, Extreme Event I and Extreme Event II are often referred to in the engineering community as noncalibrated load com- binations. With regard to the Extreme Event II load combi- nation, the most similar scenario to the Redundancy I load combination, the load factors for dead loads are the same as in the Strength I load combination in the AASHTO LRFD BDS at a reliability level of 3.5. In addition, all of the strength and extreme event load combinations in which vehicular live load is included as a load effect have the same load factors for permanent loads. Moreover, NCHRP Report 489: Design of Highway Bridges for Extreme Events, developed six proposed load combinations for various extreme events in which the dead load factors were the same as in the Strength I load com- bination (Ghosn et al. 2003). This approach actually makes sense, as the variability and bias associated with dead load is not related to the Extreme Event II load and live load. Thus, to ensure a consistent level of reliability (i.e., 3.5), the load factors for dead load remain unchanged. As the rationale supporting the specification of the dead load factors of the Strength I load combination for the Extreme Event II load combination is followed, the dead load factors calculated for the Redundancy II load combination can also be used in the corresponding Redundancy I load combination. Although a formal analytical procedure is not employed, it is reasonable that the actual dead loads of the bridge, the uncertainty on the estimate of such dead loads, and the target reliability level of the analysis do not change during the redundancy evaluation. Therefore, the dead load factors should not be different for Redundancy I and Redun- dancy II load combinations. However, to complete the Redundancy I load combination a load factor for live load is required. This is not a straight- forward issue, as live load is neither noncorrelated nor fully correlated with the occurrence of primary member failure. As previously discussed, an engineer will probably feel that the failure event is most likely associated with a sudden overload of the bridge. But experience has shown that there are several factors that are as critical—if not more critical—in provok- ing fracture as the live load level (e.g., material toughness, the degree of constraint, magnitude of residual stresses, tempera- ture, detailing, corrosion, and effectiveness of inspection). On the other hand, it cannot be defended that live load will have no effect on the occurrence of the member failure. Therefore, as live load is partially correlated with member failure, the live load factor should be between that of a noncorrelated and a fully correlated event. In the AASHTO LRFD BDS, the live load factor in the Extreme Event II load combination is 0.5 because accord- ing to the commentary, it “signifies a low probability of the concurrence of the maximum vehicular live load and the extreme events,” which is also recommended in Extreme Event I, as it is “reasonable for a wide range of values of aver- age daily truck traffic.” NCHRP Report 489 suggests a live load factor equal to 0.25 when evaluating the load effects of earthquake (Extreme Event I) and vessel collision (Extreme Event II), which are noncorrelated with the level of live load. In the current research, for the live load factor in the Redun- dancy I load combination, a conservative lower bound of 0.5 was selected, while the upper bound is set at 1.30 for a bridge built to Section 12 of the AWS D1.5 and 1.50 when it is not, as discussed above. As it is not possible to establish a level of correlation between the failure event and the level of live load to formally calculate the load factor, the middle of the intervals was selected, resulting in a live load factor of 0.90 for bridges built to Section 12 of the AWS D1.5 and 1.00 for all other bridges. However, for bridges built to the modern FCP, designers typically employ other factors that are not explicitly included in D1.5 and that would reduce the likelihood of fracture, such as improved fatigue detailing and designing for infinite fatigue life that are not readily quantifiable with regard to

36 reliability. To account for these benefits, the live load factor for bridges built to Section 12 of the AWS D1.5 was reduced from 0.90 to 0.85. The final resulting load factors for Redun- dancy I are as follows: [ ] [ ] ( )( ) ( )( ) + + + + + +       : 1 1.15 1.25 1.00 1.5, .12 1 1.05 1.05 0.85 1.5, .12 Redundancy I DA DC DW LL NOT built to AWS D Sec DA DC DW LL built to AWS D Sec R R Although the actual level of correlation between live load and the occurrence of the member failure is unknown, it is unlikely that the load effects during the failure of a pri- mary member will exceed those estimated by the Redun- dancy I load combination for the following reasons: First, the dynamic amplification factor DAR, recommended for redun- dancy evaluations when the user does not perform his own nonlinear dynamic analysis, is 0.40. This is an upper bound estimate, based on the values calculated as described in Sec- tion 3.4, and is conservatively applied to both dead load and live loads. Second, in the analysis it is assumed that the vehi- cle remains in the worst position (i.e., parked) at the exact moment in which the member fails, though in all likelihood it would be moving. Third, the presence of vehicles will likely reduce the dynamic effects of the fracture because of load stiffening and the damping effects of the contact interaction between the vehicles and the slab. Nevertheless, this conser- vative approach was intentionally taken as the actual level of correlation between live load, and the occurrence of the member failure is unknown. 3.2.4 Vehicular Live Load Model and Application The vehicular live load (LL) to be used in the Redundancy I and Redundancy II load combinations is the HL-93 live load model as described in Section 3.6.1.1 of the AASHTO LRFD BDS. This load model was selected because it is the reference load model used in NCHRP Report 368, in which data were employed in the calculation of load factors. While other live loads—such as those commonly used for ratings—could be considered, it is reasonable to use the HL-93 as described above for the specific provisions contained in the current report. However, the proposed specifications state that own- ers may use other vehicles that they have determined to be more appropriate. With regard to the vehicular dynamic load allowance (IM), 15% of the static effects of the design truck or tandem com- ponent of the HL-93 live load model is prescribed, which shall only be applied in the Redundancy II load combina- tion. The dynamic effects of interest in the Redundancy I load combination are those resulting from sudden member failure, which are far greater than the dynamic vehicular load allowance since they are applied to both dead and live loads. Thirty-three percent is prescribed in the AASHTO LRFD BDS for all limit states other than fatigue and fracture; but 33% is considered an overly conservative vehicular dynamic load allowance for the purposes of a redundancy evaluation. Moreover, in NCHRP Report 368, it is mentioned that for two side-by-side trucks, the mean vehicular dynamic load allow- ance is 10%; for one truck, it is 15%, both with a coefficient of variation of 0.8. To use the Redundancy I and Redundancy II load com- binations, it is necessary to specify the longitudinal and the transverse positioning of the HL-93 live load model in its application to FEA. The definitions of these are based on the intent of the load combinations developed herein. As previously stated, the Redundancy I load combina- tion aims to describe the point-in-time load scenario that takes place during the failure of a primary member, which implies a live load level that is between an average and a long return period. On the other hand, the Redundancy II load combination is effectively an evaluation of strength in the faulted against for normal vehicular use of the bridge without wind. For the Redundancy I load combination—regardless of the number of normal travel lanes carried by the bridge—it was deemed to be unrealistic to position vehicles outside of the striped lanes, despite occasional positioning of vehicles out- side the striped lanes. This approach was also recommended by the participants of the aforementioned consensus meeting described. Therefore, the maximum number of lanes that are required to be analyzed in the Redundancy I load combination is the number of striped lanes, and the HL-93 load model shall be centered within the lane. This is consistent with the ratio- nale used in the development of the Redundancy I load com- bination and the characteristics of a level of live load partially correlated with the occurrence of primary member failure. If the bridge is re-striped for additional lanes, the analysis needs to be repeated so that the new lane configuration is evaluated. The transverse positioning of live load in the Redun- dancy II load combination needs to be considered differently because of the longer exposure period associated with this load combination. Hence, the transverse position of live load in the Redundancy II load combination shall be as stipulated in Section 3.6.1.1.1 of the AASHTO LRFD BDS. This requires that the number of lanes is as described with the design truck and the 10.0-ft loaded width within each lane transversely positioned to produce the largest demands on the remaining intact components of the bridge. The longitudinal positioning of the truck must also be considered in Redundancy I and Redundancy II. The objec- tive (in both Redundancy I and Redundancy II load com- binations) is to maximize the load effects on the remaining

37 intact components of the bridge in the faulted state. When positive flexure is being evaluated, the centroid of the truck component of the HL-93 live load model shall be positioned longitudinally coincident with the location of the primary member failure under consideration, with a proposed fixed axle spacing of 14 ft. When the intent is to evaluate negative flexure, the provisions described in the third bullet of Sec- tion 3.6.1.3.1 of the AASHTO LRFD BDS are proposed: “For negative moment between points of contraflexure under a uniform load on all spans, and reaction at interior piers only, 90% of the effect of two design trucks spaced a minimum of 50.0 ft between the lead axle of one truck and the rear axle of the other truck, combined with 90% of the effect of the design lane load. The distance between the 32.0-kip axles of each truck shall be taken as 14.0 ft. The two design trucks shall be placed in adjacent spans to produce maximum force effects.” Finally, with regard to multiple presence of live load, the provisions described in Section 3.6.1.1.2 in the AASHTO LRFD BDS need to be applied in both Redundancy I and Redundancy II load combinations. The main reason for the application of multiple presence factors is that the statis- tical parameters with regard to vehicular live load use in the calibration of the AASHTO LRFD BDS were obtained by instrumenting mostly two-lane bridges. Further work concluded that a single vehicle 20% heavier than each of a pair of vehicles may have the same probability. Conversely, as the number of lanes increases, the probability of simul- taneous presence of heavy vehicles in every lane decreases. Nevertheless, in the aforementioned consensus meeting conducted with owners, fabricators, designers, and other researchers, the use of multiple presence factors was sup- ported. Despite the recommendations with regard to mul- tiple presence of live load, the authors found that it may not be necessary to evaluate for more than two lanes, as discussed in Section 3.2.4.1. 3.2.4.1 Considerations with Regard to Application of Vehicular Live Load Model The authors of the current research strived to develop a load model that reliably characterizes the loading condi- tions required to evaluate redundancy in steel bridges, while maintaining consistency with current governing AASHTO provisions that have been shown to be successful over the years. During the development of the proposed load model, multiple discussions were held with owners, fabricators, designers, and other researchers. Their inputs were taken into account, and the load model previously described is widely acknowledged by these expert stakeholders. However, more recent data regarding simultaneous presence of truck loads may prompt further discussion with regard to the number of lanes to be considered in the analysis and the use of multiple presence factors. For example, after reviewing the development of the current multiple presence factor criteria, more recent data and research on multiple presence were also reviewed. Based on the results of the SHRP 2 Report S2-R19B-RW-1: Bridges for Service Life Beyond 100 Years: Service Limit State Design (Modjeski and Masters, Inc. 2015), NCHRP Report 721: Fatigue Evaluation of Steel Bridges (Bowman et al. 2012), work by Fu et al. (2013), as well as other references, the likelihood of two fully corre- lated trucks positioned side by side on a two-lane bridge is a relatively rare loading condition. Further, the occurrence of two fully correlated trucks placed side by side on a single-lane bridge is much rarer. Thus, the SHRP 2 Report S2-R19B-RW-1 suggested that a single lane of the HL-93 vehicle be used for the service-limit state, even for bridges that are intended to carry more than one lane. However, the SHRP 2 Report S2-R19B- RW-1 was focused on serviceability-limit states. This was a reasonable recommendation for the service-limit states since multiple trucks can occur many times with no measureable impact on serviceability. However, for limit states that consider strength, it must also be recognized that it only takes one such scenario to compromise resistance. Hence, more than one lane should be considered for the Redundancy I and Redundancy II load combinations on bridges that are intended to carry more than one lane (i.e., when there are two or more striped lanes). The conclusion of SHRP 2 Report S2-R19B-RW-1 (Modjeski and Masters, Inc. 2015), NCHRP Report 721 (Bowman et al. 2012), and Fu et al. (2013) support that no more than two lanes should be considered in strength evaluations. The work by Fu et al. (2013) included weigh-in-motion data of about 68 million trucks gathered over a period of 436 months at 43 sites. The study demonstrated that the current multiple presence factors in the AASHTO LRFD BDS are very con- servative. This study resulted in the following equations for calculation of multiple presence factor MPF, as a func- tion of span length, average daily truck traffic ADTT, and number of lanes N for the evaluation of fatigue and strength (Fu et al. 2013): 0.988 6.87 10 4.01 10 0.0107 1 0.081 1.08 10 1.33 10 2.10 1 5 5 1 3 4 1 = + + + > = − + + + > − − − − − − • • • • MPF span length ADTT N N N MPF span length ADTT N N N fatigue strength For illustrative purposes MPFstrength was calculated based on the number of lanes and typical average daily truck traffic for a span of 100 ft. Note that the ADTT increases with the num- ber of loaded lanes (N), since the number of trucks crossing the span can also increase. The total equivalent number of trucks

38 distributed across the bridge is determined by the number of lanes multiplied by MPFstrength. The results for two, three, and four lanes loaded for the strength limit state are shown in Table 5. From the calculated multiple presence factors presented in Table 5, it can be deduced that there is no reason to consider the effects of more than two side-by-side trucks. As can be seen, as the number of lanes increases, the actual demand decreases since the reduced number of total trucks is distrib- uted among a greater number of lanes. Therefore, a maximum of two loaded lanes appears sufficient for the Redundancy I and II load combinations when the bridge is intended to carry two or more lanes. Moreover, bridges intended to carry a single lane, such as single-lane ramp structures, may not require being evaluated for two fully loaded lanes in the faulted state. In conclusion, it appears that more recent data may sup- port the use of lower multiple presence factors or number of lanes fully loaded. However, since these more recent data have not been incorporated into the current AASHTO LRFD BDS nor were they used in its calibration, the authors of the current research elected to remain consistent with the current provisions with regard to multiple presence of live load. 3.3 Minimum Performance Requirements in the Faulted State Once the results from FEA carried out per the recommen- dations of Section 3.1 under the loads described in Section 3.2, it is necessary to establish whether the structure has adequate capacity (i.e., redundancy) after the sudden failure of a given tension member. Thus, minimum strength and serviceability performance requirements have been developed, as described below. The requirements are to be checked against the results from analysis using the Redundancy I and/or Redundancy II loading combinations, depending on the specific criterion under study. 3.3.1 Minimum Strength Requirements The primary objective of these criteria is to determine if the bridge possesses sufficient strength and reliability in the faulted state. The focus is on primary steel members, the concrete slab, and the substructure/bearings. In general, the system demonstrates that it has sufficient strength after the failure of a tension member if it is able to carry the loads in both the Redundancy I and Redundancy II load combina- tions. In other words, the bridge does have adequate strength in the faulted state. The results from three-dimensional nonlinear FEA include nodal displacements and element stresses and strains, including the von Mises Stress, to establish yielding and post-yielding plastic strains (if they occur). These FEA results cannot be linked directly to nominal member capaci- ties calculated using AASHTO specifications, which presents a challenge when developing acceptance criteria for the FEA methodology. This is true regardless of whether the bridge is in the faulted state. For example, consider a two-girder com- posite simple-span bridge. In the undamaged state, a typical line-girder analysis results in moments and shear demands calculated at various locations along the girder. These force effects are then compared to nominal moment or shear capaci- ties calculated using code provisions. For the composite girder, both the demand and capacity calculations include simplifica- tions with regard to effective slab width, stiffness, and strain distribution through the girder, among other aspects. In con- trast, when using three-dimensional nonlinear FEA to ana- lyze the same bridge, the approach focuses on solid mechanics, and the output is with regard to nodal displacements and element stresses and strains; not member level moment or shear demands. Since there are no assumptions that are made with regard to effective slab width, stiffness, and so on in FEA, it is not possible to make direct comparisons with the nominal mem- ber capacities calculated using code provisions. To obtain some form of an equivalent moment carried by each girder, one would need to integrate the stresses over the cross sec- tion. In the case of inelastic behavior or when damage occurs, such as shear stud pullout or loss of bond in noncomposite sections, this becomes even more complicated and, in many cases, meaningless. Additionally, FEA will always predict a greater system strength, since the behavior, mechanics, and force redistribution are directly incorporated into the model. In short, direct comparison to code-based nominal member capacities using estimated force effects from nonlinear FEA is inappropriate to assess the state of the structure. For con- sistency, it would be more appropriate to compare the results from the analysis to accepted failure criteria that are based on Number of Loaded Lanes (N) Span Length ADTT MPFstrength Total Trucks 2 100 ft 5,000 0.87 1.74 3 100 ft 6,000 0.51 1.53 4 100 ft 7,000 0.37 1.48 Table 5. Multiple presence factors for evaluation of strength and equivalent number of trucks for different number of lanes (calculated according to Fu et al. [2013]).

39 stress and strain and then to evaluate the overall strength of the system. Therefore, the acceptance criteria for strength pre- sented below were developed with this in mind while relying on established limits currently contained in AASHTO specifi- cations, where feasible. 3.3.1.1 Structural Steel It is possible that, once the analysis is completed, the nomi- nal yield stress, ultimate stress, or the failure strain is reached at some location(s) in the structural steel. This is acceptable if the regions where this has taken place are in secondary components while primary members perform adequately. Analysis and in-service performance have shown that failure of secondary members does not necessarily compromise the overall ability of the bridge to carry vertical loads. Second- ary members such as stringers or floor beams carry a small portion of the load that is easily redistributable among other components. Other secondary elements (e.g., braces) are not designed for the direct transfer of vertical loads, and failure of a particular component is not likely to result in any perceiv- able reduction of strength. This is not to say that the analytical procedures do not need to accurately predict the capacity and failure of these members. Following the proposed procedures ensures that failure of these members is predicted by the FEA. It is simply saying that failure of such secondary members is not reason to conclude that the system has failed. In addition, if the failure of a particular bracing member results in overall structural instability, it will be revealed in the analysis, pre- suming it is performed in accordance with the methodology proposed herein. For primary members, however, specific criteria have been developed to ensure sufficient system performance in the faulted stated. The following are considered to be primary steel members, as discussed herein: • Webs and flanges in plate girders and tub-girder systems; • Chords, diagonals, and verticals in truss systems; • Cross girders or bents supporting girders or trusses; • Arch ribs, truss arches, and tie girders; and • Members deemed as such by the engineer/owner. Four basic strength criteria for primary steel components must be met when the bridge is subjected to the Redundancy I and II load combinations. These criteria are as follows: • The average strain across a component of a cross section (e.g., a flange) must be below 5 times the yield strain (ey), or 0.01 strain, whichever is smaller. For example, the aver- age strain in a tension flange must remain below 5ey, or 0.01. The proposed strain limit is based on strain levels characteristic of yielding related limit states. • The maximum stress anywhere in a primary member shall not exceed the nominal ultimate strength, and the strain anywhere in a primary component shall not exceed 0.05. The effects of strain concentrations, such as at reentrant corners or terminations of details need not be consid- ered for this limit. The proposed stress and strain limit is based on stress and strain levels that may result in fracture (rupture)-related limit states. • The compression stress must remain below the criti- cal buckling stress of the component in cases where the FEA does not account for the buckling mode. This must be checked to avoid member failures in instability-related limit states. • The system shall demonstrate a reserve margin of at least 15% of the applied live load in the Redundancy I and II load combinations, as determined by the owner. Effectively, this requirement ensures the slope of the load versus displace- ment curve for the system structure remains positive, thus guaranteeing the stability of the structural system in the faulted state. This requirement is not specific to the steel components alone, but rather the entire bridge as a system; it is nevertheless included with the requirements for steel. 3.3.1.2 Reinforced Concrete The nominal compressive strength of concrete may be exceeded in the analysis, as well. This is acceptable if the regions where this has taken place are in the barriers or haunches, and the system is able to sustain the factored loads. Concrete crush- ing in the aforementioned regions is not expected to result in appreciable reduction of strength, based on the results of the analysis and in situ performance of bridges where tension members have failed. However, if concrete crushing takes place in a significant portion of the slab, the structure should not be considered as redundant, as passage of traffic and environ- mental conditions will rapidly deteriorate the slab to a point where capacity may further be reduced. In other words, if the portion of the slab where a compressive strain of 0.003—based on ACI 318-14 analysis procedures for flexural members (ACI 2014)—has been exceeded is large enough to compromise the overall system load-carrying capacity, or if significant hinging occurs, the structure should not be considered as sufficiently redundant. In such cases, slope of the load–deflection curve would be flat or negative. In general, this will be captured by the analysis if the guidance on modeling developed during this research is followed. 3.3.1.3 Substructure In addition to checking the strength of the superstructure, the substructure must also be analyzed. Although the sub- structure many not be explicitly included in the finite element

40 model in all load combinations, the displacements and reac- tion forces at support locations are calculated in the analy- sis. These should be taken as the factored demands that the substructure must be able to safely sustain. Given the existing variety of support systems, the authors cannot specify any uni- versal criteria; for instance, the same force demand may result in stability issues for one pier but not affect the performance of a different pier. Transverse and longitudinal displacements at support locations should be considered, as a member may lose support, particularly at supports that allow for expansion. Hence, it should be verified that these horizontal displace- ments can be accommodated by the support. 3.3.2 Minimum Serviceability Requirements Several aspects related to the serviceability of the bridge of the faulted state have been considered during the develop- ment of the proposed guide specifications. These were not only concerned with safe passage of traffic but also with sta- bility of the structure itself. If the results of the analysis com- ply with the serviceability requirements described below, it is unlikely that the structure will be without any overall stability issues. The reason is that deflections necessary to compromise the stability of the structure are typically larger than the deflec- tion serviceability limits proposed. The main serviceability criterion was the vertical deflection of the superstructure; but considerations with regard to changes in the cross slope of the slab, uplift at supports beneath deck joints, and detectability of primary steel member failure are also discussed. It is possible that a structure may meet the strength require- ments while not meeting the serviceability requirements. Although the FCM on which failure is modeled cannot be redesignated as an SRM, in such cases it may be possible to alter the inspection strategy because the bridge would have sufficient strength in the faulted state. Nevertheless, the deci- sion to relax the inspection requirements in such circum- stance must be agreed upon by the owner and the supervisory agency (e.g., FHWA). 3.3.2.1 Vertical Deflection of the Superstructure Current deflection limits in the AASHTO LRFD BDS do not contain any serviceability requirements for the case of member failure, as expected. Nevertheless, the AASHTO pro- visions were reviewed to ascertain if any existing criteria could be applicable or modified for evaluating the acceptable deflec- tion in the faulted state. One requirement that appeared to possibly be applicable is the requirement related to deflections of cantilevers. In the case of a girder bridge (I or tub), the fractured girder resembles a cantilever to some degree. This is not as apparent for a truss span, but it is still believed to be a reasonable starting point. At present, AASHTO LRFD BDS limits the deflection of a cantilever because of the application of vehicular load to the span length divided by 300, when consideration of pedes- trian comfort is not required (AASHTO 2014). However, the objectives of the AASHTO LRFD BDS deflection criteria were often based on and intended for controlling vibration under live load and not deflections, per se. Rather than require an engineer to perform the dynamic analysis, the same objec- tive was achieved by limiting static deflections, which were much easier to calculate. In the faulted state, the deflection (i.e., vibrations) caused by live load is not as important as the overall deflection of the bridge caused by dead load only in the faulted state. Thus, it was concluded that the existing criteria were not applicable. Alternatively, existing bridges where fractures or pier settlements have resulted in significant dead load deflection were also examined. Specifically, four cases were exam- ined where deflections were visually apparent but traffic was known to continue crossing the bridge prior to closure by the authorities. All four bridges also carry interstate traffic. The results are summarized in Table 6, which provides the bridge name, cause of the observed deflection, amount of the deflection, and corresponding L-over value. The span length (L) is taken as the actual span length or, in the case of pier settle- ment, the summation of the length of each span on either side of the pier. In each case, traffic had safely crossed the bridge with the roadway in this severely distorted condition. Based on the cases observed, it appears that substantial verti- cal deflections can be tolerated by motorists without impact to the traveling public’s safety, at least in the short term. As can be seen, the L-over values range from about L/60 to L/277. Two of the three girders completely fractured in the Hoan Bridge, resulting in the large deflection of L/62. How- ever, even in this condition, interstate traffic continued to cross the bridge prior to closure by the police, and there were no issues (i.e., vehicular accidents) caused by the sag. The L-over values used to establish the maximum deflection cri- terion in redundancy evaluations may appear large in com- parison with existing deflection limits in the AASHTO LRFD BDS. However, as previously mentioned, these are intended to reduce vibrations, while the aim of the current research is to identify what constitutes a deflection that prevents safe passage of traffic. Therefore, a deflection limit of L/50—based on adapting the largest deflection value from Table 6 to the dead load factors—is recommended. The deflection limit of L/50 is to be checked for the Redundancy II load combination, and under dead loads (DC and DW) only. Given the lack of data available to establish the faulted state maximum deflec- tion limit, the proposed criterion was consulted with own- ers, fabricators, designers, and other researchers during an expert consensus meeting. The overall agreement was that

41 a deflection limit of L/50 after the failure of a primary steel tension member was deemed reasonable. A major advantage of ignoring live load ensures that designers will not attempt to strengthen the bridge to meet a deflection limit in the faulted state. Doing so would have the unintended consequence of reducing the deflection caused by dead load only, thereby potentially making it more difficult to detect the faulted condition. Further, the deflections caused by dead load only would already have been calculated, since the fault must be present prior to the application of the live load in the Redundancy II load combination. Thus, the data would be readily available in the analysis. Lastly, it must be recognized that the live load statistical parameters used in the Redundancy II load combination cor- respond to a loading event with a return period of 75 years and may include several lanes of factored HL-93 load models simultaneously located in the worst longitudinal position. While this load case is appropriate to check the strength of the bridge in the faulted state, it may actually never occur. Hence, including live load would result in overly conservative Bridge Failure Mode Vertical Deflection (ft) Equivalent Span (ft) Equivalent L-Over Value Photograph I-794 Hoan Bridge Fracture 3.5 217 L/62 I-65 over Wildcat Creek Pier settlement 0.75 170 L/227 I-43 Leo Frigo Bridge Pier settlement 2.0 400 L/200 I-495 over Christina River Pier settlement 1.57 436 L/277 Table 6. Examples of faulted bridges where vertical deflections were easily detected by the public and traffic continued to safely traverse the bridge.

42 limits on deflections that result in strengthening of a given structure to meet deflection limits and again, the unintended consequence of making it more difficult to detect the faulted condition. The proposed limit meets the objective of maxi- mizing the likelihood that the deflection will be sufficiently large to alert the authorities or the motoring public. 3.3.2.2 Considerations on Deflections of the Superstructure The maximum deflection criterion described is intended to avoid excessive deflections that could result in danger to vehicular traffic following failure of a tension member. How- ever, there are many documented cases where a complete girder has fractured and there was little, if any, perceptible deflection. The Diefenbaker Bridge (Saskatchewan, Canada), Neville Island Bridge (Pennsylvania), Dan Ryan Transit Elevated Structure (Illinois), US 422 over the Schuylkill River (Pennsylvania), Lafayette Bridge (Minnesota), and Milton–Madison Bridge (Indiana) are just a few such struc- tures where deflections following complete member failure were imperceptible. While this suggests there is consider- able reserve strength, this not ideal as the failed component may go undetected for a considerable period. In such a con- dition, the bridge is severely “wounded,” yet, it must still carry traffic, possibly until the failure is detected through inspection. While the Redundancy II load combination will ensure that the bridge has sufficient reserve strength in the faulted condition, it presumes no other damage occurs during the interval between failure and detection. Realistically, fatigue damage will be accelerated in the faulted state because of rou- tine traffic. This could be of concern in cases where there is little perceptible visual evidence, depending on the interval between failure and the inspection or detection of the dam- age. In contrast, if the visual evidence is clearly noticeable— such as on the structures described in Table 6—traffic will be detoured promptly, and there would be no concern with regard to fatigue. In light of the above, it may necessary for the fatigue to be checked in the faulted state if the deflections are below some threshold. Based on an evaluation of the bridges cited previously, deflection limits were in the range of L/850 to approaching L/5000. In these cases, the public continued to cross the bridge with no visual perception of the damaged condition of the bridge. The literature review did not iden- tify any reliable data on what level of deflection is necessary for the motoring public to perceive that the bridge is in the faulted condition. If the calculated vertical deflection in the faulted state is considered to be too low, a fatigue evaluation could be per- formed to estimate the remaining fatigue life. The calculation must include the effect of any fatigue damage that may have occurred prior to the failure. Procedures provided in NCHRP Report 721 can be used to examine various levels of reliability associated with the remaining life calculation (Bowman et al. 2012). Once the level of fatigue reliability is determined, the remaining life following the failure may be calculated. That remaining life may be used to establish an inspection strat- egy, given that the failure of the primary steel member under evaluation will not result in bridge collapse or loss of service- ability but that the member failure will likely not be detected by conventional means. In addition to vertical deflections, the effect of the failure on the cross slope of the slab may need to be taken into account, in some cases (e.g., on bridges with significant curvature). A significant change in cross slope will negatively affect vehicle control, requiring the driver to make adjustments to correct the trajectory of the car and/or limit the likelihood of over- turning. Discussions with experts in highway design indicate that to properly define a limit on the change of cross slope, several factors must be considered, such as the design speed of the bridge, radius of the curve, lane width, average daily truck traffic, and road superelevation (AASHTO 2011B). Since no experimental data or analytical studies exist on this topic (as related to member failures), a unique rational approach would need to be developed that incorporates all of these factors. Therefore, the engineer may need to construct a specific criterion only applicable to the bridge under evalu- ation when deemed that such serviceability criterion should be employed. Based on experience, however, this would rarely need to be done. A final additional serviceability consideration is uplift at deck joints, which may occur in the faulted state. If the com- puted uplift were excessive and were to occur at a location of a deck joint, it would negatively impact driving, possibly even preventing traffic from entering the bridge. Literature review and consultations with transportation engineers resulted in varying opinions with respect to what constitutes an accept- able amount of uplift. When selecting serviceability criteria additional to the maximum deflection criterion in Section 3.3.2.1, one must take into account that the load factors in the Redundancy I and Redundancy II load combinations were calculated to evaluate strength in the faulted state. Therefore, a particu- lar limit or threshold may need to be adjusted for the load combinations developed in the current proposed guide specifications, since the evaluation may be overly conser- vative (e.g., evaluation of cross slope of the slab) or non- conservative (e.g., detectability of primary steel member failure), depending on the criterion evaluated. Supplemen- tal load combinations or evaluation procedures may also be necessary (e.g., a fatigue evaluation of the bridge in the faulted condition should follow the analysis procedures

43 described in the AASHTO LRFD BDS for the Fatigue II load combination). 3.4 Dynamic Load Amplification Requirements For all bridges modeled, the global dynamic amplification was calculated to be between about 0.15 and 0.40, as shown in Table 1. Amplification factors of 0.30 and 0.36 were also mea- sured during the full-scale testing of the University of Texas tub-girder bridge (Neuman 2009) and the Milton–Madison Truss Bridge (Cha et al. 2014), respectively. For example, dur- ing the FEA study of the Milton–Madison Truss Bridge, the peak calculated dynamic amplification factor in the main truss members was 0.36. As discussed herein, the dynamic amplification is calculated as the ratio of the peak stress in a given member in free vibration following the sudden fracture to the stress in that member after the structure comes to rest. This factor is referred to as DAR. A DAR was calculated for each bridge based on the response of primary members, such as the girders, truss members, and primary bracing members. While higher factors were calculated in some secondary members, such as a cross frame carrying little force, these members were not deemed to be critical. Further, the peak dynamic response observed in the main members was used in the development of DAR. It is also important to recognize that the amplification is not uniform across an entire bridge. For example, on the Neville Island Bridge, the peak dynamic amplification in the main girder immediately adjacent to the fracture is greater than the peak dynamic amplification in one of the back spans or even just a few panel points from the failure. To simplify the evaluation process, it is conservatively assumed that the peak amplification is applied throughout the entire bridge. Further, although the DAR was not calculated based on maximum response of secondary members, but rather primary members, the factor is applied to the entire bridge and, hence, secondary members, as well. Thus, while the response may be underestimated in a specific bracing mem- ber, the effect is somewhat—if not entirely—offset, since the amplification is conservatively assumed to apply to the entire bridge. The value of DAR was developed with the bridge subjected to the effects of gravity or dead load, as gravity dominates the dynamic response of the bridge. However, live load was also applied, though it was assumed to have no mass and, hence, no contribution to the dynamic properties of the bridge. Attempting to include the mass and dynamic characteristics (suspension stiffness and damping characteristics) of the HS-20 load and the distributed lane load of the HL-93 adds unnecessary complications. Thus, the force effects of the live load are included in the DAR values proposed. To avoid the need for complex dynamic analysis to be required for future evaluations, one would simply (1) increase the applied factored dead load by DAR (it should be taken into account that the structure is subjected to factored dead load before the failure of the steel tension member), and (2) apply factored live loads amplified by DAR after the failure of the steel tension member when subjected to the Redundancy I load combination. Thus, the dynamic ampli- fication is applied to both dead and live loads in the Redun- dancy I load combination. At this time, the amplification caused by the inertial effects following failure of a tension member will be conservatively set at 0.40. This amplification is introduced through the DAR, used for redundancy evaluation, which is different from the common dynamic load allowance (IM) that is applied to account for dynamic impact caused by moving trucks. Owners may perform their own dynamic analysis using the proce- dures developed herein to calculate a unique value for DAR for a given bridge, however. 3.5 Detailing for New Bridges This section summarizes the development of guidelines for detailing of new bridges that would be traditionally consid- ered to contain FCMs, but through the application of the FEA methodology, the bridge is shown to have redundant capacity in the faulted state and to contain SRMs. Based on informa- tion available in the literature, the research conducted to date, and experience of the authors, it appears that current good steel detailing practices appear sufficient in ensuring ade- quate system performance. The screening criteria discussed in Section 3.6 for excluding a bridge from system analysis can also be used to identify desired steel details for new design. Following generally accepted current design practices will also likely result in acceptable performances. Specific infor- mation related to detailing and some general design aspects are provided below. 3.5.1 Structural Steel Design Details The following criteria are proposed with regard to detailing new bridges that are to be evaluated using the FEA methodol- ogy developed herein: • Details susceptible to constraint-induced fracture must be prohibited. Current AASHTO provisions prevent the use of such details, and no new guidance is required with regard to this matter. • Category C is the lowest fatigue category that should be used. Category C details are highly fatigue resistant in high- way bridges subjected to truck loading. No cases of cracking at Category C details subjected to nominal design stresses

44 were described in the reviewed literature. Because the con- stant amplitude fatigue limit for Category C is 10 ksi, actual in-service stress ranges rarely, if ever, will exceed this value. • Details susceptible to out-of-plane cracking shall not be permitted. Current provisions in the AASHTO LRFD BDS also prohibit the use of such details. Hence, current guidance appears adequate with regard to this issue. 3.5.2 Concrete Deck Design Details The following criteria are proposed with regard to detail- ing the concrete deck of new bridges that are to be evaluated using the FEA methodology developed herein: • The concrete deck shall be fully composite with steel pri- mary superstructure components, such as the main lon- gitudinal plate girders or tub girders. Stringers and floor beams should also be made composite with the deck to ensure full engagement of the floor system when the sys- tems are used in plate girder or truss construction. • Shear studs must be detailed to ensure that they extend above the bottom layer of deck reinforcement. When a sig- nificant haunch is used, the studs must extend into the deck and be able to fully engage the slab reinforcement. This is to ensure that the tension capacity of the studs is fully avail- able. The tension and shear capacity of the studs can be determined using the provisions developed in this research. 3.5.3 General Design Guidance The review of previous case studies where fractures have occurred—as well as during the course of the research con- ducted as part of this study—has revealed additional good practices that should be incorporated in design. These are listed below. • To ensure that the likelihood of cracking caused by fatigue is effectively eliminated, new structures being consid- ered for system analysis shall be designed for infinite fatigue life. • High-performance steels possess superior toughness com- pared to traditional steels. It is highly encouraged to select these grades of steel for members that would traditionally be classified as FCMs. Members of the structure that have been shown not to meet the definition of an FCM—per the provisions developed in this research—should still be designed and fabricated to meet the requirements of Sec- tion 12 of the AASHTO/AWS D1.5M/D1.5 Bridge Welding Code (AWS D1.5) (AASHTO and AWS 2015). The memo- randum issued by FHWA dated June 20, 2012, included a new term for such members: a system-redundant member, or SRM (Lwin 2012). 3.6 Limitations of Finite Element Analysis Methodology The developed FEA methodology assumes that all of the components included in finite element models are in suf- ficiently good condition to carry applied loading using the nominal strength of the material and geometry of the com- ponent. Localized attributes that negatively affect the perfor- mance of a structure—such as severe corrosion or presence of fatigue cracks—cannot be explicitly implemented into an analysis methodology in which a certain target reliability is to be achieved. First, the developed material models would need to be modified to include the effect of these negative attri- butes, and such data are not readily available. For example, corroded steel may not be able to reach its nominal ultimate strength and may behave in a more brittle manner (François et al. 2013). Second, as these negative attributes are local- ized, their inclusion in a finite element model would require a large number of modifications to the geometry and mesh. For example, modeling an existing crack would require inclu- sion of a seam, mesh refinement around the crack front, and modification of elements around the crack tip to capture the stress singularity (Fenves et al. 1973). Additionally, it would be a misuse to use FEA to justify redundancy in a bridge that has observable maintenance issues. Since the structure under evaluation not only relies on structural indetermination to provide redundant post-failure capacity but also alternative load paths, the performance of the members providing the alternative load paths cannot be under- mined by factors that cannot be reliably characterized. The redundant capacity of a corroded or cracked bridge after the failure of a steel tension member cannot be properly assessed by the developed finite element methodology. Prior to perform- ing an FEA, it is paramount that an assessment (which includes FCM hands-on inspection) of the structure using the criteria described below is performed to identify whether any negative predetermined characteristics exist. Bridges that possess any of the negative attributes enumerated in the screening phase do not qualify to have SRMs, regardless of the results of the FEA (should it still be performed). A screening criteria is presented hereafter to establish whether the FEA methodology is appli- cable to a steel structure for evaluation of primary steel tension members. If a structure that contains FCMs fails the screening, those members are to remain as FCMs and cannot be desig- nated as SRMs through analysis unless the factors are mitigated in some way. 3.6.1 Screening Criteria for Existing Bridges The objective of the screening process is to ensure that bridges with certain characteristics are not—under any cir-

45 cumstance—permitted to be considered capable of providing reliable redundant capacity in the faulted state. The criteria were established based on the work by Parr et al. (2010) and, subsequently, refined during the current study. Most of the screening criteria apply to the FCMs and not necessarily the entire bridge. However, while the results of system analysis are not likely available during the screen- ing phase, consideration should be given to the anticipated response of the bridge in the faulted state (i.e., after the initial fracture). Therefore, some members that would not be classi- fied as FCMs, such as compression members in trusses, may carry tension after failure of one of the members classified as an FCM on the design plans because of load redistribu- tion. The screening criteria and principles discussed below are also intended to be applied to these members, in some circumstances. Bridges that possess any of the negative attributes enu- merated in the criteria examined in the screening phase are not acceptable candidates to be analyzed, per the proposed finite element methodology. Nine screening criteria have been identified. However, owners should consider including additional criteria that are specific to their region or inven- tory, specific to a structural configuration under evaluation, or based on their experience. The screening criteria include the following: • New/recently retrofitted or rehabilitated FCM, • Presence of pin and hanger details, • Presence of nonredundant eyebars, • Presence of plug welds or discontinuous backup bar splices, • Presence of active fatigue cracks, • Susceptibility to constraint-induced fracture, • Presence of existing maintenance problems/load posting, • Unreliable or unavailable field inspection data, and • FCMs rated at National Bridge Inventory Condition Level 4 (or below) or Element-Level Condition State 4 (or pos- sibly 3) for members integral to the system performance in the faulted state. As previously stated, if a structure contains members that may be considered as FCMs fails the screening, those mem- bers are to remain FCMs; they cannot be designated as SRMs. The engineer simply needs to determine if any one of the screening criteria is met. The rationale for including each of the screening criteria and guidance on how to assess or evaluate a structure for the associated criteria is provided in Appendix C. The screening criteria is codified in the pro- posed guide specification in Appendix E. 3.7 Application of Research to Bridge Engineering Practice The findings of the research conducted for NCHRP 12-87A were used to develop a stand-alone document that can be used by AASHTO as a guide specification. The proposed specifica- tion is complementary to the AASHTO LRFD BDS and the AASHTO MBE. Hence, it is not to be used as the primary basis for design or load rating of steel bridges. The primary intent of the proposed specification is to provide engineers with a reliable analytical tool to evaluate the redundancy of steel bridges with primary steel tension members that may not be redundant. The system analysis procedures described throughout the current report were used to classify primary steel tension members as FCMs or SRMs. Additionally, design and detailing guidelines for new bridges are included in the proposed specification. It is envisioned that engineers will use current valid design and load rating procedures, and when a decision is to be made regarding designation of FCMs and SRMs the proposed guide specification will be used. The pro- posed guide specification in included in Appendix E. It must be noted that the analytical requirements of the pro- posed guide specification are more advanced than typically used in the design office, even for complex curved or skewed structures. The engineer in charge of conducting redundancy evaluations, per the developed FEA methodology, needs to have proven previous experience with finite element models of multipart assemblies and be familiar with material non- linearity, large deformation theory, and contact modeling. An effort was made to write a specification as clear and simple as possible, but it is understood that additional guidance may be required; hence, a set of examples in which the system analysis has been applied was developed. These can be found in Appen- dix F. Additionally, extended information with regard to shear stud modeling (Appendix A), screening criteria for determining eligibility for system analysis (Appendix C), and requirements for system analysis (Appendix D) will accompany the proposed guide specification and the application examples. Given the unique characteristics of the proposed guide specification, it is recommended that—after adoption of the specification—workshops will be needed to illustrate the use of system analysis to perform redundancy evaluations. Although no inspection interval calculation procedures are included in the current work, the proposed system analysis methodology may be used in conjunction with existing methodologies to set future FCM hands-on inspection intervals. The works by Parr et al. (2010) and NCHRP Report 782 (Washer et al. 2014) are recommended and may be used at the discretion of the owner or engineer under approval of the pertinent agency.