Bridge System Safety and Redundancy (2014)

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12 2.1 Introduction Redundancy is defined as the capability of a bridge sys- tem to continue to carry load after the failure of one of its members. This means that the system has additional reserve strength such that the failure of one member does not result in the collapse of the entire structure or a significant portion of it. The initial member failure can be either brittle or ductile. It can be caused by the application of large overloads, extreme loads, or the loss in the load carrying capacity of one element due to damaging events such as fatigue, brittle fracture, mem- ber deterioration, or an accident such as a collision by a truck, ship, or debris. For the case of damaged bridges that may have lost an ini- tial member, the object of this project is not to perform a dynamic progressive collapse analysis to check whether the bridge will be able to survive the hazard that causes an ini- tial sudden failure of the element. Instead, the object of the redundancy analysis performed in this study is to investigate whether a damaged bridge system that has survived a damag- ing event will be able to continue to carry some traffic load for a limited period of time that would allow the traffic to clear the bridge and maintain the bridge’s ability to continue to carry some load until the damage is noticed and reported to the proper authorities so corrective actions (i.e., bridge closure, repair, or replacement) are undertaken. Traditionally, bridges have been designed on a member-by- member basis, and the interaction of the members and their ability to provide different levels of redundancy following the damage to, or the failure of, one or several members have not been directly considered. A convenient method to take into consideration the redundancy of bridge systems would con- sist of developing a set of system factors that can be included as specifications in bridge design and evaluation manuals. The system factors would be applicable for routinely checking the safety of typical bridge configurations so that the mem- bers of bridges with low levels of redundancy are required to have higher safety levels than bridges with high levels of redundancy. Alternatively, for a more precise evaluation of a bridge’s redundancy, a direct analysis approach should be used. The direct analysis would consist of using a structural model and a finite element analysis program that consider the elastic and inelastic behavior of the bridge system. This pro- gram would be used to evaluate the load carrying capacity of an initially intact bridge, as well as the behavior of the bridge under different damage scenarios. The program would check the structure to verify whether its behavior is acceptable, with sufficiently high levels of safety and functionality during the application of expected large loads. Many safety-related decisions must be made in order to develop the system factors or to use the direct analysis approach. These include (1) the limit states that should be checked, (2) the level of loads that must be carried by the structure before the limit states are reached, (3) the type of damage conditions that must be borne by the structure, and (4) the inclusion of uncer- tainties in the analysis model. This report reviews the results of the analyses of bridge sys- tems including superstructures and substructures, as well as combined systems conducted during this current project and during NCHRP Report 406 and NCHRP Report 458. The goal is to develop a set of system factors that can be included in bridge specifications to evaluate the safety of existing bridges and to design new bridges taking into consideration their lev- els of redundancy. This chapter gives a general overview of the concepts of bridge safety and redundancy and describes the procedure adopted to develop a set of system factors that can be incor- porated into the design and safety-check equations in order to account for bridge system redundancy during the design of new bridges and the evaluation of existing bridges. Specifi- cally, Section 2.2 of this chapter gives a general overview of bridge performance under externally applied actions and the general concept of bridge safety. Section 2.3 proposes deter- ministic measures of redundancy. Section 2.4 gives a review C H A P T E R 2 General Concepts of Bridge Redundancy

13 of reliability theory and its application for calibrating bridge design and safety assessment codes. Section 2.5 develops the reliability model used in this study for the probabilistic eval- uation of bridge redundancy and the calibration of system factors. The concepts presented in this chapter are adapted in Chapter 3 for calibrating system factors that account for the redundancy of bridge systems subjected to lateral actions when bridge system safety is checked using the displacement- based method. Chapter 4 uses the concepts presented in this chapter to develop system factors that account for the redun- dancy of bridge systems subjected to lateral loads when their safety is checked using force-based analysis methods. Chap- ter 5 uses the concepts presented in this chapter to develop system factors that account for the redundancy of bridge sys- tems subjected to vertical loads. 2.2 Bridge System Behavior Current bridge design and evaluation techniques deal with individual members and use procedures that do not fully account for the effect of the complete structural system. As currently performed, the safety check verifies that the strength of each member is greater than the effects of applied forces by a “comfortable” safety margin. Member forces are calcu- lated using an elastic analysis while member capacity (when appropriate) is calculated using inelastic member behavior. In current and previous bridge design and evaluation speci- fications, the safety margin for strength is provided through the application of safety factors (load and/or resistance fac- tors) that are calibrated on the basis of experience and engi- neering judgment (ASD and LFD) or on a combination of experience and structural reliability methods (LRFD). Although this traditional member-oriented approach has been used successfully for years, it does not provide an ade- quate representation of the safety of the complete bridge sys- tem. In many instances, the failure of an individual member does not lead to the collapse of the complete bridge system. Current specifications do not directly differentiate between bridges that would collapse when one member fails and those that will be able to continue to carry loads after the failure of one member. Because of this difference in the consequences of a member’s failure, it is reasonable for the specifications to require that members of non-redundant bridges be designed to higher standards than members of redundant bridges. This goal can be achieved by applying system factors in the safety- check equations or as currently stipulated in the AASHTO LRFD Bridge Design Specifications by applying load modi- fiers. Because the load modifiers specified in the current LRFD were based on the judgment of the code writers rather than an evaluation of bridge redundancy, the object of this NCHRP 12-86 project is to propose an approach to account for bridge system redundancy based on measurable criteria. A first step in the process of evaluating bridge redundancy is to have a good understanding of the behavior of bridge systems under applied loads. The performance of a bridge system can be represented as shown in Figure 2.1, which gives a conceptual representation of the response of a structure to different levels of applied loads and the different criteria that should be considered when evaluating member safety or sys- tem safety as well as system redundancy. The model is valid for representing the behavior of systems under vertical loads or for systems under lateral loads. The green line in Figure 2.1 labeled “Intact system” may represent the applied load versus maximum displacement of a ductile bridge system when subjected to different levels of load. In this case, a load capacity evaluation is performed to study the behavior of an intact system that was not previously subjected to any damaging load or event. LFd LF1 LF LFu Ultimate capacity of intact system Load Factor Intact system Damaged bridge Assumed linear behavior f First member failure Bridge Response Ultimate capacity of damaged system Loss of functionality Figure 2.1. Representation of typical behavior of bridge systems.

14 To perform the load capacity analysis, the bridge is first loaded by the dead load and then the transient load is incre- mentally applied. The first structural member will fail when the transient load reaches LF1. LF1 would then be related to member safety. LF1 may represent the actual load or the mul- tiple of a basic load such as the number of design trucks that the system can carry before the first member reaches its limit capacity. Although LF1 should be evaluated using the actual response of the bridge accounting for material nonlinearity, it has been common in structural design practice to assume linear-elastic response while evaluating the ability of the sys- tem to resist the failure of the most critical member as indi- cated using the bilinear brown curve in Figure 2.1. Generally, the system will be able to carry additional load after LF1 is reached and the ultimate capacity of the entire bridge is not reached until the transient load reaches LFu. LFu would give an evaluation of system safety. Large deforma- tions rendering the bridge unfit for use are reached when the transient load reaches LFf. LFf gives a measure of system functionality. A bridge that has been loaded up to this point is said to have lost its functionality. Damage to bridge members leading to the loss in member and system capacity is also a concern. Bridge members are often subjected to fatigue stresses that may lead to the fracture and loss of the load carrying capacity of a main member. In addi- tion, deterioration and corrosion, fire, or an accident, such as a collision by a truck, ship, or debris, could cause the reduction in the load carrying capacity of one or several main members. To ensure the safety of the public, bridges should be able to sustain these damages and still operate at a sufficient level of capacity. Although a damaged bridge cannot be expected to have the same capacity of an intact system, an adequately redundant system should still be able to carry its own weight and some level of transient load to allow for clearing the bridge before closure and the undertaking of necessary repairs. Therefore, in addition to verifying the safety of the intact structure, the evalu- ation of a bridge’s safety and redundancy should consider the consequences of the failure of a critical bridge member. If the bridge has sustained major damage due to the brittle failure of one or more of its members, its behavior can be represented by the blue curve labeled “Damaged bridge” in Figure 2.1. The ultimate capacity of the damaged bridge is reached when the transient load applied after the application of the dead load reaches LFd. LFd would give a measure of the remaining safety of a damaged system. In summary, based on the bridge performance curve pre- sented in Figure 2.1, a bridge can be considered safe if it • Provides a reasonable safety level against first member failure, • Provides an adequate level of safety before it reaches its ulti- mate system capacity under extreme loading conditions, • Does not produce large deformations under expected heavy transient loads, and • Is able to carry a sufficient level of traffic load after damage to or the loss of a component. Although the explanations provided in this section are pre- sented in terms of the loads or load factors, labeled “LF” on the vertical axis of the graph, the same criteria can be expressed in terms of the response of the bridge or the horizontal axis of the graph. Most typically, the response of the bridge is rep- resented by the maximum displacement at a critical section. The load factor values that a system can support before a limit state is reached represent the system safety of the bridge. Redundancy is a measure of the relationship between the overall system performance and that of its most critical member as will be explained next. 2.3 Measures of Bridge Redundancy Traditionally, bridge engineers have defined redundancy of bridges in terms of the availability of alternate load paths that can redistribute the load should one of the main mem- bers fail. Commonly, the availability of alternate load paths has been associated with the number of supporting elements. Thus, according to current practice a multi-girder bridge superstructure would be considered redundant if it is formed by four or more parallel elements (although some engineers have defined bridges with three parallel girders as redundant). Variations in bridge cross-section configurations including the type of girders, beam spacing, and boundary conditions are not usually considered. Furthermore, a system is currently classified as either redundant or non-redundant and no con- sideration is given to the degree of redundancy. Accordingly, a system formed by two spread box-girders is usually considered to be similar to a system formed by two I-girders in terms of their redundancies independent of the spacing between the I-girders or the webs of the boxes or whether the spans are con- tinuous or simply supported. The current approach ignores the additional torsional rigidity of the boxes that may improve the load distribution and would change the redundancy level of the system when compared to that of I-girder bridges. In a first attempt at providing a method to explicitly incor- porate redundancy criteria during the bridge design pro- cess, the AASHTO LRFD Bridge Design Specifications apply load modifiers in the design-check equations to account for redundancy during the design of new bridges. The method is based on the recommendation of Frangopol and Nakib (1991). Specifically, the AASHTO LRFD recommends using a load modifier, hR, depending on the level of bridge redun- dancy with hR taking values equal to 0.95, 1.0, or 1.05. Two other load modifiers hD and hI also are used to account for member ductility and the importance of the structure in

15 terms of defense/security considerations. However, the LRFD specifications do not explain how to identify which bridges have low and high redundancy or how to define low and high ductility. As explained in the LRFD Commentary, the recom- mended values for hR have been subjectively assigned based on judgment pending additional research. The Canadian Code CAN/CSA-S6-06 (2006) directs bridge engineers to use different load factors for different target reli- ability index values that are selected based on the redundancy of the bridge system and the ductility of the member being evaluated. However, like the LRFD, this approach relies on the judgment of the engineer in deciding which bridges are redun- dant and in judging the consequence of a member’s failure. Perhaps the most specific approach for evaluating bridge redundancy is provided in the Pennsylvania (PennDOT) bridge design specifications. According to PennDOT, three- girder bridges without floorbeams and stringers are consid- ered as non-redundant and should be avoided if possible. A 3-D redundancy analysis is required for the evaluation of all non-redundant structures to check whether the failure of any tension component or other critical component will not cause collapse. To that effect, two new Extreme Event Limit States labeled as III and IV are added to the LRFD specifica- tions. Extreme event III is meant to check that the failure of one element of a component will not lead to the failure of the component. Extreme event IV is meant to check that the failure of one component will not lead to the failure of the structure. These limit states require the analysis of a dam- aged bridge with the HL-93 design load along with a reduced dead load factor equal to 1.05 and live load factors of 1.30 and 1.15, for extreme events III and IV respectively, when using the HL-93 live load in all of the lanes. When a permit load is used in the governing lane and the HL-93 in other lanes, the live load factors on the permit load are 1.10 and 1.05 for extreme events III and IV, respectively. The analysis of the damaged bridge is performed with these load combinations, which the structure should be able to carry even if they cause large deformations as long as they do not lead to collapse. No justification is provided for the selection of the live load fac- tors for the new Extreme Event Limit States. From the above three examples and other studies reported in the review of the literature undertaken during the course of this project and summarized in Appendix C (available on the TRB website by searching for NCHRP Report 776), it is clear that accounting for bridge redundancy during the safety analysis of new or existing bridges is of primary importance. However, the mechanisms and the criteria that should be used to quantify bridge redundancy and consider it during the evaluation of bridge safety still have not been fully established. The aim of this project is to develop methods to quantify bridge redundancy and propose a set of non-subjective criteria that are imple- mentable in bridge design and safety analysis processes. Because redundancy is defined as the capability of a struc- ture to continue to carry loads after the failure of one main member, a comparison between the overall capacity of origi- nally intact and damaged bridge systems as represented by LFu, LFf, LFd, in Figure 2.1, compared to the capacity of the most critical member represented by LF1, would provide a measure of the level of bridge redundancy. In this context, the researchers define a “system reserve ratio” or “redundancy ratio” for the ultimate limit state as Ru. For the serviceability limit state, the redundancy ratio is defined as Rf. For the dam- aged bridge condition, the redundancy ratio is defined as Rd, where (2.1) 1 1 1 R LF LF R LF LF R LF LF u u f f d d = = = The redundancy ratios, Ru, Rf, and Rd, provide non- subjective deterministic measures of bridge redundancy. For example, when the ratio Ru is equal to 1.0 (LFu = LF1), the ultimate capacity of the bridge system is equal to the capac- ity of the bridge to resist failure of its most critical member; such a bridge is non-redundant. As Ru increases, the level of bridge redundancy increases. Similar observations can be made about Rf and Rd. Although the redundancy ratio Ru cannot fall below 1.0, the two ratios Rf and Rd may, under certain circumstances, have values less than 1.0. A value of Rf less than 1.0 means that the bridge will exhibit large deformations at a load level smaller than the load that will cause the first member failure. This situation might occur in certain bridges because LF1 is calculated with a linear-elastic model, whereas LFf accounts for the nonlinear behavior of the bridge. A value for Rd less than 1.0 means that a damaged bridge may fail at a lower live load than the load that will cause the first member failure in the originally intact linear-elastic system. Thus, the mini- mum value that Ru can take is normally 1.0, indicating that some bridge systems may collapse when only one member reaches its load carrying capacity. However, Rd can be as low as 0.0, indicating that a bridge system may collapse under its own dead weight if a certain damage scenario takes place. The measures given in Equation 2.1 indicate that structural sys- tems are associated with different levels of redundancy. This is different than current convention that stipulates that a system is either redundant or non-redundant. The measures of redundancy set in Equation 2.1 are nor- malized, which makes them independent of the bridge spec- ifications being followed and whether the bridge system is

16 overdesigned or under designed. This makes the proposed measures valid for the evaluation of existing bridges as well as new designs. The measures also are valid whether the bridge is deficient or up to standards. To check whether a bridge system has adequate levels of redundancy, it is sufficient to use a nonlinear structural analy- sis program to calculate LFu, LFf, LFd, and LF1, and to verify that Ru, Rf, and Rd are adequate. If the system configuration does not provide sufficient levels of redundancy, the bridge configuration may need to be changed. Note that even if the levels of redundancy Ru, Rf, and Rd are lower than expected, the bridge may still have high overall levels of member and system safety with high values for LFu, LFf, LFd, and LF1. Alternatively, a redundant system with high Ru, Rf, and Rd values may have low overall system safety levels. Thus, a bridge with adequate redundancy levels may still be unsafe for certain applications if its member safety level LF1 is too low. Therefore, the goal of any bridge design specifications should not be limited to providing adequate redundancy levels but to assure adequate system safety levels. Thus, if a bridge system does not provide an adequate level of redundancy, the bridge members could be conservatively designed to increase LF1 as well as LFu, LFf, and LFd, and reduce the probability of member failures and, more importantly, reduce the probability of system collapse. The evaluation of member and system safety can be per- formed using a direct nonlinear analysis of the system to obtain LF1, LFu, LFf, and LFd and verifying that they are ade- quate. However, executing a direct analysis of system safety and redundancy involves advanced analysis tools and exper- tise that may not be readily available for day-to-day evalua- tion of common types of bridges. For this reason, it has been proposed that the evaluation of the redundancy of common- type bridges be simplified by developing system factors that can be applied during the design and safety evaluation pro- cess to strengthen the members of bridges that are not suf- ficiently redundant. As part of this and previous NCHRP projects, hundreds of bridge superstructure and substructure configurations have been evaluated to study the relationship between LFu, LFf, LFd, and LF1 and the redundancy ratios Ru, Rf, and Rd defined in Equation 2.1. The results of these analyses are used in this study to calibrate system factors that account for bridge redundancy during the design and safety evaluation of bridge systems. The calibration of load and resistance factors in modern structural design and evaluation codes and specifications has been based on probabilistic methods to ensure that the codes pro- vide consistent levels of safety considering the uncertainties associated with estimating the strength of structural members and systems, those associated with predicting the maximum loads that the structure will be subjected to within its service life, and the response of the bridge structure to these applied loads. Therefore, the calibration of system factors that should be implemented in the next generation of structural codes to account for structural redundancy also should be based on the same probabilistic principles. The theory of structural reli- ability as developed over the past decades provides the basic tools necessary for performing such calibrations, as will be explained in Section 2.4. 2.4 Overview of Structural Reliability Theoretical Background The aim of structural reliability theory is to account for the uncertainties encountered while evaluating the safety of structural members and systems or during the calibration of load and resistance factors for structural design and evalua- tion codes. To account for the uncertainties associated with predicting the load carrying capacity of a structure, the inten- sities of the expected loads, the effects of these loads, as well as the capacity of structural members may be represented by random variables. The value that a random variable can take is described by a probability density function (PDF). That is, a random variable may take a specific value with a certain probability and the ensemble of these values and their probabilities are described by the PDF. The most important statistical characteristics of a random variable are its mean value or average, and the stan- dard deviation that gives a measure of dispersion or a measure of the uncertainty in estimating the variable. The standard deviation of a random variable R with a mean – R is represented by sR. A dimensionless measure of the uncertainty is the coef- ficient of variation (COV), which is the ratio of the standard deviation divided by the mean value. For example, the COV of the random variable R is represented by VR such that (2.2)V R R R = σ Structural codes and specifications often specify nominal or characteristic values for the variables used in design or load rating equations. These nominal values are related to the means through bias values. The bias is defined as the ratio of the mean to the nominal value used during the design or evaluation process. For example, if R is the member resis- tance, the mean of R, namely, – R can be related to the nominal or design value, Rn, through a bias factor, br, such that b R (2.3)r nR = where br is the resistance bias, and Rn is the nominal value as specified by the design code. For example, A50 steel has a nominal design yield stress of 50 ksi but coupon tests show an actual average value close to 56 ksi. Hence, the bias of the yield stress is 56/50 or 1.12.

17 In structural analysis, safety may be described as the situa- tion where capacity (member strength or resistance, the maxi- mum strain that a structural material can take before rupturing or crushing, ductility capacity) exceeds demand (applied load, applied moment, applied stresses, applied strains, or ductility demand). Probability of limit state exceedance (i.e., probabil- ity that capacity is less than applied load effects which is often referred to as probability of failure) may be formally calcu- lated; however, its accuracy depends upon detailed data on the probability distributions of the variables that represent the capacity and the demand. Since such data often are not avail- able, approximate models are used for calculation. In the application of structural reliability theory for the evaluation of the safety of a structural member or system, the reserve margin of safety of a bridge component is often defined as, Z, such that Z R S (2.4)= − where R is the resistance or member capacity and S is the total load effect. Probability of limit state exceedance, Pf, is the probability that the resistance R is less than or equal to the total applied load effect S or the probability that Z is less than or equal to zero. This is symbolized by the following equation: P Pr Z 0 Pr R S (2.5)f [ ] [ ]= ≤ = ≤ where Pr is used to denote the term probability. The calibration of the AASHTO LRFD and LRFR specifi- cations was based on evaluating the member resistance and the effects of the applied loads on the member. However, the same approach can be followed if the safety check consists of comparing the capacity of the entire system. If the strength capacity R and the load demand S follow independent normal (Gaussian) distributions, then the prob- ability of limit state exceedance can be obtained based on the mean of Z and its standard deviation, which can be calculated from the mean of R and S and their standard deviations as 0 (2.6) 2 2 P Z R S f z R S = Φ − σ     = Φ − − σ + σ     where F is the normal probability function that gives the probability that the normalized random variable is below a given value, Z _ is the mean safety margin, and sZ is the stan- dard deviation of the safety margin. Thus, Equation 2.6 gives the probability that Z is less than 0 (or R less than S). The reliability index, b, is defined such that (2.7)Pf ( )= Φ −β For example, if the reliability index is b = 3.5, then the implied probability of limit state exceedance is obtained from the normal distribution tables given in most books on statis- tics as Pf = 2.326x10-4. If b = 2.5 then Pf = 6.21x10-3. A b = 2.0 implies that Pf = 2.23x10-2. One should note that these Pf val- ues are only notional measures of risk that are used to com- pare different structural design and load capacity evaluation methodologies but are not actuarial values related to the col- lapse of the structure because of the different meaning that failure can take. For example, if the strain at one critical point of the structure exceeds the ultimate strain for this material, the failure is only localized and will not necessarily entail an actual failure in the sense that the structure will necessarily collapse. Furthermore, because of the many assumptions used in assembling the statistics of the applied load effects and in the definition of the capacity of a member or system, the cal- culated probabilities can only be treated as notional proba- bilistic measures of safety. Therefore, and in order to avoid referring to Pf as the probability of failure, it is most common to use the reliability index, b, as a measure of safety in struc- tural applications. If both the capacity and the demand that are represented by the resistance R and the load S can be modeled by normal distributions, the reliability index is obtained from (2.8) 2 2 Z R S Z R S β = σ = − σ + σ where R and S are assumed statistically independent. Thus, the reliability index, b, which is often used as a measure of structural safety, gives in this case the number of standard deviations that the mean margin of safety falls on the safe side as represented in Figure 2.2. Because it gives a measure of safety in terms of a number of standard devia- tions, it is more practical to use the reliability index b which may range between 0.0 to 6.0 to assess the safety of structures rather than using probability values which may range from Figure 2.2. Graphical representation of reliability index.

18 values as high as 50% to values as low as 10-9 or lower without giving an intuitive understanding of the corresponding level of safety. The reliability index, b, defined in Equations 2.7 and 2.8 provides an exact evaluation of the probability of exceedance if R and S follow normal distributions. Although b was origi- nally developed for normal distributions, similar calculations can be made if R and S are lognormally distributed (i.e., when the logarithms of the basic variables follow normal distribu- tions). In this case, the reliability index can be calculated as ln 1 1 ln 1 1 (2.9) 2 2 2 2 R S V V V V S R R S[ ]( )( ) β = + +     + + which, for values of VR and VS on the order of 20% or less can be approximated as ln (2.10) 2 2 R S V VR S β =   + Experience shows that Equation 2.10 provides a good approximation to the reliability index in many practical applications. For example, when evaluating the reliability levels implied in current and proposed load rating proce- dures during their work on NCHRP Project 20-07 Task 285, Ghosn et al. (2012) found that Equation 2.10 gives a very good approximation to the reliability index b even when the load effect, S, did not exactly follow a lognormal distribution. Approximate iterative methods have been developed to obtain the reliability index for the cases when the basic vari- ables are neither normal nor lognormal. A commonly used approach that has been shown to provide good approxima- tions for the reliability index, b, for common structural reli- ability problems is the First Order Reliability Method (FORM). FORM uses an iterative calculation to obtain an estimate of the probability of limit state exceedance. This is accomplished by approximating the failure equation (i.e., when Z = 0) by a tangent multi-dimensional plane at the point on the failure surface closest to the mean value and mapping non-normal probability distribution functions into equivalent normal functions. For example, during the calibration of the AASHTO LRFD Bridge Design Specifications, Nowak (1999) used the FORM algorithm developed by Rackwitz and Fiessler (1978) to calculate the reliability index for the case when R is assumed to follow a lognormal distribution and S is a normal random variable. More advanced techniques including SORM (Sec- ond Order Reliability Methods) also have been developed. However, Monte Carlo simulations can be used to provide estimates of the probability of exceeding a structure’s limit state. Monte Carlo simulations are suitable for any random variable distribution type and limit state equation. In essence, a Monte Carlo simulation creates a large number of “experi- ments” through the random generation of sets of resistance and load variables. Estimates of the probability of exceedance are obtained by comparing the number of experiments where the load exceeds the resistance to the total number of gener- ated experiments. Given values of the probability of exceed- ance, Pf, the reliability index, b, is calculated from Equation 2.7 and used as a measure of structural safety even for non-normal distributions. Kulicki et al. (2007) used the Monte Carlo simu- lation while reviewing the code calibration effort reported by Nowak (1999) and verified that the results of the FORM method with the Rackwitz-Fiessler algorithm, and those of the Monte Carlo simulation are essentially similar. More detailed explanations of the principles discussed in this section can be found in published texts on structural reliability (e.g., Thoft- Christensen and Baker, 1982; Melchers, 1999). The member reliability index has been used by many code writing groups throughout the world as a measure of struc- tural safety. Reliability index values in the range of b = 2 to 4 are usually specified for individual members depending on the type of member and the structural application. For exam- ple, the calibration of the Strength I limit state in the AASHTO LRFD Bridge Design Specifications aimed to achieve a uni- form target reliability index btarget = 3.5 for a range of typical bridge span lengths, beam spacing, and materials (Nowak, 1999). A reliability index btarget = 2.5 was used by Moses (2001) for the calibration of the legal load rating in the AASHTO LRFR. These values usually correspond to the failure of a single component. If there is adequate redundancy, overall system reliability indices will be higher. Although the AASHTO LRFD and LRFR bridge specifica- tions were calibrated based on satisfying member reliability criteria, the same concepts can be used to assess the reliabil- ity of a complete bridge system. Difficulties arise in system reliability evaluations because the system resistance R is a function of the resistances of the individual members of the system and their interaction. When an explicit closed-form formulation of this relationship is not possible, evaluations of the system capacity can be performed for specific samples of the member resistances using advanced finite element analy- sis (FEA) programs. The system reliability can then be evalu- ated using analytically derived margin of safety equations, Z, obtained by fitting approximate functions through the results of the finite element analysis. Using an iterative approach, the approximate function is fitted to the samples that lie very close to the actual failure surface. This method is known as the Response Surface Method (RSM) (Melchers, 1999). An important consideration during the reliability analysis process is the type of probability density function that each random variable follows and the accuracy of the simplified

19 equations in determining the reliability index b. Saydam and Frangopol (2013) compared the results that would be obtained when Equation 2.10 is used instead of the more exact Equa- tion 2.9, assuming that R and S follow lognormal distribu- tions. They also compared the results from both Equations 2.9 and 2.10 to those of the FORM if the probability distribution of the live load in actuality follows an extreme value type I Gumbel distribution rather than the assumed lognormal distribution. They found that the percent error introduced by using the simple expressions in system reliability analysis depends not only on the COV of the resistance and load effect but also on the ratio between the mean values of resistance and load effect. While the percent error is high when the reli- ability index is small and the ratio of R/S is close to 1.0, the dif- ferences in the reliability indices remain relatively close when the COV is within the 20% to 30% range. However, when the reliability index is on the high side, above 4.5, the effect of the probability distribution of the load becomes important. Reliability-Based Code Calibration Approach The reliability index b is seldom used in practice for mak- ing decisions related to ensuring structural safety during the design of a new bridge or the evaluation of an existing structure, but it is mostly used by code writing groups for rec- ommending appropriate load and resistance factors for use during the structural design process or when evaluating speci- fications. One commonly used calibration approach is based on the principle that the members of all types of structures should have uniform or consistent reliability levels over the full range of applications. For example, load and resistance factors should be chosen to produce similar member reliabil- ity index b values for steel and concrete bridges of different span lengths, number of lanes, number of beams and beam spacing, simple or continuous spans, and roadway categories. Thus, in a traditional code, a single target b must be achieved for all applications. More recently, researchers and code groups have been suggesting that higher values of b should be used for members of more important structures such as bridges with longer spans, bridges that carry more traffic, or bridges that, according to AASHTO, are classified as critical for “social/survival or security/defense requirements” and for non-redundant configurations. This is based on concepts that structures should provide uniform risk rather than uniform member reliabilities where risk takes into consideration the consequences of failure should a bridge member exceed its limit state. Since higher b levels would require higher con- struction costs, the justification should be based on a cost- benefit analysis whereby target b values are chosen to provide a balance between cost and risk (Aktas, Moses, and Ghosn, 2001). This type of reasoning has been informally used to jus- tify the adoption of a member reliability index b = 2.5 for the 5-year service life during the load rating of existing bridges as compared to a reliability index b = 3.5 for a 75-year design life when designing the members of new bridges or to apply a load modifier to increase the reliability level of non-ductile mem- bers and members of non-redundant bridge configurations. Because it is difficult to estimate the lifecycle costs and assess the consequences of failure including the direct, indirect, and user costs that ensue when a bridge member exceeds its limit, a formal risk analysis or cost-benefit analysis is seldom used in practice. Instead, recent codes have adopted informal methods based on the perception of risk. This informal risk- inspired process is currently complementing the approach taken by previous codes that generally used the reliability index of previous safe designs to decide on the reliability cri- teria that new codes should achieve. In most cases, appropri- ate target b values are deduced based on the reliability levels of a sample population of satisfactorily performing existing designs. That is, if the safety performance of bridges designed (or rated) according to current standards has generally been found satisfactory, then the average reliability index obtained from current designs is used as the target that the new code should satisfy. The aim of the calibration procedure is to mini- mize designs that deviate from the target reliability index. The calibration based on past performance has been found to be robust in the sense that it minimizes the effects of any inadequacies in the database as reported by Ghosn and Moses (1986). Ghosn and Moses (1986) found that the load and resis- tance factors obtained following a calibration based on “safe existing designs” are relatively insensitive to errors in the sta- tistical database as long as the same statistical data and criteria used to find the target reliability index also are used to calcu- late the load and resistance factors for the new code. In fact, a change in the load and resistance statistical properties (e.g., in the COV) would affect the computed b values for all the bridges in the selected sample population of existing bridges and consequently their average b value. Assuming that the per- formance history of these bridges is satisfactory, then the target reliability index would be changed to the new “average” and the calibrated load and resistance factors that would be used for new designs would remain approximately the same. The calibration of resistance and live load factors for a new bridge code is usually executed by code writing groups as follows: • A representative sample of bridges that have been designed to efficiently satisfy existing codes and that have shown good safety record is assembled. • Reliability indices are calculated for each bridge of the rep- resentative sample set. The calculation is based on statisti- cal information about the randomness of the strength of members, the statistics of load intensities, and their effects on the structures.

20 • In general, there will be considerable scatter in such com- puted reliability indices. A target b is selected to corre- spond to the average reliability index of the representative bridge sample set. • For the development of the new code, load and resistance factors as well as nominal loads are selected by an itera- tive optimization process to satisfy the target b as closely as possible for the whole range of applications. The above approach was followed by Nowak (1999) and Moses (2001) to calibrate the appropriate live load factors for the AASHTO LRFD and LRFR Legal Load Ratings in order to meet a target member reliability index b = 3.50 for the 75-year service life of new bridge design and a member reli- ability index b = 2.5 for the 5-year rating period of existing bridges. The reliability calibration of the LRFD and LRFR was based on maintaining the same reliability level on the indi- vidual main beam members in shear and bending. No direct consideration was made during the calibration of the AASHTO LRFD and LRFR for system redundancy or system reliability. Load modifiers, h, have been included in the LRFD design-check equations to account for bridge redundancy, member ductility, and importance of the struc- ture. However, as stated in the commentary, the assigned val- ues were based on judgment and were not calibrated to meet specific reliability index targets. The concept of applying a load modifier is based on the perception of risk in terms of considering the consequence of the failure of a non-ductile member of a non-redundant bridge, or the failure of a mem- ber of an important structure. However, the specifications do not provide clear guidelines to help engineers decide when a bridge can be defined to have low levels of redundancy requiring the use of a load modifier greater than 1.0. In a similar vein, the AASHTO LRFR recommends the applica- tion of system factors less than 1.0 placed on the resistance side of members of bridges that are known to have low lev- els of redundancy based on the number of beams and beam spacing, or for some connection types that are known to have low levels of ductility. Using a more directly risk-inspired approach, the Cana- dian Code (CAN/CSA-S6-06) recommends different target reliability index b values depending on the failure mode, the system behavior, as well as the element behavior, and the inspectability of bridges. The recommended target reliabili- ties vary between b = 2.50 and 4.0 as shown in Table 2.1 and the member resistance factor is changed based on the target reliability level. It is also noted that it is not clear how the target reliability levels in Table 2.1 were determined and how an engineer can evaluate whether one element failure leads to total collapse or will only lead to local failure, and how to determine whether the failure will be sudden or gradual if no direct nonlinear analysis is performed. Summary This section reviewed reliability analysis methods and how current codes are calibrated to satisfy target reliability crite- ria. The review also explains how current recommendations for including redundancy in bridge design and evaluation are mostly based on the judgment of the code writers and the perception of risk rather than on non-subjective, reliability- based criteria. To meet the objectives of this study, and fol- lowing the format adopted in the AASHTO LRFD Bridge Design Specifications and the Canadian bridge code, it is recommended that different design criteria be established for bridges based on their levels of redundancy. This project proposes that the said objectives be achieved by applying a system factor in the safety-check equation such that bridges with low levels of redundancy be required to have higher member resistances than those of bridges with high levels of redundancy. Although requiring a higher capacity for the members of non-redundant bridges will not make them redundant, it will increase the overall system reliability so that non-redundant bridge systems would have similar sys- tem reliability levels as those of redundant systems. This can be done by using a member safety-check equation of the form ( )φ φ = γ + γ +1 (2.11)R D L Is N d n l n where fs is the system factor which is defined as a statisti- cally based multiplier relating to the safety and redundancy of the complete system. The system factor is applied to the factored nominal member resistance. The proposed system factor replaces the load modifier h used in Section 1.3.2 of the LRFD specifications. The system factor is placed on the left side of the equation because the system factor is related Table 2.1. Target reliability index for normal traffic in Canadian Code. System Behavior Element Behavior Inspection Level INSP1 INSP2 INSP3 S1 E1 4.00 3.75 3.75 E2 3.75 3.50 3.25 E3 3.50 3.25 3.00 S2 E1 3.75 3.50 3.50 E2 3.50 3.25 3.00 E3 3.25 3.00 2.75 S3 E1 3.50 3.25 3.25 E2 3.25 3.00 2.75 E3 3.00 2.75 2.50 Notes: S1 - element failure leads to total collapse; S2 - element failure does not cause total collapse; S3 - local failure only; E1 - sudden loss of capacity with no warning; E2 – sudden failure with no warning but with some post-failure capacity; E3 – gradual failure; INSP1 – component not inspectable; INSP2 – inspection records available to the evaluator; INSP3 – inspections of the critical and substandard members directed by the evaluator.

21 to the capacity of the system and should be placed on the resistance side of the equation as is the norm in reliability- based LRFD codes. f is the member resistance factor, RN is the required resistance capacity of the member accounting for the redundancy of the system, gd is the dead load factor, Dn is the dead load effect, gl is the live load factor, Ln is the live load effect on an individual member, and I is the dynamic amplification factor. When a system factor fs equal to 1.0 is used, Equation 2.11 becomes the same as the current design equation. If fs is greater than 1.0, this indicates that the system’s configuration provides a sufficient level of redundancy and that a redundant bridge could have lower member capacities than those of a non-redundant bridge and yet have a sufficiently high level of system safety. When fs is less than 1.0, then the level of redun- dancy is not sufficient and the bridge members will have to be designed to produce higher member capacities to account for the consequence of a member failure on the system’s safety. The next section describes an approach that builds on the method originally proposed in NCHRP Report 406 to cali- brate system factors for typical bridge configurations based on reliability criteria. 2.5 Reliability Calibration of System Factors This section describes the approach used in this study to calibrate the system factors that can be used in the design- check equation to account for bridge redundancy during the design of new bridges and the safety evaluation of existing bridges. The calibration is executed using reliability princi- ples in keeping with modern code calibration practice. The general procedure presented in this section is formulated for bridges subjected to vertical traffic loads. Chapters 3, 4, and 5 extend the procedure and provide specific details that are applied to calibrate the system factors for systems under lateral, as well as vertical, loads. As observed in Section 2.4, the reliability analysis of a bridge member or system requires as input statistical data on the member or system capacity as well as the loads that will be applied on the bridge structure and how the effect of these loads is distributed throughout the structure. As explained in Section 2.3, the evaluation of a bridge’s safety should verify that the bridge (1) will provide a reasonable safety level against first member failure, (2) will not reach its ultimate system capacity under extreme loading conditions, (3) will not undergo excessive deformations under expected traffic load conditions, and (4) will be able to carry some traf- fic loads after damage to, or the loss of, a component. As explained in Section 2.3, a bridge member’s capacity can be evaluated using the parameter LF1 while the originally intact system capacity can be evaluated using the parameters LFu and LFf, which assess the ultimate capacity and the func- tionality of the originally intact system. A damaged bridge’s capacity can be evaluated using the parameter LFd. Although NCHRP Report 406 and NCHRP Report 458 considered the redundancy analysis for the functionality limit state sepa- rately, a review of the NCHRP Report 406 data shows that LFf and LFu are highly correlated, which leads to similar system factors for both limit states. As an example, Figures 2.3 and 2.4 show the relation between LFf and LFu for simple-span steel I-girder and prestressed I-girder bridges where LFf is selected to be the load factor at which the bridge under verti- cal load reaches a maximum vertical displacement equal to span length/100. The fact that the trend lines in both figures pass through the origin, and the slope is exactly the same, indicate that the correlation between the two variables is very strong, which implies that there is no need to analyze a bridge’s ultimate limit state and its functionality limit state separately. For this reason, in this report the evaluation of the capacity of intact bridges will be based solely on the ultimate limit as represented by LFu. The criteria for evaluating the safety and redundancy of bridge structures as explained in Section 2.3 are based on the LFf= 0.74 LFu R² = 0.90 0 2 4 6 8 10 12 0 2 4 6 8 10 12 LF f LFu Figure 2.3. LFf versus LFu for steel I-girder bridges. Figure 2.4. LFf versus LFu for prestressed I-girder bridges. LFf= 0.74LFu R² = 0.90 0 1 2 3 4 5 6 7 8 0 2 4 6 8 10 LF f LFu

22 deterministic measures defined in Equation 2.1 that do not take into consideration the uncertainties in evaluating the sys- tem capacity and the loading. As explained in Section 2.4, the safety of a bridge member, that of the originally intact entire system, or that of a damaged system, should be assessed using reliability criteria. In addition to the probabilistic models for member and system capacities, the reliability analysis requires as input information on the expected live loads. The member and system capacities of an originally intact system should be able to sustain the maximum load expected over the bridge’s entire design life. The maximum load is herein defined as the extreme loading condition. However, a major damage to a bridge is not expected to remain unnoticed for a long period after damage given that all bridges are inspected on a 2-year cycle. Therefore, it is suggested that the vertical vehicular load that a damaged bridge should be able to sustain should corre- spond to the maximum load over the 2-year inspection cycle. This load is herein defined as regular loading. The reliability formulation of the problem is presented next. Safety Margin Equations for Bridges under Vertical Loads The first necessary step for performing the reliability anal- ysis of a bridge under the effect of vertical vehicular load con- sists of setting up the safety margin equations that compare the resistance of the member or the system to the applied loads. These safety margin equations would be used in Equa- tions 2.5 through 2.8 to find the probability of limit state exceedance and the reliability index. Using the results of an incremental analysis that traces the performance curve shown in Figure 2.1, and ignoring the functionality limit state that, as explained earlier, is highly correlated to the ultimate limit state, the following safety margin equations are obtained: For first member failure: For the Z LF LL1 1 75= − ultimate capacity of an originally intact system: For the capacity Z LF LLu u= − 75 2 12( . ) of a damaged system: Z LF LLd d= − 2 where LL75 gives the maximum load expected in a 75-year design life expressed in terms of the number of AASHTO HS-20 trucks. The 75-year load is used in Equation 2.12 for the member and ultimate limit states because the AASHTO LRFD specifies that bridges should be designed for a 75-yr design life. LL2 gives the maximum load expected in a 2-year inspection cycle expressed in terms of the number of AASHTO HS-20 trucks. The 2-year load is used for the damaged bridge because the damage should be detected during or before the 2-year inspection. LF1, LFu, and LFd give the multiples of the HS-20 trucks needed to cause the failure of the first member, reach the ultimate capacity of the originally intact system, and cause the collapse of a damaged system. The HS-20 truck is used to express the maximum load in order to keep the same basis as that used for finding LF1, LFu, LFf, and LFd. LL75 and LL2 must include the total live load on the bridge includ- ing the dynamic amplification factor. Additional discussions on these random variables are provided next. Loading Models Extreme Live Loading Conditions In addition to carrying its dead load, a bridge should be able to carry the maximum truck load expected to be applied on it during its design life without reaching its ultimate capacity. This maximum expected live load is a statistical variable that depends on the number of trucks that simultaneously cross the bridge, the positions of the trucks on the bridge deck, the weights of the trucks, the distribution of the weights to the individual axles, and the trucks’ axle configurations. In addi- tion, the load is a function of the dynamic amplification caused by the interaction between the moving loads and the structure. According to the AASHTO LRFD Bridge Design Specifica- tions, the design life of a bridge is normally equal to 75 years. For longer, expected bridge lifespans, there is a higher prob- ability of having heavier trucks simultaneously on the bridge. The 75-year lifespan, however, seems to provide an asymptotic limit beyond which the increase in the maximum expected load is practically negligible (Nowak, 1999). The 75-year expo- sure period was used in the LRFD to calibrate the live load factors for the reliability analysis of bridge members and the calibration of the resistance and load factors. In this study, the same expected live load is used for the ultimate limit state of the system as well as the first member failure limit state. In the latter case, the 75-year exposure period is consistent with the basis for the AASHTO LRFD specifications (Nowak, 1999). As indicated by Nowak (1999), the maximum expected life- time load can be expressed in terms of equivalent AASHTO HS-20 loads. For example, the maximum live load effect in a 75-year period, labeled LL75, is defined as the multiple of the HS-20 loads needed to produce the same load effect as the maximum expected 75-year load. Table 2.2 gives the expected LL75 values for simple-span bridges of different span lengths for two-lane loadings as well as one-lane load- ing as provided by Nowak (1999). In the two-lane case, the values are lower than for the one-lane case because they mul- tiply two HS-20 loads. In this study, the two-lane loading is used as the reference load configuration. The LL75 values in Table 2.2 are the same values used by Nowak (1999) in the calibration of AASHTO’s LRFD Bridge Design Specifications and include an average dynamic amplification factor equal to 1.10. Table 2.2 also shows the COV of the LL75 values. The

23 COV is the ratio of the standard deviation of LL75 divided by the mean value. The COV values in Table 2.2 also are taken from Nowak (1999). These COV values are used in this study to be consistent with the values used during the calibration of the AASHTO LRFD specifications. Specific information on the probability distribution of the static live load or the dynamic amplification factor was not provided by Nowak (1999), who assumed that the combined effect of the dead loads plus live load follows a normal probability distribution. Ghosn et al. (2012), however, observed that the maximum live load may be approximated by a lognormal distribution because it is obtained from the product of several random variables. The Central Limit Theorem states that the product of a large number of random variables should approach a lognormal distribution. Regular Live Load Conditions A damaged bridge system should still be able to continue to carry some load after the occurrence of a damaging event that reduces the load carrying capacity of a main member. This would allow (1) for clearing the traffic that is on the bridge, (2) the safe crossing of regular traffic if the damage goes un noticed for a limited period of time, and (3) the passage of emergency vehicles during disaster recovery. It is herein proposed that in addition to carrying its dead load, a damaged bridge should be able to carry the maximum live loads expected during a 2-year exposure period. The maximum live load expected over the 2-year exposure period is defined as “regular traffic load.” The 2-year exposure period is chosen because it corresponds to the biennial mandatory bridge inspection cycle. Thus, the 2-year exposure period is selected based on the premise that even if damage such as a fatigue fracture goes unnoticed for a short period of time, it is bound to be discovered during inspec- tion. LL2 is defined as the multiple of the HS-20 loads needed to produce the same load effect as that expected under regular truck traffic conditions. Table 2.2 gives the expected 2-year loads expressed as multipliers of the effect of two AASHTO HS-20 trucks for two-lane loadings and the expected 2-year loads expressed as multipliers of the effect of one AASHTO HS-20 for one- lane loadings. The LL2 values in Table 2.2 are adopted from the work of Nowak (1999). They include the mean dynamic amplification factor of 1.13. Because no statistical data on the COV of LL2 are available, the same COVs are assumed to be valid for both LL75 and LL2, although it is generally known that the shorter period is normally associated with slightly higher COVs. The same COV values are used because a break- down of the sources of uncertainty in the load model was not provided by Nowak (1999). Using the same COV follows the approximation made by Moses (2001) who assumed that the COV for the 5-year and 75-year live loads are the same. This assumption should not alter the final results because the reli- ability calibration process is known to be a robust process in the sense that errors in the database do not influence the final results as long as the new design or evaluation procedures are calibrated to match current acceptable practice (Ghosn and Moses, 1986). Following the same logic discussed for the 75-year loading, it may be reasonable to assume that the 2-year loading approaches a lognormal distribution. Resistance Model According to Nowak (1999), the load carrying capacities or resistances R of structural members may be modeled as lognormal variables. For prestressed concrete members, the mean moment capacity is about 1.05 times the nominal value obtained from typical code-specified methods with a COV of 7.5%. For composite steel members, the bias between the mean moment capacity and that obtained using code- specified methods is estimated at 1.12 with a COV of 10%. Nowak (1999) also gives bias and COV values for the dead load effect that are on the order of 1.05 and a COV of about 10% for cast-in-place members with slightly lower bias and COV for factory-made members and higher COV for pavement. The load factors LF1, LFu, and LFd are related to the bridge members’ resistances, dead weights, and the effect of the applied live loads. Specifically, the calculation of the load Span Length () Two Lane Loading One Lane Loading COV Distribuon Type LL75 LL2 LL75 LL2 VLL lognormal 45 1.67 1.53 1.97 1.81 19% 60 1.72 1.6 2.02 1.86 19% 80 1.81 1.67 2.14 1.98 19% 100 1.89 1.75 2.26 2.08 19% 120 1.98 1.84 2.35 2.17 19% 150 2.01 1.87 2.37 2.19 19% Includes 1.10 Dynamic Amplificaon Includes 1.13 Dynamic Amplificaon Table 2.2. Mean and COV of applied live loads as function of HS-20 trucks.

24 factor for first member failure LF1 can be calculated from the capacity of the most critical member of the bridge, R, the dead load effect on that member, D, and L1, which is the load effect on the member calculated due to the HS-20 trucks. Specifically, LF1 can be obtained from . . (2.13)1 1 20 LF R D L R D D F LLHS = − = − × − where L1 is expressed in terms of the fraction of the HS-20 truck that is carried by the most critical member, which is obtained by multiplying the load effect of the HS-20 by the distribution factor, D.F. Although L1 is calculated in this study using a structural model of the entire bridge, an approxima- tion for L1 can be obtained using an accurate estimation of the distribution factors such as those of the LRFD specifications. Equation 2.13 is identical to the rating factor equation used when evaluating the load carrying capacity of existing bridges without the load and resistance factors or the impact factor. Equation 2.13 serves the same purpose as the rating factor in that it provides a measure of the load carrying capacity of the system up to first member failure but on the basis of the nominal unfactored resistances and loads. In the analyses performed in this study, the capacity of a bridge system also is expressed in terms of the load factors LFu and LFd, which represent the differences between the over- all capacities of the originally intact system and that of the damaged system minus the effect of the dead load as demon- strated in Equation 2.13 for LF1. Using the statistical models of resistance and dead load from Nowak (1999) along with the resistance and dead loads of the simple-span I-girder bridges analyzed in this study and in NCHRP Report 406, the biases and COV for the load factor LF1 are obtained as shown in Table 2.3. Wang et al. (2011) performed a reliability analysis of several bridge systems using a push down analysis similar to the one used in this study. They observed that the load capac- ity evaluation will lead to capacities that approach lognormal probability distributions with an overall COV on the same order of that provided by Nowak (1999) for individual mem- bers. Based on this observation, the biases and COVs obtained in Table 2.3 will be used for the member capacities expressed in terms of LF1 as well as the system capacities of the intact and damaged systems expressed in terms of LFu and LFd. Simplified Reliability Evaluation As explained in Section 2.3, bridge safety should be assessed in terms of the three limit states identified as (1) member fail- ure, (2) ultimate limit state, and (3) damaged condition limit state. These three limit states should be checked to ensure the satisfactory and safe performance of any bridge system under extreme or regular loading conditions. “Adequate” safety mar- gins also should be provided to account for the uncertainties associated with determining bridge system capacity as well as the uncertainties associated with determining the live load lev- els that will be applied on the bridge. Following current state- of-the-art practice in bridge code calibration, adequate safety margins should be established based on structural reliability criteria similar to those used in the development of AASHTO’s LRFD and LRFR specifications, as well as the Canadian Codes (Nowak, 1999; Moses, 2001; and CAN/CSA-S6-06). The measure of safety used in the development of struc- tural design and evaluation codes is the reliability index b (Nowak, 1999). The reliability index can be used as a measure of the reliability of structural members as well as structural systems. The reliability index accounts for both the margin of safety implied by the design procedure and the uncertainties in estimating member strengths and applied loads. The reli- ability index can be related to the probability of a limit state exceedance as shown in Equations 2.5 and 2.7. Assuming that the member resistance of a bridge under ver- tical load represented by the load factor LF1, and the applied maximum lifetime live load represented by the factor LL75 are random variables that follow lognormal distributions, then the reliability index bmember for the failure of the first member can be expressed based on the safety margin Equation 2.12 using the simplified lognormal format of Equation 2.10, which is reason- able for the cases when the COVs are less than 20%. Accordingly ln ln 20 20 ln (2.14) 2 2 1 75 2 2 1 75 2 2 R S V V LF HS LL HS V V LF LL V V member R S LF LL LF LL β =   + =     + =     + where LF1 is the mean value of the load factor that will cause the first member failure in the bridge, assuming elastic analy- sis. As explained through Equation 2.13, LF1 is related to the unfactored live load margin (R-D). Thus, LF1, which is Bridge Type Bias for Member Resistance COV for Member Resistance Bias for LF COV for LF Distribuon Type Prestressed concrete 1.05 7.5% 1.05 13.7% LognormalComposite steel 1.12 10% 1.14 13% Table 2.3. Mean and COV of load capacity as function of HS-20 trucks.

25 the mean value of LF1, relates to the strength capacity of the member represented by the resistance R and the dead load D. LL75 is the mean value of the maximum expected 75-year live load, including dynamic load allowance effect. VLF is the COV of LF1, while VLL is the COV of the maximum expected live load LL75. The denominator in Equation 2.14 gives an overall measure of the uncertainty in estimating the resistance, the dead load, and the live load including dynamic amplification. Using equations similar to Equation 2.14 has been common during the reliability analysis of bridge members. However, the reliability analysis of an entire bridge system involves a much more complicated process than the one normally used for the analysis of individual members. This is because the analysis must take into consideration the interaction of all the bridge members throughout the entire loading process from the initiation of loading until collapse during which each member may undergo different levels of deformations includ- ing linear-elastic and nonlinear strains. The most effective approach to account for all the random factors that control a system’s behavior, including the uncertainties in the linear and nonlinear material modeling, is through simulations. System reliability simulation programs require extensive computa- tional effort, which cannot be accommodated within the con- straints of this project. For this reason, a simplified reliability formulation is adopted in this study, which uses a lognormal model for the system capacity analogous to the model used for individual members. A few comparisons have been performed as part of this study to verify that the proposed simplified approach gives results that fall within a reasonably acceptable range as those from advanced simulation techniques. The analyses of Wang et al. (2011) also suggest that the simplified reliability model should be applicable. The proposed simplified approach to find the reliability index for the ultimate capacity of a bridge system assuming that the load factor LFu and the live load factor LL75 follow lognormal distributions will lead to a reliability index of the system for the ultimate limit state that can be defined as ln ln 20 20 ln (2.15) 2 2 75 2 2 75 2 2 R S V V LF HS LL HS V V LF LL V V ultimate R S u LF LL u LF LL β =   + =     + =     + where LFu is the mean value of the load factor corresponding to the ultimate limit state. LFu relates to the strength capac- ity of the system and the dead load. LL75 and VLL are the mean of the 75-year live load and its COV and are the same val- ues used to calculate bmember. The limited data on the COV of bridge systems suggest that LFu and LFd have COV VLF similar to those used for LF1. The theory of reliability of structural systems demonstrates that, in general, the COV of a system is smaller than the COV of the individual members. How- ever, this observation is based on the assumption that the structural model used during the nonlinear analysis is exact. In this study, the COVs of the intact and damaged system capacity (LFu and LFd) are assumed to be equal to the COV of the member capacity (LF1) to account for the modeling uncertainties. As done for the evaluation of the reliability of the individual members, Equation 2.15 also assumes that the live load (LL75) follows a lognormal distribution. Following the same logic, the system’s ability to sustain loads after damage can be expressed as a system reliability index for damaged conditions, bdamaged, defined as ln ln 20 20 ln (2.16) 2 2 2 2 2 75 75 2 2 2 R S V V LF HS LL HS V V LF LL LF LL V V damaged R S D LF LL d LF LL β =   + =     + = ×     + where LFd is the mean load factor to reach the ultimate capac- ity of the damaged system. LFd is the capacity of the system to carry live load after one member is damaged. A 2-year expo- sure period is used for the damaged condition. The mean live load for the 2-year period is expressed as LL2, which is a multiplier of the effects of two HS-20 vehicles. Equation 2.16 assumes that the capacity of the damaged bridge system, as well as the load, follow lognormal distributions. The reliability index formulations of Equations 2.15 and 2.16 are conservative as they assume that the bridge members’ strengths are fully correlated. Probabilistic Measures of Bridge Redundancy Equations 2.14, 2.15, and 2.16 provide reliability measures that evaluate the safety of bridge members and systems. How- ever, as explained in Section 2.3, bridge redundancy is not a direct measure of the overall system capacity or overall sys- tem safety, but a measure of the additional safety provided by the system relative to that of a member. Equation 2.1 pro- vides a set of redundancy measures based on a deterministic evaluation of the additional safety that the system can pro- vide beyond its capacity to resist the failure of a critical mem- ber. Alternatively, and in order to take into consideration the uncertainties in estimating the system and member capacities as well as the applied loads, probabilistic measures of redun- dancy can be defined. In this context, a probabilistic evalua- tion of system redundancy would entail an examination of the differences between the reliability of the intact and dam- aged system expressed in terms of bultimate and bdamaged and the reliability of the most critical member, bmember, using different methods (Frangopol and Curley, 1987; Frangopol and Nakib, 1991; Hendawi and Frangopol, 1994). Specifically, Ghosn and Moses (1998) defined a set of “reliability index margins” Dbu,

26 Dbd that compare the reliability indices for the ultimate and damaged limit states to that of the most critical member. The reliability index margins Dbu, Dbd are defined as (2.17) u ultimate member d damaged member ∆β = β − β ∆β = β − β Using the simplified lognormal reliability model of Equa- tions 2.14 through 2.16 for a superstructure under the effect of vertical live loading and assuming that the COV of LFu, LFd, and LF1 are all equal to the same value, VLF, the proba- bilistic and deterministic measures of redundancy are found to be directly related to each other as shown in the following: ln ln ln ln (2.18) ln ln ln ln 75 1 75 2 2 1 2 2 2 2 2 1 75 2 2 1 75 2 2 2 75 2 2 2 LF LL LF LL V V LF LF V V R V V LF LL LF LL V V LF LF LL LL V V R LL LL V V u ultimate member u LF LL u LF LL u LF LL d damaged member d LF LL f LF LL d LF LL ( ) ∆β = β − β =     −     + =     + = + ∆β = β − β =     −     + =     + =     + The relationships in Equation 2.18 show that for a log- normal model, and assuming that the bias in the load factor is the same, the reliability index margins are directly related to the redundancy ratios Ru = LFu/LF1 and Rd = LFd/LF1, as defined in Equation 2.1. Accordingly, a bridge system will provide adequate levels of system redundancy if the reliability index margins defined in Equation 2.18 are adequate. Establishing Reliability-Based Redundancy Criteria Modern-day structural design codes are calibrated to pro- vide uniform levels of member reliability indices. The AASHTO LRFD specifications were calibrated to provide a member reli- ability index, b = 3.5, assuming that the system provides suf- ficient levels of redundancy. Bridges that are not sufficiently redundant are “penalized” by requiring that their members be more “conservatively designed” so that their member reliability indices are higher than b = 3.5. This is effectively executed by applying the load modifiers h specified in the AASHTO LRFD Bridge Design Specifications. Mertz (2008) explains that apply- ing a load modifier h = 1.10 will effectively raise the member reliability index to a value b = 4.0. This is essentially done to compensate for the lower system reliability level that a non- redundant bridge with bmember = 3.5 has compared to the system reliability level of a redundant bridge with the same member reli- ability index. However, the member reliability index is reduced to a value b = 3.0 for bridges that are redundant when they are assigned the load modifier h = 0.95. This essentially “rewards” the members of redundant bridges by allowing a lower mem- ber reliability index. This is essentially done because redundant bridges with b = 3.0 will provide sufficiently high system reli- ability levels comparable to those of non-redundant bridges with bmember = 4.0. One can observe from this interpretation, that although not explicitly stated in the AASHTO LRFD, the rationale behind the application of the load modifiers is con- sistent with the concept of providing adequate spread between the reliability indices of the system and that of the members. Systems where the spread is large can be allowed lower values of member reliability levels; those where the spread is small will have to be assigned higher member reliability levels. By using target reliability indices ranging between b = 2.5 and 4.0 as shown in Table 2.1, the Canadian CAN/CSA-S6-06 code achieves the same goals by explicitly changing the target member reliability index rather than using the preset load modifier values of the AASHTO LRFD. The rationale behind the AASHTO LRFD and the Cana- dian CAN/CSA-S6-06 code methods for accounting for bridge redundancy is consistent with the concept proposed in this project to evaluate the system redundancy based on the reliabil- ity index margins of Equation 2.18. However, the final decision on what values should be assigned to the AASHTO load modi- fiers or what member reliability indices should be used in the Canadian Code, are left up to the bridge engineer who should use his/her judgment to decide how to classify a bridge in terms of its level of redundancy and to assess the consequences of a member’s failure. Furthermore, the proposed AASHTO LRFD load modifier values were arbitrarily assigned by the code writers and the target reliability values in CAN/CSA-S6-06 were selected arbitrarily in a manner that is not consistent with modern-day code calibration methods. Following the procedures proposed in the AASHTO LRFD and the Canadian bridge code, it is recommended that differ- ent design criteria be established for bridges based on their levels of redundancy. This can be achieved by applying a sys- tem factor in the safety-check equation such that bridges with low levels of redundancy be required to have higher member resistances than those of bridges with high levels of redun- dancy. For bridges under vertical live loads, this goal can be achieved by using a member safety-check equation of the form provided in Equation 2.11, which is repeated below. ( )φ φ = γ + γ +1 (2.11)R D L Is N d n l n The system factors to be applied in conjunction with Equa- tion 2.11 should be calibrated using a reliability model such

27 that a system factor equal to 1.0 indicates that the bridge is sufficiently redundant and that the reliability index of the sys- tem is higher than that of the member by an amount equal to a target value. Following common reliability-based code calibration processes, the target values can be established by studying the reliability of bridge systems that have histori- cally shown adequate levels of redundancy in the sense that one of their critical members has failed, and the system did not undergo collapse. Determination of Reliability-Based Redundancy Criteria Equation 2.18 provides non-subjective reliability-based measures to evaluate the redundancy of bridge systems. Bridges whose reliability index margins are adequate should be con- sidered to be redundant. In Section 2.4, it was explained that the determination of the reliability criteria that bridge mem- bers should satisfy was established based on matching a tar- get reliability index equal to the average reliability index of bridges that have historically been known to have performed well. To remain consistent with modern code calibration methods, it is herein proposed that bridges that have reliabil- ity index margins Dbu and Dbd equal to or above target values be classified as redundant. The determination of the mini- mum target Dbu target and Dbd target values that a bridge should satisfy must be established based on a review of the perfor- mance of existing redundant designs. This section describes how this selection process is performed for a system under vertical loads. To establish the target reliabilities, a large number of common-type simple-span prestressed I-girder and steel I-girder bridges having span lengths varying between 45 ft and 150 ft were analyzed in NCHRP Report 406. The bridges have 4 to 10 beams with spacing varying between 4 ft and 12 ft. The results of these analyses are given in Appendixes B and C of NCHRP Report 406. For each bridge configuration, the load multipliers, LF1, LFu, and LFd, were calculated using a non linear incremental bridge analysis program. A grillage model is used for the incremental analysis of the super structure assuming two side-by-side HS-20 trucks as the base live load model. The validity of the grillage model for this type of analysis was extensively tested as part of NCHRP Report 406 and further verified during the course of this current project. Given the load factors LF1, LFu, and LFd, the reliability indices bmember, bultimate, and bdamaged were calculated for each bridge configuration using Equations 2.14, 2.15, and 2.16. Several comparisons were performed with results of FORM algorithms to verify that the application of the simplified lognormal equations gives reasonably accurate results for the purposes of this study. The results of the reliability index calculations are then used to establish the reliability-based redundancy criteria and the target reliability index margins necessary for calibrating the system factors. Ultimate Limit State Criteria In current practice, all two-girder bridges, and according to several opinions, even three-girder bridges, are defined as non-redundant. On the other hand, all bridges with four or more beams are always classified as redundant and experi- mental investigations have demonstrated that when one girder of four-girder bridges has been overloaded, the bridge has been able to sustain considerable traffic load without collapsing. Therefore, it is recommended to use the average Dbu value obtained from four-girder bridges as the target reliability index margin that all bridges should satisfy to be considered as adequately redundant. The calculations per- formed in NCHRP Report 406 show that typical two-lane, simple-span, steel and prestressed concrete I-beam bridges with four beams at spacings equal to or greater than 4 ft have Dbu average value of 0.85. The range of Dbu is between 0.04 and 1.23 for steel bridges and between 0.02 and 1.53 for pre- stressed concrete bridges. Generally, it is observed that for the same span length and beam spacing, the prestressed con- crete bridges have higher dead weight, which results in lower values of LF1, leading to slightly higher Ru = LFu/LF1 ratios and Dbu values. To account for all possible differences in the four-beam bridges, NCHRP Report 406 proposed to use a Dbu target = 0.85 as the target reliability index margin that bridges deemed to be adequately redundant should satisfy for the ultimate limit state. Damaged Condition Damaged bridges are assumed to have lost the load carrying capacity of an external girder. NCHRP Report 406 also ana- lyzed the large set of simple-span steel and prestressed concrete I-girder bridge configurations that it assembled assuming that the external girder is no longer capable of carrying any load while its dead load as well as the total live load must be carried by the remaining members. Damaged two-lane bridges with four beams analyzed in NCHRP Report 406 gave an average value for Dbd equal to -2.70. Based on these results, it is rec- ommended to use a target Dbd target = -2.70 as the target rela- tive reliability index for the damaged condition. Calibration of System Factors The calibration of the system factor fs that should be applied in Equation 2.11 can be executed using the reliability formulation presented above so that the reliability index of the members and of the systems for bridges that are not suf- ficiently redundant is increased. That is, when the available

28 reliability index margins Dbu and Dbd are lower than the tar- get values Dbu target = 0.85 and Dbd target = -2.70, a system factor fs < 1.0 should be used in Equation 2.11. However, fs should serve to lower the reliability index for the member and the system when the available Dbu and Dbd are higher than the target values. The amount by which the reliability indexes of the systems bultimate and bdamage should be increased should be equal to the deficit in the available Dbu and Dbd when compared to the target values. The goal is to ensure that non-redundant bridge configurations will still provide bultimate and bdamage values sim- ilar to those of redundant bridges. Because the design process controls the member capacities, achieving higher bultimate and bdamage values can be done by imposing higher bmember through the application of a system factor fs into the design and safety evaluation Equation 2.11. The formulation can be summarized as described in this section for the ultimate limit state for bridges under vertical loads. The same exact procedure is also valid for finding the system factor for the damaged condition limit state and for calibrating system factors for systems under lateral loads, as will be explained in Chapters 3, 4, and 5. Some of the equa- tions presented in earlier sections of this chapter are repeated for consolidating the derivation. The reliability index for a bridge member is calculated using the equation ln (2.19) 1 75 2 2 LF LL V V member LF LL β =     + where LF1 is the mean value of LF1, which is calculated from the linear structural analysis of the bridge up until the first member fails. LF1 gives the number of HS-20 trucks that the bridge member can carry in addition to the dead load. It can be expressed as (2.20)1 1 LF R D L = − where R is the bridge member capacity, D is the dead load effect and L1 is the load effect on that member due to the application of one set of HS-20 trucks on the bridge. If LF1 is found based on the nominal values of R and D, then the mean LF1 is related to the nominal value of LF1 through a bias bLF such that (2.21) 1 1 b LF LF LF = The results of the nonlinear analysis of the entire system will serve to find the load factor LFu, which also is used to find the reliability index for the ultimate limit state. ln (2.22) 75 2 2 LF LL V V ultimate u LF LL β =     + The reliability index margin is found from (2.23)u ultimate member∆β = β − β The calculated reliability index margin is compared to the target value and the deficit is found as (2.24)u deficit u target u u target ultimate member( )∆β = ∆β − ∆β = ∆β − β − β A negative Dbu deficit indicates that the redundancy level of the system is more than adequate, while a positive Dbu deficit indicates that the redundancy of the system is not sufficient. The system factor should serve to change the resistances of the bridge members so that a system that is adequately redundant could be allowed to have lower member resistances while the member resistances of a system that is not adequately redun- dant should be increased. The change in the member resis- tance should be sufficient to offset the deficit in the reliability index margin defined as Dbu deficit, so that the modified bridge will produce a modified system reliability index bNultimate higher than the current reliability index for non-redundant systems or lower for redundant systems such that (2.25) ultimate N ultimate u deficit ultimate u target ultimate member ultimate N u target member ( ) β = β + ∆β = β + ∆β − β − β β = ∆β + β Thus, the new ultimate system capacity is related to the reliability index by ln (2.26) 75 2 2 LF LL V V ultimate N u N LF LL β =     + where LFuN is the mean value that the new system ultimate capacity should reach. Also, ln (2.27) 75 2 2 LF LL V V u N LF LL u target member     + = ∆β + β and (2.28)75 2 2 LF LL euN u target member VLF VLL= ( )∆β +β + and the nominal system capacity is obtained from (2.29)LF LF b u N u N LF =

29 The required member capacity can be inferred from the relationship established between LFu and LF1 for the typi- cal bridge configurations analyzed in this study. For exam- ple, in NCHRP Report 406 it was observed that the ratio Ru = LFu/LF1 is approximately constant for I-girder bridges designed to exactly satisfy the AASHTO design specifications. Further analysis of the NCHRP Report 406 data and addi- tional analyses performed as part of this project show that a better approximation for the relationship between LFu and LF1 for deficient and overdesigned I-girder and box bridges is obtained from an equation of the form 1.16 0.75 (2.30)1LF LFu = × + Therefore, the required load factor for first member failure can be obtained from 0.75 1.16 (2.31)1LF LF N u N = − If the load factor for first member is related to the member resistance as shown in Equation 2.20 by ,1 1 LF R D L N N = − the required member resistance is (2.32)1 1R LF L DN N= × + and the system factor associated with this bridge system con- figuration is obtained as (2.33) R R s N φ = so that if the original design equation based on the member reliability is given as 1 (2.34)R D L Id n l N ( )φ = γ + γ + then 1 (2.35) R R D L I R s N d n l N N ( )φ = = γ + γ +φ leading to Equation 2.11 given as 1 (2.36)R D L Is N d n l N ( )φ φ = γ + γ + 2.6 Summary This chapter presents a method to consider redundancy during the design and load capacity evaluation of bridge systems. In keeping with the concept of NCHRP Report 406, which also is consistent with the AASHTO LRFD, AASHTO LRFR, and the Canadian Code, the method developed in this study consists of penalizing less redundant designs by requir- ing more conservative member capacities than required by current member-oriented specifications. However, bridges with redundant configurations can be rewarded by allow- ing a lower level of safety factors on their member strength capacities. This is achieved by applying a system factor in the design-check equation where a system factor fS > 1.0 applied in combination with the resistance factor indicate that the bridge is redundant while a system factor fS < 1.0 is used for less redundant configurations. The chapter describes the reliability-based calibration pro- cedure that should be followed to determine the appropri- ate system factors. The procedure is explained using bridges under vertical loads, but the same approach can be followed for bridges under lateral loads as will be explained in Chap- ters 3 and 4. The implementation of the procedure to calibrate systems factors for systems subjected to lateral or vertical loads is described in Chapters 3, 4, and 5. References AASHTO (2012) LRFD Bridge Design Specifications, 6th ed, Washing- ton, D.C. AASHTO MBE-2-M (2011) Manual for Bridge Evaluation, 2nd ed, Washington, D.C. AASHTO (2002) Standard Specifications for Highway Bridges, 17th ed, Washington, D.C. Aktas, E., Moses, F., and Ghosn, M. (2001) “Cost and Safety Optimi- zation of Structural Design Specifications,” Journal of Reliability Engineering and System Safety 73(3), Sept. 2001, 205–212. CAN/CSA-S6-06 (2006) Canadian Highway Bridge Design Code, Cana- dian Standards Association, Canada. Frangopol, D. M. and Curley, J. P. (1987) “Effects of Damage and Redundancy on Structural Reliability,” Journal of Structural Engi- neering 113 (7) 1533–1549. Frangopol, D. M. and Nakib, R. (1991) “Redundancy in Highway Bridges,” Engineering Journal, American Institute of Steel Construction, 28(1), 45–50. Ghosn, M. and Moses, F. (1986) “Reliability Calibration of a Bridge Design Code,” ASCE Journal of Structural Engineering, April. Ghosn, M., and Moses, F. (1998) NCHRP Report 406: Redundancy in Highway Bridge Superstructures, Transportation Research Board, National Research Council, Washington D.C. Ghosn, M., Sivakumar, B., and Miao, F. (2012) “Development of State- Specific Load and Resistance Factor Rating Method,” in press, ASCE Journal of Bridge Engineering, posted ahead of print Febru- ary 1, 2012. doi:10.1061/(ASCE)BE.1943-5592.0000382. Hendawi, S. and Frangopol, D. M. (1994) “System Reliability and Redundancy in Structural Design and Evaluation,” Structural Safety 16 (1+2), 47–71. Kulicki, J. M., Mertz, D. R., and Nowak, A. S. (2007) “Updating the Cali- bration Report for AASHTO LRFD Code,” Final Report, NCHRP Project 20-7/186, Transportation Research Board, National Research Council, Washington, D.C. Liu, D., et al. (2001) NCHRP Report 458: Redundancy in Highway Bridge Substructures, Transportation Research Board, National Research Council, Washington, D.C.

30 Melchers, R. E. (1999) Structural Reliability Analysis and Prediction, 2nd ed, New York: Wiley. Mertz, D. R. (2008) “Load Modifier for Ductility, Redundancy, and Operational Importance,” Aspire Magazine, Issue 08, fall. Moses, F. and Ghosn, M. (1985) “A Comprehensive Study of Bridge Loads and Reliability,” Final Report to Ohio DOT. Moses, F. (2001) NCHRP Report 454: Calibration of Load Factors for LRFR Bridge Evaluation, Transportation Research Board, National Research Council, Washington, D.C. Nowak, A. S. (1999) NCHRP Report 368: Calibration of LRFD Bridge Design Code, Transportation Research Board, National Research Council, Washington, D.C. Rackwitz, R. and Fiessler, B. (1978) “Structural Reliability under Com- bined Random Load Sequences,” Computers and Structures 9, 489–494. Saydam, D. and Frangopol, D. M. (2013) “Applicability of Simple Expressions for Bridge System Reliability Assessment,” Computers & Structures, Vol. 114–115, 59–71. Thoft-Christensen, P. and Baker, M. J. (1982) Structural Reliability and its Applications. New York: Springer-Verlag. Wang, N., Ellingwood, B. R., and Zureick, A. H. (2011) “Bridge Rating Using System Reliability Assessment II: Improvements to Bridge Rating Practices,” ASCE Journal of Bridge Engineering, 16(6), Nov/ Dec 863–871.