Bridge System Safety and Redundancy (2014)

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81 5.1 Measures of System Safety and Redundancy Figure 5.1 gives a conceptual representation of the perfor- mance of a structure under increasing loads and the different levels that should be considered when evaluating member safety, system safety, and system redundancy. For example, the green line labeled “Intact system” may represent the applied load ver- sus maximum vertical displacement of a ductile multi-girder bridge superstructure. In this case, the load is incremented to study the behavior of an intact system that was not previously subjected to any damaging load or event when the system is subjected to increasing live loads. The bilinear brown line rep- resents the behavior assumed using traditional linear-elastic analysis methods. The blue line labeled “Damaged bridge” rep- resents the response of a bridge system that has been previously damaged by deterioration, overloading, or an extreme event. To obtain the response of the originally intact system, it is assumed that the vertical live load applied on the structure has the configuration of the AASHTO HS-20 vehicle. The bridge is first loaded by the dead load and then the HS-20 load is applied. Usually, due to the presence of safety factors, no fail- ure occurs after the application of the dead load plus the HS-20 load. Using traditional safety evaluation procedures, the first structural member is assumed to fail when the HS-20 truck weight is multiplied by a factor LF1. LF1 would then be related to member safety. Note that if the bridge is under designed, or has major structural deficiencies, it is possible to have LF1 less than 1.0. Although the analysis can be performed using any basic truck model, the HS-20 truck configuration is used because it is the standard truck in the AASHTO specifications. Generally, the actual behavior follows the green curve and the ultimate capacity of the entire bridge system is not reached until the HS-20 truck weight is multiplied by a factor LFu. LFu would give an evaluation of system safety. Large vertical defor- mations rendering the bridge unfit for use are reached when the HS-20 truck weight is multiplied by a factor 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. If the bridge has sustained major damage due to the failure of one or more of its members, its behavior is represented by the curve labeled “Damaged bridge.” Examples of dam- aged bridges include structures that may have lost or suffered reduced member capacities in one or several members due to an extreme event such as collisions, fire, blast, fatigue fracture, or major degradation of member capacity caused by corro- sion or deterioration. In these cases, the ultimate capacity of the damaged bridge is reached when the weight of the HS-20 truck is multiplied by a factor LFd. LFd would give a measure of the remaining safety of a damaged system. The load multipliers, LF1, LFf, LFu, and LFd provide deter- ministic estimates of critical limit states that describe the safety of a structural system in its original intact state and its damaged state. These load multipliers are usually obtained by performing an incremental nonlinear finite element analysis of the structure. Because of the presence of large uncertain- ties in estimating the parameters that control member prop- erties, the bridge response, and the applied loads, the safety of the bridge members or system may be represented by the probability of failure, Pf, or the reliability index, b. Both Pf and b can be evaluated for each of the four critical limit states identified in Figure 5.1. Assuming that the load carrying capacity and the load follow lognormal distribu- tions, the relationship between the reliability index and the load multipliers, LF, for a bridge superstructure subjected to multiples of the HS-20 truck loading can be approximated as ln ln ln 20 20 ln (5.1) 2 2 2 2 2 2 2 2 R D P V V R P V V LF HS LL HS V V LF LL V V R D P R P LF LL LF LL β = −  + = ′  + = × ×     + =   + − ′ where R′ = R - D in this case represents the ability of the sys- tem to carry live load or the strength in the system beyond the dead load. R′ is related to the load multiplier obtained from C H A P T E R 5 Calibration of System Factors for Bridges under Vertical Load

82 the incremental analysis by R′ = LF × HS20. The applied live load P = LL × HS20 is the expected maximum live load that will be applied on the superstructure within the appropriate service period. HS20 is the load effect of the nominal HS-20 design truck. VLF is the COV of the bridge resistance defined as the standard deviation divided by the mean value. VLL is the COV of the applied live load. Equation 5.1 gives a good approximation to the actual reliability index b as long as the COV VLF and VLL remain below 20%, which is generally the case for bridge superstructures subjected to vertical live loads. Several sensitivity analyses have indicated that the lognormal model provides a reasonable model for system reliability index calculations in bridge engineering (Ghosn et al., 2010, 2012). In Equation 5.1, both the resistance and the applied live load are expressed as a function of the HS-20 truck load effect, which can then be factored out. The same formulation can be executed if the analysis is performed using a different nominal load such as the HL-93 design load or the AASHTO legal trucks. If redundancy is defined as the capability of a structure to continue to carry loads after the failure of the most critical member, then comparisons between the load multipliers LFu, LFf, LFd, and LF1 would provide non-subjective and quanti- fiable measures of system redundancy. Thus, the following three deterministic measures of system redundancy may be defined in terms of the ratio of the system’s capacity as com- pared to the most critical member’s capacity: R LF LF R LF LF (5.2) R LF LF u u 1 f f 1 d d 1 = = = where Ru = system reserve ratio for the ultimate limit state, Rf = system reserve ratio for the functionality limit state, Rd = system reserve ratio for the damage condition. Ru, Rf, and Rd can thus be used as measures of system redundancy as they represent the ability of a system to carry load beyond the failure of the most critical member. The system reserve ratios of Equation 5.2, as defined in NCHRP Report 406 and NCHRP Report 458, provide nominal deterministic measures of bridge redundancy. For example, when the ratio Ru is equal to 1.0 (LFu = LF1), the ultimate capacity of the system is equal to the capacity of the bridge to resist failure of its most critical member. Such a bridge is non-redundant. As Ru increases, the level of bridge redun- dancy increases. A redundant bridge also should be able to function with- out leading to high levels of deformations as its members undergo large nonlinear deformations. Thus, Rf provides another measure of redundancy. Similarly, a robust bridge structure should be able to carry some load after damage to one or more of its members, and Rd would provide another quantifiable non-subjective mea- sure of structural redundancy. During the course of this study and upon the review of NCHRP Report 406 results, it was established that a strong correlation exists between LFf and LFu obviating the need to use both of these measures. The strong correlation between LFf and LFu has been discussed in Chapter 2 and presented in Figures 2.3 and 2.4. This strong correlation led the calcula- tions in NCHRP Report 406 to produce similar system fac- tors for the ultimate capacity and functionality limit states. For this reason, the analyses performed in this chapter are based on the ultimate capacity LFu, and the analysis of system redundancy will be represented by Ru for the originally intact system and Rd for a damaged system. Load Factor Intact system Damaged bridge Assumed linear behavior Bridge ResponseFirst member failure LFd LF1 LFf LFu Ultimate capacity of damaged system Loss of functionality Ultimate capacity of intact system Figure 5.1. Representation of typical behavior of bridge systems.

83 The load multipliers, LFi, and the system reserve ratios in Equation 5.2 provide deterministic estimates of system safety and redundancy while Equation 5.1 can be used to determine the reliability index, b, for any member or system limit state. The reliability indices corresponding to the load multipliers LF1, LFu, or LFd of Figure 5.1 may be expressed respectively as bmember, bultimate, and bdamaged. The relationship between these three reliability indices can be investigated by studying the differences between them. This is achieved by defining Dbu and Dbd to be respectively the reliability index margins for the system’s ultimate and damaged limit states as ∆ = − ∆ = − β β β β β β u ultimate member d damaged member (5.3) The reliability index margins of Equation 5.3 give probabi- listic measures of redundancy as they represent the additional safety provided by the system as compared to the safety of the most critical bridge member. These reliability measures are directly related to the deterministic measures defined in Equation 5.2. As an example, using the simplified lognormal reliability model of Equation 5.1 for a bridge system 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 relation between the probabilistic and deterministic mea- sures of redundancy are obtained from ∆ = − =     −β β βu ultimate member uLF LL LF ln ln 75 1 LL V V LF LF V V LF LL u LF LL 75 2 2 1 2     + =     + ln 2 2 2 = ( ) + ∆ = − ln ( R V V u LF LL d damaged member 5.4) β β β =     −     + = ln ln l LF LL LF LL V V d LF LL 2 1 75 2 2 n ln LF LF LL LL V V R LL LL f LF LL d 1 75 2 2 2 75 2     + =     +V VLF LL2 2 In Equation 5.4, the live load for the ultimate and first mem- ber failure limit states is taken as the 75-year maximum live load to remain consistent with the AASHTO LRFD specifica- tions that assume that a bridge structure should have a design life equal to 75 years. However, for damaged bridges under the effect of live load, the calculation of the reliability index for the damaged system is executed using the 2-year maximum load represented by the load multiplier, LL2, rather than the maxi- mum load for the 75-year design life. The use of the 2-year load is based on the assumption that any major damage to a bridge should, in a worst case scenario, be detected during the mandatory biennial inspection cycle and thus no bridge is expected to remain damaged for more than 2 years. Because bridge engineers are not expected to perform incremental load analyses or reliability analyses and use the reliability index margins to ascertain the level of redundancy of typical bridge configurations, NCHRP Report 406 pro- posed to calibrate system factors that can be directly used in the design-check equation to account for the redundancy of typical bridge structural systems. The process followed to establish the system factors is described in Chapter 2 and is expanded in Section 5.3 for bridge systems under verti- cal loads. The calibration of the system factors requires the results of the analyses of typical bridge configurations that have been designed to meet current design standards, those that do not meet the standards, as well as those that may be overdesigned. The analysis compares the ultimate system capacity of originally intact systems as well as the capacity of damaged systems. Section 5.2 summarizes the results for the originally intact systems analyzed in this study and in NCHRP Report 406. Section 5.4 calibrates the system factors for the originally intact systems. Section 5.5 summarizes the results of the analysis for the damaged systems. Section 5.6 calibrates the system factors for the damaged bridges. 5.2 Summary of Bridge Analysis and Results for Originally Intact Systems The bridges analyzed in this study consist of continuous three-span composite steel I-girder bridges with two bents supported by three columns each, simple-span and continu- ous three-span composite steel tub girders, and simple-span and continuous span prestressed concrete spread box girders. The results of these analyses are supplemented by the results of simple-span and two-span continuous composite steel I-girder bridges and simple-span and two-span continuous prestressed concrete I-girder bridges performed in NCHRP Report 406. Validation of the results of the nonlinear analyses was made by comparing analytical results of representative bridges using the simplified space frame models adopted in this study and the results of more advanced finite element models and experimental test results. These comparisons have demonstrated that the 3-D space frame models were reason- ably accurate and can be used for the purposes of this study. Prestressed Concrete I-Girder Bridges Many simple-span and continuous-span prestressed I-girder bridges were analyzed in NCHRP Report 406. The results of these analyses were extracted for this project to study how the redun- dancy of these bridges varies with the number of beams, beam spacing, and span length. Specifically, over 100 simple-span

84 bridges varying in length between 45-ft and 150-ft with a com- posite concrete deck supported by 4, 6, 8, and 10 beams spaced at 4-ft, 6-ft, 8-ft, 10-ft, and 12-ft are analyzed. Also, over 50 pre- stressed concrete I-girder bridges with two continuous spans varying in length between 100-ft and 150-ft supported by 4, 6, 8 and 10 beams spaced at 4-ft, 6-ft, 8-ft, 10-ft, and 12-ft are inves- tigated. The bridges’ concrete slabs varied in depth between 7.5-in. and 8.5-in. depending on the beam spacing. The beams are assumed to have a compressive concrete strength f ′c = 5 ksi while the deck’s strength is equal to f ′c = 3.5 ksi. The prestress- ing tendons are assumed to be 270-ksi steel. The bridges were designed to exactly satisfy the strength requirements of the AASHTO LRFD design specifications. This was done even though normally it is the serviceability criteria that govern the design of prestressed members because bridge redundancy is related to member strength and ultimate system capacity. Sen- sitivity analyses also were performed to investigate the effect of changes in member strength, slab strength, and dead weight, as well as other parameters. The material data were used to obtain the moment-curvature relationships for the steel beams using analytical methods. The moment-curvature relationships were then used to perform the nonlinear analysis of the bridge. For the prestressed concrete bridges, the material data were used to obtain the moment-curvature relationships for the beams using analytical methods. The moment-curvature relationships were then used to perform the nonlinear analysis of the bridge. The results of the analysis of the prestressed concrete I-girder bridges are separated into two groups: narrow bridges are defined as those that have four beams at 4-ft spacing, four beams at 6-ft spacing, and six beams at 4-ft spacing; all the other bridges are defined as wide bridges. The results of the wide bridges are used to compare the ultimate capacity of the originally intact bridge represented by LFu to the capacity to resist first member failure represented by LF1. The compari- son showed that LFu is highly dependent on LF1 with a relation- ship that can be well represented by an equation of the form = + γ1.16 0.75 (5.5)1LF LFu where LFu is the ultimate capacity of the originally intact system expressed in terms of the number of side-by-side HS-20 trucks that the bridge can carry when the ultimate capacity is reached. LF1 is the load capacity at first member failure expressed in terms of the number of side-by-side HS-20 trucks that the sys- tem can carry before the first member reaches its load carry- ing capacity. g is a normalized stiffness ratio that represents the stiffness of the non-composite beam member compared to the stiffness of the deck. For I-girder bridges, g = 1. The validity of Equation 5.5 for the response of simple-span and continuous prestressed concrete I-girder bridges is verified in Figure 5.2, which plots the results obtained from Equation 5.5 versus the results obtained from the nonlinear analysis in blue for simple-span and in red for continuous span bridges. These results are for bridges loaded by two side-by-side trucks in the middle of one span. These data exclude the results for narrow bridges consisting of four beams at 4-ft and 6-ft spac- ing and for six beams at 4-ft spacing. The plots show how the results follow a consistent trend that, for the wide bridges, is largely independent of span length, number of beams, or beam spacing. Also, the trend is largely independent of the beam 0 2 4 6 8 10 12 0 2 4 6 8 10 12 LF u fr om Eq .5 .5 LFu from numerical analyses Simple span Continuous span COV=4.53% y = 0.97x R2=0.92 Figure 5.2. Verification of Equation 5.5 for prestressed concrete I-girder bridges.

85 strength and the dead load. The plot shows that the points lie close to a 45° line. The equation in the figure gives the equation of the trend line, which describes the relationship between the predicted LFu obtained from Equation 5.5 and the calculated LFu obtained from the nonlinear pushdown analysis of actual bridge systems. The trend line shows a slope equal to 0.97 and a coefficient of regression R2 = 0.92. A perfect match would lead to a trend line having an equation of the form y = 1.0 x with a coefficient of regression R2 = 1.0. The COV of the error between the predicted values and the analytical results is 4.53%. Composite Steel I-Girder Bridges Numerous simple-span and continuous-span composite- steel I-girder bridges were analyzed in NCHRP Report 406. The results of these analyses were extracted for this project to study how the redundancy of these bridges varies with the number of beams, beam spacing, and span length. Specifically, over 100 simple-span bridges varying in length between 45-ft and 150-ft with a composite concrete deck supported by 4, 6, 8, and 10 beams spaced at 4-ft, 6-ft, 8-ft, 10-ft, and 12-ft are analyzed. Also, over 30 composite steel I-girder bridges with two 120-ft continuous spans supported by 4, 6, 8, and 10 beams spaced at 4-ft, 6-ft, 8-ft, 10-ft, and 12-ft are inves- tigated. The bridges’ concrete slabs varied in depth between 7.5-in. and 8.5-in. depending on the beam spacing. The beams are assumed to be A-36 steel while the deck’s strength is equal to f ′c = 3.5 ksi. The bridges were designed to exactly satisfy the strength requirements of the AASHTO LRFD design speci- fications. Sensitivity analyses also were performed to investigate the effect of changes in member strength, slab strength, and dead weight, as well as other parameters. The moment-rotation relationships for the bridges analyzed in NCHRP Report 406 were obtained using existing empirical models based on test results as described in the appendices of NCHRP Report 406. The moment-rotation relationships were then used to per- form the nonlinear analysis of the bridge. The analyses were performed assuming that the sections in negative bending are compact and the results are compared to the cases where the sections in negative bending are noncompact. The results in NCHRP Report 406 were supplemented by the results of the analysis of three-span continuous bridges with span lengths 50-ft, 80-ft, and 50-ft. The bridges were assumed to have 4, 5, or 6 beams at 8-ft spacing. The bridges were analyzed for different strengths and beam stiffness by assuming that they have different values for ultimate moment capacities and moments of inertia. The results of the analysis of the wide composite steel I-girder bridges that compare the ultimate capacity of the originally intact bridge system represented by LFu to the capacity to resist first member failure represented by LF1 also were found to be well represented by Equation 5.5. The validity of Equation 5.5 with g = 1 for the response of simple-span and continuous composite steel I-girder bridges is verified in Figure 5.3, which plots the results obtained from Equation 5.5 versus the results obtained from the nonlinear analysis in blue for simple-span Figure 5.3. Verification of Equation 5.5 for composite steel I-girder bridges. 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 LF u fr om Eq .5 .5 LFu from numerical analyses Simple span Continuous compact Continuous non compact COV=7.70% y = 1.05x R2=0.92

86 and in red for continuous span bridges. These results are for bridges loaded by two side-by-side trucks in the middle of one span. These data exclude the results for narrow bridges consist- ing of four beams at 4-ft and 6-ft spacing and for six beams at 4-ft spacing. The plots show how the results follow a consistent trend that, for the wide bridges, is largely independent of span length, number of beams, or beam spacing. Also, the trend is largely independent of the beam strength and the dead load. The plot shows that the points lie along a 45° line. The equa- tion in the figure gives the equation of the trend line, which describes the relationship between the estimated LFu obtained from Equation 5.5 and the calculated LFu obtained from the nonlinear pushdown analysis of actual bridge systems. The trend line shows a slope equal to 1.05 and a coefficient of regres- sion R2 = 0.92. A perfect match would lead to a trend line having an equation of the form y = 1.0 x with a coefficient of regression R2 = 1.0. The plot shows that Equation 5.5 under predicts the ultimate capacity of continuous bridges with noncompact sec- tions by a very slight amount and those of continuous bridges with compact sections by a little more. However, the differences in the behavior of simple-span, continuous noncompact and continuous compact bridges are very small and can be ignored for the sake of keeping the prediction model as simple as pos- sible. The COV of the error between the predicted values and the analytical results is 7.70%, which is quite reasonable given the large variations in the bridge geometries. Prestressed Concrete Box-Girder Bridges A 120-ft simply supported bridge consisting of twin pre- stressed concrete box girders was analyzed during this course of study. The spacing between the two boxes from center to center is 14′9″. The bridge width is 37′ with 10″ depth concrete slab and the box’s depth is 6′10.5″. The boxes are assumed to have a compressive concrete strength f ′c = 7.35 ksi while the deck’s strength is equal to f ′c = 4.35 ksi. Mander’s model ( Mander, 1984) was adopted for the stress-strain curve for concrete. The prestressing tendons are assumed to be 270-ksi steel. Sensitivity analyses also were performed to investigate the effect of changes in member strength, dead load, and span continuity. The baseline continuous bridge has three spans at 80-ft, 120-ft, 80-ft. The analysis was performed for the narrow bridge configuration with one lane loaded. The results of the analysis of the prestressed concrete box-girder bridges that compare the ultimate capacity of the originally intact bridge represented by LFu to the capacity to resist first member failure represented by LF1 also were found to be well represented by Equation 5.5. However, for continuous box-girder bridges, the value of g was found to depend on the stiffness of the box near the support as described in Equation 5.6. The modification to g reflects the ability of continuous bridges with very high box stiffness to slab ratio to transfer the load longitudinally to the adjacent spans when the loaded span experiences non- linear deformations as opposed to simple-span bridges and bridges with relatively low beam stiffness which would tend to transfer the load laterally to the other beams within the loaded span. The modified g is expressed as 120 12 38 for continuous box-girder bridges 1 for other bridges (5.6) 3 EI E t box slab s( ) γ =     γ = where EIbox is the stiffness of the non-composite main lon- gitudinal girder equal to the modulus of elasticity Egirder in lb/in2 times the moment of inertia Ibox in in4, 120 12 3 E t slab s( ) is the stiffness of a 120-in. segment of the slab with a modulus of elasticity Eslab in lb/in2 and a slab thickness ts in inches. The value in the denominator is a typical stiffness ratio for I-girder bridges used as a baseline. The moment of inertia Ibox is for the cracked section that ignores the portion of the concrete in tension. The validity of Equation 5.5 with the modified g of Equa- tion 5.6 for the response of simple-span and continuous pre- stressed concrete box-girder bridges is verified in Figure 5.4, which plots the results obtained from Equation 5.5 versus the results obtained from the nonlinear analysis in blue for simple-span and in red for continuous span bridges. These results are for bridges loaded by one HS-20 truck in the middle of one span. The plots show how the results follow a consistent trend that is largely independent of the beam strength and the dead load. The plot shows that the points lie along a 45° line. The equation in the figure gives the equation of the trend line, which describes the relationship between the estimated LFu obtained from Equation 5.5 and the calculated LFu obtained from the nonlinear pushdown analysis of actual bridge sys- tems. The trend line shows a slope equal to 1.01 and a coef- ficient of regression R2 = 0.98. A perfect match would lead to a trend line having an equation of the form y = 1.0 x with a coef- ficient of regression R2 = 1.0. The COV of the error between the predicted values and the analytical results is 6.27%. In the final recommendation it is suggested that a g = 2 be used as a conservative value to keep the approach simple. Composite Steel Box-Girder Bridges A twin steel tub girder bridge with sections having a box with 9″ plate thickness supporting a three-span continuous bridge was also analyzed as a baseline for studying the behavior of steel box-girder bridges. The bridge span configuration is 100-ft, 120-ft, 100-ft. The analyses also were performed on variations of the bridge assuming different member strengths. The results were supplemented with those of simple-span steel box bridges with different member strengths, span lengths, dimension of

87 box sections, number of boxes, and box spacing. During the analyses of the bridges, wide bridges were loaded by two side- by-side trucks; narrow bridges were loaded by only one truck. The results of the analysis of the steel box-girder bridges, which compare the ultimate capacity of the originally intact bridge represented by LFu to the capacity to resist first mem- ber failure represented by LF1, also were found to be well rep- resented by Equation 5.5 with the modified g of Equation 5.6. The validity of Equation 5.5 for the response of simple-span and continuous steel box-girder bridges is verified in Fig- ure 5.5, which plots the results obtained from Equation 5.5 versus the results obtained from the nonlinear analysis in blue for simple-span and in red for continuous span bridges. These results are for wide bridges loaded by two side-by-side trucks and narrow bridges loaded by only one truck. The plot shows that the points lie along a 45° line. The equation in the figure gives the equation of the trend line that describes the relationship between the estimated LFu obtained from Equation 5.5 and the calculated LFu obtained from the nonlinear pushdown analysis of actual bridge systems. The trend line shows a slope equal to 1.02 and a coefficient of regression R2 = 0.99. A perfect match would lead to a trend 0 5 10 15 20 25 30 0 5 10 15 20 25 30 LF u fr om Eq . 5 .5 LFu from numerical analyses Simple span Continuous span COV=6.27% y = 1.01x R2=0.98 Figure 5.4. Verification of Equation 5.5 for P/S concrete box bridges. 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 LF u fr om Eq . 5 .5 LFu from numerical analyses Simple span Continuous span COV=4.11% y = 1.02x R2=0.99 Figure 5.5. Verification for steel box-girder bridges.

88 line having an equation of the form y = 1.0 x with a coef- ficient of regression R2 = 1.0. The COV of the error between the predicted values and the analytical results is 4.11%. Model Modification for Continuous Box-Girder Bridges Modified Model The sensitivity analyses described in this report demon- strated that continuous box-girder bridges provide signifi- cantly higher redundancy levels than other types of bridges. This was attributed to the high stiffness of the boxes near the interior supports, which improved their ability to transfer the load to the adjacent spans as represented by the term g in Equa- tion 5.5. Additional observations indicated that the moment capacity of the box girder in the negative bending region also plays an important role in allowing for the transfer of the load to the other spans. To account for this effect, a large number of three-span and two-span composite steel box-girder bridges, as well as three-span prestressed concrete box-girder bridges, were analyzed for different stiffness and strength values. In particular, the additional sensitivity analysis studied the effect of the stiffness of the boxes, the moment capacity of the boxes in the negative bending region, the moment capacity in the positive bending region, dead load magnitude, thickness of the slab, and the moment capacity of the slab. The results of the additional analyses performed indicated that the originally proposed Equations 5.5 and 5.6 can be fur- ther modified to account for the effect of the negative bending capacity and the stiffness of the box girders. An updated unified equation that would estimate the load factor LFu for the ulti- mate limit state of originally intact bridges in function of the load factor LF1 corresponding to the first member failure taking account the span continuity for box girders is presented as = + γ1.16 0.75 (5.7)1LF LFu where LF LF R D L LF LF LF LF 1 1 1 1 1 1 1 1 0= = − ≥ = = + + + + − + − when . R D L LF LF − − − − + − < 1 1 1 1 0when 5.8) . ( That is, LF1 in Equation 5.7 represents the ability of the weakest section of the beam, which can be either the positive bending section or the negative bending section depending on the moment capacity in each region (R), the dead load moment in each region (D), and the effect of the applied live load moment on the most critical beam (L1) where the live load represents two side-by-side HS-20 trucks applied at the middle of the span. The positive superscript in Equation 5.8 is for the positive bending region; the negative superscript is for the negative bending region. Furthermore, the value of g in Equation 5.7 should be modified as shown in Equation 5.9. ( ) ( )γ = ≥ −     + <           ≤ = − + − + − + 12 38 for continuous boxes with 1.75 1 1.75 12 38 1 1 for continuous boxes with 1.75 1.0 for other bridges. 8.0 120 (5.9) transverse slab 3 1 1 transverse slab 3 1 1 1 1 EI E b t LF LF EI E b t LF LF LF LF Use b in box s s box s s s The modification factor g is thus adjusted to take into account the stiffness of continuous box-girder bridges relative to slab stiffness as well as the negative bending strength capac- ity of the box. EIbox is the stiffness for the cracked section of the box girder in negative bending that ignores the portion of the concrete in tension. Etransverse slab is the modulus of elasticity for the slab between the boxes, bs = 120 in gives the width of the slab assuming the stiffness is calculated based on a 120-in. wide slab section having a depth ts, = − + + + +1 1 LF R D L is the load factor in the positive bending region due to two side-by-side HS-20 trucks applied in the middle of the span and R+, D+, and L1+ are the moment resistance, dead load, and maximum live load effect of the most critical beam in the positive bending region. LF1- is the load factor in the most critical member in negative bending where = − − − − − 1 1 LF R D L obtained for the two side-by- side HS-20 trucks applied at the middle of the span and R-, D-, and L1- are the moment capacity, dead load moment, and live load moment in the most critical negative bending section. The value of 38 is used to normalize the equation and is based on the stiffness of typical steel I-girder bridges designed to exactly satisfy the specifications’ strength criteria. Verification of Modified Model To verify the validity of the proposed modified model, a sen- sitivity analysis is performed by analyzing several three-span and two-span steel box-girder bridges, three-span prestressed concrete box-girder bridges, as well as three-span steel I-girder bridges. Figures 5.6, 5.7, 5.8, and 5.9 plot the predicted load fac- tor LFu obtained from Equations 5.7, 5.8, and 5.9 versus the

89 Figure 5.6. Verification of model for three-span continuous steel box-girder bridges. y = 1.04x R² = 0.98 0 10 20 30 40 50 0 10 20 30 40 50 LFu from SAP2000 COV=3.34% LF u fr om Eq . 5 .7 LFu value obtained from the nonlinear analysis performed using SAP2000. The plots are for three-span continuous steel box-girder bridges (Figure 5.6), two-span continuous steel box-girder bridges (Figure 5.7), three-span continuous pre- stressed concrete box-girder bridges (Figure 5.8) and three- span continuous I-girder bridges (Figure 5.9). All the trend lines in the figures have slopes close to 1.0 and coefficients of regression R2 also close to 1.0. This serves to confirm that Equation 5.7 provides a good model for estimating the ulti- mate capacity of bridge systems subjected to vertical live load. y = 0.99x R² = 0.95 0 10 20 30 40 50 0 10 20 30 40 50 LF u f ro m E q. 5 .7 LFu from SAP2000 COV=4.44% Figure 5.7. Verification of model for two-span continuous steel box-girder bridges. The maximum COV for the error between the predicted and the SAP2000 values is on the order of 5.5%. Narrow Simple-Span I-Girder Bridges The load factors for simple-span prestressed concrete and steel I-girder bridges having 4 beams at 4-ft spacing and four beams at 6-ft spacing or six beams at 4-ft spacing are plotted in Figures 5.10 and 5.11. These bridges are considered to be nar- row bridges. The figures show a different trend in the behavior

90 Figure 5.8. Verification of model for three-span continuous prestressed concrete box-girder bridges. y = 1.01x R² = 0.95 0 10 20 30 40 50 0 10 20 30 40 50 LF u f ro m E q. 5 .7 LFu from SAP2000 COV=5.44% of simple-span narrow bridges where the ultimate capacity of the originally intact systems is considerably lower than that of wide bridges. No discernible difference is observed between steel I-girder and prestressed concrete I-girder bridges. Simple-span bridges with four beams at 4-ft spacing are not redundant for the ultimate limit state with LFu = LF1. The actual slope of the trend line for these cases obtained in Fig- ure 5.10 is 1.01. That is, when one beam fails, the entire bridge will collapse. Bridges with four beams at 6-ft spacing are slightly more redundant showing that LFu is approximately equal to 1.11xLF1. The relationship between LFu and LF1 for bridges with six beams at 4-ft spacing is LFu = 1.18xLF1. Narrow Continuous Span I-Girder Bridges The load factors for narrow continuous span prestressed concrete bridges having 4 beams at 4-ft spacing and four beams at 6-ft spacing or six beams at 4-ft spacing are plotted in Fig- ure 5.11. These bridges are considered to be narrow bridges. The plot shows LFu versus LF1 obtained of the narrow continu- ous bridges and compares them to those of wide continuous span bridges. The figure demonstrates that bridge continuity helps improve bridge redundancy for the four beams at 6-ft and the six beams at 4-ft bridges placing them within the same range as the continuous wide bridges. However, the continuity Figure 5.9. Verification of model for three-span continuous I-girder bridges. y = 1.03x R² = 0.91 0 2 4 6 8 10 12 0 2 4 6 8 10 12 LF u f ro m E q. 5 .7 LFu from SAP2000 COV=5.09%

91 did not sufficiently improve the redundancy of four beams at 4-ft which remained at a lower level of redundancy. Only four narrow continuous steel I-girder bridges were analyzed having 4 beams at 60-ft and six beams at 4-ft spacing with compact and noncompact sections in negative bending. These data points were also plotted in Figure 5.11 but no discernible difference is observed between the narrow steel I-girder and prestressed concrete I-girder bridges. These observations are made when the bridge is loaded by two side-by-side trucks in the same span. The results of the analysis as plotted in Figure 5.11 dem- onstrate that for narrow continuous bridges, Equation 5.7 is applicable with g = 0 for 4 beams at 4-ft spacing and g = 1.0 for all other number of beams and beam spacing. Narrow Box-Girder Bridges The load factors for narrow continuous box-girder bridges having two boxes each 6-ft wide spaced at 12-ft center on cen- ter for a bridge width of 24-ft are plotted in Figure 5.12. These bridges are considered to be narrow bridges. Three plots are given. Two plots are for simple-span concrete and steel bridges loaded by a single lane of traffic. The third plot shows the results when the bridge is loaded by two side-by-side HS-20 trucks. The plots show LFu versus LF1 obtained of the narrow simple-span bridges loaded in two lanes and compare them to those loaded in a single lane. The figure demonstrates that narrow box-girder bridges loaded by a single lane behave like y = 1.01x R² = 1.00 y = 1.11x R² = 0.89 y = 1.18x R² = 0.97 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 LF u LF1 P/s_4 beams at 4 ft Steel_4 beams at 4 ft P/s_4 beams at 6 ft Steel_4 beams at 6 ft P/s_6 beams at 4 ft Steel_6 beams at 4 ft Figure 5.10. Plot of LFu vs. LF1 for narrow simple-span I-girder bridges. Figure 5.11. Plot of LFu vs. LF1 for narrow continuous span I-girder bridges. 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 LF u LF1 Wide P/s P/s_4 beams at 4 ft P/s_4 beams at 6 ft P/s_6 beams at 4 ft Steel_4 beams at 4 ft Steel_4 beams at 6ft Steel_6 beams at 4ft LFu=1.16LF1+0.75 LFu=1.16LF1

92 wide bridges while those loaded in two lanes show practically no redundancy with LFu = 1.03 LF1. Continuous Span I-Girder Bridges with Noncompact Sections When the section in negative bending is noncompact, the dominant failure mechanism may be due to trucks in differ- ent contiguous spans of the bridge rather than the two side- by-side trucks in the same span. The analysis of noncompact bridges loaded by a single HS-20 truck in each of two con- tiguous spans was performed in NCHRP Report 406 and aug- mented by additional analyses performed during the course of this study. These analyses are used to supplement the analysis of the bridges loaded by two side-by-side trucks in one span. The minimum value of LFu from both loading scenarios is then compared to the minimum of LF1+ and LF1- as defined in Equation 5.8. The results of the analysis show that, generally speaking, when first member failure takes place in negative bending with LF1- less than LF1+, the loading of the two spans governs the ultimate capacity of the system and the bridge reaches its ultimate capacity soon after the noncompact mem- ber in negative bending reaches its limiting capacity. These bridges show essentially no redundancy and LFu = LF1-. Gen- erally, when LF1+ is lower than LF1-, failure is controlled by the system loaded by two side-by-side trucks in a single span and Equation 5.7 is valid. Figure 5.13 shows the results obtained from noncompact bridges for the two failure modes governed by Equation 5.7 and LFu = LF1-. Figure 5.14 serves to verify that the proposed approach for predicting the ultimate load capacity of continuous I-girder bridges with noncompact sec- tions in the negative bending region is reasonably accurate. Figure 5.12. Plot of LFu vs. LF1 for narrow box-girder bridges. y = 1.20x R² = 1.00 y = 1.22x R² = 0.98 y = 1.03x R² = 1.00 0 5 10 15 20 25 30 0 5 10 15 20 25 LF U LF1 P/s concrete_one lane loaded Steel_one lane loaded Steel_two lanes loaded Figure 5.13. Plot of LFu vs. LF1 for continuous span I-girder bridges with compact and noncompact sections in negative bending. 0 2 4 6 8 10 12 0 2 4 6 8 10 M in LF u Min LF1 Compact_NCHRP406 Compact_NCHRP12 86 Noncompact_NCHRP406 Noncompact_NCHRP12 86 LFu=1.16LF1+0.75 LFu=1.00LF1

93 Continuous-Span Steel Box-Girder Bridges with Noncompact Negative Sections The AASHTO LRFD Bridge Design Specifications indicate that steel box-girder sections in negative bending should be considered to be noncompact. For this reason, several analy- ses were performed assuming that the boxes in the negative bending regions have no ductility and that they will fail as soon as their moment capacity is first reached. The bridges are analyzed under the effect of two trucks placed in different contiguous spans of the bridge, and the results are compared to those obtained when two side-by-side trucks are placed in the same span. The minimum value of LFu from both load- ing scenarios is then compared to the minimum of LF1+ and LF1- as defined in Equation 5.8. The results of the analysis show that, generally speaking, first member failure takes place in negative bending when LF1- is considerably lower than LF1+ and the loading of the two spans governs the ultimate capacity of the system. For these cases, the bridge reaches its ultimate capacity soon after the noncompact member in negative bending reaches its limiting capacity. These bridges show some level of redundancy and LFu = 1.16LF1-. Generally, when LF1+ is lower than LF1-, failure is controlled by the system loaded by two side-by-side trucks in a single span and Equa- tion 5.7 is valid. Figure 5.15 shows the results obtained from noncompact bridges for the two failure modes governed by Equation 5.7 and LFu = 1.16LF1-. A sensitivity analysis shows that the transition between the two curves takes places when Figure 5.14. Verification of model for continuous I-girder bridges with noncompact sections in negative bending. y = 0.95x R² = 0.96 0 2 4 6 8 10 0 2 4 6 8 10 12 LF u pr ed ic te d LFu from analysis Figure 5.15. LFu versus LF1 for continuous box-girder bridges with noncompact sections in negative bending.

94 LF1-<1.75LF1+. The transition between the two curves describ- ing the two failure modes is not sudden and can be expressed using the parameter g defined in Equation 5.9. Continuous-Span Narrow Box-Girder Bridges To investigate the behavior of originally intact narrow con- tinuous bridges under overload, several analyses were per- formed for prestressed concrete and steel bridges covering bridges with compact and noncompact sections in negative bending. Narrow box-girder bridges are defined as those hav- ing two boxes with a total travel width equal to 24-ft. The analyses are performed for bridges composed of two boxes each 6-ft wide, spaced at 12-ft center to center. The bridges are analyzed under the effect of two trucks placed in different contiguous spans of the bridge and the results are compared to those obtained when two side-by-side trucks are placed in the same span. The minimum value of LFu from both load- ing scenarios is then compared to the minimum of LF1+ and LF1- as defined in Equation 5.8. The analysis was performed for bridges with compact sections in negative bending and also sections that are noncompact in negative bending. It is observed that the results closely follow the model established in Equation 5.7 as demonstrated in Figures 5.16 and 5.17. Summary The analyses performed during the course of this project and NCHRP Report 406 have produced results on the ulti- 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 LF u LF1 Steel box_two lanes loaded P/s box_two lanes loaded LFu=1.16LF1+0.75 Figure 5.16. LFu vs. LF1 for narrow continuous box-girder bridges. Figure 5.17. Verification of model for narrow continuous box-girder bridges. y = 0.98x R² = 0.99 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 LF u pr ed ic te d LFu from analysis

95 mate load carrying capacities of originally intact typical bridge configurations and compared these to the load capac- ity of critical members. Consistent patterns for the relation- ship between these parameters describing the performance of bridges are possible for evaluating the redundancy of bridge superstructures using a few characteristic parameters that, as described in Equations 5.7, 5.8, and 5.9, consist of the member moment resistance in positive and negative bending regions, the applied dead load moment, the moment from live load applied on the most critical member and, for con- tinuous bridges, the parameter g which gives the relative stiff- ness of the beam near the support to the stiffness of the slab. The parameter g reflects the ability of continuous bridges with stiff sections near the support to spread the load to the adjacent spans as the loaded span undergoes nonlinear defor- mations. The results of the relationship between the ultimate capacity of originally intact systems under vertical load and the load carrying capacity of the most critical member are summarized in Table 5.1. The analyses performed in this section addressed the ultimate capacities of originally intact bridges. Section 5.5 addresses damaged bridge systems that may have lost the load carrying capacity of a critical member due to various possible deterioration or extreme events. That is, LF1 in Equation 5.7 represents the load carrying capacity of the weakest section of the beam, which can be either the positive bending section or the negative bending section depending on the moment capacity in each region (R), the dead load moment in each region (D), and the effect of the applied live load moment on the most critical beam (L1) where the live load represents two side-by-side HS-20 trucks applied at the middle of the span or two trucks in one lane applied in each of two contiguous spans. The positive superscript in R, D, and L1 is for the positive bending region, the negative superscript is for the negative bending region. Furthermore, the value of g for the box-girder bridges in Table 5.1 is obtained from 12 38 for continuous boxes with 1.75 1 1.75 12 38 1 1 for continuous boxes with 1.75 1.0 for other bridges. 8.0 120 transverse slab 3 1 1 transverse slab 3 1 1 1 1 EI E b t LF LF EI E b t LF LF LF LF Use b in box s s box s s s ( ) ( )γ = ≥ −     + <           ≤ = − + − + − + Table 5.1. Summary of LFu vs. LF1 for originally intact systems under vertical loads. where + + 1 1 1 + 11 when 1.0LFR DLF LF L LF − + + − = = ≥ 1 1 1 11 = when 1.0LFR DLF LF L LF −− − − +− − = < Bridge Cross-Section Type Simple-span 4 I-beams at 4-ft LFu=1.01 LF1 Simple-span 4 I-beams at 6-ft LFu=1.11 LF1 Simple-span 6 I-beams at 4-ft LFu=1.18 LF1 Continuous span 4 I-beams at 4-ft with compact members LFu=1.16 LF1 Continuous steel I-girder bridges with noncompact negative bending sections and 1 11.16 0.75LF LF 1 1.00uLF LF All other simple-span and continuous I-beam bridges LFu=1.16 LF1+0.75 Narrow simple-span box-girder bridges less than 24-ft wide LFu=1.03 LF1 All other simple-span box-girder bridges LFu=1.16 LF1+0.75 Narrow continuous box-girder bridges less than 24-ft wide LFu=1.16 LF1+0.75 Continuous steel box-girder bridges with noncompact negative bending sections and 1 1 1.75LF LF LFu=1.16 LF1+0.75 All other continuous box-girder bridges LFu=1.16 LF1+0.75

96 The modification factor g takes into account the stiffness of continuous box-girder bridges relative to slab stiffness as well as the negative bending strength capacity of the box. EIbox is the stiffness for the cracked section of the box girder in negative bending, which ignores the portion of the concrete in tension. Etransverse slab is the modulus of elasticity for the slab between the boxes, bs = 120 in. gives the width of the slab assuming the stiffness is calculated based on a 120-in. wide slab sec- tion having a depth ts, = − + + + +1 1 LF R D L is the load factor in the positive bending region due to two side-by-side HS-20 trucks applied in the middle of the span or two trucks in one lane applied in each of two contiguous spans; and R+, D+, and L1+ are the moment resistance, dead load, and maximum live load effect of the most critical beam in the positive bending region. LF1- is the load factor in the most critical member in negative bending where = − − − − − 1 1 LF R D L obtained for the two side-by- side HS-20 trucks applied at the middle of the span or two trucks in one lane applied in each of two contiguous spans and R-, D-, and L1- are the moment capacity, dead load moment, and live load moment in the most critical nega- tive bending section. The value of 38 is used to normalize the equation and is based on the stiffness of typical steel I-girder bridges designed to exactly satisfy the specifications’ strength criteria. 5.3 Calibration of System Factors for Bridges under Vertical Loads Concept of System Factors Following the procedures proposed in the AASHTO LRFD Bridge Design Specifications and the Canadian bridge code, it is recommended that different member design criteria be established for bridges based on their levels of redundancy. This can be achieved by applying a system factor in the safety- check equation such that bridges with low levels of redun- dancy be required to have higher member resistances than those of bridges with high levels of redundancy. The system factor can be implemented in the member safety-check equa- tion that takes the form 1 (5.10)R D L Is nN d n l n ( )φ φ = γ + γ + where fs is the system factor that is defined as a statistically based multiplier relating to the safety and redundancy of the complete system. The system factor is applied to the fac- tored nominal member resistance RNn that would be needed to meet the required factored loads accounting for the system’s redundancy. 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 to the capacity of the sys- tem and should be used as a resistance multiplier as is the norm in reliability-based LRFD codes. f is the member resis- tance factor, 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 fs is equal to 1.0, Equation 5.10 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 thus the members of the bridge can be designed to have lower strengths than those of bridges with low levels of redundancy. When it is less than 1.0, then the level of redundancy is not sufficient and the bridge mem- bers must be designed to have higher strengths than members of bridges with high levels of redundancy. The system factors should be calibrated using a reliability model such that a system factor equal to 1.0 indicates that the reliability index of the system is higher than that of the mem- ber by an amount equal to a target value. Specifically, the reli- ability index for the ultimate limit state of the originally intact system, bultimate, is higher than the reliability index of the mem- ber, bmember by a margin reliability index Dbu equal to the target reliability margin Dbu target. For bridges susceptible to local dam- age, the reliability index for the ultimate limit state of the dam- aged system, bdamaged, is higher than the reliability index of the member of the originally intact bridge, bmember by a margin reli- ability index Dbd equal to the target reliability margin Dbd target. Based on the analysis of typical four-girder steel and pre- stressed concrete bridges, NCHRP Report 406 recommended that the target reliability index margins be set at Dbu equal to 0.85 and Dbd equal to -2.70. These targets were selected to match the average reliability index margins of typical four- girder bridges because these bridges have been traditionally accepted as providing sufficient levels of redundancy. In this chapter, system factors are calibrated so that bridge configu- rations that produce reliability index margins equal to the target values are assigned a system factor fs = 1.0. If the mar- gin is less than the target value, then the system factor will serve to increase the reliability of the system by an amount equal to the difference. Thus, a system factor less than 1.0 is assigned. If the reliability index margin is higher than the tar- get, then a system factor greater than 1.0 may be used to lower the reliability index of the system by an amount equal to the difference between the available margin and the target value. Calibration Approach The calibration of the system factor fs can be executed using Equations 5.1 through 5.4 so that the reliability index for the intact system bultimate and that of the damaged system bdamage are increased when the available Dbu and Dbd are lower than the target values set at Dbu target = 0.85 and Dbd target = -2.70. On the

97 other hand, fs should serve to lower the reliability index for the system when the available Dbu and Dbd are higher than the tar- get values. The amount by which bultimate and bdamage should be increased should be equal to the deficit in the available Dbu and Dbd when compared to the target values while the amount by which bultimate and bdamage should be decreased should be equal to the surplus in the available Dbu and Dbd when compared to the target values. The formulation can be summarized as described next for the ultimate limit state of originally intact systems. The same exact procedure also is valid for finding the system factor for the damaged condition limit state. The reliability index for a bridge member is calculated using the lognormal model as β =   + ln (5.11) 1 75 2 2 LF LL V V member LF LL where 1LF 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 (5.12)1 1 LF R D L = − where R is the bridge member capacity, D is the dead load effect, and L1 is the effect on that member due to the applica- tion 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 1LF is related to the nominal value of LF1 through a bias bLF such that = (5.13) 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 β =   + ln (5.14)75 2 2 LF LL V V ultimate u LF LL The reliability index margin is found from (5.15)u ultimate member∆β = β − β The calculated reliability index margin is compared to the target value and the deficit is found as ( )∆β = ∆β − ∆β = ∆β − β − β (5.16)u deficit u target u u target ultimate member 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 redun- dant could be allowed to have lower member resistances while the member resistances of a system that is not adequately redundant should be increased. The change in the member resistance should be sufficient to offset the deficit in the reli- ability index margin defined as Dbu deficit, so that the modified bridge will produce a modified system reliability index bNultimate. A bridge that is non-redundant should have a higher system reliability bNultimate value than a system designed using current methods. The higher system reliability index should serve to compensate for the deficit in the reliability margin so that ( ) β = β + ∆β = β + ∆β − β − β = ∆β +β (5.17) ultimate N ultimate u deficit ultimate u target ultimate member u target member The new ultimate system capacity is related to the higher reliability index by β =   + ln (5.18)75 2 2 LF LL V V ultimate N u N LF LL where LFuN is the mean value that the new system ultimate capacity should reach. Substituting Equation 5.18 into Equa- tion 5.17 gives   + = ∆β + β ln (5.19)75 2 2 LF LL V V u N LF LL u target member Equation 5.19 can be used to solve for the mean value of the required new system capacity using = ( )∆β +β + (5.20)75 2 2LF LL euN V Vu target member LF LL Given the mean value LFuN , the nominal required system capacity is obtained from = (5.21)LF LF b u N u N LF The required member capacity associated with a system having an ultimate capacity LFuN can be inferred from the rela- tionship established between LFu and LF1 for typical bridge configurations. For example, in NCHRP Report 406 it was observed that the ratio Ru = LFu/LF1 is approximately con- stant. This assumption was found valid when LF1 remained

98 within the range of typical new designs. Further review of the NCHRP Report 406 data augmented by the results of the analyses performed as part of this project show that a better approximation for the relationship between LFu and LF1 for all simple-span and continuous I-girder bridges is obtained from an equation of the form = × +1.16 0.75 (5.22)1LF LFu Therefore, the required load factor for first member failure can be obtained from = − 0.75 1.16 (5.23)1LF LF N u N Using Equation 5.8, the load factor for first member failure is related to the nominal member capacity by = − .1 1 LF R D L N N Thus, the required member resistance is = × + (5.24)1 1R LF L DN N The system factor associated with this bridge system con- figuration should serve to increase R to a new value RN. Thus, the system factor, fs can be obtained from (5.25) R R s N φ = If the traditional design gives a value for the resistance ( )[ ] = γ + γ + φ 1 ,R D L I n d n l n then the modified member design equation becomes the same as that in Equation 5.10, repeated below. 1R D L Is nN d n l n ( )φ φ = γ + γ + The process described above also can be used to derive an algebraic expression that gives the system factor directly as a function of the coefficient in Equation 5.7 and the dead load to member resistance ratio. The closed-form expression is given as [ ]η = −   + − − φ = η ξ∆β 1 1 1 (5.26) 1 2 1 1 e D R C D R C C LF D R red red red s T where Cred1 and Cred2 are respectively the slopes and intercepts of the redundancy equations in Table 5.1. As an example, in Equa- tion 5.7 Cred1 = 1.16 and Cred2 = 0.75g. D/R in Equation 5.26 is the dead load to resistance ratio, LF1 is the live load capacity of the most critical member as defined in Equation 5.8, DbT is the target reliability index margin set at DbT = 0.85, and 2 2V VLF LLξ = + is the dispersion coefficient defined as the square root of the sum of the square of the COV for the live load carrying capacity LF and the maximum applied live load LL. Calibration Example A numerical example is presented to illustrate the calibra- tion of system factors. The example uses a three-span con- tinuous composite steel I-girder bridge having span lengths of 50-ft, 80-ft, 50-ft. The bridge is composed of six parallel members at 8-ft spacing. The ultimate moment capacity in the positive bending region was found to be R = 49,730 kip-in. The dead load effect at the midpoint of the center span is found to be D = 4,860 kip-in. The moment at the midpoint of the external girder of the center span due to the two side- by-side HS-20 AASHTO trucks is L1 = 6,450 kip-in. The nonlinear push down analysis showed that the ulti- mate capacity is reached when the weights of the HS-20 trucks are multiplied by a factor LFu = 8.70. A traditional check of member safety assuming no resis- tance or load factors gives the load factor for first member failure as shown in Equation 5.8 by = − = − = 49,730 4,860 6,450 6.961 1 LF R D L Using the data provided by Nowak (1999), the bias in the estimated value of LF1 is obtained as bLF = 1.13. The COV for the live load and the load factors are estimated as VLL = 19%, VLF = 13.5%. It is assumed that an adequately redundant sys- tem should have a reliability index margin Dbtarget = 0.85. a. Calculation of bmember Table 5.2 gives the mean value for the applied load as extracted from Nowak (1999) for simple-span bridges; because no such data is available for continuous bridges in positive bending the same values are used in this analy- sis. Accordingly mean maximum applied live load for the Two Lane Loading One Lane Loading Span LL75 LL2 LL75 LL2 45  1.67 1.53 1.97 1.81 60  1.72 1.60 2.02 1.86 80  1.81 1.67 2.14 1.98 100  1.89 1.75 2.26 2.08 120  1.98 1.84 2.35 2.17 150  2.01 1.87 2.37 2.19 Table 5.2. Mean of applied loads as function of the effect of AASHTO HS-20 trucks.

99 design life of the structure LL75 for the 80-ft span loaded in two lanes is 1.81. Assuming a lognormal model, the reliability index bmember for the failure of the first member can be expressed as ln LF LL ln 1.13 6.96 1.81 0.19 0.135 6.31 1 75 2 2 2 2V V member LL LF β = + = × + = b. Calculation of bultimate Assuming that the load factor LFu and the live load factor LL75 follow lognormal distributions, the reliability index of the system for the ultimate limit state can be found as ln LF LL ln 1.13 8.70 1.81 0.19 0.135 7.2675 2 2 2 2V V ultimate u LL LF β = + = × + = Thus, the reliability index margin for this bridge con- figuration is found to be larger than the target value. 7.26 6.31 0.95 0.85u ultimate member u target∆β = β − β = − = > ∆β = c. Calculating LFN1 Because the reliability index margin is greater than the target value, it would be allowed to lower the required member capacities of the main bridge members, so that the system reliability is reduced. The target system reliabil- ity index for the ultimate limit state can then be reduced by the difference between the existing ( ) ( ) β =   + = β − ∆β − ∆β = − − = ln 7.26 0.95 0.85 7.16 75 2 2 LF LL V V ultimate N u N LF LL ultimate u u target = × = × =× + × +Thus, 1.81 9.607.16 75 7.16 0.19 0.135 2 2 2 2LF e LL euN V VLL LF = = = 9.60 1.13 8.50LF LF b u N u N LF The analysis of a large number of I-girder bridges has established an empirical relationship between LFu and LF1 given as 1.16 0.751LF LFu = × + γ Thus, for g = 1, LF1N = (LFNu - 0.75)/1.16 = (8.50 - 0.75)/1.16 = 6.68 which would provide the required nom- inal member capacity from 1 1 LF R D L N N = − or RN = LF1N × L1 + D = 6.68 × 6,450 + 4,860 = 47,946 kip–in d. Calculating System Factor fsystem The system factor for this bridge configuration is obtained as 49,730 47,946 1.04 R R system N φ = = = e. Alternate Calculation of System Factor fsystem from Equation 5.26 Given the redundancy coefficients Cred1 = 1.16 and Cred2 = 0.75, a dead load to resistance ratio D/R = 4860/ 49730 = 0.098, LF1 = 6.96, DbT = 0.85 and ξ = + =2 2V VLF LL 0.19 0.135 0.233,2 2+ = the load modifier is obtained as [ ] η = −   + − −  = −    + − × − = ξ∆β × 1 1 1 0.098 1.16 0.098 0.75 1.16 6.96 1 0.098 0.962 1 2 1 1 0.85 0.233 e D R C D R C C LF D R e red red red T The system factor for this bridge configuration is obtained as 1 1 0.962 1.04systemφ = η = = which is the same value obtained going through the pro- cess (a) through (d). This result indicates that this bridge configuration is redun- dant providing a sufficient margin of safety against collapse should the most critical girder of the bridge reach its ultimate capacity. Therefore, it would be possible to reduce the member capacity by a factor of 1.04 and have a bridge system capacity sufficiently high to safely carry the maximum expected live load during the service life of the bridge. The results of the implementation of the calibration process for the ultimate limit state of originally intact typi- cal bridge configurations analyzed in NCHRP Report 406 and during the course of this project are summarized in Section 5.4. 5.4 System Factors for Ultimate Limit State of Originally Intact Bridges Implementation of System Factors in Bridge Specifications Table 5.1 presented a set of equations that were calibrated to describe the relationship between member load carrying capacity and the ultimate load carrying capacity of bridge

100 systems under vertical load. The relationships in Table 5.1 can be presented by equations of the form = + (5.27)1 1 2LF C LF Cu red red The relationships in Table 5.1 can be used to calibrate the system factors to be incorporated into the member design equation to account for bridge system redundancy as pre- sented in Equation 5.10 and repeated below. ( )φ φ = γ + γ +1R D L Is nN d n l n As derived in Section 5.3, the calibration of the system fac- tor would lead to an expression that gives the system factor directly as a function of the coefficients of Equation 5.27 as listed in Table 5.1, and the dead load to member resistance ratio. The closed-form expression for the system factors, fs, is given in terms of a load modifier, h, as was shown [ ]η = −   + − − φ = η ξ∆β 1 1 1 (5.26) 1 2 1 1 e D R C D R C C LF D R red red red s T where Cred1 and Cred2 are respectively the slopes and intercepts of the redundancy equations in Table 5.1. As an example, Equa- tion 5.7 would give Cred1 = 1.16 and Cred2 = 0.75g. D/R in Equa- tion 5.26 is the dead load to resistance ratio, LF1 is the live load capacity of the most critical member as defined in Equation 5.8, DbT is the target reliability index margin set at DbT = 0.85 and 2 2V VLF LLξ = + is the dispersion coefficient defined as the square root of the sum of the square of the COV for the live load carrying capacity LF and the maximum applied live load LL. An investigation of the COV for LF shows that VLF is about 15% and VLL = 20% so that 15% 20% 25%.2 2ξ = + = The implementation of Equation 5.26 for different val- ues of LF1 and D/R for bridges that satisfy the relationship LFu = 1.16LF1+0.75g where Cred1 = 1.16 and Cred2 = 0.75g leads to plots similar to those presented in Figures 5.18 and 5.19. Although Equation 5.26 should be relatively easy to apply, the data points were found to be well represented by the red curves, which follow an equation of the form ( )φ = + − + γ1 1 1.5 1 (5.28) 2 1 2 2 D R LF s The range of error in fs from Equations 5.26 and 5.28 was found to be -0.03 and 0.01 when g = 1.0 and from -0.03 to 0.05 when g = 4. This range of error is deemed acceptable. When Cred2 in Equation 5.26 is equal to zero, the system factor equation reduces to ( ) ( ) φ = − −     × = + −   ×     − −     × = + −   × − −     × ≈ + −   × ∆β×ξ ∆β×ξ ∆β×ξ ∆β×ξ ∆β×ξ ∆β×ξ ∆β×ξ ∆β×ξ ∆β×ξ ∆β×ξ ∆β×ξ ∆β×ξ ∆β×ξ 1 e e 1 e e 1 e e 1 e e 1 e 1 1 e e e 1 e (5.29) 1 1 1 1 1 2 1 2 2 1 1 1 1 2 2 1 1 1 C C D R C C D R C C D R C C C D R C D R C C C D R s red red red red red red red red red red red red red which can be implemented to find the system factors for the cases where LFu = Cred1xLF1. The implementation of Equations 5.28 and 5.29 into the bridge categories of Table 5.1 leads to the set of system factors listed in Table 5.3. That is, LF1 in Table 5.3 represents the load carrying capac- ity of the weakest section of the beam, which either can be the positive bending section or the negative bending section Figure 5.18. Plot of system factor vs. LF1 for Equations 5.26 and 5.28 with g  1. 0.95 1 1.05 1.1 1.15 1.2 1.25 0 2 4 6 8 10 12 sy st em fa ct or s LF1 Eq. 5.26 D/R=0.20 Eq. 5.26 D/R=0.50 Eq. 5.26 D/R=0.70 Fit

101 Figure 5.19. Plot of system factor vs. LF1 for Equations 5.26 and 5.28 with g  4. 0.95 1 1.05 1.1 1.15 1.2 1.25 0 2 4 6 8 10 12 sy st em fa ct or s LF1 Eq. 5.26 D/R=0.20 Eq. 5.26 D/R=0.50 Eq. 5.26 D/R=0.70 Fit where + + 1 1 1 + 11 when 1.0LFR DLF LF L LF − + + − = = ≥ 1 1 1 11 = when 1.0LFR DLF LF L LF −− − − +− − = < Bridge Cross-Section Type System Factor Simple-span 4 I-beams at 4-ft 0.80 0.16s D R Simple-span 4 I-beams at 6-ft 0.90 0.09s D R Simple-span 6 I-beams at 4-ft 0.95 0.05s D R Continuous span 4 I-beams at 4-ft with compact members 0.93 0.07s D R Continuous steel I-girder bridges with noncompact negative bending sections and 1 11.16 0.75LF LF 0.80 0.16s D R All other simple-span and continuous I-beam bridges 2 2 1 1 1.5 / 1 1s D R LF Narrow simple-span box-girder bridges less than 24-ft wide 0.83 0.14s D R All other simple-span box-girder bridges 2 2 1 1 1.5 / 1 1s D R LF Narrow continuous box-girder bridges less than 24-ft wide 2 2 1 1 1.5 / 1 1s D R LF Continuous steel box-girder bridges with noncompact negative bending sections and 1 11.75LF LF 2 2 1 1 1.5 / 1 1s D R LF All other continuous box-girder bridges 2 2 2 1 1 1.5 / 1 1s D R LF Table 5.3. System factors for originally intact systems under vertical loads.

102 depending on the moment capacity in each region (R), the dead load moment in each region (D), and the effect of the applied live load moment on the most critical beam (L1) where the live load represents two side-by-side HS-20 trucks applied at the middle of the span or two trucks in one lane applied in each of two contiguous spans. The positive superscript in R, D, and L1 is for the positive bending region, the negative superscript is for the negative bending region. The parameter L1 gives the live load applied on the most critical member, which is defined as the member that fails first. It can be calculated as . . . . 1.10 1L D F LL D F LLactual AASHTO LRFD Table = × = × where D.F. is the distribution factor and LL is the effect of the HL-93 truck load with no impact factor and no lane load. Because the distribution factors in the AASHTO LRFD speci- fications are conservative, a correction factor = 1.10 is applied when D.F. is taken from the AASHTO table. The bias = 1.10 was applied during the development of the AASHTO LRFD load distribution tables for conservative designs. In the case of estimating the redundancy of a bridge system, keeping the bias would lead to unconservative estimates of the system fac- tor. For the analysis of a single lane of traffic load, the mul- tiple presence factor M.P. = 1.2 should also be removed from the tabulated D.F. values. Furthermore, it is proposed to use a factor equal to g = 2 as a conservative value. In actuality, the value of g for the con- tinuous box-girder bridges in Table 5.2 is obtained as ( ) ( )γ = ≥ −     + <           ≤ = − + − + − + 12 38 for continuous boxes with 1.75 1 1.75 12 38 1 1 for continuous boxes with 1.75 1.0 for other bridges. 8.0 120 transverse slab 3 1 1 transverse slab 3 1 1 1 1 EI E b t LF LF EI E b t LF LF LF LF Use b in box s s box s s s The modification factor g takes into account the stiffness of continuous box-girder bridges relative to slab stiffness as well as the negative bending strength capacity of the box. EIbox is the stiffness for the cracked section of the box girder in nega- tive bending, which ignores the portion of the concrete in ten- sion. Etransverse slab is the modulus of elasticity for the slab between the boxes, bs = 120 in. gives the width of the slab assuming the stiffness is calculated based on a 120-in.-wide slab section hav- ing a depth ts, = − + + + +1 1 LF R D L is the load factor in the positive bending region due to two side-by-side HS-20 trucks applied in the middle of the span or two trucks in one lane applied in each of two contiguous spans, and R+, D+, and L1+ are the moment resistance, dead load, and maximum live load effect of the most critical beam in the positive bending region. LF1- is the load factor in the most critical member in negative bend- ing where = − − − − − 1 1 LF R D L obtained for the two side-by-side HS-20 trucks applied in the middle of the span or due to two trucks in one lane applied in each of two contiguous spans, and R-, D-, and L1- are the moment capacity, dead load moment, and live load moment in the most critical negative bending section. The value of 38 is used to normalize the equation and is based on the stiffness of typical steel I-girder bridges designed to exactly satisfy the specifications’ strength criteria. The implementation of the system factors for I-girder bridges presented in this chapter is straightforward so that engineers could apply the concept in a routine manner. For example, during the rating process, the engineer can calcu- late the resistance R, the dead load effect D, and the AASHTO LRFD load distribution factor D.F., and find Mtruck, which is the moment of the design truck load of the HL-93 (which is the same as the HS-20 truck). The effect on one beam is cal- culated as L1 = D.F.xMtruck where D.F. is the distribution factor. Next, the engineer finds the load factor that causes the first member to fail LF1 from the equation = − 1 1 LF R D L Although L1 is calculated in this study using a structural model of the entire bridge, an approximation for L1 can be obtained using the distribution factors of the LRFD speci- fications. Because of their lack of accuracy, using the sim- plified AASHTO distribution factors given in the standard specifications is not recommended. The equation to find LF1 is the rating factor equation without safety factors or impact. Given LF1, the engineer can find the system factor from Equa- tion 5.26 as illustrated in the following example. Numerical Example for Implementation A numerical example is provided to illustrate the proce- dure for using the system factor during the rating of an exist- ing bridge. In this example the researchers assume a hypothetical case where the bending moment capacity of a 120-ft prestressed concrete bridge with six beams at 8-ft spacing was found

103 to be R = 7200 kip-ft. The dead load effect is found to be 3500 kip-ft. The moment due to the AASHTO truck load alone is 1880 kip-ft. The moment for the AASHTO 3S-2 legal load is 1682 kip-ft. The distribution factor from the AASHTO LRFD is 0.75 and the impact factor is 1.33. The LRFR Operating Rating for a site where the average daily truck traffic is ADTT>5,000 is obtained as = φ − γ γ = × − × × × × =. . 1.0 7200 1.25 3500 1.80 1682 0.75 1.33 0.94R F R D L n D n L n This rating factor value based on individual member capac- ity implies that the bridge should be closed, posted, immediately rehabilitated, or replaced. Given the redundant configuration of the bridge, the bridge owners may choose to delay such actions if the bridge system capacity is found to be sufficiently high so that the bridge would be able to withstand the potential over- loading of a main girder. To assess the entire system’s load carrying capacity, the sys- tem factor is calculated based on Equation 5.26. A first step would require the calculation of LF1 according to Equation 5.8 = − × = − × = . . 7200 3500 0.75 1880 1.10 2.891 20 LF R D D F LLHS where the 1.10 bias is applied because the D.F. is taken from the AASHTO LRFD tables. If the bridge’s six beams are spaced at more than 4-ft then Equation 5.26 is used to find the sys- tem factor with = =3500 7200 0.49. D R [ ] η = −      + − γ −   = −    + − × − = ξ∆β ∗ 1 1.16 0.75 1.16 1 1 0.49 1.16 0.49 0.75 1.16 2.89 1 0.49 0.92 1 0.25 0.85 e D R D R LF D R e T 1 1.09sφ = η = The adjusted system rating of the bridge is then executed using . . 1.09 1.0 7200 1.25 3500 1.80 1682 0.75 1.33 1.15R F R D L s n D n L n = φ φ − γ γ = × × − × × × × = where the system factor fs is 1.09 and the other factors and variables are those that are normally used during the usual rating process. The adjusted rating factor R.F. = 1.15, which is higher than 1.0, implies that the bridge system will still be able to sup- port the applied loads should one member reach its limit- ing capacity because of the bridge’s redundant configuration. The bridge’s redundancy is making the system’s capacity sig- nificantly higher than the capacities of the individual mem- bers. Thus, even if one member reaches its limiting capacity, the bridge system will not collapse. In calculating LF1, the researchers have assumed in this exercise that the distribution factors in the AASHTO LRFD are on the average conservative by a factor of about 1.10 for I-girder bridges. Accounting for this difference leads to a sys- tem factor lower than estimated from the use of the AASHTO LRFD distribution factors and a lower rating than would have been obtained if the AASHTO table is used without the cor- rection. If, for the sake of simplicity, it is assumed that the load distribution factor in the AASHTO tables is accurate, then calculated value for LF1 is approximated as = − × = − × = . . 7200 3500 0.75 1880 3.181 20 LF R D D F LLHS leading to a system factor fs [ ] η = −      + − γ −   = −    + − × − = ξ∆β ∗ 1 1.16 0.75 1.16 1 1 0.49 1.16 0.49 0.75 1.16 3.18 1 0.49 0.93 1 0.25 0.85 e D R D R LF D R e T 1 1.075sφ = η = The adjusted system rating of the bridge is then executed leading to a slightly more conservative rating factor. . . 1.075 1.0 7200 1.25 3500 1.80 1682 0.75 1.33 1.11R F R D L s n D n L n = φ φ − γ γ = × × − × × × × = The proposed approach is believed to be superior to those of the current AASHTO LRFD specifications and the Cana- dian bridge code because the proposed approach eliminates the subjectivity in deciding which load modifier or reliabil- ity index the engineer should use. The proposed approach is based on non-subjective parameters and criteria that are avail- able to the engineer from the bridge cross section (number of beams and beam spacing) and the resistance and applied loads that are calculated during traditional bridge design and rating processes. 5.5 Summary of Bridge Analysis and Results for Damaged Bridges This section summarizes the results of the analyses of dam- aged I-girder and box-girder bridges. The damage scenario consisted of removing an entire I-beam from the multi-girder

104 bridges. For the box-girder bridges, the main damage sce- nario also consisted of removing an entire web. In addition, the analyses considered an alternate damage scenario consist- ing of removing a 6-in. segment of both webs and the bottom flange of one steel box girder. Prestressed Concrete and Composite Steel I-Girder Bridges Numerous simple-span and continuous-span composite- steel I-girder bridges and prestressed concrete I-girder bridges were analyzed in NCHRP Report 406. The results of these analyses are extracted for the purposes of this project to study how the redundancy of these bridges varies with the number of beams, beam spacing, and span length. The simple-span bridges varied in length between 45-ft and 150-ft with a com- posite concrete deck supported by 4, 6, 8, and 10 beams spaced at 4-ft, 6-ft, 8-ft, 10-ft, and 12-ft. Also, the NCHRP Report 406 sample included composite steel I-girder bridges with two 120-ft continuous spans supported by 4, 6, 8, and 10 beams spaced at 4-ft, 6-ft, 8-ft, 10-ft, and 12-ft. The bridges’ concrete slabs varied in depth between 7.5-in. and 8.5-in. depending on the beam spacing. The beams are assumed to be A-36 steel while the deck’s strength is equal to f ′c = 3.5 ksi. Also, over 50 prestressed concrete I-girder bridges with two continuous spans varying in length between 100-ft and 150-ft supported by 4, 6, 8, and 10 beams spaced at 4-ft, 6-ft, 8-ft, 10-ft, and 12-ft were investigated. The bridges’ concrete slabs varied in depth between 7.5-in. and 8.5-in. depending on the beam spacing. The beams are assumed to have a compressive concrete strength f ′c = 5 ksi while the deck’s strength is equal to f ′c = 3.5 ksi. The prestressing tendons are assumed to be 270-ksi steel. The NCHRP Report 406 bridges were designed to exactly satisfy the strength requirements of the AASHTO LRFD design specifications. Sensitivity analyses also were per- formed to investigate the effect of changes in member strength, slab strength, dead weight, as well as other parameters. The moment-rotation relationships for the steel bridges analyzed in NCHRP Report 406 were obtained using existing empirical models based on test results as described in the appendices of NCHRP Report 406. The moment-rotation relationships were then used to perform the nonlinear analysis of the bridge. The analyses were performed assuming that the sections in nega- tive bending are compact and the results are compared to the cases where the sections in negative bending are noncompact. The results provided in NCHRP Report 406 were supple- mented by the results of the analysis of three-span contin- uous bridges with span lengths 50-ft, 80-ft, and 50-ft. The bridges were assumed to have 4, 5, or 6 beams at 8-ft spacing. The bridges were analyzed for different strengths and beam stiffness by assuming that they have different values for ulti- mate moment capacities and moments of inertia. The results of the analysis of the wide composite steel I-girder bridges that compare the damaged system capacity of the bridge system represented by LFd to the capacity to resist first member failure represented by LF1 were found to be highly dependent on the beam spacing while the number of beams had generally little influence on the results except for the case where the bridge had only four beams at 4-ft center to center. The results are plotted in Figure 5.20 for the results averaged over their span lengths. The figure shows a clear dependence on beam spacing while the number of beams are only important for the bridge with four beams at 4-ft. The LFd/LF1 ratio was not related to the span length as illustrated in Figure 5.21. The prestressed concrete bridges showed similar trends for the effect of beam spacing as shown in Figure 5.22. However, LFd/LF1 generally dropped as the span length increased as shown in Figure 5.23. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0 2 4 6 8 10 12 14 LF d/ LF 1 beam spacing (ft) 4 beams 6 beams 8 beams 10 beams Figure 5.20. Plot of LFd /LF1 vs. beam spacing for steel I-girder bridges.

105 Figure 5.21. Effect of span length on LFd/LF1 for steel I-girder bridges. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0 50 100 150 200 LF d/ LF 1 span length (ft) 4 ft 6 ft 8 ft 10 ft 12 ft Figure 5.22. Plot of LFd /LF1 vs. beam spacing for damaged prestressed concrete I-girder bridges. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0 2 4 6 8 10 12 14 LF d/ LF 1 beam spacing (ft) 4 beams 6 beams 8 beams 10 beams Figure 5.23. Effect of span length on LFd/LF1 for damaged prestressed concrete I-girder bridges. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0 20 40 60 80 100 120 140 160 LF d/ LF 1 Span length (ft) 4 ft 6 ft 8 ft 10 ft 12 ft

106 The plot of LFd versus LF1 for a set of simple-span bridges is presented in Figure 5.24. The results show a strong linear rela- tionship between LFd and LF1 with an equation of the form = 0.60 (5.30)1LF LFd The same analysis was performed on a set of continu- ous steel girder bridges. The analysis compared the results of compact and noncompact sections for the case when the damaged bridge was loaded with two lanes in the middle of the span to the case when two trucks are in one lane, one truck placed in each of two consecutive spans. The results are plotted in Figure 5.25 by taking the minimum value of LFd from the two loading cases and comparing it to the minimum value of LF1. The plot shows that a lower bound for the results that would cover compact and noncompact sections can be established using a linear relationship of the form = 0.59 (5.31)1LF LFd This linear relationship is obtained from the regression analysis of the results of bridges with noncompact sections in negative bending. The analyses in Figures 5.24 and 5.25 were for bridges with four beams at 8-ft spacing and six beams at 8-ft spacing, respectively. As noted, the relationship between LFd and LF1 is also affected by the beam spacing. Additional analyses also investigated the effect of the deck slab strength and the weight of the damaged beam. Box-Girder Bridges with Severe Damage to One Web Numerous steel and concrete box-girder bridges were ana- lyzed assuming two damage scenarios. The first damage sce- nario assumes severe damage to an external web representing the consequences of collisions or severe corrosion to the pre- stressing tendons of prestressed concrete or the corrosion of steel box-girder bridges. In these models, it is assumed that the y = 0.60x R² = 0.96 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 1 2 3 4 5 6 7 LF d LF1 Figure 5.24. Relation between LFd and LF1 for simple-span steel I-girder bridges. Figure 5.25. Relationship between LFd and LF1 for continuous steel I-girder bridges. y = 0.59x R² = 0.99 0 1 2 3 4 5 6 0 2 4 6 8 10 12 LF d LF1 3 span_compact 3 span_noncompact

107 torsional rigidity of the box is not affected by the damage. The results of these analyses are summarized in Figure 5.26. The fig- ure shows that the load carrying capacity of damaged systems is related to the ability of the system to resist first member failure with a relationship of the form = 0.72 (5.32)1LF LFd The relationship, which is somewhat similar to that of con- tinuous I-girder bridges, holds for narrow simple-span box- girder bridges, wide simple-span box-girder bridges, as well as three-span and two-span continuous box-girder bridges as long as the damaged box maintains a sufficient level of torsional rigidity. However, if the damaged box of a narrow simple-span bridge loses its torsional capacity altogether, the capacity of the system degrades to close to half its load carrying capacity. This is because the torsional capacity of a damaged narrow box will help redistribute the load to the remaining three box webs and this ability is lost if the tor- sional capacity is lost. However, the same is not necessarily true for wide boxes that have a much larger web spacing and spacing between the boxes. In this case, the torsional capac- ity of the box will not add a significant ability to redistribute the load to the undamaged box. This is because the torsional capacity may help redistribute the load locally but, globally, the contribution of the torsion is offset by the longer distance to the undamaged portions of the bridge. The load carrying capacity of damaged narrow simple box-girder bridges that lose one box’s torsional capacity along with one web will fol- low an equation of the form = 0.46 (5.33)1LF LFd Although the AASHTO specifications assume that all steel box girders in negative bending should be considered to have noncompact sections, in this project, several continuous steel boxes with compact sections in negative bending were analyzed to study how box ductility would affect the dam- aged bridge system’s capacity assuming severe damage to an external web. Continuous prestressed concrete boxes also are investigated. The results of these analyses are summarized in Figure 5.27. The figure shows that the load carrying capacity of damaged systems is related to the ability of the system to resist first member failure with a relationship of the form = +0.59 4.50 (5.34)1LF LFd The relationship is different from that of three-span and two-span continuous box-girder bridges with noncompact sections over the supports. The data in Figure 5.27 include wide and narrow prestressed concrete boxes loaded by two lanes of traffic or separately by one lane loaded by one truck, one in each span. The wide steel boxes also were loaded by two trucks side by side or separately by one truck in each span. Two cases are considered for the effect of torsional rigidity for the damaged continuous steel boxes. One case assumes that the torsional rigidity is not affected by the damage, and in another case the damage is assumed to also eliminate the tor- sional rigidity. Equation 5.34 is found to give a lower bound to all the cases considered. Steel Box-Girder Bridges with Fractured Box Several steel simple-span and continuous box-girder bridges were analyzed assuming that one box was damaged due to Figure 5.26. Summary of the results for damaged box-girder bridges. y = 0.46x R² = 0.93 0 5 10 15 20 25 0 5 10 15 20 25 30 35 LF d LF1 Narrow simple span_w/ torsion_one lane Narrow simple span_w/ torsion_two lanes Narrow simple span_open box_two lanes Wide Simple span_w/torsion Wide Simple span_open box simple span P/s box w/ torsion Continuous box_noncompact LFd=0.72LF1 LFd=0.46LF1 y = 0.72x R2 = 0.98

108 fatigue fracture. This is done by removing about a 1-ft segment in each of the webs and the bottom flange at the midpoint along the length of one box. In these models, it is assumed that the torsional rigidity of the remaining portions of the box is not affected by the damage. The results of these analyses are summarized in Figure 5.28. The figure shows that the load carrying capacity of simple-span damaged systems also is related to the ability of the system to resist first mem- ber failure with a relationship of the form 0.82 4.14 (5.35)1LF LFd = − However, continuous bridges show remarkably strong ability to continue to carry significant loads after damage. This is due to the cantilevering effect that continuous bridges provide allowing for the effect of the damage to be reason- ably low. Effect of Beam Spacing The results of simple-span bridges provided in NCHRP Report 406 demonstrate a clear dependence on beam spac- ing, as shown in Figure 5.20. However, the load factor LF1 for the bridges analyzed during NCHRP Report 406 had a very limited range with LF1 varying between 2.2 and 4.7. This made it difficult to understand how the damaged bridge load factor LFd varies as the first member failure load fac- tor LF1 changes. Therefore, additional bridge models having load factors LF1 ranging between 3.0 and 8.0 were analyzed. In the analysis, the dead load applied along the length of each beam is assumed to be 0.97 kip/ft and the deck’s capac- ity to carry moment in the transverse direction 13.5 kip-ft/ft (or 1615 kip-in. for each 10-ft-wide slab segment). This value for moment capacity of the deck is similar to the one used in NCHRP Report 406. Figure 5.29 plots the damaged system capacity of the bridge system represented by LFd versus the capacity to resist first member failure represented by LF1. The results show a consistent trend of increasing damaged bridge capacity as the capacity of the first member represented by LF1 increases. The relationship between LFd and LF1 is depicted in Figure 5.30. The figure also shows y = 0.59x + 4.50 R² = 1.00 0.00 5.00 10.00 15.00 20.00 25.00 0 5 10 15 20 25 30 35 LF d LF1 Steel_w/ torsion Steel_open box P/s box_w/ torsion Figure 5.27. Summary of the results for damaged continuous box-girder bridges with compact section at supports. Figure 5.28. Plot of LFd vs. LF1 for fractured box-girder bridges. y = 0.82x 4.14 R² = 0.99 0 5 10 15 20 25 0 10 20 30 40 LF d LF1 Wide simple span_Partial damage Wide continuous_partial damage

109 how LFd decreases for the same LF1 as the beam spacing increases. The trend with beam spacing is similar to that depicted in Figure 5.20 with LFd = 0.48 LF1 for the systems analyzed in this case with beams at 8-ft, LFd = 0.43 LF1 for bridges with beams at 6-ft, and LFd = 0.37 LF1 for bridges with beams at 12-ft. As depicted in Figure 5.30, which includes the data from NCHRP Report 406 and additional data obtained during the course of this project, the relationship that, on average, describes the ratio of LFd/LF1 as a function of the beam spac- ing is found to be ( )= −1 0.056 (5.36) 1 LF LF S d where S is the beam spacing in feet. The relationship is for the average beam with a distributed weight along the length of the bridge of 1.0 kip/ft per beam. The equation gives the general trend with some variation around the trend based on the properties of the girders and the span length. For continuous two-span steel bridges, NCHRP Report 406 considered two negative bending section types: those that are compact and those with noncompact sections. Also, two load cases are considered. The first load case consisted of two trucks side-by-side in one span. The second load case placed one truck in each span. The load factor that causes the first failure of a member assuming linear-elastic behavior was used to determine LF1. LFd was the lowest load carrying capacity of the damaged system after removing one entire girder from one span. The results for the steel I-girder bridges are presented in Figure 5.31. The results for continuous pre- stressed concrete I-girder bridges are given in Figure 5.32. The plot shows that the trend line is very similar to that of the compact steel bridges. Therefore, the same equation would be applicable for both the continuous prestressed concrete and compact steel bridges. Figure 5.29. Effect of beam spacing on simple-span damaged bridges. y = 0.48x R² = 0.97 y = 0.43x R² = 0.96 y = 0.37x R² = 0.92 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 2.0 4.0 6.0 8.0 10.0 LF d LF1 S=8 ft S=10 ft S=12 ft Figure 5.30. Effect of beam spacing on damaged bridges. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 2 4 6 8 10 12 14 16 LF d/ LF 1 beam spacing (ft) Steel_4 beams Steel_6 beams Steel_8 beams Steel_10 beams P/s_4 beams P/s_6 beams P/s_8 beams P/s_10 beams 100ft_4 beams LFd/LF1=(1 0.0562*S)

110 Effect of Dead Weight of Damaged Beam The results of simple-span bridges provided in NCHRP Report 406 also demonstrate that bridge redundancy for the damaged limit state depends on the weight of the damaged beam that must be transferred to the adjacent undamaged beams as the bridge is loaded. To better establish this relation- ship, additional bridge models having a load factor LF1 rang- ing from 3.0 and up to 8.0 were analyzed. In the analysis, the spacing for each beam was fixed at 8-ft and the deck capacity is kept at 13.5 kip-ft/ft of slab. Figure 5.33 plots the damaged system capacity of the bridge systems analyzed in this case represented by LFd versus the capacity to resist first member failure represented by LF1.The results show that a doubling of the beam weight will reduce the slope of LFd versus LF1 by 23%. As depicted in Figure 5.33, the relationship that, on aver- age, describes the ratio of LFd/LF1 as a function of the beam spacing is found to be ( )= − ω0.60 0.12 (5.37) 1 LF LF d where w is the weight of the beam per unit length expressed in kip/ft. The relationship is for the average beam with a distributed weight along the length of the bridge of 1.0 kip/ft per beam. The equation gives the general trend with some variation around the trend based on the properties of the girders and the span length. Alternatively, a correction factor can be used on the origi- nal Equations 5.30 through 5.36 to reflect the reduction in Figure 5.31. Effect of beam spacing on damaged continuous steel I-girder bridges. y = 0.081x + 1.05 R² = 0.86 y = 0.081x + 1.35 R² = 0.96 0.00 0.20 0.40 0.60 0.80 1.00 1.20 0 2 4 6 8 10 12 14 LF d/ LF 1 Spacing /ft Non compact Compact Figure 5.32. Effect of beam spacing on damaged continuous prestressed concrete I-girder bridges. y = 0.069x + 0.95 R² = 0.72 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0 2 4 6 8 10 12 14 LF d/ LF 1 Spacing /ft

111 the redundancy ratio. The weight correction factor is defined as gweight 1.23 0.23 (5.38)d modified LF LF kip ftweight d beam ( )γ = = − ω where wbeam is the total dead weight applied on each beam expressed in kip per unit length. Effect of Deck Capacity The effect of the deck’s bending moment capacity also is investigated to study how bridge redundancy varies when the deck’s bending moment capacity is reduced. In the analysis performed in this section, the spacing between the beams is set at 8-ft or 10-ft, and the dead load on each beam is 0.97 kip/ft. Figure 5.34 plots the damaged system capacity of bridge systems represented by LFd versus the capacity to resist first member failure represented by LF1 for systems analyzed in this sensitivity analysis. The results show that damaged bridges will carry less live load as the deck’s moment capacity decreases. The trend in Figure 5.34 also demonstrates that the increase in LFd as the deck capacity increases will be smaller if the bridge member capacity represented by LF1 itself is small. As LF1 increases, the deck’s contribution to increasing the overall damaged bridge’s capacity increases. The improvement due to deck capacity for damaged bridges can be depicted as shown in Figure 5.35. Furthermore, a Figure 5.33. Effect of dead weight of damaged beam. y = 0.48x R² = 0.97 y = 0.43x R² = 0.99 y = 0.37x R² = 1.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 2.0 4.0 6.0 8.0 10.0 LF d LF1 S=8 ft, Mu slab=13.5 kip ft/ft w=0.97 kip/ft w=1.46 kip/ft w=1.95 kip/ft 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0.0 2.0 4.0 6.0 8.0 10.0 LF d LF1 S=8 ft, W=0.97kip/ft Mu=3230 kip in Mu=1615 kip in Mu=1357 kip in Mu=1098 kip in Mu=800 kip in Figure 5.34. Effect of deck capacity Mu deck on capacity of damaged bridges.

112 verification of the effect of diaphragm capacity with dif- ferent spacing is shown in Figure 5.36. The base line is the deck with a transverse bending moment capacity equal to Mtransverse = 13.5 kip-ft/ft (or 1615 kip-in. for each 10-ft-wide slab segment). The load capacity of the bridge system varies almost linearly as a function of Mtransverse reaching an upper limit for improvement at about 1.10 to 1.20. The presence of diaphragms and cross beams can be considered in terms of the contribution they make to improve the transverse bend- ing capacity of the system. Of course, this assumes that the cross beams are distributed along the length of the bridge and especially near the middle of the span. It also assumes that they are firmly attached to the longitudinal beams to resist the large bending moments that will develop as the damaged sys- tem is overloaded as shown in Figure 5.35. The equation that describes the effect of the transverse bending capacity is given in terms of a slab effect correction factor, gtransverse, as γ = = + ≤0.50 13.5 . 0.50 1.10 (5.39) modifiedLF LF M kip ft ft transverse d d transverse (5.40)M M Mtransverse slab br L= + where Mtransverse = combined moment capacity for lateral load transverse expressed in kip-ft per unit slab width, Mslab = moment capacity of slab per unit width, and Mbr/L = contribution of the bracing and diaphragms to transverse moment capacity calcu- lated using Equation 5.41 or 5.42. Equivalent transverse moment capacity for cross brac- ing as defined in the FHWA Steel Bridge Design Handbook: Bracing System Design (2012) (5.41)M F h L br L br b b = Equivalent transverse moment capacity for diaphragm (5.42)M M L br L br b = where Mbr = moment capacity of diaphragms contributing to lateral transverse distribution of vertical load between 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0 2 4 6 8 10 LF d/ LF d_ ba se Mtransverse/Mtransverse_base 4 beams at 8 ft spacing 4 beams at 10 ft spacing Modification curve 0.50 0.50 13.5 . / transverse transverse M kip ft ft Upper limit of correction factor Figure 5.35. Modification curve for deck capacity Mtransverse. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0 2 4 6 8 10 LF d/ LF d_ ba se Mtransverse/Mtransverse_base Diaphragm spacing at 10ft Diaphragm spacing at 20ft Diaphragm spacing at 30ft Modification curve 0.50 0.50 13.5 . / transverse transverse M kip ft ft Figure 5.36. Verification for diaphragm and deck capacity Mtransverse.

113 adjacent main bridge girders; Fbr = bracing chord force deter- mined from the applicable limit state for the bolts (see AISC Steel Construction Manual, 2011, Part 7), welds (see AISC, 2011, Part 8), and connecting elements (see AISC, 2011, Part 9); Lb = spacing of the cross frames or diaphragms; and hb = distance between the bracing top and bottom chords. Summary The redundancy of damaged bridge systems that may have lost the load carrying capacity of a critical member due to various possible deterioration mechanisms or extreme events is a function of the beam spacing, the slab strength, and the dead load that the damaged beam needs to release to the remaining intact members. The results of damaged bridges show a clear correlation between LFd and LF1 represented by the redundancy ratio defined as 1 R LF LF d d = . Table 5.4 lists the relationship obtained for Rd as a function of span length and the load carrying capacity of the most critical member LF1 accounting for the variation in Rd with slab strength and dead weight applied on the damaged member for the bridge types analyzed in this study. 5.6 System Factors for Damaged Bridges The calibration of the system factor for bridges suscep- tible to major damage to main load carrying members was performed using the same procedure outlined for the ulti- mate limit state, and the target reliability margin was set at Dbd target = -2.70. This target value was selected because it cor- responds to the average margin obtained for typical 4-beam I-girder bridges, which have traditionally been accepted as providing acceptable levels of redundancy. The calibration of the system factor can be performed eas- ily using the expression in Equation 5.26. In most of the cases Bridge Cross-Section Type Equation for Redundancy Ratio 1 d d LFR LF Simple-span and continuous prestressed concrete I- beam bridges with four beams at 4-ft 0.56d transverse weightR Simple-span and continuous compact steel I-girder bridges with four beams at 4-ft 0.64d transverse weightR Continuous noncompact steel I-girder bridges with four beams at 4-ft 0.58d transverse weightR Wide simple-span I-girder bridges 1 0.056d transverse weightR S Wide continuous compact steel and prestressed concrete I-girder bridges 1.35 0.08d transverseR S Wide continuous noncompact steel I-girder bridges 1.00 0.08d transverseR S Narrow simple-span steel box-girder bridges less than 24-ft, open box no torsional rigidity 0.46d transverseR Fractured simple-span steel box-girder bridges less than 24-ft wide 1 4.140.82d transverseR LF All other simple-span box-girder bridges 0.72d transverseR Continuous steel box-girder bridges with noncompact negative bending sections and 1 1 1.75LF LF 0.72d transverseR All other continuous box-girder bridges 1 4.500.59d transverseR LF where S = beam spacing in feet. d modified 1.23 0.23 ( / )weight beam d LF kip ft LF γ = = − ω mod 0.50 0.50 1.10 13.5 . / d ified transverse transverse d LF M LF kip ft ftγ = = + ≤ and ωbeam is the total dead weight applied on each beam expressed in kip per unit length. Mtransverse is the combined moment capacity of the slab and transverse members including diaphragms expressed in kip-ft per unit slab width. Table 5.4. Summary of LFd vs. LF1 for damaged systems under vertical loads.

114 listed in Table 5.3, the redundancy equation can be repre- sented in the form =LF C LF (5.43)d red1 1 where the parameter Cred1 is equal to the redundancy ratio Rd. The algebraic manipulation of Equation 5.26 is given as η = −      + − − ξ∆β 1 1 1 2 1 1 e D R C D R C C LF D R red red red T φ = η 1 s with Cred1 = Rd and Cred2 = 0 leads to an expression for the sys- tem factor given as ( ) φ = − −0.47 0.47 (5.44) R R D R s d d where 0.47 is obtained from 0.93e(ξDbT) where the COV for the evaluation of the capacity of the bridge system is assumed to be ξ = 25% and 0.93 is the live load bias LL LL 2 75 = in Equation 2.16 that accounts for the maximum 2-year live load as compared to the 75-year live load. NCHRP Report 406 recommends that the calibration of the system factor for the damaged bridge limit state be based on a 2-year live load, which coincides with the normal bridge inspection period at which point in time the damage to the bridge will certainly be detected and appro- priate actions taken to close, post, or rehabilitate it. The process of determining the system factor for the dam- aged bridge limit state can then be executed using Equa- tion 5.44, where appropriate values for Rd are obtained for the particular bridge configuration from Table 5.4. A sum- mary of the proposed system factors for damaged I-girder and box-girder bridges is provided in Tables 5.5 and 5.6 as a function of the redundancy ratio = 1 R LF LF d d , which gives the capacity of a damaged bridge system that has previously lost the load carrying capacity of a main member given as LFd and the ability of the originally intact bridge to resist first member failure, which is represented by the variable, LF1, defined in Equation 5.8. For the box-girder bridges, three different dam- age scenarios are considered. In the first scenario, one box is assumed to have been exposed to a fatigue type fracture that sliced through the entire bottom flange and two webs. The second scenario assumed major damage to one web while maintaining the torsional capacity of the box. The third sce- nario considered that the failure of the web also led to the loss of the torsional rigidity of the box. Tables 5.5 and 5.6 list the expressions for Rd as a function of beam spacing, slab strength, and the dead weight applied on the damaged member for the bridge types analyzed in this study. 5.7 Conclusions This chapter summarized the results of the redundancy analysis of simple-span and continuous steel and prestressed concrete I-girder and box-girder bridges. The redundancy was evaluated for the ultimate limit state of the originally intact Bridge Cross-Section Type Redundancy Ratio 1 d d LFR LF System Factor Simple-span and continuous prestressed concrete I-girder bridges with four beams at 4-ft 0.56d transverse weightR 0.47 (0.47 ) d s d R DR R Simple-span and continuous compact steel I- girder bridges with four beams at 4-ft 0.64d transverse weightR All other simple-span I-girder bridges 1 0.056d transverse weightR S Continuous noncompact steel I-girder bridges with four beams at 4-ft 0.58d transverseR All other continuous noncompact steel I-girder bridges 1.00 0.08d transverseR S All other continuous compact steel and prestressed concrete I- girder bridges 1.35 0.08d transverseR S Table 5.5. System factors for damaged I-girder bridges under vertical loads.

115 bridges, as well as bridges that have suffered major damage to the most critical bridge member. The results were fitted into equations that best described the redundancy based on a set of simple parameters that describe the bridge geometry and main load carrying characteristics. These parameters include the number of beams and beam spacing; the load carrying capacity of the main members; the dead over live load ratio; and, in the case of damaged bridges, the dead weight that was carried by the damaged beam prior to damage; and the trans- verse load carrying capacity of the bridge expressed in terms of the maximum moment that the slab and diaphragms can carry. The analyses performed as part of this study and NCHRP Report 406 highlight the difficulty of analyzing the behavior and load carrying capacities of damaged bridges. The results for damaged bridges are highly sensitive to many parameters, especially the modeling of the damage scenario and the sharp discontinuities that these create in the structural model, the torsional capacity of the remaining members of the dam- aged system, and the contributions of the slab and secondary members, including diaphragm and bracings. A large num- ber of sensitivity analyses were performed to understand the interaction between these parameters. However, more sen- sitivity analyses are needed to better evaluate the range of variations and the upper limits for the proposed models. References 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. Ghosn, M., Sivakumar, B., and Miao, F. (2010) “Calibration of Load and Resistance Factor Rating Methodology in New York State,” Bridge Engineering 2010, Volume 1, Monograph Accession #:01321766, 81–89 Transportation Research Record 2200, Journal of the Trans- portation Research Board, Transportation Research Board, ISSN: 0361-1981. Hunley, T. C. and Harik, I. E. (2012) “Structural Redundancy Evalu- ation of Steel Tub Girder Bridges,” Journal of Bridge Engineering 17(3) May 1. FHWA Steel Bridge Design Handbook: Bracing System Design (2012) Publication No. FHWA-IF-12-052, Vol. 13, U.S. Department of Transportation, FHWA, Washington D.C. AISC Steel Construction Manual, 14th ed (2011) American Institute of Steel Construction, Chicago, IL. Table 5.6. System factors for damaged box-girder bridges under vertical loads. Bridge Cross-Section Type Redundancy Ratio 1 d d LFR LF System Factor Fractured simple-span steel box-girder bridges less than 24-ft wide Non-redundant s=0.80 Narrow simple-span steel box-girder bridges less than 24-ft with no torsional rigidity 0.46d transverseR 0.47 (0.47 ) d s d R DR R All other simple-span box- girder bridges 0.72d transverseR Continuous steel box-girder bridges with noncompact negative bending sections and 1 11.75LF LF 0.72d transverseR All other continuous box- girder bridges 1 4.500.59d transverseR LF where S = beam spacing in feet. 1.23 0.23 ( / )weight beam kip ftγ = − ω ωbeam = total dead weight on the damaged beam in kip per unit length. 0.50 0.50 13.5 . / transverse transverse M kip ft ftγ = + Mtransverse = combined moment capacity of the slab and transverse members including diaphragms expressed in kip-ft per unit slab width. The range of applicability of transverseγ has been verified for I-girder bridges for up to a range of transverseγ =1.10.