Calibration of AASHTO LRFD Concrete Bridge Design Specifications for Serviceability (2014)

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5 CALIBRATION RESULTS 5.1 Cracking of Reinforced Concrete Components Service I Limit State – Annual Probability Traditionally, reinforced concrete components are designed to satisfy the requirements of the strength limit state and then they are checked for the Service I limit state load combination to ensure that the crack width under service conditions does not exceed a certain value. However, the specifications provisions are written in a form emphasizing reinforcement details, i.e. limiting bar spacing rather than crack width. Satisfying the Service I limit state for crack control through the distribution of reinforcement may require a reduction in the reinforcement spacing. This may require the use of smaller bar diameters or, if the smallest allowed bar diameters are already being used, an increase in the number of reinforcement bars leading to an increase in the reinforcement area. Two exposure classifications exist in AASHTO LRFD: Class 1 exposure condition and Class 2 exposure condition. Class 1 relates to an estimated maximum crack width of 0.017 in. while Class 2 relates to an estimated maximum crack width of 0.01275 in. Class 2 is typically used for situations where the concrete is subjected to severe corrosion conditions such as bridge decks exposed to deicing salts and substructures exposed to water. Class 1 is used for less corrosive conditions and could be thought of as an upper bound in regards to crack width for appearance and corrosion. Previous research indicates that there appears to be little or no correlation between crack width and corrosion. However, the different classes of exposure conditions have been so defined in the design specifications in order to provide flexibility in the application of these provisions to meet the needs of the bridge owner. The load factors for dead load (DL) and live load (LL) specified for the Service I load combination are as follows: DL load factor = 1.0 LL load factor = 1.0 When designing reinforced concrete bridge decks using the conventional design method, most designers follow a similar approach in selecting the deck thickness and reinforcement. The thickness is typically selected as the minimum acceptable thickness, often based on the owner’s standards. The choice of main reinforcement bar diameter is typically limited to #5 and #6 bars and the designer does not switch to #6 bars unless #5 bars result in bar spacing less than the minimum spacing allowed. This limits the number of possible variations and allows the development of a deck database that can be used in the calibration. For decks designed using the empirical method, not determined based on a calculated design load, the reinforcement does not change with the change in girder spacing resulting in varying crack resistance. As the statistical parameters for both the load effect and the resistance are required to perform the calibration, a meaningful calibration of decks designed using the empirical design method could not be performed. For other components, including prestressed decks, designers may select different member dimensions resulting in different reinforcement area. Even for the same reinforcement area, the designer may use bars or strands of different diameter and spacing and, consequently, result in different crack resistance and a different reliability index for each 83

possible variation. The variation in the cracking behavior of the same component with the change in the selected reinforcement prohibits the performance of a meaningful calibration for such components. Due to the reasons indicated above, the calibration for Service I limit state for crack control through the distribution of reinforcement was limited to reinforced concrete decks designed using the conventional design method. The decks are assumed to be supported on parallel longitudinal girders. 5.1.1 Live Load Model Reinforced concrete decks designed using the conventional method are designed for the heavy axles of the design truck. This required developing the statistical parameters of the axle loads of the trucks in the WIM data. The statistical parameters for the axle loads are presented in Section 4.3.4. Statistical parameters corresponding to a one year return period were assumed in the reliability analysis. ADTTs of 1000, 2500, 5000, and 10,000 were considered, however, an ADTT of 5000 was used as the basis for the calibration. 5.1.2 Target Reliability Index 5.1.2.1 Limit State Function For the control of cracking of reinforced concrete through the distribution of reinforcement, the limiting criteria are the calculated crack widths, assumed to be 0.017 in. and 0.01275 in for Class 1 and Class 2, respectively. Due to the lack of clear consequences for violating the limiting crack width, there was no basis to change the nature or the limiting values of the limit state function, i.e. the crack width criteria. The work was based on maintaining the current crack width values and calibrating the limit state to produce a uniform reliability index similar to the average reliability index produced by the current designs. 5.1.2.2 Statistical Parameters of Variables Included in the Design Several variables affect the resistance of prestressed components. Table 5-1 shows a list of variables that were considered to be random variables during the performance of the reliability analyses. These variables represent a summary of the information based on previous research studies by Siriaksorn and Naaman (1980) and Nowak (2008). 84

Table 5-1 Summary of Statistical Information for Variables used in the Calibration of Service I Limit State for Crack Control Variable Distribution Mean COV Remarks sA normal 0.9 sA 0.015 Siriaksorn and Naaman (1980) b normal nb 0.04 Siriaksorn and Naaman (1980) cE C normal 33.6 0.1217 Siriaksorn and Naaman (1980) d normal 0.99 nd 0.04 Nowak (2008) cd normal cnd 0.04 Nowak (2008) sE normal snE 0.024 Siriaksorn and Naaman (1980) cf ′ lognormal cE = cEC γ c 1.5 fc ' (psi) 3000:1.31 cnf ′ 3500:1.27 cnf ′ 4000:1.24 cnf ′ 4500:1.21 cnf ′ 5000:1.19 cnf ′ 3000:0.17 3500:0.16 4000:0.15 4500:0.14 5000:0.135 Siriaksorn and Naaman (1980) yf lognormal 1.13 ynf 0.03 Nowak (2008) h normal nh ( )1/ 6.4µ Siriaksorn and Naaman (1980) cγ normal 150 0.03 Siriaksorn and Naaman (1980) sA = area of steel rebar, in2 b = the width of equivalent transverse strip of concrete deck, in. EcC = constant parameter for concrete elasticity modulus. = effective depth of concrete section, in. cd = bottom cover measured from center of lowest bar, in sE = modulus of elasticity of steel reinforcement, psi cf ′ = specified compressive strength of concrete, psi yf = yield strength of steel reinforcement, psi h = the thickness of the deck, in. cγ = unit weight of concrete, pcf 5.1.2.3 Database of Reinforced Concrete Decks A database consisting of 15 reinforced concrete decks designed using the conventional method of deck design was developed. As typical in deck design, #5 bars were used unless they resulted in bar spacing less than 5 in.; the minimum spacing many jurisdictions allow in 85

deck design. If #5 bars result in a bar spacing less than 5 in., #6 bars were used. No maximum bar spacing was considered in the design to ensure that all decks produced a calculated crack width equal to the maximum allowed crack width allowed by the specifications. The designs were not checked for other limit states as the purpose was to calibrate the Service I limit state. The design of the 15 decks was repeated twice, once assuming Class 1 exposure conditions and another time assuming Class 2 exposure conditions. Table 5-2 presents the summary information of 15 designed bridge decks. The top and bottom concrete cover assumed in the design were 2.5 in. and 1.0 in., respectively. Table 5-2 Summary Information of 15 Bridge Decks Designed using AASHTO LRFD Conventional Deck Design Method Deck Group # Girder Spacing (ft.) Deck Thickness (in.) 1 6 7.0 7.5 8.0 2 8 7.5 8.0 8.5 3 10 8.0 8.5 9.0 9.5 4 12 8.0 8.5 9.0 9.5 10.0 5.1.2.4 Selection of the Target Reliability Index Monte Carlo simulation was used to obtain the statistical parameters of resistance (or capacity) and dead load while the statistical parameters for live load were taken from Section 4.3.4. The reliability indices for various ADTTs and exposure conditions for the 15 decks are summarized in Table 5-3. Due to the difference in positive and negative moment (bottom and top) reinforcement of the deck, the reliability index was calculated separately for the positive and negative moment reinforcement. 86

Table 5-3 Summary of Reliability Indices for Concrete Decks Designed According to AASHTO LRFD (2012) ADTT Positive Moment Region Negative Moment Region Reliability Index (Class 1) Reliability Index (Class 2) Reliability Index (Class 1) Reliability Index (Class 2) 1000 2.44 1.54 2.37 1.77 2500 1.95 1.07 1.79 1.27 5000 1.66 0.85 1.61 1.05 10000 1.39 0.33 1.02 0.5 Avg. 1.86 0.95 1.70 1.15 Max. 2.44 1.54 2.37 1.77 Min. 1.39 0.33 1.02 0.50 Std Dev. 0.45 0.50 0.56 0.53 COV 24% 53% 33% 46% It should be noted that even though the design for Class 2 resulted in more reinforcement than for Class 1 exposure conditions, the reliability index for Class 2 is lower than that for Class 1 due to the more stringent limiting criteria (narrower crack width). Current practices rarely result in the deck positive moment reinforcement being controlled by the Service I limit state due to the smaller bottom concrete cover. When Strength I limit state is considered, more positive moment reinforcement is typically required than by Service I. The additional reinforcement results in actual reliability indices for the positive moment region higher than those shown in Table 5-3. For the negative moment region, the design is often controlled by the Service I limit state. Thus, the reliability indices shown for the negative moment region in Table 5-3 are considered representative of the actual reliability indices that would be calculated when all limit states, including Strength I, are considered in the design. Therefore, it is recommended that the target reliability index be based on the reliability index for the negative moment region. Since the Class 2 case is the more common case for negative moment reinforcement of decks, the reliability index for Class 2 was used as the basis for selecting the target reliability index. The reliability index for Class 1 was assumed to represent a relaxation of the base requirements. The case of ADTT=5000 was also considered as the base case on which the reliability analysis was performed. Table 5-4 shows the inherent reliability indices for the negative moment region of decks designed for the current AASHTO LRFD Specifications. Based on the values shown in Table 5-4, target reliability indices of 1.6 and 1.0 were selected for Class 1 and Class 2, respectively, based on ADTT=5000. 87

Table 5-4 Reliability Indices of Existing Bridges based on 1-year Return Period ADTT Reliability Index Current Practice (Class 1, Negative) Current Practice (Class 2, Negative) 1000 2.37 1.77 2500 1.79 1.27 5000 1.61 1.05 10,000 1.02 0.50 5.1.3 Calibration Result The basic steps of the calibration process are shown below as they relate to the Service I calibration. 5.1.3.1 Step 1: Formulate the Limit State Function and Identify Basic Variables The limit state function considered is the limit on the estimated crack width. In the absence of information suggesting that the current criteria based on a crack width of 0.017 in. and 0.01275 in. for Class 1 and Class 2, respectively, is not adequate, the current crack widths were maintained as the limiting criteria. A discussion of crack width equations in the literature is included in Appendix A. 5.1.3.2 Step 2: Identify and Select Representative Structural Types and Design Cases The database of decks used in this study is described in Section 5.1.2.3. 5.1.3.3 Step 3: Determine Load and Resistance Parameters for the Selected Design Cases The variables include the dimension of the cross-section and the material properties. The statistical information includes the probability distribution and statistical parameters such as mean, µ, and standard deviation, σ. 5.1.3.4 Step 4: Develop Statistical Models for Load and Resistance The variables affecting the load and resistance were identified. These include live load, and resistance including the dimensions of the cross-section, the material properties, etc. The statistical information includes the probability distribution and statistical parameters for axle loads presented in Section 4.3.4 and for other variables affecting the resistance presented in Section 5.1.2.2. 5.1.3.5 Step 5: Develop the Reliability Analysis Procedure The statistical information of all the required variables is used to determine the statistical parameters of the resistance by using Monte Carlo simulation. For each deck, Monte Carlo simulation was performed for each random variable associated with the calculation of the resistance and dead load. One thousand simulations were performed. For each random variable 1000 values were generated independently based on the statistics and distribution of that random variable. For each simulation, the dead load 88

and the resistance were calculated using one of the 1000 sets of values of the random variable, i.e. the nth simulation used the nth value of each random variable where n varied from 1 to 1000. This process resulted in 1000 values of the dead load and the resistance. The mean and standard deviation of the dead load and the resistance were then calculated based on the 1000 simulations. 5.1.3.6 Step 6: Calculate the Reliability Indices for Current Design Code and Current Practice Using the statistics of the dead load and the resistance, calculated from the Monte Carlo simulation as described above, and the statistics of the live load as derived from the WIM data as described in Chapter 4, the reliability index was calculated for each deck. The reliability index was calculated using the following equation: 2 2 R Q R Q µ − µ β = σ + σ (5-1) where β = reliability index. Rµ = mean value of the resistance Qµ = mean value of the applied loads Rσ = standard deviation of the resistance Qσ = standard deviation of the applied loads. The calculated reliability indices of the decks in the database are shown in Table 5-3 for both positive and negative moment reinforcement and for Class 1 and Class 2 exposure conditions. 5.1.3.7 Step 7: Review the Results and Select the Target Reliability Index, βT The initial target reliability index was determined as shown in Table 5-4. 5.1.3.8 Step 8: Select Potential Load and Resistance Factors for Service I, Crack Control through the Distribution of Reinforcement The load factors for dead loads and live loads for Service I limit state in the AASHTO LRFD (2012) are 1.0. The existing specifications do not explicitly include a resistance factor for the distribution of the control of cracking through the distribution of reinforcement. This results in an implied resistance factor of 1.0. The load and resistance factors were maintained for the initial reliability index calculations. For Class 1 exposure condition (maximum crack width of 0.017 in), Figure 5-1 and Figure 5-2 present the reliability indices for the bridge decks in the database designed using a live load factor of 1.0 over a one year period for an ADTT of 5000. As indicated in Table 5-3, 89

the average values of the reliability index are 1.66 and 1.61 for positive and negative moment regions, respectively. Figure 5-1 Reliability Indices of Various Bridge Decks Designed Using a 1.0 Live Load Factor Over A 1 Year Return Period (ADTT=5000), Positive Moment Region, Class 1 Exposure Figure 5-2 Reliability Indices Of Various Bridge Decks Designed Using A 1.0 Live Load Factor Over A 1 Year Return Period (ADTT=5000), Negative Moment Region, Class 1 Exposure For Class 2 exposure condition (maximum crack width of 0.01275 in), Figure 5-3 and Figure 5-4 present the reliability indices for the bridge decks in the database designed using a live load factor of 1.0 over one year period for an ADTT of 5000. As indicated in Table 5-3, the 90

average values of the reliability index are 0.85 and 1.05 for positive and negative moment regions, respectively. Figure 5-3 Reliability Indices Of Various Bridge Decks Designed Using A 1.0 Live Load Factor Over A 1 Year Return Period (ADTT=5000), Positive Moment Region, Class 2 Exposure Figure 5-4 Reliability Indices Of Various Bridge Decks Designed Using A 1.0 Live Load Factor Over A 1 Year Return Period (ADTT=5000), Negative Moment Region, Class 2 Exposure As discussed in Section 5.1.2.4, for positive moment (bottom) reinforcement, Strength I limit state requirements typically result in more reinforcement than needed to satisfy Service I 91

requirements and the reliability index for cracking at the bottom will be higher than shown in Figure 5-1 and Figure 5-3. This resulted in the recommendation that the reliability index should be based on the negative moment (top) reinforcement. 5.1.3.9 Step 9: Calculate Reliability Indices As shown in Figure 5-2 and Figure 5-4, the reliability index associated with cracking at the top of the deck appears to be very uniform across the range of girder spacings considered. It was concluded that there was no need to redesign the decks for different load and/or resistance factors to improve the uniformity of the results. With this conclusion, the reliability indices are the same as shown in Table 5-3 and Table 5-4 and in Figure 5-2 and Figure 5-4. 5.1.3.10 Summary and Recommendations for Service I Limit State, Crack Control through the Distribution of Reinforcement The following conclusions are drawn based on the reported reliability analyses: • Assessment of current practice leads to recommended target reliability indices of 1.6 proposed for the base case (Class 1 exposure) and 1.0 for the enhanced requirements, i.e. smaller maximum crack width, for Class 2 exposure conditions. These values correspond to an ADTT of 5000. • The current requirements in the specifications produce uniform reliability across the range of girder spacing considered, so there is no need to change the load or the resistance factors. 5.1.4 Proposed AASHTO LRFD Revisions As indicated above, no revisions to applicable AASHTO LRFD provisions related to control of cracking by distributed reinforcement in reinforced concrete components are warranted by the results of this research. 5.2 Tension in Prestressed Concrete Beams Service III Limit State – Annual Probability Traditionally, prestressed concrete beams are proportioned for the service limit state such that the concrete tensile and compressive stresses immediately after transfer and at the final stage are within certain stress limits defined in the specifications. Under the current AASHTO LRFD Specifications (2012), two service limit state load combinations are used to calculate the stresses in prestressed concrete components: the Service Load I and Service Load III load combinations. The two service load combinations are described as: • Service I—Load combination relating to the normal operational use of the bridge with a 55 mph wind and all loads taken at their nominal values. Also related to deflection control in buried metal structures, tunnel liner plate, and thermoplastic pipe, to control crack width in reinforced concrete structures, and for transverse analysis relating to tension in concrete segmental girders. This load combination should also be used for the investigation of slope stability. • Service III—Load combination for longitudinal analysis relating to tension in prestressed concrete superstructures with the objective of crack control and to the principal tension in the webs of segmental concrete girders. 92

The load factors for DL and LL specified for the two load combinations are as follows: Service I: DL load factor = 1.0 LL load factor = 1.0 Service III: DL load factor = 1.0 LL load factor = 0.8 Based on the definition of the two limit states, Service I limit state is used for calculating all service stresses in the superstructure and substructure components at all stages except that Service III limit state is used to calculate the tensile stresses in the superstructure components under full service loads and the principal tension in webs of segmental concrete. Stresses immediately after transfer are independent of the live loads. At the final stage, typically the design is controlled by the tensile stress in the concrete and not by the compressive stresses on the opposite side of the girders. As such, the calibration for prestressed concrete superstructures was performed for Service III limit state and no calibration was performed for Service I limit state. In addition to designing prestressed concrete components for the service limit state, all prestressed components are also checked for the strength limit state. For typical precast prestressed superstructure beams, e.g. I-shapes, bulb tees and adjacent and spread box beams, the controlling case of the design is usually the service limit state. The service limit state stresses are calculated assuming an uncracked section. As such, the concrete is assumed to be subjected to tensile stresses. However, due to the relatively low load factors used for the service limit states, it is highly probable that the structure is subjected to heavy trucks that produce live load effects higher than those produced by the design factored service loads. When a heavy truck causes the tensile stress in the concrete to exceed the modulus of rupture, the concrete is expected to crack. Once the load passes, the prestressing force will cause the crack to close and it will remain closed as long as the concrete at the crack location remains under compression. However, if a truck heavy enough to cause the concrete stress calculated based on the uncracked section basis to be tensile, the crack will reopen. Successful past performance of prestressed concrete components suggests that past design requirements result in a frequency of the crack opening sufficiently small so as to not produce adverse strand fatigue problems at crack locations. 5.2.1 History of Major Relevant Design Provisions and Revisions to AASHTO LRFD Specifications 5.2.1.1 Load Factor for Live Load in Service III Load Combination During the early stages of the development of the AASHTO LRFD Specifications in the early 1990s, only Service I load combination was considered for calculating all stresses in prestressed concrete components. The load factor for live load was 1.0 which is the same load factor used for service loads under the AASHTO Standard Specifications; the predecessor to the AASHTO LRFD Specifications. The design live load specified in the AASHTO LRFD Specifications produces higher unfactored, undistributed load effects than that specified in the AASHTO Standard 93

Specifications. The girder distribution factors, particularly for interior girders, for many typical girder systems in the AASHTO LRFD Specifications are lower than that in the Standard Specifications thus reducing the difference between the unfactored distributed load effects in the two specifications. Even with the smaller distribution factor, the unfactored distributed load effects from the AASHTO LRFD Specifications were higher for most girder systems. Using the same load factor for service limit state (1.0) resulted in higher design factored load effects for the AASHTO LRFD designs than for those designed to the AASHTO Standard Specifications. The results from the trial designs conducted during the development of the AASHTO LRFD Specifications indicated a larger number of strands than required by the AASHTO Standard Specifications. This would suggest that designs performed under the AASHTO Standard Specifications resulted in under-designed components that should have shown signs of cracking. In the absence of widespread cracking, the load factor for live load was decreased to 0.8 and the Service III load combination was created and was specified for tension in prestressed concrete components. This resulted in a similar number of strands for the designs conducted using both AASHTO Standard and AASHTO LRFD Specifications. 5.2.1.2 Method of Calculating Prestressing Losses The AASHTO LRFD Bridge Design Specifications (2012) includes three methods for determining the time-dependent prestressing losses. These three methods are: • Approximate method: Currently, this method is termed: “Approximate Estimate of Time-Dependent Losses”. and is the least-detailed. It requires limited calculations to estimate the time-dependent losses. Prior to 2005, the specifications included a simpler approximate method which was termed: “Approximate Lump Sum Estimate of Time-Dependent Losses”. The lump-sum method allowed selecting a value for the time-dependent losses from a table. The value varied based on the type of girders and the type and grade of prestressing steel. Some concrete compressive strength requirements were required to be allowed to use this method. • Refined Estimates of Time-Dependent Losses: This method is more detailed than the approximate method. More details on this method are presented below. • Time-Step method: This method is highly detailed and is based on tracking the changes in the material properties with time. The loss calculations are based on the time of the application of loads and the material properties at the time of the load application. This method is required to be used in the design of post- tensioned segmental bridges. It may also be used for other types of bridges; however, due to the level of effort required, it is typically limited to segmental bridges. Throughout the remainder of Section 5.2, unless explicitly indicated otherwise, the time- dependent losses are calculated using the “Refined Estimates of Time-Dependent Losses” in AASHTO LRFD. Originally, the method of calculating prestressing force losses in AASHTO LRFD Specifications (the “pre-2005” method) was the same method used in AASHTO Standard Specifications. A new method of loss calculations (the “post-2005” method) first appeared in the 2005 Interim to the Third Edition of AASHTO LRFD Specifications. The post-2005 method is thought to produce a more accurate estimate of the losses. The post-2005 method has new equations for calculating the time-dependent prestressing losses and it also introduced the concept of “elastic gain.” After the initial prestressing loss at transfer, when load components 94

that produce tensile stresses in the concrete at the strand locations are applied to the girder, the strands are subjected to an additional tensile strain equal to the strain in the surrounding concrete due to the application of the loads. This results in an increase in the force in the strands. The increase in the force in the strands was termed “elastic gain” and the post-2005 prestressing loss method allows including the elastic gain to be used to offset some of the losses. When the “elastic gain” was considered, the post-2005 prestress loss method produced lower prestressing force losses than the earlier method. The reduction in prestressing losses resulted in fewer strands than what was required under the AASHTO Standard Specifications and under earlier editions of AASHTO LRFD Specifications. This raised some concern as some practitioners and researchers thought that the higher prestressing losses calculated using the pre-2005 loss method compensated for the lower live load effects caused by the lower design live load used in the AASHTO Standard Specifications or the lower load factor used for Service III load combination of AASHTO LRFD Specifications. Some of the work presented in the following sections was intended to investigate the effect of different loss methods and different design specifications on the reliability index for Service III load combination. 5.2.2 Live Load Model Traditionally, prestressed concrete components are designed for the number of traffic lanes, including multiple presence factors, that produces the highest load effects. This was assumed to continue in the future and all sections designed as part of this study utilized this approach. However, as indicated in Section 4.2.4, the presence of heavy loads in adjacent traffic lanes simultaneously is not likely. As such, the load side of the limit state function in the reliability analysis was calculated assuming the live load existed in only one lane and no multiple presence factor was included. The design truck, tandem, and uniform lane load specified in the AASHTO LRFD were used unless otherwise noted. The live load distribution factors specified in the AASHTO LRFD Specifications were used in distributing the design loads. The dynamic load allowance used in the original calibration of the strength limit state in AASHTO LRFD (10%) was applied to the load side. The return period considered in the calibration of the Service III limit state was one year. This return period was selected due to the fact that the live load statistics were developed based on 1 year of reliable WIM data from various WIM sites. Furthermore, since only 3 out of 32 WIM sites have an ADTT larger than 5000 and only 1 out of 32 WIM sites have an ADTT larger than 8000, an ADTT of 5000 was used for the bulk of the calibration. The bias and COV of live load were taken as shown in Table 5-5 through Table 5-9. 5.2.3 Methods of Analysis for Study Bridges Unless explicitly indicated otherwise, the methods of analysis used in designing and analyzing the study bridges throughout Section 5.2.4 through Section 5.2.6 are as follows: For bridges designed or analyzed using the post-2005 prestressing loss method: • The time-dependent prestressing loss method used is the method designated in the AASHTO LRFD (2012) as the “Refined Estimates of Time-Dependent Losses,” 95

• The section properties used in the analysis are based on the gross section of the concrete, and • The calculations of prestressing losses consider the effects of the “elastic gain” as allowed by the current design provisions. Regardless of the method of design used in designing a girder, the stresses in the girder used as part of the reliability index calculations were determined by analyzing the girder using the above assumptions. For bridges designed using the pre-2005 prestressing loss method: • The time-dependent prestressing loss method used is the method designated in the AASHTO LRFD editions prior to 2005 as the “Refined Estimates of Time- Dependent Losses,” • The section properties used in the analysis are based on the gross section of the concrete, and • The calculations neglect the effects of the “elastic gain.” 5.2.4 Target Reliability Index In the development of the AASHTO LRFD Specifications, the target reliability index for the strength limit states was 3.5. The limit state was assumed to be violated when the applied load effects exceeded the resistance which was in turn assumed to be equal to the design factored load. The definition of “failure” under the strength limit state is well defined as it relates to a certain criteria related to the properties of the materials used, such as steel yield stress or concrete compressive strength, or a behavior criteria where violation may lead to the instability of the component, such as local or global buckling. Due to the lack of clear consequences for violating the limiting stress specified for the concrete in a prestressed concrete component, selecting the limit state function required investigating different possible alternatives. 5.2.4.1 Limit State Functions Investigated The following three different limit state functions were investigated: • Decompression Limit State: This limit state assumes that the “failure” occurs when the stress in the concrete on the tension face calculated based on the uncracked section under the combined effect of factored dead load and live load ceases to be compression. • Stress Limit State: This limit state assumes that the “failure” occurs when the tensile stress in the concrete on the tension face calculated based on the uncracked section under the combined effect of factored dead load and live load exceeds a certain tensile stress limit calculated based on the uncracked section properties regardless of whether the section has been previously cracked or not. Stress limits of 0.0948t cf f ′= , 0.19t cf f ′= and 0.25t cf f ′= were initially considered in the reliability analysis, however, a stress limit of 0.19t cf f ′= was used for the final calibration. • Crack Width Limit State: This limit state assumes that the “failure” occurs when the previously formed crack in the concrete opens and the crack width reaches a certain pre-specified crack width. Crack widths of 0.008, 0.012, and 0.016 inches 96

were initially considered in the reliability analysis, however, none produced uniform reliability. The bulk of the calibration was performed using a crack width of 0.016 inches. The differentiation between different environments is accounted for in the calibration through the use of different reliability indices in association with the same crack width. For each girder, the design was performed based on certain stress limits as is conventionally done and the girder section and number of strands were determined. The reliability index was determined for each of the three limit state functions described above using the same girder design, i.e. the same girder section and same number of strands. Each of the limit state functions requires a different level of loading before the criteria is violated. As such, the frequency at which any of the three limit states is violated and the corresponding reliability index depend on the level of loading required to cause the limit state to be violated. For a specific cross-section with a specific prestressing steel area and force, reaching the decompression limit state requires less applied load than reaching a specified tensile stress which in turn requires less load than that required to reach a specific crack width. Requiring higher load to violate a specific limit state means that the section resistance is higher and this results in higher reliability index. Table 5-5 shows the required load and the corresponding reliability index for the three limit states relative to each other. Table 5-5 Relation Between Limiting Criteria and Reliability Index for a Given Girder Limiting Criteria Live Load required to violate the limiting criterion Frequency of exceeding the limiting criterion Reliability Index Decompression Lowest Highest Lowest Maximum allowable tensile stress limit Middle Middle Middle Maximum allowable crack width limit state Highest Lowest Highest With the target reliability index depending on the definition of the limit state, selecting the target reliability index required investigating all three criteria and selecting the one which provides more uniform reliability across a wide range of bridge geometrical characteristics. The process of calibration is illustrated for prestressed I-beams sections in Section 5.2.4.3 through Section 5.2.6. The final results for other types of sections are shown in Section 5.3.5. More details on the work on all types of sections are given in Appendix C and Appendix D. 5.2.4.2 Statistical Parameters of Variables Included in the Design Several variables affect the resistance of prestressed components. Table 5-6 shows a list of variables that were considered to be random variables during the performance of the reliability analyses. These variables represent a summary of the information based on previous research studies by Siriaksorn and Naaman (1980) and Nowak, et al. (2008). 97

Table 5-6 Random Variables and the Value of Their Statistical Parameters Variables Distribution Mean, μ COV, Ω Remarks As normal 0.9Asn* 0.015 Siriaksorn and Naaman (1980) Aps normal 1.01176Apsn 0.0125 Siriaksorn and Naaman (1980) b, b0, b1, bw normal bn 0.04 Siriaksorn and Naaman (1980) CEc normal 33.6 0.1217 Siriaksorn and Naaman (1980), nominal=33 ( )′= ⋅1.5/ γcE c c cC E f Cfci normal 0.6445 0.073 nominal=0.8 Cfci = fci / f′c dp, ds normal dpn, dsn 0.04 Siriaksorn and Naaman (1980) e1 normal e0n 0.04 Siriaksorn and Naaman (1980) Eps normal 1.011Epsn 0.01 Siriaksorn and Naaman (1980) Epsn = 29000 ksi Es normal Esn 0.024 Siriaksorn and Naaman (1980) f′c lognormal 1.11f′cn 0.11 Nowak (2008) fpu lognormal 1.03fpun 0.015 Nowak (2008) fpun = 270 ksi fsi normal 0.97fsin 0.08 developed based on Gross and Burns (2000) fy lognormal 1.13fyn 0.03 Nowak (2008) h, hf, hf1, hf2 normal hn, hfn, hf1n, hf2n 0.025 Siriaksorn and Naaman (1980) l normal ln 11 / (32µ) Siriaksorn and Naaman (1980) γc normal γcn = 150 0.03 Siriaksorn and Naaman (1980) Δfs normal 1.05Δfsn 0.10 developed based on Gross and Burns (2000) and Tadros, et al. (2003) Σ0 normal Σ0n 0.03 Siriaksorn and Naaman (1980) *Subscript of “n” refers to nominal values Notations: As = area of non-prestressing steel, in2 Aps = area of prestressing steel in tension zone, in2 b = prestressed beam top flange width, in. b0 = deck width transformed to the beam material, in. b1 = prestressed beam bottom flange width, in. bw = web thickness, in. c = depth of neutral axis from the extreme compression fiber, in Cfci = fci / f′c dp = distance from extreme compression fiber to centroid of prestressing steel, in. ds = distance from extreme compression fiber to centroid of non-prestressing steel, in. e1 = eccentricity of the prestressing force with respect to the centroid of the section at mid-span, in. Eps = modulus of elasticity of prestressing steel, psi Es = modulus of elasticity of non-prestressing steel, psi 98

f′c = specified compressive strength of concrete, psi fpu = specified tensile strength of prestressing steel, psi fsi = initial stress in prestressing steel, psi fy = yield strength of non-prestressing steel, psi h = girder depth, in. hf = deck thickness, in. hf1 = top flange thickness, in. hf2 = bottom flange thickness, in. l = clear span length of the beam members, ft. γc = unit weight of concrete, pcf Σ0 = sum of reinforcing element circumferences, in. Δfs = prestress losses, psi 5.2.4.3 Database of Existing Bridges A database of existing prestressed concrete girder bridges was extracted from the database of bridges used in the NCHRP 12-78 project (Mlynarski, et al. 2011). The database used in this study included 30 I- and bulb-T girder bridges, 31 adjacent box girder bridges, and 36 spread box girder bridges. The geometric characteristics of the bridges are included in Appendix C. Depending on the environmental exposure conditions, both the AASHTO Standard Specifications and the AASHTO LRFD Specifications allow designing conventional prestressed components for maximum concrete tensile stress of 0.0948t cf f ′= or 0.19t cf f ′= for severe corrosion conditions or no worse than moderate corrosion conditions, respectively. When either specifications are applied without owner’s exceptions, most bridges are designed for 0.19t cf f ′= with a small number of bridges in coastal areas designed for 0.0948t cf f ′= . It was not known what stress limit each bridge in the database was designed for. As the percentage of bridges designed for severe corrosive conditions is small, it was assumed that most bridges in the database were likely designed for the higher limit. Based on the construction date of these bridges, it is likely that all existing bridges considered were designed using the prestressing loss provisions method that existed in both the AASHTO Standard Specifications and in the pre-2005 AASHTO LRFD. The database of existing bridges was used to estimate the reliability index inherent in the existing bridge system and used this as a starting point for the calibration. 5.2.4.4 Estimated Reliability Index of Existing Bridges Table 5-7 summarizes the average reliability indices for the existing I- and bulb T girder bridges database. For example, the average reliability indices at decompression level, maximum allowable tensile stress limit under service loads of 0.19t cf f ′= , and maximum allowable crack width limit of 0.016 inches are 0.74, 1.05, and 2.69, respectively, for an ADTT of 5000 and a return period of one year. 99

Table 5-7 Summary of Reliability Indices for Existing I- and Bulb T Girder Bridges with One Lane Loaded and Return Period of 1 Year Performance Levels ADTT ADTT=1000 ADTT=2500 ADTT=5000 ADTT=10000 Decompression 0.95 0.85 0.74 0.61 Maximum Tensile Stress Limit 0.0948t cf f ′= 1.15 1.01 0.94 0.82 0.19t cf f ′= 1.24 1.14 1.05 0.95 0.25t cf f ′= 1.40 1.27 1.19 1.07 Maximum Crack Width 0.008 in 2.29 2.21 1.99 1.85 0.012 in 2.65 2.60 2.37 2.22 0.016 in 3.06 2.89 2.69 2.56 5.2.4.5 Database of Simulated Bridges A database of simulated simple span bridges was designed using AASHTO I-girder sections for four different cases. The simulated bridges have span lengths of 30, 60, 80, 100, and 140 ft. and girder spacing of 6, 8, 10, and 12 ft. This database was analyzed to determine the effect of the change in the method of estimating prestressing losses (pre-2005 and post- 2005 methods) and the design environment (“severe corrosive conditions” and “normal” or “not worse than moderate corrosion conditions”). The two environmental conditions are signified by the maximum concrete tensile stress limit ( 0.0948t cf f ′= or 0.19t cf f ′= ) used in the design. The four cases of design considered were: Case 1: AASHTO LRFD with maximum concrete tensile stress of 0.0948t cf f ′= and pre-2005 prestress loss method Case 2: AASHTO LRFD with maximum concrete tensile stress of 0.0948t cf f ′= and post-2005 prestress loss method Case 3: AASHTO LRFD with maximum concrete tensile stress of 0.19t cf f ′= and pre-2005 prestress loss method Case 4: AASHTO LRFD with maximum concrete tensile stress of 0.19t cf f ′= and post-2005 prestress loss method Table 5-8 and Table 5-9 show the span length and girder spacing along with the calculated reliability indices for I-girder bridges designed for maximum concrete tensile stress 0.0948t cf f ′= (Case 1 and Case 2) and 0.19t cf f ′= (Case 3 and Case 4), respectively, for ADTT=5000. In performing the design, the cases using post-2005 prestress loss method (Case 2 and Case 4) were designed using the smallest possible AASHTO girder size. To facilitate the comparisons, where possible, Case 1 and Case 3 were then designed using the same AASHTO section used for Case 2 and Case 4, respectively. For the cases where the section used for Case 2 or Case 4 was too small to be used for the corresponding Case 1 or Case 3, no design 100

is shown in Table 5-8 and Table 5-9 for Case 1 and Case 3. For the 140 ft. span bridges with 12 ft. girder spacing, no AASHTO I-girder section was sufficient. Bridges designed for Case 1 and Case 3 are also thought to be similar to those designed using AASHTO Standard specifications for the two environmental conditions. The reliability indices calculated for Case 1 and Case 3 represent the inherent reliability of bridges currently on the system as most of them were designed before 2005. Case 2 and Case 4 generally represent the inherent reliability of newer bridges designed using the 2005 and later versions of AASHTO LRFD for severe and normal environmental conditions, respectively. Comparing Case 1 to Case 2 and Case 3 to Case 4 shows the effect of changing the prestressing loss method. Using the post-2005 prestress loss method resulted in smaller number of strands than the pre-2005 loss method. As shown in Table 5-8 and Table 5-9, the lower number of strands resulted in lower reliability index for bridges designed using the post-2005 prestress loss method. As shown in Table 5-8 and Table 5-9, regardless of the loss method and/or the limit state used, the reliability indices for each case varied significantly. This suggested the need to calibrate the limit state to develop a combination of load and resistance factors that produce a more uniform reliability index across the range of different span lengths and girder spacings. 101

Table 5-8 Summary of the Reliability Indices of Simulated Bridges Designed Using AASHTO Girders with ADTT=5000 and 0.0948t cf f ′= Case 1 Case 2 Cases Section Type Span Length (ft.) Spacin g (ft.) Designed Using Pre-2005 Loss Method Designed Using Post-2005 Loss Method Decomp. Max. Tensile Max. Crack Decomp. Max. Tensile Max. Crack 1 AASHTO I 30 6 1.05 1.49 2.92 1.03 1.51 2.55 2 AASHTO I 30 8 0.90 0.94 2.41 0.93 1.00 2.32 3 AASHTO I 30 10 1.16 1.68 2.87 1.28 1.67 2.82 4 AASHTO I 30 12 1.28 1.67 2.91 0.63 0.97 2.29 Average for 30 ft. Span 1.10 1.45 2.78 0.97 1.29 2.50 5 AASHTO II 60 6 0.66 1.01 3.35 0.23 0.61 2.47 6 AASHTO II 60 8 — — — 0.73 1.04 2.42 7 AASHTO III 60 10 1.22 1.62 3.01 0.43 0.76 1.97 8 AASHTO III 60 12 1.57 1.96 3.68 0.73 0.99 2.51 Average for 60 ft. Span 1.15 1.53 3.35 0.53 0.85 2.34 9 AASHTO III 80 6 1.35 1.66 4.1 0.61 0.92 3.07 10 AASHTO III 80 8 1.8 2.14 5.23 0.82 1.13 3.64 11 AASHTO III 80 10 — — — 0.90 1.19 2.93 12 AASHTO IV 80 12 2.2 2.49 5.11 0.83 1.17 3.32 Average for 80 ft. Span 1.78 2.10 4.81 0.79 1.10 3.24 13 AASHTO III 100 6 — — — 1.45 1.85 3.51 14 AASHTO IV 100 8 1.86 2.00 3.86 1.33 1.43 3.44 15 AASHTO IV 100 10 — — — 1.33 1.65 3.37 16 AASHTO V 100 12 1.68 1.99 4.08 0.93 1.24 3.33 Average for 100 ft. Span 1.77 2.00 3.97 1.26 1.54 3.41 17 AASHTO IV 120 6 — — — 1.32 1.76 3.81 18 AASHTO V 120 8 1.54 2.05 3.65 0.92 1.4 3.14 19 AASHTO V 120 10 — — — 0.95 1.46 3.02 20 AASHTO VI 120 12 1.82 2.26 3.88 0.9 1.35 3.38 Average for 120 ft. Span 1.68 2.16 3.77 1.02 1.49 3.34 21 AASHTO VI 140 6 1.48 1.99 3.91 0.86 1.36 2.32 22 AASHTO VI 140 8 — — — 0.99 1.47 2.79 23 AASHTO VI 140 10 — — — 1.05 1.53 3.22 24 — 140 12 — — — — — — Average for 140 ft. Span 1.48 1.99 3.91 0.97 1.45 2.78 Average for All Spans 1.44 1.80 3.66 0.92 1.28 2.94 102

Table 5-9 Summary of the Reliability Indices of Simulated Bridges Designed Using AASHTO Girders with ADTT=5000 and 0.19t cf f ′= Case 3 Case 4 Cases Section Type Span Length (ft.) Spacing (ft.) Designed Using Pre-2005 Loss Method Designed Using Post-2005 Loss Method Decomp. Max. Tensile Max. Crack Decomp. Max. Tensile Max. Crack 1 AASHTO I 30 6 1.00 1.55 2.39 0.97 1.55 2.46 2 AASHTO I 30 8 0.94 0.92 2.35 0.91 1.00 2.16 3 AASHTO I 30 10 1.29 1.66 2.91 1.18 1.66 2.79 4 AASHTO I 30 12 1.30 1.72 3.02 1.26 1.70 2.91 Average for 30 ft. Span 1.13 1.46 2.67 1.08 1.48 2.58 5 AASHTO II 60 6 0.74 1.13 3.11 0.18 0.58 2.41 6 AASHTO II 60 8 1.04 1.39 2.82 0.28 0.66 1.91 7 AASHTO III 60 10 0.42 0.79 2.05 0.42 0.78 2.07 8 AASHTO III 60 12 0.66 1.00 2.5 0.68 0.96 2.53 Average for 60 ft. Span 0.72 1.08 2.62 0.39 0.75 2.23 9 AASHTO III 80 6 0.56 0.97 3.13 0.13 0.51 2.53 10 AASHTO III 80 8 1.06 1.46 3.43 0.42 0.78 3.2 11 AASHTO III 80 10 1.58 1.84 3.65 0.37 0.65 2.72 12 AASHTO IV 80 12 0.83 1.15 3.72 0.51 0.87 3.11 Average for 80 ft. Span 1.01 1.36 3.48 0.36 0.70 2.89 13 AASHTO III 100 6 — — — 0.82 1.23 3.44 14 AASHTO IV 100 8 1.31 1.42 3.60 0.69 0.76 2.76 15 AASHTO IV 100 10 1.80 1.98 3.67 0.75 1.04 3.12 16 AASHTO V 100 12 1.08 1.37 3.43 0.40 0.72 2.55 Average for 100 ft. Span 1.40 1.59 3.57 0.67 0.94 2.97 17 AASHTO IV 120 6 1.53 1.98 3.71 0.70 1.28 3.10 18 AASHTO V 120 8 0.90 1.30 3.31 0.46 0.85 2.55 19 AASHTO V 120 10 1.25 1.65 3.35 0.26 0.78 2.68 20 AASHTO VI 120 12 1.19 1.66 3.37 0.47 0.91 2.69 Average for 120 ft. Span 1.22 1.65 3.44 0.47 0.96 2.76 21 AASHTO VI 140 6 0.84 1.41 3.23 0.28 0.82 2.41 22 AASHTO VI 140 8 1.22 1.68 3.30 0.53 0.98 3.04 23 AASHTO VI 140 10 — — — 0.62 1.08 2.46 24 — 140 12 — — — — — — Average for 140 ft. Span 1.03 1.55 3.27 0.48 0.96 2.64 Average for All Spans 1.07 1.43 3.15 0.58 0.96 2.68 103

5.2.4.6 Selection of the Target Reliability Index The target reliability indices were selected based on the calculated average values of the reliability levels of existing bridges and previous practices with some consideration given to experiences from other Codes (Eurocode and International Organization for Standardization (ISO) 2394 Document). As indicated earlier, a return period of 1 year was selected and an ADTT equal to 5000 was used. Table 5-10 shows the target reliability indices selected in this study as well as the reliability indices for the existing and simulated bridge databases. Notice that the environmental condition for existing bridges was not known and that the two columns showing the reliability indices of the simulated bridges are for cases where the pre-2005 prestressing loss method was used as these are thought to better represent the bridges currently on the system. For example, the reliability index of existing bridges, simulated bridges designed for severe environments, and simulated bridges designed for normal environments, at the decompression performance level is around 0.74, 1.44 and 1.07, respectively (See Table 5-7 through Table 5-10). Therefore, a target reliability index of 1.2 and 1.0 was selected for the decompression performance level for bridges designed for severe environments and bridges designed for normal environments, respectively. The reliability index of 1.0 means that 15 out of 100 bridges will probably have the bottom of the girder decompress in any given year. Table 5-10 Reliability Indices for Existing and Simulated Bridges (Return Period of 1 Year and ADTT 5000) 5.2.5 Calibration Result The basic steps of the calibration process are shown below as they relate to the Service III calibration. 5.2.5.1 Step 1: Formulate the Limit State Function and Identify Basic Variables The three limit state functions that were investigated are listed in Section 5.2.4.1. The limit state function is formulated by deriving an expression for the resistance prediction Performance Level Reliability Index Average β for Existing Bridges in the NCHRP 12-78 Average β for Simulated bridges designed for 0.0948t cf f ′= and pre-2005 loss method Average β for Simulated bridges designed for 0.19t cf f ′= and pre-2005 loss method Proposed Target β for bridges in severe environment Proposed Target β for bridges in normal environment Decompression 0.74 1.44 1.07 1.20 1.00 Maximum Allowable Tensile Stress of 0.19t cf f ′= 1.05 1.80 1.43 1.50 1.25 Maximum Allowable Crack Width of 0.016 in. 2.69 3.68 3.15 3.30 3.10 104

equation. For the decompression and tensile stress limits, the stress in the concrete is calculated as it is usually done for the design of prestressed concrete components. For the crack width limit state, Appendix E presents a detailed derivation of the resistance prediction equation for a typical prestressed concrete bridge girder. The derived equation considers uncracked and cracked section behavior in a general format by including the crack width equation. In lieu of setting the stress to zero, the resistance for the decompression limit state can also be derived by setting the crack width to zero in the general equation for crack width. The majority of the equations for the prediction of the maximum crack width are given in terms of the stress in the steel. Various maximum crack width prediction equations were evaluated using test data available in the literature. Appendix F presents a comparison and evaluation of maximum crack width prediction equations for prestressed concrete members. 5.2.5.2 Step 2: Identify and Select Representative Structural Types and Design Cases Various design cases for span lengths ranging from 30 to 140 ft. were designed as shown in Section 5.2.4.5. For maximum crack width limit state, a crack width of 0.016 in. is considered. For the maximum allowable stress limit state, the stress considered is as stated in the discussion included in the following sections. 5.2.5.3 Step 3: Determine Load and Resistance Parameters for the Selected Design Cases The variables include the dimension of the cross-section and the material properties. The statistical information includes the probability distribution and statistical parameters such as mean, µ, and standard deviation, σ. 5.2.5.4 Step 4: Develop Statistical Models for Load and Resistance The variables affecting the load and resistance were identified. These include live load; those affecting resistance include the dimensions of the cross-section, the material properties, etc. The statistical information includes the probability distribution and statistical parameters for live load presented in Section 4.3.2 and for other variables affecting the resistance presented in Section 5.2.4.2. 5.2.5.5 Step 5: Develop the Reliability Analysis Procedure The statistical information of all the required variables is used to determine the statistical parameters of the resistance by using Monte Carlo simulation. Monte Carlo simulation is useful in generating a large number of random cases that are used in defining the mean and standard deviation of the resistance. For each girder, Monte Carlo simulation was performed for each random variable associated with calculation of the resistance and dead load. One thousand simulations were performed. For each random variable, 1000 values were generated independently based on the statistics and distribution of that random variable. For each simulation, the dead load and the resistance were calculated using one of the 1000 sets of values of each random variable resulting in 1000 values of the dead load and the resistance. The mean and standard deviation of the dead load and the resistance were then calculated based on the 1000 simulations. 105

5.2.5.6 Step 6: Calculate the Reliability Indices for Current Design Code and Current Practice Using the statistics of the dead load and the resistance, calculated from Monte Carlo simulation as described above, and the statistics of the live load as derived from the WIM data as described in Section 4, the reliability index was calculated for each girder. The reliability index was calculated using the following equation: 2 2 R Q R Q µ − µ β = σ + σ (5-2) where β = reliability Index Rµ = mean value of the resistance Qµ = mean value of the applied loads Rσ = standard deviation of the resistance Qσ = standard deviation of the applied loads The calculated reliability indices of existing and simulated bridges are shown in Table 5-7 through Table 5-9. 5.2.5.7 Step 7: Review the Results and Select the Target Reliability Index βT The initial target reliability index was determined as shown in Table 5-10. 5.2.5.8 Step 8: Select Potential Load and Resistance Factors for Service III For all steps, the resistance factor was assumed to be the same as in the current AASHTO LRFD Specifications (2012), i.e. equal to 1.0. The Service III limit state resistance is affected by the tensile stress limit used in the design. Therefore, in addition to trying different load factors, different stress limits for the design were also investigated. Maximum concrete design tensile stress of 0.0948t cf f ′= , 0.19t cf f ′= and 0.25t cf f ′= were considered. In addition, the simulated bridge database used in determining the target resistance factor was further expanded to allow longer spans. Due to having three different concrete tensile stress limits, Step 8 is repeated three times below and are designated 8a, 8b, and 8c. For this step, the range of span lengths was increased to 220 feet. 106

5.2.5.8.1 Step 8a: Select Potential Load and Resistance Factors for Service III - Bridges Designed for Maximum Concrete Tensile Stress of 0.0948t cf f ′= In this section, the calibration for a selected bridge database (shown in Table 5-11) was performed assuming an ADTT of 5000 and maximum concrete design tensile stress of 0.0948t cf f ′= . 1. Calculate the reliability level of designs according to AASHTO LRFD Specifications (2012) (Figure 5-5 through Figure 5-7) Figure 5-5 through Figure 5-7 show the reliability indices for the bridges designed using AASHTO-type girders according to AASHTO LRFD Specifications (2012), including a load factor of 0.8 for Service III limit state, and assuming a maximum concrete tensile stress of 0.0948t cf f ′= . The geometric characteristics of the bridges are shown in Table 5-11. It was observed that the average reliability index for the decompression limit state, maximum allowable tensile stress limit state, and maximum allowable crack width limit state are 0.97, 1.31, and 3.06, respectively. Since the reliability indices are lower than the target reliability indices and that the reliability indices are not uniform across different spans, modifications to the load factor are applied in the next step in an attempt to achieve higher, and more uniform, reliability indices. 107

Table 5-11 Summary Information of Bridges Designed with γLL=0.8, ( 0.0948t cf f ′= ) Cases Section Type Span Length (ft.) Girder Spacin g (ft.) Aps (in2) # of Strands 1 AASHTO I 30 6 1.224 8 2 AASHTO I 30 8 1.530 10 3 AASHTO I 30 10 1.836 12 4 AASHTO I 30 12 2.142 14 5 AASHTO II 60 6 2.448 16 6 AASHTO II 60 8 3.366 22 7 AASHTO III 60 10 3.060 20 8 AASHTO III 60 12 3.672 24 9 AASHTO III 80 6 3.672 24 10 AASHTO III 80 8 4.590 30 11 AASHTO III 80 10 5.508 36 12 AASHTO IV 80 12 5.202 34 13 AASHTO III 100 6 6.120 40 14 AASHTO IV 100 8 6.426 42 15 AASHTO IV 100 10 7.344 48 16 AASHTO V 100 12 7.038 46 17 AASHTO IV 120 6 7.956 52 18 AASHTO V 120 8 7.956 52 19 AASHTO V 120 10 9.180 60 20 AASHTO VI 120 12 8.874 58 21 AASHTO VI 140 6 8.262 54 22 AASHTO VI 140 8 9.792 64 23 AASHTO VI 140 10 11.322 74 24 AASHTO VI 140 12 - - 25 FIB-96 160 6 5.508 36 26 FIB-96 160 8 6.426 42 27 FIB-96 160 10 7.344 48 28 FIB-96 160 12 - - 29 FIB-96 180 6 7.344 48 30 Mod. BT-72 180 9 16.218 106 31 Mod. AASHTO VI 180 9 15.912 104 32 Mod. AASHTO VI 200 9 20.502 134 33 Mod. NEBT-2200 200 9 16.830 110 34 Mod. W95PTMG 200 9 16.830 110 35 Mod. NEBT-2200 220 9 20.808 136 108

Figure 5-5 Reliability indices for bridges at decompression limit state (ADTT=5000), γLL=0.8, ( 0.0948t cf f ′= ). Figure 5-6 Reliability indices for bridges at maximum allowable tensile stress limit state (ADTT=5000), γLL=0.8, ( 0.0948t cf f ′= ). 109

Figure 5-7 Reliability Indices for bridges at maximum allowable crack width limit state (ADTT=5000), γLL=0.8, ( 0.0948t cf f ′= ). 2. Redesign the bridges with live load factor of 1.0 In this step, the bridges have been redesigned using a live load factor of 1.0 and the dead load and resistance factors were kept the same during the redesign. Table 5-12 shows the design geometric characteristics of the redesigned bridges. Figure 5-8 through Figure 5-10 show the reliability indices for the redesigned bridges using a live load factor of 1.0. The average reliability index for the decompression limit state, the maximum allowable tensile stress limit state, and the maximum allowable crack width limit state are 1.33, 1.70, and 3.32, respectively. It was observed that the reliability level of bridges became more uniform than for the case of using a live load factor of 0.8, particularly for the decompression and maximum tensile stress limit states. Therefore, a live load factor of 1.0 was proposed to be used if the tensile stress is limited to 0.0948t cf f ′= . 110

Table 5-12 Summary Information of Bridges Designed with γLL=1.0, ( 0.0948t cf f ′= ) Section Type Span Length (ft.) Girder Spacing (ft.) Aps (in2) # of Strands 1 AASHTO I 30 6 1.224 8 2 AASHTO I 30 8 1.530 10 3 AASHTO I 30 10 1.836 12 4 AASHTO I 30 12 2.142 14 5 AASHTO II 60 6 3.06 20 6 AASHTO II 60 8 3.978 26 7 AASHTO III 60 10 3.366 22 8 AASHTO III 60 12 4.284 28 9 AASHTO III 80 6 4.284 28 10 AASHTO III 80 8 5.202 34 11 AASHTO III 80 10 6.120 40 12 AASHTO IV 80 12 5.814 38 13 AASHTO III 100 6 7.038 46 14 AASHTO IV 100 8 7.038 46 15 AASHTO IV 100 10 8.262 54 16 AASHTO V 100 12 7.650 50 17 AASHTO IV 120 6 8.874 58 18 AASHTO V 120 8 8.874 58 19 AASHTO V 120 10 10.404 68 20 AASHTO VI 120 12 9.792 64 21 AASHTO VI 140 6 8.874 58 22 AASHTO VI 140 8 10.710 70 23 AASHTO VI 140 10 - - 24 AASHTO VI 140 12 - - 25 FIB-96 160 6 5.814 38 26 FIB-96 160 8 7.344 48 27 FIB-96 160 10 7.956 52 28 FIB-96 160 12 - - 29 FIB-96 180 6 7.956 52 30 Mod. BT-72 180 9 17.442 114 31 Mod. AASHTO VI 180 9 17.442 114 32 Mod. AASHTO 200 9 22.032 144 33 Mod. NEBT-2200 200 9 18.360 120 34 Mod. W95PTMG 200 9 18.360 120 35 Mod. NEBT- 220 9 22.338 146 111

Figure 5-8 Reliability indices for bridges at decompression limit state (ADTT=5000), γLL=1.0 ( 0.0948t cf f ′= ). Figure 5-9 Reliability indices for bridges at maximum allowable tensile stress limit state (ADTT=5000), γLL=1.0 ( 0.0948t cf f ′= ). 112

Figure 5-10 Reliability indices for bridges at maximum allowable crack width limit state (ADTT=5000), γLL=1.0 ( 0.0948t cf f ′= ). 5.2.5.8.2 Step 8b: Select Potential Load and Resistance Factors for Service III - Bridges Designed for Maximum Concrete Tensile Stress of 0.19t cf f ′= In this section, the work described under Step 8a above was repeated except that the girders were redesigned assuming maximum concrete tensile stress of 0.19t cf f ′= . 1. Calculate the reliability level of designs according to AASHTO LRFD Specifications (2012) with maximum concrete tensile stress for design 0.19t cf f ′= (Figure 5-11 through Figure 5-13). Figure 5-11 Reliability indices for bridges at decompression limit state (ADTT=5000), γLL=0.8 ( 0.19t cf f ′= ). 113

Figure 5-12 Reliability indices for bridges at maximum allowable tensile stress limit state (ADTT=5000), γLL=0.8 ( 0.19t cf f ′= ). Figure 5-13 Reliability indices for bridges at maximum allowable crack width limit state (ADTT=5000), γLL=0.8 ( 0.19t cf f ′= ). 2. Redesign the bridges with live load factor of 1.0 Figure 5-14 through Figure 5-16 show the reliability indices for the redesigned bridges using a live load factor of 1.0 and 0.19t cf f ′= . Similar to bridges designed for maximum concrete tensile stress of 0.0948t cf f ′= , it was observed that the reliability level of bridges became more uniform than the case of using a live load factor of 0.8, particularly for the decompression and maximum tensile stress limit states. Therefore, a live load factor of 1.0 was proposed to be used if the maximum tensile stress is limited to 0.19 cf ′ . 114

Figure 5-14 Reliability indices for bridges at decompression limit state (ADTT=5000), γLL=1.0 ( 0.19t cf f ′= ). Figure 5-15 Reliability indices for bridges at maximum tensile stress limit state (ADTT=5000), γLL=1.0 ( 0.19t cf f ′= ). 115

Figure 5-16 Reliability indices for bridges at maximum crack width limit state (ADTT=5000), γLL=1.0 ( 0.19t cf f ′= ). 5.2.5.8.3 Step 8c: Select Potential Load and Resistance Factors for Service III – Bridges Designed for Maximum Concrete Tensile Stress of 0.25t cf f ′= In this section, the work described under Step 8a and Step 8b above was repeated except that the girders were redesigned assuming maximum concrete tensile stress of 0.25t cf f ′= . 1. Calculate the reliability level of designs according to AASHTO LRFD Specifications (2010) with maximum concrete tensile stress for design 0.25t cf f ′= (Figure 5-17 through Figure 5-19. Figure 5-17 Reliability indices for bridges at decompression limit state (ADTT=5000), γLL=0.8 ( 0.25t cf f ′= ). 116

Figure 5-18 Reliability indices for bridges at maximum allowable tensile stress limit state (ADTT=5000), γLL=0.8 ( 0.25t cf f ′= ). Figure 5-19 Reliability indices for bridges at maximum allowable crack width limit state (ADTT=5000), γLL=0.8 ( 0.25t cf f ′= ). 2. Redesign the bridges with live load factor of 1.0 Figure 5-20 through Figure 5-22 show the reliability indices for the redesigned bridges using a live load factor of 1.0 and 0.25t cf f ′= . Similar to bridges designed for maximum concrete tensile stress of 0.0948t cf f ′= and 0.16t cf f ′= , it was observed that the reliability level of bridges became more uniform than the case of using a live load factor of 0.8, particularly for the decompression and maximum tensile stress limit states. Therefore, a live load factor of 1.0 was proposed to be used if the maximum tensile stress is limited to 0.25t cf f ′= . 117

Figure 5-20 Reliability indices for bridges at decompression limit state (ADTT=5000), γLL=1.0 ( 0.25t cf f ′= ). Figure 5-21 Reliability indices for bridges at maximum tensile stress limit state (ADTT=5000), γLL=1.0 ( 0.25t cf f ′= ). 118

Figure 5-22 Reliability indices for bridges at maximum crack width limit state (ADTT=5000), γLL=1.0 ( 0.25t cf f ′= ). 5.2.5.9 Step 9: Calculate Reliability Indices The reliability indices were calculated for three different cases as shown in Step 8 above. In Step 9, the calculated values were reviewed to determine whether they are close to the target reliability index and whether they are uniform across the range of spans considered. If they are not, the load factors, resistance factors, and/or the concrete tensile stress limit used for design would need to be changed and Step 8 would be repeated. The limit state function to be used as the basis for the calibration is also determined in Step 9. 5.2.5.10 Summary of Target Reliability Indices for Different Design and Performance Levels A summary of the average reliability indices calculated for the different cases is given in Table 5-13 through Table 5-15. Regardless of the maximum tensile stress limit used in the design, the limiting criteria for the maximum tensile stress when determining the reliability index was taken as 0.19t cf f ′= . Table 5-13 Summary of Reliability Indices for Simulated Bridges Designed for 0.0948t cf f ′= ADTT Live Load Factor=0.8 Live Load Factor=1.0 Decompression Max Tensile Stress Limit Crack Width Decompression Max Tensile Stress Limit Crack Width 1000 1.05 1.41 3.16 1.42 1.79 3.36 2500 1.01 1.35 3.11 1.38 1.75 3.33 5000 0.97 1.31 3.06 1.33 1.70 3.32 10000 0.94 1.30 3.00 1.32 1.66 3.28 119

Table 5-14 Summary of Reliability Indices for Simulated Bridges Designed for 0.19t cf f ′= ADTT Live Load Factor=0.8 Live Load Factor=1.0 Decompression Max Tensile Stress Limit Crack Width Decompression Max Tensile Stress Limit Crack Width 1000 0.84 1.27 2.92 1.11 1.53 3.25 2500 0.70 1.15 2.87 1.04 1.46 3.17 5000 0.68 1.10 2.82 1.00 1.41 3.14 10000 0.64 1.07 2.78 0.98 1.34 3.11 Table 5-15 Summary of Reliability Indices for Simulated Bridges Designed for 0.25t cf f ′= ADTT Live Load Factor=0.8 Live Load Factor=1.0 Decompression Max Tensile Stress Limit Crack Width Decompression Max Tensile Stress Limit Crack Width 1000 0.20 0.55 2.83 0.93 1.29 3.03 2500 0.08 0.49 2.77 0.89 1.27 2.95 5000 0.06 0.44 2.72 0.85 1.23 2.92 10000 0.02 0.41 2.66 0.82 1.20 2.88 As indicated earlier, the calibration of the specifications are based on an ADTT of 5000. It was observed that for this ADTT, the reliability indices obtained assuming the bridges are designed for maximum stress limit of 0.0948t cf f ′= and 0.19t cf f ′= (see the outlined cells Table 5-13 and Table 5-14) are very close to the target reliability indices shown in Table 5-10. 5.2.5.11 Effect of Proposed Changes on Design To investigate the effect of the proposed change in the load factor, the number of strands required for different design cases was compared (see Table 5-16). The comparison indicated that when a live load factor of 0.8 is used in both cases, the post-2005 prestress loss method results in smaller number of strands than when the pre-2005 prestress loss method is used. It also indicated that when the post-2005 loss method is used with a load factor of 1.0, the required number of strands is similar to that required when a load factor of 0.8 is used in conjunction with the pre-2005 prestress loss method, i.e. designs similar between pre-2005 and post-2005 methods. 120

5.2.5.12 Summary and Recommendations for Service III Limit State For typical I-girders designed using the post-2005 prestress loss method and the assumptions listed in Section 5.2.3, comparing the target reliability indices shown in Table 5-10 and the calculated reliability indices for different design criteria, load factors, and design live load as shown in Table 5-13 through Table 5-15 and Figure 5-5 through Figure 5-22, the following conclusions were drawn and summarized: 1. For a specific girder of known cross-section and specific number and arrangement of prestressing strands, the reliability index varies based on: • The design maximum concrete tensile stress (a maximum tensile stress of 0.0948t cf f ′= and 0.19t cf f ′= is currently shown in AASHTO LRFD (2012) and is proposed to remain the same), • The limit state function, i.e. decompression, tensile stress of a certain value (assumed to be 0.19t cf f ′= in the work shown above), or a crack width of a certain value (assumed to be 0.016 in.), and • ADTT. The effect of different factors can be deduced from Table 5-13 through Table 5-15. 2. The target reliability index can be achieved uniformly across various span lengths using the load factor developed following the proposed calibration procedure. The level of uniformity varies with the limiting criteria. The decompression limit state showed the highest level of uniformity and is recommended to be used as the basis for the reliability analysis, i.e. the determination of the load and resistance factors and associated design criteria. 3. It is recommended that the reliability indices corresponding to ADTT of 5000 be used as the basis for the calibration. The reliability index is not highly sensitive to changes in the ADTT so there is no need to use different load factor for ADTTs up to 10000. 121

Table 5-16 Comparison of number of strands required for different design assumptions Cases Section Type Span Length (ft.) Girder Spacing (ft.) 0.0948t cf f ′= , γLL=0.8, Pre-2005 losses 0.0948t cf f ′= , γLL=0.8, Post-2005 losses 0.0948t cf f ′= , γLL=1.0, Post-2005 losses 0.19t cf f ′= , γLL=0.8, Pre-2005 losses 0.19t cf f ′= , γLL=0.8, Post-2005 losses 0.19t cf f ′= , γLL=1.0, Post-2005 losses 1 AASHTO I 30 6 8 8 8 8 8 8 2 AASHTO I 30 8 10 10 10 10 10 10 3 AASHTO I 30 10 12 12 12 12 12 12 4 AASHTO I 30 12 14 14 14 14 14 14 5 AASHTO II 60 6 20 16 20 18 16 16 6 AASHTO II 60 8 - 22 26 24 20 22 7 AASHTO III 60 10 22 20 22 20 20 20 8 AASHTO III 60 12 28 24 28 24 24 24 9 AASHTO III 80 6 28 24 28 24 22 24 10 AASHTO III 80 8 38 30 34 32 28 30 11 AASHTO III 80 10 - 36 40 42 32 38 12 AASHTO IV 80 12 40 34 38 34 32 34 13 AASHTO III 100 6 - 40 46 - 38 42 14 AASHTO IV 100 8 50 42 46 44 38 42 15 AASHTO IV 100 10 - 48 54 56 44 50 16 AASHTO V 100 12 56 46 50 48 42 46 17 AASHTO IV 120 6 - 52 58 58 48 52 18 AASHTO V 120 8 62 52 58 54 48 52 19 AASHTO V 120 10 - 60 68 68 54 60 20 AASHTO VI 120 12 74 58 64 64 54 58 21 AASHTO VI 140 6 62 54 58 54 48 52 22 AASHTO VI 140 8 - 64 70 68 58 64 23 AASHTO VI 140 10 - 74 - - 68 74 24 140 12 - - - - - - 122

4. With satisfactory past performance of prestressed beams, the target reliability index is selected to be similar to the average inherent reliability index of the bridges on the system. There is no scientific reason to substantiate targeting a different, higher or lower, reliability index. 5. The recommended target reliability index for the decompression limit state is 1.0 for bridges designed for no worse than moderate corrosion conditions and 1.2 for bridges designed for severe corrosion conditions. Based on the study of WIM data, the reliability index is determined assuming live load exists in single lane and without applying the multiple presence factor. This would appear on the “load side” of the limit state function. 6. Based on the reliability indices calculated for different design and load scenarios, to achieve the target reliability index, it is recommended that the following be used for designing for Service III limit state: • Live load factor of 1.0. • Maximum concrete tensile stress of 0.0948t cf f ′= and 0.19t cf f ′= for bridges in severe corrosion conditions and for bridges in no worse than moderate corrosion conditions, respectively. • Girders to be designed following conventional design methods and assuming the live loads exist in single lane or multiple lanes, whichever produces higher load effects. The appropriate multiple presence factor applies. These design parameters would appear on the “resistance side” of the limit state function during calibration. 7. The results of the calibration demonstrated that girders designed using the conventional design methods and the controlling number of loaded traffic lanes produce uniform reliability approximately equal to the target reliability index provided that the load factor is based on a reliability index calculated using the decompression criteria and assuming one lane of traffic. 5.2.6 Results for Adjacent Box Beams, Spread Box Beams, and American Segmental Box Institute (ASBI) Boxes Work similar to that described above for I-beams was performed for adjacent box beams, spread box beams, and ASBI box beams. The details of the work are shown in Appendix D. The final results assuming the decompression limit state, ADTT of 5000, return period of 1 year, and a load factor of 1.0 for live load are shown in Figure 5-23 through Figure 5-28. Table 5-17 shows the average reliability indices represented graphically in Figure 5-23 through Figure 5-28. 123

Figure 5-23- Adjacent box beams, reliability indices for bridges at decompression limit state (ADTT=5000), γLL=1.0 ( 0.0948t cf f ′= ). Figure 5-24 Adjacent box beams, reliability indices for bridges at decompression limit state (ADTT=5000), γLL=1.0 ( 0.19t cf f ′= ). 124

Figure 5-25 Spread box beams, reliability indices for bridges at decompression limit state (ADTT=5000), γLL=1.0 ( 0.0948t cf f ′= ). Figure 5-26 Spread box beams, reliability indices for bridges at decompression limit state (ADTT=5000), γLL=1.0 ( 0.19t cf f ′= ). 125

Figure 5-27 ASBI box beams, reliability indices for bridges at decompression limit state (ADTT=5000), γLL=1.0 ( 0.0948t cf f ′= ). Figure 5-28 ASBI box beams, reliability indices for bridges at decompression limit state (ADTT=5000), γLL=1.0 ( 0.19t cf f ′= ). 126

Table 5-17 Average Reliability Indices for Different Types of Girders Type of Section Maximum tensile stress used in design (ksi) 0.0948t cf f ′= 0.19t cf f ′= I- and Bulb T Girders 1.33 1.00 Adjacent Box Beams 1.85 1.31 Spread Box Beams 1.45 1.01 ASBI Box Beams 1.41 1.00 The results shown in Figure 5-23 through Figure 5-28 indicate that the reliability indices for each type of girder is reasonably uniform across the range of spans considered. With the exception of the adjacent box beams, the average reliability index for other section types is very close to each other and to the target reliability index. For adjacent box beams, the average reliability index is slightly higher. However, the difference does not warrant incorporating measures to reduce the resistance of the beams such as revising the distribution factor equations or use of lower load factors for adjacent box beams. 5.2.7 Sections Designed Using Other Methods of Determining Prestressing Time-Dependent Losses and/or Section Properties As indicated in Section 5.2.3, the calibration of Service III limit states presented above assumes that the sections are designed using the “Refined Estimates of Time-Dependent Losses”. AASHTO LRFD requires the time-dependent losses for segmental bridges to be determined using detailed time step methods. As such, the 2005 revisions to the “Refined Estimates of Time-Dependent Losses” did not affect the time-dependent prestressing loss calculations for segmental bridges. Historically, segmental bridges have also been designed using gross-section properties, not transformed section properties, and neglected the effects of the “elastic gain”. The time step method may also be used to design prestressed concrete components other than segmental bridges if approved by the owner. However, the level of effort required to perform time step analysis typically precluded this method for non-segmental construction. The proposed increase in the load factor for live load for Service III limit state from 0.8 to 1.0 is based on comparing sections designed using the AASHTO LRFD pre-2005 provisions and the post-2005 provisions without making any exceptions to the specifications requirements and assuming that the method termed in the AASHTO LRFD as the: “Refined Estimates of Time-Dependent Losses” was used for calculating the time-dependent losses. The development of the method termed as the “Approximate Estimate of Time-Dependent Losses” in AASHTO LRFD was based on producing prestress losses similar results to those produced by the method termed “Refined Estimates of Time-Dependent Losses”. Therefore, the change in the load factor should also be applied to the former method. The changed in the prestress loss methods in 2005 did not affect the time step method. Therefore, the increase in the load factor should not be applied to sections designed using the time step method. These sections have to satisfy the following conditions to continue using the 0.8 load factor for live load: • Time-dependent losses are determined using time step method, 127

• Gross sections properties are used for the calculations, and • The calculations of the force in the prestressing steel neglects the effects of the elastic gain. 5.2.8 Proposed AASHTO LRFD Revisions In AASHTO (2012), Article 5.9.4.2.2 (“Tension Stresses” for stresses in fully prestressed components at service limit state after losses) is the Article containing the design stress limits that are affected by the calibration of the Service III limit state. Due to the lack of changes to the design stress limits, no revisions to this section are required. The only required revisions to the specifications based on the calibration of the limit state for tension in prestressed concrete presented above are those in Article 3.4.1 to specify the load factor for live load as 0.8 or 1.0 depending on the design procedure used. 5.3 Fatigue Limit State – Lifetime 5.3.1 Formulate the Limit State Function While two limit states for load-induced fatigue are defined in AASHTO LRFD Article 3.4.1; only Fatigue I, related to infinite load-induced fatigue life; is applicable to concrete members as they are always designed for infinite life. For load-induced fatigue considerations, according to AASHTO LRFD Article 5.5.3.1, concrete members shall satisfy: ( ) ( )nγ Δf ΔF≤ (5-3) where γ = load factor (Δf) = force effect, live load stress range due to the passage of the fatigue load (ΔF)TH = constant-amplitude fatigue threshold This general limit state function is used for the calibration of the fatigue limit states for concrete members. The fatigue load of AASHTO LRFD Article 3.7.1.4 and the fatigue live-load load factors of AASHTO LRFD Table 3.4.1-1 are based upon extensive research of structural-steel highway bridges. The fatigue load is the AASHTO LRFD design truck (HS20-44 truck of AASHTO Standard Specifications, 2002), but with a fixed rear-axle spacing of 30 feet. The live-load load factors for the fatigue limit state load combinations are summarized in Table 5-18. 128

Table 5-18 Current Fatigue Load Factors Fatigue Limit State LL Load Factor Fatigue I 1.5 Fatigue II (used for steel structures only) 0.75 The load factor for the Fatigue I load combination reflects load levels found to be representative of the maximum stress range of the truck population for infinite fatigue-life design. The factor was chosen on the assumption that the maximum stress range in the random variable spectrum is twice the effective stress range caused by Fatigue II load combination. The load factor for the Fatigue II load combination reflects a load level found to be representative of the effective stress range of the truck population with respect to a small number of stress range cycles and to their cumulative effects in steel elements, components, and connections for finite fatigue-life design. Information on the Fatigue II Limit state is included for reference and they were based on work done on steel components by Kulicki et. al. (2013) The resistance factors for the fatigue limit states, ϕ, are inherently taken as unity and hence do not appear in Eq. (5-3). 5.3.1.1 Select Structural Types and Design Cases The available data suggested that two fatigue limit states for concrete members could be rationally calibrated based upon current practice and the available data: steel reinforcement in tension (AASHTO LRFD Article 5.5.3.2) and concrete in compression (AASHTO LRFD Article 5.5.3.1). 5.3.1.2 Determine Load and Resistance Parameters for Selected Design Cases 5.3.1.2.1 Steel Reinforcement in Tension Steel reinforcement considered herein includes straight reinforcing bars and welded-wire reinforcement. AASHTO LRFD Article 5.5.3.2 specifies the fatigue resistance of these types of reinforcement. The fatigue resistance of straight reinforcing bars and welded-wire reinforcement without a cross weld in the high-stress region (defined as one-third of the span on each side of the section of maximum moment) is specified as: ( )Δ 24 0.33 minTHF f= − (5-4) where fmin is the minimum stress. 129

For welded-wire reinforcement with a cross weld in the high-stress region, the fatigue resistance is specified as: ( )Δ 16 0.33 minTHF f= − (5-5) Both of these equations implicitly assume a ratio of radius to height (in other words, r/h) of the rolled-in transverse bar deformations of 0.3. These fatigue resistances are defined as constant-amplitude fatigue thresholds in AASHTO LRFD. ACI Committee Report ACI 215R-74 and the supporting literature indicate that steel reinforcement exhibits a constant-amplitude fatigue threshold. ACI 215R-74 suggests that the resistances are “a conservative lower bound of all available test results.” In other words, a horizontal constant-amplitude threshold has been drawn beneath all of the curves. The studies used to define the fatigue resistance of steel reinforcement (Fisher and Viest, 1961; Pfister and Hognestad, 1964; Burton and Hognestad, 1967; Hanson et al., 1968; Helgason et al., 1976; Lash, 1969; MacGregor et al., 1971; Amorn et al., 2007) were reanalyzed to estimate constant-amplitude fatigue thresholds for every case that can be identified in the research to determine their uncertainty, in terms of bias, mean, and COV. The various thresholds were grouped together to make design practical. 5.3.1.2.2 Concrete in Compression The compressive stress limit of 0.40 fc′ for fully prestressed components in other than segmentally constructed bridges of AASHTO LRFD Article 5.5.3.1 applies to a combination of the live load specified in the Fatigue I limit state load combination plus one-half the sum of the effective prestress and permanent loads after losses, i.e. a load combination derived from a modified Goodman diagram. This suggests that it represents an infinite-life check as the Fatigue I limit state load combination corresponds with infinite fatigue life. For this study, the research used to define these S-N curves, Hilsdorf and Kesler (1966) was re-evaluated to estimate the constant-amplitude fatigue threshold, the infinite-life fatigue resistance. The uncertainty of the fatigue resistance was quantified in terms of bias, mean, and coefficient of variation. 5.3.1.3 Develop Statistical Models for Loads and Resistances 5.3.1.3.1 Load Uncertainties Based on the analysis of WIM data discussed in Chapter 4, it is suggested that the current load factor of 1.5 for the Fatigue I limit state be increased to 2.0 to account for current and projected truck loads. Similarly, based on work by Kulicki et. al. (2013), it is proposed that the load factor of 0.75 for the Fatigue II limit state, which is not used for concrete structures and is mentioned here for reference only, be increased to a value of 0.80. The mean values and COV’s from Chapter 4 are tabulated in Table 5-19. 130

Table 5-19 Load Uncertainties Limit State Mean COV Fatigue I 2.0 0.12 Fatigue II (used for steel structures only) 0.8 0.07 5.3.1.3.2 Resistance Uncertainties The collection of the fatigue data was statistically analyzed using normal probability plots as the data best fits the normal distribution which is explained in further detail later. The normal probability plot is a graphical technique used in determining the statistical parameters of a normally distributed data set. The data points are plotted against a theoretical normal distribution and form an approximate straight line. Points that deviate from the straight line indicate deviation from normality. In other words if the observed data is normally distributed the points should form a straight line. The horizontal axis of the normal probability plot represents the values of the data set while the vertical axis is the set of standard normal values or Z-scores. These standard normal values are representative of the cumulative distribution function of the standard normal distribution. Thus an ordered pair plotted within the normal probability plots of this project has an abscissa of the new fatigue parameter and an ordinate of the corresponding standard normal value. The fatigue data was then filtered to include the data that most accurately reflects the fatigue behavior of each type of component, i.e. reinforcement in tension or concrete in compression. In other words, the data was truncated based on the nature of the curve within each normal probability plot to include the pertinent fatigue data. In general, the majority of the lower portion of each curve was selected for each detail category. This lower tail of the data was selected because it is the portion of the curve that fits the normal distribution, as it is the straight portion of the normal probability plot. Moreover, the lower portion of the fatigue data represents the range of values that fatigue cracking is expected to occur within when analyzed for the fatigue limit states load combinations using the Monte Carlo simulation approach.. Failure occurs when the load exceeds the resistance; thus the higher portions of the fatigue data sets represent fatigue resistance data that are very unlikely to be exceeded by the fatigue loads used within this study and therefore are insignificant. Different approaches for selecting the cutoff values in which different cutoff values were used for each category were investigated to determine the sensitivity of the resulting reliability indices. It was determined that the relative difference of the results determined from the different techniques were negligible. Other techniques used to determine the cutoff values included the use of constant cutoff values for all of the various components, i.e. reinforcement in tension and concrete in compression as well as manually inserting best-fit lines by different analysts. Determining the statistical parameters of the data is relatively straightforward once the data was filtered and fitted with a line of best fit using Microsoft Excel software. The mean value is simply the intersection of the best fit line with the horizontal axis. The standard deviation of the data is taken as the inverse of the slope of the best fit line. More simply stated it is the change in horizontal coordinates divided by the change in the vertical coordinates. Moreover, the COV is the ratio of the standard deviation divided by the mean of the data. The resulting 131

statistical parameters can be seen in Table 5-20. The probability plots of the fatigue data and corresponding truncated data for steel reinforcement in tension and concrete in compression can be seen in Appendix G. Figure 5-29 and Figure 5-30 show the normal probability plots of the full fatigue data set and the truncated data fatigue resistance of steel reinforcement in tension, respectively. Figure 5-29 Normal Probability Plot of Fatigue Resistance Data for Steel Reinforcement in Tension Figure 5-30 Normal Probability Plot of Truncated Fatigue Resistance Data with Best-Fit Line for Steel Reinforcement in Tension 132

Table 5-20 Resistance Uncertainties Resistance Standard Deviation COV Bias Mean Nominal Cutoff Standard Normal Variable steel reinforcement in tension 769.23 0.24 1.94 3261.54 1681.21 2 concrete in compression 117.65 0.45 1.74 260.35 149.66 2 5.3.1.4 Develop the Reliability Analysis Procedure 5.3.1.4.1 General In the code calibration it is necessary to develop a process by which to express the structural reliability or the probability of the loads on the member being greater than its resistance; in other words, failure of the criteria. The reliability analysis performed within this project is an iterative process that consists of Monte Carlo simulations to select load and resistance factors that achieve reliability close to the target reliability index. The Monte Carlo technique samples load and resistance parameters from selected statistical distributions, such as a normal distribution. The reliability is measured in terms of a reliability index, or safety index, β. β is defined as a function of the probability of failure, PF, using the following equation. Thus β is the number of standard deviations that the mean safety margin falls on the safe side. The higher the β value, the higher the reliability. β = - Φ-1(PF) where Φ-1 = the inverse standard normal distribution function 5.3.1.4.2 Monte Carlo Simulation The following is a description of the calibration procedure applied to bridge structures. The distribution of loads is typically assumed to be normally distributed as the loads are a summation of force effects. The fatigue resistance has also been assumed to follow normal distributions. These distributions for load and resistance are developed using determined statistical parameters from the available data. The minimum statistical parameters needed for each random variable is the COV and the bias (λ). The COV is a measure of the scatter of the variable and the bias is the ratio of the mean to the nominal value. The simulation is then run by selecting random values from both the load and resistance distributions and comparing them using the appropriate limit-state function. If the result from the evaluation of the limit-state function is equal to or greater than zero, the function is satisfied and no failure results. Conversely, if the result is negative then a failure is recorded. This process is repeated over a large number of iterations and the number of failures is counted to determine the failure rate. Finally the reliability index is determined by taking the inverse of the standard normal cumulative distribution function using the determined failure rate. 5.3.1.5 Calculate the Reliability Indices for Current Design Code or Current Practice Monte Carlo simulation was used to estimate the current inherent reliability indices by comparing the distribution of fatigue load with the distribution of fatigue resistance, based upon the uncertainties of load and resistance. 133

The simulations for both limit states were completed using a total of 10,000 replicates to achieve a sufficient number of failures. For steel reinforcement in reinforced concrete members, the inherent β is approximately 2.0, but the inherent β for compression of concrete members is approximately 1.0. Both of these fatigue limit states are based upon the Fatigue I limit state and design for infinite life. The calculated inherent β’s are given below in Table 5-21. Table 5-21 Current Reliability Indices for the AASHTO LRFD Fatigue I Limit States Resistance β Steel reinforcement in tension 1.9 Concrete in compression 0.9 5.3.1.6 Select the Target Reliability Index, βT Philosophically, the target reliability index should be identical for all members and all fatigue limit states. As such, the work on reinforcement and concrete fatigue was performed concurrently with, and is compared to, the work on structural steel fatigue. As such, we propose a constant target reliability index, βT, of 1.0 for steel reinforcement in tension, concrete in compression and structural steel members. This proposed target reflects the inherent reliability of the current Fatigue I limit state for concrete in compression and the Fatigue I and II limit states for structural steel members shown in Table 5-22 for comparison. This proposed target reduces the reliability of steel reinforcement in tension to levels consistent with the other calibrated fatigue limit states for which unity was chosen as the target reliability index. Table 5-22 Current Reliability Indices for Steel Members Using AASHTO Fatigue I and Fatigue II Limit States Detail Category β Fatigue I Fatigue II A 1.2 1.0 B 1.1 0.9 B' 1.5 1.0 C 1.2 0.9 C' 1.2 0.9 D 2.0 1.3 E 0.9 0.7 E' 1.7 1.4 5.3.1.7 Select Potential Load and Resistance Factors Proposed resistance factors for the Fatigue I limit state are given in Table 5-23. Resistance factors other than the current values of unity are shown in boldface. 134

Table 5-23 Proposed Fatigue I Limit-State Resistance Factors Resistance Proposed Resistance Factor, Φ Reliability Index, β Steel reinforcement in tension 0.8 1.1 Concrete in compression 1.0 0.9 5.3.1.8 Calculate Reliability Indices With the proposed resistance factors, the reliability indices are all within ± 0.1 of the target reliability index of 1.0. The resultant reliability indices tabulated above can also be achieved by revising the AASHTO LRFD constant-amplitude fatigue thresholds for steel reinforcement in tension. This may be a better solution than including a resistance factor other than unity for only one of the concrete member fatigue limit states. The required revision to the AASHTO LRFD equations for the thresholds are given below. The revised fatigue resistance of straight reinforcing bars and welded-wire reinforcement without a cross weld in the high-stress region would be specified as: ( ) minTHΔF 19 0.26f= − (5-6) where fmin is the minimum stress. For welded-wire reinforcement with a cross weld in the high-stress region, the fatigue resistance would be specified as: ( ) minTHΔF 13 – 0.26f= (5-7) 5.3.2 Proposed AASHTO LRFD Revisions In AASHTO LRFD (2012), the Fatigue Limit State applicable to concrete structures is addressed in Sections 3 and 5. The Articles that will require modification to implement the revisions recommended herein are indicated in Table 5-24. 135

Table 5-24 Summary of Relevant Articles in AASHTO LRFD for Fatigue Article (See Note) Title Relates To 3.4.1, Table 3.4.1-1 Load Factors and Load Combinations Fatigue I and II* 5.5.3.2 Reinforcing Bars Fatigue Threshold 5.5.3.3 Prestressing Tendons Fatigue Threshold * All concrete-related fatigue issues utilize Fatigue I limit state. Other revisions related to Fatigue II limit state for use in designing steel structures are detailed in Kulicki et. al. (2013) SHRP R19B report The proposed revisions are detailed in Chapters 6 and 7. 5.4 Service Design for Overload One of the goals of this project was to develop a service limit state for permit (overload) vehicles for concrete structures akin to Service II limit state in the current AASHTO LRFD which is applicable only to steel structures. The Service II limit state is intended to prevent changes in ride quality and appearance of steel structures resulting from permanent deflections due to yielding under service loads. For steel structures, the limit state function and the consequences are well defined; the yielding of the steel component and the permanent deformations associated with yielding. For all concrete structures service limit states, the limit state function and the consequences of exceeding the limit state are not well defined. For service limit states discussed above for concrete structures, the calibration was based on obtaining uniform reliability at a level similar to the average reliability inherent in past designs. The consequence of exceeding the limit state was not part of the calibration as it was not possible to quantify. In the absence of past requirements for designing concrete components for service level overloads, there is no basis to what might be a reasonable level of reliability under overloads. With the reliability indices selected for the service limit states are the range of 1.0 to 1.6 (15.9% and 5.5% probability of exceeding the limit state during one year, respectively), it is expected that the service limit states calibrated earlier will be exceeded when heavy permit vehicles cross a bridge. The question changes from the frequency of exceeding the limit state to the consequences of significantly exceeding the limit state. For a meaningful calibration under overloads, the following needs to be available: • Adequate information on the frequency of permit vehicles on the highways in comparison to other vehicles. • Adequate information on the actual weights of permit vehicles and how these weights compare to the permitted loads. • Quantifying the consequences of exceeding the limit states (required for all service limit states calibrated in this report other than fatigue for which the consequences are known) and quantifying the consequences of significantly exceeding the limit states. For example, in addition to quantifying the consequences of exceeding the compression limit state for prestressed 136

components under normal service loads, the consequences of significantly exceeding the decompression, i.e. opening a wide crack, also need to be quantified. The latter can then be used for calibration under permit loads. The following conclusions were drawn from the study of heavy vehicles in the WIM data, the Louisiana truck citation data and New Jersey Permit data (see Section 4): • The frequency of vehicles producing load effects exceeding HL-93 varies significantly from one site to another. Generally, the average frequency of vehicles producing load effects exceeding HL-93 is small and is dependent on the span length. Vehicles producing load effects exceeding HL-93 are generally assumed to be permit vehicles. • The available information on how actual loads compare to the permitted loads is limited. This information is from one source (Louisiana) and is incomplete in that the truck configuration and individual axle loads are not given. • The percentage of vehicles in each GVW category based on the permitted GVW’s varies significantly between New Jersey and Louisiana. With no information on the actual GVW’s in New Jersey and how they compare to the permitted GVW’s, it is not known how the actual GVW’s in New Jersey compare to those in Louisiana. The lack of correlation between the two states also indicates that variation between states exist and makes any assumption regarding other states’ permit vehicles unjustifiable. It was concluded that the available information is not adequate to produce meaningful calibration for the concrete limit states under overloads. 137