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Bridges for Service Life Beyond 100 Years: Service Limit State Design (2014)

Chapter: Chapter 6 - Calibration Results

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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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Suggested Citation:"Chapter 6 - Calibration Results." National Academies of Sciences, Engineering, and Medicine. 2014. Bridges for Service Life Beyond 100 Years: Service Limit State Design. Washington, DC: The National Academies Press. doi: 10.17226/22441.
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145 6.1 Foundation Deformations, Service I: Lifetime The geotechnical limit states for serviceability of a bridge struc- ture relate to foundation deformations. Within the context of foundation deformation, the geotechnical limit states can be broadly categorized into vertical and horizontal deformations for any foundation type (e.g., spread footings, driven piles, drilled shafts, micropiles). Table 6.1 summarizes the various relevant articles in AASHTO LRFD (2012) that address vertical (settlement) and horizontal deformations for various types of structural foundations. This section describes procedures that can be used for calibrating service limit states (SLSs) to evaluate the effect of vertical or horizontal deformations of all structural foundation types such as footings, drilled shafts, and driven piles. The procedure is demonstrated by using the case of immediate settlements of spread footings, and the effect of foundation deformations on bridge superstructures is discussed in the context of construction stages. 6.1.1 Target Reliability Index For strength limit states, reliability index values in the range of 3.09 to 3.54 are used. Strength (or ultimate) limit states pertain to structural safety and the loss of load-carrying capacity. In contrast, SLSs are user-defined limiting conditions that affect the function of the structure under expected service conditions. Violation of SLSs occurs at loads much smaller than those for strength limit states. As there is no danger of collapse if an SLS is violated, a smaller value of target reliability index may be used for SLSs. In the case of foundation deformation, such as settlement, the structural load effect is manifested in terms of increased moments and potential cracking. The load effect due to settlement relative to the load effect due to dead and live loads would generally be small because in the Service I limit state the load factor gSE, which represents the uncertainty in estimated settlement, is only one of many load factors. Further- more, the primary moments due to the dead and live loads are much larger than the additional (secondary) moments due to settlement. Because of these considerations and based on a consideration of the reversible and irreversible SLSs for bridge superstructures described earlier in this report, a target reliability index (bT) in the range of 0.50 to 1.00 for the calibration of load factor gSE for foundation deformation in the Service I limit state is used. 6.1.2 Calculation Models Evaluation of an SLS involves consideration of the deformation aspects of the structure or members of the structure. The load deformation characteristics of the structure or its member are important to understand because resistance must now be quantified as a function of the deformation. This section first discusses the extension of the AASHTO LRFD frame- work to incorporate the load deformation behavior, after which a calibration framework for SLSs for foundation defor- mations is presented. The proposed step-by-step procedure for calibration is described in Section 6.1.2.5, which leads to a load factor for deformations based on the target reliability index that was discussed in Section 6.1.1. The proposed pro- cedure is demonstrated by an example for immediate settle- ments of spread footings by using various analytical methods in Section 6.1.3. 6.1.2.1 Incorporation of Load Effect Deformation (Q-d) Characteristics in the AASHTO LRFD Framework The basic AASHTO LRFD framework in terms of distributions of load effects and resistances is shown in Figure 6.1, where Q = load; Qmean = mean load; C h a p t e r 6 Calibration Results

146 g = load factor; R = resistance; Rmean = mean resistance; ϕ = resistance factor; Qn = nominal load; lQ = bias factor for load; f = frequency; Rn = nominal resistance; and lR = bias factor for resistance. Details of the AASHTO LRFD framework can be found in Nowak and Collins (2013). Strength limit states were evaluated by using this framework. Because determination of defor- mation is a necessary part of the evaluation of serviceability, for the evaluation of SLS, the basic AASHTO LRFD framework shown in Figure 6.1 needs to be modified to include load effect deformation, or Q-d behavior. The Q-d behavior can be consid- ered as another dimension of the basic AASHTO LRFD frame- work, as shown in Figure 6.2, where d = deformation; dS = deformation at nominal load effect (Qn); Table 6.1. Summary of AASHTO LRFD (2012) Articles for Estimation of Vertical and Horizontal Deformation of Structural Foundations AASHTO LRFD Article Comment 10.6.2.4 Settlement Analyses for Spread Footings Article 10.6.2.4 presents methods to estimate the settlement of spread footings. Settlement analysis is based on the elastic and semiempirical Hough (1959) method for immediate settlement and the one-dimensional consolidation method for long-term settlement. 10.7.2.3 Settlement (related to driven pile groups) 10.8.2.2 Settlement (related to drilled shaft groups) 10.9.2.3 Settlement (related to micropile groups) The procedures in these articles (10.7.2.3, 10.8.2.2, and 10.9.2.3) refer to settlement analysis for an equivalent spread footing; see AASHTO LRFD (2012, Figure 10.7.2.3.1-1). 10.7.2.4 Horizontal Pile Foundation Movement 10.8.2.4 Horizontal Movement of Shaft and Shaft Groups 10.9.2.4 Horizontal Micropile Foundation Movement Lateral analysis based on the p-y method and strain wedge method is included in AASHTO LRFD (2012) for estimating horizontal (lateral) deformations of deep foundations. Note: Section 11 (Abutments, Piers and Walls) Article 11.6.2 refers to the various articles noted in the left column of this table. Thus, the articles noted in this table also apply to fill retaining walls and their foundations. Qmean Qn Rnf(R, Q) Q, R Qn Rn Rmean RQ Figure 6.1. Basic AASHTO LRFD framework for load effects and resistances. PDF of curve Mean curve Q, R f(R, Q) Qn Rn Qmean Rmean N F SδS λQ λR γQn φRn δF δN δ Figure 6.2. Incorporation of Q-d mechanism into the basic AASHTO LRFD framework.

147 dF = deformation at factored load effect [QF = g(Qn)]; and dN = deformation at load corresponding to nominal resis- tance (Rn). Although Q-d curves can have many shapes, for illustration purposes, a strain-hardening curve is shown in Figure 6.2. For discussion purposes, the mean Q-d curve is shown, and the spread of the Q-d data about the mean curve is represented sche- matically by a probability distribution function (PDF) that is discussed later in this chapter. The various relevant load effect and deformation quantities shown in the Q-d space in Figure 6.2 are shown in the regular first quadrant of the two-dimensional plot in Figure 6.3. Note that the nominal resistance is equated to a load effect that would correspond to this resistance. Figure 6.2 combines different aspects of material behavior that cover both load effects and resistances. It is important to understand the interrelationships among the various param- eters displayed on the curves. To that end the following points are made: • The load effect deformation (Q-d) curves shown in Fig- ure 6.2 and Figure 6.3 represent the measured mean curves based on field measurements. • Field measurements have upper and lower bounds with respect to the mean of the measured data. These bounds are shown schematically in Figure 6.4 and also in Figure 6.2 and Figure 6.3 through a PDF. Although PDFs for nor- mal distributions are shown, the spread of the data about the mean may be represented by normal or nonnormal distributions, as appropriate. In general, the spread of the data around the mean curve increases with increasing deformations. • Many theoretical methods are available to predict load effect deformation behavior. The theoretical models may predict a stiffer or softer material response compared with the actual response. For the purpose of discussion, a softer material behavior is shown in Figure 6.5. Because the bias factor is defined as the ratio of measured mean to predicted values, O δ δ δ δ S F N QF QS QN S F N Load Effect, Q Deformation, SL S St re n gt h Li m it St a te N o m in al R e sis ta n ce Le ve l Figure 6.3. Significant points of interest on the mean Q-d curve. Q δ Upper Bound Lower Bound Measured Mean C D Figure 6.4. Significant points of interest on the Q-d curve. Q δ Theoretical Prediction Measured Mean C D Figure 6.5. Relationship of measured mean with theoretical prediction.

148 the bias factor for deformations (ld) will vary over the full range of the Q-d curve. 6.1.2.2 Consideration of Bias Factor in Calibration of SLS A varying bias factor along the Q-d curve, although a reality, can be cumbersome to handle in the calibration process. However, the problem is made easier by realizing that for calibration of SLS the load effects between Points O and S, as shown in Figure 6.3, are of primary interest. Point S represents the service load effects, and the deformation corresponding to this point is of primary interest. As the bias factor will generally increase with increasing deformations, the value of the bias factor at Point S will be the maximum between Points O and S. Thus, use of the bias factor at Point S will be conservative. In this context, the bias factor at Point S is most relevant and, at a minimum, field data under full service loads are of importance in geotechnical SLS calibrations. The data most particularly needed for SLS evaluations are the full range of incremental loads and deformations measured on in-service structures from the beginning of construction of the first element (e.g., the foundation) to the completion of the road- way and beyond. Such data will help in the evaluation of the variability in predicted deformations for structural, as well as geotechnical, features. At present, these types of data are not routinely available; however, programs such as the Federal Highway Administration’s (FHWA) Long-Term Bridge Perfor- mance Program may offer a good avenue to collect such data. 6.1.2.3 Application of Q-d Curves in the LRFD Framework The AASHTO LRFD calibration of the strength limit state was performed by using the general concepts in Figure 6.1. This approach presumes that statistical data are available to quantify the spread of the load effects and resistances. In the context of deformations, tolerable deformations (dT) can be considered as resistances, and predicted deformations (dP) can be considered as loads. Thus, a limit state function (g) can be written as shown by Equation 6.1: (6.1)g T P= δ − δ Once the deformations are expressed in the form of a limit state, probabilistic calibration processes similar to those used for the strength limit state can be used. For strength limit states, Monte Carlo analysis is often used for calibrations. One of the assumptions of the Monte Carlo procedure is that PDFs for both the load (Q) and resistance (R) are available. However, for geotechnical SLS calibration, there are practical limitations to this approach. Although the statistical data for modeling the uncertainty in predicted deformations (dP) are available, the same is not true for tolerable deformations (dT). Some attempts have been made (e.g., Zhang and Ng 2005) to evaluate the distribution of tolerable deformations, but from a geotechnical viewpoint, it may not be possible to obtain a PDF for tolerable deformation that is applicable to the vari- ous structural SLSs discussed in other sections of this report. This is largely because it is virtually impossible to identify a consistent tolerable deformation across all elements of a structure. Many variables can affect the value of tolerable deformation for a given element. To bypass these difficulties, a single deterministic value of dT is often used for comparison against the potential spread of data for dP. In practical terms, a bridge engineer often assumes a deterministic tolerable deformation that would limit deformations according to the type of bridge structure being designed. In this case, the con- ventional calibration processes, such as the Monte Carlo pro- cedure, would not be necessary as there would be a PDF for load (Q), but a deterministic value for resistance (R). To use Monte Carlo in this situation an arbitrarily small value of standard deviation or coefficient of variation (CV) would have to be used. Although theoretically possible, this process could lead to spurious results. Thus, an alternative approach to calibration of SLSs for geotechnical features is necessary. When a deterministic value for dT is used, then by using Figure 6.1 as the basis, the resistance PDF is reduced to a single value while the load effect PDF can be used to represent the predicted deformations. This modified treatment for defor- mations is shown in Figure 6.6. In this approach, the probabil- ity of exceedance (Pe) for the predicted deformations to exceed the tolerable deformation is given by the area of the overlap of the two curves (the shaded zone shown in Figure 6.6). As the goal is to prevent serviceability-related problems, Pe can P f(R, Q) Q, R δ δT Probability of Exceedance, Pe Figure 6.6. Relationship of deterministic value of tolerable deformation (dT) and a PDF for predicted deformation (dP). Q = load effect; R = resistance; dP = predicted deformations (load effect); and dT = deterministic value of tolerable deformation (resistance).

149 be selected on the basis of the acceptable value of the target reliability index (bT). The ratio dT/dP can be thought of as a load factor for deformations for a given Pe, corresponding to bT. The PDF for the predicted deformations shown in Figure 6.6 is obtained from the data at Point S shown in Figure 6.2 and Figure 6.3. This is where the concept of the Q-d curve fits into the framework to calibrate the SLS on the basis of on defor- mations. Thus, any model that can predict a Q-d curve can be used in the conventional AASHTO LRFD framework as long as the data at Point S corresponding to SLS load effects are available through field measurements. The effect of material brittleness (or ductility) can now be introduced in the AASHTO LRFD framework through the use of an appropriate Q-d model. Examples of Q-d models are stress–strain curves, vertical load- settlement curves for foundations, p-y (lateral load–lateral displacement) curves for laterally loaded piles, shear force– shear strain curves, moment–curvature curves, and so forth. The proposed framework can incorporate any Q-d model and is therefore a general framework that is applicable to structural or geotechnical aspects. 6.1.2.4 Deterioration of Foundation and Wall Elements Most, if not all, foundation elements are buried in geomaterials. This is also true for most earth-retaining structures. Thus, the long-term performance of the foundation and wall elements can be affected by the corrosion or degradation potential of the geomaterials. The term corrosion applies to metal components, and degradation applies to nonmetal components such as poly- meric soil reinforcements in mechanically stabilized earth (MSE) walls. If the geomaterials have significant corrosion or degradation potential, then the sectional properties of the foundation and wall elements will deteriorate by reduction in the section or loss of strength, or both. The AASHTO LRFD Specifications clearly recognizes this mode of deterioration and provides definitive guidelines. For example, Articles 10.7.5 and 10.9.5 of Section 10 (Foundations) provide guidelines for evaluation of corrosion and deterioration of driven piles and micropiles, respectively. Similarly, Section 11 (Abutments, Piers, and Walls) provides guidance in Article 11.8.7 for nongravity cantilevered walls, Article 11.9.7 for anchored walls, and Articles 11.10.2.3.3 and 11.10.6.4 for MSE walls. Supplementary guidance can be found in Elias et al. (2009) and Fishman and Withiam (2011). The AASHTO, Elias et al., and Fishman and Withiam docu- ments cross reference various publications that discuss the corrosion or degradation potential of geomaterials. In general, the AASHTO articles and other documents cited above provide guidance for testing frequencies and protocols to evaluate the corrosion or degradation potential of various geomaterials. It is assumed that the foundation and wall designer will perform the necessary tests and, as appropriate, implement the necessary mitigation measures to minimize the inevitable effects of corrosion or degradation on the foun- dation and wall elements and the structures these elements support. The most common approach is to estimate the rate of corrosion or degradation over the design life of the struc- ture and provide additional sectional or strength properties (or both) that will permit the structure to perform within its strength and serviceability requirements. For example, metal elements are often provided additional section based on the anticipated loss of metal over the design life of the structure. Concrete deterioration due to sulfate attack is often mitigated by use of an appropriate type of cement. 6.1.2.5 Determination of Load Factor for Deformations The concept presented in Figure 6.6 assumes that the designer has unique (fixed) values of tolerable deformation (dT) and predicted deformation (dP). However, these values are func- tions of many parameters for a given element and the mode of deformation being evaluated. Thus, it is more practical to express the load factor for deformation as a function of the value of dP. The load factor is more conveniently determined by using an alternative form of the concept, as shown in Figure 6.7, in which the cumulative distribution function (CDF) is used instead of the PDF. In this concept it is more convenient to use the data based on the inverse of the bias factor because the predicted (calculated) deformation is plotted on the x-axis. The format shown in Figure 6.7 is used as follows: 1. Obtain data for predicted (dP) and measured (dM) deforma- tions for the deformation mode of interest (e.g., immediate settlement of spread footings). Recognize that the value of dM can be considered as resistance and equivalent to the tolerable settlement (dT). 2. Modify the data to be expressed in terms of the ratio dP/dT. In geotechnical literature (e.g., Tan and Duncan 1991) this ratio is often referred to as accuracy. Label this ratio as X. X is a random variable that can now be modeled by an appropriate PDF. Develop the appropriate statistics, and select a suitable distribution function. Express the data in terms of a CDF. 3. As shown in Figure 6.7, plot a family of CDF curves for a range of values of tolerable deformation (e.g., dT1 > dT2 > dT3) that permits the determination of values of the probability of exceedance (Pe) for a range of dP. The CDFs are generated by multiplying the CDF for accuracy (i.e., X = dP/dT) or by selected values of tolerable deformations (dT1, dT2, dT3). The plot shown in Figure 6.7 is referred to as a probability exceedance chart (PEC).

150 4. Select the design value of probability of exceedance (PeT) corresponding to the target reliability index (beT), and determine the value of dT for a given value of dP, as shown in Figure 6.7. 5. Compute the value of the deformation load factor (g = dT/dP), as shown in Figure 6.7. The benefit of this approach is that once the designer computes (predicts) a deformation for any given deformation mechanism, then the designer simply multiplies the computed value by the deformation load factor corresponding to that value of deformation and uses the factored value for evaluation at the applicable SLS load combination. This concept is valid whether structural or geotechnical deformation mechanisms are evaluated. This concept is demonstrated in the next sec- tion, in which immediate settlements for spread footings are evaluated. 6.1.3 Calibration Results The proposed procedure for calibration described in Sec- tion 6.1.2.5 is demonstrated by an example for immediate settlement of spread footings. The calibration results are presented in a step-by-step format that is generally consistent with other similar results presented in this report. 6.1.3.1 Step 1: Formulate the Limit State Function and Identify Basic Variables In the context of deformations, tolerable deformations (dT) can be considered as resistances, and predicted deformations (dP) can be considered as loads. Thus, a limit state function (g) can be given by Equation 6.2 (first introduced as Equation 6.1): (6.2)g T P= δ − δ For SLS calibration for foundation deformation, the limit state g expressed as a ratio is more appropriate, as given by Equation 6.3: (6.3)g P T= δ δ 6.1.3.2 Step 2: Identify and Select Representative Structural Types and Design Cases In general, the vertical and lateral deformations for all structural foundation types (e.g., footings, drilled shafts, and driven piles) can be calibrated using the process described in this example. For the purpose of demonstration of the calibration process, immediate vertical settlement of spread footings is used as a design case. 6.1.3.3 Step 3: Determine Load and Resistance Parameters for the Selected Design Cases The load and resistance parameters for the selected design case of immediate vertical settlement of spread footings are as follows. Load is predicted (or calculated) immediate vertical settlement (dP) and resistance is tolerable (or limiting or mea- sured) immediate vertical settlement (dT). 6.1.3.4 Step 4: Develop Statistical Models for Load and Resistance Table 6.2 shows a data set for spread footings based on vertical settlements of footings measured at 20 footings for 10 instru- mented bridges in the northeastern United States (Gifford et al. 1987). The bridges included five simple-span and five continuous-beam structures. Each of the footing designations in Table 6.2 represents a footing supporting a single sub- structure unit (abutment or pier). Four of the instrumented Pr ob ab ili ty o f E xc ee da nc e, P e Predicted (Calculated) Deformation, δ δ δ δγ δ δ δ δ P T1 T2 T3 PeT P T Deformation Load Factor, = T / P Figure 6.7. PEC for evaluation of load factor for a target probability of exceedance (PeT) at the applicable SLS combination.

151 bridges were single-span structures. Two two-span and three four-span bridges were also monitored, in addition to a single five-span structure. Nine of the structures were designed to carry highway traffic, and one four-span bridge carried railroad traffic across an Interstate highway. Additional infor- mation on the instrumentation and data collection at the 10 bridges can be found in Gifford et al. (1987). Similar and more extensive databases are available for spread footings (e.g., Sargand et al. 1999; Sargand and Masada 2006; Akbas and Kulhawy 2009; Samtani et al. 2010) and other foundation types, such as driven piles and drilled shafts. Similar data- bases are also available for lateral load behavior. However, for this report, the calibration concepts for SLS evaluations are demonstrated by use of the limited data set for spread footings shown in Table 6.2. Although spread footings are used as an example, all the concepts discussed here are applicable to other foundation types and deformation patterns. Figure 6.8 shows a plot of the data in Table 6.2 and the spread of the data about the 1:1 diagonal line, which defines the case for which the predicted and measured values are equal. Such a plot provides a visual frame of reference to judge the accuracy of the prediction method. If the data points align closely with the 1:1 diagonal line, then the predictions based on the analytical method being evaluated are close to the measured values and are more accurate than the case for which the data points do not align closely with the 1:1 diagonal line. In the geotechnical literature (e.g., Tan and Duncan 1991), accuracy is defined as the mean value of the ratio of the pre- dicted (calculated) to the measured settlements. Table 6.3 shows the values of accuracy (denoted by X, where X = dP/dM) for each footing based on the data in Table 6.2. As noted in Step 3 of the calibration process, the value of dM can be considered as the resistance and equivalent to the tolerable settlement (dT). The accuracy (i.e., X = dP/dM Table 6.2. Data for Measured Settlement (dM ) and Calculated Settlement (dP) Shown in Figure 6.8 Site Measured Settlement (in.) Calculated Settlement (in.) Schmertmann et al. (1978) Hough (1959) D’Appolonia et al. (1968) Peck and Bazaraa (1969) Burland and Burbridge (1984) S1 0.35 0.79 0.75 0.65 0.29 0.30 S2 0.67 1.85 0.94 0.39 0.16 0.12 S3 0.94 0.86 1.21 0.30 0.19 0.13 S4 0.76 0.46 1.46 0.58 0.36 0.39 S5 0.61 0.30 0.98 0.38 0.42 0.57 S6 0.42 0.52 0.61 0.50 0.17 0.34 S7 0.61 0.18 0.40 0.19 0.30 0.19 S8 0.28 0.30 0.60 0.26 0.16 0.14 S9 0.26 0.18 0.53 0.20 0.16 0.11 S10 0.29 0.29 0.40 0.23 0.16 0.09 S11 0.25 0.36 0.47 0.29 0.16 0.06 S14 0.46 0.41 1.27 0.57 0.50 0.40 S15 0.34 1.57 1.46 0.74 1.36 1.61 S16 0.23 0.26 0.74 0.39 0.17 0.17 S17 0.44 0.40 0.82 0.46 0.28 0.23 S20 0.64 1.21 0.33 0.10 0.07 0.65 S21 0.46 0.29 1.05 0.49 0.21 0.54 S22 0.66 0.54 0.84 0.56 0.52 0.31 S23 0.61 1.02 1.39 0.61 0.34 0.64 S24 0.28 0.64 0.99 0.59 0.33 0.44 Note: Gifford et al. (1987), the source for the table, note that data for Footings S12, S13, and S18 were not included because construction problems at these sites resulted in disturbance of the subgrade soils, and short-term settlement was increased. Data for Footing S19 appear to be anomalous and have been excluded in this table and Figure 6.8.

152 Figure 6.8. Comparison of measured and calculated (predicted) settlements based on service load data in Table 6.2. Table 6.3. Accuracy (X = dP/dM) Values Based on Data Shown in Table 6.2 Site Schmertmann et al. (1978) Hough (1959) D’Appolonia et al. (1968) Peck and Bazaraa (1969) Burland and Burbridge (1984) S1 2.257 2.143 1.857 0.829 0.857 S2 2.761 1.403 0.582 0.239 0.179 S3 0.915 1.287 0.319 0.202 0.138 S4 0.605 1.921 0.763 0.474 0.513 S5 0.492 1.607 0.623 0.689 0.934 S6 1.238 1.452 1.190 0.405 0.810 S7 0.295 0.656 0.311 0.492 0.311 S8 1.071 2.143 0.929 0.571 0.500 S9 0.692 2.038 0.769 0.615 0.423 S10 1.000 1.379 0.793 0.552 0.310 S11 1.440 1.880 1.160 0.640 0.240 S14 0.891 2.761 1.239 1.087 0.870 S15 4.618 4.294 2.176 4.000 4.735 S16 1.130 3.217 1.696 0.739 0.739 S17 0.909 1.864 1.045 0.636 0.523 S20 1.891 1.641 0.766 0.328 0.844 S21 0.630 1.826 1.217 1.130 0.674 S22 0.818 2.106 0.924 0.515 0.970 S23 1.672 1.623 0.967 0.541 0.721 S24 2.286 2.179 1.286 0.893 1.286

153 [or dP/dT]) is a random variable that can now be modeled by an appropriate PDF. To develop an appropriate PDF, an eval- uation of the data spread around the mean value is needed. This evaluation involves statistical analysis and the develop- ment of histograms. Table 6.4 presents the arithmetic mean (µ) and standard deviation (s) values for various methods. AASHTO LRFD recommends the use of Hough’s (1959) method, which has the smallest CV, for calculating immediate settlement. However, the Hough method is conservative by a factor of approximately two (see mean value in Table 6.4), which leads to an unnecessary use of deep foundations instead of spread footings. FHWA (Samtani and Nowatzki 2006; Samtani et al. 2010) recommends the use of the method proposed by Schmertmann et al. (1978) because it is a rational method that considers not only the applied stress and its associated strain influence distribution with depth for various footing shapes, but also the elastic prop- erties of the foundation soils, even if they are layered. Even though FHWA and AASHTO recommend the Schmertmann et al. (1978) and Hough (1959) methods, respec- tively, all the methods noted in Table 6.2 to Table 6.4 were evaluated as part of the calibration process because some agen- cies may use one of the remaining three methods as a result of past successful local practice. As noted earlier, accuracy (X = dP/dM) is a random variable that can be modeled by an appropriate PDF. The data for X in Table 6.3 were used to develop histograms. The histograms of the data for X taken from Columns 2 to 6 of Table 6.3 are shown in Figure 6.9a to Figure 6.13a, respectively. None of the histograms resembles a classic bell shape character- istic of normally distributed data. Thus, nonnormal distribu- tions would be more appropriate in these cases. To evaluate the deviation of the data from a classic normal PDF, the data for the value of X in Table 6.3 were plotted against the standard normal variable (z) to generate CDFs, as shown in Figure 6.9b to Fig- ure 6.13b. See Allen et al. (2005, Chapter 5) for a definition of z and procedures to develop the lower graphs (b) in Figures 6.9– 6.13. The beneficial attributes of this probability plot are discussed above. As Figure 6.9b to Figure 6.13b show, the data points based on Table 6.3 do not plot on the straight line, which confirms the observation of nonnormal distributions made on the basis of the histograms in Figure 6.9a to Figure 6.13a. By using procedures described in Allen et al. (2005), a log- normal distribution was used to evaluate the nonnormal data. As seen in Figure 6.9b to Figure 6.13b, the lognormal distribution fits the data better than the normal distribution. The lognormal distribution, which is valid between values of 0 and +∞, is used in these figures because (1) immediate settlement cannot have negative values, and (2) lognormal PDFs have been used in the past for nonnormal distributions during calibration of the strength limit state for geotechnical, as well as structural, features in the AASHTO LRFD frame- work. For SLS, a PDF with an upper bound and lower bound (e.g., beta distribution) instead of open tail(s) (e.g., normal or lognormal distribution) may be more appropriate because the conditions represented by an open-tail PDF are not physi- cally possible when one considers foundation deformations. As noted, the lognormal PDF is used here to be consistent with the PDFs that have been used in LRFD calibration pro- cesses to date. Guidance for the selection of an appropriate PDF and development of the distribution parameters shown in Table 6.5 is provided in Nowak and Collins (2013) or other similar books that deal with probabilistic methods. Values of the lognormal mean and lognormal standard devi- ation are needed to use the lognormal PDF. These values can be obtained by using correlations with the mean and standard deviation values for normal distribution or calculated directly from the natural logarithm (ln) of the values of the data points. Table 6.5 presents the values for correlated mean (µLNC) and correlated standard deviation (sLNC). Table 6.6 shows the log- normal of accuracy values of data in Table 6.3, and Table 6.7 presents the values for arithmetic mean (µLNA) and arithmetic standard deviation (sLNA) based on the ln(X) values in Table 6.6. The correlated and arithmetic values of the mean (µLNC and µLNA, respectively) and standard deviation (sLNC and sLNA, Table 6.4. Statistics of Accuracy (X) Values Based on Data Shown in Table 6.3 Statistic Schmertmann et al. (1978) Hough (1959) D’Appolonia et al. (1968) Peck and Bazaraa (1969) Burland and Burbridge (1984) Count 20 20 20 20 20 Minimum 0.295 0.656 0.311 0.202 0.138 Maximum 4.618 4.294 2.176 4.000 4.735 µ 1.381 1.971 1.031 0.779 0.829 s 1.006 0.769 0.476 0.796 0.968 CV 0.729 0.390 0.462 1.022 1.168 Note: CV = s/µ. (text continues on page 160)

154 (a) (b) 0 1 2 3 4 5 6 7 8 9 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Fr eq ue nc y of O cc ur re nc e Accuracy, X (Predicted/Measured) -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 St an da rd N or m al V ar ia bl e, z Accuracy, X (Predicted/Measured) Data Points Predicted LN Fitted from Normal Statistics Predicted Normal Distribution Accuracy Data for Schmertmann et al. (1978) - Full Data Fit Figure 6.9. Schmertmann et al. (1978) method: (a) histograms for accuracy (X) and (b) plot of standard normal variable (z) as a function of X.

155 (a) (b) 0 1 2 3 4 5 6 7 8 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Fr eq ue nc y of O cc ur re nc e Accuracy, X (Predicted/Measured) -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 St an da rd N or m al V ar ia bl e, z Accuracy, X (Predicted/Measured) Data Points Predicted LN Fitted from Normal Statistics Predicted Normal Distribution Accuracy Data for Hough (1959) - Full Data Fit Figure 6.10. Hough (1959) method: (a) histograms for accuracy (X) and (b) plot of standard normal variable (z) as a function of X.

156 (a) (b) 0 1 2 3 4 5 6 7 8 9 10 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Fr eq ue nc y of O cc ur re nc e Accuracy, X (Predicted/Measured) -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 St an da rd N or m al V ar ia bl e, z Accuracy, X (Predicted/Measured) Data Points Predicted LN Fitted from Normal Statistics Predicted Normal Distribution Accuracy Data for D'Appolonia et al. (1968) - Full Data Fit Figure 6.11. D’Appolonia et al. (1968) method: (a) histograms for accuracy (X) and (b) plot of standard normal variable (z) as a function of X.

157 (a) (b) 0 2 4 6 8 10 12 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Fr eq ue nc y of O cc ur re nc e Accuracy, X (Predicted/Measured) -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 St an da rd N or m al V ar ia bl e, z Accuracy, X (Predicted/Measured) Data Points Predicted LN Fitted from Normal Statistics Predicted Normal Distribution Accuracy Data for Peck and Bazaraa (1969) - Full Data Fit Figure 6.12. Peck and Bazaraa (1969) method: (a) histograms for accuracy (X) and (b) plot of standard normal variable (z) as a function of X.

158 (a) (b) 0 2 4 6 8 10 12 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Fr eq ue nc y of O cc ur re nc e Accuracy, X (Predicted/Measured) -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 St an da rd N or m al V ar ia bl e, z Accuracy, X (Predicted/Measured) Data Points Predicted LN Fitted from Normal Statistics Predicted Normal Distribution Accuracy Data for Burland and Burbridge (1984) - Full Data Fit Figure 6.13. Burland and Burbridge (1984) method: (a) histograms for accuracy (X) and (b) plot of standard normal variable (z) as a function of X.

159 Table 6.7. Statistics of ln(X) Values Based on Data Shown in Table 6.6 Statistic Schmertmann et al. (1978) Hough (1959) D’Appolonia et al. (1968) Peck and Bazaraa (1969) Burland and Burbridge (1984) Count 20 20 20 20 20 Minimum -1.2205 -0.4220 -1.1664 -1.5989 -1.9783 Maximum 1.5299 1.4572 0.7777 1.3863 1.5550 µLNA 0.1173 0.6114 -0.0793 -0.4854 -0.5161 sLNA 0.6479 0.3807 0.5029 0.6226 0.7731 Note: µLNA = arithmetic mean of ln(X) values; sLNA = arithmetic standard deviation of ln(X ) values. Table 6.5. Correlated Statistics of Accuracy (X) for Lognormal PDFs Statistic Schmertmann et al. (1978) Hough (1959) D’Appolonia et al. (1968) Peck and Bazaraa (1969) Burland and Burbridge (1984) µLNC 0.1095 0.6076 -0.0665 -0.6078 -0.6177 sLNC 0.6528 0.3766 0.4398 0.8459 0.9274 Note: The µLNC and sLNC values for lognormal distribution were calculated from the normal (arithmetic) mean and standard deviation values in Table 6.4, respectively, by using the following equations based on idealized normal and lognormal PDFs: µLNC = ln(µ) - 0.50(sLNC)2; and sLNC = [ln{(s/µ)2 + 1}]0.5. Table 6.6. Lognormal of Accuracy Values [ln(X)] Based on Data Shown in Table 6.3 Site Schmertmann et al. (1978) Hough (1959) D’Appolonia et al. (1968) Peck and Bazaraa (1969) Burland and Burbridge (1984) S1 0.8141 0.7621 0.6190 -0.1881 -0.1542 S2 1.0157 0.3386 -0.5411 -1.4321 -1.7198 S3 -0.0889 0.2525 -1.1421 -1.5989 -1.9783 S4 -0.5021 0.6529 -0.2703 -0.7472 -0.6672 S5 -0.7097 0.4741 -0.4733 -0.3732 -0.0678 S6 0.2136 0.3732 0.1744 -0.9045 -0.2113 S7 -1.2205 -0.4220 -1.1664 -0.7097 -1.1664 S8 0.0690 0.7621 -0.0741 -0.5596 -0.6931 S9 -0.3677 0.7122 -0.2624 -0.4855 -0.8602 S10 0.0000 0.3216 -0.2318 -0.5947 -1.1701 S11 0.3646 0.6313 0.1484 -0.4463 -1.4271 S14 -0.1151 1.0155 0.2144 0.0834 -0.1398 S15 1.5299 1.4572 0.7777 1.3863 1.5550 S16 0.1226 1.1686 0.5281 -0.3023 -0.3023 S17 -0.0953 0.6225 0.0445 -0.4520 -0.6487 S20 0.6369 0.4951 -0.2671 -1.1144 -0.1699 S21 -0.4613 0.6022 0.1967 0.1226 -0.3947 S22 -0.2007 0.7448 -0.0788 -0.6633 -0.0308 S23 0.5141 0.4842 -0.0333 -0.6144 -0.3267 S24 0.8267 0.7787 0.2513 -0.1133 0.2513

160 respectively) for lognormal distributions are not equal. This is because the correlated values were based on derivations for an idealized lognormal distribution and not a sample dis- tribution from actual data, which may not necessarily fit an idealized lognormal distribution. In contrast, the arithmetic values were obtained by taking the arithmetic mean and stan- dard deviation directly from the ln(X) value of each data point noted in Columns 2 to 6 of Table 6.3. It is important to use the appropriate values of mean and standard deviation based on the syntax for a lognormal dis- tribution function used by a particular computational program. For example, if one is using the @RISK program by Palisade Corporation, then the RISKLOGNORM function in that pro- gram is based on arithmetic values (µ and s) of the normal distribution. In contrast, the Microsoft Excel LOGNORMDIST (or LOGNORM.DIST) function uses the arithmetic mean (µLNA) and standard deviation (sLNA) values of ln(X). Use of improper values of mean and standard deviation can lead to drastically different results. This issue is of critical importance because calibration in this report, as mentioned earlier, is based on Microsoft Excel. Figure 6.14 shows the CDFs for accuracy (X) for various analytical methods based on the LOGNORM.DIST function in the 2010 version of Microsoft Excel using the µLNA and sLNA values noted in Table 6.7. These CDFs can now be used to develop the PEC discussed in Section 6.1.2 for various analyti- cal methods. Figure 6.15 shows the PEC for the method by Schmertmann et al. (1978). The probability of exceedance corresponding to a given predicted settlement can now be readily determined. For example, assume that the geotechnical engineer has pre- dicted a settlement of 0.85 in. The probability of exceedance of 1 in. in this case is approximately 32%. This can be found by drawing line AB, finding the intersection of the line with the curve for 1 in., drawing line BC, and reading the value from the ordinate of the PEC in Figure 6.15. Four additional curves for settlements of 1.5, 2, 2.5, and 3 in. are shown in Figure 6.15. Using the procedure demonstrated for the example above (see dashed arrows in Figure 6.15), if the predicted (calculated) value is 0.85 in., then the probability of the measured value being greater than 1.5, 2, 2.5, and 3 in. is approximately 14%, 6%, 3%, and 2%, respectively. A load factor for settlement (gSE) can be determined using the procedure in Section 6.1.2.5. For example, assume the predicted settlement is 1 in. To determine the value of gSE for a 25% target probability of exceedance (PeT), draw a horizontal line from Point D on the ordinate corresponding to a value of 25%. Next, draw a vertical line from Point E on the abscissa corresponding to a value of 1 in. Locate the point of inter- section, F, which lies between the curves for 1 in. and 1.5 in. Interpolating between the two curves leads to a value of approx- imately 1.35 in. Based on the definition of gSE noted above, the value of gSE is equal to 1.35 in./1.0 in., or 1.35. PECs for other analytical methods noted in Figure 6.14 are given in Figure 6.16 to Figure 6.19. Those PECs can be used in a similar manner as demonstrated for the PEC for the Schmertmann et al. (1978) method. A PEC chart is essentially a representation of the CDF of accuracy, or X. Similar charts are referred to as probabilistic design charts by Das and Sivakugan (2007) and Sivakugan and Johnson (2002, 2004) and artificial neural network charts by Shahin et al. (2002) and Musso and Provenzano (2003). Although not specifically in chart format, similar concepts are presented in Tan and Duncan (1991) and Duncan (2000). The specific format of PEC that is developed and used here is amenable to correlation to the AASHTO LRFD–based concept of target reliability index, as explained in Step 5. Figure 6.14. CDFs for various analytical methods for estimation of immediate settlement of spread footings. D C A E F B Figure 6.15. PEC for the Schmertmann et al. (1978) method. (continued from page 153)

161 6.1.3.5 Step 5: Develop Reliability Analysis Procedure The estimation of load factor for settlement (gSE) in terms of probability of exceedance (Pe) was demonstrated in the previ- ous step. In the AASHTO LRFD framework, calibrations are expressed in terms of a reliability index (b). b can be expressed in terms of Pe of a predicted value by using Equation 6.4, which applies to normally distributed data. As observed from Step 4, lognormal distributions have been used. Fur- thermore, the CV values noted in Table 6.4 are rather large. For a normal random variable, the relationship between b and Pe depends only on CV (i.e., one parameter), but for a lognormal distribution, it depends on the mean and stan- dard deviation or the mean and CV (i.e., two parameters). Therefore, strictly speaking, b should be based on a lognor- mal function. However, for b <2.0 there is not a significant practical difference in the Pe values for data that are normally or lognormally distributed for the wide range of CVs noted in Table 6.4. An assumption of a normal distribution is gen- erally conservative in the sense that for a given b it gives a larger Pe compared with a lognormal distribution. Further- more, conventionally the normal distribution has been assumed for strength limit states in AASHTO LRFD (as well as other international codes), which has b values larger than 2.0. The key consideration is that the type of distribution is not as important as being consistent and not mixing different distributions while comparing b values. When these various issues are taken into account, the Microsoft Excel formula that assumes normally distributed data is considered to be acceptable for SLS calibrations. Table 6.8 and Figure 6.20 were generated by using Equa- tion 6.4: NORMSINV 1 (6.4)Pe)(β = − Figure 6.16. PEC for the Hough (1959) method. Figure 6.17. PEC for the D’Appolonia et al. (1968) method. Figure 6.18. PEC for the Peck and Bazaraa (1969) method. Figure 6.19. PEC for the Burland and Burbridge (1984) method.

162 Table 6.8. Values of b and Corresponding Pe Based on Normally Distributed Data b Pe (%) b Pe (%) b Pe (%) b Pe (%) 2.00 2.28 1.50 6.68 1.00 15.87 0.50 30.85 1.95 2.56 1.45 7.35 0.95 17.11 0.45 32.64 1.90 2.87 1.40 8.08 0.90 18.41 0.40 34.46 1.85 3.22 1.35 8.85 0.85 19.77 0.35 36.32 1.80 3.59 1.30 9.68 0.80 21.19 0.30 38.21 1.75 4.01 1.25 10.56 0.75 22.66 0.25 40.13 1.70 4.46 1.20 11.51 0.70 24.20 0.20 42.07 1.65 4.95 1.15 12.51 0.65 25.78 0.15 44.04 1.60 5.48 1.10 13.57 0.60 27.43 0.10 46.02 1.55 6.06 1.05 14.69 0.55 29.12 0.05 48.01 0.00 50.00 Note: Linear interpolation may be used as an approximation for intermediate values. 1 in 1 1 in 10 1 in 1001% 10% 100% 0.0 0.5 1.0 1.5 2.0 P ,ec nadeec xE f o ytilibab orP e Reliability Index, Figure 6.20. Relationship between b and Pe for the case of a single load and single resistance. The correlation between b and Pe can now be used to rephrase the earlier discussion with respect to Figure 6.15. In that discussion, as an example, it was assumed that the geo- technical engineer has predicted a settlement of 0.85 in. From Figure 6.15, it was determined that the probability of exceed- ance of 1, 1.5, 2, 2.5, and 3 in. was approximately 32%, 14%, 6%, 3%, and 2%, respectively. Using Table 6.8 (or Figure 6.20 or Equation 6.3), the results can now be expressed in terms of reliability index values. Thus, it can be stated that if the pre- dicted settlement is 0.85 in., then the assumption of tolerable settlement values of 1, 1.5, 2, 2.5, and 3 in. means a reliability index of approximately 0.45, 1.10, 1.55, 1.90, and >2.00, respectively. In the example, the geotechnical engineer has predicted settlement dP = 0.85 in. by using the Schmertmann et al. (1978) method. The owner has specified that the SLS design for the bridge should be performed using a reliability index of 0.50. What is the value of gSE and the tolerable settlement that the bridge designer should use? The load factor (gSE) is a function of the probability of exceedance (Pe) of the foundation deformation under consid- eration, which in this example is the immediate settlement of spread footings calculated by using the analytical method of Schmertmann et al. (1978). By using either Equation 6.4 or Table 6.8, a value of Pe ≈ 0.3085 (or 30.85%) is obtained for b = 0.50. Equation 6.5 is the formula used in Microsoft Excel to deter- mine a value of accuracy (X) in terms of Pe, the mean value (µLNA), and the standard deviation (sLNA) of the lognormal dis- tribution function as computed in Step 4. The value of X repre- sents the probability of the accuracy value (dP/dT) being less than a specified value. P Xe ( )= µ σLOGNORMDIST , , (6.5)LNA LNA From Table 6.7, for the Schmertmann et al. (1978) method, µLNA = 0.1173 and sLNA = 0.6479. The goal is to determine the value of X that gives Pe = 0.3085. Thus, for this example, the expression for Pe can be written as shown by Equation 6.6: P Xe ( )= =LOGNORMDIST , 0.1173, 0.6479 0.3085 or 30.85% (6.6) Using Goal Seek in Microsoft Excel, X (i.e., dP/dT) ≈ 0.813. Note that in the 2010 version of Microsoft Excel, another function, LOGNORM.DIST, can also be used. In this case, the same result (X ≈ 0.813) is obtained by using the follow- ing syntax and using the Goal Seek function to determine X (TRUE indicates the use of a CDF): P Xe ( )= =LOGNORM.DIST , 0.1173, 0.6479, TRUE 0.3085. In the context of the AASHTO LRFD framework, the load factor (gSE) is the reciprocal of X. Thus, for immediate settle- ment of spread footings based on the method of Schmertmann et al. (1978), gSE = 1/0.813 ≈ 1.23. As per the AASHTO LRFD framework, the load factor is rounded up to the nearest 0.05, and thus gSE = 1.25 should be used. Therefore, in the bridge design example, the bridge designer should use a settlement value of (gSE)(dP) = (1.25)(0.85 in.) = 1.06 in. to assess the effect of settlement on the structure. This value can also be obtained using the graphic technique explained earlier with respect to Figure 6.15. The example demonstrated with respect to Figure 6.15 also assumed a toler- able settlement of 0.85 in., and it was found that a settlement of 1 in. would imply a 32% probability of exceedance. These

163 values are close to the value of 1.06 in. for a 30.85% probability of exceedance obtained here. Given that the load factor is rounded to the nearest 0.05, the result from the graphic tech- nique is sufficiently accurate. The procedure demonstrated in the above example can be used to develop values of gSE for any desired b by using the lognormal distribution of X for the method of Schmertmann et al. (1978). A similar approach can be used for other analytical methods and distributions. Table 6.9 presents the values of gSE results for the various analytical methods shown in Figure 6.8 and Table 6.2. Obvi- ously, gSE values less than 1.0 should not be allowed to prevent the risk of bridges being underdesigned. Furthermore, the values of gSE should be rounded to the nearest 0.05, because not doing so implies a level of confidence that is not justified by the available data. Table 6.10 presents values of gSE that are bounded by 1.0 and rounded to the nearest 0.05. 6.1.3.6 Step 6: Review Results and Selection of Load Factor for Settlement Figure 6.21 shows a plot of gSE versus b based on the data shown in Table 6.10. The current practice based on AASHTO LRFD (2012) is as follows: 1. Use the Hough (1959) method to estimate immediate settlements. 2. Use gSE = 1.0. The data in Table 6.10 and the graph in Figure 6.21 imply that b ≈ 1.65 corresponds to the current practice noted above. b ≈ 1.65 is based on the data set in Table 6.2. If additional data were included, or if a different regional data set were to be used, then the value of b may be different. However, based on a review of state practices performed as part of Samtani and Table 6.9. Computed Values of gSE for Various Methods to Estimate Immediate Settlement of Spread Footings on Cohesionless Soils Reliability Index (b) Schmertmann et al. (1978) Hough (1959) D’Appolonia et al. (1968) Peck and Bazaraa (1969) Burland and Burbridge (1984) 0.00 0.89 0.54 1.08 1.62 1.68 0.50 1.23 0.66 1.39 2.22 2.47 1.00 1.70 0.79 1.79 3.03 3.63 1.50 2.35 0.96 2.30 4.13 5.34 2.00 3.25 1.16 2.96 5.64 7.86 2.50 4.49 1.41 3.81 7.71 11.58 3.00 6.21 1.70 4.89 10.52 17.04 3.50 8.59 2.06 6.29 14.36 25.08 Table 6.10. Proposed Values of gSE for Various Methods to Estimate Immediate Settlement of Spread Footings on Cohesionless Soils Reliability Index (b) Schmertmann et al. (1978) Hough (1959) D’Appolonia et al. (1968) Peck and Bazaraa (1969) Burland and Burbridge (1984) 0.00 1.00 1.00 1.10 1.60 1.70 0.50 1.25 1.00 1.40 2.20 2.45 1.00 1.70 1.00 1.80 3.05 3.65 1.50 2.35 1.00 2.30 4.15 5.35 2.00 3.25 1.15 2.95 5.65 7.85 2.50 4.50 1.40 3.80 7.70 11.60 3.00 6.20 1.70 4.90 10.50 17.05 3.50 8.60 2.05 6.30 14.35 25.10 Note: The values of gSE have been rounded to the nearest 0.05 and limited to 1.00 or larger.

164 Figure 6.21. Evaluation of gSE based on current and target reliability indices. Nowatzki (2006) and Samtani et al. (2010), it is anticipated that, based on its inherent conservatism, the value of b is anticipated to be large and greater than 1.0 for the Hough (1959) method and gSE = 1.0. The majority of the data points for the Hough method plot below gSE = 1.0, which suggests a significant conservatism in the Hough method. This is con- sistent with the earlier observation that the Hough method is conservative (overpredicts) by a factor of approximately two (see Table 6.4), which leads to an unnecessary use of deep foundations instead of spread footings. Based on a consider- ation of reversible and irreversible SLSs for bridge super- structures, as shown earlier, a target reliability index (bT) in the range of 0.50 to 1.00 for calibration of load factor gSE for foundation deformation in the Service I limit state is accept- able. Settlement is clearly an irreversible limit state with respect to the foundation elements, but it may be reversible through intervention with respect to the superstructure. This type of logic would lead to consideration of 0.50 as the bT for calibration of immediate settlements under spread footings on cohesionless soils. In Figure 6.21, the horizontal bold dashed line corresponds to b = 0.50 for SLS evaluation. The boxes around the markers for various methods represent the spread of predicted values for the five methods evaluated here. For b = 0.50, if gSE = 1.25 is adopted, then it would encompass three of the five methods. The value of gSE = 1.25 includes the Schmertmann et al. (1978) method, which is currently recommended by Samtani and Nowatzki (2006) and Samtani et al. (2010) and is commonly used in U.S. practice. Based on these observations, gSE = 1.25 is recommended. 6.1.3.7 Step 7: Select gSE for Service I Limit State As demonstrated in Steps 5 and 6, the gSE value can be deter- mined for any reliability index (b) for various analytical methods. Use of the format shown in Figure 6.21 will lead to better regional practices in the sense that owners desiring to calibrate their local practices can readily see the implication of a certain method on the selection and cost of a foundation system. This is because the chart in Figure 6.21 shows the reliability of various methods and permits selection of an appropriate method that would lead to selection of a proper foundation system for a given set of b and gSE values (i.e., not using a deep foundation system when a spread foundation would be feasible). The agency that calibrates a value of gSE based on a locally accepted analytical method must ensure that the chosen value of gSE is consistent with the serviceability of the substructure and superstructure design, as discussed in Step 6. 6.1.4 Meaning and Use of gSE The meaning and use of gSE must be understood in the specific context of structural implications within the AASHTO LRFD framework. In particular, the value of gSE is used to assess structural implications, such as the generation of additional (secondary) moments within a given span because of settle- ment of one of the support elements and effect on the riding surface, and conceivably even appearance and roadway damage issues. If taken literally, the value of gSE = 1.25 in the example could be interpreted to mean that the settlement predicted (dP)

165 by the analytical method of Schmertmann et al. (1978) needs to be increased by 25% to limit the probability of exceedance (Pe) of the tolerable settlement (dT) to less than 30.85%, cor- responding to a target reliability index (bT) of 0.50. However, this literal interpretation is not entirely correct because the value of gSE (1.25 in this case) is just one of many load factors in the Service I limit state load combinations within the overall AASHTO LRFD framework. In addition to the SLS, settlements need to be considered at applicable strength limit states, because although settlements can cause serviceability problems, they can also have a signifi- cant effect on moments in continuous superstructures that can result in increased member sizes. Settlement is handled more explicitly in AASHTO LRFD (in which it is listed among the loads in Table 3.4.1-1, Load Combinations and Load Factors) than it was in the Standard Specifications for Highway Bridges. It appears in four of the five strength load combinations and three of the four service load combinations. This emphasis may appear to be a departure from past practice, as exemplified by AASHTO’s Standard Specifications for Highway Bridges, in that settlement does not appear in those load combinations. But settlement is mentioned in Article 3.3.2.1 of Standard Specifications (2002), which states “If differential settlement is anticipated in a structure, consideration should be given to stresses resulting from this settlement.” The parent article (3.3, Dead Load) implies that settlement effects should be considered wherever dead load appears in the allowable stress design or load factor design (LFD) load combinations. The additional moments due to the effect of settlement are very dependent on the stiffness of the bridge, as well as the angular rotation (i.e., differential settlement normalized with respect to span length, as discussed in Chapter 2 in the section on tolerable vertical deformation criteria). A limited study (Schopen 2010) of several two- and three-span steel and pre- stressed concrete continuous bridges selected from the NCHRP Project 12-78 database showed that allowing the full angular distortion suggested in Table 2.2 could result in an increase in the factored Strength I moments on the general order of as little as 10% for the more flexible units considered to more than double the moment from only the factored dead and live load moments for the stiffer units. These order of magnitude estimates are based on elastic analysis without consideration of creep, which could significantly reduce the moments, especially for relatively stiff concrete bridges. For example, a W 36 × 194 rolled beam with a 10 × 17⁄8-in. bottom cover plate composite with a 96 × 7 3⁄4-in. deck is presented in Brockenbrough and Merritt (2011). The computed moments of inertia for the basic beam and short-term composite and long-term composite sections were in the approximate ratio 1:2:3. This indicates that consideration of construction sequence, an appropriate choice of section properties, and possibly even a time-dependent calculation of creep effects could be beneficial in some cases. Use of the construction point concept (see next section) would also mitigate the settlement moments. Nevertheless, Schopen’s results suggest that the use of permissible angular distortions approaching those currently allowed by AASHTO LRFD requires careful consideration of the particular bridge and its design objectives. As the predicted (estimated or calculated) settlement (dP) is based on the Service I load combination and the load factor (gSE) is used to modify the Service I load combination, the use of gSE can potentially lead to a circular reference in the bridge design process that may require significant iterations. The following procedure is recommended to avoid a circular reference: 1. Assume zero settlement and determine the service load by using the Service I load combination. When settlement is assumed to be zero, the value of gSE is irrelevant. 2. Determine dP based on the method that has been calibrated using the procedure described here. 3. Multiply dP by gSE = 1.25 to determine the tolerable (lim- iting) settlement (dT) that should be incorporated into bridge design through use of the d-0 angular distortion and construction point concept described below. The use of the d-0 angular distortion and construction point con- cept incorporates the span lengths, differential settlements between support elements, and the various stages of con- struction into the bridge design process. 6.1.5 Effect of Foundation Deformations on Bridge Superstructures Uneven displacements of bridge abutments and pier founda- tions often lead to costly maintenance and repair measures associated with the structural distress that the bridge super- structure and substructure might experience. The bridge superstructure and substructure displacements can be due to a variety of reasons, including foundation deformations. The foundation deformations need to be evaluated in the context of span lengths and various construction steps to understand their effect on the bridge superstructures. These aspects and the concepts of angular distortion and construction points are discussed in this section. The application of load factors due to deformation (e.g., gSE) is also presented. For all bridges, stiffness should be appropriate to the con- sidered limit state. Similarly, the effects of continuity with the substructure should be considered. In assessing the structural implications of foundation deformations of concrete bridges, the determination of the stiffness of the bridge components should consider the effects of cracking, creep, and other inelastic responses.

166 6.1.5.1 d-0 Concept for Vertical Deformations (Settlements) Because of the inherent variability of geomaterials, the verti- cal deformations at the support elements of a given bridge (i.e., piers and abutments) will generally be different. This is true regardless of whether deep foundations or spread footings are used. Figure 6.22 shows the hypothetical case of a four-span bridge structure with five support elements (two abutments and three piers) for which the calculated settlement (d) at each support is different (the figure assumes rigid substructure units between the foundations and bridge superstructure). Differential settlements induce bending moments and shear in the bridge superstructure when spans are continuous over supports and potentially cause structural damage. To a lesser extent, they can also cause damage to a simple-span bridge. However, the major concern with simple-span bridges is the quality of the riding surface and aesthetics. Due to a lack of continuity over the supports, the changes in slope of the riding surface near the supports of a simple-span bridge induced by differential settlements may be more severe than those in a continuous-span bridge. Depending on factors such as the type of superstructure, the connections between the superstructure and substructure units, and the span lengths and widths, the magnitudes of differential settlement that can cause damage to the bridge structure can vary significantly. For example, the damage to the bridge structure due to a differential settlement of 2 in. over a 50-ft span is likely to be more severe than the same amount of differential settlement over a 150-ft span. Various studies, including Grant et al. (1974) and Skempton and MacDonald (1956), have determined that the severity of differential settlement on structures is roughly proportional to the angular distortion (A), which is a normalized measure of differential settlement that includes the distance over which the differential settlement occurs. Angular distortion is defined as the difference in settlement between two points (Dd) divided by the distance between the two points (L), as shown in Fig- ure 6.22. Angular distortion is a dimensionless quantity that is expressed as an angle in radians. Theoretically, the ratio Dd/L represents the tangent of the angle of distortion, but for small values of the tangent, the angles are also very small. Thus, the tangents (i.e., A) are shown as angles in Figure 6.22. For bridge structures, the two points used to evaluate the differential settlement are commonly selected as the distance between adjacent support elements (see Figure 6.22). Although all analytical methods for estimating settlements have a certain degree of uncertainty, the uncertainty of the calculated differential settlement is larger than the uncertainty of the calculated total settlement at each of the two support elements used to calculate the differential settlement (e.g., between an abutment and a pier or between two adjacent piers). For example, if one support element settles less than the amount calculated, and the other support element settles the amount calculated, the actual differential settlement will be larger than the difference between the two values of calculated settlement at the support elements. On the basis of these considerations and guidance in Samtani and Nowatzki (2006) and Barker et al. (1991), the following limit state criteria are suggested to estimate a realistic value of differential settle- ment and angular distortion: • The actual settlement of any support element could be as large as the value calculated by using a given method. Support Element Settlement Span Differential Settlement Angular Distortion Abutment 1 A1 1 | A1 – P1| A1 = (| A1 – P1|)/L1 Pier 1 P1 2 | P1 – P2| A2 = (| P1 – P2|)/L2 Pier 2 P2 3 | P2 – P3| A3 = (| P2 – P3|)/L3 Pier 3 P3 4 | P3 – A2| A4 = (| P3 – A2|)/L4 Abutment 2 A2 A1 P1 P2 P3 A2 A1 A2 A3 A4 L1 Span 1 L2 Span 2 L3 Span 3 L4 Span 4 Abutment 1 Pier 1 Pier 2 Pier 3 Abutment 2 Figure 6.22. Concept of settlement and angular distortion in bridges.

167 • At the same time, the actual settlement of the adjacent sup- port element could be zero instead of the value calculated by using the same given method. The concept outlined above is referred to as the d-0 concept, with a value of d representing full calculated settlement at one support of a span and a value of 0 representing zero settle- ment at an adjacent support. Use of the d-0 approach would result in an estimated maximum possible differential settle- ment between two adjacent supports equal to the larger of the two total settlements calculated at either end of any span. Thus, with respect to Figure 6.22, where dA1 < dP1 > dP2 < dP3 < dA2 represents the relative magnitudes of the total settlement at each support point, the differential settlements and angular distortion for design are evaluated as shown in Table 6.11. The values in Table 6.11 represent the maximum values for each span according to the criteria above and should be used for design. The hypothetical settlement profile assumed for computation of the design angular distortion for each span is represented by the dashed lines in Figure 6.23. It should not be confused with the calculated total settlement profile, which is represented by the solid lines. From the viewpoint of the damage to the bridge superstructure, the concept shown in Figure 6.23 is more important for continuous-span structures than single-span structures because of the ability of the latter to permit larger movements at support elements. 6.1.5.2 Construction Point Concept Most designers analyze foundation deformations as if a weightless bridge structure is instantaneously set in place and all the loads are applied at the same time. In reality, loads are applied gradually as construction proceeds. Consequently, settlements also occur gradually as construction proceeds. Several critical construction points or stages during construc- tion should be evaluated separately by the designer. Figure 6.3 shows the critical construction stages and their associated load- displacement behavior. The baseline format of Figure 6.24 is the same as that of Figure 6.3 except that the figure considers vertical load and vertical displacement (i.e., settlement). For- mulation of settlements in the manner shown in Figure 6.24 would permit an assessment of settlements up to that point that can affect the bridge superstructure. For example, the settlements that occur before placement of the superstructure may not be relevant to the design of the superstructure. Thus, the settlements between application of loads X and Z are the most relevant. The percentage of settlement between the placement of beams and the end of construction is generally in the range of 25% to 75%, depending on the type of superstructure and the construction sequence. With respect to the example of the four-span bridge and the angular distortions in Table 6.11, the use of the construction point concept would result in smaller angular distortions to be considered in the structural design. This will be true of any bridge evaluation. Using Figure 6.22 as a reference, Figure 6.25 compares the profiles of the calcu- lated settlements (solid lines), hypothetical maximum angular distortions (dashed lines), and the range of actual angular Table 6.11. Estimation of Design Differential Settlements and Design Angular Distortions for Hypothetical Case Shown in Figure 6.22 Span Design Differential Settlement Design Angular Distortion 1 dP1 = dP1 (assume dA1 = 0) A1 = dP1/L1 2 dP1 = dP1 (assume dP2 = 0) A2 = dP1/L2 3 dP3 = dP3 (assume dP2 = 0) A3 = dP3/L3 4 dA2 = dA2 (assume dP3 = 0) A4 = dA2/L4 A1 P1 P2 P3 A2 DA1 DA2 DA3 DA4 L1 Span 1 L2 Span 2 L3 Span 3 L4 Span 4 Abutment 1 Pier 1 Pier 2 Pier 3 Abutment 2 Legend: Calculated settlement profile (refer to Figure 6.22) Hypothetical settlement profile assumed for computation of maximum angular distortion Figure 6.23. Estimation of maximum angular distortion in bridges.

168 distortions (hatched-pattern zones) based on the construction point concept. The angular distortions shown in Figure 6.25 should be compared with the limit state criteria for angular distortions provided in Article 10.5.2.2 of AASHTO LRFD (2012) and Table 6.2 here. On the basis of the discussions above, it is recommended that the limit state of vertical deformations (i.e., settlements) should be evaluated in terms of angular distortions using the construction point concept. While using the construction point concept, it is important to recognize that the various quantities are being measured at discrete construction stages, and the associated settlements are considered to be immediate. However, the evaluation of total settlement and the maximum (design) angular distortion must also account for long-term Legend: W load after foundation construction δW displacement under load W X load after pier column and wall construction δX displacement under load X Y load after superstructure construction δY displacement under load Y Z load after wearing surface construction δZ displacement under load Z S service load (or limit) state (SLS) F factored load (strength limit state) (a) (b) X Z Y W Foundation (shallow or deep) Substructure Superstructure Wearing Surface Figure 6.24. Construction point concept for a bridge pier: (a) identification of critical construction points and (b) conceptual load-displacement pattern for a given foundation. A1 P1 P2 P3 A2 L1 Span 1 L2 Span 2 L3 Span 3 L4 Span 4 Abutment 1 Pier 1 Pier 2 Pier 3 Abutment 2 Legend: Calculated settlement profile (refer to Figure 6.22) Hypothetical settlement profile assumed for computation of maximum angular distortion Range of relevant angular distortions using construction point concept Figure 6.25. Angular distortion in bridges based on construction point concept.

169 settlements. For example, significant long-term settlements may occur if foundations are founded on saturated clay depos- its or if a layer of saturated clay falls within the zone of stress influence below the foundation, even though the foundation itself is founded on competent soil. In such cases, long-term settlements will continue under the total construction load (Z), as shown by the dashed line in Figure 6.24. Continued settlements during the service life of the structure will tend to reduce the vertical clearance under the bridge, which may cause problems when large vehicles pass below the bridge superstructure. As a result, the geotechnical specialist must estimate and report to the structural specialist the magnitude of the long-term settlement that will occur during the design life of the bridge. A key point in evaluating settlements at critical construction points is the close coordination required between the structural and geotechnical specialists. 6.1.5.3 Foundations Proportioned for Equal Settlement Often geotechnical and structural specialists will try to pro- portion foundations for equal settlement. In this case, the argument is made that there will be no differential settlement. Although this concept may work for a building structure because the footprint is localized, it is a fallacy to assume zero differential settlement for a long linear highway structure such as a bridge or a wall due to the inevitable variation of the properties of geomaterials along the length of the structure. Furthermore, as noted earlier, the prediction of settlements from any given method is uncertain in itself. Thus for high- way structures, even when the foundations are proportioned for equal settlement, it is advisable to evaluate differential settlement assuming that the settlement of any support ele- ment could be as large as the value calculated by using a given method, while at the same time, the settlement of the adjacent support element could be zero. 6.1.5.4 Horizontal Deformations Horizontal deformations generally occur due to sliding or rotation (or both) of the foundation. Horizontal deformations cause more severe and widespread problems than do equal magnitudes of vertical movement (Moulton et al. 1985). The most common location of horizontal deformations is at the abutments, which are subject to lateral earth pressure. Horizontal movements can also occur at the piers as a result of lateral loads and moments at the top of the substructure unit. The estimation of the magnitudes of horizontal move- ments should take into account the movements associated with lateral squeeze, as discussed in Samtani and Nowatzki (2006) and Samtani et al. (2010). Lateral movements due to lateral squeeze can be estimated by geotechnical specialists, and lateral movements due to sliding or lateral deformations of deep foundations can be estimated by structural specialists using input from geotechnical specialists. The limiting hori- zontal movements are strongly dependent on the type of superstructure and the connection with substructure and are therefore project specific. 6.1.6 Practical AASHTO LRFD Application of the Load Factor for Deformation Using the Construction Point Concept Article 10.5.2.2 of AASHTO LRFD (2012) addresses the topic of tolerable movements and movement criteria. This section is intended to provide additional guidance to incorporate the concept of the load factor for deformation and the construc- tion point concept into LRFD Article 10.5.2.2. The following steps should be followed to estimate a practical value of angular distortion of the superstructure on the basis of foundation settlement; a similar approach can be applied and is recom- mended for evaluation of horizontal movement and rotation of foundations. 6.1.6.1 Vertical Deformations (Settlement) 1. Compute total foundation settlement at each support ele- ment by using an owner-approved method for the assumed foundation type (e.g., spread footings, driven piles, drilled shafts) as follows: a. Determine dta, the total foundation settlement using all applicable loads in the Service I load combination. b. Determine dtp, the total foundation settlement before construction of bridge superstructure. This settlement would generally be as a result of all applicable sub- structure loads computed in accordance with a Service I load combination. c. Determine dtr, relevant total settlement: dtr = dta - dtp. 2. At a given support element assume that the actual relevant settlement could be as large as the value calculated by the chosen method. At the same time, assume that the settle- ment of the adjacent support element could be zero instead of the relevant settlement value calculated by the same chosen method. Thus, differential settlement (dd) within a given bridge span is equal to the larger of the relevant settle- ment at each of two supports of a bridge span. Compute angular distortion (Ad) as the ratio of dd to span length (Ls), where Ad is measured in radians. The discussion with respect to Table 6.11 and Figure 6.23 in Section 6.1.5 is applicable to this step. 3. Compute modified angular distortion (Adm) by multiplying Ad from Step 2 with the gSE values for settlement using the approach discussed in Section 6.1.4 based on the analytical method used for computing the total settlement value.

170 4. Compare the Adm value with owner-specified angular dis- tortion criteria. If owner-specified criteria are not available, then use 0.008 radians for the case of simple spans and 0.004 radians for the case of continuous spans as the limit- ing angular distortions. This was discussed in Chapter 2 in the section on tolerable vertical deformation criteria. Other angular distortion limits may be appropriate after consideration of • Cost of mitigation through larger foundations, realign- ment, or surcharge; • Rideability; • Vertical clearance; • Tolerable limits of deformation of other structures associated with the bridge (e.g., approach slabs, wing- walls, pavement structures, drainage grades, utilities on the bridge); • Roadway drainage; • Aesthetics; and • Safety. 5. Evaluate the structural ramifications of the computed angular distortions that are within acceptable limits as per Step 4. Modify foundation design as appropriate based on structural ramifications. 6. The above procedure should also be used for cases in which the foundations of various support elements are proportioned for equal total settlement, because the pre- diction of settlements from any given method is in itself uncertain. 6.1.6.2 Lateral Deformations Using procedures similar to settlement evaluation specified for vertical deformation in the previous subsection, lateral (horizontal) movement at the foundation level can also be evaluated. Horizontal movement criteria should be estab- lished at the top of the foundation based on the tolerance of the structure to lateral movement, with consideration of the column length and stiffness. The above guidance should take into account the following guidance from Article C10.5.2.2 of AASHTO LRFD (2012): Rotation movements should be evaluated at the top of the substructure unit in plan location and at the deck elevation. Tolerance of the superstructure to lateral movement will depend on bridge seat or joint widths, bearing type(s), struc- ture type, and load distribution effects. 6.1.6.3 Walls The procedure for computing angular distortions can also be applied for evaluating angular distortions along and trans- verse to retaining walls, as well as the junction of the approach walls to abutment walls. The angular distortion values along a retaining wall can be used to select an appropriate wall type (e.g., MSE walls can tolerate larger angular distortions than cast-in-place walls). 6.1.6.4 General Comments The following guidance from AASHTO LRFD Article 10.5.2.2 should be followed while implementing the recommenda- tions made in previous sections: Foundation movement criteria shall be consistent with the function and type of structure, anticipated service life, and consequences of unacceptable movements on structure perfor- mance. Foundation movement shall include vertical, horizontal, and rotational movements. The tolerable movement criteria shall be established by either empirical procedures or structural analyses, or by consideration of both. Foundation settlement shall be investigated using all applicable loads in the Service I load combination specified in Table 3.4.1-1. Transient loads may be omitted from settlement analyses for foundations bearing on or in cohesive soil deposits that are subject to time-dependent consolidation settlements. All applicable service limit state load combinations in Table 3.4.1-1 shall be used for evaluating horizontal move- ment and rotation of foundations. Additional guidance is provided below: • All foundation deformation evaluations should be based on the geomaterial information obtained in accordance with Article 10.4 of AASHTO LRFD (2012). • The bridge engineer should add deformations from the sub- structure (elements between foundation and superstructure) as appropriate in evaluation of angular distortions at the deck elevation. • Although the angular distortion is generally applied in the longitudinal direction of a bridge, similar analyses should be performed in the transverse direction based on consid- eration of bridge width and stiffness. 6.1.7 Proposed AASHTO LRFD Provisions In AASHTO LRFD (2012), Article 10.5.2 (Service Limit States) in Section 10 (Foundations) is the primary article that provides guidance for SLS design for bridge foundations in terms of tolerable movements. Article 10.5.2 is referenced in other articles, as indicated in Table 6.12. The changes based on geotechnical considerations are pri- marily needed in Article 10.5.2. As Article 10.5.2 references Article 3.4, changes are also needed in that article. These changes are provided in Chapter 7.

171 6.2 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, after which 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 provi- sions 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 distribu- tion of reinforcement may require a reduction in the reinforce- ment 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., and Class 2 relates to an estimated maximum crack width of 0.01275 in. Class 2 is typically used for situations in which 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 regard 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 con- ditions have been so defined in the design specifications in order to provide flexibility in the application of these provi- sions to meet the needs of the bridge owner. The load factors for dead load (DL) and live load (LL) spec- ified for the Service I load combination are as follows: DL load factor = 1.0 and 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 No. 5 and No. 6 bars, and the designer does not switch to No. 6 bars unless No. 5 bars result in bar spacing less than the minimum spacing allowed. This practice 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 deter- mined on the basis of a calculated design load, the reinforce- ment does not change with the change in girder spacing, which results 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, design- ers may select different member dimensions, resulting in dif- ferent reinforcement areas. Even for the same reinforcement area, the designer may use bars or strands of different diam- eters and spacing and, consequently, obtain different crack resistance and a different reliability index for each 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 the 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. 6.2.1 Live Load Model Reinforced concrete decks designed using the conventional method have been designed for the heavy axles of the design Table 6.12. Summary of Relevant Articles in AASHTO LRFD for Foundation Deformations Article Title Relates to 10.6.2.2 Tolerable Movements Spread footings 10.6.2.5 Overall Stability Spread footings 10.7.2.2 Tolerable Movements Driven piles 10.7.2.4 Horizontal Pile Foundation Movement Driven piles C10.7.2.5 Commentary to Settlement Due to Downdrag Driven piles 10.8.2.1 Service Limit State Drilled shafts 10.8.2.2.1 General Drilled shafts 10.8.2.3 Horizontal Movements of Shafts and Shaft Groups Drilled shafts 10.9.2.2 Tolerable Movements Micropiles 10.9.2.4 Horizontal Micropile Foundation Movement Micropiles C11.10.11 Commentary to MSE Abutments MSE walls 14.5.2.1 Number of Joints Joints and bearings Note: Article 10.5.2 and its subarticles are frequently referenced in the articles noted in the left-hand column and their corresponding commentary portion. In this table, the article number is based on the first occurrence of the reference to Article 10.5.2.

172 truck. This practice required developing the statistical param- eters of the axle loads of the trucks in the weigh-in-motion (WIM) data. The statistical parameters for the axle loads are presented in Chapter 5. Statistical parameters corresponding to a 1-year return period were assumed in the reliability analy- sis. Average daily truck traffic (ADTT) of 1,000, 2,500, 5,000, and 10,000 were considered; however, an ADTT of 5,000 was used as the basis for the calibration. 6.2.2 Target Reliability Index 6.2.2.1 Limit State Function For the control of cracking of reinforced concrete through the distribution of reinforcement, the limiting criteria are the cal- culated crack widths, assumed to be 0.017 and 0.01275 in. for Class 1 and Class 2, respectively. Due to the lack of clear con- sequences 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. 6.2.2.2 Statistical Parameters of Variables Included in the Design Several variables affect the resistance of prestressed compo- nents. Table 6.13 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 infor- mation based on research studies by Siriaksorn and Naaman (1980) and Nowak et al. (2008). 6.2.2.3 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, No. 5 bars were used unless they resulted in bar spacing less than 5 in., the minimum spacing many jurisdictions allow in deck design. If No. 5 bars resulted in a bar spacing less than 5 in., No. 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 maxi- mum allowed crack width allowed by the specifications. The designs were not checked for other limit states because the purpose was to calibrate the Service I limit state. The design of the 15 decks was repeated twice, once assuming Class 1 Table 6.13. Summary of Statistical Information for Variables Used in the Calibration of Service I Limit State for Crack Control Variable Distribution Mean CV Source As Normal 0.9As 0.015 Siriaksorn and Naaman (1980) b Normal bn 0.04 Siriaksorn and Naaman (1980) CEc Normal 33.6 0.1217 Siriaksorn and Naaman (1980) d Normal 0.99dn 0.04 Nowak et al. (2008) dc Normal dcn 0.04 Nowak et al. (2008) Es Normal Esn 0.024 Siriaksorn and Naaman (1980) f ′c Lognormal Ec = CEc g c1.5 fc′ 3,000 psi:1.31 f ′cn 3,500 psi:1.27 f ′cn 4,000 psi:1.24 f ′cn 4,500 psi:1.21 f ′cn 5,000 psi:1.19 f ′cn 3,000:0.17 3,500:0.16 4,000:0.15 4,500:0.14 5,000:0.135 Siriaksorn and Naaman (1980) fy Lognormal 1.13fyn 0.03 Nowak et al. (2008) h Normal hn 1/(6.4µ) Siriaksorn and Naaman (1980) gc Normal 150 0.03 Siriaksorn and Naaman (1980) As = area of steel rebar (in.2); b = width of equivalent transverse strip of concrete deck (in.); CEc = constant parameter for concrete elasticity modulus; d = effective depth of concrete section (in.); dc = bottom cover measured from center of lowest bar (in.); Es = modulus of elasticity of steel reinforcement (psi); f ′c = specified compressive strength of concrete (psi); fy = yield strength of steel reinforcement (psi); h = deck thickness (in.); and gc = unit weight of concrete (lb/ft3).

173 exposure conditions and a second time assuming Class 2 exposure conditions. Table 6.14 presents the summary information of the 15 designed bridge decks. 6.2.2.4 Selection of Target Reliability Index Monte Carlo simulation was used to obtain the statistical parameters of resistance (or capacity) and dead load, and the statistical parameters for live load were taken from Sec- tion 5.3.4. The reliability indices for various ADTTs and expo- sure conditions for the 15 decks are summarized in Table 6.15. 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. Even though the design for Class 2 resulted in more rein- forcement 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 small bottom concrete cover. When Strength I limit state is considered, more positive moment reinforcement is typically required than by Service I. The additional reinforce- ment results in reliability indices for the positive moment region higher than those shown in Table 6.15. For the negative moment region, the design is often con- trolled by the Service I limit state. Thus, the reliability indices shown for the negative moment region in Table 6.15 are considered representative of the 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. Because the Class 2 case is the more common case for 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 = 5,000 was also considered as the base case on which the reliability analysis was performed. Table 6.16 shows the inherent reliability indices for the negative Table 6.14. Summary Information of 15 Bridge Decks Designed Using AASHTO LRFD Conventional Deck Design Method Deck Group No. Girder Spacing (ft) Deck Thickness (in.) 1 6 7 7.5 8 2 8 7.5 8 8.5 3 10 8 8.5 9 9.5 4 12 8 8.5 9 9.5 10 Table 6.15. 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) 1,000 2.44 1.54 2.37 1.77 2,500 1.95 1.07 1.79 1.27 5,000 1.66 0.85 1.61 1.05 10,000 1.39 0.33 1.02 0.5 Average 1.86 0.95 1.70 1.15 Maximum 2.44 1.54 2.37 1.77 Minimum 1.39 0.33 1.02 0.50 Standard deviation 0.45 0.50 0.56 0.53 CV 24% 53% 33% 46%

174 moment region of decks designed for the current AASHTO LRFD. The selected target reliability indices are 1.6 and 1.0 for Class 1 and Class 2, respectively, based on ADTT = 5,000. 6.2.3 Calibration Result The basic steps of the calibration process are shown below as they relate to the Service I calibration. 6.2.3.1 Step 1: Formulate Limit State Function and Identify Basic Variables The limit state function considered was the limit on the esti- mated crack width. In the absence of information suggesting that the current criteria (based on a crack width of 0.017 and 0.01275 in. for Class 1 and Class 2, respectively) are not ade- quate, the current crack widths were maintained as the limiting criteria. A discussion of crack width equations in the literature is included in Appendix C. 6.2.3.2 Step 2: Identify and Select Representative Structural Types and Design Cases The database of decks used in this study is described in Sec- tion 6.2.2.3. 6.2.3.3 Step 3: Determine Load and Resistance Parameters for 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 (s). 6.2.3.4 Step 4: Develop Statistical Models for Load and Resistance The variables affecting the load and resistance were identified. These include live load; resistance, including the dimensions of the cross section; and the material properties. The statistical information includes the probability distribution and statisti- cal parameters for axle loads presented in Section 5.3.4 in Chapter 5 and for other variables affecting the resistance pre- sented in Section 6.2.2.2. 6.2.3.5 Step 5: Develop Reliability Analysis Procedure The statistical information of all the required variables was 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 per- formed. For each random variable 1,000 values were generated independently on the basis of the statistics and distribution of that random variable. For each simulation, the dead load and the resistance were calculated using one of the 1,000 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 1,000). This process resulted in 1,000 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 1,000 simulations. 6.2.3.6 Step 6: Calculate Reliability Indices for Current Design Code and Current Practice Using the statistics of the dead load and the resistance, calcu- lated 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 5, the reliability index (b) was calculated for each deck by using Equation 6.7: R Q R Q (6.7) 2 2 β = µ − µ σ + σ where µR = mean value of the resistance; µQ = mean value of the applied loads; sR = standard deviation of the resistance; and sQ = standard deviation of the applied loads. The calculated reliability indices of the decks in the data- base are shown in Table 6.15 for both positive and negative moment reinforcement and for Class 1 and Class 2 exposure conditions. 6.2.3.7 Step 7: Review Results and Select Target Reliability Index The initial target reliability index (bT) was determined as shown in Table 6.16. Table 6.16. Reliability Indices of Existing Bridges Based on 1-Year Return Period ADTT Current Practice (Class 1, Negative) Current Practice (Class 2, Negative) 1,000 2.37 1.77 2,500 1.79 1.27 5,000 1.61 1.05 10,000 1.02 0.50

175 6.2.3.8 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 the 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 distri- bution of reinforcement. This omission results in an implied resistance factor of 1.0. The load and resistance factors were maintained for the initial reliability index calculations. For a Class 1 exposure condition (maximum crack width of 0.017 in.), Figure 6.26 and Figure 6.27 present the reliability indices for the bridge decks in the database designed using a live load factor of 1.0 over a 1-year return period for an ADTT of 5,000. As indicated in Table 6.15, the average values of the reliability index are 1.66 and 1.61 for positive and negative moment regions, respectively. For a Class 2 exposure condition (maximum crack width of 0.01275 in.), Figure 6.28 and Figure 6.29 present the reliability indices for the bridge decks in the database designed using a live load factor of 1.0 over a 1-year return period for an ADTT of 5,000. As indicated in Table 6.15, the average values of the reliability index are 0.85 and 1.05 for positive and negative moment regions, respectively. As discussed above, for positive moment (bottom) reinforce- ment, Strength I limit state requirements typically result in more reinforcement than needed to satisfy Service I require- ments, and the reliability index for cracking at the bottom will be higher than shown in Figure 6.26 and Figure 6.28. This difference in required reinforcement resulted in the recommen- dation that the reliability index should be based on the negative moment (top) reinforcement. Figure 6.26. Reliability indices of various bridge decks designed using a 1.0 live load factor over a 1-year return period (ADTT  5,000) for positive moment region, Class 1 exposure. Figure 6.27. Reliability indices of various bridge decks designed using a 1.0 live load factor over a 1-year return period (ADTT  5,000) for negative moment region, Class 1 exposure. Figure 6.28. Reliability indices of various bridge decks designed using a 1.0 live load factor over a 1-year return period (ADTT  5,000) for positive moment region, Class 2 exposure. Figure 6.29. Reliability indices of various bridge decks designed using a 1.0 live load factor over a 1-year return period (ADTT  5,000) for negative moment region, Class 2 exposure.

176 6.2.3.9 Step 9: Calculate Reliability Indices As shown in Figure 6.27 and Figure 6.29, 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 or resistance factors to improve the unifor- mity of the results. With this conclusion, the reliability indices are the same as shown in Table 6.15 and Table 6.16 and in Figure 6.27 and Figure 6.29. 6.2.3.10 Summary and Recommendations for Service I Limit State, Crack Control Through the Distribution of Reinforcement The following conclusions are based on the reported reliabil- ity analyses: • Assessment of current practice led to recommended target reliability indices of 1.6 for the base case (Class 1 exposure) and 1.0 for the enhanced requirements (i.e., smaller maxi- mum crack width) for Class 2 exposure conditions. These values correspond to an ADTT of 5,000. • The current requirements in the specifications produce uniform reliability across the range of girder spacings con- sidered, so there is no need to change the load or the resistance factors. 6.2.4 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 war- ranted by the results of this research. 6.3 Live Load Deflections, Service I: annual probability 6.3.1 Proposed Resistance Criteria The background and state of the art of SLS for live load deflec- tion was reviewed in Chapter 2. There is considerable dis- agreement in the literature as to whether live load deflection alone is an effective measure of dynamic response or a principal contributor to deck deterioration. Human beings are more sen- sitive to acceleration than displacement per se, especially when stationary on a bridge. The combined frequency–displacement criteria in the Canadian Highway Bridge Design Code (CHBDC) (2006), based on a comparison of computed values to Figure 2.1, appear to have more aspects of response accounted for than merely comparing a computed live load deflection to span length divided by a constant (L/N criterion), which is the case in AASHTO LRFD. A direct comparison to Canadian practice (CHBDC 2006) requires consideration of the magnitude of the design live load, dynamic load allowance, load factors, and analysis assumptions. For consideration of vibrations, the CHBDC uses 90% of one CL-625 truck loading without a dynamic load allowance. The CL-625 truck has six axles totaling about 140 kips and a wheel base of about 58 ft. AASHTO LRFD uses the larger of the design truck without the uniform load or 25% of the design truck with the uniform load. The dynamic load allowance is included for this purpose; the current load factor is 1.0. A comparison of one lane of CHBDC and AASHTO LRFD loadings for this limit state is shown in Figure 6.30. Note that the cur- rent provisions of AASHTO LRFD consider live load in each design lane. To determine whether the L/N criterion captures the dynamic response criteria in Figure 2.1 sufficiently to be “deemed to 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 0 50 100 150 200 250 300R ati o: (1 .0 0* 0. 90 *C L- 62 5) / (1 .0 0* AA SH TO LR FD D efl ec ti on Li m it Lo ad in g) Base Span Length (ft) Mss-100 MAX: (HL-93 Ratios) CL-625 Ratios of: Centerline Moment for Simple Span Figure 6.30. Comparison of CHBDC and AASHTO LRFD service load moments.

177 satisfy” a need for minimum global stiffness, the live load deflec- tion for 41 bridges chosen from the NCHRP Project 12-78 data- base (Mlynarski et al. 2011) was calculated using VIRTIS and plotted against frequency, as shown in Figure 6.31. The fre- quency was calculated using the method of Barth and Wu (2007) presented in Section 2.3.1.2 in Chapter 2, with l = 1.0. The 41 bridges contained simple-span and continuous bridges and steel and prestressed concrete bridges, which sat- isfy all applicable requirements, not just live load deflection. The general trend of results was similar to the criteria curves in the figure and fell mostly between the curves for bridges with sidewalks and frequent pedestrian use and bridges with side- walks and occasional pedestrian usage. It is reasonable to assume that most, if not all, of the 41 sample bridges were essentially highway bridges without sidewalks. As shown in Figure 6.31, concrete bridges tend to be stiffer than steel bridges, but some of the concrete sample bridges exhibited responses relatively close to the CHBDC acceptance curves. For this purpose, the comparisons were made between the provisions of AASHTO LRFD and acceptable CHBDC response as indicated by the criteria curves. As the 41 bridges satisfied all applicable requirements, it is reasonable to suspect they are overdesigned with respect to just the live load deflection require- ment. For calibration purposes, the considered database should satisfy the limit state under investigation but not necessarily other limit states. To accomplish this, a set of simply supported bridges was designed to satisfy all applicable limit states by using the load and resistance factor design of the Pennsylvania Department of Transportation’s girder design program and then forced to satisfy only the AASHTO LRFD L/800 criteria. If the designs forced to satisfy L/800 satisfied any other criteria, that outcome was unintentional. The considered span lengths were 60, 90, 120, 160, 200, and 300 ft, and girder spacings were 9 and 12 ft. Deflections and frequencies were calculated, plotted, and compared with the CHBDC criteria. The results are shown in Figure 6.32. When the girders were forced to meet the L/800 criteria, the frequencies for each pair of spacings for a given span were so similar that the solid circles in the figure cannot be distinguished. For calibration purposes, the CHBDC curves were treated as deterministic (i.e., the bias was assumed to be 1.0, and the CV was assumed to be 0.0). This is analogous to the geotechnical Figure 6.31. Comparison of CHBDC requirements and various bridges satisfying all AASHTO LRFD design requirements. Existing Spread Box Girders Existing Adjacent Box Girders Existing I-Girders Existing Steel Girders 0 1 2 3 4 5 6 7 98 10 without sidewalks with sidewalks, occasional pedestrian use with sidewalks, frequent pedestrian use first flexural frequency, Hz 1000 500 200 100 50 20.0 10.0 5.0 2.0 1.0 st at ic d ef le ct io n, m m ACCEPTABLE UNACCEPTABLE Deflection Limitations for Highway Bridge Superstructure Vibration

178 calibration using a deterministic value of tolerable deformation, as presented in Section 6.1.2.3 and illustrated in Figure 6.6. 6.3.2 Calibration Results 6.3.2.1 Formulate Limit State Function The live load deflection limit state function is merely the sum of the factored loads and must be less than or equal to the factored resistance, as shown by Equation 6.8: (6.8)Q Ri i i ∑γ ≤ ϕ The basic load effect for live load deflection is obviously the deflection due to live load. The resistance was taken as the appropriate deflection limit from Figure 2.1, as discussed in Chapter 2. Substituting these variables into the general limit state function yields Equation 6.9: (6.9)limitLL LLγ ∆ ≤ ∆ 6.3.2.2 Select Structural Types and Design Cases All structural types and materials were considered for this limit state. 6.3.2.3 Determine Load and Resistance Parameters for Selected Design Cases As discussed above, the load currently used for deflection calculations in AASHTO LRFD was maintained with a bias of 1.35 and a CV of 0.12 per Section 5.5.2, which is representa- tive of the 1-year live load results for ADTT = 5,000 shown in Simulated Steel Bridges Designed to Satisfy AASHTO LRFD Deflection Limits Only Simulated Steel Bridges Designed to Satisfy AASHTO LRFD Specifications 0 1 2 3 4 5 6 7 8 9 10 without sidewalks with sidewalks, occasional pedestrian use with sidewalks, frequent pedestrian use first flexural frequency, Hz 1000 500 200 100 50 20.0 10.0 5.0 2.0 1.0 st at ic d ef le ct io n, m m ACCEPTABLE UNACCEPTABLE Deflection Limitations for Highway Bridge Superstructure Vibration Figure 6.32. Comparison of CHBDC requirements and various steel bridges satisfying all AASHTO LRFD design requirements and similar bridges satisfying only L/800 criteria.

179 Table 5.8. The resistance was taken from the CHBDC curves, which were treated as deterministic (i.e., the bias was assumed to be 1.0, and the CV was assumed to be 0.0). 6.3.2.4 Develop Statistical Models for Loads, Load Combinations, and Resistance Variables Uncertainties of Load From Tables 5.6 to 5.9, the uncertainties of live load moment were taken as approximately 0.12 for the CV and 1.35 for the bias. It was assumed that the uncertainty of deflections is the same as the uncertainty of moments. Uncertainties of resistance The live load deflection limits in Figure 2.1 were taken as invariant with a bias of unity and a CV of zero. Thus resistance was set equal to the curves shown in Figure 2.1 for the purpose of calibration. 6.3.2.5 Develop Reliability Analysis Procedure As discussed in Chapter 3, Monte Carlo simulation using MS Excel formed the basis of the reliability analysis procedure for the live load deflection limit state. 6.3.2.6 Calculate Reliability Indices for Current Design Code or Current Practice Using the current AASHTO LRFD Service I limit state with load factors equal to unity, the probability of failure is almost 100%. Thus, load and resistance factors other than unity must be considered to achieve reliability indices comparable to the other SLSs considered. 6.3.2.7 Review Results and Select Target Reliability Index Using the SLS reliability indices developed above and a reli- ability analysis that showed current practice for most fatigue limit states yields a reliability index of 1.0, a target reliability index (bT) of about 1.0 was chosen for the calibration of the live load deflection limit state to be consistent with other reversible SLSs. Thus, the proposed specifications maintain the historic level of reliability. 6.3.2.8 Select Potential Load and Resistance Factors Through trial-and-error testing, the live load load factor for the deflection limit state was selected as 1.50, along with a resistance factor of unity. 6.3.2.9 Calculate Reliability Indices A Monte Carlo simulation was again performed for the live load deflection limit state. With the live load deflection limit considered invariant, all cases yield a reliability index of about 1.0 using a live load load factor of 1.5 and a resistance factor of 1.0. 6.3.3 Potential LRFD Revisions 6.3.3.1 Theoretical Conclusions The live load deflection limit state provisions could be modified from those of the AASHTO LRFD to satisfy frequency, percep- tion, and deflection by adopting the CHBDC provisions. A rec- ommendation to use user comfort was made by Roeder et al. (2002). The provisions of Article 2.5.2.6 could be revised to include Figure 2.1 and remove the L/N criterion for steel, alu- minum, and concrete vehicular bridges. If this were done, the live load deflection limit state should be as defined in AASHTO LRFD Table 3.4.1-1 as Service V, with the load factor for live load given as 1.50. Dead load would also be needed in the load combination for use in calculating frequency. Descriptive text and commentary would be needed in Article 3.4.1. 6.3.3.2 Practical Assessment of Results The use of the CHBDC criteria appears to be a more realistic approach in that it incorporates both deflection and frequency and compares them with a set of human factors response curves. The current (and historic) AASHTO requirement includes only stiffness and compares the result to a criterion, L/N, whose background is reviewed in Article 2.3.1.2. The survey of owners (Barker and Barth 2007) indicated that the range of applications of the current criteria could produce results that differ more than mere consideration of the vari- ability of loads would indicate. Despite the obvious limitations of the current AASHTO LRFD criteria, Figure 6.31 and Figure 6.32 indicate that the two criteria are somewhat similar. A review of the 12 steel bridge designs that satisfied all the relevant AASHTO LRFD require- ments showed that they would not be changed if the deflection criteria were based on a load factor of 1.40, which resulted from the calibration described above. Stated more simply, other criteria still controlled those 12 bridges. 6.3.3.3 Recommendations Based on this research, there does not appear to be a compel- ling need to change the current AASHTO LRFD provision for live load response (i.e., load deflection) in Article 2.5.6.2. Such an approach is basically “deemed to satisfy.” However,

180 if AASHTO chooses to adopt the more complete approach of combining frequency, displacement, and perception, a pos- sible set of revisions to accomplish that change are proposed in Chapter 7. Other considerations could include basing the deter- mination of deflection on the fatigue truck of Article 3.6.1.4.1, as its longer wheel base is more representative of actual traffic. The fatigue truck for orthotropic decks could also be used, but the substitution of tandem axles for the 32K single axle is not apt to be significant except for very short spans. 6.4 Overload, Service II: annual probability 6.4.1 Basis of Limit State The basis for this limit state was presented in Section 2.3.1.5. Several questions regarding the criteria for control of perma- nent deformations arose: 1. Was the LFD requirement a good target? 2. What is the current level of experience with these pro- visions? That is, has there been a significant issue with per- manent set in girder bridges that affect appearance or rideability? 3. What reliability index is provided in current designs for which the overload provision controlled? 4. How often can these criteria be exceeded without creating a significant permanent deflection of the structure? 5. Is another choice of load factor, other than the 1.30 used in the current Service II load combination, equally valid? 6. Should this requirement be applied to multilane loading? 7. If it is used as a single-lane criterion, should the multiple presence factor (MPF) of 1.2 currently used for single- lane loading in the AASHTO LRFD be applicable to this condition? The LFD criteria resulted from an assessment of experience at the AASHO Road Test (1962); these criteria are summarized in Figure 2.3. The six data points shown, three for composite structures and three for noncomposite structures, comprise the full data set known to the research team. Most of the per- manent set occurred during the early repetitions of load, as would be expected. Because the weight of the test trucks did not vary during most of the circuits of the test track, the per- manent set accumulated slowly after the period of initial load cycles. This behavior was expected. Two of the questions to be explored in regard to the control of permanent deformations noted above were Question 1, “Was the LFD requirement a good target?” and Question 2, “What is the current level of experience with these provi- sions? That is, has there been a significant issue with perma- nent deformations in girder bridges that affect appearance or rideability?” This subject was discussed with the AASHTO Sub- committee on Bridges and Structures Steel Structures Technical Committee. Several states are represented on that committee, but the meeting was open to researchers and practitioners. No one present offered any evidence, either documented or anecdotal, that there has been any significant issue with perma- nent deformations. The research team is aware of some anec- dotal discussion of permanent set in stringers in the floor systems of some long-span bridges, but that has not been docu- mented. In summary, the current provisions appear to be serv- ing well but just how well has not been quantified statistically. Question 3, “What reliability index is provided in current designs for which the overload provision controlled?” is dis- cussed in Section 6.4.3. As detailed below, it is possible to provide some insight into how often the factored live load is exceeded, which bears on Question 4, “How often can this criteria be exceeded without creating a significant permanent deflection of the structure?” 6.4.2 Load Model The application of WIM data to the development of load models for the SLSs was discussed in Chapter 5. The location of WIM sites, the number of days of measurements, and the number of trucks after filtering are shown in Figure 5.1. The indicated number of filtered truck records includes the trucks thought to be permit trucks, as discussed in Section 5.2.3. Thus the trucks used in this study include the permit trucks, which were filtered for consideration of other limit states. Table 6.17 shows the number of times bending moments from the trucks in the database exceeded 1.0, 1.1, 1.2, and 1.3 times the HL-93 moment for simple spans of 30, 60, 90, 120, and 200 ft for each WIM site scaled to 1 year of data. Note that results for the WIM site on Florida US-29 look much different from the other sites. It was determined that trucks from other parallel routes were being diverted to Route 29, creating an unusual situation. This site was excluded from the discussion below. Figure 6.33 shows the average exceedance for the 31 remain- ing WIM sites by HL-93 ratio for each span length considered. Figure 6.34 shows the same information by span length for each HL-93 ratio considered. The reduction in the rate of exceedance with increasing HL-93 ratio is clearly evident. These data do not show how much a given HL-93 ratio was exceeded in terms of stress, but this limit state has historically been based on infrequently exceeding the criteria. The rate of exceedance of 1.3 HL-93, the current criteria, is seen to be quite small. A more meaningful assessment of the exceedance rate is presented in Table 6.18, Figure 6.35, and Figure 6.36. In this case, the exceedance data have been scaled to an assumed ADTT of 2,500 at each site assuming that the distribution of trucks (text continues on page 186)

181 Table 6.17. Bending Moment Exceedances per Year Site Ratio Truck/HL-93 ≥1.0 Ratio Truck/HL-93 ≥1.1 Ratio Truck/HL-93 ≥1.2 Ratio Truck/HL-93 ≥1.3 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft Arizona (SPS-1) 4 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Arizona (SPS-2) 0 2 6 5 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 Arkansas (SPS-2) 14 10 17 10 0 2 7 3 0 0 0 3 0 0 0 0 0 0 0 0 Colorado (SPS-2) 0 5 6 6 2 0 2 5 4 0 0 0 2 0 0 0 0 0 0 0 Delaware (SPS-1) 140 48 33 27 1 36 33 22 11 0 10 22 10 1 0 1 11 1 0 0 Illinois (SPS-6) 1 3 4 4 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 Indiana (SPS-6) 27 32 24 19 14 5 19 19 17 3 3 7 9 7 0 0 0 2 0 0 Kansas (SPS-2) 42 47 80 96 10 16 33 35 31 2 7 16 17 7 0 6 7 6 0 0 Louisiana (SPS-1) 76 16 25 30 13 44 6 12 14 7 26 6 7 7 0 6 6 5 4 0 Maine (SPS-5) 6 7 8 7 1 4 4 5 2 0 0 4 2 0 0 0 2 0 0 0 Maryland (SPS-5) 25 8 8 2 1 5 6 2 2 0 0 1 1 0 0 0 1 0 0 0 Minnesota (SPS-5) 9 8 18 19 2 7 5 6 5 0 4 2 2 1 0 2 1 1 0 0 New Mexico (SPS-1) 1 1 1 3 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 New Mexico (SPS-5) 12 7 7 9 4 4 1 1 3 0 3 0 0 0 0 0 0 0 0 0 Pennsylvania (SPS-6) 155 45 22 21 1 32 22 17 14 1 13 17 13 1 0 3 13 2 0 0 Tennessee (SPS-6) 2,085 29 8 7 0 53 4 4 0 0 5 1 0 0 0 1 0 0 0 0 Virginia (SPS-1) 7 10 1 2 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 (continued on next page)

182 Table 6.17. Bending Moment Exceedances per Year (continued) Site Ratio Truck/HL-93 ≥1.0 Ratio Truck/HL-93 ≥1.1 Ratio Truck/HL-93 ≥1.2 Ratio Truck/HL-93 ≥1.3 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft Wisconsin (SPS-1) 6 3 5 4 2 1 0 3 3 1 0 0 1 1 0 0 0 0 0 0 California Antelope EB 0 13 25 31 25 0 1 0 0 7 0 0 0 0 0 0 0 0 0 0 California Antelope WB 0 30 71 100 84 0 7 6 19 40 0 0 0 1 13 0 0 0 0 1 California Bowman 0 3 3 8 16 0 0 0 3 3 0 0 0 0 3 0 0 0 0 0 California LA-710 NB 10 99 150 153 85 1 34 55 56 16 0 7 26 21 0 0 0 4 1 0 California LA-710 SB 3 62 105 111 54 1 17 45 48 14 0 3 18 19 0 0 0 1 1 0 California Lodi 0 110 137 281 417 0 5 19 55 168 0 0 1 2 38 0 0 0 0 2 Florida I-10 279 141 159 264 152 81 41 47 77 38 23 16 14 18 5 10 5 4 5 2 Florida I-95 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Mississippi I-10 41 48 53 53 44 26 24 34 36 24 8 2 11 21 2 2 2 2 2 1 Mississippi I-55UI 0 4 5 11 8 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 Mississippi I-55R 142 100 255 349 89 20 31 50 61 33 7 8 17 22 20 2 3 5 8 9 Mississippi US-49 0 3 11 13 7 0 0 2 1 0 0 0 0 0 0 0 0 0 0 0 Mississippi US-61 0 1 5 8 6 0 0 1 2 1 0 0 1 1 0 0 0 0 0 0 Florida US-29 1,291 995 651 496 204 673 510 332 253 109 371 274 179 123 53 183 165 85 61 22 Annual Average 99.6 28.9 40.4 53.4 33.6 11.0 9.8 12.8 15.1 11.7 3.5 3.7 4.9 4.2 2.6 1.1 1.7 1.1 0.7 0.5 Note: EB = eastbound; WB = westbound; NB = northbound; SB = southbound.

183 Figure 6.33. Annual average exceedances versus span. 0 20 40 60 80 100 120 30 ft 60 ft 90 ft 120 ft 200 ft An nu al A ve ra ge Span (ft.) >= 1.0HL93 >= 1.1HL93 >= 1.2HL93 >= 1.3HL93 0 20 40 60 80 100 120 >= 1.0HL93 >= 1.1HL93 >= 1.2HL93 >= 1.3HL93 An nu al A ve ra ge Ratio Truck/HL93 30 ft 60 ft 90 ft 120 ft 200 ft Figure 6.34. Annual average exceedances versus ratio truck/HL-93.

184 Table 6.18. Events per Year Scaled to ADTT  2,500 Site Ratio Truck/HL-93 ≥1.0 Ratio Truck/HL-93 ≥1.1 Ratio Truck/HL-93 ≥1.2 Ratio Truck/HL-93 ≥1.3 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft Arizona (SPS-1) 103 0 0 26 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Arizona (SPS-2) 0 1 4 3 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 Arkansas (SPS-2) 8 5 9 5 0 1 4 2 0 0 0 2 0 0 0 0 0 0 0 0 Colorado (SPS-2) 0 13 16 16 5 0 5 13 11 0 0 0 5 0 0 0 0 0 0 0 Delaware (SPS-1) 633 217 149 122 5 163 149 100 50 0 45 100 45 5 0 5 50 5 0 0 Illinois (SPS-6) 1 3 4 4 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 Indiana (SPS-6) 79 94 69 54 39 15 54 54 49 10 10 20 25 20 0 0 0 5 0 0 Kansas (SPS-2) 80 90 153 183 19 31 63 67 59 4 13 31 32 13 0 11 13 11 0 0 Louisiana (SPS-1) 808 170 266 319 138 468 64 128 149 74 277 64 74 74 0 64 64 53 43 0 Maine (SPS-5) 30 35 40 35 5 20 20 25 10 0 0 20 10 0 0 0 10 0 0 0 Maryland (SPS-5) 139 44 44 11 6 28 33 11 11 0 0 6 6 0 0 0 6 0 0 0 Minnesota (SPS-5) 148 131 296 312 33 115 82 99 82 0 66 33 33 16 0 33 16 16 0 0 New Mexico (SPS-1) 8 8 8 16 0 0 8 8 8 0 0 0 0 0 0 0 0 0 0 0 New Mexico (SPS-5) 12 8 8 9 5 5 2 2 3 0 3 0 0 0 0 0 0 0 0 0 Pennsylvania (SPS-6) 95 27 13 13 1 20 13 10 9 1 8 10 8 1 0 2 8 1 0 0 Tennessee (SPS-6) 1,173 16 4 4 0 30 2 2 0 0 3 1 0 0 0 1 0 0 0 0 Virginia (SPS-1) 25 35 4 7 4 0 0 4 4 0 0 0 0 0 0 0 0 0 0 0 (continued on next page)

185 Wisconsin (SPS-1) 24 12 20 16 8 4 0 12 12 4 0 0 4 4 0 0 0 0 0 0 California Antelope EB 0 10 20 24 20 0 1 0 0 5 0 0 0 0 0 0 0 0 0 0 California Antelope WB 0 20 48 68 57 0 5 4 13 27 0 0 0 1 9 0 0 0 0 1 California Bowman 0 1 1 4 8 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 California LA-710 NB 2 20 31 31 17 0 7 11 11 3 0 1 5 4 0 0 0 1 0 0 California LA-710 SB 1 12 21 22 11 0 3 9 9 3 0 1 4 4 0 0 0 0 0 0 California Lodi 0 25 32 65 96 0 1 4 13 39 0 0 0 1 9 0 0 0 0 1 Florida I-10 151 76 86 142 82 44 22 26 42 21 12 9 8 9 3 6 3 2 3 1 Florida I-95 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Mississippi I-10 0 2 3 6 4 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 Mississippi I-55UI 0 2 3 6 4 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 Mississippi I-55R 93 66 167 229 58 13 21 33 40 22 5 5 11 14 13 1 2 3 5 6 Mississippi US-49 0 2 8 10 5 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 Mississippi US-61 0 6 23 40 29 0 0 6 11 6 0 0 6 6 0 0 0 0 0 0 Florida US-29 2,922 2,252 1,473 1,122 462 1,524 1,155 751 572 247 840 621 406 278 119 413 373 191 138 49 Annual Average 117.0 37.8 50.6 58.7 21.7 32.0 18.4 20.8 19.8 7.5 14.3 9.7 9.1 5.8 1.2 4.0 5.6 3.2 1.7 0.3 Table 6.18. Events per Year Scaled to ADTT  2,500 (continued) Site Ratio Truck/HL-93 ≥1.0 Ratio Truck/HL-93 ≥1.1 Ratio Truck/HL-93 ≥1.2 Ratio Truck/HL-93 ≥1.3 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft

186 0 20 40 60 80 100 120 30 ft 60 ft 90 ft 120 ft 200 ft An nu al A ve ra ge Span (ft.) >= 1.0HL93 >= 1.1HL93 >= 1.2HL93 >= 1.3HL93 Figure 6.35. Annual average events scaled to ADTT  2,500 versus span. 0 20 40 60 80 100 120 >= 1.0HL93 >= 1.1HL93 >= 1.2HL93 >= 1.3HL93 An nu al A ve ra ge Ratio Truck/HL93 30 ft 60 ft 90 ft 120 ft 200 ft Figure 6.36. Annual average events scaled to ADTT  2,500 versus ratio truck/HL-93. is the same (i.e., the data are scalable). The average rate of exceedance in Table 6.18 is higher than in Table 6.17 because many of the WIM sites were on roads with ADTTs less than 2,500. Nevertheless, the rate at which 1.3 HL-93 was exceeded remains quite low. The values in Table 6.18 can be scaled for locations with an ADTT other than 2,500 with the same assumption of scalability. Question 5 asked, “Is another choice of load factor, other than the 1.30 used in the current Service II load combination, equally valid?” Given the relatively low number of exceedances in Table 6.18, it is difficult to rationalize the need for a national load factor higher than the current value of 1.30 except pos- sibly for locations with extraordinary levels of truck traffic. For example, for a location with an ADTT of 7,500, over a 100-year service life the average exceedance would be on the order of 1,000 events, although one of the sites shown in Table 6.18 could see about 10 times that number. However, slightly over half the sites recorded no events when the moments for the spans indicated in Table 6.17 exceeded 1.3 HL-93 dur- ing the recording period. Given the lack of field evidence of a significant number of bridges with a permanent deformation due to overloads, it was not possible to establish an ADTT criterion at which the load factor should be increased. (continued from page 180)

187 Judgment and experience are still necessary. This issue is also clouded by the issues in Questions 6 and 7, “Should this require- ment be applied to multilane loading?” and “If it is used as a single-lane criterion, should the MPF of 1.2 currently used for single-lane loading in the AASHTO LRFD be applicable to this condition?” The issue of number of loaded lanes is discussed in Sec- tion 5.2.4. For the WIM sites where data were recorded in two lanes, and given the definition of correlated events in that discussion, it was shown that the number of events of multiple lanes loaded with correlated trucks was quite small, and the histograms of gross vehicle weight showed that the number of events of two heavy trucks was even smaller. It was concluded that multiple lanes of heavy trucks need not be considered for the SLSs. Thus, it was concluded that in most cases, design for control of permanent distortions need not be based on multiple lanes of overload (i.e., 1.3 HL-93). In the calibration process described in Section 6.4.3, a single-lane loading with no MPF was used on the load side of the limit state function. To summarize, based on a review of the WIM data • There is little basis for lowering the current Service II load factor. • Site-specific consideration of sites with unusually high volumes of heavy trucks is warranted. • Design for a single-lane loading is justified by this study. • Elimination of the single-lane MPF of 1.20 for Service II is justified by this study. 6.4.3 Calibration Procedure and Results 6.4.3.1 Formulate Limit State Function The Service II limit state function requires that the sum of the factored loads must be less than or equal to the factored resistance, as shown by Equation 6.10: (6.10)Q Ri i i ∑γ ≤ ϕ The two basic loads for Service II are dead load (DC + DW) and live load (LL); DC is the load factor for structural com- ponents and attachments, and DW is the load factor for wear- ing surfaces and utilities. Currently, the resistance is taken as 0.95 Fy for composite sections and 0.80 Fy for noncomposite sections with resistance factors of unity for both. The limited basis for these criteria was presented in Section 2.3.1.5. Sub- stituting these variables and the current Service II load factors of AASHTO LRFD Table 3.4.1-1 yields Equation 6.11 and Equation 6.12: Fy( )+ ≤1.3 LL +1.0 DC DW 0.95 for composite sections (6.11) Fy( )+ ≤1.3 LL +1.0 DC DW 0.80 for noncomposite sections (6.12) These are the current Service II limit state functions for investigation. 6.4.3.2 Select Structural Types and Design Cases The Service II limit state is currently intended only for steel superstructures and governs only for composite and compact sections in the positive moment region. For these regions of composite and compact sections, the Service II limit state often governs the design over the Strength I limit state. Non- composite sections are not typically compact. Thus, the structure types being considered are positive moment regions of steel girder superstructures that were modeled as simple spans, and the design case is a composite girder, which is the governing case. 6.4.3.3 Determine Load and Resistance Parameters for Selected Design Cases As discussed in the previous subsection, the Service II limit state can be investigated by concentrating on simple spans of composite steel girders. A set of 41 simple-span composite steel girder bridges was extracted from Mlynarski et al. (2011). The flexural resistances of the interior girders of these bridges were used to study the Service II limit states. The documenta- tion of the 41 bridges is given in Appendix F. It was established in Chapter 5 that although the Service II limit state is evaluated assuming multiple lanes loaded, the WIM study suggests that the Service II live load does not occur often enough to warrant design for multiple lanes. 6.4.3.4 Develop Statistical Models for Loads, Load Combinations, and Resistance Variables Uncertainties of Load The uncertainties of the various components of dead load were investigated previously with the results documented in Kulicki et al. (2007) and reproduced in Table 3.2. From Table 5.5 to Table 5.9, the uncertainties of live load are taken as approximately 0.12 for the CV and 1.35 for the bias. Uncertainties of resistance The uncertainties of the flexural resistance of composite steel girders have also been investigated and similarly documented in Kulicki et al. (2007) and reproduced in Table 3.1. 6.4.3.5 Develop Reliability Analysis Procedure Monte Carlo simulation using Microsoft Excel formed the basis of the reliability analysis procedure for the Service II

188 limit states. Use of the Monte Carlo analysis is presented in Section 3.2.3. 6.4.3.6 Calculate Reliability Indices for Current Design Code or Current Practice The Service II limit state was first introduced with LFD in the AASHTO Standard Specifications for Highway Bridges (2002) as “overload” provisions. In the development of the AASHTO LRFD, a simple calibration was made in an attempt to yield similar member proportions as with the Standard Specifications; this attempt is discussed in Section 2.3.1.5. The calculation of the inherent reliability indices for cur- rent design practice was based on LFD overload provisions. Monte Carlo simulations were performed for the 41 bridges by using a single lane of AASHTO LFD live load times the over- load load factor of 5/3 compared with the flexural resistance consistent with multiple lanes of the LFD live load. This is the requirement for which most of today’s in-service bridges were designed. The results are summarized in Table 6.19. In addi- tion, similar simulations were made assuming that the load was multiple lanes of AASHTO LRFD live load, as the original conceivers of the limit state assumed traffic to be. These results suggest that the current inherent reliability index associated with the Service II limit state is on average about 2.0. The originally assumed multiple lanes of loading suggest lower reliability but, as discussed, this loading is not very probable. 6.4.3.7 Review Results and Select Target Reliability Index Using the values in Table 6.19, a target reliability index (bT) of about 2.0 was chosen for the calibration of the Service II limit state. 6.4.3.8 Select Potential Load and Resistance Factors As an initial trial, the Service II load factors of the AASHTO LRFD (1.3 for live load and 1.0 for dead load) were selected, along with resistance factors of unity. 6.4.3.9 Calibrate Reliability Indices Monte Carlo simulations were again performed for the 41 bridges by using a single lane of AASHTO LRFD live load (which represents the load today as suggested by the WIM studies) and dead load times the AASHTO LRFD load factors compared with the flexural resistance consistent with multiple lanes of the AASHTO LRFD live load, as most of today’s new bridges will be designed. This assumes that design for multiple lanes of live load will continue. An average reliability index of 1.8 with a CV of 0.09 resulted from using the current AASHTO LRFD load and resistance factors. Thus, the reliability is comparable to the inherent reliability of current practice, but with much more unifor- mity and with a low CV compared with the original overload provisions. 6.4.4 Proposed AASHTO LRFD Revisions The Service II limit state provisions do not require any modification from those of the AASHTO LRFD. Thus, the Service II limit state will continue as defined in AASHTO LRFD Table 3.4.1-1. Given the limited background on this limit state, the possibility of reducing the demand was discussed informally with three of the AASHTO Technical Committees: T-5, T-10, and T-14, although in the case of T-14 fewer than half the members were present. Most of the members of these three committees expressed reservations about decreasing the cur- rent design requirement, citing the increasing numbers of trucks on the roads and the continual pressure to increase legal loads. At the time of this work, an increase of about 20% in legal gross vehicle weight is under discussion. 6.5 tension in prestressed Concrete Beams, Service III Limit State: annual probability Traditionally, prestressed concrete beams are proportioned for the SLS 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 (2012), two SLS load combinations are used to calculate the stresses in prestressed concrete components: the Service I and Service III load com- binations. The two service load combinations are described as follows: • Service I—Load combination relating to the normal opera- tional use of the bridge with a 55 mph wind and all loads taken at their nominal values. Service I is also related to Table 6.19. Inherent Reliability Indices Live Load b CV Single lane (reality) 1.8 0.32 Multiple lane (assumed) 1.6 0.92

189 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. The load factors for DL and LL specified for the two load combinations are as follows. • Service I: DL load factor = 1.0, and LL load factor = 1.0. • Service III: DL load factor = 1.0, and LL load factor = 0.8. On the basis of the definition of the two limit states, the Service I limit state is used for calculating all service stresses in the superstructure and substructure components at all stages, with the exception that the 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. Thus, the calibration for prestressed concrete superstructures was performed for the Service III limit state, and no calibration was performed for the Service I limit state. In addition to being designed for the SLS, all prestressed concrete components are checked for the strength limit state. For typical precast prestressed superstructure beams (e.g., I-shapes, bulb-T shapes, and adjacent and spread box beams), the controlling case of the design is usually the SLS. The SLS stresses are calculated assuming an uncracked section. The concrete is assumed to be subjected to tensile stresses. However, due to the relatively low load factors used for the SLSs, 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 on the basis of the uncracked section to be tensile, the crack will reopen. Successful past performance of prestressed concrete com- ponents suggests that past design requirements result in a frequency of the crack opening being sufficiently small that adverse strand fatigue problems at crack locations are not produced. 6.5.1 History of Major Relevant Design Provisions and Revisions to AASHTO LRFD 6.5.1.1 Load Factor for Live Load in Service III Load Combination During the early stages of the development of AASHTO LRFD in the early 1990s, only the 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 for Highway Bridges, the predecessor to AASHTO LRFD. The design live load specified in AASHTO LRFD produces higher unfactored, undistributed load effects than that specified in the AASHTO Standard Specifications. The girder distribution factors, particularly for interior girders, for many typical girder systems in AASHTO LRFD are lower than those in Standard Specifications, thus reducing the difference between the unfac- tored distributed load effects in the two specifications. Even with the smaller distribution factor, the unfactored distributed load effects from AASHTO LRFD were higher for most girder systems. Using the same load factor for SLS (1.0) resulted in higher design-factored load effects for the AASHTO LRFD designs than for those designed to the AASHTO Standard Speci- fications requirements. The results from the trial designs con- ducted during the development of AASHTO LRFD indicated a larger number of strands than required by AASHTO Stan- dard Specifications. This finding would suggest that designs performed under AASHTO Standard Specifications resulted in underdesigned 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 pre- stressed concrete components. This resulted in a similar num- ber of strands for the designs conducted using both AASHTO Standard Specifications and AASHTO LRFD. 6.5.1.2 Method of Calculating Prestressing Losses AASHTO LRFD (2012) includes three methods for determining the time-dependent prestressing losses. These three methods are as follows: 1. Approximate method—This method is termed approxi- mate estimate of time-dependent losses and is the least detailed method. It requires limited calculations to estimate

190 the time-dependent losses. Before 2005, the specifications included a simpler approximate method termed approxi- mate 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 according to the type of girders and the type and grade of prestressing steel. Some concrete compressive strength requirements were allowed to use this method. 2. Refined estimates of time-dependent losses—This method is more detailed than the approximate method. More details on this method are presented below. 3. 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 posttensioned 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 this section, unless explicitly indicated otherwise, time-dependent losses are calculated using the techniques outlined in Refined Estimates of Time- Dependent Losses in AASHTO LRFD. Originally, the method of calculating prestressing force losses in AASHTO LRFD (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. 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 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 surround- ing 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 pre- stressing loss method allows including elastic gain to be used to offset some of the losses. When the elastic gain was considered, the post-2005 pre- stress 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 AASHTO Standard Specifications and under earlier editions of AASHTO LRFD. This change raised some concern as some practitioners and researchers thought that the higher pre- stressing losses calculated using the pre-2005 loss method compensated for the lower live load effects caused by the lower design live load used in AASHTO Standard Specifications or the lower load factor used for the Service III load combination of AASHTO LRFD. Some of the work presented in the follow- ing subsections was intended to investigate the effect of dif- ferent loss methods and different design specifications on the reliability index for the Service III load combination. 6.5.2 Live Load Model Traditionally, prestressed concrete components have been designed for the number of traffic lanes, including MPFs, which produced the highest load effects. This was assumed to continue in the future, and all sections designed as part of this study used this approach. However, as indicated in Section 5.2.4, the presence of heavy loads in adjacent traffic lanes simultaneously is not likely. Thus, 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 MPF was included. The design truck, tandem, and uniform lane load specified in AASHTO LRFD were used unless otherwise noted. The live load distribution factors specified in AASHTO LRFD were used in distributing the design loads. The dynamic load allowance (10%) used in the original calibration of the strength limit state in AASHTO LRFD was applied to the load side. The return period considered in the calibration of the Ser- vice III limit state was 1 year. This return period was selected because the live load statistics were developed based on 1 year of reliable WIM data from various WIM sites. Furthermore, as only three of the 32 WIM sites had an ADTT larger than 5,000, and only one of the 32 WIM sites had an ADTT larger than 8,000, an ADTT of 5,000 was used for the bulk of the calibration. The bias and CV of live load were taken as shown in Table 5.5 to Table 5.9. 6.5.3 Methods of Analysis for Study Bridges Unless explicitly indicated otherwise, the methods of analysis used in designing and analyzing the study bridges throughout Section 6.5.4 and Section 6.5.5 are listed below. For bridges designed or analyzed using the post-2005 pre- stressing loss method, • The time-dependent prestressing loss method used is the method designated in AASHTO LRFD (2012) as the Refined Estimates of Time-Dependent Losses. • The section properties used in the analysis were based on the gross section of the concrete. • The calculations of prestressing losses considered the effects of 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

191 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 pre-2005 AASHTO LRFD editions as the Refined Estimates of Time-Dependent Losses. • The section properties used in the analysis were based on the gross section of the concrete. • The calculations neglected the effects of elastic gain. 6.5.4 Target Reliability Index In the development of AASHTO LRFD, 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. Failure under the strength limit state is well defined as it relates to a certain criterion related to the properties of the materials used (such as steel yield stress or concrete compressive strength) or to a behavior criterion that, if violated, may lead to instability of the component (such as local or global buckling). Due to the lack of clear conse- quences for violating the limiting stress specified for the con- crete in a prestressed concrete component, selecting the limit state function required investigating different alternatives. 6.5.4.1 Limit State Functions Investigated The following three limit state functions were investigated: • Decompression limit state—This limit state assumes that failure occurs when the stress in the concrete on the tension face calculated on the basis of the uncracked section under the combined effect of factored dead load and live load ceases to be in compression. • Stress limit state—This limit state assumes that failure occurs when the tensile stress in the concrete on the ten- sion face calculated on the basis of the uncracked section under the combined effect of factored dead load and live load exceeds a certain tensile stress limit calculated on the basis of the uncracked section properties regardless of whether the section has previously been cracked. Stress limits of ft = 0.0948 fc′, ft = 0.19 fc′, and ft = 0.25 fc′ were initially considered in the reliability analysis; ft = 0.19 fc′ was used for the final calibration. • Crack width limit state—This limit state assumes that fail- ure occurs when a previously formed crack in the concrete opens, and the crack width reaches a certain prespecified crack width. Crack widths of 0.008, 0.012, and 0.016 in. were initially considered in the reliability analysis, but none produced uniform reliability. The bulk of the calibration was performed using a crack width of 0.016 in. The differen- tiation between different environments was accounted for through the use of different reliability indices in association with the same crack width. For each girder, the design was performed according to certain stress limits, as is conventionally done, and the girder section and number of strands were determined. The reli- ability index was determined for each of the three limit state functions described above by 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 are violated. The frequency at which any of the three limit states will be 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 area and force, reach- ing 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 a higher load to violate a specific limit state means that the section resistance is higher, which would cause the curve representing the resistance in Figure 6.1 to be shifted to the right. This results in a higher reliability index. Table 6.20 shows the required load and the corresponding reliability index for the three limit states relative to each other. With the target reliability index dependent on the definition of the limit state, selecting the target reliability index required investigating all three criteria and selecting the one that pro- vided more uniform reliability across a wide range of bridge geometrical characteristics. 6.5.4.2 Statistical Parameters of Variables Included in the Design Several variables affect the resistance of prestressed compo- nents. Table 6.21 shows a list of variables considered to be Table 6.20. Relation Between Limiting Criteria and Reliability Index for a Given Girder Limiting Criterion 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

192 Table 6.21. Random Variables and the Value of Their Statistical Parameters Variable Distribution Mean () CV () 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 CEc = Ec/(gc1.5 z fc′ ) 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 = 29,000 ksi Es Normal Esn 0.024 Siriaksorn and Naaman (1980) f ′c Lognormal 1.11f ′cn 0.11 Nowak et al. (2008) fpu Lognormal 1.03fpun 0.015 Nowak et al. (2008); fpun = 270 ksi fsi Normal 0.97fsin 0.08 Developed from Gross and Burns (2000) fy Lognormal 1.13fyn 0.03 Nowak et al. (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) gc Normal gcn = 150 0.03 Siriaksorn and Naaman (1980) Dfs Normal 1.05Dfsn 0.10 Developed from Gross and Burns (2000) and Tadros et al. (2003) S0 Normal S0n 0.03 Siriaksorn and Naaman (1980) Note: Subscript n refers to nominal values. Notations: As = area of nonprestressing steel (in.2); Aps = area of prestressing steel in tension zone (in.2); b = prestressed beam top flange width (in.); b0 = deck width transformed to beam material (in.); b1 = prestressed beam bottom flange width (in.); bw = web thickness (in.); c = depth of neutral axis from 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 nonprestressing steel (in.); e1 = eccentricity of prestressing force with respect to centroid of the section at midspan (in.); Eps = modulus of elasticity of prestressing steel (psi); Es = modulus of elasticity of nonprestressing steel (psi); 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 nonprestressing 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 beam members (ft); gc = unit weight of concrete (lb/ft3); Dfs = prestress losses (psi); and S0 = sum of reinforcing element circumferences (in.).

193 random during the performance of the reliability analyses. These variables represent a summary of the information from research studies by Siriaksorn and Naaman (1980) and Nowak et al. (2008). 6.5.4.3 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 adja- cent box girder bridges, and 36 spread box girder bridges. The geometric characteristics of the bridges are included in Appendix F. Depending on the environmental exposure conditions, both AASHTO Standard Specifications and AASHTO LRFD allow designing conventional prestressed components for a maxi- mum concrete tensile stress of ft = 0.0948 fc′ or ft = 0.19 fc′ for severe corrosion conditions or no worse than moderate corrosion conditions, respectively. When either specification is applied without owner’s exceptions, most bridges are designed for ft = 0.19 fc′, with a small number of bridges in coastal areas designed for ft = 0.0948 fc′. The stress limit for which each bridge in the database was designed was unknown. As the per- centage of bridges designed for severe corrosive conditions is small, it was assumed that most bridges in the database were likely to have been designed for the higher limit. The construction dates of the bridges considered suggest that they were all designed using the prestressing loss provisions method that existed in both the AASHTO Standard Specifica- tions and 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 the starting point for the calibration. 6.5.4.4 Estimated Reliability Index of Existing Bridges Table 6.22 summarizes the average reliability indices for the existing I- and bulb-T girder bridges database. For example, the average reliability indices at decompression level, maxi- mum allowable tensile stress limit under service loads of ft = 0.19 fc′, and maximum allowable crack width limit of 0.016 in. are 0.74, 1.05, and 2.69, respectively, for an ADTT of 5,000 and a return period of 1 year. 6.5.4.5 Database of Simulated Bridges A database of simulated simple-span bridges was designed using AASHTO I-girder sections for four cases. The simu- lated 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 estimat- ing 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 were signified by the maximum concrete tensile stress limit ( ft = 0.0948 fc′ or ft = 0.19 fc′) used in the design. The four cases of design considered were as follows: Case 1: AASHTO LRFD with maximum concrete tensile stress of ft = 0.0948 fc′ and pre-2005 prestress loss method; Case 2: AASHTO LRFD with maximum concrete tensile stress of ft = 0.0948 fc′ and post-2005 prestress loss method; Case 3: AASHTO LRFD with maximum concrete tensile stress of ft = 0.19 fc′ and pre-2005 prestress loss method; and Case 4: AASHTO LRFD with maximum concrete tensile stress of ft = 0.19 fc′ and post-2005 prestress loss method. Table 6.23 and Table 6.24, respectively, show the span length and girder spacing along with the calculated reliability indices for I-girder bridges designed for maximum concrete tensile stress ft = 0.0948 fc′ (Case 1 and Case 2) and ft = 0.19 fc′ (Case 3 and Case 4) for ADTT = 5,000. In performing the design, the cases using the post-2005 prestress loss method (Case 2 and Case 4) were designed using the smallest possible AASHTO girder size. To facilitate the com- parisons, when 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 for which 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 is shown in Table 6.23 for Case 1 or in Table 6.24 for Case 3. For the 140-ft span bridges with 12-ft girder spacing, no AASHTO I-girder section was sufficient. Table 6.22. Summary of Reliability Indices for Existing I- and Bulb-T Girder Bridges with One Lane Loaded and Return Period of 1 Year Performance Level ADTT 1,000 2,500 5,000 10,000 Decompression 0.95 0.85 0.74 0.61 Maximum tensile stress limit ft = 0.0948 fc′ 1.15 1.01 0.94 0.82 ft = 0.19 fc′ 1.24 1.14 1.05 0.95 ft = 0.25 fc′ 1.40 1.27 1.19 1.07 Maximum crack width (in.) 0.008 2.29 2.21 1.99 1.85 0.012 2.65 2.60 2.37 2.22 0.016 3.06 2.89 2.69 2.56

194 Table 6.23. Summary of Reliability Indices of Simulated Bridges Designed Using AASHTO Girders with ADTT  5,000 and ft  0.0948 fc′ Case Section Type Span Length (ft) Spacing (ft) Case 1 Case 2 Designed Using Pre-2005 Loss Method Designed Using Post-2005 Loss Method Decompression Maximum Tensile Maximum Crack Decompression Maximum Tensile Maximum 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 na 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 Note: — = there is no design; na = not applicable because no AASHTO I-girder was sufficient.

195 Table 6.24. Summary of Reliability Indices of Simulated Bridges Designed Using AASHTO Girders with ADTT  5,000 and ft  0.19 fc′ Case Section Type Span Length (ft) Spacing (ft) Case 3 Case 4 Designed Using Pre-2005 Loss Method Designed Using Post-2005 Loss Method Decompression Maximum Tensile Maximum Crack Decompression Maximum Tensile Maximum 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 na 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 Note: — = there is no design; na = not applicable because no AASHTO I-girder was sufficient.

196 Bridges designed for Case 1 and Case 3 are also thought to be similar to those designed using AASHTO Standard Speci- fications 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 repre- sent 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 with Case 2 and Case 3 with Case 4 shows the effect of changing the prestressing loss method. Using the post-2005 prestress loss method resulted in a smaller number of strands than the pre-2005 loss method. As shown in Table 6.23 and Table 6.24, the lower number of strands resulted in lower reliability indices for bridges designed using the post-2005 prestress loss method. As shown in Table 6.23 and Table 6.24, regardless of the loss method and/or the limit state used, the reliability indices for each case varied significantly. This variation in values suggested the need to calibrate the limit state to develop a combination of load and resistance factors to produce a more uniform reliability index across the range of different span lengths and girder spacings. 6.5.4.6 Selection of Target Reliability Index The target reliability indices were selected on the basis of the calculated average values of the reliability levels of exist- ing bridges and previous practice, with some consideration given to experiences from other codes (Eurocode and ISO 2394 document). A return period of 1 year and an ADTT equal to 5,000 were used. Table 6.25 shows the target reliability indices selected in this study, as well as the reliability indices for the existing and simulated bridge databases. Note 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 for which 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 at the decompression performance level for existing bridges, simulated bridges designed for severe environments, and simulated bridges designed for normal environments was around 0.74, 1.44, and 1.07, respectively (see Table 6.22 to Table 6.25). Consequently, 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 envi- ronments, respectively. A reliability index of 1.0 means that 15 of 100 bridges will probably have the bottom of the girder decompress in any given year. 6.5.5 Calibration Result The basic steps of the calibration process are shown below as they relate to the Service III calibration. 6.5.5.1 Step 1: Formulate Limit State Function and Identify Basic Variables The three limit state functions that were investigated are listed in Section 6.5.4.1. The limit state function is formulated by deriving an expression for the resistance prediction 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 D presents a detailed derivation of the resis- tance 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 Table 6.25. Reliability Indices (b) for Existing and Simulated Bridges with Return Period of 1 Year and ADTT  5,000 Performance Level Average b for Proposed Target b for Existing Bridges in NCHRP 12-78 Simulated Bridges Designed for ft  0.0948 fc′ and Pre-2005 Loss Method Simulated Bridges Designed for ft  0.19 fc′ and Pre-2005 Loss Method Bridges in Severe Environment Bridges in Normal Environment Decompression 0.74 1.44 1.07 1.20 1.00 Maximum allowable tensile stress of ft = 0.19 fc′ 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

197 derived by setting the crack width to zero in the general equa- tion 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. Appen- dix C presents a comparison and evaluation of maximum crack width prediction equations for prestressed concrete members. 6.5.5.2 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 6.5.4.5. For a maximum crack width limit state, a crack width of 0.016 in. was consid- ered. For the maximum allowable stress limit state, the stress considered is as stated in the discussion included in the fol- lowing sections. 6.5.5.3 Step 3: Determine Load and Resistance Parameters for Selected Design Cases The variables included the dimension of the cross section and the material properties. The statistical information included the probability distribution and statistical parameters, such as mean (µ) and standard deviation (s). 6.5.5.4 Step 4: Develop Statistical Models for Load and Resistance The variables affecting the load and resistance were identified. These variables included live load; those affecting resistance, such as the dimensions of the cross section; and the material properties. The statistical information included the probability distribution and statistical parameters for live load presented in Section 5.3.2 and for other variables affecting the resis- tance presented in Section 6.5.4.2. 6.5.5.5 Step 5: Develop Reliability Analysis Procedure The statistical information of all the required variables was 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, 1,000 values were generated independently on the basis of the statistics and dis- tribution of that random variable. For each simulation, the dead load and the resistance were calculated using one of the 1,000 sets of values of each random variable, resulting in 1,000 values of the dead load and the resistance. The mean and standard deviation of the dead load and the resistance were then calculated on the basis of the 1,000 simulations. 6.5.5.6 Step 6: Calculate Reliability Indices for Current Design Code and Current Practice Using the statistics of the dead load and the resistance calcu- lated from Monte Carlo simulation (as described above) and the statistics of the live load as derived from the WIM data (as described in Section 5), the reliability index was calculated for each girder. The reliability index (b) was calculated using Equation 6.13: R Q R Q (6.13) 2 2 β = µ − µ σ + σ where µR = mean value of resistance; µQ = mean value of applied loads; sR = standard deviation of resistance; and sQ = standard deviation of applied loads. The calculated reliability indices of existing and simulated bridges are shown in Table 6.22 to Table 6.24. 6.5.5.7 Step 7: Review Results and Select Target Reliability Index The initial target reliability index (bT) was determined as shown in Table 6.25. 6.5.5.8 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 (2012) (i.e., equal to 1.0). The Service III limit state resistance is affected by the ten- sile 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 ten- sile stresses of ft = 0.0948 fc′, ft = 0.19 fc′, and ft = 0.25 fc′ were considered. In addition, the simulated bridge database used in determining the target resistance factor was expanded to allow longer spans. Because there were three concrete tensile stress limits, Step 8 is divided into three repetitions designated 8a, 8b, and 8c. For this step, the range of span lengths was increased to 220 ft.

198 step 8a: seLect potentiaL Load and resistance factors for service iii: Bridges designed for MaxiMUM concrete tensiLe stress of ft = 0.0948 ′fc The calibration for a selected bridge database (shown in Table 6.26) was performed assuming an ADTT of 5,000 and a maximum concrete design tensile stress of ft = 0.0948 fc′. 1. Calculate the reliability level of designs according to AASHTO LRFD (2012) (Figure 6.37 to Figure 6.39). Figures 6.37 to 6.39 show the reliability indices for the bridges designed using AASHTO type girders according to AASHTO LRFD (2012), including a load factor of 0.8 for the Service III limit state, and assuming a maximum concrete tensile stress of ft = 0.0948 fc′. The geometric characteristics of the bridges are shown in Table 6.26. The average reliability index for the decompression limit state, maximum allowable tensile stress limit state, and maximum allowable crack width limit state were 0.97, 1.31, and 3.06, respectively. As the reli- ability indices were lower than the target reliability indices and were not uniform across different spans, modifications to the load factor were applied in the next step in an attempt to achieve higher, and more uniform, reliability indices. 2. Redesign the bridges with a live load factor of 1.0. In this step, the bridges were redesigned using a live load factor of 1.0, and the dead load and resistance factors were kept the same. Table 6.27 shows the design geometric charac- teristics of the redesigned bridges. Table 6.26. Summary Information of Bridges Designed with gLL  0.8 and ft  0.0948 fc′ Case Section Type Span Length (ft) Girder Spacing (ft) Aps (in.2) No. 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 (continued on next page)

199 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 Note: — = a practical solution was not found. Table 6.26. Summary Information of Bridges Designed with gLL  0.8 and ft  0.0948 fc′ (continued) Case Section Type Span Length (ft) Girder Spacing (ft) Aps (in.2) No. of Strands R el ia bi lit y In de x Span Length (ft.) Figure 6.37. Reliability indices for bridges at decompression limit state (ADTT  5,000, gLL  0.8, and ft  0.0948 fc′ ). Span Length (ft.) R el ia bi lit y In de x Figure 6.38. Reliability indices for bridges at maximum allowable tensile stress limit state (ADTT  5,000, gLL  0.8, and ft  0.0948 fc′ ). R el ia bi lit y In de x Span Length (ft.) Figure 6.39. Reliability indices for bridges at maximum allowable crack width limit state (ADTT  5,000, gLL  0.8, and ft  0.0948 fc′ ). Figure 6.40 to Figure 6.42 show the reliability indices for the redesigned bridges using a live load factor of 1.0. The average reliability indices for the decompression limit state, the maximum allowable tensile stress limit state, and the maximum allowable crack width limit state were 1.33, 1.70, and 3.32, respectively. 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. Consequently, a live load factor of 1.0 was proposed if the tensile stress was limited to ft = 0.0948 fc′. step 8B: seLect potentiaL Load and resistance factors for service iii: Bridges designed for MaxiMUM concrete tensiLe stress of ft = 0.19 ′fc The work described under Step 8a was repeated, except the girders were redesigned assuming a maximum concrete tensile stress of ft = 0.19 fc′. 1. Calculate the reliability level of designs according to AASHTO LRFD (2012) with a maximum concrete tensile stress for design of ft = 0.19 fc′ (Figure 6.43 to Figure 6.45). 2. Redesign the bridges with a live load factor of 1.0. Figure 6.46 to Figure 6.48 show the reliability indices for the redesigned bridges using a live load factor of 1.0 and ft = 0.19

200 Table 6.27. Summary Information of Bridges Designed with gLL  1.0 and ft  0.0948 fc′ Case Section Type Span Length (ft) Girder Spacing (ft) Aps (in.2) No. 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 VI 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-2200 220 9 22.338 146 Note: — = a practical solution was not found.

201 R el ia bi lit y In de x Span Length (ft.) Figure 6.40. Reliability indices for bridges at decompression limit state (ADTT  5,000, gLL  1.0, and ft  0.0948 fc′ ). R el ia bi lit y In de x Span Length (ft.) Figure 6.41. Reliability indices for bridges at maximum allowable tensile stress limit state (ADTT  5,000, gLL  1.0, and ft  0.0948 fc′ ). R el ia bi lit y In de x Span Length (ft.) Figure 6.42. Reliability indices for bridges at maximum allowable crack width limit state (ADTT  5,000, gLL  1.0, and ft  0.0948 fc′ ). R el ia bi lit y In de x Span Length (ft.) Figure 6.43. Reliability indices for bridges at decompression limit state (ADTT  5,000, gLL  0.8, and ft  0.19 fc′ ). R el ia bi lit y In de x Span Length (ft.) Figure 6.44. Reliability indices for bridges at maximum allowable tensile stress limit state (ADTT  5,000, gLL  0.8, and ft  0.19 fc′ ). R el ia bi lit y In de x Span Length (ft.) Figure 6.45. Reliability indices for bridges at maximum allowable crack width limit state (ADTT  5,000, gLL  0.8, and ft  0.19 fc′ ). R el ia bi lit y In de x Span Length (ft.) Figure 6.46. Reliability indices for bridges at decompression limit state (ADTT  5,000, gLL  1.0, and ft  0.19 fc′ ). R el ia bi lit y In de x Span Length (ft.) Figure 6.47. Reliability indices for bridges at maximum tensile stress limit state (ADTT  5,000, gLL  1.0, and ft  0.19 fc′ ).

202 fc′. Similar to the case of bridges designed for a maximum concrete tensile stress of ft = 0.0948 fc′, 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. Consequently, a live load factor of 1.0 was proposed if the maximum tensile stress was limited to 0.19 fc′. step 8c: seLect potentiaL Load and resistance factors for service iii: Bridges designed for MaxiMUM concrete tensiLe stress of ft = 0.25 ′fc The work described under Step 8a and Step 8b was repeated, except the girders were redesigned assuming a maximum concrete tensile stress of ft = 0.25 fc′. 1. Calculate the reliability level of designs according to AASHTO LRFD (2010) with a maximum concrete tensile stress for design of ft = 0.25 fc′ (Figure 6.49 to Figure 6.51). 2. Redesign the bridges with a live load factor of 1.0. Figure 6.52 to Figure 6.54 show the reliability indices for the redesigned bridges using a live load factor of 1.0 and ft = 0.25 fc′. Similar to the case of bridges designed for maxi- mum concrete tensile stresses of ft = 0.0948 fc′ and ft = 0.16 fc′, the reliability level of bridges became more uni- form than the case of using a live load factor of 0.8, particu- larly for the decompression and maximum tensile stress limit states. Consequently, a live load factor of 1.0 was proposed if the maximum tensile stress was limited to ft = 0.25 fc′. 6.5.5.9 Step 9: Calculate Reliability Indices The reliability indices were calculated for three cases, as shown in Step 8. In Step 9, the calculated values were reviewed to determine whether they were close to the target reliability R el ia bi lit y In de x Span Length (ft.) Figure 6.48. Reliability indices for bridges at maximum crack width limit state (ADTT  5,000, gLL  1.0, and ft  0.19 fc′ ). R el ia bi lit y In de x Span Length (ft.) Figure 6.49. Reliability indices for bridges at decompression limit state (ADTT  5,000, gLL  0.8, and ft  0.25 fc′ ). R el ia bi lit y In de x Span Length (ft.) Figure 6.50. Reliability indices for bridges at maximum allowable tensile stress limit state (ADTT  5,000, gLL  0.8, and ft  0.25 fc′ ). R el ia bi lit y In de x Span Length (ft.) Figure 6.51. Reliability indices for bridges at maximum allowable crack width limit state (ADTT  5,000, gLL  0.8, and ft  0.25 fc′ ). R el ia bi lit y In de x Span Length (ft.) Figure 6.52. Reliability indices for bridges at decompression limit state (ADTT  5,000, gLL  1.0, and ft  0.25 fc′ ).

203 index and whether they were uniform across the range of spans considered. If they were not, the load factors, resistance factors, and/or the concrete tensile stress limit used for design were changed, and Step 8 was repeated. The limit state function to be used as the basis for the calibration was also determined in Step 9. 6.5.5.10 Summary of Target Reliability Indices for Different Design and Performance Levels Summaries of the average reliability indices calculated for the different cases are given in Table 6.28 to Table 6.30. Regardless of the maximum tensile stress limits used in the design, the limiting criterion for the maximum tensile stress when deter- mining the reliability index was taken as ft = 0.19 fc′. As indicated earlier, the calibration of the specifications were based on an ADTT of 5,000. For this ADTT, the reli- ability indices obtained assuming the bridges were designed for maximum stress limits of ft = 0.0948 fc′ and ft = 0.19 fc′ (see the bold outlined cells below in Table 6.28 and Table 6.29, respectively) are very close to the target reliability indices shown in Table 6.25. 6.5.5.11 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 R el ia bi lit y In de x Span Length (ft.) Figure 6.53. Reliability indices for bridges at maximum tensile stress limit state (ADTT  5,000, gLL  1.0, and ft  0.25 fc′ ). R el ia bi lit y In de x Span Length (ft.) Figure 6.54. Reliability indices for bridges at maximum crack width limit state (ADTT  5,000, gLL  1.0, and ft  0.25 fc′ ). Table 6.28. Summary of Reliability Indices for Simulated Bridges Designed for ft  0.0948 fc′ Live Load Factor  0.8 Live Load Factor  1.0 ADTT Decompression Maximum Tensile Stress Limit Crack Width (in.) Decompression Maximum Tensile Stress Limit Crack Width (in.) 1,000 1.05 1.41 3.16 1.42 1.79 3.36 2,500 1.01 1.35 3.11 1.38 1.75 3.33 5,000 0.97 1.31 3.06 1.33 1.70 3.32 10,000 0.94 1.30 3.00 1.32 1.66 3.28 Table 6.29. Summary of Reliability Indices for Simulated Bridges Designed for ft  0.19 fc′ Live Load Factor  0.8 Live Load Factor  1.0 ADTT Decompression Maximum Tensile Stress Limit Crack Width (in.) Decompression Maximum Tensile Stress Limit Crack Width (in.) 1,000 0.84 1.27 2.92 1.11 1.53 3.25 2,500 0.70 1.15 2.87 1.04 1.46 3.17 5,000 0.68 1.10 2.82 1.00 1.41 3.14 10,000 0.64 1.07 2.78 0.98 1.34 3.11

204 was compared (see Table 6.31). The comparison indicated that when a live load factor of 0.8 was used in both cases, the post-2005 prestress loss method resulted in a smaller number of strands than when the pre-2005 prestress loss method was used. It also indicated that when the post-2005 loss method was used with a load factor of 1.0, the required number of strands was similar to that required when a load factor of 0.8 was used in conjunction with the pre-2005 prestress loss method (i.e., the designs were similar between the pre-2005 and post-2005 methods). 6.5.5.12 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 6.5.3, and comparing the target reliability indices shown in Table 6.25 and the calculated reliability indices for different design crite- ria, load factors, and design live load as shown in Table 6.28 to Table 6.30 and Figure 6.37 to Figure 6.54, 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 reli- ability index varies on the basis of the following: • The design maximum concrete tensile stress [maximum tensile stresses of ft = 0.0948 fc′ and ft = 0.19 fc′ are currently shown in AASHTO LRFD (2012) and are pro- posed to remain the same]; • The limit state function [i.e., decompression, tensile stress of a certain value (assumed to be ft = 0.19 fc′ 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 6.28 to Table 6.30. 2. The target reliability index can be achieved uniformly across various span lengths by using the load factor devel- oped by 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 correspond- ing to an ADTT of 5,000 be used as the basis for the cali- bration. The reliability index is not highly sensitive to changes in the ADTT, so there is no need to use different load factors for ADTTs up to 10,000. 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 support targeting a different (either higher or lower) reliability index. 5. The recommended target reliability index for the decom- pression 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. The reliability index, which was based on the study of the WIM data, is determined assuming live load exists in a single lane and without applying the MPF. 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 parameters be used for designing for the Service III limit state: • Live load factor of 1.0; • Maximum concrete tensile stress of ft = 0.0948 fc′ and ft = 0.19 fc′ for bridges in severe corrosion conditions and for bridges in no worse than moderate corrosion conditions, respectively; and • Girders to be designed following conventional design methods and assuming that live loads exist in single lane or multiple lanes, whichever produces higher load effects. The appropriate MPF 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 Table 6.30. Summary of Reliability Indices for Simulated Bridges Designed for ft  0.25 fc′ Live Load Factor  0.8 Live Load Factor  1.0 ADTT Decompression Maximum Tensile Stress Limit Crack Width (in.) Decompression Maximum Tensile Stress Limit Crack Width (in.) 1,000 0.20 0.55 2.83 0.93 1.29 3.03 2,500 0.08 0.49 2.77 0.89 1.27 2.95 5,000 0.06 0.44 2.72 0.85 1.23 2.92 10,000 0.02 0.41 2.66 0.82 1.20 2.88

205 Table 6.31. Comparison of Number of Strands Required for Different Design Assumptions Case Section Type Span Length (ft) Girder Spacing (ft) ft  0.0948 fc′ , gLL  0.8, Pre-2005 Losses ft  0.0948 fc′ , gLL  0.8, Post-2005 Losses ft  0.0948 fc′ , gLL  1.0, Post-2005 Losses ft  0.19 fc′ , gLL  0.8, Pre-2005 Losses ft  0.19 fc′ , gLL  0.8, Post-2005 Losses ft  0.19 fc′ , gLL  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 na 140 12 — — — — — — Note: — = a practical solution was not found; na = not applicable.

206 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. 6.5.6 Results for Adjacent Box Beams, Spread Box Beams, and American Segmental Box Institute Box Beams Work similar to that described above for I-beams was per- formed for adjacent box beams, spread box beams, and American Segmental Box Institute (ASBI) box beams. The details of the work are shown in Appendix D. The final results assuming the decompression limit state, ADTT of 5,000, return period of 1 year, and a load factor of 1.0 for live load are shown in Figure 6.55 to Figure 6.60. Table 6.32 shows the average reliability indices represented graphically in Figure 6.55 to Figure 6.60. Figure 6.55. Adjacent box beams, reliability indices for bridges at decompression limit state (ADTT  5,000, gLL  1.0, and ft  0.0948 fc′ ). R el ia bi lit y In de x Span Length (ft.) R el ia bi lit y In de x Span Length (ft.) Figure 6.56. Adjacent box beams, reliability indices for bridges at decompression limit state (ADTT  5,000, gLL  1.0, and ft  0.19 fc′ ). R el ia bi lit y In de x Span Length (ft.) Figure 6.57. Spread box beams, reliability indices for bridges at decompression limit state (ADTT  5,000, gLL  1.0, and ft  0.0948 fc′ ). R el ia bi lit y In de x Span Length (ft.) Figure 6.58. Spread box beams, reliability indices for bridges at decompression limit state (ADTT  5,000, gLL  1.0, and ft  0.19 fc′ ). R el ia bi lit y In de x Span Length (ft.) Figure 6.59. ASBI box beams, reliability indices for bridges at decompression limit state (ADTT  5,000, gLL  1.0, and ft  0.0948 fc′ ).

207 The results shown in Figure 6.55 to Figure 6.60 indicate that the reliability indices for each type of girder are reason- ably uniform across the range of spans considered. With the exception of the adjacent box beams, the average reliability indices for other section types were very close to each other and to the target reliability index. For adjacent box beams, the average reliability index was slightly higher. However, the difference did not warrant incorporating measures to reduce the resistance of the beams, such as revising the distri- bution factor equations or using lower load factors for adjacent box beams. 6.5.7 Sections Designed Using Other Methods of Determining Prestressing Time-Dependent Losses or Section Properties As indicated in Section 6.5.3, the calibration of Service III limit states assumed that the sections were designed using the AASHTO LRFD Refined Estimates of Time-Dependent Losses. AASHTO LRFD requires the time-dependent losses for segmental bridges to be determined using detailed time- step methods. The 2005 revisions to Refined Estimates of Time-Dependent Losses did not affect the time-dependent prestressing loss calculations for segmental bridges. Historically, segmental bridges have been designed using gross section properties, not transformed section properties, and the effects of elastic gain have been neglected. If approved by the owner, the time-step method may also be used to design prestressed concrete components other than segmental bridges. However, the level of effort required to perform time-step analysis typi- cally precludes this method for nonsegmental construction. The proposed increase in the load factor for live load for the Service III limit state from 0.8 to 1.0 is based on comparing sections designed using the AASHTO LRFD pre-2005 pro- visions and the post-2005 provisions without making any exceptions to the specifications requirements and assuming that the Refined Estimates of Time-Dependent Losses method in the AASHTO LRFD was used for calculating the time- dependent losses. The development of the method termed Approximate Esti- mate of Time-Dependent Losses in AASHTO LRFD was based on producing prestress losses similar to those produced by the Refined Estimates of Time-Dependent Losses method. Thus, the change in the load factor should also be applied to the former method. Because the changes in the prestress loss methods in 2005 did not affect the time-step method, 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 the time-step method. • Gross sections properties are used for the calculations. • The calculations of the force in the prestressing steel neglects the effects of the elastic gain. 6.5.8 Proposed AASHTO LRFD Revisions AASHTO LRFD (2012) Article 5.9.4.2.2 (Tension Stresses, which discusses stresses in fully prestressed components at SLS after losses) contains 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. With respect to the calibration of the limit state for tension in prestressed concrete presented above, the only required revisions to the specifications are those in Article 3.4.1, which should specify the load factor for live load as 0.8 or 1.0 depend- ing on the design procedure used. Table 6.32. Average Reliability Indices for Different Types of Girders Maximum Tensile Stress Used in Design (ksi) Type of Section ft  0.0948 fc′ ft  0.19 fc′ 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 R el ia bi lit y In de x Span Length (ft.) Figure 6.60. ASBI box beams, reliability indices for bridges at decompression limit state (ADTT  5,000, gLL  1.0, and ft  0.19 fc′ ).

208 6.6 Fatigue Limit States: Lifetime 6.6.1 Steel Members 6.6.1.1 Formulate Limit State Function Two limit states for load-induced fatigue of steel details are defined in AASHTO LRFD Article 3.4.1: Fatigue I, related to infinite load-induced fatigue life; and Fatigue II, related to finite load-induced fatigue life. For load-induced fatigue considerations, according to AASHTO LRFD Article 6.6.1.2.2, each steel detail should satisfy Equation 6.14: (6.14)f F n( ) ( )γ ∆ ≤ ∆ where g = load factor; Df = force effect (live load stress range due to the passage of the fatigue load); and (DF)n = nominal fatigue resistance. This general limit state function is used for the calibration of the fatigue limit states. 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 on extensive research of structural steel highway bridges. The fatigue load is the AASHTO LRFD design truck [the HS20-44 truck of AASHTO Standard Specifications (2002)], but with a fixed rear-axle spacing of 30 ft. The live load load factors for the fatigue limit state load combinations are summarized in Table 6.33. 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 effec- tive stress range caused by the Fatigue II load combination. The load factor for the Fatigue II load combination reflects load levels 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. The resistance factors for the fatigue limit states (f) are inherently taken as unity and hence do not appear in Equa- tion 6.14. 6.6.1.2 Select Structural Types and Design Cases Components and details susceptible to load-induced fatigue cracking have been grouped into eight groups, called detail categories, by fatigue resistance. AASHTO LRFD Table 6.6.1.2.3-1 illustrates many common details found in steel bridge construction and identifies potential crack sites for each detail. Figure 6.61 shows the current AASHTO LRFD fatigue design curves with the eight detail categories ranging from A to E′. Table 6.33. Current Fatigue Load Factors Fatigue Limit State Live Load Load Factor Fatigue I 1.5 Fatigue II 0.75 Source: AASHTO LRFD (2012). Figure 6.61. AASHTO fatigue design curves: stress range versus number of cycles. Used with permission of the American Association of State Highway and Transportation Officials.

209 6.6.1.3 Determine Load and Resistance Parameters for Selected Design Cases A comprehensive database containing constant and variable amplitude fatigue test results for various welded steel bridge detail types was developed by Keating and Fisher (1986). This database includes the test data from various NCHRP test pro- grams and other available data. These data are presented in Appendix F. The fatigue data include the detail type of each specimen, the minimum and maximum stress values, and the number of cycles observed until fatigue failure was evident. From these data, the stress range was taken as the only significant param- eter in the determination of the fatigue life, and a relationship between the stress range and number of cycles to failure was developed for the combined fatigue data (Keating and Fisher 1986). The regression analysis performed on the stress range versus cycle relation showed that this relation was log-log in nature. The curves of the data plotted in log form are charac- terized by Equation 6.15: log log log (6.15)N A B Sr= − or in exponential form as shown in Equation 6.16 N ASr B (6.16)= − where N = number of cycles to failure; Sr = constant amplitude stress range (ksi); log A = log-N-axis intercept of S-N curve (a constant taken from AASHTO LRFD Table 6.6.1.2.5-1 for the vari- ous detail categories); and B = slope of the curve. The combined fatigue data for each detail type were placed in the eight detail categories on the basis of the fatigue performance of the details as specified by AASHTO LRFD Article 6.6.1.2.3. Fatigue design curves were then determined for each of the fatigue categories. The design curves represent allowable stress range values that are based on a 98% confi- dence limit or lower bound of fatigue resistance. Thus, for a particular detail type, most of the fatigue data fall above the design curve, and the test data should not deviate significantly from the curve. The slopes of all the design curves were deter- mined to be very close to a constant value of –3.0, as shown through the use of regression analysis (Keating and Fisher 1986). Thus, a constant slope of -3.0 was imposed on the equations in the regression analysis. Figure 6.61 showed the current AASHTO LRFD fatigue design curves with the eight detail categories. 6.6.1.4 Develop Statistical Models for Loads and Resistances Load Uncertainties On the basis of the analysis of WIM data discussed in Chap- ter 5, 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, it is proposed that the load factor of 0.75 for the Fatigue II limit state be increased to 0.80. The mean values and CVs from Chapter 5 are shown in Table 6.34. resistance Uncertainties Fatigue Damage Parameter To properly calibrate the fatigue limit states of the AASHTO LRFD, it was necessary to determine the statistical param- eters of the fatigue test data used in the bridge fatigue resis- tance model. These parameters include the bias and the CV of the fatigue test data. As previously described, the fatigue data are commonly presented in terms of the stress range and number of cycles to failure, or S-N curves in log-log space. The use of this relationship with the given constant ampli- tude fatigue test data, however, causes difficulty in accurately determining the statistical parameters. The available data were not sufficiently distributed along the S-N curves in log- log space for a regression analysis; the data were often gath- ered over a small increment of stress ranges and were limited in number. Any number of regression lines could have been used to describe this relationship between the stress range and fatigue life. To better analyze the fatigue data, a different relationship between the number of cycles and stress range was developed. The test data were arranged to couple the number of cycles and stress range in the form of an effective stress range for each test specimen. The effective stress range as presented in Article 6.6.2.2 of AASHTO LRFD (2010) was taken as the cube root of the sum of the cubes of the measured stress ranges, as seen in Equation 6.17. The effective stress range is an accepted means to compare variable amplitude fatigue data with con- stant amplitude fatigue test data. S Sr i ri (6.17)eff 3 1 3( )( ) = Σγ Table 6.34. Load Uncertainties Limit State Mean CV Fatigue I 2.0 0.12 Fatigue II 0.8 0.07

210 where (Sr)eff = effective constant amplitude stress range; gi = percentage of cycles at a particular stress range; and Sri = constant amplitude stress range for a group of cycles (ksi). The formula describing the parameter used for the test data follows the form of Equation 6.17; however, this equation is applied to each of the test specimens. Thus, the percentage term is equated to a value of one and is subsequently multi- plied by the number of cycles (N) to yield Equation 6.18: pS Sfi riN (6.18) 3 1 3( )= where Sfi is a fatigue damage parameter. The fatigue damage parameter is taken as a normally distrib- uted random variable in order to determine the bias and CV of the fatigue resistance for each of the detail categories. The data were fitted to many of the typical distributions commonly used, and it was determined that the normal distribution best characterized the nature of the fatigue data. The bias is a ratio of the mean value of the test data to the nominal value described in the specifications. The calculation of the nominal, mean, and CV values are described in the following subsections. Probability Paper to Determine Statistical Parameters The collection of the fatigue data in terms of the new fatigue parameter for each detail category was statistically analyzed using normal probability paper, as the data best fit the normal distribution. The use of normal probability paper is explained in Chapter 3. The fatigue data for each detail category were filtered to include the data that most accurately reflected the fatigue behavior of each category. In other words, the data were trun- cated 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. The lower tail of the data was selected because it was the portion of the curve that fit the nor- mal distribution (i.e., it was the straight portion of the normal probability plot). Moreover, the lower portion of the fatigue data represented the range of values within which fatigue crack- ing was expected to occur when analyzed for the fatigue limit state load combinations using the Monte Carlo simulation approach, which is discussed in more detail below. Failure occurs when load exceeds resistance; thus, the higher portions of the fatigue data sets represented fatigue resistance data that were very unlikely to be exceeded by the fatigue loads used within this study and were therefore considered insignificant. Different approaches for selecting the cutoff values for each category were investigated to determine the sensitivity of the resulting reliability indices. It was determined that the relative differences of the results determined from the different tech- niques were negligible. Other techniques used to determine the cutoff values included the use of constant cutoff values for all the detail categories and having different analysts manually insert best-fit lines. Table 6.35 shows the resulting cutoff values for the standard normal variable. Figure 6.62 and Figure 6.63 show the normal probability plots of the full fatigue data set and the truncated data for categories C and C′, respectively. Determining the statistical parameters of the data was relatively straightforward once the data for each detail cate- gory were filtered and fitted with a line of best fit by using Microsoft Excel software. The mean value of the stress param- eter 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, the standard deviation is the change in the horizontal coordinates divided by the change in the vertical coordinates. CV is the ratio of the standard deviation to the mean of the data. The resulting statistical parameters are given in Table 6.35. The probability plots of the fatigue data and corresponding trun- cated data for all detail categories can be seen in Appendix E. Table 6.35. Resistance Uncertainties Category Standard Deviation CV Bias Sf_Mean Sf_AASHTO Cutoff Standard Normal Variable A 1,000.0 0.24 1.43 4,167.40 2,924 1 B 666.7 0.22 1.34 3,077.47 2,289 1 B9 250.0 0.11 1.28 2,336.10 1,827 1 C and C9 454.6 0.21 1.35 2,210.77 1,638 1 D 185.2 0.10 1.36 1,773.69 1,300 1 E 140.9 0.12 1.17 1,207.41 1,032 1 E9 232.6 0.20 1.56 1,140.28 730 1

211 Determination of Nominal Fatigue Parameter and Bias Values CV and the mean of the fatigue resistance data were deter- mined as described in the previous subsection. These values, along with the nominal fatigue resistance, were needed to determine the bias of the data. The nominal value of the cho- sen fatigue parameter was calculated using AASHTO LRFD Equation 6.6.1.2.5-2 and rearranged to achieve the relation- ship in terms of the desired fatigue damage parameter, as seen in Equation 6.19. The resulting nominal resistance values can be seen in Table 6.35. pS N S Af r (6.19)_ AASHTO 3 1 3 1 3( )= = where Sf_AASHTO is the nominal value of the fatigue parameter using AASHTO LRFD specifications for each detail category, and A is a constant taken from AASHTO LRFD Table 6.6.1.2.5-1 for the various detail categories. The bias value for each category was determined by taking the ratio of the mean value to the nominal value of the fatigue parameter, as seen in Equation 6.20; the results are shown in Table 6.35. S Sf fBias (6.20)_Mean _AASHTO= where Sf_Mean is the mean value of the fatigue parameter using the fatigue data for each detail category. 6.6.1.5 Develop Reliability Analysis Procedure In 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 was an iterative process that consisted of Monte Carlo simulations to select load and Figure 6.62. Normal probability plot of detail categories C and C fatigue data. Figure 6.63. Normal probability plot of detail categories C and C truncated fatigue data with best-fit line.

212 resistance factors that would achieve reliability close to the target reliability index. The Monte Carlo technique samples load and resistance parameters from selected statistical distri- butions, such as a normal distribution. Reliability is mea- sured in terms of b, the reliability index or safety index. b is defined as a function of the probability of failure by using Equation 6.21. Thus b is the number of standard deviations that the mean safety margin falls on the safe side. The higher the b value, the higher the reliability. (6.21)1 Pf( )β = − Φ− where F-1 is the inverse standard normal distribution func- tion, and Pf is the probability of failure. Monte carLo siMULation The Monte Carlo analysis is described more fully in Chapter 3. A step-by-step outline of the Monte Carlo simulation using Microsoft Excel is included in Appendix F. The distribution of loads was assumed to be normally distributed as the loads are a summation of force effects. The fatigue resistance was also assumed to follow normal dis- tributions. These distributions for load and resistance were developed using determined statistical parameters from the available data. 6.6.1.6 Calculate Reliability Indices for Current Design Code or Current Practice The current reliability indices inherent for the various fatigue detail categories were determined using the Monte Carlo simu- lation technique with the provisions for the Fatigue I and II limit states as specified in AASHTO LRFD. The Fatigue I limit state uses a load factor of 1.5, which is common to all the detail types. The CV for the Fatigue I limit state was determined to be 0.12 through the work discussed in Chapter 5. The resistance parameters for the Monte Carlo simulation were determined by equating the nominal load and resistance values and then applying the statistical parameters for each of the detail categories (see Table 6.35). Insufficient fatigue data exist to determine the constant amplitude fatigue threshold por- tions of the fatigue design curves for finite fatigue design life (Fatigue II limit state). In consultation with AASHTO Technical Committee T-14 and the American Iron and Steel Institute Bridge Task Force, the research team deemed it acceptable to use the statistical parameters for the sloping portions of these curves for the constant amplitude fatigue thresholds of the different bridge detail categories. In AASHTO LRFD, the Fatigue II limit state currently has a load factor of 0.75 for all the detail categories and a CV of 0.07. The nominal resistance values were determined using Equation 6.22, which resulted from setting the AASHTO LRFD fatigue resistance Equation 6.6.1.2.5-2 equal to the design fatigue load, which was normalized to a stress range equal to 1 ksi. R A 0.75 (6.22)3= where R is resistance, and A is a constant taken from AASHTO LRFD Table 6.6.1.2.5-1 for the various detail categories. The simulations for both limit states were completed using a total of 10,000 replicates to achieve a sufficient number of failures. The resulting reliability indices for each of the eight detail categories are reported in Table 6.36. 6.6.1.7 Select Target Reliability Index Target reliability indices (bT) were based on the inherent reliability of the current specifications (see Table 6.36). The fatigue limit states were harmonized by selecting a single, common target reliability index for both steel and concrete members equal to 1.0. This proposed target was selected to best reflect the inherent reliability of the Fatigue I and II limit states for structural steel members and the Fatigue I limit state for reinforcement and concrete. 6.6.1.8 Select Potential Load and Resistance Factors When the proposed load factors of 2.0 and 0.8 for the Fatigue I and Fatigue II limit states, respectively, and the inherent resis- tance factor of 1.0 were applied along with the statistical data, the reliability indices for each detail category remained essen- tially unchanged from those reported in Table 6.36. Accepting a range of ±0.2 on the reliability index, three Fatigue I limit state reliability indices appear to be too large: detail Category B′ at b = 1.5, detail Category D at b = 2.0, and detail Category E′ Table 6.36. Current Reliability Indices (b) Using AASHTO LRFD Fatigue I and Fatigue II Limit States Category Fatigue I Fatigue II A 1.2 1.0 B 1.1 0.9 B9 1.5 1.0 C 1.2 0.9 C9 1.2 0.9 D 2.0 1.3 E 0.9 0.7 E9 1.7 1.4

213 at b = 1.7. Similarly, two Fatigue II limit state reliability indi- ces appear to be too large (detail Category D at b = 1.3 and detail Category E′ at b = 1.4) and one appears to be too small (detail Category E at b = 0.7). Proposed resistance factors for the Fatigue I limit state and the Fatigue II limit state are given in Table 6.37 and Table 6.38, respectively. Resistance factors other than the current values of unity are shown in boldface. Reliability index values using the proposed resistance factors are shown in the right-hand column. 6.6.1.9 Calculate Reliability Indices With the proposed resistance factors, the reliability indices were all within ±0.2 of the target reliability index of 1.0. The reliability indices shown in Tables 6.37 and 6.38 can also be achieved by revising the AASHTO LRFD fatigue resistance equations for steel members. This may be a better solution than including resistance factors for only a few of the detail categories and in some cases greater than unity. The required revisions to the AASHTO LRFD tables are given in Table 6.39 and Table 6.40 with changes shown in boldface. 6.6.2 Concrete Members This section deals with concrete and reinforcing steel. Pre- stressing strand is not covered as there are currently no design checks required for fully prestressed components, as explained in Chapter 2. 6.6.2.1 Formulate Limit State Function Two limit states for load-induced fatigue are defined in AASHTO LRFD Article 3.4.1; however, only Fatigue I, related to infinite load-induced fatigue life, is valid for 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 Table 6.37. Proposed Fatigue I Limit State Resistance Factors Detail Category Proposed Resistance Factor () Reliability Index (b) A 1.0 1.2 B 1.0 1.1 B9 1.10 0.9 C 1.0 1.2 C9 1.0 1.2 D 1.15 1.1 E 1.0 0.9 E9 1.20 1.0 Table 6.38. Proposed Fatigue II Limit State Resistance Factors Detail Category Proposed Resistance Factor () Reliability Index (b) A 1.0 1.0 B 1.0 0.9 B9 1.0 1.0 C 1.0 0.9 C9 1.0 0.9 D 0.95 1.0 E 1.10 1.0 E9 0.90 1.0 Table 6.39. Proposed Revisions to AASHTO LRFD Table 6.6.1.2.5-1 Detail Category Current Constant A 108 Proposed Constant A 108 A 250 250 B 120 120 B9 61 61 C 44 44 C9 44 44 D 22 21 E 11 12 E9 3.9 3.5 Table 6.40. Proposed Revisions to AASHTO LRFD Table 6.6.1.2.5-3 Detail Category Current Constant Amplitude Fatigue Threshold (ksi) Proposed Constant Amplitude Fatigue Threshold (ksi) A 24 24 B 16 16 B9 12 13 C 10 10 C9 12 12 D 7 8.0 E 4.5 4.5 E9 2.6 3.1

214 Equation 6.23, which is seen as a variation of Equation 6.14 applicable only to infinite life: f F (6.23)TH( ) ( )γ ∆ ≤ ∆ where g = load factor; Df = force effect (live load stress range due to the passage of the fatigue load); and (DF)TH = constant amplitude fatigue threshold. The general limit state function given by Equation 6.23 will be used for the calibration of the fatigue limit states for con- crete members. As discussed in Section 6.6.1.1, the Fatigue I limit state load factor is currently 1.5, and all resistance factors are inherently unity for the fatigue limit states. 6.6.2.2 Select Structural Types and Design Cases Two fatigue limit states for concrete members can be ratio- nally calibrated on the basis of 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). 6.6.2.3 Determine Load and Resistance Parameters for Selected Design Cases steeL reinforceMent in tension Steel reinforcement as it is considered here 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 by Equa- tion 6.24: F f f y( )∆ = −24 20 (6.24)TH min where fmin is the minimum stress. For welded-wire reinforcement with a cross weld in the high-stress region, the fatigue resistance is specified by Equa- tion 6.25: F f16 0.33 (6.25)TH min( )∆ = − Equations 6.24 and 6.25 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 hori- zontal constant amplitude threshold has been drawn beneath all 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 could be identified in the research to determine their uncertainty in terms of bias, mean, and CV. The various thresholds were grouped together to make design practical. concrete in coMpression The compressive stress limit of 0.40 f ′c 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 combi- nation 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 the compressive stress limit 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 reevaluated to estimate the constant amplitude fatigue threshold, which is the infinite life fatigue resistance. The uncertainty of the fatigue resistance was quantified in terms of bias, mean, and CV. 6.6.2.4 Develop Statistical Models for Loads and Resistances Load Uncertainties The distribution of fatigue loads was determined on the basis of studies conducted on the WIM data, as described in Chapter 5. The fatigue load uncertainties in terms of the mean values and CVs are tabulated in Table 6.34. resistance Uncertainties As discussed in Section 6.6.1.4.2b, the lower tail of the fatigue resistance data plots was used to characterize the uncertainties, biases, and CVs. Figure 6.64 and Figure 6.65, respectively, show the normal probability plots of the full fatigue data set and the truncated data for fatigue resistance of steel reinforcement in tension. The resulting statistical parameters, along with the cutoff scores, are given in Table 6.41. The probability plots of the fatigue data and corresponding truncated data for both steel reinforcement in tension and concrete in compression can be seen in Appendix E.

215 Figure 6.65. Normal probability plot of truncated fatigue resistance data with best-fit line for steel reinforcement in tension. Figure 6.64. Normal probability plot of fatigue resistance data for steel reinforcement in tension. Table 6.41. Resistance Uncertainties Resistance Standard Deviation CV Bias Mean Nominal Cutoff Standard Normal Variable Steel reinforcement in tension 769.23 0.24 1.94 3,261.54 1,681.21 2 Concrete in compression 117.65 0.45 1.74 260.35 149.66 2 6.6.2.5 Develop Reliability Analysis Procedure As discussed in Section 6.6.1.5, Monte Carlo simulation using Microsoft Excel formed the basis of the reliability analysis procedure for fatigue of concrete members. 6.6.2.6 Calculate 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 on the basis of the uncertainties of load and resistance. For steel reinforcement in reinforced concrete members, the current inherent b is approximately 2.0, and the current inherent b for compression of concrete members is approxi- mately 1.0. Both fatigue limit states are based on the Fatigue I limit state and design for infinite life. The calculated inherent values of b are given in Table 6.42. 6.6.2.7 Select Target Reliability Index Theoretically, the target reliability index (bT) should be identical for all members and all fatigue limit states. Thus, the work on reinforcement and concrete fatigue was performed concurrently with, and was compared with, the work on structural steel fatigue. It is proposed to use a constant bT of 1.0 for steel reinforce- ment in tension, concrete in compression, and structural steel members. This proposed target reflects the inherent reliability

216 of the current Fatigue I limit state for concrete in compression and the Fatigue I and II limit states for structural steel members. This proposed target reduces the reliability of steel reinforce- ment in tension to levels consistent with the three other cali- brated fatigue limit states. 6.6.2.8 Select Potential Load and Resistance Factors Proposed resistance factors for the Fatigue I limit state are given in Table 6.43. Resistance factors other than the current values of unity are shown in boldface. 6.6.2.9 Calculate Reliability Indices With the proposed resistance factors, the reliability indices were all within ±0.1 of the target reliability index of 1.0. The reliability indices shown in Table 6.43 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 revisions 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 by Equation 6.26: F f f y( )∆ = −30 25 (6.26)TH min 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 by Equation 6.27: F f( )∆ = −20 0.41 (6.27)TH min 6.6.3 Proposed AASHTO LRFD Revisions In AASHTO LRFD (2012), the fatigue limit state is addressed in Sections 3, 5, and 6. The articles that require modification to implement the revisions recommended here are indicated in Table 6.44. Table 6.42. Current Reliability Indices for AASHTO LRFD Fatigue I Limit States Resistance b Steel reinforcement in tension 1.9 Concrete in compression 0.9 Table 6.43. Proposed Fatigue I Limit State Resistance Factors Resistance Proposed Resistance Factor () Reliability Index (b) Steel reinforcement in tension 1.25 1.1 Concrete in compression 1.0 0.9 Table 6.44. Summary of Relevant Articles in AASHTO LRFD for Foundation Fatigue Deformations Article 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 6.6.1.2.3 Detail Categories, Table 6.6.1.2.3-1 Constant A 6.6.1.2.5 Fatigue Resistance, Table 6.6.1.2.5-1 Constant A 6.6.1.2.5 Fatigue Resistance, Table 6.6.1.2.5-2 Cycle parameter (n) 6.6.1.2.5 Fatigue Resistance, Table 6.6.1.2.5-3 Constant amplitude fatigue threshold Note: The proposed article revisions are detailed in Chapter 7.

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