Calibration of AASHTO LRFD Concrete Bridge Design Specifications for Serviceability (2014) / Chapter Skim
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From page 85... ...
5 CALIBRATION RESULTS 5.1 Cracking of Reinforced Concrete Components Service I Limit State – Annual Probability Traditionally, reinforced concrete components are designed to satisfy the requirements of the strength limit state and then they are checked for the Service I limit state load combination to ensure that the crack width under service conditions does not exceed a certain value. However, the specifications provisions are written in a form emphasizing reinforcement details, i.e.
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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.
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Table 5-1 Summary of Statistical Information for Variables used in the Calibration of Service I Limit State for Crack Control Variable Distribution Mean COV Remarks sA normal 0.9 sA 0.015 Siriaksorn and Naaman (1980) b normal nb 0.04 Siriaksorn and Naaman (1980)
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deck design. If #5 bars result in a bar spacing less than 5 in., #6 bars were used.
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Table 5-3 Summary of Reliability Indices for Concrete Decks Designed According to AASHTO LRFD (2012) ADTT Positive Moment Region Negative Moment Region Reliability Index (Class 1)
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Table 5-4 Reliability Indices of Existing Bridges based on 1-year Return Period ADTT Reliability Index Current Practice (Class 1, Negative) Current Practice (Class 2, Negative)
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and the resistance were calculated using one of the 1000 sets of values of the random variable, i.e. the nth simulation used the nth value of each random variable where n varied from 1 to 1000.
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the average values of the reliability index are 1.66 and 1.61 for positive and negative moment regions, respectively. Figure 5-1 Reliability Indices of Various Bridge Decks Designed Using a 1.0 Live Load Factor Over A 1 Year Return Period (ADTT=5000)
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average values of the reliability index are 0.85 and 1.05 for positive and negative moment regions, respectively. Figure 5-3 Reliability Indices Of Various Bridge Decks Designed Using A 1.0 Live Load Factor Over A 1 Year Return Period (ADTT=5000)
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requirements and the reliability index for cracking at the bottom will be higher than shown in Figure 5-1 and Figure 5-3. This resulted in the recommendation that the reliability index should be based on the negative moment (top)
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The load factors for DL and LL specified for the two load combinations are as follows: Service I: DL load factor = 1.0 LL load factor = 1.0 Service III: DL load factor = 1.0 LL load factor = 0.8 Based on the definition of the two limit states, Service I limit state is used for calculating all service stresses in the superstructure and substructure components at all stages except that Service III limit state is used to calculate the tensile stresses in the superstructure components under full service loads and the principal tension in webs of segmental concrete. Stresses immediately after transfer are independent of the live loads.
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Specifications. The girder distribution factors, particularly for interior girders, for many typical girder systems in the AASHTO LRFD Specifications are lower than that in the Standard Specifications thus reducing the difference between the unfactored distributed load effects in the two specifications.
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that produce tensile stresses in the concrete at the strand locations are applied to the girder, the strands are subjected to an additional tensile strain equal to the strain in the surrounding concrete due to the application of the loads. This results in an increase in the force in the strands.
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• The section properties used in the analysis are based on the gross section of the concrete, and • The calculations of prestressing losses consider the effects of the "elastic gain" as allowed by the current design provisions. Regardless of the method of design used in designing a girder, the stresses in the girder used as part of the reliability index calculations were determined by analyzing the girder using the above assumptions.
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were initially considered in the reliability analysis, however, none produced uniform reliability. The bulk of the calibration was performed using a crack width of 0.016 inches.
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Table 5-6 Random Variables and the Value of Their Statistical Parameters Variables Distribution Mean, μ COV, Ω Remarks As normal 0.9Asn* 0.015 Siriaksorn and Naaman (1980)
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f′c = specified compressive strength of concrete, psi fpu = specified tensile strength of prestressing steel, psi fsi = initial stress in prestressing steel, psi fy = yield strength of non-prestressing steel, psi h = girder depth, in. hf = deck thickness, in.
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Table 5-7 Summary of Reliability Indices for Existing I- and Bulb T Girder Bridges with One Lane Loaded and Return Period of 1 Year Performance Levels ADTT ADTT=1000 ADTT=2500 ADTT=5000 ADTT=10000 Decompression 0.95 0.85 0.74 0.61 Maximum Tensile Stress Limit 0.0948t cf f ′= 1.15 1.01 0.94 0.82 0.19t cf f ′= 1.24 1.14 1.05 0.95 0.25t cf f ′= 1.40 1.27 1.19 1.07 Maximum Crack Width 0.008 in 2.29 2.21 1.99 1.85 0.012 in 2.65 2.60 2.37 2.22 0.016 in 3.06 2.89 2.69 2.56 5.2.4.5 Database of Simulated Bridges A database of simulated simple span bridges was designed using AASHTO I-girder sections for four different cases. The simulated bridges have span lengths of 30, 60, 80, 100, and 140 ft.
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is shown in Table 5-8 and Table 5-9 for Case 1 and Case 3.
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Table 5-8 Summary of the Reliability Indices of Simulated Bridges Designed Using AASHTO Girders with ADTT=5000 and 0.0948t cf f ′= Case 1 Case 2 Cases Section Type Span Length (ft.) Spacin g (ft.)
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Table 5-9 Summary of the Reliability Indices of Simulated Bridges Designed Using AASHTO Girders with ADTT=5000 and 0.19t cf f ′= Case 3 Case 4 Cases Section Type Span Length (ft.) Spacing (ft.)
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5.2.4.6 Selection of the Target Reliability Index The target reliability indices were selected based on the calculated average values of the reliability levels of existing bridges and previous practices with some consideration given to experiences from other Codes (Eurocode and International Organization for Standardization (ISO) 2394 Document)
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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.
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5.2.5.6 Step 6: Calculate the Reliability Indices for Current Design Code and Current Practice Using the statistics of the dead load and the resistance, calculated from Monte Carlo simulation as described above, and the statistics of the live load as derived from the WIM data as described in Section 4, the reliability index was calculated for each girder. The reliability index was calculated using the following equation: 2 2 R Q R Q µ − µ β = σ + σ (5-2)
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5.2.5.8.1 Step 8a: Select Potential Load and Resistance Factors for Service III - Bridges Designed for Maximum Concrete Tensile Stress of 0.0948t cf f ′= In this section, the calibration for a selected bridge database (shown in Table 5-11) was performed assuming an ADTT of 5000 and maximum concrete design tensile stress of 0.0948t cf f ′= .
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Table 5-11 Summary Information of Bridges Designed with γLL=0.8, ( 0.0948t cf f ′= ) Cases Section Type Span Length (ft.)
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Figure 5-5 Reliability indices for bridges at decompression limit state (ADTT=5000)
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Figure 5-7 Reliability Indices for bridges at maximum allowable crack width limit state (ADTT=5000)
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Table 5-12 Summary Information of Bridges Designed with γLL=1.0, ( 0.0948t cf f ′= ) Section Type Span Length (ft.)
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Figure 5-8 Reliability indices for bridges at decompression limit state (ADTT=5000)
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Figure 5-10 Reliability indices for bridges at maximum allowable crack width limit state (ADTT=5000)
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Figure 5-12 Reliability indices for bridges at maximum allowable tensile stress limit state (ADTT=5000)
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Figure 5-14 Reliability indices for bridges at decompression limit state (ADTT=5000)
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Figure 5-16 Reliability indices for bridges at maximum crack width limit state (ADTT=5000)
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Figure 5-18 Reliability indices for bridges at maximum allowable tensile stress limit state (ADTT=5000)
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Figure 5-20 Reliability indices for bridges at decompression limit state (ADTT=5000)
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Figure 5-22 Reliability indices for bridges at maximum crack width limit state (ADTT=5000)
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Table 5-14 Summary of Reliability Indices for Simulated Bridges Designed for 0.19t cf f ′= ADTT Live Load Factor=0.8 Live Load Factor=1.0 Decompression Max Tensile Stress Limit Crack Width Decompression Max Tensile Stress Limit Crack Width 1000 0.84 1.27 2.92 1.11 1.53 3.25 2500 0.70 1.15 2.87 1.04 1.46 3.17 5000 0.68 1.10 2.82 1.00 1.41 3.14 10000 0.64 1.07 2.78 0.98 1.34 3.11 Table 5-15 Summary of Reliability Indices for Simulated Bridges Designed for 0.25t cf f ′= ADTT Live Load Factor=0.8 Live Load Factor=1.0 Decompression Max Tensile Stress Limit Crack Width Decompression Max Tensile Stress Limit Crack Width 1000 0.20 0.55 2.83 0.93 1.29 3.03 2500 0.08 0.49 2.77 0.89 1.27 2.95 5000 0.06 0.44 2.72 0.85 1.23 2.92 10000 0.02 0.41 2.66 0.82 1.20 2.88 As indicated earlier, the calibration of the specifications are based on an ADTT of 5000. It was observed that for this ADTT, the reliability indices obtained assuming the bridges are designed for maximum stress limit of 0.0948t cf f ′= and 0.19t cf f ′= (see the outlined cells Table 5-13 and Table 5-14)
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5.2.5.12 Summary and Recommendations for Service III Limit State For typical I-girders designed using the post-2005 prestress loss method and the assumptions listed in Section 5.2.3, comparing the target reliability indices shown in Table 5-10 and the calculated reliability indices for different design criteria, load factors, and design live load as shown in Table 5-13 through Table 5-15 and Figure 5-5 through Figure 5-22, the following conclusions were drawn and summarized: 1. For a specific girder of known cross-section and specific number and arrangement of prestressing strands, the reliability index varies based on: • The design maximum concrete tensile stress (a maximum tensile stress of 0.0948t cf f ′= and 0.19t cf f ′= is currently shown in AASHTO LRFD (2012)
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Table 5-16 Comparison of number of strands required for different design assumptions Cases Section Type Span Length (ft.) Girder Spacing (ft.)
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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.
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Figure 5-23- Adjacent box beams, reliability indices for bridges at decompression limit state (ADTT=5000)
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Figure 5-25 Spread box beams, reliability indices for bridges at decompression limit state (ADTT=5000)
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Figure 5-27 ASBI box beams, reliability indices for bridges at decompression limit state (ADTT=5000)
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Table 5-17 Average Reliability Indices for Different Types of Girders Type of Section Maximum tensile stress used in design (ksi) 0.0948t cf f ′= 0.19t cf f ′= I- and Bulb T Girders 1.33 1.00 Adjacent Box Beams 1.85 1.31 Spread Box Beams 1.45 1.01 ASBI Box Beams 1.41 1.00 The results shown in Figure 5-23 through Figure 5-28 indicate that the reliability indices for each type of girder is reasonably uniform across the range of spans considered.
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• Gross sections properties are used for the calculations, and • The calculations of the force in the prestressing steel neglects the effects of the elastic gain. 5.2.8 Proposed AASHTO LRFD Revisions In AASHTO (2012)
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Table 5-18 Current Fatigue Load Factors Fatigue Limit State LL Load Factor Fatigue I 1.5 Fatigue II (used for steel structures only) 0.75 The load factor for the Fatigue I load combination reflects load levels found to be representative of the maximum stress range of the truck population for infinite fatigue-life design.
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For welded-wire reinforcement with a cross weld in the high-stress region, the fatigue resistance is specified as: ( )
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Table 5-19 Load Uncertainties Limit State Mean COV Fatigue I 2.0 0.12 Fatigue II (used for steel structures only) 0.8 0.07 5.3.1.3.2 Resistance Uncertainties The collection of the fatigue data was statistically analyzed using normal probability plots as the data best fits the normal distribution which is explained in further detail later.
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statistical parameters can be seen in Table 5-20. The probability plots of the fatigue data and corresponding truncated data for steel reinforcement in tension and concrete in compression can be seen in Appendix G
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Table 5-20 Resistance Uncertainties Resistance Standard Deviation COV Bias Mean Nominal Cutoff Standard Normal Variable steel reinforcement in tension 769.23 0.24 1.94 3261.54 1681.21 2 concrete in compression 117.65 0.45 1.74 260.35 149.66 2 5.3.1.4 Develop the Reliability Analysis Procedure 5.3.1.4.1 General In the code calibration it is necessary to develop a process by which to express the structural reliability or the probability of the loads on the member being greater than its resistance; in other words, failure of the criteria. The reliability analysis performed within this project is an iterative process that consists of Monte Carlo simulations to select load and resistance factors that achieve reliability close to the target reliability index.
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The simulations for both limit states were completed using a total of 10,000 replicates to achieve a sufficient number of failures. For steel reinforcement in reinforced concrete members, the inherent β is approximately 2.0, but the inherent β for compression of concrete members is approximately 1.0.
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Table 5-23 Proposed Fatigue I Limit-State Resistance Factors Resistance Proposed Resistance Factor, Φ Reliability Index, β Steel reinforcement in tension 0.8 1.1 Concrete in compression 1.0 0.9 5.3.1.8 Calculate Reliability Indices With the proposed resistance factors, the reliability indices are all within ± 0.1 of the target reliability index of 1.0. The resultant reliability indices tabulated above can also be achieved by revising the AASHTO LRFD constant-amplitude fatigue thresholds for steel reinforcement in tension.
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Table 5-24 Summary of Relevant Articles in AASHTO LRFD for Fatigue Article (See Note) Title Relates To 3.4.1, Table 3.4.1-1 Load Factors and Load Combinations Fatigue I and II*
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components under normal service loads, the consequences of significantly exceeding the decompression, i.e. opening a wide crack, also need to be quantified.
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