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

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3 OVERVIEW OF CALIBRATION PROCESS 3.1 Introduction The new generation of bridge design codes is based on probabilistic methods. Load and resistance (load carrying capacity) parameters are treated as random variables, and structural performance is quantified in terms of the reliability index (Nowak and Collins, 2013). This approach allows for a rational comparison of different materials and load combinations. Increased degree of uncertainty causes a reduction in the reliability and strict control of structural parameters results in a safer structure. The probabilistic analysis requires statistical models of load and resistance parameters. The load models for bridges can be based on truck surveys and other field tests. Resistance models for structural components (e.g. bridge girders) can be derived from material tests, lab tests, and analytical simulations. With the advent of limit states design methodology in North American design specifications, there has been increasing demand to obtain statistical data to assess the reliability of designs. Reliability depends on load and resistance factors that are determined through calibration procedures using available statistical data. Methodologies that can be used to determine load and resistance factors are described in NCHRP Report 368 (Nowak, 1999) and Transportation Research Board (TRB) Circular E-C079 (Allen, et al., 2005), including the basic reliability concepts and detailed procedures that can be used to characterize data to develop the statistics and functions needed for reliability analysis. The code calibration procedure can include closed-form solutions for estimating load and resistance factors that can be used for simple cases, as well as more rigorous probabilistic analysis methods such as the Monte Carlo method which is described in Section 3.2.3. There are three levels of probabilistic design: Levels I, II, and III (Nowak and Collins, 2013). The Level I method is the least accurate while Level III is the only fully probabilistic method. However, Level III requires complex statistical data beyond what is generally available in engineering practice. Level I and Level II probabilistic methods are more viable approaches for structural design. In Level I design methods, safety is measured in terms of a safety factor, or the ratio of nominal (design) resistance to nominal (design) load. In Level II, safety is expressed in terms of the reliability index, β. The Level II approach generally requires iterative techniques best performed using computer algorithms. For simpler cases, closed-form solutions to estimate β are available. Closed-form analytical procedures to estimate load and resistance factors should be considered approximate, with the exception of very simple cases where an exact closed-form solution exists. Alternatively, spreadsheet programs can be used to estimate load and resistance factors using the more rigorous and adaptable Monte Carlo simulation technique, which in turn can be used to accomplish a Level II probabilistic analysis. The goal of Level I or II analyses is to develop factors that increase the nominal load or decrease the nominal resistance to give a design with an acceptable and consistent reliability. To accomplish this, an equation that incorporates and relates all of the variables that affect the potential for failure of the structure or structural component must be developed for each limit state. For LRFD calibration purposes, statistical characterization should focus on the prediction of load or resistance relative to what is actually measured in a structure. Therefore, this statistical characterization is typically applied to the bias, the ratio of the measured to predicted value. The predicted (nominal) value is calculated using the design model being investigated. 12

The degree of variation is measured in terms of the coefficient of variation, which is the ratio of standard deviation to the mean value. Regardless of the level of probabilistic design used to perform LRFD calibration, the steps needed to conduct a calibration are as follows: • Develop the limit state equation to be evaluated, so that the correct random variables are considered. Each limit state equation must be developed based on a prescribed failure mechanism. The limit state equation should include all the parameters that describe the failure mechanism and that would normally be used to carry out a deterministic design of the structure or structural component. • Statistically characterize the data upon which the calibration is based (i.e., the data that statistically represent each random variable in the limit state equation being calibrated). Key parameters include the mean, standard deviation, and coefficient of variation (COV) as well as the type of distribution that best fits the data (i.e. often normal or lognormal). • Select a target reliability value based on the margin of safety implied in current designs, considering the need for consistency with reliability values used in the development of other AASHTO LRFD specifications, the consequence of exceeding the limit state, cost and the levels of reliability for design as reported in the literature for similar structures. If the performance of existing structures that were designed using the current code provisions is acceptable, then there is no need to increase safety margin in the newly developed code. Furthermore, the acceptable safety level can be taken as corresponding to the lower tail of distribution of betas. • Determine load and resistance factors using reliability theory consistent with the selected target reliability. It is recognized that the accuracy of the results of a reliability theory analysis is directly dependent on the adequacy, in terms of quantity and quality, of the input data used. The final decision made regarding the magnitude of the load and resistance factors selected for a given limit state must consider the adequacy of the data. If the adequacy of the input data is questionable, the final load and resistance factor combination selected should be weighted toward a level of safety that is consistent with past successful design practice, using the reliability theory results to gain insight as to whether or not past practice is conservative or unconservative. The calibration procedure can be different depending on the type of limit state. In the case of serviceability limit states, it is much more complex mostly due to difficulties in formulation of the limit state equation. The parameters of load and resistance are determined not only by magnitude, as is the case with strength limit states, but also frequency of occurrence (e.g. crack opening) and as a function of time (e.g. corrosion rate, chloride penetration rate). Acceptability criteria are not well defined as they are subjective (e.g. deflection limit, allowable tensile stress) and the code-specified limit state function does not necessarily have a physical meaning (e.g. allowable compression stress in concrete). 3.2 Calibration by Determination of Reliability Indices 3.2.1 Basic Framework Expanding on the four basic steps outlined above, the framework for calibration of SLS using reliability indices is summarized as follows: 13

Step 1: Formulate the Limit State Function and Identify Basic Variables. Identify the load and resistance parameters and formulate the limit state function. For each considered limit state, the acceptability criteria were established. In most cases, it was not possible to select a deterministic boundary between what is acceptable and unacceptable. Some of the code-specified limit state functions do not have a physical meaning (e.g. allowable compression stress in concrete). Step 2: Identify and Select Representative Structural Types and Design Cases. Select the representative components and structures to be considered in the development of code provisions for the SLSs. Step 3: Determine Load and Resistance Parameters for the Selected Design Cases. Identify the design parameters based on typical structural types, loads, and locations (climate, exposure to harsh environment). For each considered element and structure, values of typical load components must be determined. Step 4: Develop Statistical Models for Load and Resistance. Gather statistical information about the performance of the considered types and models, in selected representative locations and traffic. Gather statistical information about quality of workmanship. Ideally, for given location, and traffic, the required data includes: general assessment of performance, assumed time to initiation of deterioration, assumed deterioration rate as a function of time, maintenance, and repair (frequency and extent). Develop statistical load and resistance models (as a minimum, determine the bias factors and coefficients of variation). The parameters of load and resistance are determined not only by magnitude, as is the case with strength limit states, but also frequency of occurrence (e.g. crack opening) and as a function of time (e.g. corrosion rate, chloride penetration rate). The available statistical parameters were utilized. However, the database is rather limited and for some serviceability limit states, there is a need to assess, develop, and/or derive the statistical parameters. The parameters of time-varying loads were determined for various time periods. The analyses were performed for various traffic parameters (average daily truck traffic (ADTT), legal loads, multiple presence, traffic patterns). The load frequencies serve as a basis for determination of acceptability criteria. Step 5: Develop the Reliability Analysis Procedure. The reliability can be calculated using either a closed-form formula or Monte Carlo method. The reliability index for each case can be calculated using closed-formulas available for particular types of probability distribution functions (PDFs) in the literature or Monte Carlo method. In this study, all of the reliability calculations were based on Monte Carlo analysis. The Monte Carlo method is a stochastic technique that is based on the use of random numbers and probability statistics to simulate a large number of computer-based experiments. The outcome of the simulation is a large number of solutions that takes into account all the random variables in the resistance equation. Step 6: Calculate the Reliability Indices for Current Design Code and Current Practice. Calculate the reliability indices for selected representative bridge components corresponding to current design and practice. Step 7: Review the Results and Select the Target Reliability Index, βT. Based on the calculated reliability indices, select the target reliability index, βT. Select the acceptability criteria, i.e., performance parameters, that are acceptable, and performance parameters that are not acceptable. 14

Step 8: Select Potential Load and Resistance Factors. Prepare a recommended set of load and resistance factors. The objective is that the design parameters (load and resistance factors) have to meet the acceptability criteria for the considered design situations (location and traffic). The design parameters should provide reliability that is consistent, uniform, and conceivably close to the target level. Step 9: Calculate Reliability Indices. Calculate the reliability indices corresponding to the recommended set of load and resistance factors for verification. If the design parameters do not provide consistent safety levels, modify the parameters and repeat Step 8. Figure 3-1 presents the flowchart for the basic calibration framework described above. 15

Figure 3-1 Basic calibration framework – flowchart. Step 4 above requires the analysis of data describing load and resistance. Normal probability paper is a special scale that facilitates the statistical interpretation of data. The Calibration framework Calculate the reliability indices for current design code or current practice Select structural types and design cases Yes Determine load parameters for selected design cases Determine resistance parameters for selected design cases Develop statistical models for loads and load combinations Develop statistical models for resistance variables Statistical parameters of loads Statistical parameters of resistance Formulate the limit state function Experimental data Engineering judgment β ≈ βT End of the calibration procedure Load survey Experimental data Observations Develop the reliability analysis procedure Review the results and select the target reliability index, βT Select potential load and resistance factors Calculate reliability indices Modify potential load and resistance factors List obtained load and resistance factors No Yes 16

horizontal axis represents the variable for which the cumulative distribution function (CDF) is plotted, e.g. gross vehicle weight (GVW), mid-span moment or shear. The vertical axis represents the number of standard deviations from the mean value. This is often referred to as the Standard Normal Variable” or the “Z-Score.” The vertical axis can also be interpreted as probability of being exceeded and, for example, one standard deviation corresponds to 0.159 probability of being exceeded. The most important property of normal probability paper is that the CDF of a normal random variable is represented by a straight line. The straighter the plot of data, the more accurately it can be represented as a normal distribution. In addition, the curve representing the CDF of any other type of random variable can be evaluated and its shape can provide an indication about the statistical parameters such as the maximum value, type of distribution for the whole CDF or, if needed, only for the upper or lower tail of the CDF. Furthermore, the intersection of the CDF with horizontal axis (zero on vertical scale), corresponds to the mean. The slope of CDF determines the standard deviation, σx as shown in Figure 3-2. A steeper CDF on probability paper indicates a smaller standard deviation. Further information about construction and use of the probability paper can be found in textbooks (e.g. Nowak and Collins, 2013). Figure 3-2 Use of normal probability paper. σx μx σx Normal probability scale S ta nd ar d no rm al v ar ia bl e 3 2 1 0 -1 -2 -3 0.005 0.010 0.020 0.050 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 0.950 0.980 0.990 0.995 0.841 0.159 0 1 2 3 4 5 6 7 17

3.2.2 Closed-form Solutions The reliability index, β, is defined as ( )β Φ 1 fP−= (3-1) where Φ 1− is the inverse of the standard normal distribution and Pf is the probability of failure. If the limit state function can be expressed in terms of two random variables, R representing the resistance and Q representing the load effect, g R Q= − (3-2) then the probability of failure is P = Prob (g < 0)f (3-3) Then, the reliability index, β, can be calculated using a closed-form formula in two cases: when both R and Q are normal random variables or when both R and Q are lognormal random variables. In all other cases, the available procedures produce approximate results. In the case of R and Q both being normal random variables, the reliability index, β, can be calculated using the following formula, 2 2σ σ β R Q R Q− = + (3-4) where R = mean or expected value of the distribution of resistance Q = mean or expected value of the distribution of load Rσ = standard deviation of the distribution of resistance Qσ = standard deviation of the distribution of load Sometimes, QR − is termed M , the margin of safety. Using this terminology the equation becomes, β σM M = (3-5) For the case when both distributions are lognormally distributed, a more complete derivation of the closed-form solutions and how they can be applied to LRFD calibration is shown by Allen, et al. (2005). While closed-form solutions are useful for illustrative purposes, in practice either load or resistance or both are not normally distributed which limits the use of closed-form solutions in code calibration. 18

3.2.3 Using Monte Carlo Simulation in the Calibration Process The typical application of Monte Carlo simulation, referenced in Step 5 above for bridge- structural reliability, as reported in the literature (Allen, et al., 2005; Nowak and Collins, 2013) is well known. Application of Monte Carlo simulation follows the steps below: • It is assumed that dead load is normally distributed and live load CDF is as shown on the probability paper (directly from WIM data). The statistical parameters of live load depend on the time period. For longer time period the statistical parameters are obtained by extrapolation of the available WIM data. The total load is a sum of dead load and live load and, therefore, in practice it can be treated as a normal variable. This assumption is partly justified by the Central Limit Theorem, and is acceptable if the load components are of similar magnitude (Nowak and Collins 2013). • Resistance is assumed to have lognormal distribution. The resistance side of the LRFD equation is a product of terms. • The minimum statistical parameters needed for each random variable are the coefficient of variation, V, and the bias, λ. Using the reported statistics of load and resistance along with computer-generated random numbers, the distributions of load and resistance are developed and values chosen randomly from these distributions. For example for the simple load combination of dead load plus live load, random values of dead load and live load are chosen from the normal distributions fitted in the region of interest. A random value of resistance is chosen from the lognormal distribution of resistance. • The simulation is run by selecting random values from both the load and resistance distributions. The limit state function, Ri – (Di + Li) is calculated for each set of random variables. If the value is equal to or greater than zero, the function is satisfied and the individual case is safe. If the value is negative, the criterion is not satisfied and the case represents a failure. • After a large number of iterations, the failures are counted and the failure rate determined. For the sampling to be significant at least ten failures should be observed, otherwise, more iteration is necessary. If the expected probability of failure is very low, then the number of iterations can be prohibitively large. Therefore, an alternative way to determine the reliability index is to generate a smaller number of limit state function values, plot the results on the normal probability paper, and extrapolate the obtained lower tail of the distribution function. The extrapolated lower tail will then allow for assessment of the reliability index and probability of failure (or failure rate). • Using the failure rate, the reliability index is determined as the inverse of the standard normal cumulative distribution. 3.2.4 Statistical Parameters for Resistance and Other Loads (Excerpted from Kulicki, et al. 2007) 3.2.4.1 Resistance Models The resistance was considered as a product of a nominal resistance, nR , and three factors: M = material factor (strength of material, modulus of elasticity), F = fabrication factor (geometry, dimensions), and P = professional factor (use of approximate resistance models, e.g. the Whitney stress block, idealized stress and strain distribution model). 19

PFMRR n ⋅⋅⋅= (3-6) The mean value, Rµ , and the coefficient of variation, RV , of resistance, R , may be approximated by the following accepted equations for the range of values that were considered: PFMnR R µµµµ ⋅⋅⋅= (3-7) 2 2 2 R M F PV V V V= + + (3-8) The statistical parameters of resistance of reinforced concrete and prestressed concrete were determined using the test results available prior to 1990, special simulations, and engineering judgment. They were developed for reinforced concrete T-beams and prestressed concrete AASHTO-type girders. Bias factors and coefficients of variation were determined for material factor, M, fabrication factor, F, and analysis factor, P. Factors M and F were combined. For concrete components, the material parameters were taken from Ellingwood, et al. (1980). Only the statistical parameters were obtained but no raw test data. The basis for these parameters was research by Mirza and MacGregor (1979), The data included mean value and coefficient of variation for the compressive strength of concrete, yield strength of reinforcing bars, and prestressing strands. In addition, the data included the statistical parameters of fabrication factor and professional factor. The material data, combined with statistical parameters of the fabrication factor and professional factor, were used in Monte Carlo simulations that resulted in the statistical parameters of resistance for reinforced concrete T-beams and prestressed concrete girders, for moment and shear, as shown in Table 3-1 (Nowak, 1999). The statistical parameters include three factors representing uncertainty in materials, dimensions and geometry, and analytical model. It was assumed that resistance is a lognormal random variable. Table 3-1 Statistical Parameters of Component Resistance (Used with permission of the Transportation Research Board of the National Academies) Type of Structure Material and Fabrication factors, FM Professional factor, P Resistance, R λ V λ V λ V Reinforced concrete Moment 1.12 0.12 1.02 0.06 1.14 0.13 Shear w/steel 1.13 0.12 1.075 0.10 1.20 0.155 Shear no steel 1.165 0.135 1.20 0.10 1.40 0.17 Prestressed concrete Moment 1.04 0.045 1.01 0.06 1.05 0.075 Shear w/steel 1.07 0.10 1.075 0.10 1.15 0.14 20

3.2.4.2 Statistics of Loads Other Than Live Load The data presented below were developed in support of strength calibrations but are equally applicable to load calculations related to SLS calibration. The bias factors for DL1 and DL2 were provided by the Ontario Ministry of Transportation based on surveys of actual bridges in conjunction with calibration of the Ontario Highway Bridge Design Code (OHBDC) (OHBDC, 1979; Lind and Nowak, 1978). The coefficients of variation provided by the Ministry of Transportation for dead load were 0.04 and 0.08 for DL1 and DL2, respectively (Lind and Nowak, 1978). However, there is no report available to support this data. The coefficients of variation used in calibration were taken from the National Bureau of Standards (NBS) Special Publication 577 (Ellingwood, et al. 1980) and include other uncertainties (also human error). The parameters of DL3 are calculated using the survey data provided by the Ontario Ministry of Transportation in conjunction with calibration of the OHBDC (1979). Table 3-2 Statistical Parameters of Dead Load Dead Load Component Bias Factor Coefficient of Variation Factory made members, DL1 1.03 0.08 Cast-in-place, DL2 1.05 0.10 Wearing surface, DL3 1.00 (for 3 in. mean thickness) 0.25 Miscellaneous, DL4 1.03 ~ 1.05 0.08 ~ 0.10 3.3 “Deemed to Satisfy” “When you can measure what you are speaking about, and express it in numbers, you know something about it, when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely, in your thoughts advanced to the stage of science.” William Thomson, Lord Kelvin The least rigorous process for establishing design requirements, and load and resistance factors in particular, is referred to as “deemed to satisfy.” In this process, experience and empirical observations are used to define the boundary between satisfactory performance and unsatisfactory performance. It provides no quantifiable way of assessing the provided margin of adequacy such as safety or reliability. Since there is no way to quantify the performance margin, there is no way to assess the benefit of a change in requirement other than a general knowledge that changing a certain parameter should move in the direction of higher performance. The obvious corollary is that cost/benefit cannot be quantified. An example of “Deemed to Satisfy” is the specification of concrete cover requirements in U.S. practice which is based only on experience and has no consistent mathematical basis. The above notwithstanding, “deemed to satisfy” has a place in the pantheon of engineering tools. It is often the basis of detailing requirements and may serve as the beginning of design specification development as in “experience shows that if we do (or don’t do) this or that the results are generally acceptable.” Expert elicitation (Delphi Process) or an experimental program may provide insight into the adequacy of deemed to satisfy. 21