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

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

48 C h a p t e r 3 3.1 Introduction The new generation of bridge design codes is based on proba- bilistic methods. Load and resistance (load-carrying capac- ity) 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 ratio- nal comparison of different materials and load combinations. An increased degree of uncertainty causes a reduction in reli- ability, 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 ana- lytical simulations. With the advent of limit states design methodology in North American design specifications, there has been an 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, including the basic reliability con- cepts and detailed procedures that can be used to characterize data to develop the statistics and functions needed for reliabil- ity analysis, are described in NCHRP Report 368 (Nowak 1999) and TRB Circular E-C079 (Allen et al. 2005). 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 probabilis- tic analysis methods such as the Monte Carlo method, which is described in Section 3.2.3. There are three levels of proba- bilistic design: Levels I, II, and III (Nowak and Collins 2013). The Level I method is the least accurate, and Level III is the only fully probabilistic method. However, Level III requires complex statistical data beyond what are 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 a reli- ability index (b). The Level II approach generally requires iterative techniques best performed using computer algo- rithms. For simpler cases, closed-form solutions to estimate b are available. Closed-form analytical procedures to estimate load and resistance factors should be considered approxi- mate, with the exception of very simple cases for which 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 simula- tion 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 the variables that affect the potential for failure of the structure or structural component must be developed for each limit state. For load and resistance factor design (LRFD) calibration purposes, statistical characterization should focus on the pre- diction of load or resistance relative to what is actually mea- sured in a structure. Thus, this statistical characterization is typically applied to the bias, the ratio of the measured to pre- dicted value. The predicted (nominal) value is calculated using the design model being investigated. The degree of variation is measured in terms of the coefficient of variation (CV), which is the ratio of the standard deviation to the mean value. Regardless of the level of probabilistic design used to per- form LRFD calibration, the steps needed to conduct a cali- bration are as follows: • Develop the limit state equation to be evaluated so that the correct random variables are considered. Each limit Overview of Calibration Process

49 state equation must be developed on the basis of a pre- scribed 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 on which the calibration is based (i.e., the data that statistically represent each ran- dom variable in the limit state equation being calibrated). Key parameters include the mean, standard deviation, and CV, as well as the type of distribution that best fits the data (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 struc- tures. If the performance of existing structures that were designed using the current code provisions is acceptable, then there is no need to increase the safety margin in the newly developed code. Furthermore, the acceptable safety level can be taken as corresponding to the lower tail of dis- tribution of betas. • Determine load and resistance factors by using reliability theory consistent with the selected target reliability. 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 past practice is conserva- tive 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 by frequency of occurrence (e.g., crack opening) and as a function of time (e.g., corrosion rate, chloride penetration rate). Accept- ability 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 frame- work for calibration of service limit state (SLS) using reliabil- ity indices is summarized as follows: 1. Formulate the limit state function and identify basic vari- ables. Identify the load and resistance parameters and for- mulate 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 bound- ary 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). 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. 3. Determine load and resistance parameters for the selected design cases. Identify the design parameters on the basis of typical structural types, loads, and locations (climate, exposure to harsh environment). For each considered ele- ment and structure, the values of typical load components must be determined. 4. Develop statistical models for load and resistance. Gather sta- tistical information about the performance of the consid- ered types and models in selected representative locations and traffic. Gather statistical information about quality of workmanship. Ideally, for a given location and traffic, the required data include general assessment of performance, assumed time to initiation of deterioration, assumed dete- rioration rate as a function of time, maintenance, and repair (frequency and extent). Develop statistical load and resis- tance models (as a minimum, determine the bias factors and CVs). The parameters of load and resistance are deter- mined not only by magnitude, as is the case with strength limit states, but also by frequency of occurrence (e.g., crack opening) and as a function of time (e.g., corrosion rate, chloride penetration rate). The available statistical param- eters were used, but the database is limited, and for some serviceability limit states there is a need to assess, develop, 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, legal loads, multiple presence, traffic patterns). The load frequencies served as a basis for determination of acceptability criteria.

50 5. Develop the reliability analysis procedure. Reliability can be calculated using either a closed-form formula or the Monte Carlo method. The reliability index for each case can be calculated using closed formulas available for par- ticular types of probability distribution functions in the literature or the Monte Carlo method. In this study, all the reliability calculations were based on Monte Carlo analy- sis. 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 ran- dom variables in the resistance equation. 6. Calculate the reliability indices for current design code and current practice. Calculate the reliability indices for selected representative bridge components corresponding to cur- rent design and practice. 7. Review the results and select the target reliability index. Use the calculated reliability indices to select the target reliabil- ity index (bT). Select the acceptability criteria (i.e., perfor- mance parameters) that are acceptable and the performance parameters that are not acceptable. 8. Select potential load and resistance factors. Prepare a rec- ommended set of load and resistance factors. The objec- tive is that the design parameters (load and resistance factors) have to meet the acceptability criteria for the con- sidered design situations (location and traffic). The design parameters should provide reliability that is consistent, uniform, and conceivably close to the target level. 9. Calculate reliability indices. Calculate the reliability indices corresponding to the recommended set of load and resis- tance 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 in the nine steps above. Step 4 requires the analysis of data describing load and resistance. Normal probability paper is a special scale that facilitates the statistical interpretation of data. The horizontal axis represents the variable (e.g., gross vehicle weight, mid- span moment, or shear) for which the cumulative distribu- tion function (CDF) is plotted. The vertical axis represents the number of standard deviations from the mean value, which is often referred to as the standard normal variable, or the Z-score. The vertical axis can also be interpreted as the probability of being exceeded; 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 and type of distribution for the whole CDF or, if needed, only for the upper or lower tail of the CDF. Fur- thermore, the intersection of the CDF with the horizontal axis (zero on the vertical scale) corresponds to the mean. The slope of the CDF determines the standard deviation, or sx as shown in Figure 3.2. A steeper CDF on probability paper indicates a smaller standard deviation. Further information about the construction and use of probability paper can be found in textbooks (e.g., Nowak and Collins 2013). 3.2.2 Closed-Form Solutions The reliability index (b) is defined as shown by Equation 3.1: (3.1)1 Pf )(β = Φ− where F-1 is the inverse of the standard normal distribution, and Pf is the probability of failure. If the limit state function (g) can be expressed in terms of two random variables, R representing resistance and Q repre- senting the load effect, then g is given by Equation 3.2: (3.2)g R Q= − and the probability of failure is expressed by Equation 3.3: Prob 0 (3.3)P gf ( )= < b can then 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. If both R and Q are normal random variables, b can be calculated using Equation 3.4: (3.4) 2 2 R Q R Q β = − σ + σ where R _ = mean or expected value of the distribution of resistance; Q _ = mean or expected value of the distribution of load; sR = standard deviation of the distribution of resistance; and sQ = standard deviation of the distribution of load.

51 Calibration framework Calculate the reliability indices for current design code or current practice Select structural types and design cases 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 Figure 3.1. Flowchart of basic calibration framework.

52 Sometimes, R - Q is termed M, the margin of safety. Using this terminology, b is given by Equation 3.5: (3.5) M M β = σ For the case in which 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). Although closed-form solu- tions 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. 3.2.3 Using Monte Carlo Simulation in the Calibration Process The typical application of Monte Carlo simulation, refer- enced in Step 5 for bridge structural reliability and as reported in the literature (Allen et al. 2005; Nowak and Collins 2013), is well known. Application of Monte Carlo simulation follows these steps: • It is assumed that dead load is normally distributed and live load CDF is as shown on the probability paper [directly from WIM (weigh in motion) data]. The statistical param- eters of live load depend on the time period. For longer time periods, 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 it is acceptable if the load components are of similar magni- tude (Nowak and Collins 2013). • Resistance is assumed to be lognormally distributed. The resistance side of the LRFD equation is a product of terms. • The minimum statistical parameters needed for each random variable are the CV (V) and the bias (l). Using the reported statistics of load and resistance along with computer-generated random numbers, the distributions of load and resistance are developed, and values are chosen Figure 3.2. Use of normal probability paper. σx µx σx Normal probability scale St an da rd n o 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

53 randomly from these distributions. For example, for the simple load combination of dead load plus live load, ran- dom values of dead load and live load are chosen from the normal distributions fitted in the region of interest. A ran- dom value of resistance is chosen from the lognormal dis- tribution 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 is determined. For the sampling to be significant at least 10 failures should be observed; other- wise, more iteration is necessary. If the expected probabil- ity of failure is very low, then the number of iterations can be prohibitively large. An alternative way to determine the reliability index is to generate a smaller number of limit state function values, plot the results on normal probabil- ity paper, and extrapolate the obtained lower tail of the distribution function. The extrapolated lower tail will allow for assessment of the reliability index and probability of failure (or failure rate). • By using the failure rate, the reliability index is determined as the inverse of the standard normal cumulative distribution. A step-by-step procedure for implementing the Monte Carlo method by using statistical functions commonly avail- able in spreadsheet programs is presented in Appendix F. 3.2.4 Statistical Parameters for Resistance and Other Loads The discussion in this section is excerpted from Kulicki et al. (2007). Resistance Models Resistance was considered as a product of a nominal resis- tance (Rn) and three factors: M, or material factor (strength of material, modulus of elasticity); F, or fabrication factor (geometry, dimensions); and P, or professional factor (use of approximate resistance models; e.g., the Whitney stress block, idealized stress and strain distribution model). Resistance (R) is given by Equation 3.6: (3.6)R R M F Pn i i i= The mean value of resistance (µR) and the CV of resistance (VR) may be approximated by Equations 3.7 and 3.8, respectively, which are accepted equations for the range of values that were considered: (3.7)RR n M F Pi i iµ = µ µ µ (3.8)2 2 2V V V VR M F P= + + The statistical parameters of resistance were determined using the test results available before 1990, special simula- tions, and engineering judgment. They were developed for non composite and composite steel girders, reinforced con- crete T-beams, and prestressed concrete AASHTO-type girders. Bias factors and CVs were determined for material factor M, fabrication factor F, and analysis factor P. Factors M and F were combined. For structural steel, the statistical parameters are found in papers by Ravindra and Galambos (1978), Yura et al. (1978), Cooper et al. (1978), and Hansell et al. (1978), which are sum- marized in Ellingwood et al. (1980). The information included the mean values and CV for the yield strength of steel, tensile strength of steel, and modulus of elasticity for hot-rolled beams and plates. In addition, they provided the statistical parameters (mean value and CV) for the fabrication factor and the professional factor. In the very last phase of calibration for AASHTO LRFD, the American Iron and Steel Institute provided the upgraded bias factors and CVs for yield strength of structural steel. These values were then used in Monte Carlo simulations to determine the parameters of resistance for noncomposite and composite girders for the moment- carrying capacity and shear. [More recent data gathered after the Northridge earthquake by Dexter et al. (2000) and Dexter and Melendrez (2000), and data reported by Bartlett et al. (2003), show improved statistics, although Bartlett et al. rec- ommend no resistance factor changes until more is known. In the case of the steel SLSs calibrated in this study, the newer data could affect only the overload limit state, making the reli- ability analysis somewhat conservative. Given the paucity of resistance data on which this limit state is based, the analysis was not updated for the more recent data.] For concrete components, the material parameters were taken from Ellingwood et al. (1980). As in the case of struc- tural steel, the statistical parameters were obtained, but no raw test data. The basis for these parameters was research by Mirza and MacGregor (1979a, 1979b). The data included mean value and CV for the compressive strength of concrete, yield strength of reinforcing bars, and prestressing strands. In addition, the data included the statistical parameters of fab- rication factor and professional factor. The material data, combined with the statistical parame- ters of the fabrication factor and professional factor, were used in Monte Carlo simulations that resulted in the statisti- cal parameters of resistance for steel girders (noncomposite

54 Table 3.1. Statistical Parameters of Component Resistance Type of Structure Material and Fabrication Factors (M and F) Professional Factor (P) Resistance (R) l V l V l V Noncomposite steel girders Moment (compact) 1.095 0.075 1.02 0.06 1.12 0.10 Moment (noncompact) 1.085 0.075 1.03 0.06 1.12 0.10 Shear 1.12 0.08 1.02 0.07 1.14 0.105 Composite steel girders Moment 1.07 0.08 1.05 0.06 1.12 0.10 Shear 1.12 0.08 1.02 0.07 1.14 0.105 Reinforced concrete Moment 1.12 0.12 1.02 0.06 1.14 0.13 Shear with steel 1.13 0.12 1.075 0.10 1.20 0.155 Shear without 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 with steel 1.07 0.10 1.075 0.10 1.15 0.14 Source: Nowak (1999). Table 3.2. Statistical Parameters of Dead Load Dead Load Component Bias Factor CV Factory-made members, DL1 1.03 0.08 Cast-in-place, DL2 1.05 0.10 Wearing surface, DL3 3 in. (mean thickness) 0.25 Miscellaneous, DL4 1.03 ~ 1.05 0.08 ~ 0.10 and composite), reinforced concrete T-beams, and pre- stressed concrete girders, for moment and shear, as shown in Table 3.1 (Nowak 1999). The statistical parameters include three factors representing uncertainty in materials, dimen- sions and geometry, and analytical model. It was assumed that resistance is a lognormal random variable. Statistics of Loads Other Than Live Load The data presented below were developed in support of strength calibrations, but they are equally applicable to load calculations related to SLS calibration (see Table 3.2). The bias factors for DL1 and DL2 were provided by the Ontario Ministry of Transportation and were based on sur- veys of actual bridges in conjunction with calibration of the Ontario Highway Bridge Design Code (OHBDC 1979; Lind and Nowak 1978). The CVs 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 these data. The CVs used in calibration were taken from the National Bureau of Stan- dards Special Publication 577 (Ellingwood et al. 1980) and include other uncertainties (also human error). The parameters of DL3 were calculated using the survey data provided by the Ontario Ministry of Transportation in conjunction with calibration of the OHBDC (1979). 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 mea- ger and unsatisfactory kind; it may be the beginning of knowl- edge, but you have scarcely, in your thoughts advanced to the stage of science.—William Thomson, Lord Kelvin The least rigorous process for establishing design require- ments, and load and resistance factors in particular, is referred to as “deemed to satisfy.” In this process, experience and empirical observation are used to define the boundary between satisfactory performance and unsatisfactory perfor- mance. It provides no quantifiable way of assessing the pro- vided margin of adequacy, such as safety or reliability. As 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 this or that should move in the direction of higher performance. The obvious corollary is that cost–benefit cannot be quantified. An exam- ple 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. Nonetheless, “deemed to satisfy” has a place in the pan- theon of engineering tools. It is often the basis of detailing requirements and may serve as the beginning of design speci- fication development, as in “experience shows that if we do (or do not do) this or that the results are generally accept- able.” Expert elicitation (Delphi process) or an experimental program may provide insight into the adequacy of “deemed to satisfy.”

55 3.4 Customizing the process The data used in the calibration described in this report are provided in Appendix F. The key to providing the ability for owners to adjust the calibration of the SLSs for their own experience is to either adjust the data in Appendix F or supply state-specific information of the same type. The following attributes were identified as necessary to allow bridge owners to customize the calibration process and to develop spreadsheets for their particular needs. The ability of the process to address these issues is pro- vided as follows: • Ability of the Monte Carlo procedure to produce a probability of criteria exceedance and the associated reliability index. This ability is at the core of applying the Monte Carlo procedure. If 100,000 trial calculations of a given limit state function are produced using randomly generated loads and resistances that are consistent with the mean values and CVs for that limit state function, and the function is not satisfied 100 times, then the failure rate is 0.001 and the success rate is 0.999. The corresponding reliability index from a hand- book of probability functions or inverse standard normal CDF available in many computer applications is 3.09. • Ability to accept a user-supplied deterioration of load-carrying capacity. A possible approach to downgrading the resistance with time is discussed in Chapter 4 by using condition num- ber as a surrogate for deterioration. For example, assume that it is expected that at some point in time corrosion will have resulted in a 10% reduction in resistance of a class of bridges. Referring to the box marked Statistical Parameters of Resis- tance in Figure 3.1, the resistance would be adjusted (low- ered) by 10%. If it is determined that not only is it expected that the average resistance will be lowered, but that the values of resistance are becoming more diverse (random), then the bias and CV of resistance can also be adjusted based on that experience, as they are simply input variables. Rerunning the calculations for the affected population of the originally pro- vided bridges, or an owner-supplied set, will allow the owner to track the change in reliability indices. If one wanted to esti- mate the effect on reliability-based ratings or postings, one could keep or modify the target reliability index and repeat the lower iteration loop by using revised trial load factors until sufficient convergence of the reliability indices was found. This would essentially be a recalibration. • Ability to react to user intervention as reflected in an improved resistance (also user supplied). This is basically the opposite of the process of downgrading resistance discussed above. • Ability to accept either a user-supplied database from which the product will determine a new bias and CV or a user-supplied bias and CV from an external calculation. As discussed above, the bias and CV are input variables that a bridge owner is able to adjust. • Ability to accommodate a user-supplied resistance model. This is especially important for the geotechnical community due to the regional nature of practice in that discipline.