Target Reliability Analysis for Structures
Type of Degreedissertation
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Structural performance depends on load and resistance parameters, in particular magnitude and frequency of load components and their combinations, strength of materials, modulus of elasticity, dimensions, rate of deterioration, and so on. The problem is that these parameters are time-varying random variables and the structure is expected to perform its function in an acceptable manner. Therefore, prediction of structural performance involves uncertainty. Reliability is defined as a probability of the structure to acceptably perform its function. There is a need to determine the optimum or target reliability level, as too low reliability can result in problems (e.g. cracking, vibrations, collapse) and too high reliability can be costly. The papers in this dissertation deal with selection criteria for the target reliability. Acceptable performance has to be defined and this can be very subjective. The boundary between acceptable and unacceptable performance is called a limit state. Mathematical expression that defines a limit state is the limit state function. In general, a limit state function, g, is a function of load and resistance parameters, g(X1,…Xn) and g>0 or g=0 represents acceptable state of the structure and g<0 represents unacceptable state, or failure to perform the expected function. Therefore, the objective of reliability analysis is to determine the probability of g being negative, or probability of failure. Once the limit state function is formulated as a function of load and resistance parameters, the probability of failure and corresponding reliability index can be calculated using one of the available procedures. There are several procedures available for performing the reliability analysis and they are presented in textbook, including Nowak and Collins (2013). They vary with regard to accuracy, required effort and simplicity of form. The direct calculation of probability of failure, PF, is complicated as it involves a double integration and convolution functions. A practical procedure, allowing to avoid double integration, to determine the probability of failure and reliability is proposed in paper #1. Therefore, the major developments were directed at calculation of the so called reliability index, β. If g is treated as a normal random variable then the relationship between β and PF is β=- Ф^(-1) (PF), where Ф^(-1) is the inverse standard normal distribution. Otherwise, the formula involves a certain degree of approximation. This problem is considered in paper #1 in this dissertation. There are four types of limit states: ultimate (or strength) limit state (ULS), serviceability limit state (SLS), fatigue limit state and extreme events limit state. Each limit state provides different acceptability criteria. Examples of ULS include moment or shear carrying capacity of a beam, column buckling, or overall stability of a beam. Examples of SLS include a limit on deflection, excessive vibrations, or cracking of concrete. Fatigue limit state is a limit on stress and/or number of load cycles. Extreme events include resistance to earthquakes and vessel/vehicle collision. The definition of what is acceptable performance and what is not can be very subjective. For example, what is acceptable deflection in a bridge due to traffic load? The answer to this question is very difficult, as the answer is not just one value but values of deflection and associated frequencies of allowable occurrence. Live load representing traffic on the bridge consists of a mixture of vehicles. Extremely heavy trucks can occur relatively infrequently. By measurements and structural analysis, it is possible to develop a cumulative distribution function (CDF) of deflection. So there is a need to select acceptable/tolerable CDF of deflection. This issue is a subject of paper #4. In general, reliability has to be calculated separately for each of the considered limit states. The target reliability can also be different for each limit state as it depends mostly on consequences of failure. What happens if the limit state is exceeded? If there are serious consequences (closure for traffic, partial or total collapse, injuries, fatalities), the target reliability has to be increased to avoid major disaster. On the other hand, if not much happens, a lower reliability level can be tolerated. An example can be deflection limit state in bridges. If an extremely heavy truck crosses a bridge causing an excessive deflection, there no serious damage to the bridge, as it can still perform its function. The other major factor that affects selection of the target reliability level is economics. If increasing safety is costly, a lower reliability can be acceptable. On the other hand, if safety is cheap, a large safety margin can be economically justified. For example, bolted connections have a very high reliability because safety is cheaper in bolts compared to safety in beams and columns these bolts connect. There are also other factors that affect the selection of the target reliability such as past practice, risk perception by the society and political aspects. A practical implementation of the target reliability concepts is presented in paper #3. The objective is calculation of load and resistance factors for concrete circular tunnels. This study involved formulation of the limit state functions for the considered structures. The statistical parameters were determined for the major variables, representative for the current practice. A reliability analysis procedure was developed and applied to calculate a wide spectrum of reliability indices. Using the current practice as a benchmark, the target reliability index was selected for each limit state, including moment, shear and compression capacities. Finally, the resistance factors were selected so as to minimize the closeness to the target reliability index.