ESTIMATION OF RELIABILITY AND RISK

Great advances in computational mechanics and numerical methods have permitted the solution of complex, previously intractable nonlinear problems either in solid or fluid mechanics. Computers have now sufficient memory and efficient operational systems which allow the solution of problems with a great number of degrees of freedom in a reasonable time. On the other hand, the theoretical models, and their numerical representation, implement relationships between variables which are not necessarily known with precision. These uncertain variables introduce randomness in the results and this, in turn, affects conclusions regarding reliability and quantification of risks. This presentation discusses the need to integrate the deterministic calculations from numerical models with methods to assess the probabilistic nature of the predictions. At the same time, the presentation discusses briefly different strategies to implement the estimation of probabilities, as well as probabilistic design procedures based on the reliable satisfaction of different performance requirements. The discussion is illustrated with two applications: the collision force between an ice mass and an offshore exploration platform, and the collision force between vessels and the piers of a bridge. Vancouver, B.C., Canada V6T 1Z4, rowfa1@civil.ubc. ca COMPUTATIONAL MODELS AND UNCERTAINTIES: Mecánica Computacional Vol XXX, págs. 3-21 (artículo completo) Oscar Möller, Javier W. Signorelli, Mario A. Storti (Eds.) Rosario, Argentina, 1-4 Noviembre 2011 Copyright © 2011 Asociación Argentina de Mecánica Computacional http://www.amcaonline. org.ar 1 MODELS, UNCERTAINTIES AND RELIABILITY Models in computational mechanics normally have, as a base, a corresponding set of differential equations. For example, the foundation for models based on the theory of elasticity incorporate stresses and strains into equations for constitutive material behavior, conditions of equilibrium and compatibility of deformations. These models could represent different levels of complexity, from small deformations leading to linear behavior, to large strains for nonlinear problems. The material representation could also be linear or nonlinear and, in either case, incorporates “material characteristic constants” which have to be determined from experimental testing (e.g., the “modulus of elasticity E”). The incorporation of plastic behavior in the material equations leads to the theory of plasticity and, similarly, the consideration of time effects to the theory of viscoelasticity (either linear or nonlinear). Equilibrium equations could include acceleration and damping terms, needed for the study of dynamic problems. As a general rule, these equations do not admit closed-form solutions except in very simple cases. As a norm, the equations are then solved by numerical means, using either finite difference approximations for derivatives or implementing variational principles resulting in what we know as “finite element methods”. The theoretical models are, of course, approximate interpretations of reality and are subject to some error. The numerical form of the theoretical models incorporate a further error related, for example, to discretization techniques, time step size, convergence criteria, etc. Furthermore, the constants in the material model are not deterministic: the experimental tests required for their evaluation yield different results for each test replication. For example, the modulus of elasticity E would vary from experiment to experiment, and would show spatial variability within a mesh of finite elements. Variability in material properties appear also when considering time effects like creep or relaxation, or when incorporating effects of size. Quantities like density are not uniformly distributed within a given volume and, for example, the density variation in a composite material is dependent on its manufacturing process. Uncertain manufacturing variables like pressing temperature, pressure, rate of cooling after pressing, all influence the final shape of a composite component, a randomness which is of major importance for the use of such composites in aircraft structures. The error in the quantification of the material behavior can be reduced by further testing, and it is said to have an aleatoric base. The model error is intrinsic to our interpretation of reality and it is thus called a source of epistemic uncertainty. Any application in computational mechanics will then contain a set of intervening random variables. These uncertainties require consideration, using tools of probability theory in conjunction with the computational mechanics model. Two basic problems arise: 1) to determine the probability that an output from the models will exceed a target value, or 2) to calculate the value of a model parameter so that the probability of an output exceeding a threshold in a given performance criterion be as prescribed. The first type of problem falls into the area of reliability evaluation, while the second relates to “performance-based design”. Since uncertainties, whether epistemic or aleatoric, cannot be avoided, a complete solution of the problem not only requires a robust deterministic computational technique but also a link to a probabilistic assessment of the effect from the uncertainties. R. FOSCHI 4 Copyright © 2011 Asociación Argentina de Mecánica Computacional http://www.amcaonline. org.ar 2 PROBLEM FORMULATION AND RELIABILITY ASSESSMENT In general, the performance of an engineering system can be described by a performance or limit state function G(x) , the form of which is ) , ( ) , ( ) ( d d c c d x D d x C x G − = . (1) G(x) is a function of the intervening random variables and can always be written as the difference between two functions, a system capacity C and a demand D. The variables may be divided into two groups: one, (xc , dc ) , related to the capacity C, and a second, (xd , dd) , associated with the demand D. The vectors xc and xd include all those variables which are random or uncertain, while the vectors dc and dd include all those quantities which are either deterministic or known with sufficient certainty. For example, the depth of a beam will affect its bending capacity and it will form part of dc , while the beam span will affect the bending moment demand and will form part of dd . On the other hand, the bending strength will form part of xc while the actual applied load will be included in xd . In practical problems the random vectors xc and xd may include many random variables. Very often, neither the function C nor D can be given explicit form, and they are only known as discrete results from numerical calculations using the computational models. In this case, it is convenient to give C or D a mathematical representation by fitting the numerical results with one of different types of response surfaces (Bucher et al.,1990; Faravelli,1989). When the performance function is written as shown in Eqn. (1), non-performance will correspond to situations where the variables combine to make D > C, or G < 0. Thus, estimating the probability of non-performance is equivalent to estimating the probability of the event G < 0 . This will be the ”probability of failure” Pf , and the reliability will then be the complement 1.0 – Pf . The calculation of reliability requires statistical information for the different random variables. For example, the bending strength capacity of a composite will depend on the strength of the fibers, the properties of the matrix and the fiber volume percentage in the mix. Statistics for each of these components has to be determined from appropriate tests. Some variables may have more influence than others and, accordingly, they will require a more precise statistical description. Some demand variables will rely on available data, for example, wind speeds, river flows, depth of snow packs, quantity of rain, historical earthquake intensities, or occupancy loadings. Of course, there may be some variables for which no data or very little information are available. In this case, subjective estimates of the variability may be introduced in order to study the importance of such assumptions in the overall reliability of the system. Should a particular variable be found to be quite important but lacking detailed information, more effort should be spent in obtaining more data to avoid a corresponding penalty in the reliability estimate. Apart from obtaining the solution of a problem with a better representation of reality, the use of reliability or probabilistic concepts offers many advantages not adequately addressed by more traditional, deterministic calculations. The calculated reliability in a given limit state will decrease if the information is sketchy or if the variability is high. Conversely, a premium is received through better information on the material, better modeling, or better procedures for quality control. In this sense, probabilistic methods encourage innovation in applications and manufacturing. How do we implement, in practice, reliability calculations? One simple approach is by sampling values of the intervening variables x, from their corresponding probability distributions, and verifying whether the function G(x) results positive or negative. If G(x) < 0, then the particular sample of variables leads to system failure. By maintaining a counter of the Mecánica Computacional Vol XXX, págs. 3-21 (2011) 5 Copyright © 2011 Asociación Argentina de Mecánica Computacional http://www.amcaonline. org.ar number of failures, Nf , observed in a sequence of N sample sets, the probability of failure can be estimated as the ratio N N P f f ≅ (2) This process is known as a Monte Carlo simulation. The result will be different if the process is repeated for another sample of N selections. The variability in