Imprecise probability in engineering a case study

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1 Imprecise probability in engineering a case study Tutorial, ISIPTA 11 Innsbruck, July 25 28, 2011 Michael Oberguggenberger Unit of Engineering Mathematics University of Innsbruck, Austria

2 Contents Modeling uncertainty in engineering Failure probability Imprecise probability models An example from geotechnics: a case study Deterministic/probabilistic dimensioning Random set modeling A hybrid model Sensitivity analysis by IP methods Decision making in engineering

3 Modeling uncertainty in engineering reality correspondence rules model observations semantics state variables data interpretation parameters action computation Definition and axiomatics: How is uncertainty described and what are the combination rules? Numerics: How is uncertainty propagated through the computational model? Semantics: What is the meaning of the results what do they say about our conception of reality?

4 Model uncertainties: choice of the structural model selection of state variables and parameters choice of limit state function transitional states during construction Parameter uncertainties: random fluctuations lack of information random measurement errors systematic measurement errors fluctuations due to spatial inhomogeneity errors made by assigning parameter status to state variables parameters carry burden of model insufficiency

5 Range of available information: frequency distributions obtained from large samples values from small samples or single measurements interval bounds experts point estimates educated guesses from experience Failure probability R: all random variables describing the resistance of a structure S: all random variables describing the loads Limit state function: g(r, S) Failure probability: p f = P ( g(r, S) < 0 ) Trouble with the failure probability: Codes require p f = 10 6

6 Imprecise probability models Intervals analysis, fuzzy sets, evidence theory, random sets, sets of probability measures, lower and upper previsions, info-gap-analysis, etc. Understanding uncertainties: reflection about the choice of model and the failure mechanisms; assessing the variability of input and output variables and model parameters; sensitivity analysis; assessing the reliability of the structure; decision about acceptance or non-acceptance of the design. Practical engineering = decision making with the help of scientific tools!

7 An example from geotechnics: a case study Example: an infinite beam on a linear elastic bedding (Winkler beam). Deterministic/probabilistic dimensioning The displacement w(x) is described by the bending equation EI w IV (x) + bc w(x) = q(x), < x <, EI is the flexural rigidity of the beam (assumed deterministic) b its effective width (assumed deterministic), c the bearing coefficient of the foundation (nondeterministic), q(x) the loading (nondeterministic). Singular boundary value problem for the standardized equation: u IV (x) + 4k 4 u(x) = p(x), < x <

8 with bc/ei = 4k 4, p(x) = q(x)/ei. In case k is a constant and p(x) is an integrable function, its unique deterministic solution is u(x) = in terms of its Green function G(x, y) p(y) dy G(x, y) = 1 8k 3 e k x y ( sin k x y + cos k x y ). In case the standardized load p(x) p is constant, the displacement is constant as well and simply given by u(x) p 4k 4, w(x) q bc. Example: central moduli of k = 10 2, p = 10 8.

9 Buried cast-iron pipeline with an effective diameter of 6 [cm], covered by 100 [cm] of top soil (q = 10 [N/cm]) and bedded in loosely packed sand (c 6.7 [N/cm 3 ]). The resulting overall displacement would amount to w(x) 0.25 [cm]. Probabilistic design: input parameters as random variables. Standard engineering practice: mean values given by the deterministic design values and a coefficient of variation is assumed. Here for q (mean µ q = 10) and bc (mean µ bc = 40); we take a coefficient of variation of 10%. Type of distribution: Gaussian normal distribution assumed, i.e., q N(10, 1), bc N(40, 16). Under this assumption, we can compute the probability density of the displacement w = q/bc and read off the quantiles. For example, the probability that the displacement is larger than 0.5 [cm] is

10 Probability density (left) and distribution function (right) of displacement under constant, but random load. Probabilistic design, load as a random field: q(x), x R, varying randomly from one point x to another. Standard assumption in soil engineering: the soil load is given by a stationary Gaussian field. We need to provide the mean value, the variance and the autocorrelation function. The mean value is assumed to be µq = 10; for the field variance we took σq2 = 4.

11 A typical autocorrelation function is of the form C(ρ) = exp ρ /` at spatial lag ρ, where ` is the so-called correlation length. We assumed a moderate correlation length of ` = 100 [cm] and took bc fixed at its deterministic design value 40 [N/cm2 ]. Random field model: trajectories of load process and corresponding displacement.

12 Random field model: Trajectory of bending moment and simulation of maximal bending moment. A critical quantity for assessing the safety against failure is the maximal bending moment Mmax in the beam, given by Mmax = max(eiw00 (x)). A Monte Carlo simulation of N = 500 trajectories resulted in an estimate for the distribution of Mmax. Using a kernel smoother, we got, for example, P (Mmax > 6000)

13 Random set modeling Assuming that q and bc are normally distributed random variables is rather artificial. The available information consists of a nominal value and a coefficient of variation. One way of organizing this information is by means of a Tchebycheff random set. If the loading q, say, is preliminarily viewed as a random variable with unknown probability distribution, but with expectation value µ q and variance σq, 2 Tchebycheff s inequality asserts that the probability of the event { q µ q > σ } q α is less or equal to α, where α (0, 1]. Let Q(α) = [ µ q σ q / α, µ q + σ q / α ]. By Tchebycheff s inequality, the probability of Q(α) is greater or equal to 1 α, while the probability of its complement Q(α) c is less or equal to α.

14 Thus Q(α) may be viewed as an approximate two-sided (1 α)-fractile range for the parameter q. This is a conservative, non-parametric estimate valid for whatever distribution of the random variable q. It encodes the minimal information that can be extracted from the expectation and the variance of a random variable without further parametric assumptions. We formalize this as an infinite random set (actually a random interval) α Q(α) on the space Ω = (0, 1], equipped with the uniform probability distribution. We apply this construction to the loading q with µ q = 10 and σ q = 1.5 (from a coefficient of variation of 15%). This results in a random set Q with contour function:

15 Tchebycheff random set for load and probability box for resulting displacement. In a similar way, we construct a Tchebycheff random set BC for the variable bc, using µbc = 40 and σbc = 6. To form the joint random set (Q, BC), the dependence of Q and BC has to be modeled. To make computations easy, we settle for fuzzy set independence; that is, the joint random set is also defined on Ω = (0, 1] with focal elements Q(α) BC(α), α Ω.

16 The random set data can be propagated through the mapping that gives the displacement w(q, bc) = q/bc, resulting in a random set W with focal sets W (α) = w(q(α) BC(α)), the set of values attained when (q, bc) range in Q(α) BC(α). The evaluation of the interval bounds for W (α) requires a global optimization. It is useful to describe the output random set as a probability box, which is bounded by the lower and the upper distribution functions F (x) = P (, x], F (x) = P (, x], see figure on previous page. The probability box immediately gives information on quantile ranges. For the event A = {W 0.5}, for example, we get the probability interval [P (A), P (A)] = [0.96, 1].

17 A hybrid model Goal: combination of stochastic differential equations with random set parameters. This can be used in the dynamics of structures. Earthquake induced vibrations can be modeled by stochastic processes, like colored noise, whereas uncertain material parameters can be modeled by random intervals. Here: a hybrid model for the elastically bedded beam. The load q will be modeled as a random field, while the bedding parameter bc will be modeled as a random set. Of course, the model can be generalized to higher levels by also taking the field parameters σ q and l as random sets, etc. For the sake of simplicity, we shall take bc as an interval and the field parameters as µ q = 10, σ q = 2, l = 100. For bc we choose the interval [20, 40].

18 Hybrid model: Single interval trajectory of bending moment (left) and p-box for maximal bending moment (right). The resulting output w is a set-valued stochastic process; more precisely, each trajectory is interval-valued. In order to assess the statistical properties of the output, a sample of trajectories has to be generated. From there, one can compute, e.g., the upper and lower distribution functions of the maximal bending moments in the beam (p-box).

19 From the list of computed values (interpolated using a kernel smoother) one may obtain upper and lower probabilities that given limits are being exceeded, e.g., P (M max > 6000) , P (M max > 6000) , P (M max > 8000) 0, P (M max > 8000) Typically, the probability intervals obtained by taking interval uncertainty into account are large and possibly of a quite a different magnitude than the sharp probabilistic estimates. If the decision maker is not satisfied with the relatively wide range of probabilities, more information about the distribution of the parameters has to be gathered. If the imprecise probability model properly reflects the available data, the results have to be accepted and should not be distorted by artificial probabilistic assumptions.

20 Sensitivity analysis by IP methods We take up the simple model w = q/bc from the previous section. The probability box shows the total variability of the output. We successively freeze the variables q and bc at their nominal value, compute the displacement w and observe whether the probability box gets narrower or not. Sensitivity analysis: probability box with bc pinched (left) and with q pinched (right). The outer thin lines indicate the p-box without pinching. The figures show that q has a bigger influence on the displacement w than bc.

21 Decision making in engineering The dream of a decision maker in engineering companies is to get a single number stating that the design of the structure to be built is safe. This is also reflected in the Eurocodes that purport and require a reliability R = 1 p f of At best, such a number can be viewed as an operational probability. It is hardly credible (nor verifiable or falsifiable) that such a number says anything about the frequency of failure. Thus the designing engineer is left with the question how big the distance to failure actually is. There is no answer to this question there are many examples of structural failures due to reasons not taken into account in the model and the reliability analysis, or due to false reactions to unexpected behavior during construction. The key to a well-founded decision about the safety of a structure is an interplay between model building, numerical calculations, laboratory experiments, in-situ investigations, and uncertainty analysis in a feed-back loop. This loop extends to monitoring during construction and possible re-design.

22 The available knowledge should be properly captured by the uncertainty model thus the need for a framework going beyond probability and using various methods from imprecise probability. Generally, the basis for reliability analysis will be a sensitivity analysis of the model, identifying the most influential parameters. A complete probabilistic or non-probabilistic reliability analysis is usually beyond available computing power and time. Instead, the analysis will usually be done with the aid of a reduced model. When proceeding with an imprecise probability model, the output parameters measuring the distance to failure come with upper and lower probabilities. In view of the lack of a total ordering, the designing engineer will have to base the decision about acceptance or rejection of the design on considerations as in multiple criteria decision making. In the end, a decision based on interval-valued, but accurate estimates will be more reliable than a decision based on sharp, but false numbers. Indeed, IP methods are more and more appreciated by practicing engineers who are aware of the need of a rational analysis of uncertainty.

However, reliability analysis is not limited to calculation of the probability of failure.

However, reliability analysis is not limited to calculation of the probability of failure. Probabilistic Analysis probabilistic analysis methods, including the first and second-order reliability methods, Monte Carlo simulation, Importance sampling, Latin Hypercube sampling, and stochastic expansions