Probabilistic Methods for Structural Design

In the last 15 years several international documents have been published dealing with the basic concepts of structural design.

The different types of uncertainties are considered and their differences are identified. Various methods of statistical analysis of data are reviewed and their usefulness and domain of applicability are identified. The common methods of representing model uncertainties are indicated and several examples of assessment of model uncertainties indicate how the principles described can be applied.

We first propose to trace some streams of thought which have contributed directly to what we now call Response Surface Methodology (abbreviated to RSM). A mathematical description of RSM is given in section 2.

In many metallic structures, flaws are inherent due to, e.g., notches, welding defects and voids. Macro cracks can originate from these flaws, and under time varying loading grow to a critical size causing catastrophic failure. The conditions governing the fatigue crack growth are the geometry of the structure and crack initiation site, the material characteristics, the environmental conditions and the loading. In general, these conditions are of random nature. The appropriate analysis and design methodologies should therefore be based on probabilistic methods.

A methodology for probabilistic fatigue assessment of welded joints using the S-N and fracture mechanics approaches is proposed. A fatigue crack propagation model is presented for a semi-elliptical crack in a welded plate which accounts for the effects of weld geometry, residual stresses, stress ratio, fatigue threshold and variable amplitude loading. A simple lognormal format and a rigorous FORM/SORM approach is used for evaluating the reliability of a joint against failure by fatigue. The model accounts for the uncertainties in fatigue loading, stress analysis, stress intensity factors, initial defect size and crack growth material properties. Examples involving reliability analysis of tubular joints of offshore structures are presented.

The structural components that are subjected to compressive forces are often critical elements in many structures. This is even more important if one has in mind that compressive failure is followed by a decrease in load carrying capacity which will overload adjacent structural components, being bound to precipitate an overall failure of the structure.

Methods of evaluating the reliability index are now well known, and more and more software is available for its calculation in the classical cases. Nevertheless, most require an explicit limit state function. As failures generally appear during severe and extreme loading conditions, they are often associated with strongly non-linear mechanical behaviour. Then, it is unrealistic to calculate a large number of realizations, because the existing mechanical models are generally too time consuming.

In general, a system is understood as a technical arrangement of clearly identifiable (system—) components whose functioning depends on the proper functioning of all or a subset of its components. For a reliability analysis, a number of idealizations are convenient if not necessary. It is assumed that the components can attain only two states, i.e. one functioning (safe, working, active,...) and one failure (unsafe, defect, inactive,...) state. This is a simplification which is not always appropriate but we will maintain it throughout the text. If there is a natural multi—state description of a component or a system we shall assume that this is reduced to a two-state description in a suitable manner. In practice, this step of modeling might be not an easy task. It is, nevertheless, mandatory in practical system reliability analyses. Several attempts have been made to establish concepts for analyzing systems with multi—state components (see, for example, Caldarola, 1980; Fardis and Cornell, 1981). It should be clear that systems then have also multiple states and the definition of safe or failure states requires great care. Such relatively recent extensions of the classical concepts cannot be dealt with herein.

Classical Statistical Extremes Theory describes the random behaviour of the largest or smallest values of independent and identically distributed (i.i.d.) samples: the theory may be extended to random interdependent sequences and thus it can be used in concrete decision or design problems where the crossing of some bounds can give rise to breakdown or disaster. Large waves, gusts of wind, floods, large insurance claims etc., are examples or maxima; droughts, fatigue, rupture, failures of nuclear reactors, etc., may be connected with minima; disaster can occur if some bounds are exceeded or not attained. The design of a breakwater, of a plane, of a high antenna or of a high tower, etc., must, take in to account the risks of (random) failure and/or disaster.

When only one time-varying load acts on a structure, and failure is defined as the load process crossing some level, then the extreme value distribution of the load contains information which is sufficient for decisions about reliability. The theory of stochastic load combinations is applied in situations where a structure is subjected to two or more time-varying scalar loads acting simultaneously. The scalar loads can be components of the same load process or be components of different load processes. To evaluate the reliability of the structure, each load cannot be characterized by its extreme-value distribution alone; a more detailed characterization of the stochastic process is necessary. The reason is that the loads in general do not attain their extreme values at the same time.

Whereas theory and concepts for the computation of time-invariant reliability are now well—known and can be performed efficiently and reliably by various methods, much less theory is available for methods which are capable of handling time variant reliability problems. Time variant problems are usually present with time—variant environmental loading and possibly time—variant (deteriorating) structural properties. One needs to compute not primarily the probability that a structural system is in an adverse state at any given time. It is rather the probability that such an adverse state is reached for the first time in a given reference period. There are two important cases in which computation of so—called first passage probabilities is still possible with time—invariant methods. This is when the failure criterion is related to strictly increasing cumulative damage phenomena, for example in structural fatigue. Then, the probability of survival is equal to the probability that damage has not reached a critical value at a given I time. The other case is when all variables are time—invariant except one which then can be replaced by its extreme value, but only in the stationary case.

Several methods are currently available to solve a large number of problems in mechanics involving uncertain quantities described by stochastic processes, fields or waves. At this time, however, Monte Carlo simulation appears to be the only universal method that can provide accurate solutions for certain problems in stochastic mechanics involving nonlinearity, system stochasticity, stochastic stability, parametric excitations, large variations of uncertain parameters, etc., and that can assess the accuracy of other approximate methods such as perturbation, statistical linearization, closure techniques, stochastic averaging, etc. The major advantage of Monte Carlo simulation is that accurate solutions can be obtained for any problem whose deterministic solution (either analytical or numerical) is known. The only disadvantage of Monte Carlo simulation is that it is usually time-consuming. It is the author’s belief, however, that in the years to come, the continued evolution of digital computers will further enhance the usefulness of Monte Carlo simulation techniques in the area of engineering mechanics and structural engineering.

Until recently, stochastic structural mechanics has addressed the issue of deterministic structures subjected to random loading. With the availability of more accurate analysis and design tools, however, quantifying the sensitivity of model predictions to uncertainty in the mechanical properties of structures has become possible. It has been observed, with the help of these tools, that this type of uncertainty can be more significant to the overall predictions of a particular structural model than the more traditional fluctuations attributed to external loads. In view of that, recent procedures have been developed for representing uncertainties in the parameters of a structural model, as well as, for propagating this uncertainty to obtain the associated uncertainty in the predicted response. The stochastic finite element method is a procedure for performing such an analysis, whereby the spatial extent of the structure has been represented within the context of the finite element method. This chapter describes a recent implementation of the stochastic finite element method that combines theoretical rigor with generality and efficiency of implementation. Specifically, the Spectral Stochastic Finite Element Method (SSFEM) presented in this chapter addresses the situation where the uncertain material properties are realizations of a spatially fluctuating random field. The formulation relies on discretizing the random processes using spectral expansions, thus eliminating the correlation between the requisite mesh size to meet energy-based convergence criteria, and the scales of fluctuation of the random material properties involved.

A stochastic finite element method based on an extended response surface technique is coupled with fracture mechanics concepts to evaluate the lifetime of a cracked structural component subjected to cyclic loading. For a complex structure the relationship between the fracture mechanics parameters and the crack depth can only be obtained by numerical approaches based on the discretization of the continuum into finite elements. As a consequence, any random field describing the stochastic nature of the input parameters has to be discretized into stochastic finite elements. An extended response surface approach is used to characterize in closed form the numerical input-output stochastic relationship. This information is used to define a fatigue crack growth model for the evaluation of the lifetime probability distribution function.

The evolution of structural codes during the present century is outlined. The description is centered on the design of buildings and public works of structural concrete and mainly reflects experience in Western Europe. Particular attention is paid to the introduction and generalised use of probabilistic concepts. Although most of the specific considerations refer to structural concrete, the fundamental concepts apply to other materials used in civil engineering : steel, masonry, timber, etc.

After a brief overview of the so-called stochastic linearization methods for approaching structural dynamics problems, attention is focused on a special procedure for seismic fragility analysis which makes use of response surface techniques and level-2 reliability methods. The local amplification is considered through a boundary element idealization. The numerical examples investigate the goodness of the experiment plans adopted and, especially, the accuracy of the level-2 reliability idealization.

Risk management has become increasingly important to industry and society during the last decades. This is largely due to an urge for increased efficiency and competitiveness by industry itself, and to increasing standards regarding personnel safety and environmental preservation, imposed by society.