Ove Ditlevsen

Methods of structural systems reliability

Abstract In a condensed form this report presents the last five to ten years of development within the field of structural reliability theory with particular attention to structural systems. Besides dealing with effective specific methods of probability calculation it reflects one school of thought with respect to the interpretation of structural reliability measures. This school supports the pragmatic decision model point of view rather than the ideal of a strict physical theory. Since such thinking may not have gained general consensus the approval of the report by the committee does not commit its members to agree either on the general philosophy or on the details of the report. The main chapters are: Single Element Reliability; Series Systems Reliability; Parallel Systems and General Systems Reliability; Large Redundant Idealized Structural Systems; Modifications Needed to Deal with Variations in Time. Special emphasis is put to the often suppressed problem of formulating realistic limit state models for structural systems when considered in the night of the load history dependency problem. Attention is given to the ideal plasticity theory because it allows substantial simplifications of this problem. In fact, it seems to be the only constitutive model that can justify the time independent stable configuration type of model. Also ideal brittle systems subject to proportional random loading are analysed in some detail.

Model uncertainty in structural reliability

Abstract Structural reliability analysis is based on idealized limit state models. The level of sophistication of the models is, on the one hand, kept suitably simple in order to have mathematical operability. On the other hand, the models are believed to be reasonably predictive with respect to real failures. In order that the reliability analysis can serve as a decision tool in real life decisions in structural design, it is necessary to include elements for subjective assessment of model uncertainty information in the reliability model itself. The paper discusses how this can be done in a rational and philosophically satisfying way without losing the general level of operational simplicity of the limit state models. The philosophy and the mathematical arguments of the paper support the rules of dealing with model uncertainty that are introduced on an intuitive basis in recent international safety code recommendations.

Random fatigue crack growth—a first passage problem

Abstract The random residual in the logarithmic form of the Paris-Erdogan equation is for a single stress cycle modeled as a weighted average of a white noise material property process over the crack increment. This leads to a stochastic Paris-Erdogan equation that determines the crack increment implicitly in terms of a probability distribution of the smallest solution to the equation. This is a first passage problem in Brownian motion. For all weighting functions consistent with this model the solution of the first passage problem has the form as a randomized Paris-Erdogan equation simply with a multiplicative random variable on the right side of the equation. It is independently and identically distributed from stress cycle to stress cycle. For each of two specific weighting functions the probability distribution of this random factor is obtained. The simplest of the two is rather trivially leading to the lognormal distribution. The other and less trivial example leads to the inverse Gaussian distribution. This is interesting because it admits an exact description of a suitable transformation of the crack growth process as a function of the number of constant amplitude stress cycles in terms of an inverse Gaussian process with stationary and independent increments. Also the exact probability distribution of the number of stress cycles required to grow the crack a certain length can be calculated in this case. For any distribution of the factor the transformed crack growth process for constant amplitude loading will be asymptotically Gaussian with stationary and independent increments.

Stochastic fatigue, fracture and damage analysis

Abstract This paper reviews and summarizes the development and recent progress of methods of stochastic fatigue, fracture and damage analysis. Topics covered include structural fatigue, structural fracture, cumulative damage, maintainability and inspection and structural damage. Several “new” methods such as expert systems and fuzzy sets and their applications to damage analysis are briefly discussed. It is concluded that good methods are available for the purpose of making analysis and design. However, the fundamental mechanisms for fatigue, fracture and damage remain to be further investigated.

Directional simulation in Gaussian processes

Abstract Aiming at reducing the sampling variance of the estimated probability on a given set, a generalization is given of the method of uniform mean centred directional sampling in the standardized n- dimensional Gaussian space. Two modifications of different nature are involved. The one consists in shifting the origin to a point different from the mean. The resulting off-mean centred directional simulation method is operational due to the existence of a closed form generalized chi-square distribution function for the square of the distance from the origin to the Gaussain random point. Suitable choices of the origin give considerably reduced sampling variances in most cases. Secondly there is a possibility of variance reduction by defining the sampling distribution in such a way that the exact probability on a given half-space is obtained by a single simulation. Comparative effectivity studies for different examples are made. Finally the method is also set up for computing mean outcrossing rates for Gaussian vector processes.

Statistical analysis of the virkler data on fatigue crack growth

The extensive data set obtained by Virkler et al. for fatigue crack growth under homogeneous cyclic stressing is the object of statistical analysis. It is based on a previously published stochastic model of the crack growth. The statistical scatter of the experimental data is made up of a random “between” specimen variation and a random “within” specimen variation, the former being of the finite dimensional random vector type and the latter of random independent increment process type. The main results of the statistical analysis are 1. (1) that a random equation of the ParisErdogan type that allows for random material inhomogeneities fits very well to the data and 2. (2) that the distribution type for the number of stress cycles needed to grow a crack by a given length is convincingly described as being inverse Gaussian. Within the basic stochastic model this distribution type is asymptotically correct for large cycle number increments. Furthermore 3. (3) the random vector variation between specimens is reasonably well described by a joint normallog normal distribution. Numerical differentiation of the (basically non-difierentiable) experimental crack growth curves to obtain crack growth rates is avoided in the present model. In place the model points at direct maximum likelihood estimation of the parameters in the Paris-Erdogan equation.

Availability, reliability and downtime of systems with repairable components

Closed-form expressions are derived for the steady-state availability, mean rate of failure, mean duration of downtime and lower bound reliability of a general system with randomly and independently failing repairable components. Component failures are assumed to be homogeneous Poisson events in time and repair durations are assumed to be exponentially distributed. The results are expressed in terms of the mean rates of failure and mean durations of repair of the individual components. Closed-form expressions are also derived for the rates of change of the various probabilistic system performance measures with respect to the mean rate of failure and the mean duration of repair of each component. These expressions provide a convenient framework for identifying important components within the system and for decision-making aimed at upgrading the system availability or reliability, or reducing the mean duration of system downtime. Example applications to an electrical substation system demonstrate the use of the formulas developed in the paper.

Methods of stochastic structural dynamics

Abstract A concise review is given of the analytical methods of stochastic structural dynamics which deals with structural systems under time-varying random excitation. Included in the review are both linear and nonlinear structures and both parametric and non-parametric random excitations. Mathematically, parametric excitations appear in the coefficients for the unknowns in the equations of motion, whereas non-parametric excitations appear as inhomogeneous terms on the right hand side. Physically, random parametric excitations represent the variation of structural properties with time; therefore, they can affect the stability of structural response. Approximate methods are described for those cases for which exact solutions are presently not available.

Random fatigue crack growth with retardation

Abstract On basis of a study of the literature concerning empirical findings in fatigue crack growth in metal specimens under constant amplitude loading with occasional overloads, the paper summarizes the reported qualitative effects of the overloads. The great scatter of the observations and the difficulty of setting up a clear physical mechanism, which in deterministic terms explains the crack growth retarding effects of the overloads, motivates attempts to formulate stochastic process models of phenomenological type. The paper shows that birth processes have features that make them applicable in modelling fatigue crack growth processes. In fact, this process type allows a time transformation that reduces the case of variable amplitude loading to the case of constant amplitude loading. The mean growth curve defined as the mean time of growth to a given crack length as function of this crack length may in the constant load amplitude case be calibrated to the Paris-Erdogan law. For the case of occasional overloads it may be further calibrated to the empirical results reported in the form of the Wheeler model of crack retardation based on the concept of a strengthening plastic zone at the crack tip caused by the overload.

Solution of a class of load combination problems by directional simulation

Abstract The load combination problem of Ferry Borges-Castanheta may be effectively handled by the Rackwitz-Fiessler algorithm provided the load pulses have absolutely continuous distribution function. However, realistic modelling of extraordinary actions requires the load pulse distributions to have a concentrated probability at zero. In principle this discontinuity may be handled by conditioning such that the combination problem reduces to several problems with absolutely continuous distribution functions. With just some few extraordinary actions taking part in the combination problem this method of conditioning becomes quite cumbersome and even impracticable. If it is assumed that the single load pulses are clipped normal random variables, i.e. of the form max{0, X} where X is normal, then a combination of the RF-algorithm and directional Monte Carlo simulation technique turns out to be useful. At any given argument the directional simulation method gives not only a confidence interval for the value of the distribution function but also a confidence interval for the value of the density function of a random variable defined as a sum of clipped dependent or independent normal variables. This is just what is needed in the RF-algorithm in order to apply the principle of normal tail approximation on the distribution of the sum.