G.I. Schuëller

Random fields and stochastic finite elements

Abstract This state of the art paper identifies as the distinguishing feature of stochastic finite element analysis that it involves the discretization of the parameter space of a random field of material properties and / or loads. This discretization implies that the stochastic input consists of a vector of random variables whose covariance matrix depends on the finite element mesh. The paper provides an overview of basic concepts underlying random field theory, describes specific analytical tools to convey first- and second-order information about homogeneous random fields, and surveys available information on the space-time variation of random loads and material properties encountered in structural engineering. Stochastic finite element formulations covering a wide range of applications to both static and dynamic problems in structural engineering are examined, and a parallel approach to stochastic finite difference analysis is outlined.

Reliability-Based Optimization of structural systems

A method to carry out a Reliability-Based Optimization (RBO) of especially nonlinear structural systems is introduced. Statistical uncertainties involving both structural and loading properties are considered. The concept is based on the separation of structural reliability analyses and the optimization procedures. Two approaches are discussed, depending on the interaction of reliability analysis and mathematical programming and the way of representation of the limit state functions (LSF) of the structure. As, for cases of practical significance, the LSF is known only pointwise it is approximated by Response Surfaces (RS). For the response calculations Finite Element (FE) procedures are utilized. Failure probabilities are determined by applying variance reducing Monte Carlo simulation (MCS) techniques such as Importance Sampling (IS). Following the reliability analysis, the optimization procedure is controlled by the NLPQL algorithm. A numerical example in terms of a template ocean platform exemplifies the procedures.

On efficient computational schemes to calculate structural failure probabilities

Abstract Various methods to calculate structural failure probability are compared in the light of their efficiency and accuracy. The investigation is confined to procedures which provide error estimates. Hence it concentrates on approaches based on numerical integration and simulation where particular attention is focused on variance reduction techniques. Importance sampling procedures are recommended as they provide accurate results with acceptable computational efforts. It is shown that the range of simulation can be identified either by utilizing the design point or, even more effectively, adaptive sampling i.e. an Iterative Fast Monte Carlo (IFM) procedure. Finally, the response surface method (RSM) is recommended for larger structures where each calculation of the limit state function requires a complete Finite Element analysis.

Equivalent linearization and Monte Carlo simulation in stochastic dynamics

Equivalent linearization (EQL) and Monte Carlo Simulation (MCS) are the most important techniques in analyzing large nonlinear structural systems under random excitation. This paper reviews the development and the state-of-the-art of EQL and MCS in stochastic structural dynamics.

On advanced Monte Carlo simulation procedures in stochastic structural dynamics

Abstract As analytical methods to predict the response of MDOF-systems under stochastic loading are quite limited, numerical procedures, such as Monte Carlo simulation techniques, are frequently applied. Direct Monte Carlo simulation, however, particularly for reliability analyses, is not suitable for providing sufficient information on the tails of the distribution of the response. In this paper Monte Carlo methods, applicable to Markov processes are further advanced, i.e. by leading the generated samples towards the low probability range which is practically not accessible by direct Monte Carlo methods. In this context, notions as developed for variance reduction techniques are introduced and discussed. Based on criteria denoting those realizations which lead most likely to failure, a numerical procedure called “Double & Clump” (D&C) is discussed briefly. It is shown that the rather numerically involved D&C procedure can be simplified by a technique known as “Russian Roulette and Splitting” (RR&S). In two numerical examples, both procedures D&C and RR&S are compared with direct Monte Carlo simulation as well as the Response Surface Method (RSM) demonstrating comparable accuracy.

Random eigenvalue problems for large systems

Abstract The inherent uncertainties in geometry, material properties, etc. of engineering structures can be represented by stochastic models, where the parameters are described by probabilistic laws. Results from any analysis based on stochastic models inherit probabilistic information as well, which can be used e.g. for reliability analysis. Particularly in linear dynamics of structures the calculation and analysis of random eigenvalues and eigenvectors is crucial. A quite versatile, however computationally intensive way to analyze such systems is direct Monte Carlo simulation. In this paper procedures are shown, which allow a significant reduction of computational efforts of the simulation using a subspace iteration scheme with “optimally” selected start-vectors. As the subspace iteration procedure, although quite accurate, requires a factorization of the stiffness matrix, as an alternative, a procedure based on component mode synthesis is suggested.

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.

Efficient computational procedures for reliability estimates of MDOF-systems

Abstract —This paper presents a new numerical code for determining the reliability of multi- degree-of-freedom (MDOF) linear or non-linear models of structure Several examples of application are presented. They are discussed in context with the issue of accuracy and computational efficiency of the new code and of other approximate solution methods such as statistical linearization.

A computational procedure to estimate the stochastic dynamic response of large non-linear FE-models

Abstract An algorithm for the computation of the stochastic non-stationary non-linear response of large FE-models is presented. In stochastic analysis, the response is described by the mean and covariance function and possibly by the probability distribution of the non-linear response. To estimate the covariance matrix, the method of equivalent statistical linearization is applied for linearizing all non-linear elements. The large fully populated and symmetric covariance matrix of dimension ⩾2 n is described by the so called Karhunen–Loeve expansion representation, which allows one to employ a feasible description. In this context, an efficient procedure is suggested to determine the minimal number of Karhunen–Loeve vectors necessary to assure a sufficiently accurate representation. This method in fact allows one to employ any available deterministic integration scheme to compute the Karhunen–Loeve vectors. To increase the computational efficiency, modal coordinates are used to represent the linear sub-system. Special attention is given to the effects of mode truncation which are generally not negligible for large FE-systems. Moreover, the suggested approach has no limitation w.r.t. the size of the model in terms of degrees of freedom (DOFs) or the number of non-linear elements. The feasibility of the proposed procedure is demonstrated in the numerical examples where the methodology is applied to an office building with approximately 25, 50 and 100 thousand DOFs containing hysteretic damping elements.

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.