H.J. Pradlwarter

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.

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.

Non-linear stochastic response distributions by local statistical linearization

Abstract A novel computational approach based on the statistical equivalent linearization and Gaussian superposition is introduced. The methodology allows to extend the concept of statistical equivalent linearization to proceed from estimates of the second moment properties of the stochastic response to estimates for the non-Gaussian probability distribution. This paper introduces the concept of local equivalent linearization designed to compute a close approximation of the non-stationary probability density function p( x ,t) of a non-linear system exited by random excitation. Non-Gaussian distributions are represented as a sum of normal distributions. The suggested approach employs the well-developed equivalent linearization procedure or Gaussian closure to compute the non-Gaussian distribution of the response of a non-linear system. Since equivalent linearization is applicable for higher dimensions and finite element models, the suggested approach lends itself to providing numerical solutions for higher-dimensional cases.

Reliability of MDOF-systems

To estimate the safety and reliability of nonlinear structural (MDOF)-systems subjected to random dynamic loading the response surface methodology (RSM) and Monte Carlo Simulation (MCS) respectively appear to be the most general approaches to calculate the failure probabilities of such systems. In an effort to reduce the number of computational time consuming simulations as generally required for direct MCS, only a small part of which is important for the assessment of failure rates, a new selective MCS technique named ‘Double and Clump’ (D&C) is suggested. The main idea of this procedure is to increase for each time step the relatively low number of response samples containing higher mechanical energy. The selective MCS procedure is tested by analysing nonlinear SDOF oscillators, for which exact analytical solutions exist. A good agreement between the distributions F(x) in the tails is obtained covering a wide range, i.e. 10−7 < F(x) < 1·0–10−7. The efficiency of selective MCS is demonstrated by applying it to a six-storey frame structure subjected to earthquake excitation. A comparison with results obtained by using the RSM is presented and discussed.

On the Computation of the Stochastic Response of Highly Nonlinear Large MDOF-Systems Modeled by Finite Elements

A computational procedure to evaluate the stochastic response of a general non-linear finite element model is presented. Special attention is given to the storage of large systems and to computational efficiency by use of complex modal analysis. The stochastic equivalent linearization method is utilized in the procedure. It also allows one to use all available “deterministic” nonlinear elements. Adaptive Monte Carlo simulation is suggested to determine the stochastic equivalent linear incremental properties of the non-linear finite element.

Non-stationary response of large linear FE models under stochastic loading

Abstract A numerical procedure to compute the statistical second moment (variance) characteristics of the response of linear structures––modeled by large FE models––under stochastic loading is presented. For this purpose modal analysis including a static correction procedure is applied in order to calculate the structural response. A non-white, non-zero mean, non-stationary, distributed loading as applied is represented by the well known Karhunen–Loeve expansion, which allows to describe any type of non-white excitation and can be easily adjusted to available statistical data. In the proposed approach, step-by-step integration procedures developed for deterministic FE analysis are applied in order to compute the second moment characteristics of the stochastic response.

Equivalent linearization—a suitable tool for analyzing MDOF-systems☆

Abstract A method for the evaluation of the stochastic response of MDOF-systems, based on the equivalent linearization (EQL) method, is presented. This paper focuses on the applicability and accuracy of EQL within design procedures of realistic structures. A numerical approach for evaluating the linearization coefficients is introduced for the benefit of high flexibility, simplicity, and most realistic modeling of restoring force laws. It is shown that the generally made assumption of a normally distributed response introduces some arbitrariness in the estimated stochastic response. The source and effect of this arbitrariness on the accuracy and convergence of the equivalent linearization procedure is studied and discussed. Finally the procedure is applied to investigate the effect of damping devices assembled in an eight story building subjected to bi-directional stochastic, i.e. earthquake excitation. The stationary solution obtained by Monte Carlo simulation, conventional EQL as well as the present approach are compared and discussed.