Keywords: finite element method, FORM, probabilistic methods, reliability analysis, sampling, sensitivity, simulation, transformation method.

Mechanical modeling of physical systems is often complicated by the presence of uncertainties. The implications of these uncertainties are particularly important in the assessment of several potential regulatory options. Even though significant effort may be needed to incorporate uncertainties into the modeling process, this could potentially result in providing useful information that can aid in decision making.

For several decades, the theory of probability has been used in mechanics to model the random structural properties (materials, geometry, boundary conditions) and phenomena (turbulence, seismic wave, loads) acting on the mechanical systems. The probabilistic approach takes into account the uncertainties of the model data in order to improve the robustness of the forecasts and the optimized configuration. The structural reliability has become a discipline of international interest, addressing issues such as the performance-based cost-safety balancing [1].

Conventional methods for sensitivity analysis [2] and uncertainty propagation can be broadly classified into four categories: "sensitivity checking", analytical methods, sampling based methods, and computer algebra based methods.

Sensitivity checking involves the study of the model response for a set of changes in the model formulation, and for selected parameter combinations. Analytical methods involve either the differentiation of model equations and subsequent solution of a set of auxiliary sensitivity equations, or the reformulation of the original model using stochastic differential equations. On the other hand, the sampling based methods involve running the original model for a set of input parameter combinations and estimating the sensitivity or uncertainty using the model outputs at those points. Another sensitivity method is based on direct manipulation of the computer program, and is termed automatic differentiation.

First and second order reliability methods [3] (FORM and SORM, respectively) are approximation methods that estimate the probability of an event under consideration (typically termed "failure"). In addition, these methods provide the contribution to the probability of failure from each input random variable, at no additional computational effort. These methods are useful in uncertainty analysis of models with a single failure criterion.

In this paper, the reliability analysis of mechanical system with parameter uncertainties has been considered. The uncertainty has been considered in the material properties e.g. young modulus and in load separately. A proposed technique is presented in order to evaluate the stochastic mechanical response. The method is based on the combination of the probabilistic transformation methods [4] for a single random variable (e.g. Young's modulus or load) and the deterministic finite element method (FEM) [5]. The transformation technique evaluates the probability density function (PDF) of the system output by multiplying the input PDF by the Jacobean of the inverse mechanical function. To prove the performance of the proposed method, we analyze the reliability of a cantilever beam then the result is compared with 10,000 Monte Carlo simulations.

1

H. Procaccia, P. Morilhat. Fiabilité des structures des installations industrielles. Eyrolles 1996.

2

G. Sistla, S.T. Rao, and J. Godowitch, "Sensitivity analysis of a nested ozone air quality model", Proceedings of the AMS/AWMA Joint Conference on Applications of Air Pollution Modeling and Its Applications, New Orleans, LA, 1991.

3

M. Hohenbichler, S. Gollwitzer, W. Kruse, and R. Rackwitz. "New light on first- and second-order reliability methods", Structural Safety., 4(4):267-284, 1987. doi:10.1016/0167-4730(87)90002-6

4

Soong T., Random Differential Equations in Science and Engineering. Academic Press 1973.

5

Chateauneuf A., Comprendre les éléments finis. Ellipses 2005.

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