reliability analysis of one dimentional structures using explicit probability density function of static response in stochastic finite element

Reliability analysis, with considering the randomness of geometry, materials and loading variables, is a good guide for structural design in engineering. The stochastic finite element method is used to analyzing the structural systems, with regard to uncertainty in random parameters of structures. Solving the equations of stochastic finite element leads to the calculation of all possible structural responses taking into account all the random variables of the structural system. Due to the uncertainty effect on the response of structural systems, reliability analysis is essential. However, due to the limitations of the classical methods of reliability analysis, there is a need to calculate the reliability index based on the probability density function of response in structures. In this study, by combining perturbation method and change – of – variable method and without the limitation of the statistical type of random distribution of random variables, the probability density function of response is calculated. By calculating the probability density function of the static response of the structures, the probability of failure and the reliability index are obtained directly.</span>
It is obvious that the accuracy of the result of the stochastic finite element analysis depends on the random field element meshes. </span>For this purpose, the distributed random field is discretized over the number of elements of equal length in structural members for each random variable.
In stochastic finite element method, due to the uncertainty of the characteristics of random variables in the structure, it is necessary to define a correlation function between a random variable in different elements. The reliability index is considered as a measure of convergence by considering the scale of fluctuations and the correlation function of random variables in adjacent elements in structure. In each number of elements, the structural reliability index is calculated using the explicit probability density function of response and the structural resistance function to converge on a certain number of elements. In this study, the stochastic finite element analysis is performed in linear static mode for a simple beam and a cantilever column and the variables of geometry, materials and loading are considered randomly with real statistical distributions according to the literature review. As can be expected from the deterministic finite element method, as the number of elements increase and the meshing is smaller, the reliability index increases.Considering the lower scale of fluctuations for random variables makes the reliability index converge to a larger value. However, on a larger scale of fluctuations convergence occurs in a smaller number of elements due to the greater correlation of random variables in adjacent elements. Also, the results of the probability density function of the response with the explicit method compared to the Monte Carlo simulation method show a good match in result. The advantage of using the reliability index as a measure of convergence in meshing, the configuration of limited random components is to consider all possible structural responses instead of using the average structural response in the design of structures.