Noon Digital Resources
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STOCHASTIC FINITE ELEMENT ANALYSIS OF THE FREE VIBRATION OF FUNCTIONALLY GRADED MATERIAL PLATES
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Afeefa Shaker, Wael Abdelrahman, Mohammad Tawfik and
Edward Sadek
Abstract
The superior properties of Functionally Graded Materials (FGM) are
usually accompanied by randomness in their properties due to difficulties in
tailoring the gradients during manufacturing processes. Using the Stochastic
Finite Element Method (SFEM) proved to be a powerful tool in studying the
sensitivity of the static response of FGM plates to uncertainties in their
material properties. This tool is yet to be used in studying free vibration
of FGM plates. The aim of this work is to use a Second Order Reliability
Method (SORM), combined with a nine-noded isoparametric Lagrangian element
based on the third order shear deformation theory to investigate sensitivity
of the fundamental frequency of FGM plates to material uncertainties. These
include uncertainties in ceramic and metal Young’s modulus and Poisson’s
ratio, their densities and the ceramic volume fraction. The developed code
utilizes MATLAB capabilities to derive the derivatives of the stiffness and
mass matrices symbolically with a considerable reduction in calculation
time. Calculating the eigenvectors at the mean values of the variables and
updating them only at the last iteration significantly increases solution
speed. The results of the stochastic finite element code are compared to
published results and to the results of the well-established Monte Carlo
simulation technique with importance sampling. Results show that the
relative importance of variations in the constituents’ properties is highly
dependent on the volume fraction and is virtually independent of the
frequency ratio for practical values of solution reliability. SORM is proven
to be an excellent rapid tool in the stochastic analysis of free vibration
of FGM plates, when compared to the slower Monte Carlo simulation
techniques.
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