Stochastic Finite element analysis of the free vibration of functionally graded material plates

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 both a First Order Reliability Method (FORM) and the 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 the effect of uncertainties on both the metal and ceramic constituents. The basic random variables include ceramic and metal Young's modulus and Poisson's ratio, their densities and 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 proves to be a reasonable simplification which significantly increases solution speed. The stochastic finite element code is validated using available data in the literature, in addition to comparisons with results of the well-established Monte Carlo simulation technique with importance sampling. Results show that SORM is an excellent rapid tool in the stochastic analysis of free vibration of FGM plates, when compared to the slower Monte Carlo simulation techniques.

Stochastic vibration characteristics of finite element modelled functionally gradient plates

“…The accuracy of the present finite element formulation in conjunction with an improved higher order kinematics for random behaviour of simply supported square plates is validated by comparing the results with those available in the literature [16]. The top surface is ceramic rich and the bottom surface is metal rich.…”

“…al. [16]. The difference in the result shows the importance of quadratic variation of transverse displacement to achieve the accountability of normal strain and its derivative in calculation of transverse shear strains.…”

“…Comparison of covariance (COV) of the square of natural frequency for simply supported (Al/ZrO 2 ) square FGM plate with Shaker et al[16].…”

The Stochastic Finite Element in the Natural Frequency of Functionally Graded Material Beams

“…In most of the studies conducted, investigators dealt with the free vibration of functionally graded laminates with random material properties using first-order perturbation technique (FOPT) incorporating mixed type and semi-analytical approach to derive the standard eigenvalue problem [7]. Some of these papers presented the stochastic finite element method (SFEM) to investigate the natural frequency of functionally graded plates based on the higher-order shear deformation theory (HSDT) utilizing first-order reliability method and second-order reliability method [8]. In most cases, Jagtap et al [9] examined the stochastic nonlinear free vibration response of FGM plate using HSDT with von-Karman kinematic nonlinearity via direct iterative stochastic finite element method.…”

Stochastic finite element method for free vibration characteristics of random FGM beams

“…Onkar et al [23] studied the probabilistic characteristics of buckling strength for random composite plates considering pre-buckled stress. Shaker et al [24] dealt with the sensitivity of fundamental eigenvalue of a square FGM plate to material uncertainties using reliability method. Lal and Singh [25,26] studied the statistics of linear modal frequencies and nonlinear fundamental frequency for random composite plates having elastic supporting.…”