Abstract
In this paper, the Galerkin method is used to obtain approximate solutions for Kirchhoff plates stochastic bending problem with uncertainty over plates flexural rigidity coefficient. The uncertainty in the rigidity coefficient is represented by means of parameterized stochastic processes. A theorem of Lax-Milgram type, about existence and uniqueness of the theoretical solutions, is presented and used in selection of the approximate solution space. The Wiener-Askey scheme of generalized polynomials chaos (gPC) is used to model the stochastic behavior of the displacement solutions. The performance of the approximate Galerkin solution scheme developed herein is evaluated by comparing first and second order moments of the approximate solution with the same moments evaluated from Monte Carlo simulation. Rapid convergence of approximate Galerkin's solution to the first and second order moments is observed, for the problems studied herein. Results also show that using the developed Galerkin's scheme one gets adequate estimates for accrued probability function to a random variable generated by the stochastic process of displacement.
Highlights
The field of stochastic mechanics has been subject of extensive research and significant developments in recent years
To measure the performance of numerical solutions obtained through Galerkin method in view of the Monte Carlo simulation are defined relative error function in expected value and variance
One compares the estimates obtained from the numerical solutions and by Monte Carlo simulation for the accrued probability function for a random variable generated by the displacement stochastic process
Summary
The field of stochastic mechanics has been subject of extensive research and significant developments in recent years. Chen and Soares [16] used Kharunen-Loeve’s expansion and the gPC’s to get numerical solutions for the plates bending problem in composite material; Vanmarcke and Grigoriu [25] studied the bending of Timoshenko beams with random shear modulus; Elishakoff et al [26] employed the theory of mean square calculus to construct a solution to the boundary value problem of beam bending with stochastic bending modulus; Chakraborty and Sarkar [27] used the Neumann series and Monte Carlo simulation to obtain statistical moments of the displacements of curved beams, with uncertainty in the elasticity modulus of the foundation They present numerical solutions for stochastic beam problems, none of the papers referenced above address the matter of existence and uniqueness of the solutions. One compare still the estimates for the accrued probability function, and from Monte Carlo stimulation
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