Keywords: stochastic dynamics, uncertainty propagation, series expansion.

Uncertainties in specifying material properties, geometric parameters, boundary conditions and applied loadings are unavoidable in describing real-life engineering structural systems. Traditionally this has been catered for through the use of safety factors at the design stage. Such an approach may not be always satisfactory in today's competitive design environment, for example, in minimum weight design of aircraft structures. The situation may also arise when system safety is being jeopardized due to the lack of detailed treatment of uncertainty at the design stage, for example, the finite probability of resonance occurring is unlikely to be captured by a safety-factor based approach due to the intricate nonlinear relationships between the system parameters and the natural frequencies. For these reasons, the stochastic finite element methods (SFEM) have been developed to analyze a wide range of computational mechanics problems with uncertainty in a rigorous manner. After the discretisation of the governing stochastic partial differential equation, it is possible to express it in terms of a set of ordinary differential equations with random coefficients. This can be achieved by discretising the random fields using the Karhunen-Loeve expansion.

Several methods have been proposed to solve the set of random equations arising in stochastic mechanics. These methods include, perturbation methods, Monte Carlo simulation and projection methods. The projection methods can be applied using the polynomial chaos expansion, generalized polynomial chaos expansion, random function expansion or by using the reduced basis method. In the projection methods, the random function is projected to an infinite dimensional orthonormal basis of functions. Due to the need of employing a large number of basis functions, one has to solve a very large set of deterministic linear algebraic equations to obtain the coefficients. This can be prohibitively computationally expensive for large systems with many random variables. To address this problem, in this paper an alternative approach by projecting the solution to a finite dimensional orthonormal vector space is proposed. The existence and uniqueness of a such a solution is rigorously proved. It is shown that the response can be obtained using a finite series comprising a function of random variables and orthonormal vectors. A new computational approach to construct this series is proposed. The convergence property of this series is discussed. A hybrid analytical and simulation based approach is proposed to obtain the moments and probability density function of the response. The method is illustrated using the numerical example of a plate bending problem. The results are compared with the results obtained using the direct Monte Carlo simulation and polynomial chaos expansion.

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