Variability response function for nonlinear beam

Abstract

The concept of variability response function based on the weighted integral method and the local average method is extended to the nonlinear beam element with random elasticity. The elastic modulus is considered to be one-dimensional, homogenous, stochastic field. The stochastic stiffness matrix is calculated by using nonlinear finite element. The stochastic element stiffness matrix is represented as linear combination of deterministic element stiffness matrix and random variables (weighted integrals) with zero-mean property. The concept of the variability response function is used to compute upper bounds of the response variability. The first and second moment of the stochastic elastic modulus are used as input quantities for description of the random variables. The response variability is calculated using the first-order Taylor expansion approximation of the variability response function. The randomness of the beam deflection is expressed as the function of the randomness of the elasticity. The use of the variability response function based on the weighted integral method is compared with the use of the variability response function based on the local average method in the sense to show the influence of reducing the computational effort on the loss of accuracy. The use of local average method gives approximation with small loss of accuracy with less random variables per each finite element. Numerical examples are provided for the both methods, different wave numbers and different number of the finite elements. It has been shown the variability of the displacement as the function of the variability of the elastic modulus as input random quantity.