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Variabilty response function for the beams with random elasticity and cross section

The concept of variability response function based on the weighted integral method and the local average method is extended to the beam bending problem with random elasticity and cross section of the beam. The elastic modulus and height of the beam are considered to be one-dimensional, homogenous, non-correlated, stochastic fields. The stochastic stiffness matrix is calculated by using standard cubic finite element. The stochastic element stiffness matrix is represented as linear combination of deterministic element stifness matrix and 3 random variables (weighted integrals) with zero-mean property. The concept of variability response function is used to compute upper bounds of the response variability. The first and second moment of stochastic elastic modulus and the height of the beam 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 use of 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 only one random variable per each finite lement. Numerical examples are provided for both methods and for different boundary and loading conditions, different wave numbers and different number of finite lements. It has been shown that variability of the response displacement could be much larger than variability of each input random quantity.

response variability; stochastic stiffness matrix; weighted integral and local average method; beam bending

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