Perturbation Approach

For static problems, the linear finite element equations are

Ku = F, (28)

where K is the stiffness matrix, u is the nodal displacement vector and F is the nodal force vector. In the following we will assume that only the stiffness matrix involves randomness, i. e. the Young’s modulus E(x) is described as

E(x) = E (1 + ef(x)) (29)

with E and є as the mean and the coefficient of variation of the Young’s modulus, respectively, and f(x) as a zero-mean unit random field. Utilizing the stochastic finite element method the random field is discretized by a set of n zero-mean unit random variables Rj with covariances Cov[Rj, Rk] (j, k = 1, … ,n). In case of small to moderate coefficients of variation є of the Young’s modulus a

flrst-order perturbation approach suffices to render accurate second-moment results. Therewith, the nodal displacements can be written as

whereby K and U are the stiffness matrix and the nodal displacement vector evaluated at the means Vj = 0 of the random variables Rj. Taking expectations, the mean and the covariance of the nodal displacements are given by

E[u] = U (31)

and

respectively.

To include measured dynamic responses in the random field description – and therewith the nodal displacements – the generalized eigenvalue problem

(-M Xi + K) ei = 0 (33)

has to be solved, with M denoting the (deterministic) stiffness matrix, and Xi and ei the i-th eigen­value and eigenvector, respectively. Expressing the eigenvalues by their Rayleigh coefficients

and applying again a first-order perturbation approach, the random eigenvalues Xi can be expressed as a linear combination of the random variables Rj, i. e.

where ei are the eigenvectors evaluated at Vj = 0. Given m identified frequencies ші (Xi = &>2), this information can be included in the random field description by transforming the random variables Rj such that they are conditional on the set of measurements s = [s1,s2, … ,sm], whereby si = Xi — Xi. Consequently, also the nodal displacements are conditional on the measurements s, i. e. the conditional mean and covariance are