Stochastic Finite Element Formulation
As discussed earlier, the element stiffness matrix Ke relates the nodal forces Fe to the nodal displacements Ue:
Fe = Ke Ue (18)
in which, for the element with me nodes (j); j = 1… me as sketched in Figure 2:
Fe = [(fxj, fyj, fzj); j = 1… mef ; Ue = [(uj, vj, Wj; j = 1. ..me]T. (19)
Based on the principle of virtual work, the element stiffness matrix for a linear material law (assuming geometrical linearity as well) is obtained as
Ke = j BT(x, y)D(x, y)B(x, y)dVe. (20)
Typically, the strain interpolation matrix B(x, y) is chosen in polynomial form, i. e.
B(x, y) = Bkixkyl; k, l,r > 0. (21)
In this equation, Bkl are constant matrices. In fact, for the CST element shown above there is only one such matrix, i. e. B00. Assuming that the system randomness is described by a random elastic modulus E(x, y), the elasticity matrix D(x, y) can be written as
D(x, y) = DcE(x, y). (22)
Using the polynomial form of B(x, y), the element stiffness matrix finally becomes
Ke = E EE EBTiDcBm„ E(x, y)xkylxmyn dVe.
k+l<r m+n<r ve
The last term in this equation is a so-called weighted integral of the random field E(x, y).
X-klmn = f E(x, y)xkylxmyn dVe.
Using this representation, it is possible to achieve a description of the random variation of the element stiffness matrix in terms of the mean values and the covariance matrix of the weighted integrals.
Due to the numerical rather than analytical integration procedures as utilized in FE analysis, this weighted integral is represented by linear combinations of the values Rj; j = 1.. .n of the random field at discrete integration points.
The global stiffness matrix is then assembled by applying standard FE techniques into the form
K =£ KjRj, (25)
which can be used as a starting point for a perturbation analysis with respect to the discretized random field Rj.
The general situation of a SFE analysis in nonlinear dynamics generally requires the solution of the following matrix-vector equation
Mx + Cx + r(x) = f(t). (26)
In Equation (26), M is the mass matrix, C is the damping matrix, x denotes the vector of nodal displacements, r(x) is the vector of restoring forces depending nonlinearly on the nodal displacements, and f(t) is the applied load.
Within the FE concept, the restoring force vector r(x) is assembled from corresponding element forces, e. g. based on the principle of virtual work in the form
r(x)Sxe = f oe(c)SeedVe. (27)
In Equation (27), the subscript e refers to a particular element. Obviously, this equation implies that the randomness of the material properties immediately affects the calculated restoring forces due to the integration over the volume of the element. Consequently any randomness of these material data will be reflected in the restoring forces as well as the tangential stiffness matrix KT derived from them.
Utilizing a linearization approach at element level, classical SFE-methods as outlined above can be applied.