Finite Element Analysis of Beams with Nonlocal Foundations

In this paper, a nonlocal viscoelastic foundation model is proposed and used to analyse the dynamics of beams with different boundary conditions using the finite element method. Unlike local foundation models the reaction of the nonlocal model is obtained as weighted average of state variables over a spatial domain via convolution integrals with spatial kernel functions that depend on a distance measure. In the finite element analysis, the interpo-lating shape functions of the element displacement field are identical to those of standard two-node beam elements. However, for nonlocal elasticity or damping, nodes remote from the element do have an effect on the energy expressions, and hence the damping and stiff-ness matrices. The expressions of these direct and cross stiffness and damping matrices may be obtained explicitly for some common spatial kernel functions. Alternatively nu-merical integration may be applied to obtain solutions. Numerical results for eigenvalues and associated eigenmodes of Euler-Bernoulli beams are presented and compared (where possible) with results in literature using exact solutions, Galerkin approximations or the transfer matrix approach. The examples demonstrate that the finite element technique is efficient for the dynamic analysis of beams with nonlocal viscoelastic foundations. I