Finite Element Analysis of Dynamically Loaded Homogeneous Viscous Beams

ABSTRACT

The paper is concerned with impulsively loaded beams in which the material is treated as homogeneous viscous as an approximation of a rigid-viscoplastic constitutive relation. As opposed to the standard displacement method finite element formulation, where interpolation functions describing the velocity field across elements is given, a mixed formulation is used in which nodal velocities and nodal moments are carried as parameters. At each instant the accelerations (by the Tamuzh principle) and the rates of change of moment (by a virtual velocities formulation) are found, and velocities and moments are integrated forward independently. The properties of the mode solution are also introduced, and the forward integration is carried through only for the difference between the mode solution and the actual solution. This leads to a very efficient scheme for the numerical solution of a cantilever beam problem shown as an illustration.