Dynamic stiffness for structures with distributed deterministic or random loads

Abstract

The dynamic stiffness method applies mainly to excitations of harmonic nodal forces. For distributed loads, modal analysis is generally required. In the case of a clamped–clamped beam, the modal decomposition of a uniformly distributed load by the eigenbeam functions inherits slow convergence because the finite loads at the beam-ends cannot be represented efficiently by the zero deflection and zero slope of the clamped–clamped beam functions. The computed reactions at the supports do not converge at all. The problem is eliminated in this paper by using the finite element interpolation functions for the distributed load. If the distributed load is adequately represented, explicit exact solutions are found. Otherwise, the residual load is expanded in the modal space. As the residual modal force is much smaller and agrees well with the clamped–clamped conditions, fast convergence is achieved. By means of the principle of superposition, a structure with members having distributed loads can be analyzed by two systems: one is associated with the individual members having distributed loads and the other is associated with resulting equivalent nodal forces. The required frequency functions are given for all possible cases. The results presented are exact if the load is interpolated adequately by finite element shape functions. Both deterministic and random loads are considered. Closed-form solutions are obtained for the first time.