On consistent discretization of inertia terms for C0 structural elements with modified stiffness matrices

In the existing Þnite element calculations of dynamic problems using C0 structural elements, the inertia terms are evaluated without any reference to the modiÞcations such as reduced integration, projections etc., typically needed in the discretization of the sti¤ness terms. A di¤erent discretization of inertia is discussed here. It is based on the following two observations. First, as shown in this work (at least for the beam problems), the modiÞed sti¤ness matrix for a given C0 element can be obtained by standard, unmodiÞed approach, in which degrees of freedom remain unchanged, but the shape functions are di¤erent. Those modiÞed functions are of higher order and deÞne the translational Þeld within the element in terms of both translational and rotational parameters. Second, if standard consistent approach to the formulation of dynamic problems is to be followed, approximation of the displacement Þeld used in the unmodiÞed evaluation of the sti¤ness terms should also be used in discretization of the inertia terms. This implies that the modiÞed higher-order functions should be employed when evaluating the element mass matrix for the C0 elements with modiÞed sti¤ness matrices. As a consequence of this approach, consistency between formulation of the inertia and sti¤ness terms is restored. This leads to inertial coupling between rotational and translational degrees of freedom, which is absent in standard evaluation of inertia. It is demonstrated that this approach tends to improve accuracy of dynamic computations