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

In the existing finite element calculations of dynamic problems using C⁰ structural elements, the inertia terms are evaluated without any reference to the modifications such as reduced integration, projections etc., typically needed in the discretization of the stiffness terms. A different 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 modified stiffness matrix for a given C⁰ element can be obtained by standard, unmodified approach, in which degrees of freedom remain unchanged, but the shape functions are different. Those modified functions are of higher order and define the translational field 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 field used in the unmodified evaluation of the stiffness terms should also be used in discretization of the inertia terms. This implies that the modified higher-order functions should be employed when evaluating the element mass matrix for the C⁰ elements with modified stiffness matrices. As a consequence of this approach, consistency between formulation of the inertia and stiffness 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.