Rigid body motions in curved finite elements

Abstract

D functions used to construct the stiffness matrix of a finite element should possess the following properties. 1) Infinitesimal rigid body motions should be accurately represented. If this requirement is not met the conditions of equilibrium of the element are not satisfied.*• 2) The displacement functions should contain all the lower terms of a complete set of functions. This requirement insures monotonic convergence by mesh size reduction. 3) A minimum degree of interelement continuity must be maintained between adjacent elements. This minimum degree of compatibility must insure a perfect match for the inplane and the out of plane components of displacement. Also for the out of plane component, slopes tangent and normal to all common edges of two adjacent elements must match. This requirement then insures convergence to an exact result by mesh size reduction. The importance of the last two requirements is firmly established; however, the first requirement has been shown to be problem dependent. If the structure to be analyzed is so constrained that no element of the structure is ever going to undergo any rigid body motion, then obviously this requirement can be violated. For example, axisymmetric elements acted upon by axisymmetric loads need to have only one rigid body mode: a rigid translation parallel to the axis of symmetry. For this particular type of element a truncated cone as used by Grafton and Strome always includes a rigid body motion parallel to the longitudinal axis. However, if the axisymmetric element is to have curvature in the longitudinal direction, then all the rigid body modes are absent. Jones and Strome recognized such a deficiency and reintroduced a longitudinal translation in their element. Later, Stricklin et al. reported on a similar improved element but omitted the longitudinal rigid body motion altogether. This last element is capable of handling asymmetric loading, therefore it is not difficult to imagine a loading in which many elements would have to undergo considerable transverse motion; a cantilevered structure would lead to such a situation. Haisler and Stricklin studied the influence of longitudinal translation and observed that such a rigid motion is recuperated by mesh size reduction. For elements of rectangular aspects, Bogner, Fox, and Schmit developed a systematic method for constructing acceptable displacement fields. However, for curved cylindrical elements, only two rigid body modes are accounted for. The same authors reported on a (48 X 48) stiffness matrix and mentioned that an eigenvalue analysis of such a matrix indicated that rigid body motions were adequately represented. However, as pointed out in our study of curved cylindrical elements, rigid body motions cannot be represented by independent displacement components. In the same reference, the importance of these rigid body motions is clearly illustrated in several examples. However, the inclusion of rigid body motions was done at the expense of rigorous interelement compatibility. This compromise resulted in a significant improvement in the behavior of the element. In this paper we develop a method to include rigid body motions without comprising deformational compatibility. The method is general and can be applied without difficulties to any element, curved or flat. The improvements of a curved cylindrical element are illustrated with one example.