A new pre-loaded membrane geometric stiffness matrix with full rigid body capabilities

Abstract

Space structures, due to economic considerations, must be light-weight. Accurate prediction of the natural frequencies and mode shapes is critical for determining the structural adequacy of components, and designing a control system. The total stiffness of a member, in many cases, includes both the elastic stiffness of the material as well as additional geometric stiffness due to pre-load (initial stress stiffness). The pre-load causes serious reservations on the use of standard finite element techniques of solution. In particular, a phenomenon known as “grounding”, or false stiffening, of the stiffness matrix occurs during rigid body rotation. The author has previously shown that that the grounding of a beam element is caused by a lack of rigid body rotational capability, and is typical of beam geometric stiffness matrices formulated by others, including those with higher-order effects. Having identified the source of the problem as the force imbalance inherent in the formulations, the author developed a beam stiffness matrix from a directed force perspective, and showed that the resultant global stiffness matrix contained complete rigid body mode capability, and performed well in the diagonalization methodology customarily used in dynamic analysis. In this paper, the authors investigate the “grounding” of membrane elements, and develop a new membrane element with rigid body rotational capabilities.