On Simple and Efficient Shell and Solid Finite Elements with Rotational Degrees of Freedom

Abstract

Rotational degrees of freedom (RDOF) in shell and solid finite elements have drawn considerable amount of research effort in the 1980’s and 1990’s. Without full RDOF in the elements general rigid body motions can not be dealt with correctly and results of solution can not be obtained accurately. In particular, when the translational and rotational motions are required to be controlled in a structural system, typically encountered in aerospace and other industries, such a control can not be achieved appropriately without the incorporation of the RDOF in the finite element model. This paper is concerned with the investigation the original goal of which was to provide mixed variational principles for the development of simple and efficient finite elements with RDOF. These finite elements include: (i) isotropic and laminated composite flat triangular shell elements [1], (ii) flat triangular shell elements with embedded and distributed piezoelectric components, and (iii) lower order four-node tetrahedral solid elements. They are indispensable in the analysis of large scale aerospace structural systems which include non-periodic cellular structures. One common feature among these elements is that their translational degrees of freedom (DOF) are hinged on the hybrid strain formulation while their RDOF are displacement based. Allman’s approach [2] for drilling DOF (DDOF) and RDOF is included in the formulation. Every one of the finite elements introduced is capable of producing the correct number of rigid body modes and there are no zero-energy spurious modes that need to be suppressed. Another feature of these elements is that explicit expressions for element matrices are obtained with manual and a symbolic algebra package, MAPLE. Thus, no numerical matrix inversion and numerical integration are necessary in the derivation of element matrices. These explicit expressions are relatively much more efficient to apply in the analysis and design of large scale structural systems.