TABLE OF CONTENTS Any structural domain may be classified into one of three categories and suitably discretized by finite elements as follows. One-dimensional space: line elements, two-dimensional space: triangular and quadrilateral elements, and three-dimensional space: tetrahedral, prismatic, and hexahedral elements, or any combination thereof. The displacement values within an element are interpolated in terms of nodal values that may be displacements, or their derivatives. Depending on the pattern of the displacement distribution, problems in structural mechanics may be broadly categorized as beam, truss, and cable; plane stress and strain; axisymmetric solids; plate bending; and shells and three-dimensional solids.
Finite element force-displacement characteristics may be derived for each class of problem by applying variational or weighted-residual techniques or simply using the Gauss divergence theorem. The generation of stiffness, inertia, and other associated element matrices is considered in this chapter. The fundamental step in developing these matrices involves choosing appropriate shape functions, either in the local coordinate system or using a natural coordinate system that enables convenient use of standard numerical integration procedures. With the availability of symbolic manipulation software packages it is also quite straightforward to generate these matrices using the local coordinate system. A displacement expression must satisfy rigid-body and constant strain requirements as well as interelement compatibility. The latter requirement is satisfied if the displacement field is continuous up to the derivative one order lower than the highest derivative occurring in the strain displacement equations. Satisfaction of rigid-body and constant strain states are achieved when the displacement expansion is chosen...
