Chapter 6 - Assembly of Element Matrices and Vectors and Derivation of System Equations

Abstract

The assembly of element matrices and vectors, derivation of the system equations, and incorporation of the boundary conditions are considered. Because different local coordinate systems (for different elements) may have been used in deriving the element matrices and vectors, it is necessary to transform the element matrices and vectors of all elements into one global coordinate system. Thus a coordinate transformation matrix is to be derived that relates the nodal unknowns defined in the local coordinate system to those defined in the global coordinate system. By using a scalar characteristic, such as strain energy, that is invariant with the coordinate system, the relations for the element matrices and vectors between the local and global coordinate systems can be derived. The assembly of element matrices and vectors in the global coordinate system to derive the system matrix and system vector based on the requirement of compatibility at element nodes is presented. Three methods are presented for incorporating the boundary conditions. In addition, a penalty method is also outlined for incorporating any general boundary condition in a unified and simple manner. All the methods are presented with illustrative examples.