By Steven E. Koonin

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The discussion in this direction is more involved, and interested readers are referred to a monograph by Liu [2002]. In this book, we shall only discuss elements whose corresponding moment matrices are invertible. Note that the derivatives of the shape functions can be obtained very easily, as all the functions involved are polynomials. 28) The issues related to the compatibility of element shape functions will be addressed in Chapter 11. Note that there are many other methods for creating shape functions which do not necessarily follow the standard procedure described above.

23), we have assumed the existence of the inverse of P. There could be cases where P−1 does not exist, and the construction of shape functions will fail. The existence of P−1 depends upon (1) the basis function used, and (2) the nodal distribution of the element. The basis functions have to be chosen first from a linearly-independent set of bases, and then the inclusion of the basis terms should be based on the nodal distribution in the element. The discussion in this direction is more involved, and interested readers are referred to a monograph by Liu [2002].

The solid can also be loaded by body force f b and surface force fs in any distributed fashion in the volume of the solid. 3. On each surface, there will be the normal stress component, and two components of shearing stress. The sign convention for the subscript is that the first letter represents the surface on which the stress is acting, and the second letter represents the direction of the stress. The directions of the stresses shown in the figure are taken to be the positive directions. 1. Four common types of structural components.