Most complex deformable body problems commonly found in practice can only be solved with numerical methods. The stiffness-based finite element method is today the numerical method of choice for analyzing deformable bodies as well as many other engineering problems. In this chapter we will describe both stiffness-based and force-based finite elements. It will be shown that while the stiffness-based finite element method is based on solving equilibrium equations for the displacements, the force-based method relies on appropriately combining equilibrium and compatibility to solve directly for the forces (or stresses). We will examine both stiffness-based and force-based finite elements for axial load and bending problems. Simple examples will be used that allow us to work through the application of these finite element methods in detail and to compare results with analytical solutions. Finally, we will outline an important alternative to a finite element approach - the boundary element method – for solving stress problems numerically.
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