In this chapter we discuss the mathematical foundation for obtaining the finite element equations for a general engineering problem or a physical system. Before studying the material in this chapter, it is essential to review the concepts of matrix algebra and indicial notation discussed in Appendix A. The chapter will provide a detailed discussion of how to formulate finite element equations using variational principles and weighted residual methods. The development will start with simple one-dimensional problems and will then proceed to full three-dimensional cases. The focus will be on deriving the FE equations in linear elasticity and heat transfer applications. The equations developed and presented here will be the basis for our discussion in the following chapters: the linear elasticity and heat transfer chapters.
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