ABSTRACT
The present chapter provides a brief description of the Finite Element Method (FEM) in the framework of linear elasticity. This type of problems includes two assumptions: small displacements (with respect to the characteristic dimensions of the body) and a linear elastic stress-strain relationship (meaning that the relations between stresses and strains are given by a fixed set of constants; these relations are univocal, with no dependence on the load path). In practice, the first assumption states that geometrical changes after load application to the body are so small that they may be neglected, while the second assumption means that the effects of different loads can be superimposed on each other. In this framework, a linear proportion between causes and effects holds, as firstly stated in 1675 by Robert Hooke (1635–1703) by means of a succinct Latin anagram standing for “ut tensio, sic vis”, that is, “as the extension, so the force”. Basically, if the external load (understood as a point load or the full set of forces applied to a body) doubles, so do the effects, namely, displacements, strains and stresses.
Although linear elasticity is a simplification with respect to the goal of the book, which is nonlinear analysis, linear elasticity allows a simple presentation of the finite element method, providing the basis for further insight discussed in the next chapters. Here, applications of FEM to simple problems are also discussed and verified with respect to theoretical considerations.
