Finite element method in structural mechanics

Finite element method (FEM) is a powerful technique originally developed for numerical solution of complex problems in structural mechanics, and it remains the method of choice for complex systems. In the FEM, the structural system is modeled by a set of appropriate finite elements:

• straight or curved one-dimentional elements endowed with physical properties such as axial, bending, and torsional stiffnesses. This type of elements is suitable for modeling trusses, beams, frames, grids, and cables.
• two-dimentional elements for membrane action (plane stress, plane strain) and/or bending action (plates and shells). The elements may have a variety of shapes such as triangle or quadrilateral.
• three-dimensional elements for modeling 3-D solids such as machine components or soil mass.

While FEM can be presented in different perspectives and emphases, its development for structural analysis follows the more traditional approach via the virtual work principle or the minimum total potential energy principle. The virtual work principle approach is more general as it is applicable for both linear and non-linear behaviours.

The principle of virtual work for a deformable body expresses the mathematical identity of external and internal virtual work:

${\mbox{External virtual work}}=\int _{V}{\boldsymbol {\sigma }}\delta {\boldsymbol {\epsilon }}\,dV\qquad \mathrm {(1)}$

In the above equation, when the virtual strains $\delta {\boldsymbol {\epsilon }}$ are consistent with the virtual displacements, its use will lead to the equilibrium equation expressed in terms of displacements.

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