Finite Element Method in Structural Mechanics - Theoretical Overview of FEM-Displacement Formulation: From Elements To System To Solution

Theoretical Overview of FEM-Displacement Formulation: From Elements To System To Solution

While the theory of FEM can be presented in different perspectives or 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 to both linear and non-linear material behaviours.

The principle of virtual displacements for the structural system expresses the mathematical identity of external and internal virtual work:

The virtual internal work in the right-hand-side of the above equation may be found by summing the virtual work in the individual elements—This is the crucial step where we will need displacement functions written only for the small domain rather than over the entire system. As shown in the subsequent sections, Eq.(1) leads to the following governing equilibrium equation for the system:

where

= vector of nodal forces, representing external forces applied to the system's nodes.

= vector of system's nodal displacements, which will, by interpolation, yield displacements at any point of the finite element mesh.

= vector of equivalent nodal forces, representing all external effects other than the nodal forces which are already included in the preceding nodal force vector R. These external effects may include distributed or concentrated surface forces, body forces, thermal effects, initial stresses and strains.

= system stiffness matrix, which will be established by assembling the elements' stiffness matrices :.

Once the supports' constraints are accounted for, the nodal displacements are found by solving the system of linear equations (2), symbolically:

Subsequently, the strains and stresses in individual elements may be found as follows:

where

= vector of element's nodal displacements--a subset of the system displacement vector r that pertains to the element under consideration.

= strain-displacement matrix that transforms nodal displacements q to strains at any point in the element.

= elasticity matrix that transforms effective strains to stresses at any point in the element.

= vector of initial strains in the element.

= vector of initial stresses in the element.

By applying the virtual work equation (1) to the system, we can establish the element matrices, as well as the technique of assembling the system matrices and . Other matrices such as, and can be directly set up from data input.