The basic
finite element equation to be solved for structures experiencing static loads
can be expressed as:
Ku = P
where K is the stiffness matrix of the
structure (an assemblage of individual element stiffness matrices). The
vector u is the displacement
vector, and P is the vector of
loads applied to the structure. The above equation is the equilibrium of
external and internal forces.
The
stiffness matrix is singular, unless displacement boundary conditions are
applied to fix the rigid body degrees of freedom of the model.
The equilibrium equation is
solved either by a direct or an iterative solver. By default, the direct solver
is invoked, whereby the unknown displacements are
simultaneously solved using a Gauss elimination method that exploits the
sparseness and symmetry of the stiffness matrix, K, for computational efficiency. Alternatively, an iterative
solver using the preconditioning conjugate gradient method may be used.
While the direct solver is very robust, accurate and efficient, the
iterative solver is sometimes superior, in terms of speed, for thick-walled
solid structures.
Once the unknown displacements at the nodal
points of the elements are calculated, the stresses can be calculated by using
the constitutive relations for the material. For linear static analysis
where the deformations are in the elastic range, i.e.: the stresses, σ, are assumed to be linear
functions of the strains, ε, Hooke’s law can be used to calculate the
stresses. Hooke’s law can be stated as:
σ = Cε
with the
elasticity matrix C of the
material. The strains ε are a
function of the displacements.
