Basics of finite element analysis (FEA) - 2

What kind of formulation is used in FEA?

A system of linear equations is created to solve for displacement at each node of each element. A vector {u} which contains all the possible displacements at the node of each element is formulated.

Before formulation, we need to know what is the meaning of displacement at a node. Consider following line element 1-2 which has 2 nodes. Each node has linear displacement in the x and y direction and each node has angular displacement theta. There are a total of 6 displacements u1, v1, u2, v2, θ1, and θ2. Each of these displacements is called a Degree of Freedom. For the following case, there are 6 degrees of freedom.

The displacement vector {u} for the above element is given by,

Displacement Vector

Depending on the type of element the number of linear and angular displacements varies.

What is Hooke's law?

The spring stiffness 'k' determines how far the spring gets displaced when force 'F' is applied to the spring.

F = -k.x

where k = stiffness of the spring

x = displacement of the spring

F = Force applied to the spring

This analogy is applied to each mesh element to formulate a linear system of equations.

{F}=[k]{u}

where F = nodal forces

k = stiffness matrix for mesh element

u = displacement vector

What is the global stiffness matrix?

All the individual stiffness matrices are assembled into one big stiffness matrix. The assembly of all the local (elemental) stiffness matrices into one big stiffness matrix is called the global stiffness matrix.

Why does a global stiffness matrix called a sparse and banded matrix?

When a global stiffness matrix is formed there are a large number of cells with zero value in the matrix (because there is no relation between corresponding nodes of those elements). The non-zero elements in the global stiffness matrix are usually along the diagonal of the stiffness matrix. For example,