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2.2 Finite Element Method 15

2.2 Finite Element Method

2.2.1 Finite Element Equations
Finite element equations can be derived from the
governing differential equation by using the Method of Weighted
Residuals (MWR). The idea of the method is to transform the
differential equation into the corresponding algebraic equations by
requiring that the error is minimum. These algebraic equations
consist of numerical operations of addition, subtraction, multi-
plication and division. Such operations allow the use of calculators
to determine solutions for small problems. For larger problems, a
computer program must be developed and employed.
The derived finite element equations are normally
written in matrix form so that they can be used in computer
programming easily. The finite element equations for the truss
element are,

   u   
F
K
where   is the element stiffness matrix;   u is the column
K
matrix or vector that consists the nodal displacement unknowns;
F
and   is the column matrix or vector that contains the nodal
loads. These matrices depend on the element types used as
explained in the following section.

2.2.2 Element Types
The standard two-node truss element is shown in the
figure. The element lies in the x-coordinate direction and consists
of a node at each end. The element length is L with the cross-
sectional area of A and made from a material that has the Young’s
modulus of E . At an equilibrium condition, the forces at node 1
and 2 are F and F , causing the displacements of u and u in its
2
1
1
2
axial direction, respectively.