Page 52 - Finite Element Analysis with ANSYS Workbench

3.2 Finite Element Method 43

Since the strain is related to the displacement and deflection as,
 u  2 w
    z
x
x  x  2
Then, the stress can be determined from the deflection as,
 2 w
 x  Ez x  2

For a typical beam in a three-dimensional frame
structure, its deflection may be in a direction other than the z-
coordinate. In addition, the beam may be twisted by torsion caused
by the applied loads or affected by other members. These
influences must be considered and included for the analysis of three
dimensional beam structures.

3.2 Finite Element Method

3.2.1 Finite Element Equations
Finite element equations can be derived directly from
the beam governing differential equation by using the method of
weighted residuals. Detailed derivation can be found in many
finite element textbooks including the one written by the same
author. The derived finite element equations are in the form,
     
K 
F

where   is the element stiffness matrix;   is the element
K
vector containing nodal unknowns of deflections and slopes; and
  is the element vector containing nodal forces and moments.
F
These element matrices depend on the selected beam element types
as explained in the following section.

3.2.2 Element Types
The basic beam bending element with two nodes is
shown in the figure. Each node has two unknowns of the
deflection w and slope  .