Page 72 - Finite Element Analysis with ANSYS Workbench

4.2 Finite Element Method 63

   E  u    v         E     u    v      0
x  2(1    )   y x   y    1   2  x  y   
       

which could be solved for the two displacement components u and
v in the x- and y-direction, respectively. Determination of the
strain components    and stress components   
,
,
,
,
y
xy
xy
y
x
x
can then be followed.

4.2 Finite Element Method

4.2.1 Finite Element Equations
Finite element equations can be derived by applying the
method of weighted residuals to the partial differential equations.
Detailed derivation can be found in many finite element textbooks
including the one written by the same author.
The derived finite element equations are written in
matrix form as,
K 
     
F

where   is the element stiffness matrix;   is the element
K
vector containing the nodal displacement unknowns of u and v;
and   is the element vector containing the nodal forces in the x-
F
and y-direction. Number of equations and sizes of these element
matrices depend on the element type selected. These element
equations are formed up for every element before assembling them
together to become a large set of simultaneous equations.
Boundary conditions of the problem are then imposed before
solving them for the displacement solutions of u and v at nodes.

4.2.2 Element Types
Triangular and quadrilateral elements are the two
popular element types used in the plane stress analysis. The
triangular element may contain three or six nodes, while the
quadrilateral element may consist of four or eight nodes as shown
in the figures.