Page 113 - Finite Element Analysis with ANSYS Workbench

104 Chapter 6 Three-Dimensional Solid Analysis

w
(, , )  
wx y z NN N N  1 2 3 4   
where  v T   vv v v  
1
2
4
3
T
w
    ww w w  
2
1
3
4
Thus, the element vector of nodal unknowns contains the total of
12 unknowns as,
    T u v w u vw u v w u vw  1 1 1 2 2 2 3 3 3 4 4 4 
The element vector containing the six strain compo-
nents can be determined from,
       
B 



(6 1) (6 12) (12 1)
B
where the matrix  is called the strain-displacement matrix
which relates the six strain components with the 12 nodal
K
displacements. The element stiffness matrix   can then be
determined from,
T
     V
B
CB
K




(12 12) (12 6) (6 6) (6 12)
These element matrices are determined for all elements before
assembling them to become a large stiffness matrix of the system
equations. Boundary conditions are then applied and the system
equations are solved for all nodal displacement solutions u ,v
,
i
i
w .
i
Once all nodal solutions ,u ,v w are obtained, the
i
i
i
element stresses are determined from,
B 

        
C




(6 1) (6 6) (6 12) (12 1)
The same process is applied for the hexahedral element
but the number of equations is larger. For example, the 8-node
hexahedral element contains 24 equations while the 20-node
element consists of 60 equations. Developing a computer program
is thus a must for solving a problem. We will use the ANSYS
software through its Workbench to analyze three-dimensional solid
problems in the following sections.