Page 198 - Finite Element Analysis with ANSYS Workbench

10.2 Finite Element Method 189

Similarly, the finite element equations for solid prob-
lem can be derived by applying the method of weighted residuals to
the governing differential equations as described in chapter 6. The
finite element equations are in the form,
K 
F 
       
F
0
K
where  is the element stiffness matrix;   is the element
F
F
vector containing nodal forces, and   is the element vector
0
containing nodal forces from temperature change. In the finite
element equations, the unknowns are the displacement components
, u , w at nodes which are contained in the element vector  .

v

10.2.2 Element Types
A common finite element mesh should be employed for
both heat transfer and solid stress analyses. Nodal temperatures
obtained from heat transfer analysis can be transferred directly to
the same nodes of the solid stress analysis. The overall thermal
stress analysis thus can be performed conveniently.
The finite element equations of the solid stress problem
F
include the load vector   from the temperature change. This
0
load vector affects the solid solutions of the deformation and
F
stresses. As an example, the load vector   due to temperature
0
change for the two-node truss element as shown in the figure is,
  1 
   F 0 ( AE  ref  
)T T
 1 
where A is the truss cross-sectional area, E is the material
Young’s modulus,  is the coefficient of thermal expansion, T is
the average element temperature, and T ref is the reference
temperature for zero stress.

u 1 u 2

F 0 F
1 , A , E  2 0
x
L