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188 Chapter 10 Thermal Stress Analysis

For small deformation theory, the strain components
,
are written in forms of the displacement components ,u v w as,
u   u v 
  ;   
x
x  xy y  x 
v   u  w
  ;   
y
y  xz z  x 
 w v   w
  ;   
z
z  yz z  y 
As mentioned earlier, the heat transfer problem is
firstly analyzed for temperature solution. The computed tempera-
ture is input into the solid problem for stress analysis. Once the
stress analysis is performed and the displacement components are
obtained, the six strain components can then be computed.
Determination of the six stress components is followed to complete
the analysis of thermal stress problem.

10.2 Finite Element Method

10.2.1 Finite Element Equations

Finite element equations for heat transfer problem can
be derived by applying the method of weighted residuals to the
governing differential equation as described in chapter 9. The
finite element equations are in the form,

T
   CT     c K  K h  K r   
  c Q   Q Q    Q q    h Q   r Q

The element matrices on the left-hand side of the equations are the
capacitance, conduction, convection and radiation matrices,
respectively. The vectors on the right-hand side of the equations
are associated with conduction, internal heat generation, specified
heating, convection and radiation, respectively. Forms and sizes of
these element matrices and vectors depend on the element types.
The unknowns of the finite element equations above are the nodal
temperatures.