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Thermal analysis Besides stress and deformation
problems, thermal analysis is the second most
frequent type of problem, solved by Finite
Element Method. Thermal conduction in solid body
is described by the differential equation with
boundary conditions formulated as prescribed
surface temperature ST T T, prescribed heat
flux Sq q q, or prescribed convection q
a (T To). Here is T K temperature, To
surrounding temperature, q W m-2 heat
flux, a W m-2K-1 film coefficient, t s
time , k W m-1K-1 thermal conductivity,
c J kg-1K-1 thermal capacity, ? kg m-3
material density, Q W m-3 heat power
intensity.
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Appropriate functional, which is the basis of
variational formulation of thermal conduction
problem, has the form The primary unknown
function is temperature, which is approximated in
the same way as the displacements N is the
matrix of shape functions and dT the matrix of
unknown nodal temperatures. The difference in
comparison with displacements is, that
temperature as scalar variable is fully described
by a single value in node. For triangular element
described in Chapter 4, the matrices are Time
and space derivative of temperature are
, where
and L
.
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Inserting this into the functional, we obtain for
one finite element where k ??? BT.k. B dV
element matrix of thermal conduction, c ???
NT.?.c.N dV element matrix thermal capacity and
fQ ??? NT.Q dV, fq ?? NT.q dSq are
matrices of thermal loading from internal and
external sources. From the condition of
stationary value of the functional of the whole
body, completed from all elements in the same way
as described in Chapter 2, we have the final
form of discretised equation of thermal
conduction where are
global matrices of thermal capacity, conduction
and loading and UT is the matrix of unknown nodal
temperatures. The equation describes a
non-stationary, transient problem of thermal
conduction. As an illustrative example see the
analysis of problem of temperature distribution
in pipe-wall intersection in Example 10.1, its
time evolution can be seen here.
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Thermal stress analysis Many of machine parts are
working at high temperatures. High temperature is
the cause for thermal dilatation. Constraining
this dilatation will lead to thermal stress. To
analyse this phenomenon, we must divide the
strain field into two components stress and
thermal strain The first one is caused by stress
?? D-1.? , or ? D . ?? D . ( ? - ?T )
. The second component is caused by thermal
expansion ?T ?.?T ?.1,1,1,0,0,0T. ?T,
where ? K-1 is a thermal expansion
coefficient. Inserting this into the strain
energy expression, we can obtain the thermal
expansion loading matrix, fT ??? BT. D. ?.?T dV
. which is used as a load matrix in a
stress-displacement analysis. To formulate this
matrix and compute thermal stress, a thermal
analysis must be realised first so that we have
the value of .?T in all nodes of the mesh. It is
usual to run both analyses - thermal and stress -
on the same mesh, changing only the type of
element and analysis.
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Corresponding element types in ANSYS are given in
the Tab. 10.1. Illustration of the solution of
coupled thermal-stress problem is given in the
Example 4.3.