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Temperature gradient loading on beams and shells
It is only possible to explicitly apply top and bottom
temperatures to two and three dimensional continuum elements
because they have distinct faces to which temperature values,
T and T0, can be applied. With shell, plate and
beam elements this is not possible and must be accomplished
by applying a temperature gradient loading.
A temperature gradient loading thus applies
a differential thermal load across the top and bottom surfaces
of a surface element. The effect of this gradient being
to cause bending in the structure.
For example, a simple cantilever, modelled
with a two dimensional beam element, of thickness t, with
a top surface temperature (T) of 100 °C
and bottom surface
temperature of 20 °C , would have a through thickness temperature
gradient of
dT/dy = (100 - 20)
/ t
Where y is the local through-thickness axis
direction. By using temperature gradient loading, LUSAS
will evaluate a thermal bending strain (f)
f
= a
* (dT/dy - dT0/dy)
Where a
is the
coefficient of thermal expansion for the material. The cantilever
will deform by bending up (or down) as though the temperatures
had been applied explicitly to the top and bottom surfaces.
from this gradient as
The "direction" in which these
thermal strains are applied, again, depends on the element
type being used. In broad terms, however, the application
of a temperature gradient will generate a bending strain.
For instance, Bar elements, having only Fx, do not have
the capability to support a temperature gradient. Shell
elements will calculate fx
and fy thermal strains
which, in turn, generate bending moments Mx and
My. For further details, see the specific element
description in the LUSAS Element Reference and Theory
manuals.
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