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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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