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Analytical solution for thermal stresses in a rectangular plate

  1. Oct 9, 2012 #1
    Hello everybody,

    I am solving a 2D problem of thermal stresses in a rectangular plate in which temperature is changing only y direction. Plate has fixed displacement conditions. Could anyone help me to find out analytical solution of thermal stresses for my problem? Does anyone suggest me where I can find analytical solution?

    Thanks in advance.

    Best regards,
  2. jcsd
  3. Oct 10, 2012 #2
    What are "fixed displacement conditions"?

    Solutions could be something like a parabola or a cosine in the transverse direction, multiplied by a tanh in the expansion direction. Try a few ones.
  4. Oct 11, 2012 #3

    Thanks for the reply. I have mistakenly mention fixed displacement boundary condition but actually all surfaces of rectangle are traction free (free surfaces). Temperature change is parabolic on y-axis.

    Could you please let me know where I can get analytical solution for this problem?
  5. Oct 16, 2012 #4
    If temperature is uniform along the X axis and all edges are free to move, stress is zero everywhere, whatever the profile is along the Y axis.

    In the more general case, try a product of cos(x) and cos(y) with different harmonic periods - but a particular polynomial solution can look good.
  6. Oct 16, 2012 #5


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    I think I can see how you reached that (wrong) conclusion, but what about non-zero shear strains in the plate, and the stresses they create?

    The direct stress component normal to a free edge must be zero, but the other stress components along a free edge need not be zero. Axial tension in a rod is a simple (non-thermal) example.
  7. Oct 18, 2012 #6

    It must be something like tanh(x) times a polynomial of y.
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