Vibration and buckling of flat free–free plates under non-uniform in-plane thermal stresses

Detailed consideration is given to the modes and frequencies of a free rectangular Kirchoff plate subjected to in-plane stresses generated by prescribed non-uniform surface temperature distributions which are doubly symmetrical about the plate central axes. Physical understanding is sought of phenomena observed by previous investigators. Stress distributions corresponding to three different temperature distributions have first been studied and incorporated in a Rayleigh Ritz analysis to find natural frequencies and modes. All frequencies change as the temperature changes, some much more than others. All eventually vanish, one after the other, as the temperature reaches certain critical positive and negative values at which the plate goes into statically unstable buckling modes. Whether the frequencies rise or fall with rising temperature at the plate centre depends on the relative magnitudes of pairs of positive and negative critical temperatures. The modes of buckling at each pair of critical temperatures may differ greatly from one another and also from the vibration modes at zero temperature. The relationship between the square of the frequency and the temperature is then no longer approximately linear, although it is exactly so for certain simple in-plane stress distributions. Conditions have nevertheless been identified under which it is a very good approximation to the actual frequencies of the heated plate over wide temperature ranges.