The generalized boundary value problems in theory of plates

France Brešar

Theoretical and applied mechanics, Tome 6 (1980) no. 1, p. 3 .

Résumé

The basic results of Kirchhoff's theory of thin plates ($\sigma_x=\alpha Z$, $\sigma_y=\beta Z,\dots$, $w_0=w_0(x,y)$) are only an approximation of the actual state. Much better approximations were proposed by Stevenson in 1942. He approximated the components of the stress tensor and displacements with the polynomials up to the fifth degree, where the coefficients are defined with five analytical functions $\Omega,\omega,\varphi,\psi,P$. The function $P$ is defined with the transverse load $q$, but the other four functions with the boundary value conditions. In this thesis we show how concrete problems can be transformed into boundary value problems of the analytic functions (cf. Def. 3.2., 3.5., 3.6., 3.7., 3.8) The unique solvability of these problems has also been proven. Similar results, which are given in the present work for medium-thick plates, were obtained in 1938 by Lehnickij [2] for thin plates and in 1965 by Marušič [3] for medium-thick plates for $q=0$. In both examples only functions $\Omega$ and co appear, which describe the bending of the middle plane. In our example, functions $\varphi$ and $\omega$ are added, which show that the plate bending is simultaneously bound to the wall problem. The whole contribution is based on Stevenson's theorems and the whole represents a novelty in plate theory.