Engineering Transactions, 12, 2, pp. 339-395, 1964

### Zastosowanie Operatorów Różniczkowych do Zadań z Teorii Sprężystości

J. Sułocki
Politechnika Szczecińska
Poland

The paper is divided into nine sections. The introduction contains a brief survey of methods used for developing engineer's theories of plates (thin, medium and thick) and in general problems of elasticity.
It is pointed out that for each particular problem certain fundamental assumptions are usually introduced, on which further procedure is based.
The present paper is devoted to the idea of devising a uniform procedure for the entire class of problems based on static, geometric and physical equations of the theory of isotropic bodies. To this aim general equations are first derived, based on differential operators of infinite order, The relevant argument for the static problem of elasticity of a stratum layer is contained in Sec. 2 and is based on the existing publications [9 and 10], the mass forces being taken additionally into consideration.
In Sec. 3 is considered the problem of bending of a plate. It is found that in the existing homo- geneous solution  certain components are omitted. The correct form of the solution is found.
In Sec. 4 a similar algorithm is built up for the dynamic problems (not yet encountered in the literature).
Sec. 5 presents, in two particular problems concerning vibration of rectangular parallelepipeds, an application of a the fixed algorithm.
Sec. 6 is devoted to the construction of an algorithm composed of differential operators of finite order. Such a procedure is applied mainly to solve the problem of admissibility of using a finite number of terms in an infinite algorithm as recommended in Ref. 10.
In each particular operator of finite order are separated components constituting a symmetric table layout and additional terms of the solution perturbing the symmetry. It is found that the first correspond to operators produced by simplifying infinite operators by reducing them to a finite number of terms. The complements are a result of assuming operators of finite order.
Sec. 7 is devoted to the derivation of equations of the theory of plates. In the equations expressing the state of stress there appear also components of the homogeneous solution, similar to those found in Sec. 3.
In Sec. 8 the engineer's theory of plates is explained in a simple manner showing that if the statement is correct, it leads to a set of partial differential equations (not to a single biharmonic equation as hitherto assumed).

Full Text: PDF

#### References

E. REISSNER, The effect of transverse shear deformation on the bending of elastic plates, J. Appl. Mech., 2, 12 (1945).

E. REISSNER, On bending of elastic plates, Quart. Appl. Math., 5, 1947.

M. SCHÄFER, Über eine Verfeinerung der Klassischen Theorie dünner schwag gebogener Platten, ZAMM, B., 32, 1952.

K. GIRKMANN, Dźwigary powierzchniowe, Dodatek do I wyd. polskiego, Arkady, Warszawa 1961.

J. MOSSAKOWSKI, Równania teorii Reissnera dla płyt ortotropowych, Księga Jubil. prof. dr W. Wierzbickiego, PWN, Warszawa 1959.

S.J. MEDWADOWSKI, On a refined theory of elastic orthotropic plates, Dissert. Univer. of California, 1956.

A. E. GREEN, W. ZERNA, Theoretical Elasticity, Oxford 1954.

E. KOPPE, Die dicke Platte mit nichtlinearer Spannungsverteilung, ZAMM, B. 37, 1957.

[in Russian]

[in Russian]

S. KALISKI, Pewne problemy brzegowe dynamicznej teorii sprężystości i ciał niesprężystych, WAT, Warszawa 1957.

R. SOLECKI, Rozwiązanie ogólne prostopadłościanu ortotropowego i przykład z teoril płyt grubych, Arch. Mech. Stos., 1, 13 (1961).

[in Russian]

[in Russian]

[in Russian]