Engineering Transactions,

**13**, 2, pp. 355-380, 1965### Zastosowanie Metody Sił Zespolonych w Teorii Powłok Ortotropowych

The present paper concerns the theory of orthotropic shells assuming the relation between the section forces and the deformation components in the form (5.1). The rigidities involved in these equations are determined by equations (5.2). It is assumed that equations (3.1) derived by V. V. NOVOZHILOY for isotropic shells are valid. The equilibrium equations are assumed, after A. E. H. LOVE, according to (4.1). Next, making use of the mechanical-geometrical analogy of A. L. GOLDENVEIZER, formally determined complex and conjugate complex forces (6.3) are introduced for which the final equations (6.4)-(6.7) are obtained. Thermal influences are taken into consideration assuming linear temperature variability along the normal to the middle surface of the shell [equation (2.1)]. It is also assumed that the heat flow is steady.

Two groups of sets of equations are considered. The approximate equations (7.1) are satisfied for the first group. For the other one, equation (7.5) is satisfied additionally. By rejecting small terms a simpler form of the equations of the theory of isotropic shells is obtained, (7.9) and (7.11).

Shells of revolution are considered in Sec. 8. As a particular case, the equation of E. REISSNER is obtained.

In every particular case of orthotropy, the auxiliary functions of Table 1 should be examined. The functions satisfying the relations (7.1) or (7.5) should be rejected thus obtaining a simplified fundamental set of equations. As an example is analysed a toroidal shell (cf. [1]) working in a quasi axially symmetric state. The aim of the present paper is to give equations that can be used to solve problems similar to those of [1], but for arbitrary states of stress. The equations obtained are also valid for ribbed shells (Fig. 3a) or shells with natural orthotropy. It is also possible to use them to solve problems of shells with variable rigidity (Fig. 3a).

Two groups of sets of equations are considered. The approximate equations (7.1) are satisfied for the first group. For the other one, equation (7.5) is satisfied additionally. By rejecting small terms a simpler form of the equations of the theory of isotropic shells is obtained, (7.9) and (7.11).

Shells of revolution are considered in Sec. 8. As a particular case, the equation of E. REISSNER is obtained.

In every particular case of orthotropy, the auxiliary functions of Table 1 should be examined. The functions satisfying the relations (7.1) or (7.5) should be rejected thus obtaining a simplified fundamental set of equations. As an example is analysed a toroidal shell (cf. [1]) working in a quasi axially symmetric state. The aim of the present paper is to give equations that can be used to solve problems similar to those of [1], but for arbitrary states of stress. The equations obtained are also valid for ribbed shells (Fig. 3a) or shells with natural orthotropy. It is also possible to use them to solve problems of shells with variable rigidity (Fig. 3a).

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Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).

#### References

S. BORKOWSKI, Zginanie luków falistych, Rozpr. Inzyn., 1, 12 (1964), 137.

A. B. H. LOVE, A Treatise on the Mathematical Theory of Elasticity, Cambridge 1934.

E. REISSNER, Rotationally symmetric problems in the theory of thin elastic shells, Proc. Th. U.S. Nat. Congr. Appl. Mech., 1958, 51.

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