Engineering Transactions, 17, 1, pp. 3-21, 1969

### Pewne rozwiązanie szczególne dla wstępnie sprężonej powłoki tekstylnej

M. Wizmur
Politechnika Gdańska
Poland

The present paper is concerned with the relation between the stress and the distortion of an arbitrary membrane. The material of the shell is anisotropic. The analysis is performed in the tens or notation. The membrane which is initially plane and orthotropic is transformed into an undevelopable surface. The consequence of this is that orthotropy is replaced by curvilinear anisotropy. The displacement tensor is completed by the distortion tensor. The stress-strain relations
are expressed in the generalized coordinates of the surface, and are assumed to be linear. The components of the displacement vector at the middle Surface are infinitesimally small whilst its normal component is large. This leads to non-linear equations in w. The equations of equilibrium for a membrane are transformed into equations of displacement. Next, the hyperbolic paraboloid is considering showing on the grounds of the Tshebyshew-net theory and by assuming that the shell is shallow, that the equation obtained can be written in a simpler form. These equations are solved the Galerkin method thus finding the relation required. The theoretical considerations are illustrated by a numerical example concerning a textile membrane of 4.550 sq.m. designed as a covering for the Sopot Open Air Opera. The results show how the membrane forces vary under conditions
of distortion.

Full Text: PDF

#### References

A. E. GREEN and W. ZERNA, Theoretical Elasticity, Oxford 1963.

Cz. WOZNIAK, Nieliniowa teoria powłok, PWN, Warszawa 1966.

P. M. NAGHDI, R.P. NORDGREN, On the nonlinear theory of elastic shells under the Kirchhoff hypothesis, Quart. Appl. Math., 1, 1963.

E. BIBLEWICZ, O geometrycznie nieliniowej teorii powłok lepkosprężystych, Rozpr. Inzyn., 4, 14 (1966).

П. А. Кидчевский, Основы аналитической механики оболочек, Москва 1963.

С. Т. Лехнищшж, Теория упругости анизотропного тела, Москва 1950.

T. ŻYŁYŃSKI, Metrologia włókiennictwa, WPL, Warszawa 1965.

R. MEREDITH, J. W. HEARLE, Physical Methods of Investigating Textiles, N. York 1959.

Ф. В. Каган, Основы теории поверхностей, Москва 1948.

P. WILDE, Some problems of the theory of membranes formed by inextensibles cords, Arch. Mech. Sios., 4, 18 (1966).

M. WIZMUR, Siły wewnętrzne wywołane dystorsją w paraboloidzie hiperbolicznej, rozpatrywanej w stanie bez zgięciowym, Arch. Inżyn. Lad., 2, 14 (1968).

Ф. OTTO, P. Тростел, Пневматические строительные конструкции, Москва 1967.