Engineering Transactions,

**10**, 4, pp. 733-756, 1962### Pewne Zagadnienie Zginania Pasma Płytowego

The engineer's theory of thin isotropic plates (the Sophie Germain theory) is used to analyse the problem of the infinite cantilever plate strip of uniform thickness h, the load on the free edge being variable according to (1). On the basis of the Refs, [1] and [2] it is assumed that the deformed middle surface can be represented by means of a Fourier integral (8) and the external load of the plate - by means of the integral (9).

Next, the Cauchy residue method is used for calculating the integrals in (13)-(16). The roots of the transcendental equation (17) are collated in Tab. 1. The accurate solution of the differential equation (2) with the boundary conditions (10) are obtained in the form of the equations (20)-(27). On the basis of the equations obtained the graphs Fig. 3-7 and the tables 3, 4 and 5 have been prepared by confining ourselves to the first three terms of the series (p = 1, 2, 3). The expressions obtained constituting products of two functions one of which is independent of x and the other of y, the auxiliary quantities contained in Tab. 2 enable us to perform numerical computations of which the results are collated in Tab. 2 for every b/a ration. The graphs Figs. 8 and 9 have been drawn by rejecting the series (p = 0).

The paper illustrates also the influence of the Poisson ratio V on the values of the section forces mxx, myy, mxy and the displacement w.

Next, the Cauchy residue method is used for calculating the integrals in (13)-(16). The roots of the transcendental equation (17) are collated in Tab. 1. The accurate solution of the differential equation (2) with the boundary conditions (10) are obtained in the form of the equations (20)-(27). On the basis of the equations obtained the graphs Fig. 3-7 and the tables 3, 4 and 5 have been prepared by confining ourselves to the first three terms of the series (p = 1, 2, 3). The expressions obtained constituting products of two functions one of which is independent of x and the other of y, the auxiliary quantities contained in Tab. 2 enable us to perform numerical computations of which the results are collated in Tab. 2 for every b/a ration. The graphs Figs. 8 and 9 have been drawn by rejecting the series (p = 0).

The paper illustrates also the influence of the Poisson ratio V on the values of the section forces mxx, myy, mxy and the displacement w.

**Full Text:**PDF

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

#### References

H. JUNG, Über eine Anwendung der Fourier-transformation in der Plattenstatik, Mathematische

Nachrichten, 6, 6 (1952), Seite 343.

W. NOWACKI, Zginanie płyt ciągłych nieskończenie długich, Arch.- Mech. Stos., 2, 1 (1949).

K. GIRKMANN, Dźwigary powierzchniowe, Arkady Warszawa 1957. F

W. NOWACKI, Zagadnienia termosprężystości, PWN Warszawa 1960.

F. LÖSCH, Siebenstellige Tafeln der elementaren transzendenten Funktionen, Springer-Verlag, Berlin-1954.

F. LETA, Teoria funkcji analitycznych, PWN Warszawa 1957.