Engineering Transactions,

**1**, -, pp. 3-17, 1953### Płyty na Sprężystym Podłożu

In this paper finite differences are used to determine deformations of plates on elastic foundations. The difference equation of plate presented in Fig. 1 is given by Eq. (5), where w denotes the deflection of the plate, D its flexural ridigity, k unit reaction of the foundation (the reaction being supposed to be in the sense opposite to that of the z-axis), q unit load of the plate. Expressing the differences by the deflections of individual points according to the designations in Fig. 2 (the plate being divided in elementary squares Az = Ay) Eq. (7) is obtained, where qu ist the unit external load of the elementary area corresponding to the point k, and km the unit reaction of the foundation for the elementary area corresponding to the point k.

A" number of equations (7) are established, equal to that of unknown deflections of the plate. To have the number of unknown quantities equal to that of equations (which are linear), theoretical points of deflection outside the plate are expressed by the deflections of real points. The manner in which these relations are found is given by Eqs. (9.1)-(12). Thus we obtain finally a system of linear equations, the number of unknown deflections being equal to that of the established equations.

Egs. (7) are based on Win kler's assumptions, which means that the continuity of the foundation is not taken into account and the

deflection of the foundation is assumed to be proportional to the load. Further it is assumed that the foundation can be subjected to compressive and tensile stresses as well. These assumptions differ considerably from the conditions of really existing foundations, especially when applied to the ground. Furthermore the author has calculated the deformation of a rectangular plate loaded in the central part, assuming that the foundation can be subjected to compressive stresses only, and permitting free separation of the plate from the foundation. The method used consists in successive suppressing the tensile stresses at points where they exist. After several approximations we can obtain the real surface of deflection of a plate pressed against an elastic foundation. The corresponding graphs for discussed plate, for k = 1 kG/cm, are shown in Figs 5a, 5b, 5c and 5d. The calculation has been repeated for the modulus ten times greater (k = 10 kG/cm). The corresponding graphs are shown in Figs 9a, 9b and 9c.

The method presented permits to investigate different kinds of plates on elastic foundations of any kind (e. g. characterized by a variable modulus of elasticity). In particular it permits to calculate the deflection of a plate on an elastic foundation which can be subjected to compressive stresses only.

The method can be used, for instance, to calculate foundation beams

and plates.

A" number of equations (7) are established, equal to that of unknown deflections of the plate. To have the number of unknown quantities equal to that of equations (which are linear), theoretical points of deflection outside the plate are expressed by the deflections of real points. The manner in which these relations are found is given by Eqs. (9.1)-(12). Thus we obtain finally a system of linear equations, the number of unknown deflections being equal to that of the established equations.

Egs. (7) are based on Win kler's assumptions, which means that the continuity of the foundation is not taken into account and the

deflection of the foundation is assumed to be proportional to the load. Further it is assumed that the foundation can be subjected to compressive and tensile stresses as well. These assumptions differ considerably from the conditions of really existing foundations, especially when applied to the ground. Furthermore the author has calculated the deformation of a rectangular plate loaded in the central part, assuming that the foundation can be subjected to compressive stresses only, and permitting free separation of the plate from the foundation. The method used consists in successive suppressing the tensile stresses at points where they exist. After several approximations we can obtain the real surface of deflection of a plate pressed against an elastic foundation. The corresponding graphs for discussed plate, for k = 1 kG/cm, are shown in Figs 5a, 5b, 5c and 5d. The calculation has been repeated for the modulus ten times greater (k = 10 kG/cm). The corresponding graphs are shown in Figs 9a, 9b and 9c.

The method presented permits to investigate different kinds of plates on elastic foundations of any kind (e. g. characterized by a variable modulus of elasticity). In particular it permits to calculate the deflection of a plate on an elastic foundation which can be subjected to compressive stresses only.

The method can be used, for instance, to calculate foundation beams

and plates.

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

#### References

Woprosy stroitielnoj miechaniki, pod redakcja J. W. Urbana, Moskwa 1951.

K. Beyer, Die Statik im Eisenbetonbau, cz. I i II, Berlin 1933-1934.

W. Wierzbicki, Obliczenie płyty wspornikowej za pomocą równań różnicowych, Warszawa 1934.