Engineering Transactions,
1, -, pp. 3-39, 1953
Wyznaczenie Przybliżonej Wielkości Ugięcia Płyt na Podstawie Metody Ritza
The present paper is a development of the method used by the author in the paper Ref. [1], for approximate determination of deflection of plates.
The functions in Ritz's assumption (1.6) are chosen by confronting bending of plates with that of beams (examples of such procedure have been already given in the paper Ref. [4]). Using formally that analogy it is sufficient to take only one function (a polynome) to obtain a comparatively exact solution.
In the first part of the paper circular plates are investigated with circularly symmetrical supports and loads.
In Sec. 2 the solution is given for circular plates simply supported on the periphery and subjected to a load uniformly distributed over the whole surface (Fig. 2), or its central part (Fig. 3), or uniformly distributed along a concentric circle (Fig. 4), or linearly diminishing from the centre to the edge of the plate (Fig. 5).
In Sec. 3 circular plates are discussed, fixed along the periphery and
loaded in the same manner as in Sec. 2 (Figs 8, 9, 11 and 12).
Sec. 4 presents an investigation of circular plates simply supported or fixed with one concentric circular hole, loaded in various manners
(Figs 13, 14, 15, 16 and 17).
To calculate the deflections of circular cantilever plates (Sec. 5) the function Ritz's is taken in the form of (5.1), i. e. as for an uniformly loaded cantilever beam.
For more characteristic problems, presented in Figs 18 and 19, detailed solutions are given.
In the second part (beginning with Sec. 6) rectangular plates are
investigated, for which Ritz's assumption is expressed by Eq. (6.3). The problems solved concern Figs 21b, 22, 23, 24, 25, 26 and 27. A comparison with the solution obtained by means of the method of finite differences, [6], leads to the conclusion [problems (6.4) and (6.5)] that although the latter method is more exact, Ritz's method involves a more simple and quick calculation, presenting no major difficulties even in more complicated cases like of a rectangular plate with a hole, mentioned in [4].
The functions in Ritz's assumption (1.6) are chosen by confronting bending of plates with that of beams (examples of such procedure have been already given in the paper Ref. [4]). Using formally that analogy it is sufficient to take only one function (a polynome) to obtain a comparatively exact solution.
In the first part of the paper circular plates are investigated with circularly symmetrical supports and loads.
In Sec. 2 the solution is given for circular plates simply supported on the periphery and subjected to a load uniformly distributed over the whole surface (Fig. 2), or its central part (Fig. 3), or uniformly distributed along a concentric circle (Fig. 4), or linearly diminishing from the centre to the edge of the plate (Fig. 5).
In Sec. 3 circular plates are discussed, fixed along the periphery and
loaded in the same manner as in Sec. 2 (Figs 8, 9, 11 and 12).
Sec. 4 presents an investigation of circular plates simply supported or fixed with one concentric circular hole, loaded in various manners
(Figs 13, 14, 15, 16 and 17).
To calculate the deflections of circular cantilever plates (Sec. 5) the function Ritz's is taken in the form of (5.1), i. e. as for an uniformly loaded cantilever beam.
For more characteristic problems, presented in Figs 18 and 19, detailed solutions are given.
In the second part (beginning with Sec. 6) rectangular plates are
investigated, for which Ritz's assumption is expressed by Eq. (6.3). The problems solved concern Figs 21b, 22, 23, 24, 25, 26 and 27. A comparison with the solution obtained by means of the method of finite differences, [6], leads to the conclusion [problems (6.4) and (6.5)] that although the latter method is more exact, Ritz's method involves a more simple and quick calculation, presenting no major difficulties even in more complicated cases like of a rectangular plate with a hole, mentioned in [4].
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References
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