**9**, 4, pp. 641-708, 1961

### Badanie Wyboczenia przy Pełzaniu Płyt Kołowych w Zakresie Małych i Dużych Ugięć

and A. R. RSHANICYN. In a short period of time, a large number of papers have been published, and different methods for analysis of creep buckling, mainly of columns, have been presented. Recently, J. A. H. HULT, N. J. HOFF, I. FINNIE and W. R. HELLER have published methodical reviews of the problem. At present, the topic of creep buckling has become of interest in connection with the rapid development of modern technique as concerning engineering materials subjected to high loads and high temperatures.

There is little published information on creep buckling of plates. As far as the present author. knows, there is no published paper on creep buckling of plates in the range of large deflections. This problem seems to be very important from the point of view of practical application of the theory.

In the present paper, the problem of creep buckling has been established quite differently from the approach in other papers dealing with columns and plates. Instead of a prescribed load, we assume as given the displacement conditions or strictly speaking, the deformation rates on the: boundary. Thus, the deformation of the plate has a forced character and stresses are variable in. time. Such an approach is in agreement with real conditions for plates in steady-state creep, and facilitates the physical interpretation of the results obtained. It should be emphasized that the quantities assumed as essential are usually measured experimentally. In the first general part of the present paper, the problem and assumptions are stated for circular-plates with freely clamped edges. The physical law for an elastic and creeping incompressible material is assumed to be […] where S; is strain rate component, oik stress deviator component, G-shear modulus and t = b(o)-fluidity, being generally a function of stress intensity G;. The geometrical relations for deformation rates in the middle plane of the plate are in general:

[…].

The assumptions made, the general equations for circular plates are derived, valid for both elastic and creep deformation state, The system of three equations equation of equilibrium of stresses in the middle plane of the plate, equation of equilibrium of moments, and the conditions of compatibility, together with boundary conditions, determine deflection and total strain rates or creep strain rates (if elastic deformations are disregarded). The analysis is restricted to prediction of deflection increase in time. The condition: deflection rate equal to infinity gives the theoretically critical time point of creep buckling. In investigation, the following manner of procedure is applied: The shape of deflection function w and the form of displacement function u in the middle plane of the plate are assumed directly as functions of time. The above assumptions make it possible to satisfy the condition of compatibility. To satisfy the equations of equilibrium, the Galerkin's integral condition is applied. Thus, the deflection at the center of the plate as function of time is found from ordinary non- linear differential equations.

In the second part of the present paper, a geometrically and physically linear case is discussed when the plate undergoes compression with constant velocity on the boundary, Some particular cases of velocity proportional to deflection produced, and reciprocal of deflection, are considered and corresponding stresses found. As the second approach, the geometrical nonlinearity deflection-time relations evaluated. The solutions of non-linear of deformation rates is assumed and differential equations are obtained by using the perturbation method. When elastic deformations are disregarded, the solutions for both linear and non-linear cases are found. The influence of variability of stress intensity through the thickness of the plate is considered and discussesd. The behavior of the plate in time depends essentially on mechanical properties of the material, load, slenderness ratio and out-of-straight parameter. The influence of all these parameters, except the one out-of-straight, is expressed through the value of the coefficient *; Therefore ÷ = 0, for purely elastic plate and x = for purely creeping plate (as regards physical properties). It is found that theoretical critical time exists for small deflections of the plate if the deformation rate on the boundary is an increasing function of deflection (for 0 6% co). For the deformation rate constant in time, the theoretical critical time appears only in the small interval 0 <* < ½(3)°.

The critical time for a purely elastic plate is defined as the point of time at which the compression in the middle plane of the plate attains its critical value. There is no critical time, according to our definition, when large deflections are considered, even if the deformation rate on the boundary is an increasing function of deflection. In general, we conclude that critical time increases with increasing influence of creep. Some of the above conclusions are apparently in contradiction with general opinion. That can be explained, however, by the assumed scheme of loading of the plate. For small deflections of the plate, we do not obtain solutions valid for an initially flat form of the plate. In the case of large deflections, the solutions are valid also for an initially flat form of the plate, if only elastic deformations are taken into account. Thus, the last-mentioned conclusion does not hold for % = 00, The investigation of variability of stress intensity through the thickness of the plate is very difficult in the general case. The simplified analysis made in the case of small deflections, and for creep only, seems to show that a great influence on deflection increase is exerted when this variability occurs. Further investigation is necessary. The above theoretical and numerical results should be regarded critically, and the theoretical models of the present paper may constitute a basis for an evaluation of phenomena connected with the problem of creep buckling of circular plates.

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