Engineering Transactions, 10, 1, pp. 117-175, 1962

Stan Naprężęnia i Przemieszczenia w Grubych Płytach, Wywołany Działaniem Nieustalonego Pola Temperatury

W. Derski
Politechnika Łódźka

The paper is composed of three parts. The first is devoted to a circular plate, the second - to a rectangular plate and the third contains a discussion of the results and a numerical example. The aim of the paper is the determination of the state of stress and displacement in a thick plate, due to the action of a non-steady-state temperature field. The thermal action is determined by means of a prescribed temperature distribution. Three cases are considered. and the character of considered. In the first the region the thermal action is symmetric in relation to the middle plane of the plate Z = 0. In the second - the temperature field is determined by 'boundary conditions asymmetric in relation to the plane z = 0. In the third case the region of thermal action lies in the plane z = h only. In each of these three cases are discussed physically possible thermal boundary conditions on the part of the boundary that does not constitute the region of thermal action. To determine the stress and displacement components the potential of thermoelastic displacement is used as usually. Mechanical boundary conditions are satisfied in an accurate manner on the surfaces Z w -Eh and in the integral manner on the lateral surfaces. In the case of rectangular plates even an approximate satisfaction of the boundary conditions on the lateral surfaces leads to an infinite system of differential equations.
The solutions obtained have the form of simple double and triple trigonometric series converging slowly for small values of time. To overcome this difficulty an asymptotic solutions are given valid for small values of time.
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Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).


H. BATEMAN, Tables of Integral Transforms, Vol. 1, McGraw-Hill, New York 1954.

H. C. CARSLAW and J. C. JAEGER, Conduction of Heat in Solids, Oxford Press, 1950.

W. DERSKI, Nieustalone naprężenia cieplne w nieskończonej rurze grubościennej, Arch. Inżyn.

Lądowej, 2, 4 (1958).

W. DERSKI, Stan naprężenia przemieszczenia W grubej płycie kołowej, wywołany działaniem nieustalonego pola temperatury, Rozpr. Inz., 1, 9 (1961).

W. DERSKI, The State of Stress and Displacement in a Thick Circular Plate due to a Non- Stationary Temperature Field, Bull. Acad. Polon. Sci. Sér. Sci. Tech., 1, 7 (1959).

[in Russian]

B. E. GATEWOOD, Thermal Stresses, McGraw-Hill, New York 1957.

K. GIRKMAN, Dźwigary powierzchniowe (tłum. z niem.)., Warszawa 1958 Arkady.

S. GOLDSTEIN, Proc. Lond. Math. Soc., ser. 2, Vol. 34, 1939.

J. N. GOODIER, On the Integration of Thermo-Elastic Equations, Phil. Mag., Vol. 23, 23, 1937.

A. GRAY and Mc ROBERT, A Treatise on Bessel Functions, Londyn 1952.

M. T. HUBER, Teoria sprężystości, t. 4 i 5, Warszawa PWN 1954.

K. KNOPP, Szeregi nieskończone (tłum. z niem.), Warszawa 1956 PWN.

A. E. H. LOVE, A. Treatise on the Mathematical Theory of Elasticity, Cambridge Univ. Press, 1927.

N. W. McLACHLAN, Bessel Functions, Oxford Press, 1955.

E. MELAN, Spannungen infolge nichtstationärer Temperaturfelder, Österreich, Ing.-Archiv, 2-3, 9.1955.

E. MELAN, H. PARKUS, Wärmespannungen infoge stationürer Temperaturfelder, Wieden 1953.

A. NÁDAI, Elastische Platten, Berlin 1925.

W. NOWACKI, Stan naprężenia w grubej płycie kołowej, wywołany działaniem pola temperatury, AIL, 4, 3 (1957).

W. NOWACKI, Zagadnienia termosprężystości, Warszawa 1960 PWN,

H. PARKUS, Instationäre Wärmespannungen, Wieden 1959.

I. N. SNEDDON, Fourier Transforms, McGraw-Hill 1951.

A. TIMPE, Achsensymetrische Deformation von Umdrehungskörpern, ZAMM, 1924.

G. TOLSTOW, Szeregi Fouriera (tłum. z ros.), Warszawa 1954 PWN.

C. J. TRANTER, Integral Transforms in Mathematical Phisics, Londyn 1950.

K. W. WAGNER, Rachunek operatorowy i przekształcenie Laplace'a (tłum. z niem.), Warszawa 1960 PWN.

G. N. WATSON, The Theory of Bessel Functions, Cambridge Univ, Press. 1944.