Engineering Transactions,

**7**, 4, pp. 559-568, 1959### Pewne Zagadnienie Termosprężyste dla Ortotropowej Tarczy Prostokątnej

The object of the present paper is to determine the thermal stresses in a rectangular plate whose edges are free from stresses. It is assumed that the plate is both elastically and thermally orthotropic. The temperature at the edges is known and distributed in an arbitrary manner.

It is assumed that the heat flows across the surface of the plate at a rate proportional to the temperature difference between the plate and the ambient medium, at the point considered. Using the finite Fourier transformation, a general solution is obtained for the thermal problem (1.5) and (1.6). The case of linearly variable temperature along the edges is treated in detail. The calculation of the stress reduces in the case considered to solving the differential equation (2.5) assuming that the stress function and its derivatives in the normal direction are equal to zero along the entire edge. The general solution obtained by means of the Fourier transformation has the form determined by the Eq. (3.2), where the coefficients Em, Fm, Gn, Hn in the Eqs. (3.1) should be determined from the boundary conditions of the problem. In the case of double symmetry, the calculation of these coefficients reduces to the solution of two infinite systems of equations, (3.5.1) and (3.5.2). The solution is obtained in the case of very intense heat absorption (3.8). This solution is of for the remaining cases, and determines the possibility of obtaining the correct solution of the equations (3.5.1) and (3.5.2). In the case of weak heat absorption the argument is analogous to that of calculation of the bending moments for a clamped plate uniformly loaded, (3.13).

A numerical example is given.

It is assumed that the heat flows across the surface of the plate at a rate proportional to the temperature difference between the plate and the ambient medium, at the point considered. Using the finite Fourier transformation, a general solution is obtained for the thermal problem (1.5) and (1.6). The case of linearly variable temperature along the edges is treated in detail. The calculation of the stress reduces in the case considered to solving the differential equation (2.5) assuming that the stress function and its derivatives in the normal direction are equal to zero along the entire edge. The general solution obtained by means of the Fourier transformation has the form determined by the Eq. (3.2), where the coefficients Em, Fm, Gn, Hn in the Eqs. (3.1) should be determined from the boundary conditions of the problem. In the case of double symmetry, the calculation of these coefficients reduces to the solution of two infinite systems of equations, (3.5.1) and (3.5.2). The solution is obtained in the case of very intense heat absorption (3.8). This solution is of for the remaining cases, and determines the possibility of obtaining the correct solution of the equations (3.5.1) and (3.5.2). In the case of weak heat absorption the argument is analogous to that of calculation of the bending moments for a clamped plate uniformly loaded, (3.13).

A numerical example is given.

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

#### References

E. Melan, H. Parkus, Wärmespannungen infolge stationärer Temperaturfelder, Wien 1953.

P. Wilde, The General Solution for a Rectangular Orthotropic Plate Expressed by Double Trigonometric Series, Arch. Mech, stos., (1958).

J. Mossakowski, The State of Stress and Displacement in a Thin Anisotropic Plate Due to a Concentrated Source of Heat, Arch. Mech. stos., (1957).

[in Russian]

W. Nowacki, Drgania własne i wyboczenie płyty prostokątnej na obwodzie całkowicie utwierzonej, Arch. Mech. stos., 4, 6 (1954).