Summary
A method of determining the thermal stresses in a flat rectangular isotropic plate of constant thickness with arbitrary temperature distribution in the plane of the plate and with no variation in temperature through the thickness is presented. The thermal stress have been obtained in terms of Fourier series and integrals that satisfy the differential equation and the boundary conditions. Several examples have been presented to show the application of the method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- x, y :
-
rectangular coordinates
- σ x, σy :
-
direct stresses
- τ xy :
-
shear stress
- ø :
-
Airy's stress function
- E :
-
Young's modulus of elasticity
- α :
-
coefficient of thermal expansion
- T :
-
temperature
- ▽ 2 :
-
Laplace operator: \( - \left( {\frac{{\partial ^2 }}{{\partial x^2 }} + \frac{{\partial ^2 }}{{\partial y^2 }}} \right)\)
- ▽ 4 :
-
biharmonic operator \(\left( {\frac{{\partial ^4 }}{{\partial x^4 }} + 2\frac{{\partial ^4 }}{{\partial x^2 \partial y^2 }} + \frac{{\partial ^4 }}{{\partial y^4 }}} \right)\)
- 2a :
-
length of the plate
- 2b :
-
width of the plate
- a/b :
-
aspect ratio
- a mr, bms, cnr, dns :
-
Fourier coefficients defined in equation (6)
- α m=mπ/a m=1, 2, 3, ...:
-
β n=nπ/2a n=1, 3, 5, ...
- γ r=rπ/b r=1, 2, 3, ...:
-
δ s=sπ/2b s=1, 3, 5, ...
- A m, Bm, Cn, Dn, Er, Fr, Gs, Hs :
-
Fourier coefficients
- K rand L s :
-
Fourier coefficients defined in equation (20)
- σ∞ :
-
direct stress at infinity
- T 1(x, y):
-
temperature distribution symmetrical in x and y
- T 2(x, y):
-
temperature distribution symmetrical in x and antisymmetrical in y
- T 3(x, y):
-
temperature distribution antisymmetrical in x and symmetrical in y
- T 4(x, y):
-
temperature distribution antisymmetrical in x and y
References
Heldenfels, R. R. and W. M. Roberts, Experimental and theoretical determination of thermal stresses in flat plates, NACA TN 2769, Aug. 1952.
Horvay, G., Thermal stresses in rectangular strips-I, Proc. of Second U.S. National congress of Applied Mechanics, ASME, p. 313–322, 1954.
Horvay, G. and J. S. Born, J. Math. and Phys. 33 (1955) 360.
Born, J. S. and G. Horvay, J. Appl. Mech. 22 (1955) 401.
Argyris, J. H. and S. Kelsey, Aircraft Engineering 26 (1954) 413.
Singer, J., M. Anliker and S. Lederman, Thermal stresses and thermal buckling, Wright Air development centre, Technical report 57–69, Polytechnic Institute of Brooklyn, April 1957.
Horvay, G., J. Appl. Mech. 20 (1953) 87.
Mandelson, A. and N. Hirschberg, Analysis of elastic thermal stress in thin plate with spanwise and chordwise variation of temperature and thickness, NACA TN-3778, Nov. 1956.
Klöppel, K. and W. Schönbach, Der Stahlbau 27 (1958) 122.
Przemieniecki, J. S., Aeronautics Quart. 10 (1959) 65.
Pickett, G., J. Appl. Mech. 11 (1944) A176.
Girkmann, K., Flächentragwerke, 5th Edition, Springer Verlag, Vienna, 1959.
Teodorescu, P. P., Probleme plane in Theoria Elasticitati, Vol. 1, (Rumania, Editura Academiie Republicii Populare, 1961), p. 558.
Sundara Raja Iyengar, K. T. and K. Chandrashekhara, Brit. J. Appl. Phys. 13 (1962) 501.
Sundara Raja Iyengar, K. T., Öster. Ing. Archiv 16 (1962) 185.
Sundara Raja Iyengar, K. T., Der spannungzustand in einem elastischen Halbstreifen und seine technischen anwendungen, Dr-Ing. Thesis, Hannover, 1960.
Timoshenko, S. P. and J. N. Goodier, Theory of Elasticity, p. 401, Second edition, McGraw Hill, New York, 1951.
Gerard, G., Progress in Photothermoelasticity in “PHOTOELASTICITY” Proc. International Symposium (Pergamon Press 1963), p. 88.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Sundara Raja Iyengar, K.T., Chandrashekhara, K. Thermal stresses in rectangular plates. Appl. sci. Res. 15, 141–160 (1966). https://doi.org/10.1007/BF00411552
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00411552