By D. J. Johns
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24) derived in Chapter 2 and is of no further concern in the present bending problem. Its solution leads only to membrane stresses which do not influence the bending behaviour of the plate for small deflexions. 20) I t is clear, therefore, that eqns. 20) a r e equivalent to their isothermal forms with a normal loading of p — V 2 M r / ( l — v). Therefore all the known techniques for isothermal problems m a y be used also for the corresponding thermoelastic problems. I n the derivation above it was implicitly assumed that T = T^xv^, but, when t h e temperature varies arbitrarily in the thickness direction of the plate (z) as well as over the planform (x, y) then 52 THERMAL STRESS ANALYSES ideally the general equations of three-dimensional thermoelasticity should be used.
3, results obtained from plane stress analyses m a y be applied directly to problems of plane strain by substituting the appropriate parameters of eqn. 17). 1. 1. e. 2) This equation has been derived assuming zero body forces, a n d leads to the following forms of equation in two-dimensional problems. 2) becomes 33 THERMAL STRESS ANALYSES 34 2 2 θφ 1 + v _ 3φ — 2 + — 2 = — — aT £# djy . 3) with φ, u and y all independent of z ; and the corresponding plane stress equation is ^ + ! e. 5) gives the full solution for localized heating in an infinite plate where the deformation and stress must tend to zero at infinity.
5. One-Dimensional Thermal Stresses in a Thin Rectangular Slab T h e problem of determining the correct thermal stress distribu tions for general two-dimensional temperature distributions in an unrestrained thin slab of rectangular planform is of considerable interest and many analyses on this topic have now been published. Some of these will be examined in Chapter 3, but for present pur poses the special case of a one-dimensional temperature distribution will be considered here. T h e solution presented forms a convenient basis on which the more general two-dimensional solutions may be derived.