8-6 THERMAL EFFECTS

We examine in this section the asymmetric bending of a beam subject to temperature changes. Symmetric bending is then obtained as a special case. (For a more extensive treatment of this subject see [4, p. 307].) It is assumed that no mechanical loads are applied. Since we consider only small deformations and a linear stress-strain relationship, the principle of superposition is valid. Thus simultaneous mechanical and thermal effects can be analyzed separately and the resulting stresses and deformations added. The basis for our analysis is the stress-strain relation for a thermoelastic solid (see Chapter 4),

Following the procedure adopted in Section 8-2, the expression for normal stress due to temperature changes alone is now assumed to be

The unknown constants a _T, b _T, and c _T will be determined using the equilibrium equations, which now are

It is further assumed that as in Section 8-2, y and z are the principal centroidal axes of inertia. Thus with Eq. (8-231) we obtain the following results:

where

Consequently the thermal stress equals

and the corresponding strain follows from Eq. (8-229),

When the cross section and the change of the temperature are symmetric with respect to one of the principal axes, say Oz,

then M _Tz = 0, and the expression for the stress reduces to

When the temperature varies only along the x axis then, regardless of the geometry of the cross section,

Hence,