Thermal Stresses in Uniform Bars:

In this section we will discuss the problem of thermal stresses in bars of uniform cross- section while they are partially or totally restrained against change in length because of temperature changes. As the area of cross-section is the similar over the total length, the stress shall be the same over each cross-section.

Fully Restrained Bar

Let us assume a bar of uniform cross-sectional area A & length L and constrained at both ends as illustrated in Figure (a) and (b).

(a)                                                                   (b)

Figure

Let us enhance the temperature of the bar by ΔT. If the bar were free to expand, it would have enhanced to a length L + ΔL as in Figure (b). However, this enhance in length, ΔL, is completely restrained. Therefore, the restraining force, P, shall be equivalent to a force which would have generated the same change in length. Following these arguments

ΔL = αL ΔT = PL / AE   or P = AE α ΔT

Thus,   Thermal Strain = ΔL/ L = α ΔT

Thermal Stress in the bar = Strain × E = E α ΔT

Which is the same as P / A . As the force P is compressive, the stress developed is also compressive in nature.