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Elongation and Stress of a Member Due to Temperature

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Consider a homogenous rod AB of uniform cross section, which rests freely on a smooth horizontal surface as shown in Fig1. If the temperature of the rod is raised by ΔT, the rod elongates by an amount δ_T which is proportional to both the temperature change ΔT and the length L of the rod as shown in Fig2. Then the following equation is:

(Eq1)

δT = α(ΔT)L

α is a constant characteristic of the material, called the coefficient of thermal expansion. Since δ_T and L are both expressed in units of length, α represents a quantity per degree C, or per degree F, depending whether the temperature change is expressed in degrees Celsius or in degrees Fahrenheit.

With the deformation of δ_T must be associated a strain ε_T = δ_T/L. Recalling Eq1, it is concluded that:

(Eq2)

εT = αΔT

The strain ε_T is referred to as thermal strain, since it is caused by the change in temperature of the rod. In the case being considered here, there is no stress associated with the strain ε_T.

Now assume that the same rod AB of length L is placed between two fixed supports at a distance L from each other as shown in Fig3. Again, there is neither stress nor strain in this initial condition. If the temperature is raised by ΔT, the rod cannot elongate because of the restraints imposed on its ends; the elongation δ_T of the rod is thus zero. Since the rod is homogeneous and of uniform cross section, the strain ε_T at any point is ε_T = δ_T/L and, thus, also zero. However, the supports will exert equal and opposite forces P and P' on the rod after the temperature has been raised, to keep it from elongating as shown in Fig4. It thus follows that a state of stress (with no corresponding strain) is created in the rod.

As preparations are made to determine the stress σ created by the temperature change ΔT, it is observed that the problem to be solved is statically indeterminate. Therefore, the magnitude P of the reactions at the supports should first be computed from the condition that the elongation of the rod is zero. Using the superposition method described in the lesson Superposition Method, the rod is detached from its support B as shown in Fig5 and left to elongate freely as it undergoes the temperature change ΔT as shown in Fig6. According to Eq1, the corresponding elongation is:

δ_T = α(ΔT)L

Applying now to end B the force P representing the redundant reaction, and recalling Eq3 from the lesson Deformation Due To Axial Loading, a second deformation is obtained as shown in Fig7 and given as:

δP =

PL AE

Expressing that the total deformation δ must be zero:

δ = δT + δP = α(ΔT)L +

PL AE

= 0

from which it is concluded that:

P = −AEα(ΔT)

and that the stress in the rod due to the temperature change ΔT is:

(Eq3)

σ  = P A =  −Eα(ΔT)

It should be kept in mind that the result obtained here and the earlier remark regarding the absence of any strain in the rod apply only in the case of a homogeneous rod of uniform cross section. Any other problem involving a restrained structure undergoing a change in temperature must be analyzed on its own merits. However, the same general approach can be used; i.e., the deformation due to the temperature change and the deformation due to the redundant reaction can be considered separately and the solutions obtained can be superposed.