Two circular rods, both of length L and having the same diameter, are placed end to end between rigid supports with no initial stress in the rods. The coefficient of linear expansion and Young's modulus for rod A are αA and YA respectively; those for rod B are αB and YB respectively. Both rods are "normal" materials with α>0. The temperature of the rods is now raised by ΔT. What is the stress F/A in the rods after heating?Express the stress in terms of ?A, ?B, YA, YB, and ?T.

Explanation:

From the information given.

The formula for Youngs modulus(Y) can be illustrated as follows:

where;

F = force

A = area

ΔL / L = the fraction of the change that occurred in the length.

If we make ΔL / L the subject of the formula, the above expression can be re-written as:

However, the total fractional change that occurred in the length can be computed as:

where;

The area of both rods is said to be equal as well.

Thus;

----- Let this be equation (1)

Recall that:

in which

ΔL = change in length

L = the original length of rod before extension

∝ = coefficient of linear expansion

ΔT = temperature change

Similarly:

The total fractional change in the length is:

The sum of both the thermal and compressive stress of the combined rod is equal to the fractional change.

Mathematically:

where;

∴

From equation (1) & (2)

Making the stress the subject, we get;

the displacement produced by the body.  

the amount of force subjected to the body.  

the angle between the direction of force and displacement