# Finding Increase in Temperature for Two Rods

## Homework Statement

A steel rod and an aluminum rod of equal length and diameter are placed end to end and secured so that they cannot flex. The rods are heated to the same final temperature, and the steel is found to increase in length by one-tenth of a percent. If the total length of the rods together remains constant, find the increase in temperature for the rods and the mutual stress on the rods.

Ans: 375 oC , 700 MPa

## Homework Equations

(RATE OF HEAT FLOW)
H = ΔT/R R = L/KA

(LINEAR HEAT EXPANSION)
ΔL = L0αΔT[/B]

(STRESS)
F/A = -YαΔT

## The Attempt at a Solution

I first solved for ΔT from the linear heat of expansion equation...

Lf(steel) - li(steel) = li(steel)*α*ΔT

1.001*Li(steel) - li(steel) = li(steel)*α*ΔT

0.001 = α*ΔT where α = 1.2 *10^-5

ΔT = 83.33 C

After I find this, I though I would then plug this into the Stress equation for a given metal since they have the same change in temperature. Though this is obviously not the correct answer.

Am I missing something here?

You need to find the change in the length of aluminium rod. Do you expect it to be the same as that of steel if they are not secured together? First calculate and then visualize.

SteamKing
Staff Emeritus
Homework Helper

## Homework Statement

A steel rod and an aluminum rod of equal length and diameter are placed end to end and secured so that they cannot flex. The rods are heated to the same final temperature, and the steel is found to increase in length by one-tenth of a percent. If the total length of the rods together remains constant, find the increase in temperature for the rods and the mutual stress on the rods.

Ans: 375 oC , 700 MPa

## Homework Equations

(RATE OF HEAT FLOW)
H = ΔT/R R = L/KA

(LINEAR HEAT EXPANSION)
ΔL = L0αΔT[/B]

(STRESS)
F/A = -YαΔT

## The Attempt at a Solution

I first solved for ΔT from the linear heat of expansion equation...

Lf(steel) - li(steel) = li(steel)*α*ΔT

1.001*Li(steel) - li(steel) = li(steel)*α*ΔT

0.001 = α*ΔT where α = 1.2 *10^-5

ΔT = 83.33 C

After I find this, I though I would then plug this into the Stress equation for a given metal since they have the same change in temperature. Though this is obviously not the correct answer.

Am I missing something here?
All I see are calculations for the steel rod. What happened to the aluminum rod?

Also, you need a formula which relates mechanical stress to axial deflection in a rod.

haruspex
Homework Helper
Gold Member

## Homework Statement

A steel rod and an aluminum rod of equal length and diameter are placed end to end and secured so that they cannot flex. The rods are heated to the same final temperature, and the steel is found to increase in length by one-tenth of a percent. If the total length of the rods together remains constant, find the increase in temperature for the rods and the mutual stress on the rods.

Am I missing something here?
My guess is that you are missing a crucial piece of the statement of the question. I believe the idea is that the rods are clamped in such a way that overall length cannot change, as well as not being able to flex. Thus, the tendency to expand due to the raised temperature is exactly compensated by the compression of the rods due to the end loading.
You will need the Young's modulus of the materials, though it gets a bit messy since that may depend on the temperature, which you do not know yet. Perhaps it doesn't make enough difference to matter.