How Much Force Is Needed to Prevent a Steel I-Beam from Expanding When Heated?

[travisccook]

Homework Statement

a)A steel I-beam is 30m long, and its cross-sectional area is .001 square meters. If its temperature is increased by 80C what is its change in length?

Ysteel = 20x10^10 N/m^2
(Alpha)steel = 1.2x10^-5/C

b) How much force should be applied to keep it from expanding?

Homework Equations

(Delta)L = (Alpha)(Delta)T * L

The Attempt at a Solution

We've solved (we think) (Delta)L to be .0288.
[(Delta)L = (1.2*10^-5)(80)(30) = .0288

Our question is how to approach b part. We're unable to find much of anything in our notes,text, etc... related to these kinds of forces of expansion!

We're somewhat confident about our solution to a part, though any observations on our mistakes would be appreciated!

Thanks!

 

[AvroArrow]

For part b, the force required to keep it from expanding can be found using F = YAΔL, where Y is Young's modulus, A is the cross-sectional area of the beam, and ΔL is the change in length. Therefore, the force required to keep the beam from expanding is F = (20x10^10 N/m^2)(.001 m^2)(.0288 m) = 5.76x10^7 N.

 

[ogb p]

Your solution for the change in length (Delta)L seems to be correct. To approach part b, you need to consider the concept of thermal stress. When a material experiences a change in temperature, it expands or contracts, which can create internal forces known as thermal stress. In order to prevent the steel I-beam from expanding, an external force must be applied to counteract the thermal stress. This external force is known as the "restoring force" and it should be equal in magnitude and opposite in direction to the thermal stress.

To calculate the force needed to keep the steel I-beam from expanding, you can use the equation:

F = Y * A * (Delta)L

Where:
F = force
Y = Young's modulus of steel (given as 20x10^10 N/m^2)
A = cross-sectional area (.001 square meters)
(Delta)L = change in length (.0288 meters)

Substituting these values into the equation, you should get a force of approximately 576,000 N. This is the amount of force that should be applied to the steel I-beam to keep it from expanding when its temperature is increased by 80C.

It is important to note that this calculation assumes that the steel I-beam is perfectly constrained and cannot expand in any direction. In real life, there may be other factors at play that could affect the amount of force needed to prevent expansion, such as the type of support or the type of connection between the I-beam and other structures.