NCEES Practice Exam Question 112

I have the 2016 NCEES practice exam for Machine Design and Materials. Problem 112 shows a horizontal shaft that is cantilevered on the left. On the right, there is a spring between the shaft and a wall. The problem is to determine the "increase in force" due to a change in temperature. The young's modulus, cross-section, and coefficient of thermal expansion are given for the shaft and the spring constant is given for the spring.

My original solution for this looks something like this:



However the solution in the text begins with something like this:



I can see how Hooke's law can be used to relate the spring constant to the Young's Modulus:



But the rest of their solution makes no sense to me. Why does the RHS of their equation subtract the spring deflection from the thermal expansion?

I think if you sketch out the bar with its free length.  Then sketch a dotted length in for the thermal expansion of the free length.  Then add the spring force pushing back onto the bar back into its equilibrium length.  That should help you understand there are differences in the lengths.   

I sketched it out and derived the equation for you but the file size limit higher than what is allowed.  I took a picture of it, instead of scanning the paper.

Maybe tomorrow if you need it...

I've tried sketching this out as you suggested. I still don't understand the why the spring deflection is subtracted from the thermal expansion.

I laid this out in SMath and DWG to make it look nice.   The final formula I used is Hooke's Law for axial deformation the D=FL/AE....  You just need to interpret the right dimensions D and L and so forth.    Let the bar expand freely due to thermal expansion and then apply Hooke's Law using the Spring force.  

View attachment bar and spring.pdf

This one is a little better with added detail.

View attachment bar and spring.pdf

I don't have the full problem or diagram.  Is it a tension spring, perhaps?

Audi driver said:

I don't have the full problem or diagram.  Is it a tension spring, perhaps?

Click to expand...

I do not have that edition of the book either.  I think it is like the problem  #504 in the 2001 NCEES machine design problems.

MikeGlass1969 said:

I do not have that edition of the book either.  I think it is like the problem  #504 in the 2001 NCEES machine design problems.

Click to expand...

Ok, I had borrowed that problem set book so I don't have it either anymore.

Here ya go..

View attachment 2001 NCEES MDM 504.pdf

Ah hah, I had not accounted for the strain in the bar due to the force of the spring - an oversight on my part. Thank you for clearing this up for me.