# Simple beam - NON intermediate load



## Lily (Oct 3, 2010)

We can take problem 115 in the 2001 sample PE exam as example, we have a pulley that is located at the end of a shaft, the shaft is hold by 2 bearings, and the pulley is not between these bearings. the question is what is the maximum bending stress in the shaft. the answer given in the NCEES is Mr/I with M=Force applied by Pulley * distance to closest bearing from pulley. when I look at the deflexion for a simple beam with intermediate load the Moment given is Pab/L, but there are no equations when the load on the beam is outside the bearings. what's the basis of NCEES answer to this problem?

Thanks!


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## navyasw02 (Oct 3, 2010)

Lily said:


> We can take problem 115 in the 2001 sample PE exam as example, we have a pulley that is located at the end of a shaft, the shaft is hold by 2 bearings, and the pulley is not between these bearings. the question is what is the maximum bending stress in the shaft. the answer given in the NCEES is Mr/I with M=Force applied by Pulley * distance to closest bearing from pulley. when I look at the deflexion for a simple beam with intermediate load the Moment given is Pab/L, but there are no equations when the load on the beam is outside the bearings. what's the basis of NCEES answer to this problem?Thanks!


I'm not sure if I understand what you're asking, but you know that the sum of the moments from the right bearing is zero. Take either the left bearing reaction force times its distance or the pulley force times its distance from the right bearing and use that for the moment. The rest is just My/I


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## IL-SE (Oct 3, 2010)

I don't have the practice exam, so I don't know exactly what the question is.

I think the problem is in the first equation. Mr/I wouldn't give you the stress if it was force [lb]x distance [in]/second moment of inertia [in^4]...just looking at the units it would give you lb/in^3. I think the M is actually a moment and the equation would be My/I where M is the moment, y is the distance to the extreme fiber (d/2 if it's symmetric) and I is the second moment of inertia. The moment M would be Pab/L for a concentrated load.


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## Lily (Oct 5, 2010)

Sorry for not being very clear,

let's imagine this situation

shaft------left bearing-------shaft------right bearing------shaft-------pulley

so the pulley is at the extreme right of the shaft, the shaft is hold by 2 bearing. So the pulley is applying a load P, that is not between the bearings, but on the shaft outside the bearings on the extreme right.

The question is what is the maximum stress in the shaft.

I guess M=Pab/L cannot be applicable in this situation, because a = distance between the pulley and the left bearing, b = distance between the pulley and the left bearing = distance between bearings + distance between pulley and right bearing.

the answer in the book was: M=Force applied by the pulley multiplied by the distance to the right pulley. which is the moment formula for a load applied to a cantilever beam instead of a simply supported beam.

My question is not about the stress formula which I agree is My/I but about the moment formula P*a instead of P*a*b/L.

Hope it's clear this time


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## MA_PE (Oct 5, 2010)

Lily said:


> Sorry for not being very clear,let's imagine this situation
> 
> shaft------left bearing-------shaft------right bearing------shaft-------pulley
> 
> ...


but that portion of the shaft (from the right bearing to the pulley) is NOT a simply supported beam. It IS a cantilever from the single support (the right bearing) to the load applied at the unsupported end of the beam (the pulley).

For completeness in doing the problem, the stress also needs to be checked at the "simple beam" part of the shaft located beween the bearings. It is not immediately clear where the controlling stress is located on the shaft because the stress depends on the loads and spacings of the supports, but apparently it is controlled by the cantilevered end.

Hope that helps.


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## Lily (Oct 5, 2010)

I agree, they should have also checked the stress between the bearings, and make sure it's not higher than the stress outside the bearings applied by the pulley,

What I would have done is to draw the shear and moment diagrams for the whole shaft and find when it has a maximum value.

Thanks!


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## MA_PE (Oct 5, 2010)

but they already know the answer, and it must be controlled by the end section. The solution does not necessarily need to show all the work involved to get to the solution. If the simply supported section controlled they likely would not have included the cantilever part.


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## IL-SE (Oct 5, 2010)

For a cantilever w/backspan that has a load applied at the end of the cantilever, Mmax will always be at the support (Mmax=P*a) the moment between the supports is Pax/l and goes from Mmax at the right bearing to zero at the left support in a straight line.

Pab/l would only be appropriate for a load applied between the bearings where a is the distance from the left bearing to the load and b is the distance from the right bearing to the load.

Take a look at page 13, Fig 20 and 21: http://www.awc.org/pdf/DA6-BeamFormulas.pdf


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## Lily (Oct 5, 2010)

Thanks for your answers!

Figure 20 explains it all, thanks for the link IL-SE


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## IL-SE (Oct 5, 2010)

You're welcome. Glad we could get to the bottom of this one!

Good luck with the exam.


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