Why do you not shift the NA 7

[CorporalToe]

Hello I was doing a question in order to prep for the PE exam and I ran into this problem which I got wrong.

In this question since the cross section is being loaded axially plus a moment due to the eccentricity, that the neutral axis will shift from the centroid location. However in the solution they do not shift the NA why is this?

 

[driftLimiter]

The bending modulus S doesnt change when the NA changes.

 

[CorporalToe]

Why would it not change, isn't the section modulus S = I/c, where c is the distance from the NA to the outer most fiber that I am looking at?

 

[SWComposites]

No, the distance “e” in the above is measured from the axis around which the moment is defined.

 

[driftLimiter]

c is the distance from the centroidal axis to the extreme fiber of the bending section.

 

[phamENG]

And why is the neutral axis changing? Note that, while they might show what looks like rebar, they never call this a reinforced concrete column. It's just a rectangular column. Be careful not to read too much into these questions. They aren't as complicated as you want to think they are.

 

[CorporalToe]

In the PE Reference Handbook Section 1.6.7.2 they define c as distance from NA.

The NA axis is shifting due to a axial load being applied with a eccentricity.

 

[phamENG]

The NA axis is shifting due to a axial load being applied with a eccentricity.

And why does the eccentricity have anything to do with the location of the neutral axis?

 

[SWComposites]

Well, that handbook is wrong. Or confusing. What they should have said is “c is from the centroidal axis.”

And to prove it, think of simple superposition. Calc the extreme fiber stress separately for the axial and moment loads, then sum them. The sum should match the answer given in the first post.

 

[phamENG]

superposition

Whelp. That's what I was trying to lead you to, CorporalToe.

In general theory, yes, the neutral axis will shift because the neutral axis is the point with no tension or compressive stress. BUT...you get there using superposition. So you get the stress from pure bending (My/I) and add it to the stress from pure axial load (P/A). If you want to find the resulting point of zero stress, you can, but it would not really be useful for answering the question.

 

[CorporalToe]

The NA is where the stress in the cross section = 0. If we take the formula sigma = P/A - My/I and set it equal to 0

P/A - My/I = 0
M = Pe
then we can find that

y = I/(Ae)

This y would be where the NA axis is located with reference to the centroid. Because this is where sigma = 0, and the reason it is with reference to the centroid is because e is given with reference to the centroid.

 

[CorporalToe]

What is superposition?

 

[Celt83]

EDIT: Response is inaccurate/incorrect, see my later post


c is the extreme distance in any direction from the elastic neutral axis.

Your statement regarding P/A-My/I = 0 at the neutral axis is incorrect Axial stress P/A is constant over the cross section the elastic neutral axis is where the bending stress is 0.

 

[phamENG]

What is superposition?

The foundation upon which the practice of structural engineering is built.

To boil it down to this example:

If you have a beam in bending with no axial load, you will have flexural stress. If the same beam is actually a column and has axial load but no bending, you'll have axial stress. Now put both loads on that beam (a beam-column, if you want to be technical). Bending does not change the axial stress and axial load does not change the flexural stress, so they are additive. You can superimpose them on one another to get a total stress.​

Seriously, look that up. You won't survive long if you don't know what superposition is.

 

[canwesteng]

Celt nailed... I'll go against the grain and say your question does demonstrate a decent understanding of the term neutral axis (if a pretty poor understanding of section properties), but the working in the manual is a bit bizarre. c is really the distance from the geometric centroid to the extreme fiber, which is the neutral axis under bending. I'm struggling to think of when one would need to think of it as the neutral axis - maybe in a cracked concrete section, but you don't use S in that case. Ultimately, think of S as a geometric property independent of the load.

 

[BridgeSmith]

In the absence of defined material properties, we should not assume any, and therefore the answer is straightforward: P/A - M/S, with M = P*e. The position of the NA can be calculated, but it is irrelevant to the solution.

 

[CorporalToe]

Can you relax, I knew the concept just forgot the name of it. Your response isn't helpful at all. I was just confused due to the wording in the PE Reference Manual

 

[rb1957]

where does it state (rather than us assuming) that the load is on the CL ?

what are we to make of the circles ? re-bar ?? wouldn't that affect the solution ?

"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.

 

[Celt83]

I jumped the gun on my response and what I wrote is actually not correct.

As others stated c is the extreme distance in any direction from the axis of bending, for most situations this coincides with the centroidal axis.

The problem statement is pretty bare so make the assumption that it wants an elastic analysis:
Your statement regarding the position of the elastic neutral axis shifting with an applied axial load is correct. Your thoughts that this shift alters S or y in the formula for bending stress was incorrect.

Your formula for the equation of the y position of the elastic neutral axis is indeed correct.

 

[Eng16080]

I haven't read every response, but if we're calling the neutral axis the location of zero stress, then it definitely has to move as the eccentricity changes, otherwise equilibrium is not met. The explanation above by Celt83 shows this. In the problem solution, the dashed line isn't labeled as the neutral axis. It's probably intended to show the centroidal axis. Your assumption of it being the n.a. isn't correct.