And in doing so, you have just transferred it to moments about the centroid!
However, if you have a material like reinforced concrete or prestressed concrete where you are combining materials of different types, the plastic centroid and elastic centroids are not necessarily the same so you cannot assume the elastic centroid, you have to calculate the location of the plastic centroid and do your "transfer" as in your case, or moments about as the rest of us are calling it, in relation to the plastic centroid!
In doing what, exactly, have I "transfered" the moments to the centroid?
The beam can "feel" where I'm summing moments about?
How can so many engineers not get it?! This is one of the most fundamental concepts in statics. It does not matter AT ALL where you choose to sum moments. Equilibrium MUST be maintained no matter how you look at a problem! We choose to sum moments about certain axes for convenience, but ONLY for convenience. I can sum moments about the center stripe of Broadway Avenue and the answer will be EXACTLY the same.
I am just changing the axial load and moment to be a stress distribution. If you consider the forces on one side and the stress distribution on the other you will always get zero moment otherwise there is no equilibrium. You have to look at only the internal forces to get the internal forces.
So, in one way or another, everyone seems to be in agreement that it is on the conservative side not to include compression reinforcement in the moment capacity. Which makes sense if it's not yielding in the first place. If, however it was included most likey would not add much strenght to the situtaion but would increase ductility allowing for more tension steel if required.
Sorry Entry..EIT. We've sort of hijacked your thread.
Yeas. That's what we are saying.
StrEIT.. I'm not sure what we're arguing about. It seems to me that you have proven my point.
You are not really "changing" anything. What you are describing is the internal stresses that RESULT FROM the externally applied loads.
If you take your beam and cut a section somewhere along the length (and assuming your loads are applied at the end and self weight is neglected, the section cut may be anywhere along the length), then the resulting FBD must be in equilibrium. The resulting interior moments and compression result in the stress distribution you have described.
You may sum moments about any point you wish: centroid, PNA, my kitchen counter; the resulting "interior" moment will be the same. In this particular case, the internal moment is exactly the same as the exterior moment, as we neglected self weight. I'm really failing to see how we are disagreeing on this.
I think the part that we are disagreeing about is when there is an axial load. If you have moment only, then I agree.... you can take moments about any point in space and it won't matter because it's only a force-couple. When you add an a net axial force to the problem, then it does matter where you take moments about.
If you read my example above, I think it shows that pretty clearly. The stress distribution (0-16.67 ksi) needs to match the internal (or external forces) to be in equlibrium. If you sum moments of the stress distribution (0-16.67ksi) about the neutral axis (the bottom fiber) then you get a different moment than if you sum moments about the centroidal axis. The only reason for this is because there is a net axial load.
If you take your net axial force and apply it at the neutral axis (and account for the resulting moments), the answer will be the same.
From Macgregor (page 492),
The nominal moment capacity Mn for the assumed strain distribution is found by summing the moments of all the internal forces about the centroid of the column. The moments are summed about the centroid of the section, because this is the axis about which the moments are computed in a conventional structural analysis. In the 1950s and 1960s, the moments were sometimes calculated about the plastic centroid, the location of the resultant forces in a column strained uniformly in compression. The centroid and plastic centroid are the same point in a symmetrical column with symmetrical reinforcement
It doesn't matter what your loading is. Statics is statics. You cannot choose to sum only internal moments and disregard the free body diagram. An axial load will not change the beauty of statics.
The stress distribution is what it is as a result of the externally applied loads. In the end, what you are trying to determine is the maximum stress in the member.
Try this: apply your axial load somewhere other than the centroid and sum your moments about the centroid. You cannot disregard shear anymore, or the reactions at the support. Now your internal stresses vary depending on where along the beam you cut your section. Which axis would you sum moments about?
Here's where I think something is getting lost in translation. I'm assuming that we know the starting point to arrive at an answer simply to make my point. In reality, however, when you are trying to determine the moment capacity of a section knowing only the only strains and accompanying stresses, you have to determine the moment acting on a section based on those internal stresses (forces).
I think my statics are working out fine. When you look at the moment acting at a section of a beam - I think I just figured out EXACTLY where we are missing each other. Tell me if I'm wrong. You are thinking of a single element that has applied forces on one side and internal forces on the other side. In that case, I agree that it doesn't matter when you take moments about, they must always sum to 0 to maintain equilibrium. The point I was making earlier in this thread, and trying to reiterate here is that if you have a concrete section with a strain analysis only (such that you don't know what the applied forces are), it is critical that you take moments about the centroidal axis to get the moment acting on the section. In this case you will get a net moment and axial load, it won't sum to zero, because you have no applied forces taht you know about - only an assumed strain profile.
you are specifically referring to columns and the traditional way of computing the moment capacity of a section with a predetermined external axial load.
Summing moments about the centroid in this case certainly makes it easier, as you essentially neglect the axial load itself, as the stresses from the assumed equally loaded cross section essentially "cancel out" when summed about the centroid.
I'm not arguing that it's done traditionally this way (read one of my first few posts); I was just thrown for a loop when people started arguing that the answer (read "physical interpretation") could change depending on how you chose to view a problem. Like you, I also wrote a few Excel spreadsheets to calculate the interaction diagrams in school and now that I remember, I did sum about the centroid.
My argument is that it shouldn't and doesn't matter how you look at it. You could choose to make it easier on yourself by summing about the centroid or you could make more work for yourself by choosing to sum about some other axis. The answer is the same.
You guys need to read what each other is writing, because you are both correct. I believe this to be the case:
You may take the sum of the moments at any point and if in equilibrium they will equal zero. So to solve for an externally applied moment you must consider the externally applied loads i.e. the compressive force. However to simplify this procedure because the compressive force is unknown we can take the moment about the centroid creating a 0 length moment arm. Therefore your both correct!
SEIT:
In your summation about neutral axis, you are not accounting for the axial force (100 kip) based on which you ended up with your stress distribution. The net result will still be 200 k-in.
Quoting Cool Hand Luke.. "what we've got here is failure to communicate.."
I now understand what you're saying, and you are right. I kept on thinking in the other direction. You've assumed a profile and want to determine the forces that led to that profile.
which summation are you referring to? I think the PDF I posted is fairly clear.
frv-
I wish I got big bucks - I'd settle for middle-of-the-road bucks. I figured we were talking past each other. Once I realized that you were using the full FBD with the applied external loads then it all made sense and I agree.
"If I'm trying to determine what forces are acting on this section based on the stress profile alone then it DOES make a difference where you sum moments about"
Agreed. I was doing the summation of moments based on an axial load being applied at the centroid and moments taken about the neutral axis.
I cannot believe that there have been 35 replies to this!
Anyway, my 2 cents worth.
In a very lightly loaded beam, taking the compression reinforcement into account may actually reduce the calculated moment capacity (but not the actual capacity). I ignore it in these cases.
Where you have a highly loaded beam and the compression extends well below the level of the reinforcement then it does actually increase the moment capacity significantly.
Why would the compression reinforcement reduce the calculated moment capacity? If you are referring to a situation where the compression reinforcement may end up in tension at the top, then yes; but I have never seen a case where the neutral axis is close enough to the compression block for tis to be the case. Maybe it does.
I disagree with your second point; If you start with a beam without compression reinforcement and add some steel within the compression block, all you'll do is reduce the depth of the compression block (which under normal circumstances is a small percentage of the depth of the beam to begin with) meaning your moment arm from your tension steel (the quantity of which which is unchanged) to the centroid of your compression is slightly increased, resulting in marginally higher calculated moment capacity (certainly well below 10%).
It may be helpful to think in terms of an eccentric axial force, rather than an axial force + moment. In the case of the posted example it is immediately obvious that the eccentricity of the axial force is 8 inches above the NA, or 2 inches above the centroid, and that those two statements are saying the same thing.
This discussion reminds me of the history of beam bending theory, where Galileo's incorrect theory continued to be used for literally hundreds of years after Antoine Parent, then Bernoulli and Euler, then Coulomb had all come up with the theory we use today.
that is true and very clear IF you know the external forces. If, however, you're going on just a strain or stress profile than you cant't do that. Well maybe for a steel plate you could, but not for a RC section.
