Shear flow in concrete beam

[jeffhed]

I have a concrete beam I need to reinforce with a plate or angle scabbed on the side. Most of my shear flow calculations have involved reinforcing with a new element entirely on the bottom and/or top. So when you calculated Q, your area and ybar of the element you were attaching is entirely below or above your point of interest. Now my question is this: Say I am attaching an 8" steel plate at the bottom of the concrete beam on the side with a row of bolts along the center line of the plate, so 4" up from the bottom of the beam. When I calculate my area and ybar, it should be for the entire plate that I am attaching, right? Not just the portion of the plate that is below my point of interest, in this case the bottom 4" of the plate? I have looked in a few strength of material books at the office and can't find any examples of reinforcing beams on the side, probably because it is the least efficient. Obviously with the significantly lower area and ybar, if I am incorrectly using the entire area of the plate, this could make a big difference in the calculated shear flow, in this case it is double. Has anyone else done this before, if so, did you calculate the shear flow as I have, or did you use only the area and ybar that is below the point of interest?

 

[JAE]

I would do this:

A = the entire area of the plates

e = the distance from the centroid of the plates to the neutral axis of the combined section.

Q = A x e

By using the entire area, you are at the very least being conservative (larger A means larger Q which means larger q when q = VQ/I)

 

[jeffhed]

JAE,
Thanks JAE,
That is how I was performing my calculations. I just started wondering about doing it this way because it doesn't matter where the fasteners are in the plate. Say we are attaching the 8" plate(s) so the bottom of the plate(s) lines up with the bottom of the concrete beam, it wouldn't matter if you put the fasteners 2" , 4" or 6" up from the bottom of the plate(s). The moment of inertia of the composite section would remain the same as well as the ybar and the area of the plates. I guess it makes sense because you have to transfer the forces to that element no matter what, regardless of where the fasteners are in the plate(s). At first without thinking about it, I was expecting to see a lower attachment force at the bottom of the plate (2" up)than at the top (6" up). But for this to happen, the ybar or area has to change as you change your point of connection. In reality if there was a change, it probably wouldn't have been big enough to make a difference anyway.

 

[RFreund]

There are a couple good threads on this that I will see if I can find. They are not for concrete/steel but same principles. I'll see if I can find them.

However I do believe the position of the bolts will effect the design. (If they are bolted through the N.A. there is no shear flow)

If the plates are "side plates" this is similar to a flitch beam condition. In which case the steel and concrete will take loads in proportion to their stiffness.

If these plates are intended to add width to the flange then I believe you are back to a shear flow calc.


EIT

 

[jeffhed]

RFreund,
The position of the bolts will make a difference, but once you place the plate where you want it, the position of the bolts inside the plate cant make a difference in the situation we have talked about above. You calculate the neutral axis of cracked beam section including the original steel and the new steel side plate. You then calculate the moment of inertia of the cracked section. The area and ybar of the plate remain the same regardless of where the bolts are installed in the plate. Because the ybar, the area, and the I of the section stay the same, VQ/I cannot change. How does locating the bolts anywhere within the plate change ybar, the area, or the I in VQ/I?

 

[hokie66]

In a composite design, shear flow calulations should take area of the plate to occur at the height of the connection. But I agree with RFreund, I would just consider the loads to be taken in proportion to the stiffness of the two members, with the bolts there to effect the transfer from one to the other.

 

[jeffhed]

hokie66,
The beam is under reinforced. I am adding the plate as additional reinforcing to the concrete beam to strengthen the section enough to support the required loading. I don't think it would help enough to do it like a flitch beam scenario. I think I would need quite a bit more steel. To give a little more detail on my analysis, I first analyzed the beam flexural capacity as is. When it didn't work, I added steel until I could make it work. When I had determined how much steel I needed, I went back and did a shear flow analysis to determine the required connection force, and then the fasteners. So back to my original question, with shear flow, I should only use the area and ybar of the side plate that is outside the connection point, in this case the lower 4" since I will be bolting the plate to the concrete at the center of the plate? Or do I use the entire area of the plate and ybar of the whole plate? I ran it the second way as discussed earlier because I was having a hard time wrapping my head around why the whole area of the plate would not be included. But as I also said before, then it would not matter where you bolted the plate to the beam as your ybar and area would always be the same.

 

[ToadJones]

Well, for comparison sake, say you have a wide flange beam with a channel cap. The entire area of the channel is considered for the area and "ybar" even though a portion of the channel is close to the n.a. than the fastener.
Keep in mind you are transfecting load to the plate via shear.

 

[ToadJones]

"transfecting" ???
Transferring

 

[RFreund]

Ok,

I reread the posts and I think that you would need to design for shear flow the way JAE described. Really the shear flow is not across a horizontal plan but the vertical plane. Similar to shear flow in the flanges of a W-section.

However when you analyze the bending strength of the section would the location of the connection determine the stress in the steel (when looking at an elastic stress distribution)?

EIT

 

[hokie66]

With the plate on the side of the concrete beam, it is in combined tension and flexure. The shear "flows" at the point of connection, so that is the depth used in the shear flow calculation. I see no reason not to use the entire section in tension, but you have to check the combined stress.

 

[jeffhed]

ToadJones,
So you agree with what JAE said? As I said I ran the numbers this way at first but began to question if it was correct. My biggest problem with this approach is this: What if the point of the connection does not correspond with the ybar of the plate? It would make no difference if the attachment occured at the top of the plate or the bottom of the plate or the middle. The ybar and area of the plate remain the same regardless of where the bolts are located in the plate. For comparison say we have a plate that is the same depth as the neutral axis to the bottom of the beam. Regardless of the location if the connecting bolts, the ybar is the same and area is the same. I am still having a hard time believing. that this is correct. If we want to calculate
shear flow in a piece of tube steel would you use the entire side walls when checking shear at the corners? We would only use the area and ybars of the portion of the sidewalls that are above or below the point of interest.

 

[ToadJones]

I dont really think you have a shear flow problem here.
imagine you have your beam simply supported and the plates are simply supported on either side of the concrete beam but are not connected to the concrete beam.
If you load the concrete beam it will deflect but not translate relative to the plates.
The fasteners are simply delivering load to the plates and preventing the plates from buckling.
It's hard to grasp a shear flow mechanism because none exists in this case

 

[jeffhed]

RFreund,
Yes, very similar to shear flow in the flanges of a w section. when I analyzed the section with the new steel, I calculated the new centroid of the steel using the existing reinforcing and this new plate. The centroid of the steel group is where I am placing the bolts.

hokie66,
So are saying that this is more like a built up wood member? Bolt them together based on their stiffness? That could work but I would need much more steel than I have right now, like a C-channel. It would probably result in less bolts but more steel. The contractor really wants to use a plate in this case if at all possible which is why I am where I am at.

ToadJones,
I fail to see how we don't have shear flow here if I am trying to add reinforcing to the concrete beam. If I have a plate on the bottom I have shear flow to determine the fasteners. If I put the reinforcing on the side, then I don't?

 

[ToadJones]

jeffhed-
What if you bolted two 2x10's together to make a header.
does it matter where the bolts are?

With the plates in the arrangement you have, I believe the three beams (concrete beam + two plates) will simply carry the load proportional to their stiffness.

 

[ToadJones]

also keep in mind that if the beam is existing, and you slap plates on the side, it will do absolutely nothing if the beam is currently loaded to its service load. The plates will only help with additional load.

 

[RFreund]

I'm attaching the following. To show a few different cases and I think I'm confusing myself the more I think about this. Shear flow is the result of 2 surfaces trying to slide past each other due to normal forces having different magnitudes (due to different magnitudes of moment). But I'm not sure how to apply this. Maybe these sketches will spark some thought

EIT

 

[ToadJones]

To be clear, I am assuming that your "side plates" are the same depth as the beam. I also assumed that the n.a. of the concrete beam and the sideplates were coincident. If this is not the case, then ignore what I have said. (It would also make sense the the n.a. of the concrete beam is closer to the top of the of the beam and therefore the sideplates n.a. are not in the same place).

 

[jeffhed]

ToadJones,
I agree if the plates are the same depth as the concrete we do not have a shear flow problem. This is not the case in this instance. The steel is not the same depth as the concrete. The concrete beam is 42" deep and under reinforced, I am adding an 8" plate on the side of the concrete beam at the bottom to make up for the deficiency in steel.

RFreund,
Case 1 at the top of your drawing illustrates what I am doing. However, I am trying to use one plate due to the outside being exposed to soil and trying to avoid through bolting.

So, which is the correct way? We all agree that my point of interest is the point of connection. Do I use the entire area of the plate and the ybar of the entire plate? Or the area of the plate below the point of connection and the ybar for the portion of the plate below the connection point. I have attached my own drawing similar to RFruend. After drawing this out and thinking more about this, I believe that the correct way is as JAE said. We have to develop the entire plate through the bolts. Just as if it was attached to the bottom.

 

[jeffhed]

To add further to my post, I think Option number 2 in my drawing would only be if the plate and concrete were continuously bonded. Where we are just bolting the plate to the beam on one spot, we should be using the entire area of the plate and the ybar of the entire plate in this case.