Partially Reinforced Beam 3

[waldo459]

Can anyone explain to me how a allowable shear and moment diagram will look like for a steel beam that is partially reinforced with a C on top? The Lb is the length of the beam, the beam in reinforced in the middle third. Will the moment capacity of the unreinforced section be based on the entire length of the beam? Would the reinforced section moment capacity be based on the length of the reinforced section?

 

[Teguci]

Shear capacity shouldn't be affected. It will be controlled by the unreinforced section.

Moment will be tricky, lateral buckling will be controlled at the reinforced section but not at the ends. Perhaps a model is in order with the varying conditions. The cap channel does not provide a braced point.

Perhaps a proportion can be used? Maybe (Lend / Lr)*2+ (Lmid/ Lr) can be used to determine your limiting mode of failure?

 

[rb1957]

i'd calculate the bending stresses at various locations ... at the end for the reinforcement, within the reinforcement, at the maximum moment position, etc. I'd calculate the bending stress at several places on the X-sections the extreme fiber of the reinforcement, the extreme fiber of the un-reinforced section, etc,

then i'd integrate these stresses to find the load going into the reinforcement. then i'd look at the connection of the reinforcement to see if it could handle these shears.

 

[csd72]

I would calculate lateral torsional buckling of the stiffened shape based on the full length of the beam.

 

[kslee1000]

My simple thoughts:

If it is a simply supported beam - one end pinned, one end on roller, with gravity loads only without brace in between, then you have a simple problem.

Calculate moment along the beam as usual, let Lb = L (full length), get My/I (I varies for ends and the middle sections), then check bending stresses w/r to code. At this point, you can judge which segment/section controls the design, and determine the moment capacity of the beam.

For the reinforced segment (composite), calculate the shear force along the beam as usual, then calculate shear flow (VQ/I), and check the shear capacity of the connectors at the interface.

If you have Pin-Pin ends or fixed ends, or axial loads, it is more complicate. You may need to evaluate the effect due to eccentricity (the neutral axes of the segments are not aligned).

 

[frv]

kslee1000-

How are you accounting for LTB?

 

[JAE]

The problem is - when the channel cap only goes over part of the length of the beam, the AISC chapter F provisions don't have an answer such that you can take the channel into account for LTB.

 

[msquared48]

I'd analyze it as a continuous span section of varying I. There should be a computer program out there that can do this. I even think that I could fool BeamPro into doing this analysis.

Mike McCann
MMC Engineering

 

[miecz]

I wouldn't entrust this to a computer program. In fact, I would expect a computer program to botch this analysis.

I see this as a complex LTB problem, and I haven't seen a solution to it anywhere. First place I would look is the Guide to Stability Design Criteria for Metal Structures. AISC published a solution for stepped columns in the journal back in the seventies. Not the same thing, but similar in that it was a buckling problem on a section with variable section properties. The math got really hairy.

As a first step, I would analyze it as though it were reinforced the full length. That's not conservative, but at least you'll see where you stand. How much of the length is reinforced?

 

[msquared48]

Yep - took a look at the simple span with uniform I and the simple with varying I, both with a uniform load to keep it simple. Worked with BeamPro.

Thinking this through...

The allowable shear is just a block at the end thirds and a deeper block top and bottom at the middle third.

The allowable moment, simplistically, is a simple beam moment diagram with the middle third higher than the end thirds, a vertical jump at the third points.

The deflection diagram is that for a simple beam, but less across the entire span for the varying I condition when compared to the uniform I condition.

I will try to scan and post the resulting diagrams with my comments.

Mike McCann
MMC Engineering

 

[msquared48]

Here are the printouts with my markups, one for the uniform I, one for the varying I, both loaded identically.

The allowable lines are conceptual, what I would simplistically expect assuming similar lateral bracing situations too...



Mike McCann
MMC Engineering

 

[msquared48]

Here is the simple span analysis with uniform I for deflection comparison. Apparently I could only upload one file per post.

Mike McCann
MMC Engineering

 

[JAE]

Mike,
I guess the real question here is not about the analysis (i.e. finding the shears and moments) but about what the capacities are for shear and moment. And AISC has no answer for this directly. What you can visualize is this:

1. You first have a pure WF beam spanning L. It has a moment capacity based upon Lb (for this situation I think the condition is that Lb = L) So the [Φ]Mn is determined by AISC Chapter F provisions.

2. You then add a 1 foot long piece of channel cap on the beam at midspan. Now by observation, we're pretty sure the the [Φ]Mn is the same as in step 1 above.

3. You then look at a condition where the channel isn't 1 foot long, but rather the full length of the beam. Now, you can take AISC Section F7 (I think that's the right section) and determine [Φ]Mn based upon the composite section of the WF and the channel combined. No problem here.

4. For ANY length of channel between the 1 foot length and the full length things get a little hazy. The answer is somewhere between the [Φ]Mn in step 2 and the [Φ]Mn in step 3....we just don't know how it varies and AISC doesn't give us an answer. Therefore we are forced to use the [Φ]Mn from step 1 to be conservative.

 

[hokie66]

I agree with JAE for a beam which only has a capping channel in the middle third. For the uniform load case, the bending moment at the third point is 89% of the midspan moment, so there is not much penalty in being conservative.

If the channel went within say a foot of each end, I would tend to call it full length and use the capacity of the composite section.

 

[haynewp]

I would either:

1. Add braces where the cross section changes and use each third as a separate Lb, get phi*Mn based on the cross section and loading within each length

2. Run the channel the entire length of the beam and use the compound section for phi*Mn calculation

 

[frv]

haynewep-


In the case you describe, this would completely solve the LTB problem that several other people and I previously alluded to.

But if there are no braces at these points, this becomes a very complicated issue, as miecz points out. Again, it is only complicated by LTB. If the beam were fully braced, then this becomes very straight forward- kind of like analyzing the capacity of an OWSJ.

 

[haynewp]

frv;

That is the reason for the suggestions [if possible], so that the complicated issue would go away.

 

[JAE]

haynewp is very correctly pointing out one of the main tools of a structural engineer.

In attempting to solve a difficult problem, get rid of the difficulty and the problem can easily be solved.

 

[frv]

Agreed.

I only wish architectural constraints didn't prohibit this as frequently as they do.

 

[waldo459]

Thanks to everyone for the comments. I understand this is a difficult issue. Lb being the primary problem. Is it reasonable to use the following: For the first third, Mn based on Lb is 30' for the unreinforced beam, for the middle third, Mn is based on Lb of 30' for the reinforced section.