partially reinforced beam

huangyhg

partially reinforced beam
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?
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?

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.

i would calculate lateral torsional buckling of the stiffened shape based on the full length of the beam.
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).
kslee1000-
how are you accounting for ltb?
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.

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
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?
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
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