Analyzing structural slab with top and bottom steel

[skimboard20]

I am analyzing a structural slab that has equivalent steel at top and bottom (As = As'). This creates an issue when computing "a", where a = (As-As')*fy/(0.85*f'c*b). Surely there is a way to calculate the strength of a slab with equal top and bottom reinforcement without "a" going to zero? Any input on this?

Other relevant info:
- one-way slab (analyzing 1 ft strip as a beam)
- slab is 10" thick with #6 @ 9"oc top and bottom
- f'c - 2500 psi
- clear cover is 3/4" top and bottom
- this is an elevated slab that forms a garage roof and supports landscaping

Thanks for the help!

 

[KootK]

For a slab of those proportions, I'd normally just ignore the presence of the compression steel unless it's a desperation play for improved deflection control. The compression bars usually wind up too close to the neutral axis to be effective and some folks feel that true compression steel ought to be tied.

Alternately, ignore the concrete and call it 100% steel with respect to flexure.

 

[ClearCalcs]

That equation is only valid if your top steel is yielding in compression. With equal areas of steel that are both yielding at the same time, your concrete wouldn’t take any load and thus a = 0.

But if a = 0, then your top steel would be in tension, so it’s certainly not yielding in compression. In that case, you have to calculate the stress in your top steel from your strain profile.

Setting up your force balance equation, you get:
Ts = AsFy
Cc = beta1 * c * 0.85 * f’c * b
Cs = As’ * Es * 0.003 * (c-d’)/c

Set Ts = Cc + Cs and solve for c (it’ll be a quadratic). You can then find a = beta1 * c.

I hope that helps!

-Laurent

http://www.ClearCalcs.com

 

[Tomfh]

Ignore the top steel.

 

[ClearCalcs]

KootK, I was also thinking about just ignoring the concrete and went down a bit of a rabbit hole. Curious to hear your thoughts on this!

Considering only steel, Mn = AsFy * (d - d’). In reality, this is equivalent to saying that a/2 = d’, ie the compression steel & concrete block have coincident lines of action.

I’m reasonably sure that this is always conservative as long as a/2 < d’, since the concrete resultant would be contributing to the moment resistance.

But what happens when a/2 > d’ ? In that case the concrete resultant actually decreases the moment resistance when taking moments around d’ ! Is it even possible for a/2 to be greater than d’ with As = As’? Instinctively I’d assume not, but it’s a bit weird to visualize.

On the other hand, if we truly ignore the concrete in flexure, there still has to be shear flow through the concrete between the top and bottom steel. But for the top steel to yield in compression with As = As’, we’re probably far past the ultimate concrete strain so there’ll be some significant spalling. Can we really rely on the concrete for shear flow between bars then?

-Laurent

http://www.ClearCalcs.com

 

[RWW0002]

Compression steel does not generally have a large influence on strength, and as others have mentioned I usually ignore it for strength calculations. For the case you mention above where the area of compression steel is high, it almost certainly does not yield. You would have to use a strain compatibility solution to determine the stress in your top bars and calculate capacity from that.

However Compression steel is worth considering for deflection and section ductility.

One exception to the "ignore compression steel" recommendation is when your section is tension controlled already yet you need more capacity. Adding tension steel alone once εt<0.005 will not increase design strength. However you can add compression steel along with some tension steel and get an increased capacity.

 

[KootK]

KootK, I was also thinking about just ignoring the concrete and went down a bit of a rabbit hole. Curious to hear your thoughts on this!

I accidentally baited your here I'm afraid. I was being facetious and wouldn't, myself, seriously consider the steel only option. Fundamentally, I just don't like designs that pretend that things work in a manner which we no to be substantially false. And I feel that would be the case with this. Granted, one might still be able to argue it as a lower bound solution. Still though, like you, I find these things fun to think about. In practice, I've kicked around the idea of using steel only as an upper limit on deflections. Even that gets a bit messy, though, when one considers the impact that the ignored concrete has on the rebar in between T&s cracks etc.

Is it even possible for a/2 to be greater than d’ with As = As’?

I'm afraid that I don't know. Obviously, a/2 greater than 'd' wouldn't represent a statically admissible solution with the bars assumed fully yielded. But, as we know, they're not fully yielded. To answer this question fully, on might have to study the section paying strict attention to strain compatibility, perhaps using a stress block assumption more advanced than Whitney's, and maybe redefining what is meant by "a" in this context. Somebody like Denial, IDS, or rapt would be a good fit for this question, hopefully one or all of them stop by.

Can we really rely on the concrete for shear flow between bars then?

Another fine question. And another reason for me not to do this. In addition to trying to avoid solutions that don't mirror reality, I also try to avoid solutions that haven't been vetted through testing. And this issue would probably be one of those. Some thoughts:

1) I would think that the concrete betwen the bars would be most critical for shear flow. And everything between the bars should remain at modest strain levels.

2) If the stain beyond the bars is extreme, it may spall the concrete there and compromise bar bond. That would certainly compromise shear flow.

3) Other than cantilevers, most common design situations have low-ish shear at locations of peak moment. This might help with #2.