What might happened if f's is negative 9

[XinLok]

If we take a concrete beam with the following dimensions:
Mu = 136.77 k-ft
b = 12 in
d = 8.43 in
d' = 3.56 in
f'c = 4000 psi, fy = 60,000 psi

According to ACI code, I have to deal with a Double reinforced concrete cause M max = 58.27 k-ft < 136.77

If I continue by using excel to avoid writing everything here, I will get f's < 0 cause c - d' < 0

If I stick with the dimension above, what will happen and what does mean f's negative?

 

[XinLok]

Dear All,
First thank you all for all your great replies.

Let me ask the same question but in another way.
Considering some changes with my previous inputs above:
Mu = 136.77 k-ft, b = 12 in, d = 8.43 in
d' = 2 and (I change d' from the previous values (3.56in) to 2)
f'c = 4000 psi, fy = 60,000 psi.

by recalculating a & c, I got the following:

So, for me, things looks like normal when d' = 2.

I will do the same thing by now with d' = 3.5

Now, there is a problem.
So, there is a serious problem if the steel fixer by mistake instead of installing the compression rebars at d' = 2, he installed at 3.5. Am I right???

I want to understand theoretically what does mean A's negative of f's is negative.
So, the section while d' = 3.5 is invalid? if it is valid what shall I do with A's which is negative? Do I have to deal with this section is single reinforced concrete? but I cannot cause Mu is higher then M max.

 

[Settingsun]

Yes, the concept is marginal. You have 3" of compression concrete, ~1.5" of that is cover meaning the reinforcement has to be in its exact design position to act as intended. The negative value of f's when d'=3.5" means it is in tension and doing virtually nothing for your capacity. Having the reinforcement position wrong by 1.5" in a 12" beam is a big miss and should be rejected, but it's still too tight a tolerance to be safe in my opinion.

Basically, doubly-reinforced design in such a shallow beam isn't practical, especially when relying on it for 60% of the total capacity.

Is B really only 12" or can you count some slab? Why are you limited to 12" depth and width? Higher concrete strength available?

Is deflection a problem?

 

[Lomarandil]

To illustrate, a bar placed (as intended?) at 2" depth (#1) and at 3.5" depth (#2) behave fundamentally differently. Bar #1 is in compression, Bar #2 is in tension.

So, yes, on the analysis side, a beam with top bars at #2 would be analyzed as a singly-reinforced beam section (as you've noted, with very little strain/stress/resultant capacity in the top bars). Additional tensile steel could be added to make the beam work as singly-reinforced, steel at position #1 could be added to return to doubly-reinforced behavior, or beam dimensions could be increased.

If this is a beam which has already been poured -- I agree, that's a significant mistake by the inspector and contractor. It needs to be fixed (at contractor expense), whether that means breaking it out and pouring again, or perhaps some sort of external strengthening like a composite wrap would be an option.

I also share Steve's concern about deflection with a beam loaded this heavily.

----
just call me Lo.

 

[BAretired]

BA:
Lo is correct you said the same thing I did. If ec=0.003 and es=0.002 balanced, if ec=0.003 and es>0.002 under-reinforced, if ec=0.003 and es<0.002 over reinforced and has potential for sudden brittle failure.

You and Lo are both correct. We both said the same thing in a different way. My sincere apologies to you and Lo. My brain cells were on holiday mode.

BA

 

[Celt83]

BA
No worries we were approaching it from different angles but I think really reaching similar conclusions. I was looking at a purely strain compatibility approach and you were come in with, on rereading, a more practical approach showing the inefficiency of the section without going thru the more detailed analysis, I now have a new quick check I can do thanks to your post with the Mu balanced formula.

My Personal Open Source Structural Applications:

https://github.com/buddyd16/Structural-Engineering

Open Source Structural GitHub Group:

https://github.com/open-struct-engineer

 

[XinLok]

Thank you all for your great replies, God bless you Dears.
Things are clear now.

 

[r13]

XinLok,

Strain and stress diagrams are the best tools for reinforced concrete design. From which you can easily visualize the behaviors of a given cross section with regards to the changes in design parameters. Sketches below show the case with both steel have yield (SK1), and the case with tension steel has reached the maximum strain limit. Have fun.

Note the strain limit on SK2 corresponding to the maximum amount of reinforcing (Ast_MAX) of the cross section, under which (Ast_MAX) the cross section is considered "under-reinforced". Depending on the version of code, the strain limit can be set as 0.005 (Under -reinforced - tension steel strain > 0.005.)