Moment Capacity of Doubly Reinforced Beam to ACI318

[llocou]

Hello all,

I'm trying to find the capacity of a beam to ACI 318. Bear with me, I'm a BS/Eurocodes guide. Details of beam:

1000mm x 1000mm
50mm cover, 12 mm links, 25mm bars. 7 bars, top and bottom.
So As = As' = 3436mm^2. Here's as far as I've got with ACI method:
d' = 50+12+25/2 = 74.5mm
d= 1000-74.5 = 925.5mm
fc' = 30N/mm^2
fy = fs' (?) = 415N/mm^2

Cc = 0.85*fc'*b*a = 0.85*30*1000*a = 25500a
Cs = (fs'-0.85fc')*As' = (415 - 0.85*30)*3436 = 1338322
T = As*fy = 3436 * 415 = 1425940

Cs + Cc = T

25500a + 1338322 = 1425940
a = 3.44 mm <- here's where alarm bells first go off for me! The concrete compression zone is above the compression reinforcing steel... Carrying on anyway:

x = a/Beta = 3.44/0.85 = 4.04mm <- N.A again above reinforcing steel!?

Epsilon s' = ((x-d')*0.003)/x = ((4.04-74.5)*0.03)/4.04 = -0.052 <-negative strain!

Anyway, you can imagine how it carries on from here. Any advice? Appreciate any help, many thanks in advance!

 

[structSU10]

You have to make the assumption the compression steel does not yield. This sets up the equation of Cc + Cs = T in which 0.85*f`c*b*x + fs*As = As*fy . the stress in the compression steel is found by (x-d`)*0.003/x, so substitute that into the above equation and solve for x. It can be done iteratively, or in a solve block in something like Mathcad.

In general, the extra capacity due to doubly reinforced section is negligible, and can probably be ignored with not real effect on economy. The compression steel will help with long term deflection, so something is a plus in that regard, but you still don't have to include for strength.

 

[llocou]

You have to make the assumption the compression steel does not yield. This sets up the equation of Cc + Cs = T in which 0.85*f`c*b*x + fs*As = As*fy . the stress in the compression steel is found by (x-d`)*0.003/x, so substitute that into the above equation and solve for x. It can be done iteratively, or in a solve block in something like Mathcad.

In general, the extra capacity due to doubly reinforced section is negligible, and can probably be ignored with not real effect on economy. The compression steel will help with long term deflection, so something is a plus in that regard, but you still don't have to include for strength.

Thanks a lot for your reply. So I should be following the method laid out here in slides 29 onwards?:

http://www.assakkaf.com/Courses/ENC...35dd6db143=0&bcsi_scan_filename=CHAPTER5d.pdf

 

[AELLC]

The compression steel in a double-reinforced slab, being of small d, never yields. In beams, the compression steel may however yield, but not always.

The useful situation for double-reinforcing is where you are limited to a beam height that is too small for a single-reinforced beam to work efficiently.

For most situations, you can select the height, so structSU10 is correct with the statement that the compression steel is not significant to increase capacity.

 

[llocou]

Thanks for your replies. I'm actually not designing the beam, just checking the capacity. I've been using the 'compression steel not yielded' method, and used quadratic formula to find x:

(0.85*fc'*b*beta) x^2 + (0.003*Es*As'-As*fy) x - 0.003d'*Es*As' = 0

So using A x^2 + B x + C = 0

A = 21 675
B = 738 740
C = -161 268 660

giving x = -105mm or 70.9mm. The value of x here is absolute I'm assuming, so I need to take 70.9mm, which still places my NA above the compression reinforcement (d'=74.5mm). Can this be?

 

[PUEngineer]

Why not ignore the compression steel?

 

[AELLC]

Try taking d' from top of beam to compression steel.

 

[AELLC]

sorry, ignore above post, I see you already did.

 

[llocou]

Try taking d' from top of beam to compression steel.

That's what I've done. 50mm cover + 12mm links + 25mm/2 to centre of bar = 74.5mm.

 

[jayrod12]

Yes that can be the case where your "compression" steel is actually in tension based on your strain distibution. Since it's less than 5mm from the N.A. your strain in the steel will be minimal which will equate to almost no stress (approximately 35MPa) you are better off to neglect it completely. If you were to add more bottom steel (get closer to balanced strain case) it would drag the N.A. down putting the upper steel in compression.

 

[rapt]

If you stop using simplified formulae and do a strain compatibility analysis of the cross-section (10 minutes of hand calculation if you understand section design), then you will get the correct strains and stresses in each layer of reinforcement and you will know what is going on.

Just because reinforcement is near the compression face does not mean it is in compression. Unless your beam is very heavily reinforced in tension and is relatively deep and has no compression flange, it is unlikely your compression face reinforcement will be significantly in compression let alone be at yield in compression.

Use the knowledge you hopefully learnt at university to understand this sort of situation. Engineers should not have to ask questions like this if they understand engineering!

 

[AELLC]

If the beam is not continuous and has no negative moment in the center portion of the span, then the rebar you see at the top of the beam is shrink+temp or a nominal percentage of the bott rebar, and it would be nonsense to attempt to include it in the capacity investigation as being additive to the moment capacity based on bott rebar only.

 

[IDS]

it would be nonsense to attempt to include it in the capacity investigation as being additive to the moment capacity based on bott rebar only.

No, it is good sense to include all the steel in the calculation. In this case it is near the neutral axis so it will have little force and even less moment, so the difference it makes is entirely negligible. In other cases the top steel does make a significant difference. The easiest way to know is to plug it into the calculation, then you don't have to worry about it.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[AELLC]

Doug,
Only if you have an efficient, proven way of calculating it. Calculating for bott rebar only is much simpler, and may still be the prevalent method in most companies. At least it was decades ago when I was into heavy commercial. I have no idea what the policy is in most companies now.

 

[IDS]

Only if you have an efficient, proven way of calculating it. Calculating for bott rebar only is much simpler, and may still be the prevalent method in most companies.

I would hope that anyone designing reinforced concrete would have an efficient proven way of calculating it. It's really not difficult.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[AELLC]

Doug -

I had decades of experience as an employee. I got to be a one-man shop relatively recently. It (process of designing beams) actually could be difficult due to the policies of certain companies, believe me. Maybe not now, we have so much computer assistance.

Even years after IBM-type pc's were being widely used in small-to-medium size consulting companies, structural engineers weren't using them very much. Even I thought, Excel was only a tool for business/accountant people, hah!

Typically we were expected to work 60 hrs and be paid only for 40 hrs, and I certainly remember doing a LOT of concrete beams with bottom rebar only effective. With only a calculator and paper and pencil, it would have been more work to include As', plus the boss would have ragged on me for being "ivory-tower professional student, we are trying to make a profit on these jobs, you know"

 

[AELLC]

I also meant to say designing a double-reinforced beam is a lot more steps, and unless you have it as a canned computer solution, it is a lot more practical to just design as bott. rebar only.