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Rectangular Concrete Section - Top steel in C or T?

Rectangular Concrete Section - Top steel in C or T?

Rectangular Concrete Section - Top steel in C or T?

(OP)
I'm having an issue determining whether the top steel is in compression or tension in a MathCAD sheet. I'm limiting the scope of the problem to rectangular sections and 2-layers of reinforcement.

f`c = 6 ksi
Beta1 = .75
fy = 60 ksi
Es = 29000 ksi
max tensile steel strain: epsilon_t_max = .004 (ACI 318-08 10.3.4)

b = 12"
h = 8"

As = .8 in^2 (#4 @ 3"o.c.)
d = 6.25"
A`s = .8 in^2 (top steel - may or may not be in compression)
d` = 1.75"

With the top steel in compression:
As*fy = .85*f`c*Beta1*c*b + A`s*Es[(c-d`)/c]
Solving the quadratic, I calculate the neutral axis: c = 1.41"

With the top steel in tension:
(As+A`s)*fy = .85*f`c*Beta1*c*b
Solving the equation, I calculate the neutral axis: c = 2.09"

With the top steel in compression, c is calculated higher than d`. With the top steel in tension, c is lower than d`. This doesn't make sense.

I found a test to determine which was occurring. It said that if the rho (.008 in this case) of the bottom/tensile steel was less than rho_max, the beam is singly reinforced. Limiting tensile strain to .004, I calculated rho_max = .85*Beta1*(f`c/fy)*(.003/(.003+.004)) = .027 so the section is singly reinforced. This tells me to use c = 2.09" but that's deeper than d` which I thought was in tension. This doesn't make sense.

What am I missing?

RE: Rectangular Concrete Section - Top steel in C or T?

If top and bottom steel are stressed in tension to fy, then As.fy = 1.6*60 = 96k. In order to get a large enough compression block, c needs to be 2.09".

What this shows is that the top steel cannot carry such a large tension which means that f's < fy.

BA

RE: Rectangular Concrete Section - Top steel in C or T?

What BA is saying - your top steel isn't yielded yet and has a stress in tension below fy. Your tension equation assumes the steel yields - change As` to multiply by the Es*(c-d`/c) like it is for the other equation and you should get your answer.

RE: Rectangular Concrete Section - Top steel in C or T?

Quote (concengr)

I found a test to determine which was occurring. It said that if the rho (.008 in this case) of the bottom/tensile steel was less than rho_max, the beam is singly reinforced. Limiting tensile strain to .004, I calculated rho_max = .85*Beta1*(f`c/fy)*(.003/(.003+.004)) = .027 so the section is singly reinforced. This tells me to use c = 2.09" but that's deeper than d` which I thought was in tension. This doesn't make sense.

First, rho of the bottom steel is 0.8/(12*6.25) = 0.0107 in this case, not 0.008 as stated above.

Second, if you are limiting tensile strain to 0.004 then the strain varies from 0.003 at the top to 0.004 at the level of the bottom steel. This means that the strain in the top steel is 0.003 - 0.007d'/d = 0.00104 (compressive strain). The top steel cannot be in tension.

Neglecting the top steel, you would calculate c using As of 0.8 in2 and would find that c = 1.045", exactly one half the value which you calculated.

The only way you could have the top steel in tension is if the beam were subjected to axial tension combined with flexure.

BA

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