Doubt about Neutral axis of Reinforced concrete 1

[Yt.]

Basically, ACI assumption about the location of neutral Axis of a moment subjected member begins directly matching Tension of steel with compression of concrete and compression steel. Developing this idea a proportion is calculated on how much is equilibrated by concrete and the rest then resisted by steel.

when improved, the concrete portion occupied by steel is discounted.

Then through Navier hypothesis the deformation of steel is calculated and so the stresses. Also the neutral axis location.
Done the above, nominal moment is computed by assuming a coupled behaviour of forces. sum(Forces*distances)

My doubt is about the first equilibrium assumption and why moment is computed that way.
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The external and internal forces equilibrium of a member's section responds to:

E1, E2, respective Young's modulus, k: Navier slope
And the moment developed by the forces that resist it should be measured by sum(F*di) (As expressed above), not with (d-a/2) and (d-d') distances:

(autocad picture)

The way it's done in ACI, when multiplying forces by distance (d-a/2) and (d-d'), correspond to forces rotating at the midpoints of each one of those lines.

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Another way i have seen this can be done,

before cracks:

After cracks:

The problem of this last method is that the moment is spread only through the areas (and distances) without taking into account that the compression steel doesn't need to be as loaded as tension one,

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I don't see the need of directly match Ts=Cc+Cs, becauses their proportions respond also to distance to neutral axes. I think it should be T*d3=Cc*d2+Cs*d1 (following the autocad picture).

It is similar to the way moments are computed in interaction diagrams, but in that case distances are computed to the gravity center of cross sections when it is symmetrical

Why is ACI not taking distances to neutral axis?

 

[appot]

Ts = Cc + Cs implies that the internal forces are in equilibrium (sum of forces = 0). If they are not in equilibrium then something is accelerating.

 

[Yt.]

I know, but their are canceling with the external moment.

 

[Yt.]

I meant, it is not my intend to deny ACI at all. I have a doubt about neutral axis location, the way I see the regular procedure isn't getting my clear with the rotational equilibrium of forces

 

[jayrod12]

If you don't have Ts=Cc + Ts then your internal forces do not balance. This is day one of statics and strength of materials. Distance to the neutral axis is irrelevant.

In concrete specifically, the neutral axis depth is a best guess anyway, since we use an adjustment to change the parabolic stress block into a rectangular stress block.

 

[Yt.]

Ok, i know i can't deny that either. But i have to change some of my views, perhaps my confusion is to think that the member rotates about the neutral axis. Can be said that concrete block rotate at (d-a/2) midpoint and compression bars rotates about (d-d') midpoint?

 

[Yt.]

If that's the case, where is tension bar rotating about?

 

[Yt.]

Sorry if it is a fool question, but a can't conciliate to prevail horizontal equilibrium over rotational one, Both should be achieved and there should be a way to understand it

 

[Ingenuity]

I am not sure I really understand your question, so the following maybe irrelevant.

Without an externally applied axial load (i.e. a RC beam section NOT a RC column section), you can take moments about any point of the cut-section. Take moments about the top compression fiber, bottom tension fiber, NA, top or bottom steel location, etc - it won't matter - you will achieve the same M.

Your equilibrium concept of rotation about the NA=0 is mistaken, and maybe if you redraw your AutoCAD sketch/FBD with a missing 'M', then the equilibrium conditions maybe clearer:

∑H=0: T + C[sub]c[/sub] + C[sub]s[/sub] = 0 (no externally applied axial load)​

∑M=0: T*d[sub]3[/sub] + C[sub]c[/sub]*d[sub]2[/sub] + C[sub]s[/sub]*d[sub]1[/sub] = M​

Solve for both of these equilibrium equations and all is good.

 

[Yt.]

Thanks Ingenuity, and all of you

That let's me clear. I think that to be studying and working continuously is dizzying me.
While moment's component forces are applied on a straight line theirs directions will always be perpendicular to any point of the line, so i can take all of them as the way I see the center of rotation.