First moment of area of the reinforcement (Eurocode 2) - Avscorreia

[NTCONLINE]

Dear everyone,

I would like to have your opinions on the following issue:

I've seen someone who calculated the first moment of area of the reinforcement about the centroid of the section, using the following formula:

S = -Asc (yc - d2) + Ast (d -yc)
where:
S = first moment of area of the reinforcement
Asc = area of steel in compression
Ast = area of steel in tension
d = effective section depth (from top fibre to tension steel)
d2 = concrete cover of compression steel
yc = distance from extreme compressive fibre to neutral axis

My question is: whether the minus sign ("-") in front of Asc was correct ?
Should it be + ? If it is correct, why would we need to subtract the compression steel component ?

the formula was used by @avscorreia in his spreadsheet
(

http://files.engineering.com/getfil...-8472-b3d8589acf74&file=DefSLS_rect_v01.xlsm)

in another thread,
(

http://www.eng-tips.com/viewthread.cfm?qid=402136)

which has been closed so I could not ask him there directly,

I need to find a correct equation to calculate S to use Clause 7.4.3 (6) Eurocode 2.

Thank you in advance.

Regards
NTCONLINE

 

[Guest262653]

I honestly can't remember why I've done that in the spreadsheet. I need to start documenting these things better... I'll look into it during the weekend.
And thank you for taking the trouble to check it!

 

[Guest262653]

I've taken a closer look into it and I think it may be correct. As shrinkage curvature occurs when there's a difference between the top and bottom reinforcement, the area module that is calculated takes this effect into account by considering the effect of the top reinforcement with a negative sign.

 

[IDS]

The first moment of area of a symmetrical section about the centroidal axis is zero, so the minus sign is correct. Alternatively it could be written:
S = Asc (d2 - yc) + Ast (d -yc)
d2 should be the dimension to the centre of the bar, not the cover depth.

The Eurocode clause is for calculation of curvature due to shrinkage, and in an uncracked symmetrical section uniform shrinkage does not cause any curvature, so it makes sense that the effect of the top and bottom steel should have a different sign.

Note that in a cracked section, where the neutral axis is not at the centroid of the concrete section, symmetrical steel will generate significant curvature, even under uniform shrinkage.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[NTCONLINE]

Thank you both IDS and avscorreia !
I've also doubled checked : it is correct to have the negative sign.
My hand calculation for deflection makes sense now and I get the reasonable deflected shape.

Kind regards
NTCONLINE