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Re: Extended rules of thumb

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Mike, Roger,

	The equation As = m/(1.76*d) is not so much a "rule of thumb" as it is
an upper bound on the required reinforcing.  This equation was presented
in the old A.C.I. publication SP-3.  It is derived from the equation

	As = M/(Fs*j*d)

with appropriate numerical substitutions for Fs and j.  Mike is right
about this being a very useful equation for manual reinforced concrete
design as all of "us older guys" can confirm; but I don't think I'd use
it in a spreadsheet when we can do much better.

	For example, if we use the equations

	Mu = Q*As*Fy*(d-a/2),

	a = As*Fy/(0.85*Fc*b),

where	Q = phi (0.9 for steel)

and solve this mess for As we get

As = 0.85*Fc*b*d/Fy*[1-{1-2*Mu/(0.85*Q*Fc*b*d^2)}^0.5]

which correctly calculates the required steel (unless I've made a typo
in the translation).

	There are two things to remember about using this equation:
1.) there is no "fat" in the design; if you want any extra you have to
provide it yourself; and
2.) the range of mathematical validity of the equation is larger than
the range of physical reality of of the beam design problem.  You must
check the capacity of the concrete section and not apply Mu larger than
the capacity of the concrete section to this equation.

	Keep it up with the "rules of thumb".  They are very interesting.

				Best regards,

				H. Daryl Richardson


Roger Turk wrote:
>
> Mike Hemstad wrote:
>
> . > Working stress concrete design (OK, all you new guys,
> . > flame away): As = M / 1.76 d    (M in ft-k, d in
> . > inches), or, far more accurately (good for
> . > spreadsheets)
>
> . > Per foot width,
> . >  As = ((M, ft-k)^1.04) x 11.84/(Fs (ksi) x d (inches))
>
> And for USD: fy = 60 ksi      2500 <= fc' <= 4500
>
>    Approx.   d = sqrt{70Mu/b}       Mu in ft-kips, b in inches
>
>    Approx.   As = Mu/{4.3 * d}      Mu in ft-kips, d in inches
>
> HTH
>
> A. Roger Turk, P.E.(Structural)
> Tucson, Arizona
>
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