Pile Cap Reinf. Equation

[structbd]

Hello Fellow Structural Engineers,

I recently began reviewing a calculation regarding the required reinforcement for a pile cap foundation. In the calculation, I was reviewing I came across two equations that I would not have used if I were doing this design calc. I wanted to know if anyone has seen these equations used and where are they from? Any references would be greatly appreciated.

Thanks in advance.

 

[ProgrammingPE]

I haven't seen the formulas in that form before, but this ends up being identical to the equations that use the Flexural Resistance Factor, R.

R = Mn/(b*d²)
R = ρ*fy*(1-0.588*ρ*fy/f'c)

If you solve the second equation for ρ using the quadratic equation and simplify it, you'll get:
ρ = f'c/fy*(0.85-√(0.85²-2*0.85*R/f'c)) = f'c/fy *(0.85 - √(0.7225-1.7*R/f'c))

They are using "z" instead of "R"
z = R/f'c = Mn/(f'c*b*d²) = Mu/(Φ*f'c*b*d²)

They are using "ω" instead of "ρ"
ω = ρ*fy/f'c = As/(b*d)*fy/f'c

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