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Example 1: Determining Steel Area When Beam Dimensions Are Predetermined

The dimensions of the beam shown in Figure 1 have been selected for architectural reasons.
Determine the reinforcing steel area by each of the methods described in this section.
Figure 1










Method 1 using tables

Mu=160ft-k=1920000 lb-in
Mu/(Ф*b*d^2) =(1920000)/(0.9*21^2*16)=302.3psi

ρ from table A.12 by interpolation 
ρ --------(0.0054-0.0053)/(303.4-298.1)=(0.0054-ρ)/(303.4-302.3)
ρ=0.00538
As=ρ*b*d=0.00538*16*21=1.81in2
                                        Use 6 #5 bars (1.84 in.2)

The second method(The use of ρ formula)
Rn=Mu/(b*d^2)=302.3psi

ρ=(0.85*3000/60000*(1-√(1-(302.3/(0.85*6000))
ρ=0.00538
same as in table method

The third method(Trial and error method)

assuming a=2in

As=Mu/(Ф*fy*(d-a/2)
As=1.78in2

checking a value

a=(As*fy/(0.85*fc'*b))
a=(1.78*60000/(0.85*3000*16))
a=2.61in    far

assume a=2.6

As=Mu/(Ф*fy*(d-a/2)
As=1.81in2

a=(As*fy/(0.85*fc'*b))
a=(1.81*60000/(0.85*3000*16))
a=2.66in    close enough 







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