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Balanced Steel Percentage for beams

In balanced steel section concrete reach strain of 0.003(it is nominal capacity) and at the same time reinforcing steel yields. figure no:1 showing the strain diagram for a beam in a balanced condition. the strain for concrete equals 0.003 (εc=0.003), the strain for steel εy=(fy/Es), modulus of elasticity for steel Es=29x10^6 psi(200000 Mpa).









(c/d)=0.003/(εy+0.003)




(c/d)=0.003/((fy/Es)+0.003)

(c/d)=0.003/((fy/29X10^6)+0.003)

(C/d)=87,000/(fy+87,000)
C=(87,000*d)/(fy+87,000)

Figure 1

C is the distance from extreme compression fiber to the neutral axis. a=β1*C. if we equal the force C=T


C=T
0.85fc'*ab=As*fy
a=(As*fy)/(0.85fc'*b)
a=β1*C
β1*C=(As*fy)/(0.85fc'*b)
C=(As*fy)/(0.85fc'*b*β1)

ρ=As/(b*d)

C=(ρ*fy*d)/(0.85fc'*β1)

if we equal C from the previous equation, we can get an expression for ρb

(ρ*fy*d)/(0.85fc'*β1)=(87,000*d)/(fy+87,000)

ρb=((87,000*d)/(fy+87,000))*(0.85fc'*β1/fy)



in si units





Figure 2

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