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PROBLEM 2.35 2.35 The 1.35 m concrete post is reinforced with six steel bars, each with a 28 mm diameter. Knowing that ( E_{s}=200 ext{ GPa} ) and ( E_{c}=29 ext{ GPa} ), determine the normal stresses in the steel and in the concrete when a 1560 kN axial centric force P is applied to the post. SOLUTION Let ( P_{c}= ) portion of axial force carried by concrete ( P_{s}= ) portion carried by the six steel rods ( delta=frac{P_{c} L}{E_{c} A_{c}} quad P_{c}=frac{E_{c} A_{c} delta}{L} ) ( delta=frac{P_{s} L}{E_{s} A_{s}} quad P_{s}=frac{E_{s} A_{s} delta}{L} ) ( P=P_{c}+P_{s}=left(E_{c} A_{c}+E_{s} A_{s} ight) frac{delta}{L} ) ( varepsilon=frac{delta}{L}=frac{P}{E_{c} A_{c}+E_{s} A_{s}} ) ( A_{s}=6 frac{pi}{4} d_{s}^{2}=frac{6 pi}{4}(28)^{2}=3694.5 mathrm{~mm}^{2} ) ( A_{c}=frac{pi}{4} d_{c}^{2}-A_{s}=frac{pi}{4}(450)^{2}-3694.5=155348.6 mathrm{~mm}^{2} ) ( L=1.35 mathrm{~m}=1350 mathrm{~mm} ) ( varepsilon=frac{-1560 imes 10^{3}}{left(frac{29 imes 10^{9}}{10^{6}} ight)(155348.6)+left(frac{200 imes 10^{9}}{10^{6}} ight)(3694.5)}=-297.48 imes 10^{-6} ) ( sigma_{s}=E_{s} varepsilon=left(frac{200 imes 10^{9}}{10^{6}} ight)left(-297.48 imes 10^{-6} ight)=-59.5 mathrm{~N} / mathrm{mm}^{2}=-59.5 mathrm{MPa} ) ( sigma_{c}=E_{c} varepsilon=left(29 imes frac{10^{9}}{10^{6}} ight)left(-297.48 imes 10^{-6} ight)=-8.627 mathrm{~N} / mathrm{mm}^{2}=-8.627 mathrm{MPa} )

          PROBLEM 2.35
2.35 The 1.35 m concrete post is reinforced with six steel bars, each with a 28 mm diameter. Knowing that ( E_{s}=200 	ext{ GPa} ) and ( E_{c}=29 	ext{ GPa} ), determine the normal stresses in the steel and in the concrete when a 1560 kN axial centric force P is applied to the post.
SOLUTION
Let
( P_{c}= ) portion of axial force carried by concrete
( P_{s}= ) portion carried by the six steel rods
( delta=frac{P_{c} L}{E_{c} A_{c}} quad P_{c}=frac{E_{c} A_{c} delta}{L} )
( delta=frac{P_{s} L}{E_{s} A_{s}} quad P_{s}=frac{E_{s} A_{s} delta}{L} )
( P=P_{c}+P_{s}=left(E_{c} A_{c}+E_{s} A_{s}
ight) frac{delta}{L} )
( varepsilon=frac{delta}{L}=frac{P}{E_{c} A_{c}+E_{s} A_{s}} )
( A_{s}=6 frac{pi}{4} d_{s}^{2}=frac{6 pi}{4}(28)^{2}=3694.5 mathrm{~mm}^{2} )
( A_{c}=frac{pi}{4} d_{c}^{2}-A_{s}=frac{pi}{4}(450)^{2}-3694.5=155348.6 mathrm{~mm}^{2} )
( L=1.35 mathrm{~m}=1350 mathrm{~mm} )
( varepsilon=frac{-1560 	imes 10^{3}}{left(frac{29 	imes 10^{9}}{10^{6}}
ight)(155348.6)+left(frac{200 	imes 10^{9}}{10^{6}}
ight)(3694.5)}=-297.48 	imes 10^{-6} )
( sigma_{s}=E_{s} varepsilon=left(frac{200 	imes 10^{9}}{10^{6}}
ight)left(-297.48 	imes 10^{-6}
ight)=-59.5 mathrm{~N} / mathrm{mm}^{2}=-59.5 mathrm{MPa} )
( sigma_{c}=E_{c} varepsilon=left(29 	imes frac{10^{9}}{10^{6}}
ight)left(-297.48 	imes 10^{-6}
ight)=-8.627 mathrm{~N} / mathrm{mm}^{2}=-8.627 mathrm{MPa} )

$PROBLEM 2.35 2.35 The 1.35 m concrete post is reinforced with six steel bars, each with a 28 mm diameter. Knowing that ( Es=200 ext GPa ) and ( Ec=29 ext GPa ), determine the normal stresses in the steel and in the concrete when a 1560 kN axial centric force P is applied to the post. SOLUTION Let ( Pc= ) portion of axial force carried by concrete ( Ps= ) portion carried by the six steel rods ( delta=fracPc LEc Ac quad Pc=fracEc Ac deltaL ) ( delta=fracPs LEs As quad Ps=fracEs As deltaL ) ( P=Pc+Ps=left(Ec Ac+Es As ight) fracdeltaL ) ( varepsilon=fracdeltaL=fracPEc Ac+Es As ) ( As=6 fracpi4 ds^2=frac6 pi4(28)^2=3694.5 mathrm mm^2 ) ( Ac=fracpi4 dc^2-As=fracpi4(450)^2-3694.5=155348.6 mathrm mm^2 ) ( L=1.35 mathrm m=1350 mathrm mm ) ( varepsilon=frac-1560 imes 10^3left(frac29 imes 10^910^6 ight)(155348.6)+left(frac200 imes 10^910^6 ight)(3694.5)=-297.48 imes 10^-6 ) ( sigmas=Es varepsilon=left(frac200 imes 10^910^6 ight)left(-297.48 imes 10^-6 ight)=-59.5 mathrm N / mathrmmm^2=-59.5 mathrmMPa ) ( sigmac=Ec varepsilon=left(29 imes frac10^910^6 ight)left(-297.48 imes 10^-6 ight)=-8.627 mathrm N / mathrmmm^2=-8.627 mathrmMPa )$

Added by Ibrahim A.

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