Question #227858

a short reinforced concrete column of section 200mm Ã—220mm is to be reinforce with 4 number steel bars and is required to carry n axial load of 850kn. the stress in the concrete must not exceed 7n/mm^2 and the stress in the steel must not exceed 150n/mm^2. determine the diameter required for the bars and the subsequent stresses occuring in the concrete and the steel under the specified load.[ young's modulus; concrete =14kn/mm^2, steel =210kn/mm^2]

Expert's answer

\delta_{st}=\delta_{co}=\delta

(\frac{F\cdot l}{S\cdot E})_{co}=(\frac{F\cdot l}{S\cdot E})_{st}

\frac{\sigma_{co}\cdot l}{14\cdot10^9}=\frac{\sigma_{st}\cdot l}{210\cdot10^9}

14\cdot \sigma_{st}=210\cdot \sigma_{co}

WhenÂ \sigma_{st}=150\cdot 10^6 Pa

14\cdot 150\cdot 10^6=210\cdot \sigma_{co} \to \sigma_{co}=10\cdot 10^6 Pa>6\cdot 10^6 Pa Â (not okay!)

WhenÂ \sigma_{co}=6\cdot 10^6 Pa

14\cdot \sigma_{st}=210\cdot 6\cdot 10^6 \to \sigma_{st}=90\cdot 10^6 Pa<150\cdot 10^6 PaÂ (okay!)

UseÂ \sigma_{st}=90\cdot 10^6 Pa; \sigma_{co}=6\cdot 10^6 Pa

F_{st}+F_{co}=850000

\sigma_{st}\cdot S_{st}+\sigma_{co}\cdot S_{co}=850000

90\cdot 10^6\cdot S_{st}+6\cdot 10^6\cdot(0.2\cdot 0.22-S_{st})=850000

84\cdot S_{st}=0.85-0.264 \to S_{st}=0.006976 m^2=6976m^2

For one steel bar

S_{0st}=6976/4=1744 mm^2

\frac{\pi\cdot D^2}{4}=1744 \to D\approx 47.1 mm

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