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 Pa"Â (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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