Answer to Question #102832 in Mechanical Engineering for Tope

Question #102832
A short reinforced concrete column of section 200 mm x 220mm is to be reinforced with 4 numbered steel bars and is required to carru an axial load of 850kN. The stress in the concrete must not exceed 7N/mm^2 and the stress must not exceed 150N/mm^2. Determine the diameter required for the bars and the subsequent stresses occurring in the concrete and the steel under the specified load. (Young's moduli of concrete=14kN/mm^2 and Young's moduli of steel=210kN/mm^2. With suitable sketch
Expert's answer


"(\\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}"

When "\\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" (not okay!)

When "\\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" (okay!)

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


"\\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" Answer

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