Answer to Question #85351 in Mechanics | Relativity for ABC

Question #85351
To find the optimum shape for a cross-section of a beam against bending, beam sections of a square, circle and equilateral triangle with the same cross-sectional area (A) are considered. If the beams have the same length (L), Young’s Modulus (E) and support condition, determine which shape of the beam is the best against bending?
1
Expert's answer
2019-02-27T09:44:02-0500

Let 's consider failure in bending upon momentum "M". The stress on the top surfaces of a symmetric beam is:


"\\sigma=\\frac{M}{Z},"

So higher "Z" is better to reduce the stress. Just compare a square with side "b", circle of radius "r", triangle of side "a":


"Z_{Square}=I_{Square}\\cdot \\frac{b}{2}=\\frac{1}{6} A^{3\/2},"

"Z_{Circle}=I_{Circle}\\frac{1}{r}=\\frac{1}{4\\sqrt{\\pi}} A^{3\/2},"

"Z_{Triangle}=I_{Triangle}\\frac{2\\sqrt{3}}{a}=\\frac{1}{12\\sqrt[4]{3}} A^{3\/2}."

Thus square resists failure in bending better than circle, and circle - better than triangle. See "Materials Selection in Mechanical Design" book.


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