Answer to Question #85351 in Mechanics | Relativity for ABC

Question

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?

Answer 1

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.