FV-Example 4 - Determine the minimum weight of cantilever beam

A cantilever beam of length L and Young's modulus E is subjected to a bending at its free end. Compare the spring constants of beams with cross sections in the form of a solid circle, square, and hollow circle. Determine which of these cross sections leads to an economical design for a specified value of bending stiffness of the beam. $$ k = \frac{3EI}{l^3}$$ $$ I = \frac{kl^3}{3E} = C.$$ For a solid circular section with diameter d. $$ I_1 = \frac{\pi d^4}{64} = C $$ $$ d^4 = \frac{64 C}{\pi} \rightarrow d^2 = \sqrt{\frac{64C}{\pi}} $$ $$ G_1 = \frac{\pi d^2}{4}l = \frac{\pi l}{4}\sqrt{\frac{64C}{\pi}}$$ $$ G_1 = 3.5449 l \sqrt{C}.$$ For a hollow circular section with mean diameter \(d\) and wall thickness t=0.1d, weight of beam is equal to: $$ G_2 = \frac{\pi l}{4}(d_o^4 - d_1^4)$$ $$ G_2 = \frac{\pi l}{4}((d+t)^2 - (d-t)^2)$$ $$ G_2 = \frac{\pi l}{4}4dt = \pi dtl$$ $$ G_2 = \pi l (0.1d^2) = 0.1 \pi l \sqrt{\frac{64C}{\pi}}$$ $$4 G_2 = 1.4180 l \sqrt{C}$$ For a square section with side d, weight of the beam \(G_3\) is: $$ G_3 = d^2 l = l \sqrt{\frac{64C}{\pi}} $$ $$ G_3 = 4.5135 l \sqrt{C}$$