A bar of length L and Young's modulus is subjected to an axial force. Compare the spring constants of bars with cross sections in the form of a solid circle, square, and hollow circle. Determine which of these cross section leads to an economical desing for a specified value of axial stiffness of the bar.
The axial stiffness of the bar can be written as: $$ k = \frac{AE}{L} $$ The cross-section is solid circular with diameter d can be written as: $$ A_1 = \frac{\pi d^2}{4} $$ The cross-section is square with side \(d\) can be written as: $$ A_2 = d.$$ The cross-section is hollow circular with diameter \(d\) and the thickness \(t = 0.1\) can be written as: $$ A = \pi d t = \pi d (0.1d) = 0.1\pi d^2$$ For specified value of k, the cross-section are is constant and will be labeled as \(c\). $$ A = \frac{kL}{E} = c $$ Weight of the bar.
With solid circular section: $$ G_1 = \frac{\pi d^2}{4}L = cL $$ The hollow circular section: $$ G_3 = 0.1 \pi d^2 L$$ $$ G_3 = 0.1 \pi \left(\frac{4c}{\pi}\right) L $$ $$ G_3 = 0.4 cL = 0.4 G_1 $$ The square section: $$ G_2 = d^2 L = \frac{4c}{\pi} L $$ $$ G_2 = \frac{4}{\pi} = 1.2732 G_1 $$ The shaft with the hollow circular cross-section corresponds to minimum weight.
The axial stiffness of the bar can be written as: $$ k = \frac{AE}{L} $$ The cross-section is solid circular with diameter d can be written as: $$ A_1 = \frac{\pi d^2}{4} $$ The cross-section is square with side \(d\) can be written as: $$ A_2 = d.$$ The cross-section is hollow circular with diameter \(d\) and the thickness \(t = 0.1\) can be written as: $$ A = \pi d t = \pi d (0.1d) = 0.1\pi d^2$$ For specified value of k, the cross-section are is constant and will be labeled as \(c\). $$ A = \frac{kL}{E} = c $$ Weight of the bar.
With solid circular section: $$ G_1 = \frac{\pi d^2}{4}L = cL $$ The hollow circular section: $$ G_3 = 0.1 \pi d^2 L$$ $$ G_3 = 0.1 \pi \left(\frac{4c}{\pi}\right) L $$ $$ G_3 = 0.4 cL = 0.4 G_1 $$ The square section: $$ G_2 = d^2 L = \frac{4c}{\pi} L $$ $$ G_2 = \frac{4}{\pi} = 1.2732 G_1 $$ The shaft with the hollow circular cross-section corresponds to minimum weight.