$5-$Supl$1.$ A solid bar and a tube have the same cross$-$sectional area, meaning that they are made with the

same amount of material. The solid circular section has a radius of $c=0\mathrm{.}03m.$ The tubular section has an

outer radius of $c_o=0\mathrm{.}05m$ and an inner radius of $c_i=0\mathrm{.}04m.$

Compute and compare $T_{\text{m}\text{a}\text{x}}$ assuming the allowable shear stress of $100$ MPa $,$ for each of the two

cross sections.

Which is a better design?

Plot the shear stress distribution along a radial line in both cases when $T_{\text{m}\text{a}\text{x}}($which is different in

the two cases, based on $($a$))$ is applied.

$5-$Supl$1.$ Hints:

It is known that: ${}_{\text{a}\text{l}\text{l}\text{o}\text{w}}={}_{\text{m}\text{a}\text{x}}=\frac{T_{\text{m}\text{a}\text{x}}{P}_{\text{m}\text{a}\text{x}}}{J}$

Solid bar: ${}_{\text{m}\text{a}\text{x}}=c,J=\frac{}{2}{c}^4,$ and solve $T_{\text{m}\text{a}\text{x}}.$

Tubular bar: ${}_{\text{m}\text{a}\text{x}}=c_{0}J=\frac{}{2}(c_o^4-c_i^4),$ and solve $T_{\text{m}\text{a}\text{x}}$

Compare the results. The larger $T_{\text{m}\text{a}\text{x}}$ suggests a better design! Think why

Your plot of the stress along a radial line should look like $($not exactly the

same as$)$ the Figure on the right. It is always linear from $0$ in the center to

the max on the outer surface.