Problem $3$

Version $1.$ The fully solid cylinder in Figure $2$ is simply supported at points A and E and is manufactured with a shoulder at point B and a groove at point D $.$ Rigid arms AG and EK are fixed to the cylinder and subjected to static forces P and F $.$ With shoulder and groove radii of $6\mathrm{.}8$ mm and $P=3F,$ compute the principal stresses as a function of P and identify the critical section based on the distortion$-$energy theory $($i$.$e$.,$ calculate the von Mises effective stress$).$ You may neglect bending shear. The indication of a good design will be if all critical von Mises effective stress values are of the same order or magnitude.

Figure $3$ Fully solid cylinder with shoulder at point B and groove at point D$.$

Version $2.$ What if we try to save some weight with the same dimensions and loads, except that we now use a smaller diameter of $30$ mm between A and B $($instead of $40$ mm $).$ Calculate the new von Mises effective stress at point B$.($Note that this change does not change the shear and bending moment diagrams$).$ Is this still a good design?

Disclaimer: Normally we don't worry so much about ductile material stress concentration points, as opposed to brittle stress concentration points. However, for the purposes of these calculations, we are using a ductile material.