Problem $2$

A carbon steel ball $(E=207$GPa,$v=0\mathrm{.}292)$ with a $30-mm$ diameter is pressed against a flat

carbon steel plate with a force of $50$ N $.$

Note: In the contact area calculation, flat substrate corresponds to sphere diameter $d.$

$($a$)$ Determine the maximum contact pressure and contact size.

$($b$)$ Determine the maximum shear stress $({}_{\text{m}\text{a}\text{x}}),$ and the depth below the point of contact

where it will be experienced.