Chapter 3, Problem 46P

Consider a simply supported beam of rectangular cross section of constant width b and variable depth h. so proportioned that the maximum stress σ_x at the outer surface due to bending is constant, when subjected to a load F at a distance a from the left support and a distance c from the right support. Show that the depth h at location x is given by

$h\text{ = }\sqrt{\frac{6Fcx}{lb{\sigma }_{\max }}}$ 0 ≤ x ≤ a