Q $5($b$)$

$[10$ Marks$]$

Figure $7$ is an idealised representation of a machine member subjected to the action of an alternating force $F$ which places the member in completely reversed bending. The piece has a rectangular cross section and a thickness of $10$ mm throughout. The material is BS$080$M$50$ steel hardened and tempered to $200$ Bhn with an ultimate tensile strength of $690$ MPa $.$ All surfaces are ground. Based on $85\%$ reliability, infinite life, and a minimum factor of safety, determine the maximum value of the alternating force $F$ which can be applied.

Figure $7$: machine member $($all dimensions are millimeters$)$