PROBLEM $2\mathrm{.}3(20$ Points$)$ A cantilever beam is subject to unknown normal and tensile forces. The normal force is directed downward and has magnitude $F.$ The tensile force has magnitude $10F.$ Four identical strain gauges, each with gauge factor $2,$ are attached to the beam and connected to the arms of a Wheatstone bridge circuit. When the forces are applied, the circuit's output voltage is $0\mathrm{.}25$ mV $.$ Determine the unknown force magnitude $F$ in kN $.$

Note: The magnitude of the bending strain at the beam's surface at distance $L$ from the free end is $6F\frac{L}{E_{m}b{t}^2},$ where $b$ and $t$ are the beam's width and thickness.

Beam properties

thickness: $t=0\mathrm{.}01m$

width: $b=0\mathrm{.}04m$

elastic modulus: $60$ Gpa

Poisson's ratio: $0\mathrm{.}32$