A rectangular section $(h^{**}b)$ cantilever beam is subjected to a vertical load $($ F $)$ located at the free$-$end. The load is fluctuating and produces an upward deflection at the free end that ranges from $"$s$"$ to $"5$s$".$ A crack of initial size $"$ai$"$ have been detected at the bottom surface of the beam.

Data: $E=206$GPa,$L_1=0\mathrm{.}1m,L_2=0\mathrm{.}9m,h=60mm,b=15mm,s=1\mathrm{.}6mm,a_i=1mm,C={2\mathrm{.}00}^{**}{10}^{-12}($when $K_1$ is in MPa.m ${?}^{0\mathrm{.}5}),m=3\mathrm{.}6,K_{1c}=45$MPa.$m^{0\mathrm{.}5}.$

AF

Note: As the value of $"$ Y $"$ varies with the size of the crack... first... find the initial final crack size $"a_{f,i}"$ with the $Y_i($value of $Y$ obtained with the initial crack size $a_i)...$ then use $"a_{f,i}"$ to obtain $($again$)$ the value of $Y_f($value of $Y$ obtained with the final crack size $af,i).$ Use the $Y_{\text{a}\text{v}\text{e}\text{r}\text{a}\text{g}\text{e}}=\frac{Y_f+Y_i}{2}$ to obtain the final crack size $a_f$ and the life of the component.

Determine the number of cycles for fracture.

An edge crack is located at the mid$-$span, bottom part of a rectangular beam that is loaded by a two vertical forces $($F$)$ and one horizontal force $(10$F$)$ as shown in the figure.