.A cantilever bean made of cold drawn carbon steel of circular cross section as shown in fig. is subjected to a load which varies from -F to 3F. Determine the maximum load that this member can withstand for an indefinite life using FOS as 2. The theoretical stress concentration factor is 1.42 and the notch sensitivity is 0.9. Assume the following values. Ultimate stress = 550 $N/mm^2$, Yield stress = 470 $N/mm^2$, size factor = 0.85, surface finish factor = 0.89.

$W_{min}=-F$

$W_{max}=3F$

$FOS=2$

$K_f=2$

$q=0.9$

$\sigma _u=550N/mm^2$

$\sigma _y=470N/mm^2$

$K_b=0.85$

$K_a=0.89$

$K_c=1$.....Assume 50% reliability (Refer table in textbook)

$\sigma _{-1}'=0.5\sigma _u=0.5\times 550=275 MPa$

Actual endurance limit

$\sigma _{-1}=(0.89)(0.85)(1).\frac{275}{1.378}$

$\sigma _{-1}=150.97 MPa$

$M_{max}=3F\times 125=375F$

$M_{min}=-F\times 125=-125F$

$\sigma _{b_{max}}=\frac{M_{max}}{Z}=\frac{375F}{\frac{\pi}{32}d^3}=\frac{375F}{\frac{\pi}{32}(13)^3}=1.738F$

$\sigma _{min}=\frac{M_{min}}{Z}=\frac{-125F}{\frac{\pi}{32}(13)^3}=-0.5795F$

$\sigma _M=\frac{1.738F+(-0.5795F)}{2}=0.57925F$

$\sigma _a=\frac{1.738F-(-0.5795F)}{2}=1.159F$

Soderberg equation,

$\frac{1}{2}=\frac{0.57925F}{470}+\frac{1.159F}{150.97}$

$F=56.11N$