Answer to Question #314425 in Mechanical Engineering for Kamal

Question

A solid square rod is cantilevered at one end. The rod is 0.6 m long and supports a completely reversing transverse load at the other end of ±2 kN. The material is AISI 1080 hot-rolled steel. If the rod must support this load for 104 cycles with a factor of safety of 1.5 at 100°C, what dimension should the square cross section have? Neglect any stress concentrations at the support end.

Answer 1

$L=0.6 m,F a ​ =2 kN,n=1.5,N=10 4 cycles, S ut ​ =770 MPa,S y ​ =420 MPa$

First evaluate the fatigue strength.

$S e ′ ​ =0.5(770)=385 MPa$

$S e ′ ​ =0.5(770)=385 MPa$

ince the size is not yet known, assume a typical value of $k_{b}​ = 0.85$ and check later. All other modifiers are equal to one.

$S e ​ =$

$=0.488(0.85)(385)=160 MPa$

$S ut ​ =770/6.89=112 kpsi$

$f = 0.83$

$a=\frac{\left(f S_{u t}\right)^{2}}{S_{e}}=\frac{[0.83(770)]^{2}}{160}=2553 MPa$

$b=-\frac{1}{3} \log \left(\frac{f S_{u t}}{S_{e}}\right)=-\frac{1}{3} \log \left(\frac{0.83(770)}{160}\right)=-0.2005$

$S_{f}=a N^{b}=2553\left(10^{4}\right)^{-0.2005}=403 MPa$

Now evaluate the stress.

$M max ​ =(2000 N)(0.6 m)=1200 N⋅m$

$σ_a =σ _{max}=\frac{Mc}{I} = \frac{6M}{B^3}= \frac{7200}{b^3} Pa$

$\sigma_{max}= \frac{7200}{0.030^3}\times 10^-6= 267 Mpa$

$n_y= \frac{S_y}{\sigma_{max}}=\frac{420}{267}=1.57$