Question 5.17: A machine component is subjected to fluctuating stress that ...

\text { Given } S_{u t}=600 N / mm ^{2} \quad S_{y t}=450 N / mm ^{2} .

S_{e}=270 N / mm ^{2} \sigma_{\max .}=100 N / mm ^{2} .

\sigma_{\min .}=40 N / mm ^{2} .

Step I Permissible mean and amplitude stresses

\sigma_{a}=\frac{1}{2}(100-40)=30 N / mm ^{2} .

\sigma_{m}=\frac{1}{2}(100+40)=70 N / mm ^{2} .

S_{a}=n \sigma_{a}=30 n .

S_{m}=n \sigma_{m}=70 n .

where n is the factor of safety.
Step II Factor of safety using Gerber theory
From Eq. (5.46),

\frac{S_{a}}{S_{e}}+\left(\frac{S_{m}}{S_{u t}}\right)^{2}=1           (5.46).

\frac{S_{a}}{S_{e}}+\left(\frac{S_{m}}{S_{u t}}\right)^{2}=1 .

or      \left(\frac{30 n}{270}\right)+\left(\frac{70 n}{600}\right)^{2}=1 .

n^{2}+8.16 n-73.47=0 .

Solving the above quadratic equation,

n = 5.41                (i).

Step III Factor of safety using Soderberg line
The equation of the Soderberg line is as follows,

\frac{S_{a}}{S_{e}}+\frac{S_{m}}{S_{y t}}=1 .

\left(\frac{30 n}{270}\right)+\left(\frac{70 n}{450}\right)=1 .

n = 3.75          (ii).

Step IV Factor of safety using Goodman line
The equation of the Goodman line is as follows:

\frac{S_{a}}{S_{e}}+\frac{S_{m}}{S_{u t}}=1 .

\left(\frac{30 n}{270}\right)+\left(\frac{70 n}{600}\right)=1 .

n = 4.39                (iii).

Step V Factor of safety against static failure
The factor of safety against static failure is given by,

n=\frac{S_{y t}}{\sigma_{\max }}=\frac{450}{100}=4.5           (iv).