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A parabolic curve joining 'S_e' on the stress amplitude axis and '$S_{ut}$' on the mean stress axis is called the

**Gerber parabola**.According to the Gerber criteria, the region below this curve is considered to be safe. The equation for the Gerber parabolic curve is,

$(\frac{S_m}{S_{ut}})^2$ + $(\frac{S_a}{S_{e}})$ = 1 ............(1)

$(\frac{N_F. \sigma_m}{S_{ut}})^2$ + $(\frac{N_F. \sigma_a}{S_{e}})$ = 1 ............(2)

This equation (1) and (2) repersent the Gerber parabola.

*Note* :-

1) When load is purely static then ---> $\sigma_a$ is zero failure criteria is $S_{ut}$ or $S_{yt}$

2) When load is variable then stress is completely reversing ---> $\sigma_m$ is zero and hence, Failure criteria is endurance strength ($S_e$)

** VVIMP**:- 1) Prove Soderberg Equation :-..

Consider, any point P on line CD from similar triangular $\triangle PQD$ and $\triangle COD$, we can write,

$\frac{PQ}{CO}$ = $\frac{QD}{OD}$ = $\frac{OD-OQ}{OD}$ = 1- $\frac{OQ}{OD}$

$\therefore \frac{\sigma_a}{(\frac{S_e}{N_F})} = [1- \frac{\sigma_m}{(\frac{S_{yt}}{N_f})}]$

$\therefore \sigma_a = (\frac{S_e}{N_F}) [1- \frac{N_f \times \sigma_m}{S_{yt}}]$

$\therefore \sigma_a = S_e [\frac{1}{N_f}-\frac{\sigma_m}{S_{yt}}]$

$\frac{\sigma_a}{S_e}$ = $\frac{1}{N_f}$ - $\frac{\sigma_m}{S_{yt}}$

$\frac{1}{N_F}$ = $\frac{\sigma_a}{S_e}$ + $\frac{\sigma_m}{S_{yt}}$ ==> Known as Soderberg equation.

**2) Prove Goodman Equation:-**

Consider any point P on line CD. Similar triangles $\triangle PQD$ and $\triangle COD$, we can write:-

$\frac{PQ}{CO}$ = $\frac{QD}{OD}$ = $\frac{OD-OQ}{CO}$ = 1- $\frac{OQ}{OD}$

$\therefore \frac{\sigma_a}{(\frac{S_e}{N_F})} = [1- \frac{\sigma_m}{(\frac{S_{ut}}{N_f})}]$

$\therefore \sigma_a = S_e [\frac{1}{N_f}-\frac{\sigma_m}{S_{ut}}]$

[$\frac{\sigma_a}{S_e}$ + $\frac{\sigma_m}{S_{ut}}$ = $\frac{1}{N_F}$] ===> known as Goodman equation.

**Actual Endurance limit** :-

Actual endurance limit is defined as the maximum value of completely reversed stress that a standard specimen can sustain for an infinite no of cycles without fatigue failure.

*For Ductile Material* (i.e steel)

$\sigma_{-1}$ = 0.5 $\sigma_u$

*For Brittle Material* (i.e. cast iron)

$\sigma_{-1}$ = 0.4 $\sigma_u$

It shows that approximate relationship between the endurance limit and ultimate tensile strength.

• Endurance limit of actual component to be designed = $\sigma_{-1}$ = $K_a.K_b.K_c.(\frac{\sigma_{-1}}{K_F})$

Where, $\sigma_{-1}$ = Endurance limit for rotating beam standard specimen.

$K_a$ = surface finish factor,

$K_b$ = size factor

Dia.(mm) | $K_b$ |
---|---|

D<7.5 | 1 |

7.5<d<50</td> | 0.85 |

d>50 | 0.75 |

$K_c$ = Reliability factor

Reliability (R%) | $K_c$ |
---|---|

50% | 1 |

90% | 0.897 |

95% | 0.868 |

99% | 0.814 |

99.9% | 0.753 |

99.99% | 0.702 |

99.999% | 0.659 |

$K_f$ = Fatigue stress concentration factor

**Summary:**

- Goodman Equation :- [Based on ultimate strength]

1) $\frac{1}{n}$ = $\frac{\sigma_M}{\sigma_u}$ + $\frac{\sigma_a}{\sigma_{-1}}$

2) $\frac{\sigma_a}{(\frac{\sigma_{-1}}{FoS})}$ = 1- $\frac{\sigma_m}{(\frac{\sigma_u}{FoS})}$

Where n = F.O.S

- Soderberg Equation [Based on yield strength]

1) $\frac{1}{n}$ = $\frac{\sigma_M}{\sigma_y}$ + $\frac{\sigma_a}{\sigma_{-1}}$

2) $\frac{\sigma_a}{(\frac{\sigma_{-1}}{FoS})}$ = 1- $\frac{\sigma_m}{(\frac{\sigma_y}{FoS})}$

$K_a$ = PSG 7.17

$K_b$ = Table

$K_c$ = Table

$K_f = 1 + q(K_t - 1)$

$K_t=$ PSG 7.14

$q =$ PSG 7.8 based on r