# Fatigue Strength and S-N Diagram

 Question 1
The S-N curve from a fatigue test for steel is shown. Which one of the options gives the endurance limit? A $S_{ut}$ B $S_{2}$ C $S_{3}$ D $S_{4}$
GATE ME 2023   Machine Design
Question 1 Explanation:
For some ferrous (iron base) and titanium alloys, the S-N curve becomes horizontal at higher N values; or there is a limiting stress level, called the fatigue limit (also sometimes the endurance limit), below which fatigue will not occur. This fatigue limit represents the largest value of fluctuating stress that will not cause failure for essentially an infinite number of cycles. For many steels, fatigue limits range between 35% and 60% of the tensile strength. Question 2
The figure shows the relationship between fatigue strength (S) and fatigue life (N) of a material. The fatigue strength of the material for a life of 1000 cycles is 450 MPa, while its fatigue strength for a life of $10^6$ cycles is 150 MPa. The life of a cylindrical shaft made of this material subjected to an alternating stress of 200 MPa will then be cycles (round off to the nearest integer).
 A 163840 B 124589 C 365247 D 457812
GATE ME 2021 SET-2   Machine Design
Question 2 Explanation: Equation of line $A \bar{B}:$
\begin{aligned} y-y_{1} &=\frac{\left(y_{2}-y_{1}\right)}{\left(x_{2}-x_{1}\right)}\left[x-x_{1}\right] \\ \log _{10} 200-\log _{10} 450 &=\frac{\log _{10} 150-\log _{10} 450}{(6-3)}\left[\log _{10} N-3\right] \\ N &=163840.580 \text { cycles } \end{aligned}

 Question 3
A machine member is subjected to fluctuating stress $\sigma =\sigma _0 \cos (8 \pi t)$. The endurance limit of the material is 350 MPa. If the factor of safety used in the design is 3.5 then the maximum allowable value of $\sigma _0$ is __________ MPa (round off to 2 decimal places).
 A 68.5 B 78.2 C 122.5 D 100
GATE ME 2020 SET-2   Machine Design
Question 3 Explanation:
\begin{aligned} \text{Fluctuating stress, } \quad \sigma&=\sigma_{o} \cos (8 \pi t) \\ \sigma_{\max } &=\sigma_{o} \\ \sigma_{\min } &=-\sigma_{o} \\ \sigma_{\text {man }} &=\frac{\sigma_{\max }+\sigma_{\min }}{2}=0 \\ \sigma_{v} &=\frac{\sigma_{\max }-\sigma_{\min }}{2}=\frac{2 \sigma_{o}}{2}=\sigma_{o} \\ \sigma_{e} &=350 \mathrm{MPa} \\ \mathrm{FOS} &=3.5 \end{aligned}
From strength criteria, $\frac{\sigma_{v}}{\sigma_{e}} \leq \frac{1}{\mathrm{FOS}}$
\begin{aligned} \frac{\sigma_{o}}{350} & \leq \frac{1}{3.5} \\ \sigma_{o} & \leq 100 \mathrm{MPa} \end{aligned}
 Question 4
During a high cycle fatigue test, a metallic specimen is subjected to cyclic loading with a mean stress of +140 MPa, and a minimum stress of -70 MPa. The R-ratio (minimum stress to maximum stress) for this cyclic loading is_______ (round off to one decimal place)
 A 0 B -0.8 C -0.2 D -0.5
GATE ME 2019 SET-1   Machine Design
Question 4 Explanation:
$\begin{array}{l} \sigma_{\min }=-70 \mathrm{MPa} \\ \sigma_{\text {mean }}=140 \mathrm{MPa} \\ \Rightarrow \sigma_{\text {mean }}=\frac{\sigma_{\min }+\sigma_{\max }}{2}=140 \\ \Rightarrow \frac{-70+\sigma_{\max }}{2}=140 \\ \Rightarrow \sigma_{\max }=350 \mathrm{MPa} \\ \text { The R-ratio, } \mathrm{R}=\frac{\sigma_{\min }}{\sigma_{\max }}=-\frac{70}{350}=-0.2 \end{array}$
 Question 5
Fatigue life of a material for a fully reversed loading condition is estimated from
$\sigma_{a}=1100N^{-0.15}$
where $\sigma_{a}$
is the stress amplitude in MPa and N is the failure life in cycles.The maximum allowable stress amplitude (in MPa) for a life of $1\times10^{5}$ cycles under the same loading condition is ________(correct to two decimal places).
 A 150.25 B 195.61 C 240.36 D 280.48
GATE ME 2018 SET-2   Machine Design
Question 5 Explanation:
\begin{aligned} \frac{\sigma_{\max }-\sigma_{\min }}{2} &=1100 \mathrm{N}^{-0.15} \\ \frac{\sigma_{\max }-\left(-\sigma_{\max }\right)}{2} &=1100 \mathrm{N}^{-0.15} \\ \frac{2 \sigma_{\max }}{2} &=1100 \mathrm{N}^{-0.15} \\ \sigma_{\max } &=1100 \mathrm{N}^{-0.15}=1100 \times\left(10^{5}\right)^{-0.15} \\ &=1100 \times(10)^{-0.75}=\frac{1100}{5.62} \\ \sigma_{\max } &=195.61 \mathrm{MPa} \end{aligned}

There are 5 questions to complete.