Static Dynamic Loading and Failure Theories


Question 1
A structural member under loading has a uniform state of plane stress which in usual notations is given by \sigma _x=3P,\sigma _y=-2P,\tau _{xy}=\sqrt{2}P , where P \gt 0. The yield strength of the material is 350 MPa. If the member is designed using the maximum distortion energy theory, then the value of P at which yielding starts (according to the maximum distortion energy theory) is
A
70 Mpa
B
90 Mpa
C
120 Mpa
D
75 Mpa
GATE ME 2022 SET-2   Machine Design
Question 1 Explanation: 
Given,
\sigma _x=3P
\sigma _y=-2P
\tau =\sqrt{2}P
According to maximum distortion energy theory,
\begin{aligned} \sqrt{\sigma _x^2-\sigma _x\sigma _y+\sigma _y^2+3\tau _{xy}^2} &=\frac{S_{yt}}{FOS} \\ P\sqrt{3^2-3(-2)+(-2)^2+3(\sqrt{2})^2}&= \frac{350}{1}\\ P \times 5&=350 \\ P&=70\; MPa \end{aligned}
Question 2
The von Mises stress at a point in a body subjected to forces is proportional to the square root of the
A
total strain energy per unit volume
B
plastic strain energy per unit volume
C
dilatational strain energy per unit volume
D
distortional strain energy per unit volume
GATE ME 2021 SET-2   Machine Design
Question 2 Explanation: 
Condition for failure as per M.D.E.T.
Distortion energy per unit volume under tri-axial state of stress > Distortion energy per unit volume under uni-axial state of stress.
\begin{aligned} &\text { Hence, }\left(\frac{1+\mu}{6 E}\right)\left[\left(\sigma_{1}-\sigma_{2}\right)^{2}+\left(\sigma_{2}-\sigma_{3}\right)^{2}+\left(\sigma_{1}-\sigma_{3}\right)^{2}\right]>\left(\frac{1+\mu}{3 E}\right)\left(S_{y t}\right)^{2}\\ &\left(\sigma_{1}-\sigma_{2}\right)^{2}+\left(\sigma_{2}-\sigma_{3}\right)^{2}+\left(\sigma_{1}-\sigma_{3}\right)^{2}>2\left(S_{y t}\right)^{2}\\ &\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}+\sigma_{3}^{2}+\sigma_{1} \sigma_{2}-\sigma_{2} \sigma_{3}-\sigma_{1} \sigma_{3}}>\left(S_{y t}\right) \text { (Von Mises effective stress) } \end{aligned}
S_{y t} = Von Mises effective stress is defined as the uni-axial yield stress that would create same distortion energy created by the tri-axial state of stress.


Question 3
A machine part in the form of cantilever beam is subjected to fluctuating load as shown in the figure. The load varies from 800 N to 1600 N. The modified endurance, yield and ultimate strengths of the material are 200 MPa, 500 MPa and 600 MPa, respectively.

The factor of safety of the beam using modified Goodman criterion is _______ (round off to one decimal place).
A
1.2
B
2
C
2.5
D
2.9
GATE ME 2021 SET-1   Machine Design
Question 3 Explanation: 


A : Critical Point
\begin{aligned} \sigma_{\max , A} &=\sigma_{b, \max } \text { at } A \text { due to } 1600 \mathrm{~N}=\frac{6 \mathrm{M}}{b d^{2}}=\frac{6 \times 1600 \times 200}{12 \times(2 \sigma)^{2}} \\ \sigma_{\max } &=200 \mathrm{MPa} \\ \sigma_{\min , A} &=\sigma_{b, \max } \text { at } A \text { due to } 800 \mathrm{~N} \\ &=\frac{6 \times 800 \times 200}{12 \times(20)^{2}}=100 \mathrm{MPa} \\ \text { Modified Goodman } &=\frac{\sigma_{m}}{S_{y t}}+\frac{\sigma_{a}}{\sigma_{e}} \leq \frac{1}{N} \\ \sigma_{m} &=\left|\frac{\sigma_{\max }+\sigma_{\min }}{2}\right|=150 \mathrm{MPa} \\ \sigma_{a} &=\left|\frac{\sigma_{\max }-\sigma_{\min }}{2}\right|=50 \mathrm{MPa} \\ \frac{150}{600}+\frac{50}{200} & \leq \frac{1}{N} \\ N & \leq 2 \\ N & \approx 2\\ \text { Langer, } \qquad \frac{\sigma_{m}}{S_{y t}}+\frac{\sigma_{a}}{S_{y t}} & \leq \frac{1}{N} \\ \frac{150}{500}+\frac{50}{500} & \leq \frac{1}{N} \qquad \qquad \qquad \qquad (N\leq2.5)\\ N &=2.5 \end{aligned}
Modified Goodman = Safe result of [Goodman or Langer]
Question 4
Endurance limit of a beam subjected to pure bending decreases with
A
decrease in the surface roughness and decrease in the size of the beam
B
increase in the surface roughness and decrease in the size of the beam
C
increase in the surface roughness and increase in the size of the beam
D
decrease in the surface roughness and increase in the size of the beam
GATE ME 2019 SET-2   Machine Design
Question 4 Explanation: 
Endurance limit decreases with increase in surface roughness and with increase in size of the beam.
Question 5
A bar is subjected to a combination of a steady load of 60 kN and a load fluctuating between -10 kN and 90 kN. The corrected endurance limit of the bar is 150 MPa, the yield strength of the material is 480 MPa and the ultimate strength of the material is 600 MPa. The bar cross-section is square with side a. If the factor of safety is 2, the value of a (in mm), according to the modified Goodman's criterion, is ________ (correct to two decimal places).
A
20.96
B
45.36
C
31.62
D
52.45
GATE ME 2018 SET-2   Machine Design
Question 5 Explanation: 


\begin{array}{l} P_{m}=\frac{P_{\max }+P_{\min }}{2} \\ P_{a}=\frac{P_{\max }-P_{\min }}{2} \\ P_{m}=100 \mathrm{kN} \\ \mathrm{Pa}=50 \mathrm{kN} \\ \sigma_{\mathrm{m}}=\frac{100 \times 10^{3}}{a^{2}} \mathrm{MPa} \\ \sigma_{a}=\frac{50 \times 10^{3}}{a^{2}} \mathrm{MPa} \end{array}
Solution by Goodman Equation,
\begin{aligned} \frac{\sigma_{m}}{S_{u t}}+\frac{\sigma_{a}}{\sigma_{e}} &=\frac{1}{N} \\ 100\left[\frac{100}{a^{2} \times 600}+\frac{50}{150 a^{2}}\right] &=\frac{1}{2} \\ a^{2} &=1000 \\ a &=31.62 \mathrm{mm} \end{aligned}
Solution by Langar equation,
\begin{aligned} \frac{\sigma_{m}}{S_{y t}}+\frac{\sigma_{a}}{S_{y t}}&=\frac{1}{N}\\ 100\left[\frac{100}{480 a^{2}}+\frac{50}{480 a^{2}}\right] &=\frac{1}{2} \\ a^{2} &=625 \\ a &=25 \mathrm{mm} \end{aligned}
Hence final answer by modified Goodman's Griterion is 31.62 mm.


There are 5 questions to complete.

2 thoughts on “Static Dynamic Loading and Failure Theories”

Leave a Comment