Mohr’s Circle


Question 1
Which of the plot(s) shown is/are valid Mohr's circle representations of a plane stress state in a material? (The center of each circle is indicated by O.)

A
M1
B
M2
C
M3
D
M4
GATE ME 2023   Strength of Materials
Question 1 Explanation: 
Mohr's circle is a graphical representation of plane stress and shear stress.
Mohr's circle is always symmetrical about the x-axis.
Question 2
A linear elastic structure under plane stress condition is subjected to two sets of loading, I and II. The resulting states of stress at a point corresponding to these two loadings are as shown in the figure below. If these two sets of loading are applied simultaneously, then the net normal component of stress \sigma _{xx} is ________.

A
\frac{3\sigma }{2}
B
\sigma \left (1+\frac{1}{\sqrt{2}} \right )
C
\frac{\sigma }{2}
D
\sigma \left (1-\frac{1}{\sqrt{2}} \right )
GATE ME 2022 SET-2   Strength of Materials
Question 2 Explanation: 




\sigma _x=0,\sigma _y=0,\tau _{xy}=0,\theta =-45^{\circ}
\begin{aligned} \sigma _{xx}&=\sigma +\sigma _\theta \\ \sigma _\theta&=\left [ \frac{\sigma _x +\sigma _y}{2} \right ] +\left [ \frac{\sigma _x -\sigma _y}{2} \right ] \cos 2\theta +\tau _{xy} \sin 2\theta \\ \sigma _\theta &= \frac{0+\sigma }{2} +\left [\frac{0-\sigma }{2} \right ] \cos 2(-45)+0\\ \sigma _\theta &= \frac{\sigma }{2}\\ \sigma _{xx} &=\sigma +\sigma _\theta =\sigma +\frac{\sigma }{2}=\frac{3\sigma }{2} \end{aligned}


Question 3
The stress state at a point in a material under plane stress condition is equi-biaxial tension with a magnitude of 10 MPa. If one unit on the \sigma -\tau plane is 1 MPa, the Mohr's circle representation of the state-of-stress is given by
A
a circle with a radius equal to principal stress and its center at the origin of the \sigma -\tau plane
B
a point on the \sigma axis at a distance of 10 units from the origin
C
a circle with a radius of 10 units on the \sigma -\tau plane
D
a point on the \tau axis at a distance of 10 units from the origin
GATE ME 2020 SET-1   Strength of Materials
Question 3 Explanation: 


The given state of stress is represented by a point on \sigma -\tau graph which is located on \sigma-axis at a distance of 10 units from origin.
Question 4
The state of stress at a point in a component isrepresented by a Mohr's circle of radius 100MPa centered at 200 MPa on the normal stress axis. On a plane passing through the same point, the normal stress is 260 MPa. The magnitude of the shear stress on the same plane at the same point is ______ MPa.
A
48
B
63
C
96
D
80
GATE ME 2019 SET-2   Strength of Materials
Question 4 Explanation: 


In triangle CEF
\begin{array}{l} \mathrm{CF}^{2}=\mathrm{CE}^{2}+\mathrm{EF}^{2} \\ 100^{2}=60^{2}=\mathrm{EF}^{2}\\ \mathrm{EF}^{2}=100^{2}-60^{2}=6400 \\ \mathrm{EF}=80 \mathrm{MPa} \end{array}
\mathrm{EF} \rightarrow Represents shear stress at the same point =\mathrm{EF}=\tau=80 \mathrm{MPa}
Question 5
The state of stress at a point, for a body in plane stress, is shown in the figure below. If the minimum principal stress is 10 kPa, then the normal stress \sigma_{y} (in kPa) is
A
9.45
B
18.88
C
37.78
D
75.5
GATE ME 2018 SET-1   Strength of Materials
Question 5 Explanation: 
\begin{aligned} \sigma_{x} &=100 \mathrm{kPa}, \tau_{x y}=50 \mathrm{kPa} \\ \text { Minimum principal stress } &=\frac{\sigma_{x}+\sigma_{y}}{2}-\sqrt{\left(\frac{\sigma_{x}-\sigma_{y}}{2}\right)^{2}+\tau_{x y}^{2}} \\ 10 &=\frac{100+\sigma_{y}}{2}-\sqrt{\left(\frac{100-\sigma_{y}}{2}\right)^{2}+50^{2}} \\ \therefore \quad \sqrt{\left(50-\frac{\sigma_{y}}{2}\right)^{2}+50^{2}} &=50+\frac{\sigma_{y}}{2}-10=40+\frac{\sigma_{y}}{2} \end{aligned}
By squaring
2500+\frac{\sigma_{y}^{2}}{4}-50 \sigma_{y}+2500=1600+\frac{\sigma_{y}^{2}}{4}+40 \sigma_{y}
\begin{aligned} \therefore \quad 90 \sigma_{y} &=3400 \\ \sigma_{y} &=37.78 \mathrm{MPa} \end{aligned}


There are 5 questions to complete.

11 thoughts on “Mohr’s Circle”

  2. Question 12, Explanation given is right but Answer given is wrong. Answer would be option B (175 MPa, 175 MPa) …However Answer Provided is Opion D (0,0) which is wrong.

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