# Mechanical Metallurgy Questions and Answers – Elastic Behaviour – Mohr’s Circle of Stress in 2-Dimension

This set of Mechanical Metallurgy Multiple Choice Questions & Answers (MCQs) focuses on “Elastic Behaviour – Mohr’s Circle of Stress in 2-Dimension”.

1. Mohr’s Circle represents__________
a) the endurance limit to fatigue failure
b) the state of stress at a point on an oblique plane
c) the area of multiple circles
d) the lattice planes of the crystal

Explanation: Mohr’s circle is a graphical representation developed by O. Mohr to show the state of stress at any point on an obliques plane. Endurance limit is calculated by the S-N curve. Lattice planes are represented by stereographic projection. Area of multiple circles is calculated by basic geometry.

2. The radius of Mohr’s circle is equal to________
a) σx
b) τmax
c) σy
d) (σx+σy)/2

Explanation: The key concept behind the representation of Stress in the form of a circle is that the stress equation 1 & 2 are rearranged in the form of equation (3)
$$\sigma x’-\frac{(\sigma x+\sigma y)}{2}=\frac{\sigma x+\sigma y}{2} cos2θ+τxysin2θ$$—— (1)
$$τy’x’=\frac{\sigma x+\sigma y}{2} sin2θ+τxycos2θ$$——- (2)
$$(\frac{\sigma x’-(\sigma x+\sigma y)}{2})^2+(τx’y’)^2=(\frac{(\sigma x-\sigma y)}{2})^2+(τxy)^2$$——– (3)
General equation of a circle is:
=> (x-a)2 +(y-b)2=r2 —– (4)
So, by comparing equation 3 with 4 we can conclude that when
=> (x-a)=0
=> y=r
=> r = τmax.

3. An angle θ in the physical element is represented by ________ on Mohr’s circle.
a) 2θ
b) 3θ
c) 4θ
d) θ

Explanation: In the case of a physical element the angle between normal stress and principle stress vary from 0 to 90 degree, but in its circular representation it can vary from 0 to 180 degree. So, Mohr’s circle rotates by 2θ when the physical element is rotated by θ.

4. The X and Y- axis of Mohr’s circle represent ________
a) normal stress and shear stress
b) shear stress and normal stress
c) principal normal stress and principal shear stress
d) principal shear stress and principal normal stress

Explanation: X-axis represents the normal stress and Y-axis represents the shear stress. The point of intersection of Mohr’s circle on X-axis represents the principal normal stress. The point of intersection of Mohr’s circle on Y-axis represents the principal shear stress.

5. Draw the Mohr’s circle for the following state of stress:
Principal stresses = 50MPa, -50MPa.
Maximum shear stress = 50MPa.
a) b) c) d) Explanation: Principal normal stresses are the point of intersection of the circle on the x-axis. In this case, it is at 50MPa and -50MPa. So, the diagram with the origin of the circle at center has principal stresses at 50MPa and -50MPa. Also, as radius is same in all the case, shear stress will be 50 MPa for all the given options.

6. Determine the principal normal stress and shear stress for the following case:
σx=240 MPa
σy=-30 MPa
τ= 50 Mpa.
a) σ1=240 MPa, σ2=-30 MPa, τmax=50 MPa
b) σ1=210 MPa, σ2=-40 MPa, τmax=40 MPa
c) σ1=248.96 MPa, σ2=-38.96 MPa, τmax=140.8 MPa
d) σ1=248.96 MPa, σ2=-38.96 MPa, τmax=50 MPa

Explanation: First of all, draw a Mohr’s circle for the following problem.
On graph paper, draw x and y-axes with x-axis denoting normal stress and y-axis denoting shear stress. Locate the point (240,-50) and (-30,50) on the graph and join both points by a straight line. Point where the line intersects the x-axis is the origin of the circle as shown in the figure. Now determine the x coordinates where the y=0 and also y coordinate where x=(240-30)/2=105 MPa.
The point where y=0 MPa;
=> σ1=248.96 MPa, σ2=-38.96 Mpa (principal normal stresses)
The point where x=105 MPa;
=> y= τmax=140.8 MPa (principal shear stress).

7. The point of intersection of the Mohr’s circle on the x-axis represents principal normal stresses because the shear stress at these points is zero.
a) True
b) False

Explanation: Principal normal stress is defined as the plane of zero shear stress. Along the circumference of the circle, only 2 points are present where shear stress is zero (y=0). So, the point of intersection represents the plane of zero shear stress.

8. Is it possible to determine the principal strains by Mohr’s circle?
a) True
b) False

Explanation: Strain on a body along different direction can be represented in the form of a second-degree polynomial equation. This equation on rearrangement converts into the equation of circle. And any stress in form of circle can be represented in form of Mohr’s circle.

9. Which of the following Mohr’s Circle represent the condition of pure uniaxial tension?
a) b) c) d) Explanation: The condition for a uniaxial tension is that only one value of principal normal stress exists. Other principal stresses become zero. Only in the correct option, one of the principal stress is zero. In remaining options, both the principal stresses are non-zero.

10. Which of the following Mohr’s Circle represents the condition of pure shear?
a) b) c) d) Explanation: For pure shear, both the principal normal stresses are always equal and opposite in sign to cancel out the net effect. So only in circle with center at origin will have principal stress of equal magnitude and opposite sign.

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