Mohr’s Circle for Plane Stress and Plane Strain Notes for Civil Engineering Exams

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Updated on: September 25th, 2023

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Mohr’s circle is the locus of points representing the magnitude of normal and shear stress at the various plane in a given stress element. Graphically, a variation of normal stress and shear stress are studied with the help of Mohr’s circle.

σ₁ and σ₂ are Principal Stresses, then normal and shear stress on the lane which is inclined at angle ‘θ’ from the major principal plane, then

Mohr’s Circle for Plane Stress and Plane Strain Notes for Civil Engineering Exams

Normal stress

Shear stress

The General State of Stress at an Element:

If σ_x and σ_y are normal stress on vertical and horizontal plane respectively and this plane is accompanied by shear stress then normal stress and shear stress on the plane, which is inclined at an angle θ from the plane of

Mohr’s Circle for Plane Stress and Plane Strain Notes for Civil Engineering Exams

then,

Mohr’s Circle for Plane Stress and Plane Strain Notes for Civil Engineering Exams

Let σ_x, σ_y be two normal stresses(both tensile) and τ_xy be shear stress then:

Maximum and Minimum Principle Stresses are:
The radius of Mohr’s circle:

Strength of Materials

Mohr’s Circle for Plane Stress and Plane Strain Notes for Civil Engineering Exams

Observations from Mohr’s Circle

The following are the observations of Mohr’s circle as

At point M on circle σ_n is maximum and shear stress is zero.

∴ Maximum principal stress ≡ coordinate of M

At point N on the circle, σ_n is minimum and shear stress τ is zero.

∴ minimum principal stress ≡ coordinate of N

At point P on Circle τ is maximum.

Maximum shear stress ≡ ordinate of P(i.e. the radius of the circle)

Also, normal stress on the plane of maximum shear stress:
σ_n= abscissa of P

Where σ_n ≡ Average stress
Mohr’s circle becomes zero at a point if the radius of the circle has the following consideration.
The radius of the circle:
If σ_x = σ_y, then radius of Mohr’s circle is zero and τ_xy = 0.
The sum of normal stresses acting on perpendicular faces of a plane stress elements is constant and independent of the angle θ.

Strain analysis:

Mohr’s Circle for Plane Stress and Plane Strain Notes for Civil Engineering Exams

Relation between Principle strain and stress:

Mohr’s Circle for Plane Stress and Plane Strain Notes for Civil Engineering Exams

Strain Rosetts:

Rectangular Strain Rosette:

Equiangular Strain Rosette (delta strain rosette):

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Mohr’s Circle for Plane Stress and Plane Strain Notes for Civil Engineering Exams

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