Mohr's Circle For Plane Stress & Plane Strain | Strength of Materials (SOM) - Mechanical Engineering PDF Download

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 lane  which is inclined at angle ‘θ’ from major principal plane, then

Different Stress Diagram

Mohr's circle for plane Stress and StrainNormal Stress

Shear Stress

General State of Stress at an Element

If  σx and σy 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

then,

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

Maximum and Minimum Principal Stresses are:

The radius of Mohr’s circle:

Strength of Materials

Mohr's circle for plane stressed

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 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. radius of 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 radius of circle has the following consideration.
• Radius of 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.
  σ₁ + σ₂ = σ₁ + σ₂

Strain analysis

Principle Strain  

∈_θ + ∈_{θ + 90º} = ∈_x + ∈_y

Relation between Principle strain and stress

Strain Rosetts

• Rectangular Strain Rosette

∈_x = ∈_0º, ∈_y = ∈_90º and Ф = 2∈_45º - (∈_x + ∈_y)

• Equiangular Strain Rosette (delta strain rosette)

∈_{θº =} ∈_x