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σ_x = 30 : σ_y = 30 : τ_xy = 0,

Mohr Circle will be a point with centre (30,0) and radius 0 units

 τ_max = 0 , radius = 0

Maximum principal stress theory (Rankine’s theory)

According to this theory, permanent set takes place under a state of complex stress, when the value of maximum principal stress is equal to that of yield point stress as found in a simple tensile test.

For design criterion, the maximum principal stress (σ1) must not exceed the working stress ‘σy’ for the material.

Note: For no shear failure τ ≤ 0.57 σy

Graphical representation

For brittle material, which do not fail by yielding but fail by brittle fracture, this theory gives satisfactory result.

The graph is always square even for different values of σ1 and σ2.

Maximum principal strain theory (ST. Venant’s theory)

According to this theory, a ductile material begins to yield when the maximum principal strain reaches the strain at which yielding occurs in simple tension.

 For no failure in uni – axial loading.

 For no failure in tri – axial loading.

 For design, Here, ϵ = Principal strain

σ1, σ2 and σ3 = Principal stresses   

Graphical Representation

This story over estimate the elastic strength of ductile material.

Maximum shear stress theory

(Guest & Tresca’s Theory)

According to this theory, failure of specimen subjected to any combination of load when the maximum shearing stress at any point reaches the failure value equal to that developed at the yielding in an axial tensile or compressive test of the same material.

Graphical Representation

σ1 and σ2 are maximum and minimum principal stress respectively.

Here, τmax = Maximum shear stress

σy = permissible stress

This theory gives satisfactory result for ductile material.

Maximum strain energy theory (Haigh’s theory)

According to this theory, a body complex stress fails when the total strain energy at elastic limit in simple tension.

Graphical Representation.

This theory does not apply to brittle material for which elastic limit stress in tension and in compression are quite different.

Maximum shear strain energy / Distortion energy theory / Mises – Henky theory.

It states that inelastic action at any point in body, under any combination of stress begging, when the strain energy of distortion per unit volume absorbed at the point is equal to the strain energy of distortion absorbed per unit volume at any point in a bar stressed to the elastic limit under the state of uniaxial stress as occurs in a simple tension / compression test.

It gives very good result in ductile material.  

It cannot be applied for material under hydrostatic pressure.

All theories will give same results if loading is uniaxial.

On plane of principal stress, shear stress is always zero but in the plane of maximum shear stress, normal stress may exists.

Shear stresses on mutually perpendicular planes are equal due to moment equilibrium.

Maximum shear stress is numerically half of the difference between maximum and minimum principal stress.

Let the maximum tensile stress is σ.

According to maximum strain energy theory

But σ₂ is compressive

As per maximum shear strain energy theory,


σ_y = 90√2 MPa 

Equivalent stress

= σ₁−μσ₂

= 100 – 0.50 × 100 = 50 N/mm²

According to von Mises theory, for yielding

σ_yt = 245.76 MPa

Radius of Mohr’s circle = 100 MPa
And centre of Mohr’s circle is at a distance,
 from the origin. Here it is in origin.

As, σ_x + σ_y = 0 both the normal stress may be zero which is a pure shear case or opposite in nature which don’t exist in any of the options. So it is the pure shear case, where the radius is

Maximum and Minimum Normal stress is given by:


σ_x = 250

σ_y = 60




τ_xy = 175.99 Mpa