Shear strength of soil is the resistance to deformation by continuous shear displacement of soil particles by the action of shear stresses.

Shear stresses > Shear Strength, failure takes place

• Failure may be sinking of footing or movement of a wedge of soil behind a retaining wall forcing it to move or slide.
• Shear strength is due to friction between particles, interlocking, and cohesion.
• Principal plane – shear stress = 0
• Normal stresses acting on Principal Planes are called Principal Stresses.
• Critical stress values and Obliquities generally occur on major and minor principal planes (two-dimensional solution)

Mohr-Coulomb failure theory:

Coulomb – 1776

Mohr – 1900 (extended Coulomb’s theory)

The theory states that:

1. Materials essentially fail in shear. the critical shear stress causing shear failure depends upon the properties of the material as well as normal stress on the failure plane.
2. The shear strength is equal to shear stress at failure on a potential plane.
3. In a material subjected to three-dimensional principal stress (Ϭ1, Ϭ2, and Ϭ3), the intermediate principal stress Ϭ2 doesn’t have any influence on the strength of the material.

Mathematically,

τ = f(Ϭ)

τ = shear stress

Ϭ= normal stress

MOHR’S STRESS CIRCLE 

Ϭ – Normal stress

τ -Shear stress

Ϭ1- major principal plane

Ϭ3- minor principal plane

α = Inclination to major plane

Resolving the forces horizontally;

Ϭ3*BC = Ϭ*AC Sinα – τ *AC Cosα

Vertically;

Ϭ1*AB = Ϭ*AC Cosα + τ *AC Sinα

Dividing both sides by AC;

Ϭ3*Sinα = Ϭ*Sinα – τ *Cosα – (i)

And,

Ϭ1*Cosα = Ϭ* Cosα + τ *Sinα- (ii)

Multiplying (i) by Cosα and (ii) by Sinα and subtracting (i) from (ii)

Cosα*Sinα (Ϭ1-Ϭ3) = τ (Sin²α + Cos²α)

Therefore,

τ = {(Ϭ1-Ϭ2)Sinα}/2 – (iii)

Substituting τ in eqⁿ (i)

Ϭ3 Sinα = Ϭ Sinα – {(Ϭ1-Ϭ3)*Sin2α*Cosα}/2

Or, Ϭ3 = Ϭ – {(Ϭ1-Ϭ3)*2Cos²α}/2

Or, Ϭ3 = Ϭ – (Ϭ1-Ϭ3)*Cos²α

Or, Ϭ = Ϭ3 + (Ϭ1-Ϭ3)* Cos²α

Or, Ϭ = Ϭ3 + (Ϭ1-Ϭ3)* {(1+Cos2α)/2}

           = Ϭ3 + {(Ϭ1-Ϭ3)/2} + {(Ϭ1-Ϭ3)/2} *Cos2α

Therefore,

Ϭ = {(Ϭ1+Ϭ3)/2} + {(Ϭ1-Ϭ3)/2} *Cos2α – (iv)

MOHR’S CIRCLE

Values of Ϭ, τ for α are plotted, locus of all the points gives a circle known as Mohr Circle

OE – Normal Stress (inclined at α)

ED – Shear Stress (inclined at α)

A – Minor Principal Stress

B – Major Principal Stress

These plane intersects at A (pole)

Principle Planes inclined to the Co-ordinate Axis :

Draw MN//BP (P = Pole) (Major Plane)

                   Similarly NO//AP           (Minor Plane)

                   OM//P’P  (P’ = Pole of inclined Plane)

                   P’X = τ; OX = Ϭ_n   (for inclined)

Angle of Obliquity:

Relation between Ф and α :

Coulomb failure criterion:

Τ_f = C + Ϭ_nf * tanФ (linear function)

Mohr failure criterion :

Τ_f = f (Ϭ_nf)  (Unique function)

Mohr-Coulomb failure Criterion :

θ= 45 + Ф/2

References:

1. Terzaghi, Karl, Peck, R.B & John, Wiley (1969) Soil mechanics in engineering practice, New York.
2. Arora , K.R (2008), Soil mechanics and foundation engineering, Delhi: Standard Publisher Distribution

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