Principal Planes (Shear Strenth of Soil)

Definition of Principal Planes

A three-dimensional stress system is applied to a soil mass. The stresses in the third direction, on the other hand, are irrelevant. As a consequence, the stress system is reduced to a two-dimensional model. Ordinarily, the simple strain conditions are considered, in which the strain in the third direction is zero. Under strip footing, such conditions can occur.

There are three planes on which the shear stresses are zero at any point in a stressed body. They are mainly mentioned as principal planes.

The compressive stresses on the principal planes are at their highest or lowest levels. The plane with the greatest compressive stress (σ1) is called a major principal plane, while the plane with the least compressive stress (σ3) is called a minor principal plane.

The third principle plane is subjected to the stress which is intermediate between (σ1) and (σ3) which is of not much relevance. Only the major principal is stress (σ1) and the minor principal stress (σ3) are important.

In soil engineering, unlike in solid mechanics, compressive stresses are positive and tensile stresses are negative. Tensile stresses are uncommon in soil engineering issues, so this is solely to prevent unnecessary negative signs. principal planes diagram

Also Read : Effective Stress Concept in Soil Mechanics
Also Read : Darcy’s Law & Its Limitations

Principal Planes Formula and Calculation

Consider a soil component. A plane perpendicular to the intermediate principal plane is shown in Fig. Horizontal is the main principal plane, and vertical is the minor principal plane. Let us consider a plane (AB) which is inclined at an angle Θ to the major principal plane (AC).

Let σ (sigma) be the normal stress and τ (tau) be the shear stress on AB.

Resolving the forces acting on the wedge ABC in the x direction.

$\sigma _{3}BC=\sigma AB.sin\theta -\tau AB.cos \theta$

$\sigma _{3}\frac{BC}{AB}=\sigma sin\theta-\tau cos\theta$

But, $\frac{BC}{AB}=sin\theta$

$\sigma _{3} sin\theta=\sigma sin\theta-\tau cos\theta$ ….(eqn 1)

Resolving the forces in y direction :

$\sigma _{1} AC=\sigma AB.cos\theta+\tau AB.sin\theta$

$\sigma _{1} cos\theta=\sigma cos\theta+\tau sin\theta$ ….(eqn 2)

Multiplying equation (eqn 1) by cosΘ and (eqn 2) by sinΘ and substracting :

$(\sigma _{1}-\sigma _{3})sin\theta .cos\theta=\sigma( cos\theta. sin\theta-cos\theta. sin\theta)+\tau(sin^{2}\theta+cos^{2}\theta)$

$(\sigma _{1}-\sigma _{3})sin\theta .cos\theta=\tau$

$\tau =\frac{1}{2}(\sigma _{1}-\sigma _{3})sin2\theta$ …..(eqn 3)

Substituting in equation (eqn 1)

$\sigma_{3}sin\theta=\sigma sin\theta-\frac{\sigma_{1}-\sigma_{3}}{2}sin2\theta.cos\theta$

$\sigma=\sigma_{3}+(\sigma_{1}-\sigma_{3})\frac{1+cos2\theta}{2}$

$\sigma=\frac{\sigma_{1}+\sigma_{3}}{2}+(\frac{\sigma_{1}-\sigma_{3}}{2}) cos2\theta$ …..(eqn 4)

Equation 3 and 4 give the stresses on an inclined plane AB making an angle (Θ) with the major principal plane. The angle is measured in the anticlockwise direction.

Principal Planes (Shear Strenth of Soil)

Definition of Principal Planes

Principal Planes Formula and Calculation

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Definition of Principal Planes

Principal Planes Formula and Calculation

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