The effect of intermediate principal was studied by D.C. Drucker and W. Prager in 1952, who proposed concept of a three dimensional yield envelope.

The stress condition during plastic equilibrium can be represented by equation of Mohr-Coulomb,
\(\frac{?_1-?_3}{2}-\frac{?_1+?_3}{2} sin?=c cos?,\)
since tan??=sin?/cos?
? \(?_1=2c \,tan?(45�+\frac{?}{2})+?_3 tan^2 (45�+\frac{?}{2}).\)

The ory on which computation of stress in a state of plastic equilibrium is based is called ory of plasticity.

Since all points in auxiliary plane represent deviatorial state of stress, plane is also called deviatoric plane.

For case of passive state, major principal stress denoted by ?₁ is in horizontal direction and minor principal stress ?₃ is in vertical direction.
? \(\frac{?_1}{?_3} =\frac{?_h}{?_v} =K_p=tan^2 (45�+\frac{?}{2}). \)

Concept:

Critical depth/Unsupported vertical cut (Hc): It is depth up to which vertical excavation of cohesive soil can stand without any lateral support.

${\mathrm{H}}_{\mathrm{c}}={\displaystyle \frac{4\times c}{\gamma \times \sqrt{K_a}}}$

${\mathrm{k}}_{\mathrm{a}}={\displaystyle \frac{1-\mathrm{s}\mathrm{i}\mathrm{n}?}{1+\mathrm{s}\mathrm{i}\mathrm{n}?}}$

Where

ka = Rankine’s coefficient of active earth pressure

? = Angle of internal friction

γ = Density of soil

c = cohesion

Calculation:

? = 0  for soft clay, γ = 20 kN/m³, H = 4 m

${\mathrm{k}}_{\mathrm{a}}={\displaystyle \frac{1-\mathrm{s}\mathrm{i}\mathrm{n}?}{1+\mathrm{s}\mathrm{i}\mathrm{n}?}}$

${\mathrm{k}}_{\mathrm{a}}={\displaystyle \frac{1-\mathrm{s}\mathrm{i}\mathrm{n}0}{1+\mathrm{s}\mathrm{i}\mathrm{n}0}}=1$

${\mathrm{H}}_{\mathrm{c}}={\displaystyle \frac{4\times c}{\gamma \times \sqrt{K_a}}}$
$4={\displaystyle \frac{4\times c}{20\times 1}}$

c = 20 kN/m²

During passive state of plastic equilibrium, major principal stress denoted by ?₁ is in horizontal direction and minor principal stress ?₃ is in vertical direction. Therefore, movement of wall will be towards back fill.

Concept

Cantilever Retaining wall

The horizontal pressures on backfill side will push wall outward, which will tend to slide on its footing. The driving force from applied loads must be resisted by an opposite friction force at interface of footing base and underlying soil, produced by bearing pressure against base.

If back of wall were vertical, n force acts on wall. On or hand, if back of wall were sloping, n force acts on vertical soil plane as illustrated in Figure . The total force acts along back of wall at a height of H/3 from base of wall.

Based on assumptions of wedge ory:
i. The backfill is dry, cohesion less, homogeneous, isotropic, and elastically undeformable.
ii. The slip surface is a plane which passes through heel of wall
iii. The sliding wedge itself acts as a rigid body.

In practice, earth pressures for retaining walls of less than 6m height are obtained from graphs or tables.