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# Rankine's Earth Pressure Theory | At Rest, Active and Passive Earth Pressure Formula

Updated: Dec 19, 2020

To analyse the pressure relation between retaining wall and soil mass, the most commonly used theory is Rankine's Earth Pressure Theory. Three conditions are analysed - Wall at rest, Active Earth Pressure and Passive Earth Pressure. Rankine's theory assumes the wall to be smooth and soil cohesionless. This theory is also extended to cohesive soils.

Let us hit the basics first!

## Retaining Wall and Backfill

Every time it is not possible to place the soil until it has great shear strength. Then we construct a wall to keep it in place and will fill the soil back to it. It is called a backfill and the wall is called the retaining wall. If the wall hadn't been there, soil mass could have come out, which implies the soil mass is applying some pressure on the wall, that is called earth pressure. So we should design the retaining wall against the earth pressure.

### Design of Retaining Wall

While designing a retaining wall, the following points should be kept in mind.

Water does not have shear strength hence the stresses in the vertical and horizontal direction are the same.

Whereas the soil has some shear strength and the horizontal stress is a function of vertical stress.

### Types of Wall Movement

The wall will attain 3 conditions at any point of time. According to the wall position, earth pressure changes. Three general conditions of the wall are-

The wall does not move or the wall at rest

The wall moves away from the backfill (Active earth pressure condition)

The wall moves towards the backfill (Passive earth pressure condition)

## Wall at rest

The wall and base of the wall are rigid, which makes the wall stay at a single place without any movement.

As the wall does not move in any direction, the lateral strains (ratio of change in the area to the original area) is zero.

## Active earth pressure

Wall moves away from the backfill

The wall moves in one direction i.e. far from backfill. Wall and its base are not rigid under this case.

As the wall moves away from the soil, because of this some of the pressure of soil gets relieved, hence the shear resistance gets mobilized and it is in opposition to the wall movement.

The shear resistance is in the opposite direction, so, the pressure in the horizontal direction gets reduced.

As the wall keeps moving away, we will reach a condition where full shear resistance would be mobilized. After a point, there will not be any further resistance even if the wall moves away.

That horizontal pressure would be the minimum and it is called Active Earth Pressure.

## Passive earth pressure

Wall moves towards the backfill

Due to the mass of creating wall and the speed which it moves, the mass of the soil(backfill) tends to move upward.

The shear resistance acts in the same direction as horizontal pressure. There will be a situation where the stress can’t be maximum than that, that is the Passive Earth Pressure.

## Rankine's Earth Pressure Theory

Rankine's theory was only applicable to uniform cohesionless soil, as originally proposed. But later in 1915 Bell extended the Rankine's theory for cohesive soil as well.

All the soil mass in plastic equilibrium remains at active/passive earth pressure. This means for every point in soil, shear failure is just about to happen. In reality, only a part of soil mass will be in plastic equilibrium.

### Assumptions of Rankine's theory

Some of the important assumptions of Rankine's earth pressure theory are-

The soil is isotropic and homogenous, which means c, φ and γ have the same values everywhere, and they have the same values in all directions at every point (i.e., the strength on a vertical plane is the same as that on a horizontal plane). NOTE: This theory is also applicable for multi-layered soil having different values of c, φ and γ in different layers.

Where, c: cohesion of soil

φ: angle of internal friction

γ: angle of internal friction of soil

The wall is infinitely long hence the problem may be analyzed in only two dimensions(2D). Geotechnical engineers consider this as a plane strain condition.

The ground surface is a plane (although it does not necessarily need to be level).

There is no wall friction (𝛅 = 0)

The wall moves sufficiently to develop the active or passive condition.

The most critical shear surface is a plane. In reality, it is slightly concave up, but this is a reasonable assumption (especially for the active case) and it simplifies the analysis.

The resultant of the normal and shear forces that act on the back of the wall is inclined at an angle parallel to the ground surface.

### Rankine's earth pressure | Wall at rest

The theory of elasticity is used. Here there is zero displacements, so zero strain. The soil is homogeneous, isotropic and semi-infinite. Elastic modulus and Poisson's ratio is constant for total soil mass.

The 'coefficient of earth pressure at rest condition' or 'at rest earth pressure constant' is denoted as Ko and defined as the ratio of lateral pressure(σh) to vertical pressure(σv).

Ko = σh/σv

At any depth Z, the vertical pressure, σv = γZ

Where γ is the unit weight of soil

From elastic theory, for the soil having Poisson's ratio μ, the coefficient Ko = μ/(1-μ )

For Saturated Clay, the typical value of Poisson's ratio is 0.40-0.50 and the value of Ko is 0.67 to 1.00

For Unsaturated clay, the typical value of Poisson's ratio is 0.10-0.30 and the value of Ko is 0.11 to 0.42

For Sandy Clay, the typical value of Poisson's ratio is 0.20-0.30 and the value of Ko is 0.25 to 0.42

For Silty, the typical value of Poisson's ratio is 0.30-0.35 and the value of Ko is 0.42 to 0.54

For Dense Snad, the typical value of Poisson's ratio is 0.20-0.40 and the value of Ko is 0.25 to 0.67

For Coarse Sand, the typical value of Poisson's ratio is 0.15 and the typical value of Ko is 0.18

For Fine-Grained Soil, the typical value of Poisson's ratio is 0.25 and the typical value of Ko is 0.33

For Rock, the typical value of Poisson's ratio is 0.10-0.40 and the value of Ko is 0.11 to 0.67

According to Jaky (1944), a good approximation for Ko is Ko = 1 - Sinφ, where φ is the angle of internal friction.

Where, Ip: Plasticity Index

OCR: Over Consolidated Ratio; the ratio of pre-consolidation pressure to present effective overburden pressure

So, the lateral earth pressure at any depth Z will be, σh = Ko x σv = KoγZ

The resultant pressure at rest acts at a height of H/3 from the bottom and the pressure is denoted by Po.

Po = Total area of the triangular pressure diagram

So, Po = 0.5 x H x KoγH = 0.5KoγH^2

### Active Earth Pressure Condition

In this condition, the wall moves away from the backfill. The phrase plastic equilibrium in soil refers to the condition where every point in a soil mass is on the verge of failure.

The general formula for the lateral stress (σh) in the active condition is,

Where C: Cohesion of soil

Ka: Active earth pressure coefficient

φ: Angle of internal friction

The pressure distribution diagram can be represented as below.

At zero-depth(z=0), the expression of lateral pressure becomes,

σh=-2C√Ka

The value of σh can also be [-2C x tan(45 - φ/2)].

**For cohesionless soil(C=0), **the expression of lateral pressures is-

In this case, the negative side of pressure distribution diagram will not be there, because at Z=0, σh = 0. The pressure distribution diagram starts from the origin.

**Failure plane**

The failure planes in the soil make (45 + φ/2) degree angles with the direction of the major principal plane, that is horizontal. These are called potential slip planes.

### Passive Earth Pressure Condition

In this condition, the retaining wall moves towards the soil mass/backfill. The formula and expressions are quite similar to the active state.

Some +, - sign changes in the active state formula will do our job. This way it's very easy to remember these formulas.

The general formula for the lateral stress (σh) in the passive condition is,

The pressure distribution diagram can be represented as below.

**For cohesionless soil(C=0), **the expression of lateral pressures is-

**Failure plane**

For Rankine's passive state, the slip planes make (45 - φ/2) degree angles with the direction of the minor principal plane - that is, in the horizontal direction.

## Problem on Rankine's earth pressure theory

**Try this problem!**

This is all for this, we are planning to make another article for the special cases like submerged backfill, surcharged backfill, inclined backfill, etc. Let us know what all you want for the next one in comments.

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