# Rankines Lateral Earth Pressure

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## Introduction on Rankine's Lateral Earth Pressure :

Lateral earth pressure is the pressure that soil exerts in the horizontal direction. Retaining and sheet-pile walls, both braced and unbraced excavations, grain in silo walls and bins, and earth or rock contacting tunnel walls and other underground structures require a quantitative estimate of the lateral pressure on a structural member for either a design or stability analysis.

## Concepts and Formulas of Rankine's Lateral Earth Pressure:

The equation above relates EFFECTIVE stresses not TOTAL stresses! Thus, for calculation of horizontal total stress in presence of ground water table, first effective vertical stress must be calculated then based on At-restRankine, or Coulomb theory (whichever is appropriate), horizontal effective stress will be calculated. Summing the effective horizontal stress with pore water pressure at that point will result in total horizontal stress.

### Assumptions of Rankine earth pressure theory:

Rankine considered soil in a state of plastic equilibrium and used essentially the same assumptions as Coulomb, except that he assumed no wall friction or soil cohesion.

### Rankine earth pressure coefficients:

There are two commonly uses lateral earth pressure theories: Coulomb (1776) and Rankine (1857).

Rankine active earth pressure coefficient:

$K_{a}=cos\beta \frac{cos\beta-(cos^{2}\beta-cos^{2}\phi)^{1/2}}{cos\beta+(cos^{2}\beta-cos^{2}\phi)^{1/2}}$

For the case where beta = 0:

$K_{a}=tan^{2}(45-\frac{\phi}{2})$

Rankine passive earth pressure coefficient:

$K_{p}=cos\beta \frac{cos\beta+(cos^{2}\beta-cos^{2}\phi)^{1/2}}{cos\beta-(cos^{2}\beta-cos^{2}\phi)^{1/2}}$

For the case where beta = 0:

$K_{p}=tan^{2}(45+\frac{\phi}{2})$

Where

f is internal friction angle of the soil,

is the slope of the backfill

An illustration of differences between at-rest, active and passive states is given in At-Rest State article.

One should not use the Rankine method for Kp when b > 0, since it decreases with increasing b.

### For cohesive soils (Clays):

Neither the Coulomb nor Rankine method explicitly incorporated cohesion as an equation parameter in lateral earth pressure computations. For soils with cohesion, Bell (1915) developed an analytical solution that uses the square root of the pressure coefficient to predict the cohesion's contribution to the overall resulting pressure. These equations represent the total lateral earth pressure (not effective). The first term represents the non-cohesive contribution and the second term the cohesive contribution. The first equation is for an active situation and the second for passive situations.

$\sigma_{h}=K_{a}\sigma_{v}-2c\sqrt{K_{a}}$

$\sigma_{h}=K_{p}\sigma_{v}+2c\sqrt{K_{p}}$

## Solved sample problems of Rankine's Lateral Earth Pressure:

### Example 1: Rankine's lateral earth pressure with horizontal backfill (English units)

Given:

Height of earth at heel, H = 12 ft

Height  of earth at toe

Friction angle of soil:= 30 degree

Horizontal backfill,

Unit weight of backfill soil:g = 115 lb/ft3

Requirement:

Using Rankine's lateral earth pressure

1. determine Rankine total active force, Pa, at heel per foot width of wall

2. determine Rankine's total passive force, Pp at toe per foot width of wall

Solution:

Active earth pressure coefficient:

Ka = tan2(45-f/2) = 0.333

Total active force:

Pa = gH2Ka/2 = 2760 lb/ft               (per one ft width of wall)

Passive earth pressure coefficient:

Kp = tan2(45+f/2) = 3

Total passive force:

Pp = gH2Kp/2 = 690 lb/ft               (per one ft width of wall)

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