To begin, consider a simple case of active earth pressure on an inclined wall face with a uniformly sloping backfill. Backfill is made up of homogeneous, elastic, and isotropic cohesionless soil. Consider a unit length of the wall perpendicular to the plane of the paper. As shown in figure 1 below, the forces acting on the sliding wedge are :
- W = the weight of the soil contained in the sliding wedge,
- R = the soil reaction across the plane of sliding, and
- The active thrust ‘Pa’ against the wall, which reacts from the wall on to the sliding wedge.
Above figure 1 depicts a triangle of forces. For determining the active thrust, Pa, using the terminology of fig 1, one can proceed as follows :
Substituting and simplifying,
……..(Eq. no 1)
From the triangles of forces,
Substituting for W,
……..(Eq. no 2)
The maximum value of Pa is obtained by equating the first derivative of Pa with respect to θ to zero;
or and substituting the corresponding value of θ.
The value of Pa so obtained is written as :
……(Eq. no 3)
This is usually written as
Where,
……(Eq. no 4)
Ka being the coefficient of active earth pressure.
……(Eq. no 5)
by substituting α = 90°, β = 0°, and δ = Φ.
For a smooth vertical wall retaining a backfill with horizontal surface,
α = 90°, δ = 0, and β = 0;
Which is the same as the Rankine value. In fact, for this simple case, one may proceed from fundamentals as follows:
With reference of fig 2
……(Eq. no 6)
For maximum value of Pa,
or,
or sin Φ cos (2θ – Φ) = 0, on simplification.
or
Substituting in Eq. no 6,
……(Eq. no 7)
as obtained by substitution in the general equation. Surprisingly, this method is sometimes referred to as Rankine’s method of Trial Wedges.
A few representative values of Ka from Eq. no 4 for certain values of Φ, δ, α and β are shown in table below :
| δ↓φ→ | 20° | 30° | 40° | |
|---|---|---|---|---|
| α = 90° | β = 0° | |||
| 0° | 0.49 | 0.33 | 0.22 | |
| 10° | 0.45 | 0.32 | 0.21 | |
| 20° | 0.43 | 0.31 | 0.20 | |
| 30° | ... | 0.30 | 0.20 | |
| α = 90° | β = 10° | |||
| 0° | 0.51 | 0.37 | 0.24 | |
| 10° | 0.52 | 0.35 | 0.23 | |
| 20° | 0.52 | 0.34 | 0.22 | |
| 30° | .... | 0.33 | 0.22 | |
| α = 90° | β = 20° | |||
| 0° | 0.88 | 0.44 | 0.27 | |
| 10° | 0.90 | 0.43 | 0.26 | |
| 20° | 0.94 | 0.42 | 0.25 | |
| 30° | .... | 0.42 | 0.25 |
Also Read : Coulomb’s Wedge Theory & It’s Assumptions
The “Trial-Wedge Process” is an obvious graphical approach that comes to mind. In this process, a few trial rupture surfaces with different inclinations are assumed, with the horizontal passing through the heel of the wall.
The triangle of forces is completed with each trial surface, and the value of Pa is determined. If an adequate number of intelligently designed trial rupture surfaces are analysed, a ‘Θ – Pa’ plot should appear similar to that shown in fig no 3
The maximum value of Pa from this plot represents the expected total active thrust on the wall per lineal unit, as well as the corresponding value of theta, which represents the inclination of the most likely rupture surface.
Positive & Negative Wall Friction
A few remarks on wall friction may be appropriate at this point. In the active case, the outward stretching causes the backfill soil to move downward in relation to the wall. For the active case, such a downward shear force on the wall is referred to as “positive wall friction”. As shown in Fig. 1, this causes the active thrust exerted on the sliding wedge to incline upward. This means that the active thrust applied to the wall will have a downward inclination.
The horizontal compression in the passive case must be accompanied by an upward bulging of the soil, resulting in upward shear on the wall. For the passive case, such upward shear on the wall is referred to as “positive wall friction”. This causes the passive thrust exerted on the sliding wedge to incline downward. This means that the passive resistance exerted on the wall will incline upward.
Wall friction is almost always positive in the active case. Negative wall friction can occur under certain circumstances, such as when a portion of the backfill soil immediately behind the wall is excavated for repair and the wall is braced against the remaining earth mass of the backfill.
In the passive case, either positive or negative wall friction can develop. This sign of wall friction must be determined through an examination of the motions that are expected in each field situation. When wall friction is present, the rupture surface takes on a curved rather than a plane shape. Fig no 4 show the nature of the surface for positive and negative wall friction values, respectively.
The angle of wall friction ‘δ’, will not be greater than ‘Φ’; at the maximum it can equal Φ, for a rough wall with a loose fill. For a wall with dense fill, δ will be less than Φ. It may range from ½Φ to ¾Φ in most cases; it is usually assumed as (2/3)Φ in the absence of precise data.
The possibility of δ shifting from +Φ to –Φ in the worst case should be considered in the design of a retaining wall. The value of Ka for the case of a vertical wall retaining a fill with a level surface, in which Φ ranges from 20° to 40° and δ ranges from 0° to Φ, may be obtained from the chart given in fig no 5
In the active case of Coulomb theory, the assumption of plane failure is incorrect by only a small amount. Fellenius demonstrated that the assumption of circular arcs for failure surfaces results in active thrusts that are generally not more than 5% greater than the corresponding values from the Coulomb theory.