The obliquity angles at the wall and on the failure plane are of opposite sign in the passive case, as opposed to the active case. In the Coulomb theory, a plane failure surface is assumed for the passive case as well, but the critical plane is the one with the lowest passive thrust. As shown in Fig. 1, the failure plane is at a much smaller angle to the horizontal than in the active case.
Figure 1 depicts the force triangle. The passive resistance ‘Pp’ can be calculated using the following terminology:
…..(as in active case)
From the triangle of forces,
Substituting for W,
….(eqn 1)
The minimum value of Pp is obtained by differentiating eqn 1 with respect to θ
equating to zero, and substituting the corresponding value of θ.
The value of Pp so obtained may be written as :
where,
For a vertical wall retaining a horizontal backfill and for which the friction is equal to φ,
α = 90°, β = 0°, and δ = φ, and Kp reduces to,
For a smooth vertical wall retaining a horizontal backfill,
α = 90°, β = 0° and δ = 0°,
which is the same as the Rankine value.
As with the active case, it is possible to start with the fundamentals in this simple case.
(θ + φ) takes the place of (θ – φ) and (45° + φ/2) that of (45° – φ/2) in the work relating to
the active case.
Also Read : Culmann’s Graphical Method
Also Read : Skempton’s Pore Pressure Parameters
Coulomb’s theory with a plane surface of failure is only valid if the wall friction in relation to passive resistance is zero. When wall friction is present, the passive resistance obtained by plane failure surfaces is much higher than that obtained by assuming curved failure surfaces, which is closer to the truth. With increasing wall friction, the error increases. As a result, there are errors on the unsafe side.
Terzaghi (1943) proposed a more rigorous type of analysis based on the assumption of a curved failure surface (logarithmic spiral form) similar to those shown in Fig 2.
Coefficient of Passive Earth Pressure From Curved Failure Surfaces
| δ ↓ φ → | 10° | 20° | 30° | 40° |
|---|---|---|---|---|
| 0° | 1.42 | 2.04 | 3.00 | 4.60 |
| φ/2 | 1.56 | 2.60 | 4.80 | 10.40 |
| φ | 1.65 | 3.00 | 6.40 | 17.50 |
| -φ | 0.72 | 0.58 | 0.54 | 0.52 |
Sokolovski’s (1965) method could also be used. This produces essentially the same results as the previous method. Theoretical predictions for passive resistance with wall friction are not as well supported by experimental evidence as those for active thrust, and thus cannot be relied upon as much. In this regard, Tschebotarioff (1951) presents the results of a few large-scale laboratory tests.