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Comparison with Active and Passive Earth Pressures
1
Introduction
In this simple example, Slope/W is used to determine the theoretical active and passive earth pressures,
and the position of the critical slip surfaces for a homogeneous vertical cut slope. The solution is then
compared with Rankine’s theory of active and passive earth pressures.
Analysis method: Spencer
Two analyses
Homogeneous material
Purely frictional material
Use of Entry and Exit slip surfaces
2
Configuration and setup
Figure 1 shows the geometry of a simple homogeneous vertical cut slope. A frictional material with Phi =
30o is used. A line load representing the active or passive force is assumed to act at the lower 1/3 of the
vertical slope. Figure 1 also shows the use of the Entry and Exit slip surface option in the searching of the
critical slip surfaces. Note that a single point exit zone is used to force the slip surfaces to always exit at
the toe of the vertical slope. Pore water pressure condition is ignored in this comparison study.
Figure 1 Geometry and material property of the simple slope with vertical cut
SLOPE/W Example File: Comparison with active and passive earth pressures (pdf) (gsz)
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SLOPE/W solution – active case
Using Rankine’s active pressure theory, the active failure plane should be (45 + /2) = 60 o,, and the
active pressure force acting on the slope can be calculated as:
Active Force = Ka
 H2
2
Active Force  0.334 x
 tan2 (45 
  H2
2
)
2
20 x10 x10
 334 kN
2
By applying a line load of 334 kN at the lower 1/3 point of the vertical slope, the line load should balance
out the active force of the slope, and a factor of safety equal to 1.0 should be obtained with the critical
failure plane at 60 degrees. Figure 2 shows the solution from SLOPE/W. Note that SLOPE/W gives
essentially the identical solution as given by Rankine’s active earth pressure theory.
Figure 2 SLOPE/W solution of the active case
4
SLOPE/W solution – passive case
Using Rankine’s passive pressure theory, the passive failure plane should be (45 - /2) = 30 o,, and the
passive pressure force acting on the slope can be calculated as:
Passive Force = Kp
 H2
 tan2 (45 
  H2
)
2
2
2
20 x10 x10
Passive Force  3.0 x
 3000 kN
2
The above equation implies that when a line load of 3000 kN is applied at the lower 1/3 point of the
vertical slope, the line load should balance out the passive force of the slope, and a factor of safety equal
to 1.0 should be obtained with the critical failure plane at 30 degrees. Figure 3 shows the solution from
SLOPE/W Example File: Comparison with active and passive earth pressures (pdf) (gsz)
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SLOPE/W. Note that SLOPE/W gives essentially the identical solution as given by Rankine’s passive
earth pressure theory.
Passive Force = Kp
 H2
 tan2 (45 
  H2
)
2
2
2
20 x10 x10
Passive Force  3.0 x
 3000 kN
2
Figure 3 SLOPE/W solution of the passive case
5
Discussion
Figure 4 illustrates what happens to the factors of safety of the cut slope when the lateral line load is
changed in both the active and the passive cases. The factor of safety is less than 1.0 when the lateral
force is less than the active force. As the lateral force increases, the factor of safety increases. At a factor
of safety of 1.0, the lateral force is equal to the active force. A further increase in the lateral force results
in a further increase in the factor of safety.
As the lateral force approaches the at-rest condition, the factor of safety tends towards infinity. The reason
this happens is because the gravitational driving force is balanced by the lateral force. Since factor of
safety is a ratio of the resisting force divided by the driving force, the factor of safety approaches infinity
when the driving force is approaching zero.
Once on the passive side, a further increase in the lateral force results in a further decrease in the factor of
safety. The passive earth force occurs when the factor of safety is again equal to 1.0.
SLOPE/W Example File: Comparison with active and passive earth pressures (pdf) (gsz)
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Infinity
Force
Force
Active Case
Passive Case
1.0
Active
Force
At-Rest
Force
Passive
Force
Lateral Force
Figure 4 Factor of safety versus lateral force
As illustrated in Figure 4, note the reverse in direction in the resisting force between the active and
passive case. This reverse in direction is needed, since the slope is moving downstream in the active case,
but upstream in the passive case. In the previous version of SLOPE/W, this reverse in direction must be
modeled by using a negative cohesion and frictional angle in the passive case. In the current version, this
is no longer needed. You can simply select the “Allow passive mode” option in the KeyIn Analysis - Slip
Surface tab. SLOPE/W will handle the directional change in resistance forces for the passive case
automatically.
6
Conclusion
The factors of safety as well as the slip surface positions computed by SLOPE/W compare nicely with
Rankine’s theory on active and passive earth pressures. This simple example confirms that SLOPE/W is
formulated correctly.
SLOPE/W Example File: Comparison with active and passive earth pressures (pdf) (gsz)
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