For the Coulomb wedge in the active case, the free body diagram for no cohesion is as shown in Figure 12.20a.

**An Approximate “Coulomb Mechanism”**

For the Coulomb wedge
in the active case, the free body diagram for no cohesion is as shown in Figure
12.20a. Let *i* be the angle at which the surface of the backfill rises
above the horizontal,* *β be the angle between the wall and the plane, which is vertical to the
horizontal surface of the ground in front of the wall, and ρ* _{A}*
be the angle between the failure surface and the horizontal.

From the force
equilibrium of the failure wedge, the maximum value of *P _{A}*,
the total force acting on the wall, is:

Similarly, the corresponding
expression for the passive thrust (the free body diagram is shown in Figure
12.20b) is:

A very useful
simplification of the Prandtl or Sokolovski failure surfaces is, as shown in
Figure 12.21, to eliminate the fanshaped transition zone and thereby average
its effect concentrating the shear transfer at AC. Line AC can then be thought
of as a retaining wall with the active lateral thrust *P _{A}* from
region I pushing against the passive resistance

Similarly, considering
only the contribution of the weight of the soil below the footing (*q* =
*c* = 0), at collapse

Finally, it might be
noted that for only cohesion (* *ϕ =0),
equilibrium considerations of the Coulomb mechanism gives a value of *N _{c}*
=
6.0, which is not unreasonable compared to the exact value of 5.14.

The corresponding
staticbearing capacity factors *N _{q}* and

rather than computing
the limit load *P _{L}* using the integral of the limit pressure
distribution. Therefore, they are unnecessarily low since both the curvatures
of the actual critical load intensity computed and its eccentricity are
neglected. If we assume the distribution is parabolic, the value is ~15%
higher, which is a better representation but still conservative. This modification
is reflected in Figure 12.22 which gives the standard accepted value for
bearing capacity factors for design of shallow strip footings.

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Civil : Principles of Solid Mechanics : Slip Line Analysis : An Approximate “Coulomb Mechanism” |

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