As in the elastic range, the general solution for a general wedge at the limit state can be applied to special cases of particular interest.

**Other Special Cases: Slopes and Footings**

As in the elastic range,
the general solution for a general wedge at the limit state can be applied to
special cases of particular interest. Consider first a free slope (Figure
12.13a) where there is no wall to help restrain the sliding. Since *T *=*
*0,
cos (2ψ + 2β)*
*=*
*0
and therefore the slip surface is unique

Since there is no
closedform elasticity solution for a slope loaded by its own weight, this
result is particularly important in that at least the stress field at collapse
is known to the designer.

and, for a vertical
slope, there is little load capacity beyond first yield.

At
the other extreme, as β approaches zero (a
very gentle slope), ψ approaches

135 o and *H _{cr}* =10.28/γ.

We have also, in fact,
finally derived the exact limitload solution for the footing problem for which
upperbound estimates were made using circular and sliding block mechanisms. The
solution for the passive case for a wall at β= 0 gives it to us directly. If we assume a smooth footing (the wall) then from
Equation (12.25),

135 o giving the failure
mechanism as shown in Figure 12.14. The normal stress on the wall is the
uniform distributed limit load *p _{L}* so

This is considered the
correct solution. The “best” circular and slidingblock mechanisms from previews
pages give *p _{L}* == 5.7

**Example 12.4**

A long semicircular channel with radius *R*
is to be cut from an elastic, perfectlyplastic, incompressible halfspace.

a. Solve
for the elastic stress field before the channel is filled with water and
thereby determine the factor of safety against yield. Assume plane strain.

b. Estimate
the limit load (i.e., the critical limiting radius *R _{L}* for collapse)
and compare to the yield radius

Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail

Civil - Principles of Solid Mechanics - Slip Line Analysis : Slopes and Footings |

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