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# 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.

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 (2Žł + 2╬▓) = 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 ╬▓ approaches zero (a very gentle slope), Žł approaches

135┬ o and Hcr =10.28/╬│

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 ╬▓= 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 pL so This is considered the correct solution. The ŌĆ£bestŌĆØ circular and slidingblock mechanisms from previews pages give pL == 5.7c illustrating the upperbound theorem. The symmetry of the correct slip surface shows why the unit weight drops out of the expression for pL when ŽĢ=0.

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 RL for collapse) and compare to the yield radius RY.  Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
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