Unfortunately Kötter’s equation cannot be integrated directly for a closedform solution in the general case of a Coulomb material with weight or for nonconstant boundary loads.

**The General Case**

Unfortunately Kötter’s
equation cannot be integrated directly for a closedform solution in the general
case of a Coulomb material with weight or for nonconstant boundary loads. The
most common numerical procedure for integrating Kötter ’s equation was
developed by V.V. Sokolovski* where the integration proceeds from one boundary
to the other in the direction of the characteristic slip lines using a finitedifference
approximation. This shooting technique called “the method of characteristics”
is necessary since, as already discussed, the governing equations are
hyperbolic and, if the load is specified completely on the entire boundary, a
solution will not normally exist.

The result then may be
an “exact” solution for the slip field but only approximate the load
prescribed. Take for example the footing problem shown again in Figure 12.19.
We would like to solve for the *uniform* load intensity *p _{L}*
that will cause collapse by forcing the material to shear under the punch and
bulge upward lifting the

The resulting slipline
field and limit loading *p*(*x*) are shown in Figure 12.19b for a
smooth, surface footing (*q* = 0) with *c* = γ = 1
and ϕ = 30º. Neither the width of the footing, *b*, nor the distribution of
the limit load *p _{L}* can be specified in advance and we see that

method that we can make
as accurate as we want to solve for the wrong load! One can use an equivalent
limit load intensity

to approximate a
uniform failure loading but that will still underestimate the capacity since
the moment around 0 is less. Reversing the direction of the shooting technique
does no good in that if *p* is specified constant, then *q* becomes a
function of *x*.

If *q* is not
constant, then as shown in Figure 12.19c, the slip surface *s* with γ != 0 is neither straight, a log spiral, or circular in any zone. A variety of
solutions using the method of characteristics are presented in the literature
for the bearing capacity problem, slopes, and retaining walls for a variety of
surcharge distributions. Since these seldom correspond to realisticboundary
conditions they are primarily of academic interest.

Another approach to the
general problem is to incorporate the effect of weight using the same critical
slip surface derived by Prandtl for γ=0.
This cannot be correct, but a reasonable argument can be made that it is
conservative. This is the approach used by Terzaghi* in presenting his general
bearing capacity formula:

where *N*γ like the other two “Bearing Capacity Factors” *N _{c}* and

The power of the upperbound
approach to limit analysis, however, is supposed to be the simplicity of the
method. To return to this theme, let us conclude by introducing a simple
failure mechanism based, for the bearing capacity problem, on Coulomb wedges,
which is more versatile, easy for computation, and is intuitively satisfying.

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Civil - Principles of Solid Mechanics - Slip Line Analysis : Slip Line Analysis: The General Case |

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