A similar paradox and a clue to its source and possible resolution can be seen clearly in the solution for a uniform (isotropic) loading on each face (Figure 8.4a) of a wedge.

**Uniform Loading Cases **

A similar paradox and a clue to its
source and possible resolution can be seen clearly in the solution for a
uniform (isotropic) loading on each face (Figure 8.4a) of a wedge. From the
boundary condition that at any radius when,

*m *must equal 2 and
using only the symmetric terms

Clearly, since there is
no singularity in the solution, its nonexistence for 2ϕ = 180^{o} introduces a more
fundamental ambiguity. For wedge angles less than 90^{o} where there is no constraint on Eθ no matter what the radius, the isotropic field seems reasonable, but for 2ϕ > 90o , it seems suspect.
Certainly equilib-rium, compatibility, and the stress boundary conditions are
satisfied for any wedge angle, but there seems to be some added condition on
deformations that precludes a simple isotropic field.

The same is true for the accepted
solution for a wedge with uniform tension and compression on the faces (Figure
8.4b). Again, by similar reasoning, *m = *2, the distribution of σ* _{r}*
should be asymmetric and, therefore, we choose:

Again this solution
cannot be correct in the far field for the half-space where the horizontal
stress σ* _{y}*
is a function of v. Moreover, the
solution has the same singularity at 2ϕ=

Finally, if we add the
two previous cases together as in Figure 8.4 where *w*/2*=*p *= **q*, this gives the solution for a wedge with a uniform load along* *the
top surface (Figure 8.5).

The special cases of a vertical cut, ϕ=45o , and the half-space are shown in Figure 8.5 b and c. Again in the far field under the load, the solution is sus-pect, but near the origin it gives sensible results. An example of how wedge solutions can be combined to give a solution for a haunched beam with uni-form loading is shown in Figure 8.5d. Other examples are given in the chap-ter problems.*

**Example 8.2**

A wedge is loaded by a uniform shearing
traction, *s*, on its “upper” face as shown below. Determine the general
stress function in terms of unknown coefficients *A*, *B*, *C*,
and *D*.

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Civil : Principles of Solid Mechanics : Wedges and the Half-Space : Uniform Loading Cases |

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