A force doublet is commonly defined as two equal but opposite forces acting in an infinite medium

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A force doublet is commonly defined as two equal but opposite forces acting in an infinite medium as shown in the following figure. Develop the stress field for this problem by superimposing the solution from Example 13.1 onto that of another single force of –P acting at the point z =–d. In particular, consider the case as d→0 such that the product Pd→D, where D is a constant. This summation and limiting process yield a solution that is simply the derivative of the original Kelvin state. For example, the superposition of the radial stress component gives:

lim [o,(r,z)  o,(r,z+ d)] : = OR dor z TRO -d- D 8T (1- - v) dz The other stress components follow in an

11 Z P P d y

Data from example 13.1

Consider the problem (commonly referred to as Kelvin's problem) of a single concentrated force acting at a

where A is an arbitrary constant to be determined. We shall now show that this potential produces the correct

Clearly, these stresses (and displacements) are singular at the origin and vanish at infinity. To analyze the

P N P Resultant Boundary Condition Evaluation y

Equation 13.5.9

OR = 0, sin + Ocos+2t, sincoso % = o.sin + o,cos -2t, sincos TR = (0, -) sinocos - t, (sino- cos)

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