By Ian Naismith Sneddon (auth.), S. Flügge (eds.)
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From these equations it follows that [q;" (z) H = [ip" (z) H = [Vi' (z) J1 = o. 3) If the displacements are single-valued we have from Eq. 15) [~q;(z)- zip'(z)- 1ji(z)J1 = o. Differentiating this with respect to z we have [~q;'(z)- ip'(z)]1 = o, and comparison with Eq. 1) shows that = [q;'(z)J1 [ip'(:Z)J1 = 0. 4) We now find expressions for the components X, Y of the resultant stress and for the resultant couple, M, acting on an arc AB in the material. We have X+ i Y = J (Pxn + ipyn) ds AB 1 CoKER and FILON: Photoelasticity.
7) II. Airy stress function. 28. The biharmonic equation. 3) it is supposed that the body force components X, Yare derived 1 2 L. N. G. FILON: Phil. Trans. Roy. Soc. , Ser. A 201, 63 (1903). G. B. AIRY: Brit. Assoc. Rep. 1862. Phil. Trans. Roy. Soc. Lond. 153, 49 (1863). Sect. 28. 43 The biharmonic equation. 1) and that there exists a function X such that T xy o2x ox oy' =--- then they reduce to the forms o·-a ( - -o-x n V) =0. 2) where the function X is known as the Airy stress function. 2) should provide valid solutions of elastic problems they must not only satisfy the equations of equilibrium but also the compatibility relation for the stresses.
10) and, from Eq. 2) we see that in the absence of body forces, the corresponding stresses are ax=c3 x+d3 y, ay=a3 x+b3 y, ixy=-b 3 x-c3 y. 11) Various cases are obtained from this set of stresses by making some of the coefficients zero. If, for example, only a3 is non-zero, aY is the only non-zero stress component; but if one of b3 , c3 is the only non-zero coefficient, a shearing stress is present in addition to a normal stress. When polynomials of higher degree than the third are used, the bi-harmonic Eq.