By P. R. Lancaster, D. Mitchell

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33 - c2y + 2c33)] = 0. Therefore ex = vwc 3 3EI = vw 2E Thus the central axis is not the neutral axis. , 6 = 5wR. 2 5 5 + ~)] 2 Again, for deep plates the second term in the bracket, which is due to shear, will be significant. 5 is the stress compatibility equation in rectangular coordinates. It is often more convenient to use a system of polar coordinates. In what follows, the polar equivalents of direct and shear stresses will be derived along with the stress compatibility equation. An example will follow in which the stress/strain equations are used with their displacements derived in chapter 1.

33 - c2y + 2c33)] = 0. Therefore ex = vwc 3 3EI = vw 2E Thus the central axis is not the neutral axis. , 6 = 5wR. 2 5 5 + ~)] 2 Again, for deep plates the second term in the bracket, which is due to shear, will be significant. 5 is the stress compatibility equation in rectangular coordinates. It is often more convenient to use a system of polar coordinates. In what follows, the polar equivalents of direct and shear stresses will be derived along with the stress compatibility equation. An example will follow in which the stress/strain equations are used with their displacements derived in chapter 1.

21 giving u = ~[- ~( 1 + v) + 2 Cr ( 1 - v) + B [ (1 - J 2 ( 1 - v) ( r Q. 2ld directly f 2 (r) = Lr (2. 2lf) (L. 22b give the displacements for any radius r at any inclination e. 2le is independent of r. M and N can therefore only describe translatory movements of the whole body. Consequently M and N have no effect on the strain and can be eliminated from the displacement equation. 22b, B must be zero otherwise shearing would occur on radial planes due to the tangential displacement of a point given by (ra, e = 0, 2~, 4~, etc).