Download Advanced Solid Mechanics: Theory, worked examples and by P. R. Lancaster, D. Mitchell PDF

By P. R. Lancaster, D. Mitchell

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Example text

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). ) Thus, for a cylindrical body where the stresses are distributed symmetrically about an axis there is only a radial displacement possible.

In polar coordinates (y rsine) (2. 8. Note that a or = r has two terms %+ [%) cos2e that is, at the boundary of the circle r = b, the stresses consist of a constant radial stress(cr/2) applied to a thick walled plate (axial symmetry) and a part which varies with cos2e. For this a stress function has to be found. The stresses due to both parts for a~ r ~ b can be superimposed to solve this problem of a plate having a hole and subjected to tension. 24) where f(r) is a function of r only. 16.

T. _ l1 - .!. 14) Also a Y -- ~ a2o~. 11). 17) This is the stress compatibility equation when using polar coordinates and 41 is a function of r and e. This equation although more complicated in appearance than its equivalent rectangular equation is generally more useful in the results which may be obtained from it. The transforms from a • a and T to a • a 9 and x y xy r Tre should be remembered if conversions from one to the other are required. 4 Solid of Revolution with Axially Symmetrical Loading The stresses or and a 8 are independent of e and Tre must vanish for this particular configuration of stress and geometry, since elements on neighbouring radial planes are subjected to the same stress at radius r.

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