By P. R. Lancaster, D. Mitchell
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Extra info for Advanced Solid Mechanics: Theory, worked examples and problems
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.
7 · J Fig. 7 where the effect of the hole will be negligible at a large distance 'b' from it. Assume the distance is defined by a circle radius 'b' and let the radius of the hole be 'a'. 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.
Note that ox = -(Px)y/I = -My/I where M is the bending moment at a section x, and when y is negative·, stresses are tensile as expected. The results are the same as those found from elementary beam theory. The shear stress distribution is parabolic and is independent of x. 9c is maintained at x = 0. In practice this cannot usually be achieved but by making use of the principle of St. 9. This means that at whatever point or planes the forces are applied there will be differences in the stress distributions in the immediate vicinity of these points.
Advanced Solid Mechanics: Theory, worked examples and problems by P. R. Lancaster, D. Mitchell