06_PlateTheory_06_PolarCoordinates

06_PlateTheory_06_PolarCoordinates - Section 6.6 6.6 Plate...

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Section 6.6 Solid Mechanics Part II Kelly 163 6.6 Plate Problems in Polar Coordinates 6.6.1 Plate Equations in Polar Coordinates To examine directly plate problems in polar coordinates, one can first transform the Cartesian plate equations considered in the previous sections into ones in terms of polar coordinates. First, the definitions of the moments and forces are now + + + = = = 2 / 2 / 2 / 2 / 2 / 2 / , , h h r r h h h h rr r dz z M dz z M dz z M θ θθ σ (6.6.1) and + + = = 2 / 2 / 2 / 2 / , h h z h h zr r dz V dz V (6.6.2) The strain-curvature relations, Eqns. 6.2.27, can be transformed to polar coordinates using the transformations from Cartesian to polar coordinates detailed in §4.2 (in particular, §4.2.6). One finds that { Problem 1} + = + = = ε r w r w r z w r r w r z r w z r rr 2 2 2 2 2 2 2 1 1 1 1 (6.6.3) The moment-curvature relations 6.2.31 become { Problem 2} () + = + + = + + = ν r w r w r D M r w w r r w r D M w r r w r r w D M r r 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 (6.6.4) The governing differential equation 6.4.9 now reads D q w r r r r = + + 2 2 2 2 2 2 1 1 (6.6.5)
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Section 6.6 Solid Mechanics Part II Kelly 164 The shear forces in terms of deflection, Eqn 6.4.12, now read { Problem 3} + + = + + = 2 2 2 2 2 2 2 2 2 2 1 1 1 , 1 1 θ w r r w r r w r D V w r r w r r w r D V r (6.6.6) Finally, the stresses are { Problem 4} θθ σ r r r rr M h z M h z M h z 3 3 3 12 , 12 , 12 = = = (6.6.7) and = = 2 2 2 / 1 2 3 , 2 / 1 2 3 h z h V h z h V z r zr (6.6.8) The differential equation 6.6.5 can be solved using a method similar to the Airy stress
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06_PlateTheory_06_PolarCoordinates - Section 6.6 6.6 Plate...

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