Section 6.5 Solid Mechanics Part II Kelly 1496.5 Plate Problems in Rectangular Coordinates In this section, a number of important plate problems will be examined using Cartesian coordinates. 6.5.1 Uniform Pressure producing Bending in One Direction Consider first the case of a plate which bends in one direction only. From 6.3.11 the deflection and moments are 2222)(,)(),(dxwdDxMdxwdDxMxfwyxν−===(6.5.1) The differential equation 6.4.9 reads Dxqdxwd)(44−=(6.5.2) The corresponding equation for a beam is EIxpdxwd/)(/44=. If )(/)(xqbxp−=, with bthe depth of the beam, with 12/3bhI=, the plate will respond more stiffly than the beam by a factor of )1/(12ν−, a factor of about 10% for 3.0=ν, since ()bEIEhD22311112νν−=−=(6.5.3) The extra stiffness is due to the constraining effect of yM, which is not present in the beam. 6.5.2 Deflection of a Circular Plate by a Uniform Lateral Load A solution for a circular plate problem is presented next. This problem will be examined again in the section which follows using the more natural polar coordinates. Consider a circular plate with boundary 222ayx=+, (6.5.4) clamped at its edges and subjected to a uniform lateral load q, Fig. 6.5.1.
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