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06_PlateTheory_05_RectangularCoordinates

06_PlateTheory_05_RectangularCoordinates - Section 6.5 6.5...

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Section 6.5 Solid Mechanics Part II Kelly 149 6.5 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 2 2 2 2 ) ( , ) ( ), ( dx w d D x M dx w d D x M x f w y x ν = = = (6.5.1) The differential equation 6.4.9 reads D x q dx w d ) ( 4 4 = (6.5.2) The corresponding equation for a beam is EI x p dx w d / ) ( / 4 4 = . If ) ( / ) ( x q b x p = , with b the depth of the beam, with 12 / 3 bh I = , the plate will respond more stiffly than the beam by a factor of ) 1 /( 1 2 ν , a factor of about 10% for 3 . 0 = ν , since ( ) b EI Eh D 2 2 3 1 1 1 12 ν ν = = (6.5.3) The extra stiffness is due to the constraining effect of y M , 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 2 2 2 a y x = + , (6.5.4) clamped at its edges and subjected to a uniform lateral load q , Fig. 6.5.1.

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