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 , a factor of about 10% for 3 . 0 = , since () b EI Eh D 2 2 3 1 1 1 12 = = (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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Section 6.5 Solid Mechanics Part II Kelly 150 Figure 6.5.1: a clamped circular plate subjected to a uniform lateral load The differential equation for the problem is given by 6.4.9. The boundary conditions are that the slope and deflection are zero at the boundary: 2 2 2 along 0 , 0 , 0 a y x y w x w w = + = = = (6.5.5) It will be shown that the deflection 2 2 2 2 ) ( a y x c w + = (6.5.6) is a solution to the problem. First, this function certainly satisfies 6.5.5. Further, letting 2 2 2 ) , ( a y x y x f + = , (6.5.7) the relevant partial derivatives are () c y w c y x w c x w cy y w cx y x w cy y x w cx x w f y c y w cxy y x w f x c x w cyf y w cxf x w 24 , 8 , 24 24 , 8 , 8 , 24 2 4 , 8 , 2 4 4 , 4 4 4 2 2 4 4 4 3 3 2 3 2 3 3 3 2 2 2 2 2 2 2 = = = = = = = + = = + = = = (6.5.8) Substituting these into the differential equation now yields D q c 64 = (6.5.9) so the deflection is 2 2 2 2 ) ( 64 a y x D q w + = (6.5.10) x y a q
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Section 6.5 Solid Mechanics Part II Kelly 151 This is plotted in Fig. 6.5.2. The maximum deflection occurs at the plate centre, where D qa w 64 4 max = . (6.5.11) Figure 6.5.2: mid-plane deflection of the clamped circular plate The curvature 2 2 / x w along a radial line 0 = y is displayed in Fig. 6.5.3. The curvature is positive toward the centre of the plate (the plate curves upward) and is negative towards the edge of the plate (the plate curves downward). Figure 6.5.3: curvature in the clamped circular plate The moments occurring in the plate are, from the moment-curvature equations 6.2.31 and 6.5.8, [] xy q M a y x q M a y x q M xy y x ) 1 ( 8 ) 1 ( ) 3 ( ) 1 3 ( 16 ) 1 ( ) 1 3 ( ) 3 ( 16 2 2 2 2 2 2 ν + = + + + + = + + + + = (6.5.12) The moment x M along a radial line 0 = y is of the same character as the curvature displayed in Fig. 6.5.3.
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06_PlateTheory_05_RectangularCoordinates - Section 6.5 6.5...

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