06_PlateTheory_03_BendingTwisting

06_PlateTheory_03_BendingTwisting - Section 6.3 Solid...

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Unformatted text preview: Section 6.3 Solid Mechanics Part II Kelly 139 6.3 Plates subjected to Pure Bending and Twisting 6.3.1 Pure Bending of an Elastic Plate Consider a plate subjected to bending moments 1 M M x = and 2 M M y = , with no other loading, as shown in Fig. 6.3.1. Figure 6.3.1: A plate under Pure Bending From equilibrium considerations, these moments act at all points within the plate they are constant throughout the plate. Thus, from the moment-curvature equations 6.2.31, one has the set of coupled partial differential equations y x w x w y w D M y w x w D M = + = + = 2 2 2 2 2 2 2 2 2 2 1 , , (6.3.1) Solving for the derivatives, , ) 1 ( , ) 1 ( 2 2 1 2 2 2 2 2 1 2 2 = = = y x w D M M y w D M M x w (6.3.2) Integrating the first two equations twice gives 1 ) ( ) ( ) 1 ( 2 1 ), ( ) ( ) 1 ( 2 1 2 1 2 2 1 2 2 1 2 2 2 1 x g y x g y D M M w y f x y f x D M M w + + = + + = (6.3.3) and integrating the third shows that two of these four unknown functions are constants: B x g A y f y F y w x G x w = = = = ) ( , ) ( ) ( ), ( 1 1 (6.3.4) 1 this analysis is similar to that used to evaluate displacements in plane elastostatic problems, 1.2.4 this analysis is similar to that used to evaluate displacements in plane elastostatic problems, 1....
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This note was uploaded on 01/20/2012 for the course ENGINEERIN 2 taught by Professor Staff during the Fall '11 term at Auckland.

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06_PlateTheory_03_BendingTwisting - Section 6.3 Solid...

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