Elasticity3 - Application Solutions of Plane Elasticity...

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Unformatted text preview: Application Solutions of Plane Elasticity Professor M. H. Sadd Solutions to Plane Problems Cartesian Coordinates y x x y xy y x - = = = 2 2 2 2 2 , , Airy Representation 2 4 4 4 2 2 4 4 4 = = + + y y x x Biharmonic Governing Equation ) , ( , ) , ( y x f T y x f T y y x x = = Traction Boundary Conditions R S x y Uniaxial Tension of a Beam x y T T 2l 2c ) , ( ) , ( ) , ( , ) , ( Conditions Boundary = = = = c x y l c x T y l xy xy y x , Try 2 02 = = = = xy y x T y A ) ( ) ( 1 ) ( ) ( 1 nts Displaceme x g y E T v E T E e y v y f x E T u E T E e x u x y y y x x + - = - = - = = + = = - = = = + = = = + ) ( ) ( 2 x g y f e x v y u xy xy o o o o v x x g u y y f + = + - = ) ( ) ( ) , ( ) , ( , ) , ( Conditions Boundary nt Displaceme Overall = = = = = = o o o x v v v u u Pure Bending of a Beam x y M M 2l 2c --- = = = = = c c x c c x xy xy y M ydy y l dy y l y l c x c x ) , ( , ) , ( ) , ( ) , ( , ) , ( Condtions Boundary , 2 3 3 3 03 = = - = = xy y x y c M y A ) ( 4 3 2 3 ) ( 2 3 2 3 2 3 3 3 3 x g y Ec M v y Ec M y v y f xy Ec M u y Ec M x u + = = +- = - = o o o o v x x Ec M x g u y y f + + = + - = 2 3 4 3 ) ( ) ( nts Displaceme ] [ 2 , , Elasticity of Theory 2 2 2 l x y EI M v EI Mxy u y I M xy y x- + =- = = = - = ] [ 2 ) , ( , Materials of Strength 2 2 l x EI M x v v y I M xy y x- = = = = - = =- = ) , ( and ) , ( l u l v 3 2 4 / 3 , Ec Ml v u o o o- = = = Note Integrated Boundary Conditions Bending of a Beam by Uniform Transverse Loading x y w 2c 2l wl wl --- = = = - =- = = c c xy c c x c c x y y xy wl dy y l ydy y l dy y l w c x c x c x ) , ( , ) , ( , ) , ( ) , ( , ) , ( , ) , ( Conditions Boundary 5 23 3 2 23 3 03 2 21 2 20 5 y A y x A y A y x A x A- + + + = ) ( 2 3 2 3 2 ) 5 3 ( ) ( 2 Elasticity of Theory 2 2 3 2 3 2 3 2 2 y c x I w c y c y I w y c y I w y x l I w xy y x-- = +-- = - +- = ) ( 2 ) ( 2 Materials of Strength 2 2 2 2 y c x I w It VQ y x l I w I My xy y x-- = = = - = = c x / w- Elasticity , x / w- Strength of Materials l/c = 2...
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Elasticity3 - Application Solutions of Plane Elasticity...

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