Elasticity3

# 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 ∂ ∂ φ ∂- = τ ∂ φ ∂ = σ ∂ φ ∂ = σ 2 2 2 2 2 , , Airy Representation 2 4 4 4 2 2 4 4 4 = φ ∇ = ∂ φ ∂ + ∂ ∂ φ ∂ + ∂ φ ∂ 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 = ± τ = ± τ = ± σ = ± σ c x y l c x T y l xy xy y x , Try 2 02 = τ = σ = σ ⇒ = φ 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 + ν- = ⇒ ν- = νσ- σ = = ∂ ∂ + = ⇒ = νσ- σ = = ∂ ∂ ⇒ = ′ + ′ ⇒ = μ τ = = ∂ ∂ + ∂ ∂ ) ( ) ( 2 x g y f e x v y u xy xy o o o o v x x g u y y f + ϖ = + ϖ- = ) ( ) ( ) , ( ) , ( , ) , ( Conditions Boundary nt Displaceme Overall = ϖ ⇒ = ∂ ∂ = ⇒ = = ⇒ = o o o x v v v u u Pure Bending of a Beam x y M M 2l 2c ∫ ∫--- = ± σ = ± σ = ± τ = ± τ = ± σ 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 = τ = σ- = σ ⇒ = φ 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 + ν = ⇒ ν = ∂ ∂ +- = ⇒- = ∂ ∂ o o o o v x x Ec M x g u y y f + ϖ + = + ϖ- = 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- + ν =- = = τ = σ- = σ ] [ 2 ) , ( , Materials of Strength 2 2 l x EI M x v v y I M xy y x- = = = τ = σ- = σ ⇒ =- = ± ) , ( and ) , ( l u l v 3 2 4 / 3 , Ec Ml v u o o o- = = ϖ = Note Integrated Boundary Conditions Bending of a Beam by Uniform Transverse Loading x y w 2c 2l wl wl ∫ ∫ ∫--- = ± τ = ± σ = ± σ- =- σ = σ = ± τ 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- + + + = φ ) ( 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-- = τ +-- = σ- +- = σ ) ( 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-- = = τ = σ- = = σ c x / w- Elasticity , x / w- Strength of Materials l/c = 2...
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## This note was uploaded on 10/03/2011 for the course MCE 561 taught by Professor Sadd during the Spring '11 term at Rhode Island.

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Elasticity3 - Application Solutions of Plane Elasticity...

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