Beamplanestresscantilever - 1 Example of Solving a Two...

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Unformatted text preview: 1 Example of Solving a Two Dimensional Elasticity Problem Using Airy Stress Function Starting from 4 5 3 4 2 2 3 3 2 4 1 y d xy d y x d y x d x d + + + + = φ 3 4 2 3 2 2 3 1 y c xy c y x c x c + + + + 3 2 1 2 3 2 2 1 a y a x a y b xy b x b + + + + + + find the stresses, yy xx , σ σ and xy τ for the following beam problem. Assume plane stress and satisfy the boundary conditions on the left and right edges at least globally. a) Write the boundary conditions locally on top and bottom edge. Write the boundary conditions globally on the left and right edges. b) Find the expressions for the stresses in terms of the constants. c) Apply all boundary conditions and 4 = φ ∇ to find the constants in φ . d) Write the expression for the stresses. e) Write the expression for the strains. f) Find the expressions for the displacements. y x 2h t 2 SOLUTION Boundary Conditions a) The boundary conditions are on follows Top L x , ) h , x ( yy < < = − σ (1) L x , ) h , x ( xy < < = − τ (2) Bottom L x , ) h , x ( yy < < = σ (3) L x , ) h , x ( xy < < = τ (4) Left dy t ) y , ( : P h h xx x = σ ∫ − (5) P dy t ) y , ( : P h h xy y − = τ ∫ − (6) ∫ − = σ h h xx dy ty ) y , ( : M (7) Right dy t ) y , L ( : P h h xx x = σ ∫ − (8) P dy t ) y , L ( : P h h xy y − = τ ∫ − (9) ∫ − − = h h xx PL dy t y y L M ) , ( : σ (10) 3 Expressions for stresses b) 2 2 x y ∂ φ ∂ = σ 3 2b y 4 6c x 3 2c 2 y 5 12d xy 4 6d 2 x 3 2d + + + + + = (11) 2 2 y x ∂ φ ∂ = σ 1 2 1 2 3 2 2 1 b 2 y c 2 x c 6 y d 2 xy d 6 x d 12 +...
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Beamplanestresscantilever - 1 Example of Solving a Two...

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