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Chapter 3 less HW solution

Chapter 3 less HW solution - Chapter 3 Plates Subjected to...

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Chapter 3 Plates Subjected to In-plane Forces 1. Derivation of Airy’s Stress Function, 2-D From equations of equilibrium, we have (by crossing out the third row and column) ( 29 0 no body force 0 xy x y yx x y y x τ σ σ τ + = + = (3-1) From the 2-D compatibility equation in terms of stress, we have ( 29 ( 29 ( 29 2 2 2 2 2 2 1 0 xy y x x y x x y y τ σ μσ μ σ μσ - - + + - = ∂ ∂ (3-2) Assume arbitrary functions R and S , such that ( 29 ( 29 ( 29 ( 29 , , and , , y x yx xy S x y R x y y y S x y R x y y x σ σ τ τ = = = - = - (3-3) Since ( 29 ( 29 , , xy yx R x y S x y x y τ τ = = (3-4) Assume and R S y x φ φ = = (3-5) From Eqs. (3-4) and (3-5), it follows 2 2 x y y x φ φ = ∂ ∂ ∂ ∂ (3-6) From Eqs. (3-3), (3-4), and (3-5), it follows 1
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2 2 2 2 2 , , and x y xy y x x y φ φ φ σ σ τ = = = - ∂ ∂ (3-7) where φ is called Airy’s stress function after George Biddell Airy (1801-1892). Substituting Eq. (3-7) into Eq. (3-2), yields 4 4 4 4 2 2 4 2 0 x x y y φ φ φ + + = (3-8) Eq. (3-8) is called a biharmonic equation, sometimes written as 2 2 0 φ ∇ ∇ = . Boundary Forces and Airy’s Stress Function where , x y =boundary traction (externally applied surface load)/unit area (unit width, see Timoshenko Theory of Elasticity, 2 nd ed., page 13). Consider equilibrium of the infinitesimal triangular prism. 0 and 0 x x yx y y xy F x ds dy dx F y ds dx dy σ τ σ τ = = - - = = - - (3-9) Dividing Eq. (3-9) by ds , gives cos sin sin cos x yx x yx y xy y xy dy dx x ds ds dx dy y ds ds σ τ σ α τ α σ τ σ α τ α = + = + = + = + (3-10) dy y s y α ds xy τ ( 29 τ ( 29 σ n dx - y x yx τ α 0 x s x x x 0 x y σ x σ g g g 2
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In traversing ds from 0 to s (counter clockwise positive for the right hand coordinates system), x decreases in the first and second quadrants and dx will be negative. Substituting Eq. (3-7) into Eq. (3-10), yields 2 2 2 2 2 2 2 2 2 2 2 2 cos sin sin cos dy dx x y x y y ds x y ds dx dy y x x y x ds x y ds φ φ φ φ α α φ φ φ φ α α = - = + + ∂ ∂ ∂ ∂ = - = - - ∂ ∂ ∂ ∂ (3-11) The x and y and the perimeter coordinate s are related as ( 29 ( 29 x f s y f s = = (3-12) Recalling the following chain-rule expansion: ( 29 , dF x y F dx F dy ds x ds y ds = + Similarly, d dx dy ds y x y ds y y ds φ φ φ = + Therefore (see Timoshenko, Theory of Elasticity, 2 nd . ed., page 190), 2 2 2 2 2 2 d dx dy x ds y x y ds y ds d dx dy y ds x x ds x y ds φ φ φ φ φ φ = + = ∂ ∂ = + = - ∂ ∂ (3-13) Let h be the thickness of the plate, then (remember force/unit area x = ). Then 0 0 1 1 or s d x d xds xds C ds y h y h y h y φ φ φ φ = = = +
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