formula_sheet_2003 - Cartesian Co-ordinate System Equations...

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Unformatted text preview: Cartesian Co-ordinate System Equations Equilibrium Equations Z y x z Y z x y X z y x yz zx z yz xy y zx xy x = + ∂ τ ∂ + ∂ τ ∂ + ∂ σ ∂ = + ∂ τ ∂ + ∂ τ ∂ + ∂ σ ∂ = + ∂ τ ∂ + ∂ τ ∂ + ∂ σ ∂ Strain–Displacement Equations y w z v x w z u x v y u z w y v x u yz zx xy z y x ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ = ∂ ∂ = ∈ ∂ ∂ = ∈ ∂ ∂ = ∈ γ γ γ , , , , Hooke’s law G G G E E E E E E E E E zx zx yz yz xy xy z y x z z y x y z y x x τ γ τ γ τ γ σ νσ νσ νσ σ νσ νσ νσ σ = = = + − − = ∈ − + − = ∈ − − = ∈ , , Boundary Conditions m l n Z l n m Y n m l X yz zx z xy yz y zx xy x τ τ σ τ τ σ τ τ σ + + = + + = + + = , , Hooke’s law – Plane Stress ( ) ( ) G E E xy xy x y y y x x τ γ νσ σ νσ σ = − = ∈ − = ∈ , 1 , 1 Hooke’s law – Plane Strain ( ) [ ] ( ) [ ] . 1 1 , 1 1 G v E v E xy xy x y y y x x τ γ σ σ ν ν σ σ ν ν = − − + = ∈ − − + = ∈ Transformation of strains and stresses , ] [T = xy y x 12 2 1 ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ τ σ σ τ σ σ , /2 [T] = /2 xy y x 12 2 1 ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ γ ε ε γ ε...
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formula_sheet_2003 - Cartesian Co-ordinate System Equations...

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