exam_1_eqn - Important relations and concepts in AE 322 1)...

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Important relations and concepts in AE 322 1) Theory of elasticity 1.1) 3D theory Equilibrium equations : , 0 ij j i f σ += ; ij = ji Cauchy’s relation : T i = ji n j Kinematic : ε ij = 1 2 u i,j + u j,i () Compatibility : ij, kl + kl,ij = ik, jl + jl, ik Constitutive (isotropic) : ij = λε mm δ ij + 2 με ij λ = ν E 1 + 1 2 , μ = G = E 21 + 1.2) 2D theory (plane stress) Equilibrium : ( ) , 0; , 1, 2 ij j i ij ji fi σσ = = j Kinematic : ,, 1 2 ij i j j i uu =+ Compatibility : 2 12,12 = 11, 22 + 22,11 Constitutive (isotropic) : 11 = E 1 2 11 + νε 22 [] , 22 = E 1 2 22 + 11 , 12 = E 1 + 12 33 =− 1 11 + 22 , 33 = 0 For plane strain : replace E by E/ 1 2 ( ) ; by /1 ( ) ; 33 = 0; 33 = νσ 11 + 22 1
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2) Bending and extension of beams 2.1) Conventions for positive force and moment resultants x NN M t M t x V y V y M z M z y x V z M y z M y V z Nx () =+ σ xx x,y,z dydz A ;V y x ( ) =− xy ( ) A z x ( ) xz A M y x z xx A ;M z x ( ) y xx x, y,z ( ) A t x ( ) y xz z xy [] A 2.2) Euler-Bernoulli assumptions (beam bending) - plane cross-sections remain plane after deformation
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exam_1_eqn - Important relations and concepts in AE 322 1)...

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