ME_423_lecture_4 - ME 423 09/9/2011 Lecture 4 3D Stresses p...

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Unformatted text preview: ME 423 09/9/2011 Lecture 4 3D Stresses p Unit vectors to the (x,y,z) axes : i, j, k Unit normal vector to plane ABC: p Direction cosines of p: l = i • p ; m = j • p ; n = k • p l 2 + m2 + n2 = 1 If area ABC=1, then OBC=l, OCA=m, OAB=n If p is a principle direction lσ = lσ xx + mσ yx + nσ zx mσ = lσ xy + mσ yy + nσ zy nσ = lσ xz + mσ yz + nσ zz From a non-trivial solution for the homogeneous equation of l,m,n (σ xx − σ ) σ xy σ xz σ xy (σ yy −σ ) σ yz σ xz σ yz =0 (σ zz − σ ) 1 We have the equation for principle stresses in 3D σ 3 − I 1σ 2 + I 2σ − I 3 = 0 Stress invariants I 1 = σ xx + σ yy + σ zz 2 2 2 I 2 = σ xxσ yy + σ xxσ zz + σ yyσ zz − σ xy − σ yz − σ xz σ xx σ xy σ xz I 3 = σ xy σ yy σ yz σ xz σ yz σ zz Closed form solution for cubic solution 2 ...
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ME_423_lecture_4 - ME 423 09/9/2011 Lecture 4 3D Stresses p...

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