# HW2_Soln_r1 - 1 Find the components of the traction n T r...

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Unformatted text preview: 1. Find the components of the traction n T r on planes defined by n 1 = 1 2 , n 2 = 1 2 , n 3 = and n 1 = 1 2 , n 2 = - 1 2 , n 3 = for the following states of stress: (a) σ 11 = σ σ 12 = σ 21 = σ 13 = σ 31 = σ 22 = σ σ 23 = σ 32 = σ 33 = σ (b) σ 11 = σ σ 12 = σ 21 = σ σ 13 = σ 31 = σ 22 = σ σ 23 = σ 32 = σ 33 = Solution a) g G = ¡ ¢ ¢ ¢ £ ¤ 1/√2 1/√2 ¥ = ¤ ¢/√2 ¢/√2 ¥ g G = ¡ ¢ ¢ ¢ £ ¤ 1/√2 −1/√2 ¥ = ¤− ¢/√2 ¢/√2 ¥ b) ¡ ¢ ¢ ¢ ¢ £ ¤ 1/√2 1/√2 ¥ = ¡ ¢√2 ¢√2 £ ¡ ¢ ¢ ¢ ¢ £ ¤− 1/√2 1/√2 ¥ = ¡ £ 2. The state of stress at a point P in a material is given by: σ ij [ ] = 20 2 1 2- 15 2 1 2 3 KPa (a) Compute the components of traction n T r on the plane passing through P whose outward normal vector n r makes equal angles with the coordinate axes. Note: this is the octahedral plane, 1 2 3 1 3, 1 3, 1/ 3 n n n = = = . (b) Compute the normal nn σ and tangential ns σ components of traction on this plane. (c) Repeat the above exercise for the stress state: 10 2 1 2 15 5 1 5 3 ij KPa σ =- (a) g G = ¡ 20 2 1 2 −15 2 1 2 3 ¢ £ 1/√3 1/√3 1/√3 ¤ = £− 23/√3 11/√3 2√3 ¤ b) σ ¥¥ = T ¦ ¥ ∙ n ¨ = ©ª √ª ∙ « √ª- «« √ª ∙ « √ª + 2√3 ∙ « √ª = 6 σ ¥¬ = ­ |T ¦ ¥ | © − σ ¥¥ © = 17­ © ª = 13.88 c)...
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HW2_Soln_r1 - 1 Find the components of the traction n T r...

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