hw1 - σ X ij = T 2 1 2 2 1 2 where T is a constant in some...

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MAE 261A E NERGY AND V ARIATIONAL P RINCIPLES IN S TRUCTURAL M ECHANICS F ALL 2004 Homework Assignment No. 1 Due Wednesday, October 13, 2004. Problem 1 Let u , v , and w be vectors. Using index notation (and the Einstein summation convention), prove the following vector identities: a. ijk pqk = δ ip δ jq - δ iq δ jp b. u × ( v × w ) = v ( u · w ) - w ( u · v ) Problem 2 Let Φ( x ) and v ( x ) be twice-differentiable scalar and vector fields, respectively. Using index notation, prove the following: a. curl ( grad Φ) = ∇ × ( Φ) = 0 . b. div ( curl v ) = ∇ · ( ∇ × v ) = 0 . Problem 3 (Problem 1.2a from Shames & Dym) The matrix of components of the Cauchy stress tensor, in a frame X = { 0 ; e 1 , e 2 , e 3 } is [ σ X ij ] = 1000 200 0 200 - 6000 - 400 0 - 400 0 psi . Find the components of the surface traction vector for an interface whose normal vector is n = 0 . 11 e 1 + 0 . 35 e 2 + 0 . 93 e 3 . Problem 4 The stress tensor at a point in a solid is given by
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Unformatted text preview: [ σ X ij ] = T 2 1 2 2 1 2 where T is a constant in some appropriate set of units. Calculate T such that there will be a traction-free plane through the point, and determine the orientation (i.e., the normal vector) of this plane. Problem 5 A solid slab occupies the region-a < x 1 < a,-a < x 2 < a,-h < x 3 < h, and has the stress distribution σ 11 =-p ( x 2 1-x 2 2 ) a 2 , σ 22 = p ( x 2 1-x 2 2 ) a 2 , σ 12 = 2 px 1 x 2 a 2 , σ 13 = σ 23 = σ 33 = 0 . a. Examine whether there are any body forces within the slab. b. Calculate the resultant tractions acting on the faces x 1 = ± a, x 2 = ± a . 1...
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