hw2 - β x 3 = αx 3 which varies linearly along the shaft...

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MAE 261A E NERGY AND V ARIATIONAL P RINCIPLES IN S TRUCTURAL M ECHANICS F ALL 2004 Homework Assignment No. 2 Due Wednesday, Oct 27, 2004. Problem 1 (Problem 1.20 from Shames & Dym) For a two-dimensional (plane stress) orthotropic continuum, the constitutive law is usually given as σ 11 = C 11 11 + C 12 22 σ 22 = C 21 11 + C 22 22 σ 12 = G 12 12 . Construct an appropriate strain energy density. Also, verify that we must make C 21 = C 12 . Check your constitutive law by computing stresses σ 11 , σ 22 , and σ 12 . Problem 2 (Problem 1.26 from Shames & Dym) Show that the following function is the proper Airy stress function for a simply supported rectangular beam of unit width, length L , and height h , carrying a uniform distributed load q . (The beam is aligned with the x -axis, and the load is in the positive z -direction.) Φ = Az 3 6 + Bx 2 z 3 6 - C 2 x 2 z - qx 2 4 - B 30 z 5 where A = 12 h 3 ( qL 2 8 - qh 2 20 ) , B = - 6 q h 3 , C = - 3 q 2 h . Problem 3 A torque is applied at the end of a circular cylinder. Assuming that each cross-section rotates about the x 3 axis, relative to its original position, through an angle
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Unformatted text preview: β ( x 3 ) = αx 3 which varies linearly along the shaft. The displacement field is then u 1 = x 1 [cos β ( x 3 )-1]-x 2 sin β ( x 3 ) u 2 = x 1 sin β ( x 3 ) + x 2 [cos β ( x 3 )-1] u 3 = 0 . (i) Compute the components of the Green strain tensor E = 1 2 ( ∇ u + ∇ u T + ∇ u T · ∇ u ) . (ii) Show (in components) that when β ¿ 1 the Green strain is approximately equal to the small strain tensor ² = 1 2 ( ∇ u + ∇ u T ) . Problem 4 (Problem 2.1 from Shames & Dym) Given the functional: I [ y ( x )] = Z x 2 x 1 [3 x 2 + 2 x ( y ) 2 + 10 xy ] dx, (i) Compute the first variation δI . (ii) Find the Euler-Lagrange equation. Problem 5 (Problem 2.10 from Shames & Dym) By setting the first variation to zero δI = 0 , find the Euler-Lagrange equation for the Brachistochrone problem, with functional I [ y ( x )] = 1 √ 2 g Z x 2 x 1 s 1 + ( y ) 2 y dx. 1...
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