hw6 - Problem 3(a Derive the complementary strain energy...

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MAE 261A ENERGY AND VARIATIONAL PRINCIPLES IN STRUCTURAL MECHANICS FALL 2004 Homework Assignment No. 6 Due Monday, December 6, 2004. Problem 1 (a) Show that to lowest order, the nonlinear axial strain in an Euler-Bernoulli beam is given by e xx = u p 0 + 1 2 ( v p 0 ) 2 - yv pp 0 . (b) Show that to lowest order, the strain energy is given by W = 1 2 i l 0 [ EA ( u p 0 ) 2 + EI ( v pp 0 ) 2 - N ( v p 0 ) 2 ] dx where N = EAu p 0 is the axial force internal to the beam. (c) Incorporating the above strain energy, minimize total potential energy to derive the governing equations and boundary conditions for a beam loaded as shown on page 1 of the handout on Euler-Bernoulli beams. Note: The equilibrium equations resulting from the above exercise allow us to study buckling of Euler- Bernoulli beams. Problem 2 Derive the governing differential equations for the deflection v ( x ) and the cross-section rotation ψ ( x ) and corresponding boundary conditions of a Timoshenko beam by minimizing total potential energy.
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Unformatted text preview: Problem 3 (a) Derive the complementary strain energy for a linearly elastic Timoshenko beam. (b) Consider a cantilever beam with a vertical point load applied at its free end. Use the minimum comple-mentary energy theorem (or equivalently, Castigliano’s Second Theorem) to find the deflection of the tip of the beam. How does this compare to the same quantity for an Euler-Bernoulli beam? Problem 4 Starting from the general expression for the complementary strain energy of a linear elastic body, W * = i B 1 2 σ ij e ij dV, show that for a Kirchhoff plate this takes the form W * = i Ω 1 2 [ M 2 x-2 νM x M y + M 2 y + 2(1 + ν ) M 2 xy ] dS. Hint: make use of the generalized plane stress constitutive equations and the definitions of the Moment resultants as given in the handout on plates. 1...
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