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# Chapter32 - AAE 221 Structures II Spring 2004 Chapter 3.2...

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AAE 221 – Structures II – Spring 2004 Chapter 3.2 – Analytical solutions of static problems using energy methods 1. Consider the beam problem illustrated in fig. 1. fig. 1 The beam is homogenous and symmetric, has length L, a moment of inertia I and a Young’s modulus E. To simplify your computations, write the stiffness K of the rotational spring as L EI K a = where alpha is a non-dimensional parameter. (a) Obtain the beam deflection using Euler-Bernoulli beam theory. How does the deflection at the free end of the beam change when the length of the beam is doubled (assuming that a remains constant)? (b) Obtain the deflection at the free end of the beam using Castigliano’s second theorem. Compare with the solution found in (a). (c) Plot (schematically) the variation of the displacement of the end of the beam with respect to the parameter a. What happens when the stiffness of the torsional element tends to infinity? What about when a tends to zero?

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Chapter32 - AAE 221 Structures II Spring 2004 Chapter 3.2...

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