Chapter22 - AAE 221 Aerospace Structures II Spring 2004...

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AAE 221 – Aerospace Structures II – Spring 2004 Chapter 2.2 – Beam Torsion 1. An uniform beam with elliptical cross-section is submitted to a twisting moment of 400 Nm. The length of the beam is 4m. The beam is made of aluminum (E = 70,000 N/mm 2 ), ν = 0.30). The dimensions of the cross-section are illustrated below. Determine the total twisting angle between the two ends of the beam, the location and value of the maximum warping, and of the maximum shear stress. 2. Refer to the thin-walled circular tube illustrated below. a) Using the exact solution seen in class for the solid (i.e., not hollow) cylinder, derive the expression of the torsion constant J for the hollow cross-section as a function of the radii a i and a e . b) Compute an approximate expression for J using the fact that the thickness h is very small compared with the radii and that we can therefore assume a linear variation of the stress function Ψ through the thickness () = Ψ i e i e a r 0 k a r a r a h k for for as illustrated below.
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(Hint: first you need to find the unknown constant k using the relation seen in class = Ψ i C i A 2 n µθ , where C i is the inner contour around the inner surface A i and Ψ / n denotes the normal derivative of Ψ . Then, when you have the expression of Ψ , you can compute the torsion constant J .) c) Show how you can go directly from the exact expression of J found in part a to the approximate expression found in part b. 3. Find the approximate torsion constant and maximum stress of the open thin- walled cross-section shown below. 4. Refer to the figure below. a) Using the membrane analogy, give a qualitative assessment of the resistance in torsion of the two cross-sections shown above (assume that the open cross-section remains open at all times, i.e., that two ends do not touch). b) Compute the ratio between the torsion constants of the two cross-sections (for the closed cross-section, use the result of problem 2 when the thickness is 1/50th of
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the inner radius. Does the result correspond to your expectations obtained in part a? 5.
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This note was uploaded on 01/18/2010 for the course AE 322 taught by Professor Lambros during the Spring '04 term at University of Illinois at Urbana–Champaign.

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Chapter22 - AAE 221 Aerospace Structures II Spring 2004...

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