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Finding the deformation by Castigliano's theorem

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Can someone show me how Castigliano's Method can be used to determine the elastic displacement of the red beam resulting from a torque placed on the circle (you can assume that the angular rotation of the circle is small ~ 2 or 3 degrees).

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The expert defines the deformation by Castigliano's theorem. The torque placed on the circle are determined.

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Finding the deflection of the elastic beam by using the ...

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