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Published byJena Carleton Modified about 1 year ago

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Castigliano’s theorems Castigliano’s first theorem: For linearly elastic structure, the Castigliano’s first theorem may be defined as the first partial derivative of the strain energy of the structure with respect to any particular force gives the displacement of the point of application of that force in the direction of its line of action.

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Castigliano’s First theorem derivation Consider an elastic beam AB subjected to loads W 1 and W 2, acting at points 1 and 2 respectively

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Castigliano’s First theorem derivation If

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Castigliano’s First theorem derivation Then Similarly, Considering the work done = U i

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Now applying W 2 at Point 2 first and then applying W 1 at Point 1, Castigliano’s First theorem derivation Strain energy, U i Similarly,

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Considering equation (III) and (IV), and equating them, it can be shown that Castigliano’s First theorem derivation This is called Betti – Maxwell’s reciprocal theorem Deflection at point 2 due to a unit load at point 1 is equal to the deflection at point 1 due to a unit load at point 2.

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From Eqn. (III), Castigliano’s First theorem derivation From Eqn. (IV), This is Castigliano’s first theorem.

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Castigliano’s second theorem Similarly the energy U i can be express in terms of spring stiffnesses k 11, k 12 (or k 21 ), & k 22 and deflections δ1 and δ2; then it can be shown that This is Castigliano’s second theorem. When rotations are to be determined,

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