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The beam has young’s modulus e and second area moment i Castigliano's theorem is defined as a principle stating that for an elastic structure, the first partial derivative of the strain energy stored in the structure with respect to a generalized displacement is equal to the corresponding generalized force acting on it. Neglect shear strain energy due to bending.

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The elongation of each bar as determined from eq The theorem is very indeterminate structures. (6.6) is ∆m = cos ( θm) q1 + sin ( θm) q2 m = 1 , 2 , 3

This theorem applies to statically determined structures and system subjected to concentrated forces or moments

The distribution of bending moments can be uniquely determined from global equilibrium as function of the forces, \ (u = u (p)\). Castigliano’s theorem is a fundamental principle in the field of structural engineering and material science This theorem, pivotal in the analysis of stress and deflection in elastic bodies, was developed by italian engineer carlo alberto castigliano in the 19th century. Castigliano’s theorem for beam deflection for linearly elastic structures, the partial derivative of the strain energy with respect to an applied force (or couple) is equal to the displacement (or rotation) of the force (or couple) along its line of action.

Castigliano’s second theorem the second theorem of castigliano states that the work done by external forces in a structure will be minimum