Theorem of Castigliano

The theorem of Castigliano (of the name of Carlo Alberto Castigliano) is at the base many methods of calculating of the efforts in resistance of materials. It rests on an energy relation and allows a relatively simple calculation of the specific sizes (efforts or displacements) sought.

First statement of Castigliano

The Dérivée partial of the work of the external forces compared to a force is equal to the Déplacement point of application according to the line of action of this force. Thus the derivative partial of the work of the external forces compared to a Couple (mechanical) determines the Rotation Poutre with the right of the section where this couple applies.

$\backslash \; frac\; \{\backslash \; partial\; U\}\; \{\backslash \; partial\; q\_i\}\; =\; Q\_i$ with $i=1,\dots ,\; n$

$q\_i$… generalized displacements

$Q\_i$… generalized forces

$U\; =\; U\; (q\_1,\dots ,\; q\_n)$… elastic energy (energy of Deformation)

Second statement of Castigliano

$\backslash \; frac\; \{\backslash \; partial\; U^\; \{*\}\}\; \{\backslash \; partial\; Q\_i\}\; =\; q\_i$ with $i=1,\dots ,\; n$

$q\_i$… generalized displacements

$Q\_i$… generalized forces

$U^*\; =\; U^*\; (Q\_1,\dots ,\; Q\_n)$… internal energy, known as energy complementary to deformation

Statement of Menabrea

The second statement of Castigliano can also be made profitable for the calculation of hyperstatic efforts. The particular form under which one uses it then takes the name of equation of Menabrea.

$\backslash \; frac\; \{\backslash \; partial\; U^*\}\; \{\backslash \; partial\; X\_i\}\; =\; 0$ with $i=1,\dots ,\; n$

$X\_i$… unknown hyperstatic (which “does not work” not) $U^*\; =\; U^*\; (X\_1,\dots ,\; X\_n)$… energy interns, known as energy complementary to deformation

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