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| | Irwin Butts is what my spouse loves to call me although I don't truly like becoming called like that. South Dakota is where me and my spouse reside. Doing ceramics is what adore doing. For years he's been operating as a receptionist.<br><br>my page :: home std test kit ([http://tcciaarusha-bic.info/groups/simple-stuff-you-could-do-to-remove-candida/ this article]) |
| '''Castigliano's method''', named for [[Carlo Alberto Castigliano]], is a method for determining the displacements of a [[Linear elasticity|linear-elastic]] system based on the [[partial derivative]]s of the [[Energy]] principle structures, he is known for his two theorems. The basic concept may be easy to understand by recalling that a change in energy is equal to the causing force times the resulting displacement. Therefore, the causing force is equal to the change in energy divided by the resulting displacement. Alternatively, the resulting displacement is equal to the change in energy divided by the causing force. Partial derivatives are needed to relate causing forces and resulting displacements to the change in energy.
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| * '''Castigliano's first theorem''' – for forces in an elastic structure
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| Castigliano's method for calculating forces is an application of his first theorem, which states:
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| :''If the strain energy of an elastic structure can be expressed as a function of generalised displacement q<sub>i</sub>; then the partial derivative of the strain energy with respect to generalised displacement gives the generalised force Q<sub>i</sub>.''
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| In equation form,
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| :<math>Q_i=\frac{\partial \bold{U}}{\partial q_i}</math>
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| where U is the strain energy.
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| * '''Castigliano's second theorem''' – for displacements in a linearly elastic structure.
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| Castigliano's method for calculating displacements is an application of his second theorem, which states:
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| :''If the strain energy of a linearly elastic structure can be expressed as a function of generalised force Q<sub>i</sub>; then the partial derivative of the strain energy with respect to generalised force gives the generalised displacement q<sub>i</sub> in the direction of Q<sub>i</sub>.''
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| As above this can also be expressed as:
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| :<math>q_i=\frac{\partial \bold{U}}{\partial Q_i}.</math>
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| ==Examples==
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| For a thin, straight cantilever beam with a load P at the end, the displacement <math>\delta</math> at the end can be found by Castigliano's second theorem :
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| :<math>\delta = \frac{\partial \bold{U}}{\partial P}</math> | |
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| :<math>\delta = \frac{\partial}{\partial P}\int_0^L{\frac{[M(l)]^2}{2EI}dl}
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| = \frac{\partial}{\partial P}\int_0^L{\frac{[Pl]^2}{2EI}dl} </math>
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| where E is Young's Modulus and I is the second moment of area of the cross-section, and M(l)=P x l is the expression for the internal moment at a point at distance l from the end, therefore:
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| :<math>= \int_0^L{\frac{Pl^2}{EI}dl}</math> | |
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| :<math>= \frac{PL^3}{3EI}.</math>
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| The result is the standard formula given for cantilever beams under end loads.
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| ==External links==
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| * [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Castigliano.html Carlo Alberto Castigliano]
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| * [http://en.wikibooks.org/wiki/Structural_Engineering#Castigliano.27s_theorems Castigliano's theorems] module of Wikibooks
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| * [http://www.matheplanet.com/matheplanet/nuke/html/article.php?sid=1008&mode=&order=0 Castigliano's method: some examples] – ''German language page''
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| * [http://www.translate.google.com/translate?prev=hp&hl=en&js=y&u=http%3A%2F%2Fwww.matheplanet.com%2Fmatheplanet%2Fnuke%2Fhtml%2Farticle.php%3Fsid%3D1008%26mode%3D%26order%3D0%26ref%3Dhttp%253A%252F%252Fen.wikipedia.org%252Fwiki%252FCastigliano%2527s_method%26f%3D1%26ff%3Dy%26rd3%3D1&sl=de&tl=en&history_state0= Castigliano's method: some examples] – ''Google's translation of the above German language page''
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| [[Category:Structural analysis]]
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