The Castigliano's method (by Carlo Alberto Castigliano ) is the basis for various calculation methods in engineering mechanics . It is based on an energy approach and enables the relatively simple calculation of selected quantities.
Castigliano's Theorem The partial derivation of the deformation energy stored in a linear elastic body according to the external force results in the shift of the force application point in the direction of this force. Analogously, the partial derivative of the strain energy for a moment , the rotation of the beam at the point of this moment. In order to be able to determine the deflection at points without the application of force with the Castigliano theorem, auxiliary forces must be introduced at these points, which are set to zero after derivation.
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{\ displaystyle {\ frac {\ partial U} {\ partial F_ {k}}} = v_ {k} = \ sum _ {i = 1} ^ {n} \ int _ {l_ {i}} \ left [ {\ frac {M_ {bxi}} {(EI_ {xx}) _ {i}}} {\ frac {\ partial M_ {bxi}} {\ partial F_ {k}}} + {\ frac {M_ {byi }} {(EI_ {yy}) _ {i}}} {\ frac {\ partial M_ {byi}} {\ partial F_ {k}}} + {\ frac {M_ {ti}} {(GI_ {t }) _ {i}}} {\ frac {\ partial M_ {ti}} {\ partial F_ {k}}} + {\ frac {F_ {Li}} {(EA) _ {i}}} {\ frac {\ partial F_ {Li}} {\ partial F_ {k}}} + {\ frac {F_ {Qxi}} {(GA \ kappa _ {x}) _ {i}}} {\ frac {\ partial F_ {Qxi}} {\ partial F_ {k}}} + {\ frac {F_ {Qyi}} {(GA \ kappa _ {y}) _ {i}}} {\ frac {\ partial F_ {Qyi} } {\ partial F_ {k}}} \ right] ds_ {i}}
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{\ displaystyle {\ frac {\ partial U} {\ partial M_ {k}}} = \ varphi _ {k} = \ sum _ {i = 1} ^ {n} \ int _ {l_ {i}} \ left [{\ frac {M_ {bxi}} {(EI_ {xx}) _ {i}}} {\ frac {\ partial M_ {bxi}} {\ partial M_ {k}}} + {\ frac {M_ {byi}} {(EI_ {yy}) _ {i}}} {\ frac {\ partial M_ {byi}} {\ partial M_ {k}}} + {\ frac {M_ {ti}} {(GI_ {t}) _ {i}}} {\ frac {\ partial M_ {ti}} {\ partial M_ {k}}} + {\ frac {F_ {Li}} {(EA) _ {i}}} {\ frac {\ partial F_ {Li}} {\ partial M_ {k}}} + {\ frac {F_ {Qxi}} {(GA \ kappa _ {x}) _ {i}}} {\ frac { \ partial F_ {Qxi}} {\ partial M_ {k}}} + {\ frac {F_ {Qyi}} {(GA \ kappa _ {y}) _ {i}}} {\ frac {\ partial F_ { Qyi}} {\ partial M_ {k}}} \ right] ds_ {i}}
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Distortion energy (deformation energy)
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Shear correction factor of the respective cross-section
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Castigliano's theorem can also be used to calculate statically indeterminate quantities. In this special form it is then referred to as the Menabrea sentence. Menabrea's theorem states that the partial derivative of the deformation energy after a statically indeterminate support reaction is zero.
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internal supplementary energy
literature
Carlo Alberto Castigliano: Théorie de l'équilibre des systèmes élastiques et ses applications . Nero, Turin 1879.
Heinz Parkus : Mechanics of Solid Bodies. 2nd Edition. Springer-Verlag, Vienna 1995, ISBN 3-211-80777-2
Jens Wittenburg , Eduard Pestel : Strength theory - a text and work book. 3. Edition. Springer-Verlag, Berlin 2001, ISBN 3-540-42099-1
Dietmar Gross, Werner Hauger, Walter Schnell: Technical Mechanics 1. Statics. 6th edition. Springer Verlag, Berlin 2002, ISBN 3-540-43850-5
Herbert Balke : Introduction to technical mechanics - strength theory. 1st edition. Springer-Verlag, Berlin 2008, ISBN 978-3-540-37890-7
R. Mahnken: Textbook of Technical Mechanics - Elastostatics , 1st edition Springer, Berlin 2015, ISBN 978-3-662-44797-0
Christian Spura: Technical Mechanics 2. Elastostatics , 1st edition Springer, Wiesbaden 2019, ISBN 978-3-658-19978-4
Web links
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">
![{\ displaystyle {\ frac {\ partial U} {\ partial F_ {k}}} = v_ {k} = \ sum _ {i = 1} ^ {n} \ int _ {l_ {i}} \ left [ {\ frac {M_ {bxi}} {(EI_ {xx}) _ {i}}} {\ frac {\ partial M_ {bxi}} {\ partial F_ {k}}} + {\ frac {M_ {byi }} {(EI_ {yy}) _ {i}}} {\ frac {\ partial M_ {byi}} {\ partial F_ {k}}} + {\ frac {M_ {ti}} {(GI_ {t }) _ {i}}} {\ frac {\ partial M_ {ti}} {\ partial F_ {k}}} + {\ frac {F_ {Li}} {(EA) _ {i}}} {\ frac {\ partial F_ {Li}} {\ partial F_ {k}}} + {\ frac {F_ {Qxi}} {(GA \ kappa _ {x}) _ {i}}} {\ frac {\ partial F_ {Qxi}} {\ partial F_ {k}}} + {\ frac {F_ {Qyi}} {(GA \ kappa _ {y}) _ {i}}} {\ frac {\ partial F_ {Qyi} } {\ partial F_ {k}}} \ right] ds_ {i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95b87297e460081472ab6ab65c21f28a74e5009a)
![{\ displaystyle {\ frac {\ partial U} {\ partial M_ {k}}} = \ varphi _ {k} = \ sum _ {i = 1} ^ {n} \ int _ {l_ {i}} \ left [{\ frac {M_ {bxi}} {(EI_ {xx}) _ {i}}} {\ frac {\ partial M_ {bxi}} {\ partial M_ {k}}} + {\ frac {M_ {byi}} {(EI_ {yy}) _ {i}}} {\ frac {\ partial M_ {byi}} {\ partial M_ {k}}} + {\ frac {M_ {ti}} {(GI_ {t}) _ {i}}} {\ frac {\ partial M_ {ti}} {\ partial M_ {k}}} + {\ frac {F_ {Li}} {(EA) _ {i}}} {\ frac {\ partial F_ {Li}} {\ partial M_ {k}}} + {\ frac {F_ {Qxi}} {(GA \ kappa _ {x}) _ {i}}} {\ frac { \ partial F_ {Qxi}} {\ partial M_ {k}}} + {\ frac {F_ {Qyi}} {(GA \ kappa _ {y}) _ {i}}} {\ frac {\ partial F_ { Qyi}} {\ partial M_ {k}}} \ right] ds_ {i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd49bd957010af34c23aee616fa0c05eef1f4cec)
