Satz von Castigliano – Wikipedia

The Satz von Castigliano (according to Carlo Alberto Castigliano) is the basis for various calculation methods in technical mechanics. It is based on an energy approach and enables the relatively simple calculation of selected sizes.

The partial derivation of the formal change energy stored in a linear elastic body according to the external force results in the shift

${displaystyle v_{k}}$

of the power point in the direction of this force. Analogously, the partial derivation of the formal change energy results in the twisting after a moment

${DISPLAYSTYLE VARPHI _ {k}}$

of the bar at the point of attack of this moment. In order to be able to determine the deflection in places without the influence of Castigliano, auxiliary workers must be introduced at these areas, which are set to zero after the derivation.

${displaystyle {frac {partial U}{partial F_{k}}}=v_{k}=sum _{i=1}^{n}int _{l_{i}}left[{frac {M_{bxi}}{(EI_{xx})_{i}}}{frac {partial M_{bxi}}{partial F_{k}}}+{frac {M_{byi}}{(EI_{yy})_{i}}}{frac {partial M_{byi}}{partial F_{k}}}+{frac {M_{ti}}{(GI_{t})_{i}}}{frac {partial M_{ti}}{partial F_{k}}}+{frac {F_{Li}}{(EA)_{i}}}{frac {partial F_{Li}}{partial F_{k}}}+{frac {F_{Qxi}}{(GAkappa _{x})_{i}}}{frac {partial F_{Qxi}}{partial F_{k}}}+{frac {F_{Qyi}}{(GAkappa _{y})_{i}}}{frac {partial F_{Qyi}}{partial F_{k}}}right]ds_{i}}$

${displaystyle {frac {partial U}{partial M_{k}}}=varphi _{k}=sum _{i=1}^{n}int _{l_{i}}left[{frac {M_{bxi}}{(EI_{xx})_{i}}}{frac {partial M_{bxi}}{partial M_{k}}}+{frac {M_{byi}}{(EI_{yy})_{i}}}{frac {partial M_{byi}}{partial M_{k}}}+{frac {M_{ti}}{(GI_{t})_{i}}}{frac {partial M_{ti}}{partial M_{k}}}+{frac {F_{Li}}{(EA)_{i}}}{frac {partial F_{Li}}{partial M_{k}}}+{frac {F_{Qxi}}{(GAkappa _{x})_{i}}}{frac {partial F_{Qxi}}{partial M_{k}}}+{frac {F_{Qyi}}{(GAkappa _{y})_{i}}}{frac {partial F_{Qyi}}{partial M_{k}}}right]ds_{i}}$

${displaystyle U=U(q_{1},dots ,q_{n})}$

= Disturbance energy (formal change energy)

${displaystyle n}$

= Number of areas

${displaystyle i}$

= Index of the respective area

${displaystyle l_{i}}$

= Lengths of the areas

${displaystyle F_{k}}$

= generalized force

${displaystyle M_{k}}$

= generalized moment

${Displaystyle m_ {bxi}, m_ {byi}}$

= Bending moments

${displaystyle M_{ti}}$

= Torsionsmoment

${displaystyle F_{Li}}$

= Longing power

${Disclioughyle F_ {}}, F_ {}}$

= Transverse forces

${displaystyle kappa _{x},kappa _{y}}$

= Thrust correction factor of the respective cross -section

${displaystyle q_{i}}$

= generalized commuting paths

${displaystyle s_{i}}$

= local coordinates with

${displaystyle 0leq s_{i}leq l_{i}}$

Castigliano’s sentence can also be used statically indefinite variables for the calculation. In this special form, it is then referred to as the sentence of Menabrea. Menabrea’s sentence states that the partial derivation of the formal change energy after a statically undetermined warehouse reaction is zero.

${displaystyle {frac {partial U^{*}}{partial X_{i}}}=0}$

with

${displaystyle i=1,dots ,n}$

${displaystyle X_{i}}$

= statically indefinite variables (whose work must be zero each)

${Displaystyle u^{*} = u^{*} (x_ {1}, dots, x_ {n})}$

= inner supplementary energy

• Carlo Alberto Castigliano: Theory of equilibrium of elastic systems and its applications . Nero, Turin 1879.
• Heinz Parkus: Mechanics the solid body. 2nd Edition. Springer-Verlag, Vienna 1995, ISBN 3-211-80777-2
• Jens Wittenburg, Eduard Pestel: Strengthening- a teaching and workbook. 3. Edition. Springer-Verlag, Berlin 2001, ISBN 3-540-42099-1
• Herbert Balk: Introduction to technical mechanics – strength theory. 1st edition. Springer-Verlag, Berlin 2008, ISBN 978-3-540-37890-7
• R. Mahnken: Technical mechanics textbook – elastostatics , 1st ed. Springer, Berlin 2015, ISBN 978-3-662-44797-0
• Christian Spura: Technical mechanics 2. Elastostatics , 1st ed. Springer, Wiesbaden 2019, ISBN 978-3-658-19978-4