Virtual CdM lab 2021+ shortcut.

lecture notes, as of 2021-09-07 (big endian).

to the _v20210971 version. Sorry.

The paragraphs Once we obtained the expressions… and All the contribution of the external loads.. turned out to be subtly wrong, even if the so obtained results are in fact correct for the given test case (but not e.g. in the case of inhomogeneous imposed displacements).

The correct procedure is in general:

• not to substitute first the expression of the reaction forces as a function of the external loads within the strain energy $U$, and then to apply the Castigliano's second theorem, but instead
• to first apply the Castigliano's second theorem on the raw strain energy expression thus obtaining displacements, and then to substitute the reaction force expressions as functions of the external loads within the obtained displacement expressions.

and hence, equivalently

$$ U=U\left(P,F,X_\mathrm{A},Y_\mathrm{A},\Psi_\mathrm{A},H,I,Z_\mathrm{A},\Theta_\mathrm{A},\Phi_\mathrm{A}\right) $$

$$ \tilde{u}_\mathrm{B} \gets \frac{\partial U}{\partial F} = \tilde{u}_\mathrm{B} \left(P,F,X_\mathrm{A},Y_\mathrm{A},\Psi_\mathrm{A},H,I,Z_\mathrm{A},\Theta_\mathrm{A},\Phi_\mathrm{A}\right) $$

$$ \tilde{u}_\mathrm{B} \gets \left. \tilde{u}_\mathrm{B} \right|_{ X_\mathrm{A} \gets X_\mathrm{A}\left(P,F,H,I\right), Y_\mathrm{A}\gets Y_\mathrm{A}\left(P,F,H,I\right), \ldots}= \tilde{u}_\mathrm{B}\left(P,F,H,I\right) $$

$$ \tilde{u}_\mathrm{B} \gets \left. \tilde{u}_\mathrm{B} \right|_{F \gets 0,I \gets 0} = \tilde{u}_\mathrm{B} \left( P, H \right) $$ where “$\gets$” is the assignment operator; analogous treatise are performed to obtain $\tilde{u}_\mathrm{C}, \tilde{w}_\mathrm{C},\tilde{w}_\mathrm{D}$.

A solution scheme based on the principle of virtual works (and not on the second Castigliano theorem) is not prone to this subtle error, since the tracer virtual unit action used to obtain the consistent virtual displacement/deformation field is – since virtual – uncoupled with both the real external forces and the real parametrically defined reactions.

The beam structure centroidal axis lies on a plane, which is also a symmetry plane for the cross-sections.

Symmetric and skew-symmetric loads with respect to such a plane are called in-plane and out-of-plane loads, respectively.

If the superposition of effects holds (e.g., if the structure behaves linearly) each load set only induces an associated subset of the possible components of internal action, see

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Maxima worksheet v0001

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