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

to the _v20210971 version. Sorry.

The symmetric part of the Eq. (1.38) matrix is defined compliance matrix for the beam section (and material), and not the matrix itself. According to the Betti theorem, in fact, compliance (and stiffness) matrices are symmetric.

“those directions coincide … in the case of a twice symmetric cross section.” may be safely replaced by the wider “those directions coincide to the symmetry axis and its perpendicular direction in the case of a symmetric cross section.”.

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 firstly substitute the expression of the parametric reaction forces as a function of the external loads within the strain energy $U$, and to subsequently apply the Castigliano's second theorem, but instead
• to firstly apply the Castigliano's second theorem on the raw strain energy expression, thus obtaining displacements, and to subsequently substitute the parametric 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 $\tilde{u}_\mathrm{C}, \tilde{w}_\mathrm{C},\tilde{w}_\mathrm{D}$.

Only the expressions for the statically determinate reactions (i.e. the ones associated to the principal structure residual constraints) may – and must, if present – be replaced within $U$ by their expressions as functions of external loads and parametric reactions before the application of the Castigliano's theorem; such expressions are due to equilibrium alone, and not kinematic compliance.

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

(ipe source)

A general load is applied to a symmetric structure in a); in b), c) the loads applied on each half structure is treated separately. In d), e) the action on the loaded portion is halved, and symmetrically propagated to the other portion; those symmetric actions are accumulated in the symmetric part of the load f). In g), h) the action on the loaded portion is halved, and skew-symmetrically propagated on the unloaded portion; those skew-symmetric actions are accumulated in the skew-symmetric part of the load i).