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wikipaomffcd2022:start [2022/03/08 11:12] – [Errata corrige] ebertocchiwikipaomffcd2022:start [2022/03/08 11:38] (versione attuale) – [Errata corrige] ebertocchi
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 === pp.49-50 === === pp.49-50 ===
-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.+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: 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   *  **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.   * 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
  
 $$ U=U\left(P,F,X_\mathrm{A},Y_\mathrm{A},\Psi_\mathrm{A},H,I,Z_\mathrm{A},\Theta_\mathrm{A},\Phi_\mathrm{A}\right) $$ 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)
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 \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) \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.
 ==== Planar beam structure i.p. and o.o.p. loadings ==== ==== Planar beam structure i.p. and o.o.p. loadings ====
 The beam structure centroidal axis lies on a plane, which is also a symmetry plane for the cross-sections.  The beam structure centroidal axis lies on a plane, which is also a symmetry plane for the cross-sections. 
wikipaomffcd2022/start.1646737952.txt.gz · Ultima modifica: 2022/03/08 11:12 da ebertocchi