## 3 Classical NR Shell Theory for Moderately Thick, Linear Elastic, Shells
### Reduction from 3D to 2D Shell Elasticity by Means of the Principle of Virtual Power
#### 3.2.2 Effective External Force and Couple Resultants  
We considered boundary problems that describe the linear elastodynamics of the shell undergoing the actions of different distributed forces in the bulk \(\mathcal{V}\stackrel{{\mathrm{def}}}{{=}}\overline{\mathcal{S}} \times[-h/2,h/2]\) and on a part of its surface boundary \(\partial\mathcal{V}\). In principle, in shell analyses, _essential_ (i.e., _kinematic_) _loading conditions_ that notably involve some components of the displacement in Equation (24) can only be imposed on some part \(\overline{\mathcal{C}}_{u}\times[-h/2,h/2]\subseteq\mathcal{S}_{\ell}\), while the _natural_ (i.e., _sthenic_) _loading_ ones may arbitrary be imposed on any of portion of \(\partial\mathcal{V}\). On \(\mathcal{S}_{\ell}\), mixed (i.e., involving essential and natural) boundary conditions can also be applied; moreover, the prescribed components of displacements can still be associated with the reaction efforts, and we treat them hereafter as the other prescribed ones. As in [90] (Chapter 9), the shell is subjected then to applied body forces of density \(\mathbf{f}=f_{I}\mathbf{t}_{I}+f_{3}\mathbf{n}\) per unit volume in the bulk of \(\mathcal{V}\) and possibly boundary forces of density \(\mathbf{F}=F_{I}\mathbf{t}_{I}+F_{3}\mathbf{n}\) per unit surface on the boundary \(\partial\mathcal{V}\); on the lateral surface \(\mathcal{S}_{\ell}\), the components of that latter are more judiciously expressed by \(\mathbf{F}=F_{\tau}\mathbf{\tilde{\tau}}+F_{\psi}\mathbf{\tilde{v}}+F_{3}\mathbf{n}\) and \(F_{\
u}=\mathbf{\tilde{v}}\cdot\mathbf{F}\) with \(F_{\tau}=\mathbf{\tilde{\tau}}\cdot\mathbf{F}\). For simplicity, we assumed nevertheless that there is no concentrated load applied at the kinematical free edges and sharp corners of \(\partial\mathcal{V}\) and, so, on the kinematical free vertices \(\{\overline{s}_{\ell}^{c}\}\) of \(\overline{\mathcal{C}}\setminus\overline{\mathcal{C}}_{u}\). **D'Alembert-Lagrange's generalized internal forces \(\mathcal{Q}_{\mathfrak{q}}^{\mathrm{ext}}\) of the NR-model. ** As for the internal stress, the external loads can also be substituted by an effective system of force and couple loads considering the variational work (or virtual power) of the 3D loads \(\mathbf{f}\) and \(\mathbf{F}\) for any kinematically admissible field of variation \(\mathbf{u}\) as in Equation (26):  
\[\mathcal{P}^{\mathrm{ext}}[\mathbf{u},\mathbf{\underline{u}}](t)\stackrel{{ \mathrm{def}}}{{=}}\int_{\mathcal{V}}\mathbf{f}(\mathbf{\alpha},t)\cdot\mathbf{u}(\mathbf{ \alpha})\,d\mathcal{V}+\int_{\partial\mathcal{V}}\mathbf{F}(\mathbf{\alpha},t)\cdot \mathbf{u}(\mathbf{\alpha})\,d\mathcal{V}\,. \tag{59}\]  
The final contributions for Equation (28):  
\[\overline{\mathcal{P}}^{\mathrm{ext}}[\mathfrak{q},\mathfrak{q}](t)\equiv \overline{\mathcal{P}}^{\mathrm{ext}}_{/\overline{\mathcal{S}}}[\mathfrak{q}, \mathfrak{q}](t)+\overline{\mathcal{P}}^{\mathrm{ext}}_{/\overline{\mathcal{C }}}[\mathfrak{q},\mathfrak{q}](t)\] (60a) with \[ \begin{array}{l}\overline{\mathcal{P}}^{\mathrm{ext}}_{/ \overline{\mathcal{S}}}[\mathfrak{q},\mathfrak{q}]\stackrel{{ \mathrm{def}}}{{=}}\int_{\overline{\mathcal{S}}}[\mathbf{\tilde{p}}\cdot\mathbf{ \tilde{q}}+\mathbf{\tilde{m}}\cdot\mathbf{\tilde{\theta}}]d\overline{\mathcal{S}}\equiv \int_{\overline{\mathcal{S}}}\mathfrak{q}^{\mathrm{T}}\,\mathcal{Q}_{\mathfrak{q }}^{\mathrm{ext}}d\overline{\mathcal{S}}\\ \overline{\mathcal{P}}^{\mathrm{ext}}_{/\overline{\mathcal{C}}}[\mathfrak{q}, \mathfrak{q}]\stackrel{{\mathrm{def}}}{{=}}\int_{\overline{ \mathcal{C}}}[\overline{\lambda}(\mathbf{\tilde{s}}_{\ell})\cdot\mathbf{\tilde{u}}( \mathbf{\tilde{s}}_{\ell})+\mathbf{\tilde{\mu}}(\mathbf{\tilde{s}}_{\ell})\cdot\mathbf{\tilde{ \theta}}(\mathbf{\tilde{s}}_{\ell})]ds_{\ell}\equiv\oint_{\overline{\mathcal{C}}} \mathfrak{q}^{\mathrm{T}}\,\mathcal{Q}_{\mathfrak{q}}^{\mathrm{ext}}ds_{\ell} \end{array}