## 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
#### 3.2.1 Effective Internal Stress Force and Stress Couple Resultants
** The integration over the shell thickness of the local constitutive equations for the 3D thin shell stress fields in Equation (45) provides then effective constitutive ones for the 2D thin shell stress resultants and couples in Equation (35) (for \(I=1,2\)) as  
\[N_{II} = 2 Gh|\varepsilon_{II}|\Big{[}a_{II}a_{||I}+\frac{\
u}{1-\
u}a_{ II}|j|+\frac{hb_{II}}{4}\beta_{||I|}\Big{]}\quad\text{(in-plane normal stress resultants)} \tag{47a}\] \[|\varepsilon_{II}|N_{IJ} = Gh|\varepsilon_{II}|\Big{[}a_{||J}+a_{II}a_{J||I}+\frac{hb_{II}}{4} \beta_{||I|}\Big{]}\quad\text{(in-plane shear stress resultants)} \tag{47b}\]\[Q_{I} = \zeta Gh|\epsilon_{IJ}|a_{IJ}\big{[}a_{3||I}+\beta_{I}\big{]}\quad \text{(transverse shear stress resultants)} \tag{47c}\] \[|\epsilon_{IJ}|M_{IIJ} = \frac{Gh^{2}}{6}\epsilon_{IJ}\Big{[}3b_{IJ}\bar{u}_{||I|}+hc_{IJ} \beta_{I||I}+\frac{hv}{1-v}\beta_{I||J}\Big{]}\quad\text{(bending stress moments)}\] (47d) \[|\epsilon_{IJ}|M_{IIJ} = \frac{Gh^{2}}{12}\epsilon_{IJ}\Big{[}3b_{IJ}\bar{u}_{J||I}+h \beta_{I||I}+hc_{IJ}\beta_{J||I}\Big{]}\quad\text{(twisting stress moments)}. \tag{47e}\]  
These expressions rely on the following (zeroth, first, and second) statistical moments:  
\[a_{IJ}(\varrho,\bar{\mathbf{\kappa}}) \stackrel{{\text{def}}}{{=}} \int_{-h/2}^{h/2}\frac{A_{I}B_{I}}{A_{I}B_{I}}\frac{da_{3}}{h} \equiv\int_{-h/2}^{h/2}\frac{\frac{a_{3}}{R_{\eta_{j}}}+1}{\frac{R_{\eta_{j}} }{R_{\eta_{j}}}+1}\frac{da_{3}}{h} \tag{48a}\] \[\equiv 1-\frac{R_{\eta_{j}}^{2}}{h^{2}}\bigg{(}\frac{h}{R_{n_{ I}}}-\frac{h}{R_{n_{I}}}\bigg{)}\Bigg{[}\frac{h}{R_{n_{I}}}-\ln\Bigg{(}\frac{1+ \frac{h}{2R_{n_{I}}}}{1-\frac{h}{2R_{n_{I}}}}\Bigg{)}\Bigg{]}\] \[\sigma_{12}=G(\varepsilon_{12}^{s}+\varepsilon_{21}^{s})\equiv 2G \varepsilon_{12}^{s}\equiv\sigma_{21} \text{(general case)} \tag{45}\]0 \[\sigma_{12}=G(\varepsilon_{12}^{s}+\varepsilon_{21}^{s})\equiv 2G \varepsilon_{12}^{s}\equiv\sigma_{21} \text{(general case)} \tag{45}\]1 (48b) \[\sigma_{12}=G(\varepsilon_{12}^{s}+\varepsilon_{21}^{s})\equiv 2G \varepsilon_{12}^{s}\equiv\sigma_{21} \text{(general case)} \tag{45}\]2 \[\sigma_{12}=G(\varepsilon_{12}^{s}+\varepsilon_{21}^{s})\equiv 2G \varepsilon_{12}^{s}\equiv\sigma_{21} \text{(general case)} \tag{45}\]3 (48c) \[\sigma_{12}=G(\varepsilon_{12}^{s}+\varepsilon_{21}^{s})\equiv 2G \varepsilon_{12}^{s}\equiv\sigma_{21} \text{(general case)} \tag{45}\]4 \[\sigma_{12}=G(\varepsilon_{12}^{s}+\varepsilon_{21}^{s})\equiv 2G \varepsilon_{12}^{s}\equiv\sigma_{21} \text{(general case)} \tag{45}\]5  
As observed now in Equation (47), these functions of \(\bar{\mathbf{\kappa}}\) and \(\varrho(\bar{\mathbf{\kappa}})\) introduced in Equation (39) produce various coupling effects between all the stress resultants and stress couples due as well to the possible local \"non-flatness\" and \"non-sphericity\" of the shell principal coordinate system on its generic rest shape \(\overline{\mathcal{S}}\). Their approximations in Equation (48) constitute in a sense Byrne-Flugge-Lur'ye's approximations when used notably with the Love-Kirchhoff-Koiter kinematic constraints (see [76; 80]; [73], Section 2.10, pages 43-46). The account of at least such a second approximation order in \(\max_{J=1,2}\Bigg{\{}\Bigg{|}\frac{h}{R_{n_{J}}}\Bigg{\}}\Bigg{|}\ll 1\) (i.e., all the terms linear in \(\Bigg{\{}\frac{\alpha_{3}}{R_{n_{I}}}\,,\,\bigg{(}\frac{\alpha_{3}}{R_{n_{I}}} \bigg{)}^{2}\,,\,\frac{|\epsilon_{IJ}|\alpha_{3}^{2}}{R_{n_{J}}R_{n_{I}}}\Bigg{\}} _{I=1,2}\)) usually allows not restricting the analysis to _thin_, _shallow shells_ and explaining then the appearance of _bending moments under extension_ and, reciprocally, _extension forces under bending_--as well as the in-plane analogous _twisting-shearing_ effects induced (without adding new kinematical variables) by geometrical corrections.