# A Minimal GBT Model for Distortional-Twist Elastic Analysis of Box-Girder Bridges
## 3 A Minimal GBT Model
### Stresses
\begin{cases}-\frac{E}{2}y_{i}s_{i}\bigg{(}3+4(\alpha-1)\frac{s ^{2}}{b^{2}}-\alpha\bigg{)}\varphi^{\prime\prime}(z)&i=1,\,3\\ -\frac{E}{2}y_{i}s_{i}\bigg{(}-3+4(\alpha+1)\frac{s^{2}}{b^{2}}-\alpha\bigg{)} \varphi^{\prime\prime}(z)&i=2,\,4\end{cases} \] \[\tau^{f}(s,z) = \begin{cases}G\frac{y_{i}}{b^{2}}\big{(}b^{2}(\alpha-3)-12(\alpha -1)s_{i}^{2}\big{)}\varphi^{\prime}(z)&i=1,\,3\\ G\frac{y_{i}}{b^{2}}\big{(}h^{2}(\alpha+3)-12s_{i}^{2}(\alpha+1)\big{)} \varphi^{\prime}(z)&i=2,\,4\end{cases} \]  
with \(\alpha\) defined in Equation (16). For the torsional mode as well, the membrane reactive tangential stresses must be determined from equilibrium, leading to the following (see the Appendix D):  
\[\tau^{mr}(z,s) = \begin{cases}\bigg{(}\frac{bh}{48}\frac{2 bt_{1}+ht_{2}}{t_{1}}- \frac{1}{4}hs_{i}^{2}\bigg{)}E\varphi^{\prime\prime\prime}(z)&i=1,\,3\\ \bigg{(}-\frac{bh}{48}\frac{(bt_{1}+2 ht_{2})}{t_{2}}+\frac{1}{4}bs_{i}^{2} \bigg{)}E\varphi^{\prime\prime\prime}(z)&i=2,\,4\end{cases}  \tag{27}\]  
The relevant stress-fields (with \(max\) denoting evaluation at \(y_{i}=\pm\frac{t_{i}}{j}\)) are qualitatively depicted in Figure 8. The diagrams repeat themselves at any abscissas \(z\), being altered by a factor proportional to \(\varphi(z),\varphi^{\prime}(z),\varphi^{\prime\prime}(z)\), as stated by Equations (26) and (27). The membrane normal stresses \(\sigma_{z}^{m}\) (Figure 8a) are caused by nonuniform warping; they are proportional to \(W_{d}(s)\) (Equation (19)) and vary on \(z\) as \(\varphi^{\prime\prime}(z)\). The flexural normal stresses \(\sigma_{z}^{f}\) (maximum value in Figure 8b) are due to the longitudinal flexure of plates; they, being proportional to \(V_{d}(s)\) (Equation (17)), are cubic on the cross section and depend on \(z\) via \(\varphi^{\prime\prime}(z)\). The transverse flexural normal stresses \(\sigma_{s}^{f}\) (maximum value in Figure 8c) are generated by the flexure of the plates in the transverse direction; being proportional to \(V_{d}^{\prime\prime}(s)\), they are step-wise linear on the cross section and are modulated by \(\varphi(z)\). The membrane reactive tangential stresses \(\tau^{mr}\) (Figure 8d) are step-wise parabolic and proportional to \(\varphi^{\prime\prime\prime}(z)\). The flexural tangential stresses \(\tau^{f}\) (maximum value in Figure 8e) are related to the torsion of the plates; they depend on \(V_{d}^{\prime}(s)\), are step-wise parabolic on the cross section, and are modulated by \(\varphi^{\prime}(z)\). It is worth noticing that, according to the model presented here, the distortional tangential stresses are purely reactive, since no shear strains occur in the distortional mode; in contrast, longitudinal normal stresses are of the active type, directly related to distortional warping. The model, therefore, is different from that adopted in Reference [8], where due to the assumed step-wise constant shear with no warping, the tangential stress is active while the normal stress is reactive. As an advantage of the method followed here, the continuity of the flow of tangential stresses is assured at the joints, a condition that cannot be satisfied in the alternative approach.