# A Minimal GBT Model for Distortional-Twist Elastic Analysis of Box-Girder Bridges
## 3 A Minimal GBT Model
### Stresses  
Once the GBT equations are solved and the amplitudes \(\theta(z)\), \(\varphi(z)\) are evaluated, the active stresses can be computed by coming back to Equation (5). Reactive stresses, instead, must be determined from equilibrium. By separating the effects of the two modes, the following results are drawn (with the Poisson ratio taken zero). Moreover, from now on, \(\tau:=\tau_{\text{zS}}\) is used to simplify the notation. In the torsional mode, the active stresses are as follows:  
\[\sigma_{s}^{m}(z,s) =0 \tag{24}\] \[\sigma_{z}^{m}(z,s) = \begin{cases}\frac{E}{2}h\beta\varepsilon_{i}\theta^{\prime\prime }(z)&i=1,\,3\\ -\frac{E}{2}b\beta\varepsilon_{i}\theta^{\prime\prime}(z)&i=2,\,4\end{cases} \] \[\tau^{m}(z,s) = \begin{cases}G\frac{bht_{2}}{h\mu_{1}+b\mu_{2}}\theta^{\prime} (z)&i=1,\,3\\ G\frac{bht_{1}}{h\mu_{1}+b\mu_{2}}\theta^{\prime}(z)&i=2,\,4\end{cases} \] \[\sigma_{s}^{f}(z,s) =0\] \[\sigma_{z}^{f}(z,s) =-Ey_{i}s_{i}\theta^{\prime\prime}(z)\qquad i=1,\ldots 4\] \[\tau^{f}(z,s) =-2 Gy_{i}\theta^{\prime}(z)\qquad i=1,\ldots 4\]  
and the reactive stresses (see the Appendix A) are as follows:  
\[\tau^{mr}(z,s)= \begin{cases}-E\theta^{\prime\prime\prime}(z)h\frac{(ht_{1}-b _{2})\left(3h\left(b^{2}-4\varepsilon_{z}^{2}\right)t_{1}+b\left(b^{2}+2\left( h^{2}-6\varepsilon_{z}^{2}\right)\right)t_{2}\right)}{48\left(ht_{1}+b\mu_{2} \right)^{2}}&i=1,3\\ E\theta^{\prime\prime\prime}(z)b\frac{(ht_{1}-b_{2})\left(h\left(2b^{2}+h^{2} -2\varepsilon_{z}^{2}\right)t_{1}+3b\left(h^{2}-4\varepsilon_{z}^{2}\right)t _{2}\right)}{48\left(ht_{1}+b\mu_{2}\right)^{2}}&i=2,4\end{cases}  \tag{25}\]  
The normal stresses \(\sigma_{z}^{m}(z,s)\) coincide with those predicted by the Vlasov theory (see Equation (A3) in Appendix A), while the tangential stresses \(\tau^{m}(z,s)\) coincide with those of the [PERSON] theory (see Equation (A4) in Appendix A). The reactive tangential stresses \(\tau^{mr}\) cannot be captured by the constitutive law of the GBT model (which is based on the same Vlasov kinematic constraint on the flow). The relevant stress-fields are qualitatively depicted in Figure 7 at a generic cross-section; there, the index \(max\) denotes evaluation at the top/bottom face of the plates (i.e., at \(y_{i}=\pm\frac{\mu_{1}}{2}\)). The diagrams repeat themselves at any abscissas \(z\), being altered by a factor proportional to \(\theta^{\prime}(z),\theta^{\prime\prime}(z),\theta^{\prime\prime\prime}(z)\), as stated by Equations (24) and (25). The membrane normal stresses \(\sigma_{z}^{m}\) (Figure 7a) are caused by nonuniform warping; they are proportional to \(W_{t}(s)\) (Equation (15)) and vary on \(z\) as \(\theta^{\prime\prime}(z)\). The flexural normal stresses \(\sigma_{z}^{f}\) (maximum value in Figure 7b) are caused by the flexure of plates in the longitudinal direction; they are proportional to \(V_{t}(s)\) (Equation (13)) and depend on \(z\) via \(\theta^{\prime\prime}(z)\). The membrane active tangential stresses \(\tau^{m}\) (Figure 7c) are consistent with the Bredt behavior of the box; they are step-wise constant on the cross section and vary with \(\theta^{\prime}(z)\). The reactive component \(\tau^{mr}\) (Figure 7d) must be added to them; they are step-wise parabolic on the cross section and vary with \(\theta^{\prime\prime\prime}(z)\). The flexural tangential stresses \(\tau^{f}\) (maximum value in Figure 7e) are related to the torsion of the plates; being proportional to \(V_{t}^{\prime}(s)\), they are step-wise constant on the cross section and vary with \(\theta^{\prime}(z)\). In the distortional mode, the active stresses are as follows:  
\[\sigma_{s}^{m}(s,z) =0 \tag{26}\] \[\sigma_{z}^{m}(s,z) = \begin{cases}\frac{E}{2}hs_{i}\varphi^{\prime\prime}(z)&i=1,\,3\\ -\frac{E}{2}hs_{i}\varphi^{\prime\prime}(z)&i=2,\,4\end{cases} \] \[\tau^{m}(s,z) =0\] \[\sigma_{s}^{f}(s,z) = \begin{cases}-Ey_{i}\frac{12(\alpha-1)}{b^{2}}s_{i}\varphi(z)&i=1,\,3\\ -Ey_{i}\frac{12(\alpha+1)}{b^{2}}s_{i}\varphi(z)&i=2,\,4\end{cases} \] \[\sigma_{z}^{f}(s,z) =