# A Minimal GBT Model for Distortional-Twist Elastic Analysis of Box-Girder Bridges
## 5 Numerical Results
### Deflection Analysis
#### 5.1.2 Stress Analysis in Torsion-Distortion  
The complete analytical solution (\(\theta\
eq 0\), \(\varphi\
eq 0\)) is now considered, and relevant stresses (Equations (26) and (27)) are compared with those furnished by the unconstrained FE model. The membrane longitudinal normal stresses at the corners, \(\sigma_{z}^{m}\left(z,\frac{b}{2}\right)\), \(\sigma_{z}^{m}\left(z,\frac{b}{2}\right)\), are plotted in Figure 19a,b; they are found to be in excellent agreement with FEM analysis. Moreover, they are about three times larger that the normal stresses induced by torsion alone (Figure 17), thus highlighting the role of warping by distortion. The membrane tangential stresses at half side, \(\tau^{m}(z,s_{1}=0)\), \(\tau^{m}(z,s_{2}=0)\), are plotted in Figure 19c,d. Here, the contributions of active and reactive parts, together with their sum, are displayed. It is seen that the reactive component associated with distortion is essential, as discussed before, to achieve a good agreement with the numerical results. Finally, the maximum flexural tangential stresses at half side \(\tau^{f}\left(z,s_{1}=0,y_{1}=\frac{t_{1}}{2}\right)\) and \(\tau^{f}\left(z,s_{2}=0,y_{2}=\frac{t_{2}}{2}\right)\) are plotted in Figure 19e,f, showing, again, an excellent accordance between analytical and numerical results. The flexural component of the tangential stress is found to be comparable or even larger than the membrane component, thus revealing the importance to consider the variation in stresses inside the thickness of the cross section of reinforced concrete bridges.  
The transverse normal stresses, mainly related to the distortion, are reported in Figure 20. Stresses acting at the cross section \(z=7.5\,\)m are displayed in Figure 20a,b. The linear law predicted by the analytical model is confirmed by FEM to within an error of about 10% at corners. When the variation with \(z\) of the stresses at corners was analyzed (Figure 20c,d), a 10% error is confirmed just in the region loaded, while the error almost vanishes far from this zone. As a curiosity, while the analytical model predicts the same absolute values for the stresses at the two faces, the FEM analysis reveals slight differences, more significant on one of the two faces. The reactive transverse normal stresses, which are associated with the normal forces induced by the bending moments, were checked as an attempt to explain this discrepancy, but they were found to be negligibly small. Therefore, such a phenomenon cannot be captured by the unique distortion mode considered here. Moreover, further investigation would be needed to unfold the effects of the way the external loads are applied to the box-girder, which, according to the De Saint Venant principle, are not negligible close to the loaded region.  
As a final comment, it is important to notice that the distortional stresses in Figures 19 and 20 are of the same order of magnitude or larger than the stresses due to pure torsion in Figures 16 and 17. Therefore, distortion plays an important role in this sample system.  
Figure 19: Torsion coupled to distortion: (**a,b**) membrane longitudinal normal stresses evaluated at the corners (\(s_{i}=\frac{b}{2}\), \(\frac{b}{2}\)); (**c,d**) active (dashed black line), reactive (blue line), and total (red line) membrane tangential stresses evaluated at half side (\(s_{i}=0\)); (**e,f**) flexural tangential stresses evaluated at half side (\(s_{i}=0\)) bottom/top edge (\(y_{i}=\frac{t_{i}}{2}\)); (**a,c,e**) element 1, (**b,d,f**) element 2; Fourier analysis with \(N=50\) terms, FEM analysis (bullets).