# A Minimal GBT Model for Distortional-Twist Elastic Analysis of Box-Girder Bridges
## 5 Numerical Results
### Deflection Analysis
#### 5.1.1 Stress Analysis in Pure Torsion  
A stress analysis was carried out by referring to the Fourier solution with \(N=50\) terms. The stresses were evaluated by referring to the uncracked cross section. In order to separate the causes of discrepancies between analytical and FEM analyses, the pure torsion case (\(\theta\
eq 0\), \(\varphi\equiv 0\)) was considered first, for which the Vlasov beam model is usually assumed to be accurate. Thus, the stresses provided by the analytical model via Equations (24) and (25) were compared with those furnished by a FEM model, in which indeformability of the cross section was enforced via \"body constraints\", simulating thin diaphragms at any cross sections of the mesh. It should be remembered, that the analytical model proposed here reproduces the Vlasov model by further encapsassing for variation in the stresses within the thickness.  
The pattern of stresses on a tipical cross section was examined first. Figure 16 reports, by continuous/dashed lines, the analytical longitudinal normal stresses \(\sigma_{z,max}^{m+f}:=\sigma_{z}^{m}+\sigma_{z,max}^{f}\) and the tangential stresses \(\tau_{max}^{m+f}:=\tau^{m}+\tau_{max}^{f}\) evaluated at the middle abscissa of the load (\(z=7.5\,\mathrm{m}\)) and at bottom/top edges of the cross section. The bullets represent numerical FEM results. It appears that the \"enriched Vlasov model\" captures very well the state of stress sufficiently far from the corners of the cross section. Close to them, indeed, and probably exhalted from the abrupt change in thickness, the tangential stresses violate the Bredt law; this occurrence, in turn, entails a local disturbance on the longitudinal stresses for equilibrium reasons. The maximum differences between analytical and numerical results are of the order of 10-15%.  
The dependence of the stresses on the longitudinal abscissa was then investigated. Figure 17 shows how the normal stresses at the corners and the tangential stresses at the half side \(s_{i}=0\) depend on \(z\). It appears that the differences between the analytical and numerical models disappear at large distance from the load, thus confirming the local nature of the disturbance.  
Concerning the tangential reactive stresses, which are related to the complementary torsional moment, they are negligibly small at \(z=7.5\,\mathrm{m}\) (not shown here). However, as Figure 18 shows, they are significant at \(z=11.25\,\mathrm{m}\) (i.e., where the load has a discontinuity point) according to the diagram of \(\theta^{\prime\prime\prime}(z)\) in Figure 15e.  
Figure 15: Derivatives of deflections: (a,c,e) twist angle derivatives, (b,d,f) distortion derivatives; comparison between Fourier (\(N=50\) terms, thick red lines) and uncoupled solutions to the differential equations (thin gray lines).  
Figure 16: Torsion without distortion: longitudinal normal stresses and tangential stresses at the cross section \(z=7.5\) m vs. \(s_{i}\); bottom edge (continuous line), top edge (dashed lines); (**a,b**) \(\sigma_{z,max}^{m+j}:=\sigma_{z}^{m}+\sigma_{z,max}^{f}\); (**c,d**) \(\tau_{max}^{m+j}:=\tau^{m}+\tau_{max}^{f}\); (**a,b**) element 1, (**c,d**) element 2. Fourier analysis with \(N=50\) terms, FEM analysis (bullets).  
Figure 17: Torsion without distortion: longitudinal normal \(\sigma_{z,max}^{m+f}:=\sigma_{z}^{m}+\sigma_{z,max}^{f}\) and tangential \(\tau_{max}^{m+f}:=\tau^{m}+\tau_{max}^{f}\) stresses, at corners (\(s_{i}=\frac{b}{2}\), \(\frac{b}{2}\)) and top face (dashed lines) and bottom face (continuous lines ) vs. \(z\) ; (**a,b**) element 1, (**c,d**) element 2; Fourier analysis with \(N=50\) terms; FEM analysis (bullets).