# A Minimal GBT Model for Distortional-Twist Elastic Analysis of Box-Girder Bridges
## 5 Numerical Results
### Deflection Analysis  
First, Fourier analysis was carried out for the GBT equations. By expanding the modal loads in sinus Fourier series, the coefficients \(c_{t_{n}}=c_{d_{n}}=:c_{n}\) were found, as reported in Figure 12(a), together with the truncated series with \(N=20,50,100\) terms. The Fourier spectra for deflections, as furnished by Equation (31), together with the reconstructed series (for \(N=50\), value used from now on) are represented in Figure 12(b),c. As a general comment, distortion of the box girder is remarkable, larger than twist, and it propagates to a large distance from the loaded region. \begin{table}
\begin{tabular}{c c c c} \hline \hline (\(\hat{u}\),\(\hat{y}\)) & (\(\hat{1}\),\(\hat{1}\)) & (\(\hat{1}\),\(\hat{2}\)) & (\(\hat{2}\),\(\hat{2}\)) \\ \hline \(C_{ij}^{c}\) & \(1.18\times 10^{8}\) & \(1.70\times 10^{8}\) & \(2.44\times 10^{8}\) \\ \(C_{ij}^{f}\) & \(1.74\times 10^{6}\) & \(2.21\times 10^{6}\) & \(3.29\times 10^{6}\) \\ \(D_{ij}^{s}\) & \(1.02\times 10^{8}\) & 0 & 0 \\ \(D_{ij}^{t}\) & \(1.88\times 10^{6}\) & \(3.50\times 10^{5}\) & \(2.63\times 10^{6}\) \\ \(B_{ij}^{f}\) & 0 & 0 & \(6.80\times 10^{5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Coefficients of the Generalized Beam Theory (GBT) equations. Figure 12: Segment of distributed load, \(p_{0}=\pm\frac{p}{\Delta}\), centered at \(z=\xi\): (a) longitudinal view; (b) cross section at \(z=\xi\), dimensions in meters. The simplified method was successively applied by uncoupling the GBT equations. The integration domain was broken in three subintervals, i.e., \(\mathcal{I}_{1}:=\left(0,\xi-\frac{\Delta}{2}\right)\), \(\mathcal{I}_{2}:=\left(\xi-\frac{\Delta}{2},\xi+\frac{\Delta}{2}\right)\), \(\mathcal{I}_{3}:=\left(\xi+\frac{\Delta}{2},\ell\right)\), and the general solution (Equation (34)) was written in each of them. There, \(\bar{\theta}_{i}(z_{i})=\bar{\varphi}_{i}(z_{i})=0\) (\(i=1,3\)) and \(\bar{\theta}_{2}(z_{2})=-\frac{bp_{0}}{2\ell}z_{2}^{2}\), \(\bar{\varphi}_{2}(z_{2})=\frac{bp_{0}}{2\ell}\) are the particular solutions. The twelve arbitrary constants \(c_{k_{j}}\) appearing in each problem were determined by enforcing two boundary conditions at each \(z=0\), \(\ell\) and four conditions at each of the internal boundaries, as discussed before. The responses \(\theta(z),\varphi(z)\) thus obtained, Equation (34), are plotted in Figure 14 and compared with those of the complete Fourier analysis. Here, the exact numerical results, as provided by the FEM analysis are also reported. It appears that FEM and Fourier analysis are in excellent agreement, in spite of the stronger simplifications introduced in the analytical model. The simplified analysis also supplies satisfactory results (maximum error of about +11% on twist and about +10% on distortion), especially far from the loads. Differences with the exact results put into light the contribution of coupling, which should not be ignored, if more accurate results are desired. It is concluded that the error provided by the simplified approach is  
Figure 13: Fourier coefficients and truncated series for (**a**) loads, (**b**) twist angle, and (**c**) distortion. reasonable and that, at least in this case study, this method is conservative since it magnifies the deflection (in agreement with the conclusions of Reference [6]). The first, second, and third derivatives of both \(\theta(z)\) and \(\varphi(z)\) amplitudes, needed for stresses evaluation according to Equations (24)-(27), were then computed. In Figure 15, the Fourier (coupled) solution is compared with that of the simplified (uncoupled) method (comparison with FEM being not straightforward). It appears that, although small spurious oscillations persist with \(N=50\), the Fourier solution is quite smooth.