# A Minimal GBT Model for Distortional-Twist Elastic Analysis of Box-Girder Bridges
## 4 Algorithmic Aspects
### Solution Methods
#### 4.2.2 A Simplified Approach: The Uncoupled Equations  
A heuristic approach is attempted here, consistent with the literature, in which all out-of-diagonal terms are (arbitrarily) neglected. In this case, the differential equations and boundary conditions read as follows:  
\[\begin{split} EI_{\omega}\theta^{\prime\prime\prime\prime}-GJ \theta^{\prime\prime}&=p_{t}\\ \theta(0)=\theta^{\prime\prime}(0)=\theta(\ell)=\theta^{\prime \prime}(\ell)&=0\end{split} \tag{32}\]  
together with  
\[\begin{split} C_{22}\varphi^{\prime\prime\prime\prime}+B_{22} \varphi&=p_{d}\\ \varphi(0)=\varphi^{\prime\prime}(0)=\varphi(\ell)=\varphi^{ \prime\prime}(\ell)&=0\end{split} \tag{33}\]  
The first equation coincides with Equation (2) of the Vlasov theory, while the second one is the classic equation of a beam on Winkler soil. It should be noticed that, according to this simplified approach, distortional stresses simply add themselves to Vlasov stresses without any feedback effect.  
Integration usually calls for dividing the \((0,\ell)\) interval in subintervals in which the loads are continuous; in the generic subinterval \(i\), the general solution reads as follows:  
\[\begin{split}\theta(z_{i})&=c_{1i}e^{\lambda(z_{i} -l_{i})}+c_{2i}e^{-2z_{i}}+c_{3i}z_{i}+c_{4i}+\bar{\theta}_{i}(z_{i})\\ \varphi(z_{i})&=e^{-\mu z}\big{(}c_{1i}\sin(\mu z) +c_{2i}\cos(\mu z)\big{)}+e^{\mu z}\big{(}c_{3i}\sin(\mu z)+c_{4i}\cos(\mu z) \big{)}+\bar{\varphi}_{i}(z_{i})\end{split} \tag{34}\]  
where \(l_{i}\) is the lengths of the subintervals, \(z_{i}\) is the local abscissas spanning them, with the origin at the left end; and \(c_{k_{i}}\) is an arbitrary constant. Moreover, \(\bar{\theta}_{i}(z_{i})\), \(\bar{\varphi}_{i}(z_{i})\) are particular solutions. Finally,  
\[\lambda:=\sqrt{\frac{GJ}{EI_{\omega}}},\qquad\mu:=\sqrt[4]{\frac{B_{22}}{4\,C _{22}}} \tag{35}\]  
are wave numbers.  
The arbitrary constants \(c_{j}^{(i)}\) appearing in each problem are determined by enforcing two boundary conditions at each \(z=0\), \(\ell\) end and four conditions at each of the internal boundaries, as supplied by Equation (23). If no lumped couples are present, these read as follows:  
* for the twist problem, (i) continuity of \(\theta\), (ii) continuity of \(\theta^{\prime}\), (iii) equilibrium of forces dual of \(\delta\theta\) (entailing continuity of the bimoment \(B=EI_{\omega}\theta^{\prime\prime}\)), and (iv) equilibrium of forces dual of \(\delta\theta^{\prime}\) (entailing continuity of the total torsional moment \(\bar{M}_{t}=GJ\theta^{\prime}-EI_{\omega}\theta^{\prime\prime\prime}\));
* for the distortional problem, (i) continuity of \(\varphi\), (ii) continuity of \(\varphi^{\prime}\), (iii) equilibrium of forces dual of \(\delta\varphi\) (entailing continuity of \(\varphi^{\prime\prime}\)), and (iv) equilibrium of forces dual of \(\delta\varphi^{\prime}\) (entailing continuity of \(\varphi^{\prime\prime\prime}\)).  
By numerically solving two sets of linear algebraic problems, the arbitrary constants are evaluated and the solution (Equation (34)) is analytically obtained in step-wise form.