# A Minimal GBT Model for Distortional-Twist Elastic Analysis of Box-Girder Bridges
## 2 Background
### Beam Theory  
When a thin-walled box-girder is subject to nonuniform torsion, one can apply the extended Vlasov theory [36] for closed TWB. The theory assumes (i) that the cross section is undeformable in its own plane but warps out-of-plane and (ii) that the shear strains \(\gamma_{z\mathrm{s}}\) on the middle surface of the beam do not vanish, as for open TWB, but they satisfy the condition \(\gamma_{z\mathrm{s}}(z,s)t(s)=:\phi(z)\), i.e., the flow of the strains across the thickness, at a given \(z\), is constant along \(\Gamma\), as suggested by the Bredt theory. The combination of the two geometric constraints yields the warping \(w(z,s)=-\theta^{\prime}(z)\omega(s)\), with \(\theta^{\prime}(z)\) being the generally nonuniform torsional curvature and \(\omega(s)\) being the warping function:  
\[\omega(s)=2\left(\Omega(s)-\Omega_{0}\frac{\int_{0}^{s}\frac{ds}{t(s)}}{ \oint\frac{ds}{t(s)}}\right) \tag{1}\]  
where \(\Omega_{0}\) is the area enclosed by \(\Gamma\) and \(\Omega(s)\) is the sector area function. Equation (1) generalizes the more familiar \(\omega(s)=2\Omega(s)\), which holds for open TWB.  
When the equilibrium is expressed in terms of the twist angle \(\theta(z)\), the following field equation and boundary conditions are found:  
\[\begin{split} EL_{\omega}\theta^{\prime\prime\prime\prime}-GJ \theta^{\prime\prime}&=c_{z}(z)\\ \big{[}\big{(}GJ\theta^{\prime}-EI_{\omega}\theta^{\prime\prime \prime}\big{)}\delta\theta\big{]}_{0}^{\ell}&=0\\ \big{[}EI_{\omega}\theta^{\prime\prime}\delta\theta^{\prime} \big{]}_{0}^{\ell}&=0\end{split} \tag{2}\]  
where \(E_{z},G\) are elastic moduli; \(I_{\omega}:=\oint_{\Gamma}\omega^{2}(s)t(s)ds\) is the warping inertia moment, \(J:=\frac{4\Omega^{2}}{\oint\frac{\partial\omega}{(s)}}\) is the De Saint Venant torsion inertia moment; \(c_{z}(z)\) are distributed external torsional couples, \(\delta\) indicates a virtual displacement, and a prime denotes \(z\)-differentiation. The second equation in Equation (2) shows that the external torsional moment \(\bar{M}_{t}\) is equilibrated by two internal forces: the De Saint Venant torsional moment \(M_{t}=GJ\theta^{\prime}\) and the complementary torsional moment \(M_{t}^{*}=-EI_{\omega}\theta^{\prime\prime\prime}\), i.e., \(\bar{M}_{t}=M_{t}+M_{t}^{*}\).  
Once Equation (2) is solved and \(\theta(z)\) is determined, the normal stresses are evaluated via \(\sigma_{z}(z,s)=-\frac{B(z)}{I_{\omega}}\omega(s)\), where \(B(z)=EI_{\omega}\theta^{\prime\prime}\) is the bimoment. Evaluation of the tangential stresses is more complicated. Indeed, they consist of an _active_ and a _reactive_ component, \(\tau_{zs}=\tau_{zs}^{a}+\tau_{zs}^{r}\), with the former being associated with the uniform flow of the shear strains and the latter being associated with the geometric constraint that enforces the uniformity. This circumstance differs from that occurring in open TWB, where tangential stresses are only of reactive nature. The active stress components are evaluated by the [PERSON] formula, \(\tau_{zs}^{a}=\frac{M_{t}}{2\Omega_{0}t}\). The reactive stress components are determined by integrating the indefinite equilibrium equation averaged on the thickness, \((\tau_{zs}^{r}t)_{z}+\left(\sigma_{z}t\right)_{z}=0\), where a comma denotes partial differentiation with respect the following variable. This supplies \(\tau_{zs}^{r}(z,s)t(s)=-\frac{M_{t}^{*}(z)S_{\omega}(s)}{L_{\omega}}+q_{0}(z)\), where \(S_{\omega}(s):=\oint_{\Gamma}\omega(s)t(s)ds\) and \(q_{0}(z)\) is an arbitrary function; the latter is determined by enforcing the static equivalence between \(\tau_{zs}^{r}\) and \(M_{t}^{*}\). Details on these calculations are given in Appendix A for rectangular box-girders.