# A Minimal GBT Model for Distortional-Twist Elastic Analysis of Box-Girder Bridges
## 3 A Minimal GBT Model
### GBT Equations  
By using the torsional mode (Equations (13) and (15)) and the distortional mode (Equations (17) and (19)), the \(2\times 2\) matrices (Equation (8) appearing in the GBT equations (Equation (7)) are evaluated, for which the components are as follows:  
Figure 4: Distortional deflection: (**a**) translating joints, (**b**) rotating joints, and (**c**) superposition of displacements. Figure 5: Distortional mode: (**a**) tangential displacements, (**b**) normal displacements, and (**c**) warping. \[C^{e}_{\alpha\beta} =E\underset{T}{\oint}tW_{\alpha}W_{\beta}\,ds,\qquad C^{f}_{ \alpha\beta}=\underset{T}{\oint}DV_{\alpha}V_{\beta}\,ds \tag{21}\] \[D^{f}_{\alpha\beta} =v\underset{T}{\oint}V_{\alpha}V^{\prime\prime}_{\beta}\,ds, \qquad D^{t}_{\alpha\beta}=\frac{G}{3}\underset{T}{\oint}t^{3}V^{\prime}_{ \alpha}V^{\prime}_{\beta}\,ds\] \[D^{s}_{\alpha\beta} =G\underset{T}{\oint}t(W^{\prime}_{\alpha}+U_{\alpha})(W^{ \prime}_{\beta}+U_{\beta})\,ds\] \[B^{f}_{\alpha\beta} =\frac{E}{12}\underset{T}{\oint}t^{3}V^{\prime\prime}_{\alpha}V^{ \prime\prime}_{\beta}\,ds,\qquad B^{d}_{\alpha\beta}=E\underset{T}{\oint}tU_{ \alpha}U^{\prime}_{\beta}\,ds\] \[\alpha,\beta=t,d\]  
together with \(\mathbf{a}=(\theta,\varphi)^{T}\), \(\mathbf{p}=(c_{t},c_{d})^{T}\), in which \(c_{\alpha}\) is the couples per unit length. A careful check of these matrices reveals the following:  
* since \(U_{t}(s)\) and \(U_{d}(s)\) are step-wise constant, \(U^{\prime}_{t}=U^{\prime}_{d}=0\), and therefore \(\mathbf{B}^{d}=\mathbf{0}\) (i.e., no in-plane dilatation occurs);
* since \(V_{t}(s)\) is step-wise linear and \(V_{d}(s)\) quadratic, then \(B^{f}_{11}=B^{f}_{12}=B^{f}_{21}=0\), so that only \(B^{f}_{22}\
eq 0\) in matrix \(\mathbf{B}^{f}\);
* since \(W^{\prime}_{t}+U_{t}=Q_{t}\
eq 0\) (torsional shear different from zero) and \(W^{\prime}_{d}+U_{d}=Q_{d}=0\) (distortional shear zero), then \(D^{s}_{12}=D^{s}_{21}=D^{s}_{22}=0\), so that only \(D^{s}_{11}\
eq 0\) in matrix \(\mathbf{D}^{s}\); the implications on stresses of \(Q_{d}=0\) will be discussed ahead. Moreover, aimed at obtaining the simplest model, the Poisson ratio is taken as zero, so that \(\mathbf{D}^{f}=\mathbf{0}\). Concerning \(\mathbf{D}^{f}\), it accounts for linear distribution of the tangential stresses in the thickness. When TWBs are considered, these terms are usually negligible with respect to the Bredt stresses; however, in the case of reinforced concrete box-girder bridges, where thicknesses are not so small, they can give significant contributions, as it will be shown by numerical simulations. Therefore, matrix \(\mathbf{D}^{f}\) is retained in the analysis. As a result, the field GBT equations reduce to the following:  
\[\left( \begin{array}{cc}C_{11}&C_{12}\\ C_{21}&C_{22}\end{array} \right)\left( \begin{array}{cc}\theta^{\prime\prime \prime}\\ \varphi^{\prime\prime\prime}\end{array} \right)+\left( \begin{array}{cc}D_{11} &D_{12}\\ D_{21}&D_{22}\end{array} \right)\left( \begin{array}{cc}\theta^{\prime\prime }\\ \varphi^{\prime\prime}\end{array} \right)+\left( \begin{array}{cc}0&0\\ 0&B^{f}_{22}\end{array} \right)\left( \begin{array}{cc}\theta\\ \varphi\end{array} \right)=\left( \begin{array}{c}c_{t}\\ c_{d}\end{array} \right) \tag{22}\]  
where \(C_{ij}:=C^{e}_{ij}+C^{f}_{ij}\) and \(D_{ij}:=D^{s}_{ij}+D^{t}_{ij}\), together with the following boundary conditions:  
\[\left[\left( \begin{array}{cc}\delta\theta^{\prime}&\delta\varphi^{\prime} \end{array} \right)^{T}\left( \begin{array}{cc}C_{11}&C_{12}\\ C_{21}&C_{22}\end{array} \right)\left( \begin{array}{cc}\theta^{\prime\prime }\\ \varphi^{\prime\prime}\end{array} \right)\right]^{\ell}_{0}=\left( \begin{array}{ cc}0\\ 0\end{array} \right) \tag{23}\] \[\left[\left( \begin{array}{cc}\delta\theta&\delta\varphi^{\prime }\end{array} \right)^{T}\left(\left( \begin{array}{cc}C_{11}&C_{12}\\ C_{21}&C_{22}\end{array} \right)\left( \begin{array}{cc}\theta^{\prime \prime\prime}\\ \varphi^{\prime\prime\prime}\end{array} \right)-\left( \begin{array}{cc}D_{11} &D_{12}\\ D_{21}&D_{22}\end{array} \right)\left(