# A Minimal GBT Model for Distortional-Twist Elastic Analysis of Box-Girder Bridges
## 2 Background
### GBT Theory  
When the hypothesis of undeformability of the cross section is removed, the box-girder behaves as a plate assembly. According to the classic version of the GBT theory [23; 37] (in which only the so-called _conventional_ modes are taken into account), the displacement field is expressed as follows:  
\[ \begin{split} u(z,s)&=\sum_{k=1}^{K}U_{k}(s)\,a_{ k}(z)=\mathbf{U}^{T}\mathbf{a}\\ v(z,s)&=\sum_{k=1}^{K}V_{k}(s)\,a_{k}(z)=\mathbf{V}^{T }\mathbf{a}\\ w(z,s)&=\sum_{k=1}^{K}W_{k}(s)\,a^{\prime}{}_{k}( z)=\mathbf{W}^{T}\mathbf{a}^{\prime}\end{split}  \tag{3}\]  
where \(u,v,w\) are tangential (along the directrix), normal (along the thickness), and axial (along the axis) components, respectively. Here, \(U_{k}(s),V_{k}(s),W_{k}(s)\) are assumed to be modal shapes and \(a_{k}(z)\) are unknown amplitude functions. When the Kirchhoff hypothesis is invoked, the relevant strains \(\mathbf{\epsilon}=\left(\varepsilon_{s}^{m},\varepsilon_{z}^{m},\gamma_{zs}^{m}, \varepsilon_{s}^{f},\varepsilon_{s}^{f},\gamma_{zs}^{f}\right)^{T}\) are derived:  
\[\begin{split}\varepsilon_{s}^{m}&=\mathbf{U}^{\prime T} \mathbf{a},\qquad\varepsilon_{z}^{m}=\mathbf{W}^{T}\mathbf{a}^{\prime\prime}, \qquad\gamma_{zs}^{m}=\left(\mathbf{U}^{T}+\mathbf{W}^{\prime T}\right)\mathbf{a}^independent variable. From linear [PERSON]'s law for isotropic material, the planar stresses \(\sigma=\left(\sigma_{s}^{m},\sigma_{z}^{m},\tau_{zs}^{m},\sigma_{s}^{f},\sigma_{z} ^{f},\tau_{zs}^{f}\right)^{T}\) are found as follows:  
\[\sigma_{s}^{m}=E\mathbf{U}^{\prime T}\mathbf{a},\qquad\sigma_{z}^{m}=E\mathbf{W}^{T}\mathbf{a}^{ \prime\prime},\qquad\tau_{zs}^{m}=G\Big{(}\mathbf{U}^{T}+\mathbf{W}^{\prime T}\Big{)} \mathbf{a}^{\prime} \tag{5}\] \[\sigma_{s}^{f}=-y\frac{E}{1-\
u^{2}}\Big{(}\mathbf{V}^{\prime\prime T} \mathbf{a}+\
u\mathbf{V}^{T}\mathbf{a}^{\prime\prime}\Big{)},\qquad\sigma_{z}^{f}=-y\frac{ E}{1-\
u^{2}}\Big{(}\
u\mathbf{V}^{\prime\prime T}\mathbf{a}+\mathbf{V}^{T}\mathbf{a}^{\prime\prime} \Big{)}\] \[\tau_{zs}^{f}=-2 yGV^{\prime T}\mathbf{a}^{\prime}\]  
where \(\
u\) is the Poisson ratio. Finally, by making use of the Virtual Work Principle,  
\[\int\limits_{0}^{\ell}dz\int\limits_{-\frac{t(s)}{2}}^{\frac{t(s)}{2}}dy\oint \limits_{T}\sigma^{T}\delta\mathbf{e}ds=\int\limits_{0}^{\ell}dz\oint\limits_{T} \mathbf{f}^{T}\delta\mathbf{u}ds \tag{6}\]  
in which \(\mathbf{f}=\big{(}f_{s}(z,s),f_{y}(z,s),f_{z}(z,s)\big{)}^{T}\) are surface external forces and \(\mathbf{u}=(u(z,s),v(z,s),\)\(w(z,s))^{T}\), the following ordinary differential equations and relevant boundary conditions are obtained:  
\[ \begin{split}(\mathbf{C}^{e}+\mathbf{C}^{f})\mathbf{a}^{\prime\prime\prime \prime}+(\mathbf{D}^{f}+\mathbf{D}^{f/T}-\mathbf{D}^{s}-\mathbf{D}^{t})\mathbf{a}^{\prime\prime}+( \mathbf{B}^{f}+\mathbf{B}^{d})\mathbf{a}=\mathbf{p}\\ \Big{[}\delta\mathbf{a}^{\prime T}\Big{(}\Big{(}\mathbf{C}^{f}+\mathbf{C}^{e }\Big{)}\mathbf{a}^{\prime\prime}+\mathbf{D}^{f}\mathbf{a}\Big{)}\Big{]}_{0}^{\ell}=\mathbf{0 }\\ \Big{[}\delta\mathbf{a}^{\intercal}\Big{(}\Big{(}\mathbf{C}^{f}+\mathbf{C}^{ e}\Big{)}\mathbf{a}^{\prime\prime\prime}+\Big{(}\mathbf{D}^{f}-\mathbf{D}^{s}-\mathbf{D}^{t} \Big{)}\mathbf{a}^{\prime}\Big{)}\Big{]}_{0}\ell=\mathbf{0}\end{split}  \tag{7}\]  
where  
\[ \begin{split}\mathbf{C}^{e}=&\oint\limits_{T}Et\mathbf{W} \mathbf{W}^{T}ds,\qquad\mathbf{C}^{f}=\oint\limits_{T}D\mathbf{V}\mathbf{V}^{T}ds\\ \mathbf{D}^{f}=&\oint\limits_{T}\
u D\mathbf{V}^{\prime\prime \intercal}ds,\qquad\mathbf{D}^{t}=\oint\limits_{T}\frac{G\mathbf{t}^{3}}{3}\mathbf{V}^{ \prime}\mathbf{V}^{\prime T}ds,\\ \mathbf{D}^{s}=&\oint\limits_{T}Gt\Big{(}\mathbf{W}^{\prime T }+\mathbf{U}^{T}\Big{)}^{T}\Big{(}\mathbf{W}^{\prime T}+\mathbf{U}^{T}\Big{)}ds\\ \mathbf{B}^{f}=&\oint\limits_{T}D\mathbf{V}^{\prime\prime} \mathbf{V}^{\prime\prime T}ds,\qquad\mathbf{B}^{d}=\oint\limits_{T}Et\mathbf{U}\mathbf{U}^{ \prime\intercal}ds\end{split}