## 1 Introduction
### Torsional Effect of Continuous Twin I-Girder Bridge  
#### 2.1.1 Section Conversion for Torsion Analysis of Composite Beams  
Several basic assumptions are first made for the analysis of constrained torsion in open composite beams. These assumptions are the following:  
1. the section profile is not deformed in its plane under small deformations;
2. the shear strain on the middle face of the bar is zero;
3. the slip effect between the steel beam and the concrete is ignored (as the slip effect has a great impact on the structural stiffness [18; 19; 20]);
4. both steel and concrete are ideal linear elastic materials. Starting from the basic assumptions, the deformation and the stress state of an open thin-walled bar are discussed. The warping displacement at any point on the cross-section can be defined as: \(u(x,s)=-\varphi^{\prime}(x)\omega(s)\), where \(\varphi^{\prime}\) represents the torsion angle rate, \(\omega\) is the principal sectoral coordinate of the cross-section and \(s\) is the cross-sectional curve coordinate. Referring to the relationship between the warping displacement and the longitudinal normal strain of the micro-element and Hooke's law, the longitudinal warping normal  
Figure 1: Twin I-girder bridge (in mm): (**a**) Elevation view (first span, two spans in total); (**b**) Bird’s-eye view (first span, two spans in total); (**c**) Plane view (Section A-A); (**d**) Plane view (Section B-B). stress can be found as follows \(\sigma_{w}=-E_{1}\varphi^{\prime\prime}(x)\omega(s)\) where \(E_{1}\) is called the commuted modulus of elasticity of the thin-walled rod and is obtained from the first basic assumption and [PERSON]'s law \(E_{1}=\frac{E}{1-\mu^{2}}\), where \(E\) is [PERSON]'s modulus of the material. As \(\mu\) is generally very small, we can consider that \(E_{1}=E\). Similar to the generalized forces in primary beam theory \(M_{x}=\int\limits_{F}\sigma ydF\), the generalized force-warping bimoment is defined in constrained torsion \(B=\int\limits_{F}\sigma_{w}\cdot\omega dF\). For composite beams, one can write:  
\[B=\int_{A}\sigma_{\omega}\omega dA=\int_{A_{s}}\sigma_{s\omega}\omega dA_{s}+ \int_{A_{c}}\sigma_{c\omega}\omega dA_{c} \tag{1}\]  
where \(\sigma_{s\omega}\) and \(\sigma_{c\omega}\) represent, respectively, the warping normal stress of the steel girder and concrete deck, and \(A_{s}\) and \(A_{c}\) represent, respectively, the cross-sectional areas of the steel girder and concrete deck. Substituting the warping normal stress equation \(\sigma_{w}=-E_{1}\varphi^{\prime\prime}(x)\omega(s)\) to Equation (1), we get:  
\[B=-E_{s1}\varphi^{\prime\prime}(x)\int_{A_{s}}\omega^{2}dA_{s}-E_{c1}\varphi^{ \prime\prime}(x)\int_{A_{c}}\omega^{2}dA_{c} \tag{2}\]  
Equation (2) can be written as follows:  
\[B=-E_{s1}\varphi^{\prime\prime}(x)\biggl{(}\int_{A_{s}}\omega^{2}dA_{s}+\frac {1}{\alpha_{E}}\int_{A_{c}}\omega^{2}dA_{c}\biggr{)} \tag{3}\]  
Therefore, the principal sectoral moment of inertia of the steel-concrete section is:  
\[I_{\omega,int}=\int_{A_{s}}\omega^{2}dA_{s}+\frac{1}{\alpha_{E}}\int_{A_{c}} \omega^{2}dA_{c} \tag{4}\]  
where \(\alpha_{E}\) is the modulus of elasticity ratio \(\alpha_{E}=\frac{E_{s1}}{E_{c1}}\). However, Equation (4) can be written in another form as follows:  
\[ \begin{array}{ll}I_{\omega,int}&=\int_{A_{s}}\omega^{2}dA_{s}+\frac{1}{ \alpha_{E}}\int_{A_{c}}\omega^{2}dA_{c}\\ &=t_{s}\int_{A_{s}}\omega^{2}ds+\frac{t_{c}}{\alpha_{E}}\int_{A_{c}}\omega^{2} ds\\ &=t_{s}\int_{A_{s}}\omega^{2}ds+t_{c}^{\prime}\int_{A_{c}}\omega^{2}ds\end{array}  \tag{5}\]  
where \(t_{s}\) and \(t_{c}\) represent, respectively, the actual thickness of the steel girder and concrete deck, and \(t_{c}\)' is the equivalent thickness of the concrete deck. From the above equation, the section should be converted by fixing the width of the concrete deck slab and changing only its thickness; thus, the sectoral coordinates of the section will remain constant even after the conversion.