## 1 Introduction
### Application of the Displacement Method to the Analysis of Constrained Torsion in Continuous Beams  
In solving the internal forces of a plane rigid frame by the displacement method, there are three general problems to be solved:  
1. Determine which displacements on the structure are to be used as the basic unknown quantities;
2. Work out the internal forces of a single-span statically indeterminate bar when various displacements and loads occur at the rod ends;
3. Calculate these displacements. The following is a discussion of the differences between the application of the displacement method in thin-walled rod structures and in general structures, starting with these three problems in turn. The application of the displacement method to restrain torsion in thin-walled structures is illustrated by the example of a planar rigid frame structure as shown in Figure 3a. Since there is no torsion in the rigid frame's bars when it is subjected to loads in its plane, the torsion characteristics of the thin-walled bars are not shown; however, when the plane rigid frame is subjected to out-of-plane loads, there is also the warping deformation \(Z_{4}\) of the node section in addition to the linear displacement \(Z_{1}\) perpendicular to the plane of the rigid frame, the angular displacement \(Z_{2}\) around the \(AB\) bar, and the angular displacement \(Z_{3}\) around the \(BC\) bar (since the warping of the section is proportional to the torsion angle rate \(\varphi^{\prime}\), it can be expressed using \(\varphi^{\prime}\)). The basic system of a planar rigid frame is shown in Figure 3b; with additional constraints at node B, the basic equation of the displacement method can be obtained as shown in system (6). \[ \begin{array}{l}r_{11}Z_{1}+r_{12}Z_{2}+r_{13}Z_{3}+r_{14}Z_{4}+R_{1\text{p} }=0\\ r_{21}Z_{1}+r_{22}Z_{2}+r_{23}Z_{3}+r_{24}Z_{4}+R_{2\text{p}}=0\\ r_{31}Z_{1}+r_{32}Z_{2}+r_{33}Z_{3}+r_{34}Z_{4}+R_{3\text{p}}=0\\ r_{41}Z_{1}+r_{42}Z_{2}+r_{43}Z_{3}+r_{44}Z_{4}+R_{4\text{p}}=0\end{array}  \tag{6}\]  
The above structure is analogous to an \(n\)-span continuous twin I-girder bridge as shown in Figures 4 and 5. It can be found that, as the original structure has two bearings at the middle supports, \(Z_{1}\), \(Z_{2}\), and \(Z_{3}\) are all null at each middle support, and only the warping deformation of the section \(Z_{4}\) is produced. When an additional restraint is placed at the middle support, the middle span is a bar fixed at both ends and the two side spans are bars fixed at one end and hinged at the other. Table 1 gives a sketch of the calculation of a single-span statically indeterminate bar for different loads and boundary conditions. Figure 3: Planar rigid frame: (**a**) force diagram (**b**) basic system. For the calculation of single-span statically indeterminate bars, the force method is used for the rigid frame in-plane problem; as for the thin-walled bars subjected to torsional loads, [PERSON]'s theory is applied [22]. The torsional differential equation for an open thin-walled bar is listed as follows:  
\[E_{1}L_{w}\varphi^{4}(x)-GK\varphi^{*}(x)=m_{T}(x) \tag{7}\]  
where \(m_{T}\) is the external load torque, and \(E_{1}l_{w}\) and \(GK\) represent the uniform torsional stiffness and the non-uniform torsional stiffness, respectively. By solving the differential equations according to the initial parameter method, the forces at the rod end of the single span bars can be obtained. That is, the coefficients and the free terms in the equations of the displacement method can be used to solve the basic equations of the displacement method. After that, according to the superposition principle, the distribution of the non-uniform torsional internal forces of the two-span continuous beam, shown in Figure 5, under the concentrated torque in a span, can be identified. Figure 6 is the schematic diagram of the distribution of internal forces.