## 2 One-Dimensional Formulation
### Basis Deformation Modes  
In order to implement the interpolation process, nodal displacements should be properly defined first. Here four degrees of freedom of each node are discussed, including three translations and one rotation about the longitudinal axis. Consequently, the cross-section discretization led to a total of 4 \(\times\) (\(m_{n}+m_{i}\)) potential deformation patterns. In theory, these deformation patterns could be superposed with these nodal displacements. The problem is that the linear independency and structural meaning could not be guaranteed before a guideline was appropriately proposed. To settle the matter, a measure in GBT [28] was adopted here, which considered individually imposing a unit displacement on one of the four degrees of freedom of each node but with zero displacement on other nodes. The derived 4 \(\times\) (\(m_{n}+m_{i}\)) deformation patterns might have been of no definite physical significance, but the whole set of these patterns, which were linearly independent, could express any one of the 4 \(\times\) (\(m_{n}+m_{i}\)) potential cross-section deformation patterns with a linear superposition. In this sense, one deformation pattern resulted from the imposing of a unit displacement could be designated as one basis of the deformation mode.  
The basis function adopted in Equation (2) could be viewed as the mathematical description of a basis deformation mode. The function was to be approximated with interpolation polynomials subsequently. In consideration of their partial differential orders in Equation (11),  
Figure 2: Discretization of a prismatic thin-walled cross-section: (**a**) A rectangular hollow cross-section; (**b**) a branched hollow cross-section.  
different approximation functions were adopted. These included, a set of linear Lagrange functions for the axial (out-of-plane) and tangential (in-plane) displacements, and a set of cubic Hermite functions for the normal (in-plane) component. These basis functions vary along the midline coordinate \(s\). As mentioned above, \(\varphi_{k}(s)\) and \(\psi_{k}(s)\) were in the form of linear Lagrange functions, while \(\omega_{k}(s)\) was described with cubic Hermite functions.  
It should be pointed out that the different measures were separately taken for natural and intermediate nodes in the approximation, since they played different roles. For a natural node \(i\), a unit displacement was imposed on it, but null displacements were imposed on all the other natural nodes. The basis functions might have varied within each wall containing node \(i\) but were null in all the other walls. For an intermediate node \(j\), the basis function matched the imposition of a unit displacement on node \(j\) but matched the null displacements, on all other nodes. This means that the functions varied within the two segments adjacent to node \(j\) but were null elsewhere. According to the above guideline, \(N=4\times(m_{n}+m_{i})=32\) basis deformation modes were obtained for the cross-section in Figure 2a. As shown in Figure 3, among the thirty-two modes arrayed in a \(4\times 8\) matrix, each of the \(4\) rows were derived with an imposed unit displacement on eight nodes, while each of the eight columns represented the deformation modes that resulted from four unit displacements, separately imposed on one node. In the approximation of their deformation shapes, cubic Hermite functions were applied for the normal displacements, and linear functions for the tangential component (the 2 rd\(\sim\)4 th rows), while the normal displacements were interpolated with the linear Lagrange functions (the first row).