## 3 Identification of Deformation Modes
### Preliminary Deformation Modes  
PDM are defined on a meshed cross-section. A prismatic cross-section can be viewed as a \"network\" of a minimum number of straight walls. The nodes on these walls were divided into three types: The key, the natural and the artificial ones. The walls were bound to cross-link at intersection nodes, and the two intersection nodes closest to the wall ends were referred to as key nodes, while the other ones were classified into natural nodes. The nodes located at wall free ends were considered natural nodes. A selective number of artificial nodes may also be introduced to divide long walls into segments, when the wall is notably long. The nodes were numbered around clockwise in the sequence of key nodes, natural nodes and artificial nodes. The result is shown in Figure 2, where a set of \(N=12\) nodes, including \(n_{\text{d}}=4\) key nodes, \(n_{\text{f}}=4\) natural nodes and \(n_{\text{a}}=4\) artificial nodes, carried out the discretization of the cross-section. Figure 2a also shows the cross-section geometry, having a height of \(h=0.6\) m, a width of \(b=0.8\) m and a flange width of \(c=0.4\) m.  
The definition of PDM is related to the imposition of unit displacements on the discretization nodes. Here, four degrees of freedom at each node were considered, including three translations along three orthogonal axes and one rotation about the longitudinal axis. Thus, the discretization leads to a total of \(4\times N\) PDM. Note that different types of nodes were imposed different displacement boundaries in the procedure: For a key node \(i\), the unit displacement was imposed on it and null displacements on all other key nodes; for a natural node \(j\), the related deformation mode corresponded to the imposition of a unit displacement on it and null displacements on all key nodes and other natural nodes; and for an artificial node \(k\), the related deformation modes were obtained by imposing a unit displacement on node \(k\) and null displacements on all other nodes.  
In order to be included in the higher order model, PDM should be mathematically described with shape functions about \(s\). Here, the shape functions were defined through the interpolation of the imposed unit displacements. For convenience, these shape functions were decomposed into components being consistent with \(z,s\) and \(n\), respectively. The interpolation polynomials should satisfy the continuity condition along \(s\). To ensure the efficiency, different interpolation polynomials were adopted for different shape functions. For example, a set of linear Lagrange functions was used to describe the axial deformation \(\varphi_{k}\left(s\right)\) and tangential deformation \(\upsilon_{k}\left(s\right)\), while the normal deformation \(\omega_{k}\left(s\right)\) employs a set of cubic Hermite functions. For a better presentation, the resulted PDM of the illustrative cross-section were numbered from \(1\) to \(48\) and exhibited in a matrix form in Figure 3.  
These PDM can be divided into three families: (i) The Dominant family, including those stemming from the imposition of translational unit displacements on key nodes; (ii) Integral family, composing those involving natural nodes; and (iii) Selective family, with all other ones included. This is conducive  
Figure 2: (a) Geometry of the illustrative cross-section and (b) the cross-section discretization.  
to the separation of classic modes and higher order modes. For example, the classic modes can only be assembled with members of the Dominant family (probably revised with the members of Integral family), while the Selective family is dedicated to generate higher order modes.  
Figure 3: Preliminary deformation modes (PDM) of the illustrative cross-section stemming from the imposition of four-unit displacements on each of the 12 discretization nodes.