## 2 A Higher Order Model
### Displacement Field  
The displacement of an arbitrary point on the thin-walled structure was expressed with three components \(U\), \(V\) and \(W\) in the local coordinate system. By considering the membrane and flexural behaviours, the 3D displacement field \(\mathbf{D}=[U(s,\,n,\,z),\,V(s,\,n,\,z),\,W(s,\,n,\,z)]^{\mathrm{T}}\) can be expressed by  
\[U(s,n,z)=u(s,z)-nw_{,z}(s,z),\,\,V(s,n,z)=v(s,z)-nw_{,s}(s,z),\,\,W(s,n,z)=w(s,z) \tag{1}\]  
where (i) the subscript comms indicate differentiation with respect to the following variable, and (ii) the variables, \(u\), \(v\) and \(w\), are the mid-plane displacement components along \(z\), \(s\) and \(n\), respectively. By separating the variable dependencies on \(s\) and \(z\), these components were obtained with the linear combination of deformation modes (including \(N_{1}\) out-of-plane and \(N_{2}\) in-plane ones) as  
\[u(s,z)=\sum_{k=1}^{N_{1}}\phi_{k}(s)\chi_{k}(z),\,\,v(s,z)=\sum_{h=1}^{N_{2}} \psi_{h}(s)\chi_{N_{1}+h}(z),\,\,w(s,z)=\sum_{h=1}^{N_{2}}\omega_{h}(s)\chi_{ N_{1}+h}(z) \tag{2}\]  
where (i) \(k\) and \(h\) are the numbers of out-of-plane and in-plane deformation modes, respectively; (ii) \(\varphi\), \(\psi\), \(\omega\) are the deformation components of deformation modes along \(z\), \(s\) and \(n\), respectively; and (iii) \(\chi\) is  
Figure 1: Global (\(x\), \(y\), \(z\)) and local (\(s\), \(n\), \(z\)) coordinate systems of the illustrative thin-walled structure.  
the amplitude function varying along \(z\). Notice that the shape functions are initially defined based on PDM and finally replaced by identified deformation modes to describe the displacement field.  
Substitute Equation (2) into Equation (1), and the 3D displacement field can be rewritten in a 1D form by involving a transformation matrix \(\mathbf{H}\) as  
\[\mathbf{D}(s,n,z)=\mathbf{H}\mathbf{x},\ \mathbf{H}=\left[\begin{array}{cc} \boldsymbol{\varphi}&-n\boldsymbol{\omega}\frac{\partial}{\partial z}\\ 0&\boldsymbol{\psi}-n\boldsymbol{\omega}_{s}\\ 0&\boldsymbol{\omega}\end{array}\right] \tag{3}\]  
where (i) the amplitude function vector \(\mathbf{x}\) is spanned with amplitude functions \(\chi\), and (ii) \(\boldsymbol{\varphi}\), \(\boldsymbol{\psi}\) and \(\boldsymbol{\omega}\) are the shape function vectors consisting of \(\varphi(s)\), \(\psi(s)\) and \(\omega(s)\) in row matrices, respectively.  
The strain and stress fields were obtained under the small displacement hypothesis. By applying the Kirchhoff formulation, the strain \(\varepsilon=[\varepsilon_{zz}(s,n,z)\), \(\varepsilon_{ss}(s,n,z)\), \(\gamma_{sz}(s,n,z)]^{\mathrm{T}}\) and stress fields \(\boldsymbol{\sigma}=[\sigma_{zz}(s,\)\(n,z)\), \(\sigma_{ss}(s,n,z)\), \(\tau_{sz}(s,n,z)]^{\mathrm{T}}\) can be obtained as  
\[\varepsilon(s,n,z)=\mathbf{C}\mathbf{D},\ \mathbf{C}=\left[\begin{array}{ c c c}\frac{\partial}{\partial z}&0&0\\ 0&\frac{\partial}{\partial z}&0\\ \frac{\partial}{\partial s}&\frac{\partial}{\partial z}&0\end{array}\right] \tag{4}\]  
\[\boldsymbol{\sigma}(s,n,z)=\mathbf{E}\varepsilon,\ \mathbf{E}=\left[\begin{array}{ c c c}\frac{E}{1-\
u^{2}}&\frac{E\
u}{1-\
u^{2}}&0\\ \frac{E\
u}{1-\
u^{2}}&\frac{E}{1-\
u^{2}}&0\\ 0&0&\frac{E}{2(1+\
u)}\end{array}\right] \tag{5}\]  
where (i) \(\mathbf{C}\) and \(\mathbf{E}\) are the compatibility operator and the constitutive matrix, respectively, and (ii) \(E\) and \(\
u\) are the material Young's modulus and [PERSON]'s ratio, respectively.