## 2 One-Dimensional Formulation  
### Displacement and Deformation Fields  
Consider a prismatic thin-walled structure depicted in Figure 1. The displacement of a point on the midline of the cross-section is defined in terms of the axial \(u\), tangential \(v\), and normal \(w\) components, which are defined to be positive, along the axes of the local coordinate system (\(n\), \(s\), \(z\)), adopted on each wall, respectively. The global coordinate system \((x,y,z)\) with its origin located on the centroid of the cross-section of the beam end is also shown.  
The displacement field \(\mathbf{U}=[U_{2}(n,s,z),\,U_{S}(n,s,z),\,U_{n}(n,s,z)]^{\mathrm{T}}\) considering both membrane and flexural behaviors of thin-walled structures is defined as  
\[U_{z}(n,s,z)=u(s,z)-n\frac{\partial w(s,z)}{\partial z},\,\,U_{S}(n,s,z)=v(s,z )-n\frac{\partial w(s,z)}{\partial s},\,\,U_{n}(n,s,z)=w(s,z) \tag{1}\]  
where the displacement variables \(u(s,z)\), \(v(s,z)\), and \(w(s,z)\) are separately approximated with a set of linearly independent basis functions, defined over the cross-section. This approximation separates the variable dependencies in \(z\) and \(s\) direction. The displacement field on the cross-section midline, \(\mathbf{u}=[u(s,z),\,v(s,z),\,w(s,z)]^{\mathrm{T}}\), is thus approximated as follows:  
\[u(s,z)=\mathbf{\Psi}_{1}\mathbf{x},\,\,v(s,z)=\mathbf{\Psi}_{2}\mathbf{x},\,\,w(s,z)= \mathbf{\psi}_{3}\mathbf{x} \tag{2}\]  
where \(\mathbf{\Psi}_{1}=[\varphi_{1}(s),\,\varphi_{2}(s),\,\ldots,\,\varphi_{N}(s)]\), \(\mathbf{\Psi}_{2}=[\psi_{1}(s),\,\psi_{2}(s),\,\ldots,\,\psi_{N}(s)]\), and \(\mathbf{\Psi}_{3}=[\omega_{1}(s),\,\omega_{2}(s),\,\ldots,\,\omega_{N}(s)]\) correspond to the sets of \(N\) basis functions, while for \(\mathbf{x}=[\chi_{1}(z),\,\chi_{2}(z),\,\ldots,\,\chi_{N}(z)]^{\mathrm{T}}\), the generalized displacement \(\chi_{i}(z)\) represents the axial variation of \(N_{1}\) out-of-plane and \(N_{2}\) in-plane deformation amplitudes; \(N=N_{1}+N_{2}\).  
The three-dimensional displacement \(\mathbf{U}\) can be expressed with a transformation matrix \(\mathbf{H}=[\mathbf{H}_{1}\mathrm{{}^{T}},\,\mathbf{H}_{2}\mathrm{{}^{T}}, \,\mathbf{H}_{3}\mathrm{{}^{T}}]^{\mathrm{T}}\) as  
\[U_{2}(n,s,z)=\mathbf{H}_{1}\mathbf{u},\,\,U_{S}(n,s,z)=\mathbf{H}_{2}\mathbf{ u},\,\,U_{n}(n,s,z)=\mathbf{H}_{3}\mathbf{u} \tag{3}\]  
where \(\mathbf{H}_{1}=[0,\,1,\,-n\partial/\partial z]\), \(\mathbf{H}_{2}=[1,\,0,\,-n\partial/\partial s]\) and \(\mathbf{H}_{3}=[0,\,0,\,1]\).  
Neglecting defect and material uncertainty, the deformation field \(\epsilon=[\epsilon_{zz}(n,\,s,\,z),\,\epsilon_{ss}(n,\,s,\,z),\,\gamma_{sz}(n,s,\,z)]^{\mathrm{T}}\) and the corresponding stress field \(\mathbf{\sigma}=[\sigma_{zz}(n,\,s,\,z),\,\sigma_{ss}(n,\,s,\,z),\,\tau_{sz}(n,\, s,\,z)]^{\mathrm{T}}\) are obtained under the small displacement hypothesis:  
\[\epsilon=\mathbf{C}\mathbf{U}=\left[\begin{array}{ccc}\frac{\partial}{ \partial z}&0&0\\ 0&\frac{\partial}{\partial s}&0\\ \frac{\partial}{\partial s}&\frac{\partial}{\partial z}&0\end{array}\right] \mathbf{U},\,\,\mathbf{\sigma}=\mathbf{E}\epsilon=\left[\begin{array}{ccc}\frac {E}{1-v^{2}}&\frac{Ev}{1-v^{2}}&0\\ \frac{Ev}{1-v^{2}}&\frac{E}{1-v^{2}}&0\\ 0&0&\frac{E}{2(1+v)}\end{array}\right]\epsilon \tag{4}\]  
where \(\mathbf{C}\) is the compatibility operator and \(\mathbf{E}\) is the constitutive matrix of the plate in plane stress condition; \(E\) and \(v\) are the Young's modulus and [PERSON]'s ratio of the material, respectively.  
Figure 1: Global \((x,y,z)\) and local \((s,n,z)\) coordinate systems of a thin-walled structure with a branched cross-section profile.