## 2 Establishment of the Deformation Reconstruction Model for Beam Structure
### Inverse Finite Element Method for Beam Element  
Based on the assumption of [PERSON] beam theory, a three-dimensional Cartesian coordinate system for a beam structure of constant cross-section is established, as shown in Figure 2. Here, \(L\) represents the length of the beam element, the central axis of the beam overlaps with the x-axis of the Cartesian coordinate system, and the y-axis overlaps with the main inertia axis of the section as well. The deformation field at a point B(x,y,z) on the surface can be expressed as  
\[\left\{\begin{matrix}u_{x}(x,\ y,z)=u(x)+z\theta_{y}(x)-y\theta_{z}(x)\\ u_{y}(x,y,z)=v(x)-z\theta_{x}(x)\\ u_{z}(x,y,z)=w(x)+y\theta_{x}(x)\end{matrix}\right., \tag{1}\]  
where \(\left\{u_{x}(x,y,z),\ u_{y}(x,y,z),\ u_{z}(x,y,z)\right\}\) are the deformation displacements along the X, Y, and Z axes at point B, respectively, and \(\mathbf{u}(x,y,z)=\left\{u(x),v(x),w(x),\theta_{x}(x),\theta_{y}(x),\theta_{z}(x )\right\}\) represents the degree of freedom of the node, as shown in Figure 2.  
In the finite element framework, arbitrary nodal degrees of freedom \(\mathbf{u}(x,y,z)\) can be obtained by the dot product of the shape function \(\mathbf{N}(x)\) and discrete nodal degrees of freedom \(\mathbf{u}^{e}\):  
\[\mathbf{u}(\mathbf{x})=\mathbf{N}(\mathbf{x})\mathbf{u}^{e}, \tag{2}\]  
Then, the section strain vector \(e(\mathbf{u}^{e})\) can be calculated based on Equation (2), given by  
\[e(\mathbf{u}^{e})=\mathbf{B}(\mathbf{x})\mathbf{u}^{e}, \tag{3}\]  
where \(\mathbf{B}(\mathbf{x})\) is a matrix containing the derivative of the shape function \(\mathbf{N}(\mathbf{x})\). However, the theoretical section strain \(e(\mathbf{u}^{e})\) cannot be obtained directly from the strain sensor due to the existence of measurement strain errors. Therefore, a least squares function, \(\varphi\), is used to establish the relationship between the analytic section strain \(e(\mathbf{u}^{e})\) and measured section strain vector \(e^{e}\):  
\[\mathbf{\varphi}(\mathbf{u})=\|e(\mathbf{u}^{e})-e^{e}\|^{2}, \tag{4}\]  
where \(e^{e}\) is the calculated section strain from the measured surface strain. Expanding the above least squares error function gives the following quadratic expansion:  
\[\mathbf{\varphi}(\mathbf{u})=\frac{1}{2}(\mathbf{u}^{e})^{T}\mathbf{k}^{e}\mathbf{u}^{e}-(\mathbf{u}^ {e})^{T}\mathbf{f}^{e}+\mathbf{c}^{e}, \tag{5}\]  
where \(\mathbf{c}^{e}\) is a constant, and \(\mathbf{k}^{e}\) and \(\mathbf{f}^{e}\) are defined as follows:  
\[\left\{\begin{matrix}\mathbf{k}^{e}_{k}=\frac{L}{\eta}\sum_{i=1}^{\mathbf{n}}\left[ \mathbf{B}^{T}_{k}(\mathbf{x}_{i})\mathbf{B}_{k}(\mathbf{x}_{i})\right]\\ \mathbf{f}^{e}_{k}=\frac{L}{\eta}\sum_{d=1}^{\mathbf{n}}\left[\mathbf{B}^{T}_{k}(\mathbf{x}_{i })\mathbf{e}^{e}_{k}(\mathbf{x}_{i})\right]\end{matrix}\right., \tag{6}\]  
Figure 2: Deformation field of a Timoshenko Round Beam.  
where \(L\) is the length of the beam element; \(n\) and \(x_{i}(0<x_{i}<L)\) are the number and axial coordinates of the estimated section strain position, respectively. When the error function \(\mathbf{\varphi}^{e}(\mathbf{u})\) is minimized with respect to \(\mathbf{u}^{e}\), the result is simplified as  
\[\mathbf{k}^{e}\mathbf{u}^{e}=\mathbf{f} \tag{7}\]