## 2 Problem Description  
This paper focuses on reconstructing the displacement of Euler-Bernoulli beams and Kirchhoff plates. Figure 1 illustrates the Cartesian coordinate system used for the beam analysis in this study. In Figure 1, the \(x\) -coordinate is taken along the length of the beam; the \(z\) -coordinate is taken along the thickness (the height) of the beam. The \(\beta\) denotes the rotating angle of the cross-section and \(w\) denotes the vertical displacement along the coordinate \(z\). The beam has a length of \(l\) and a thickness of \(t\). According to Euler-Bernoulli beam theory (EBT), the normal strain at point \(P\) on the cross-section of the beam can be expressed as follows:  
\[\varepsilon_{x}^{P}=\frac{-c}{\rho} \tag{1}\]  
in which \(c\) is the perpendicular distance from the point \(P\) to the neutral axis and \(\rho\) is the radius of curvature of the neutral surface of the beam. In the case of small displacements, the curvature of the neutral surface can be expressed as follows:  
\[\frac{1}{\rho}=\pm\frac{\mathbf{w}^{\prime}}{\left(1+\mathbf{w}^{\prime 2}\right)^{3/2}} \approx\pm\mathbf{w}^{\prime\prime}=\frac{d^{2}\mathbf{w}}{dx^{2}} \tag{2}\]  
By aligning Equation (2) with Equation (1), the relationship between strain and displacement at point \(P\) can be expressed as follows:  
\[\varepsilon_{x}^{P}=-c\frac{d^{2}\mathbf{w}(x_{P})}{dx^{2}} \tag{3}\]  
in which \(x_{P}\) is the coordinate of point \(P\) along the coordinate \(x\). In practice, strain measuring points can only be arranged at the surface of the beam. For beams with regular cross-sections and uniform thickness, the distance, \(c\), from the strain measuring points to the neutral axis is half of the thickness, \(t\). Therefore, based on Equation (3), the strain-displacement relation on the upper surface of the beam can be expressed as follows:  
\[\varepsilon_{x}(x)=-0.5t\frac{d^{2}\mathbf{w}(x)}{dx^{2}},\;x\in[0,l] \tag{4}\]  
Figure 2 illustrates the Cartesian coordinate system used for the plate analysis, in which \(t\) denotes the thickness and \((l,l^{\prime})\) denotes the lengths along the co  
Figure 1: Cartesian coordinate system used for the beam analysis: (**a**) displacement of the beam and (**b**) micro-segment of the beam. According to the classical Kirchhoff plate theory (CPT), the shear strains \(\gamma_{xx}\) and \(\gamma_{zy}\) of the plate remain zero, which can be written as follows:  
\[\left\{ \begin{array}{l}\gamma_{xx}=\frac{\partial u}{\partial z}+\frac{ \partial w}{\partial x}=0\\ \gamma_{zy}=\frac{\partial w}{\partial z}+\frac{\partial w}{\partial y}=0\end{array} \right. \tag{5}\]  
where \(\left(\mathbf{u},\mathbf{v}\right)\) are the displacement components along the \(\left(x,y\right)\) coordinate directions, respectively, and \(\mathbf{w}\) is the vertical displacement along coordinate \(z\). Integrating Equation (5) while taking into account that \(\left(\mathbf{u}\right)_{z=0}=0\) and \(\left(\mathbf{v}\right)_{z=0}=0\) in CPT, the geometric relationship between \(\mathbf{u},\mathbf{v}\) and \(\mathbf{w}\) can be obtained as follows:  
\[\left\{ \begin{array}{l}\mathbf{u}=-z\frac{\partial w}{\partial x}\\ \mathbf{v}=-z\frac{\partial w}{\partial y}\end{array} \right. \tag{6}\]  
Considering that the strain measuring points can only be located on the surface of the plate in practice, we take the first derivative on both sides of Equation (6). Therefore, the strain-displacement relation on the surface of the plate can be expressed as follows:  
\[\left\{ \begin{array}{l}\epsilon_{x}(x,y)=-0.5t\frac{\partial^{2}\mathbf{w}(x,y) }{\partial x^{2}}\\ \epsilon_{y}(x,y)=-0.5t\frac{\partial^{2}\mathbf{w}(x,y)}{\partial y^{2}}\end{array}  \right.,\;(x,y)\in\left[0,l\right]\times\;\left[0,l^{\prime}\right] \tag{7}\]  
in which \(\left(\epsilon_{x},\epsilon_{y}\right)\) are the strain components along the \(\left(x,y\right)\) coordinate directions, respectively.