## 2 Introduction of Analysis Methods  
### Method of Numerical Shape Function  
Compared with simulative results, the limited data of structural responses collected from experiments or monitored from real bridges are the most direct and effective reflection of their mechanical behavioral characteristics. To some extent, these response data can reflect the working behavior or evaluate the real working state of the structure under loading. However, it still cannot provide a full expression of structural mechanical characteristics. In order to obtain detailed information about structural response based on experiments, the method of numerical shape function (NSF) was proposed [21], which can accurately interpolate/expand experimental data with clear physical significance. NSF is a new and effective interpolation method for interpolating experimental data field directly through conventional shape function [22; 23] and finite element simulation. With reference to the interpolation method of thin plate splines (TPS) [21], this method applies the finite element (FE) simulation of a specific thin-plate model to obtain the numerical shape functions of sampled points. Then, the data at non-sampled points are acquired with the interpolation calculation of numerical shape functions and the sampled data. To introduce this method, the deflection field of one cross-section in arch-A (introduced in Section 3.1) is used here as an example, and the parameters of this extracted section are also seen in Section 3.1. As shown in Figure 1a, the cross-section was constructed and meshed with element Shell 181 by the software ANSYS [24]. The deflections of four quarter points were measured as sampled points. In the regular coordinate system (\(x\), \(y\)), the element displacement field \(u(x,y)\) is expressed as a node displacement by Equation (1):  
\[u=\sum_{i=1}^{4}u_{i}N_{i}(x,y) \tag{1}\]  
where \(u_{i}\) is an element displacement at \(i\)-th node and \(N_{i}\) is the corresponding shape function. Since the FE model is divided into a certain number of elements, the relatively accurate node displacement can be calculated by establishing the overall equilibrium equation based on the principle of virtual displacement. Pursuant to the static calculation, the z-directional displacement \(\mathbf{N}_{1}\) is provided in Figure 1b by applying a unit displacement at point 1 along the \(z\)-axis, while the other three points are fixed along the \(z\)-axis to restrict rigid displacements. Similar to the calculation of \(\mathbf{N}_{1}\), other numerical shape functions \(\mathbf{N}_{i}\) can also be obtained. As an example, \(\mathbf{N}_{3}\) was calculated and depicted in Figure 1c. Due to the central symmetry of the section shape, \(\mathbf{N}_{1}\) is in rotational symmetry with \(\mathbf{N}_{3}\). Without considering large deformation or elastoplasticity, the interpolation field constructed by [PERSON]'s theorem is independent of loading paths, and linear superposition can be carried out for  
Figure 1: Finite element (FE) model and numerical shape functions: (**a**) The FE model of the cross-section; (**b**) numerical shape function \(\mathbf{N}_{1}\); (**c**) numerical shape function \(\mathbf{N}_{3}\). the simulative results with explicit physical meanings. Therefore, according to the deflection of four sampled points, the deflection field \(\mathbf{D}\) of the whole section is obtained by Equation (2):  
\[\mathbf{D}=\sum_{i=1}^{m}u_{i}\mathbf{N}_{i},\ \mathbf{N}_{i}=\left[N_{i}(x_{1}), \ N_{i}(x_{2})\cdots N_{i}(x_{j})\cdots N_{i}(x_{n})\right] \tag{2}\]  
where \(\mathbf{N}_{i}\) is the shape function of \(i\)-th point, \(N_{i}(x_{j})\) is the function value at element node \(x_{j}\), \(n\) is the total number of element nodes, and \(m=4\) is the total number of sampled points. From Figure 1, it is seen that combining the shape function (data interpolation) and numerical simulation (NSF) method can provide an estimation of experimental data at unsampled locations, especially in regions that are difficult and expensive to access.