## 4 Interpolation of Experimental Data by NSF Method
### Interpolation on Strains and Stress  
Before modeling the structural stressing state, the limited strains were interpolated to obtain the strain fields of each cross-section, as well as the stress fields based on the constitutive relation. Here, the software ANSYS was used to construct FE models of cross-sections, and shell element Shell 181 was chosen as the element type in consideration of its high precision. After dividing mesh, the strains of element nodes were interpolated based on the numerical shape functions \(\mathbf{N}_{1}\)\(\sim\)\(\mathbf{N}_{4}\) with  
Figure 4: The error results of strain data in arch-A: (**a**) Contrast curves on cross-section 4; (**b**) the validation errors of all sections. four measured strains \(\varepsilon_{1}\)-\(\varepsilon_{4}\). Under a certain load, the interpolation strain filed \(\mathbf{D}\) of one section was calculated by Equation (6):  
\[\mathbf{D}=\sum_{i=1}^{4}\varepsilon_{i}\mathbf{N}_{i} \tag{6}\]  
As shown in Figure 5, the interpolation strain field of section 11 in arch-A under 130 kN is listed as an example. The four measured strains were \(\varepsilon_{1}=-980\)\(\mu\)\(\epsilon\), \(\varepsilon_{2}=-765\)\(\mu\)\(\varepsilon\), \(\varepsilon_{3}=13\)\(\mu\)\(\epsilon\), and \(\varepsilon_{4}=-1199\)\(\mu\)\(\epsilon\), respectively. According to Equation (6), the strain field \(\mathbf{D}\) can be obtained. Based on strain fields interpolated by the NSF method, the corresponding stress fields could be also achieved through the constitutive relation of materials. The stress field of section 11 in arch A under 130 kN is also shown in Figure 5. When considering in the construction of constitutive relations, Equation (7), which reflects the uniaxial stress-strain relation based on numerous results [34; 35], was adopted for the constitutive relationship model of core concrete in compression:  
\[y=\left\{ \begin{array}{ll}2x-x^{2},&(x\leq 1)\\ \frac{x}{\beta_{0}(x-1)^{2}},&(x>1)\end{array} \right. \tag{7}\]  
where \(x=\varepsilon/\varepsilon_{0}\); \(y=\sigma/\sigma_{0}\); \(\sigma_{0}=f_{c}^{\ast}\); \(\varepsilon_{0}=\varepsilon^{\prime}+800\xi^{2}\times 10^{-6}\); \(\xi=f_{y}A_{s}/f_{dc}A_{c}\); \(\varepsilon_{c}^{\prime}=(1300+12.5f_{c}^{\prime})\times 10^{-6}\); \(\beta_{0}=\left(2.36\times 10^{-5}\right)^{\left[0.25+(\xi-0.5)^{7}\right]}\times f _{c}^{\prime 0.5}\times 0.5\geq 0.12\); \(f_{c}^{\prime}\) is the cylinder concrete compressive strength; \(f_{y}\) and \(f_{ck}\) are the yield strength of steel and the axial compressive strength of concrete; \(A_{s}\) and \(A_{c}\) are the section areas of steel and concrete;\(f_{cu}\) is the axial compressive strength of normative cube concrete; and \(f_{ck}=0.67\)\(\varepsilon_{cu}\); \(\varepsilon_{c}^{\prime}\) is the peak strain of concrete. Equation (8) shows the tensile constitutive relation of concrete [36]:  
\[y=\left\{ \begin{array}{ll}1.2x-0.2\xi^{6},&x<1\\ \frac{x}{\alpha_{t}(x-1)^{1.7}+x},&x>1\end{array} \right. \tag{8}\]  
where \(y=\sigma/f_{t}\); \(x=\varepsilon/\varepsilon_{t}\), \(\alpha_{t}=0.312f_{t}^{2}\), \(\sigma\) is the tensile stress of concrete, \(f_{t}\) is the axial tensile strength of concrete and \(f_{t}=0.26f_{cu}^{2/3}\)\(\varepsilon\) is the strain of concrete, and \(\varepsilon_{t}\) is the peak strain of concrete and \(\varepsilon_{t}=31.4f_{cu}^{0.36}\times 10^{-6}\). A five-stage stress-strain curve for steel tube [2] was adopted in calculation, as shown in Figure 6. At point \(a\), there was  
\[\varepsilon_{\mathrm{e}}=f_{\mathrm{p}}/E_{s} \tag{9}\]  
where \(f_{\mathrm{p}}\) is the proportional limit of steel, \(f_{\mathrm{p}}=0.8\,f_{y};f_{y}\) is the yield strength of steel, and \(E_{\mathrm{s}}\) is the elastic modulus of steel. At point \(b\), \(\varepsilon_{\mathrm{e1}}=1.5\varepsilon_{\mathrm{e}}\). At point \(c\), \(\varepsilon_{\mathrm{e2}}=10\varepsilon_{\mathrm{e1}}\). At point \(d\), \(\varepsilon_{\mathrm{e3}}=100\varepsilon_{\mathrm{e2}}\).