# Working State of ECC Link Slabs Used in Continuous Bridge Decks
## 4 Experimental Data Expansion Using the Response Simulating Interpolation Method  
The mid cross section of ECC link slab is the most disadvantageous position under the loading condition presented in Figure 1. Meanwhile, the sample points within the mid cross section are the most intensive. Hence it should be a good attempt to investigate the working states of ECC link slabs based on the GSED sum of mid cross sections. Importantly, these scattered measured strains in this area need to be expanded into spatially continuous strain data without loss of original information before the detection of structural working state. Space interpolation methods (SIMs) providing access to the data expansion from limited scattered sample points have been widely used in various subjects [37]. SIMs can be grouped into four categories: Global methods (trend surface analysis and regression models), local methods (natural neighbor and splines), geostatistical methods (simple kriging), and mixed methods (trend surface analysis combined with kriging) [38]. Despite the various assumptions and properties of these SIMs, nearly the general estimation formula is shared [37]:  
\[r(j)=\sum_{i=1}^{m}\omega_{i}(j)r_{i}, \tag{11}\]  
where \(r(j)\) is the simulated response value at the interpolated \(j\)th point, \(\omega_{i}\) refers to the weighting function assigned to the \(i\)th sample point at which the observed response value is \(r_{i}\), and \(m\) represents the total number of sampling points. Consequently it is a matter to determine the weighting function \(\omega_{i}\). Among the various SIMs, the response simulating interpolation method is introduced which makes full use of the important properties of numerical shape function (NSF) simulated by a physical model. According to the discretization concept and the properties of NSF, the weighting function can be derived by numerical simulation [39]. Actually, the numerical simulation generally used for estimating experimental responses can partially reflect the implicit information included in a specific physical model [40]. Hence the combination of numerical simulation and interpolation is of adequate accuracy. The data expanded by a reliable SIM is deemed to be helpful to the investigation on the working state of ECC link slab. The determination of NSF and the estimation of response field (strains) are implemented following the procedure [39]:  
1. Divide the area into a certain amount of suitable elements as shown in Figure 5. Although the shape function of each element is estimated, the simulated strain fields assembled with enough small elements can still reflect the strain characteristics of this area exactly with high order continuity. The reason is that the calculation of element stiffness matrices and the assembling of global stiffness equation are based on the virtual displacement principle and the force balance principle, respectively [41]. 2. Apply a unit strain at node 1 along the direction at right angle to this area, while the other nodes are fixed along the same direction to restrict rigid body displacement. Then, the simulated strain field solved by static finite element analysis represents the shape function \(N_{1}\), as shown in Figure 6. And it can be expressed by discrete vector N\({}_{1}\) = {\(N_{1}\) (1), \(N_{1}\) (2),..., \(N_{1}\) (\(k\)),..., \(N_{1}\) (\(n\))}, where \(N_{1}\) (\(k\)) is the function value at element node \(k\) and \(n\) is the total node number. 3. Other NSFs (N\({}_{i}\)) can be determined in the same way. Given the strain value at each measuring point and the corresponding NSF, the strain field of this area denoted by \(\varepsilon\) = {\(\varepsilon\) (1), \(\varepsilon\) (2),..., \(\varepsilon\) (\(n\))} can be constructed as following, \[\varepsilon=\sum_{i=1}^{m}\varepsilon_{i}\textbf{N}_{i},\] (12) where \(m\) is the number of measured points of strain.