## 3 Stress Measurements and Data Updating Based on Grey Theory
### Compensation for the Deviation in the Measured Stress Due to the Uneven Temperature  
To compensate for the deviation in the measured stress due to the uneven temperature, two temperature sensors were, respectively, arranged on the top and the bottom of each measurement cross-section of the fixed ends of cantilevers to measure the temperature discrepancy between the top and the bottom of the cross-section in each construction stage in real time. Then, a mathematical formulation to calculate the stress deviation due to the uneven temperature from the measured temperature discrepancy was established based on the FE analyses and the response surface (RS) method. More precisely, a 0-block of Huangsha Harbor Bridge is simulated numerically using 282,811 tetrahedral solid elements with 423,401 nodes on a commercial FE platform first (see Figure 4). The length of the selected 0-block is 10 m; its width is 12 m; its height is 3.5 m. Its detailed geometry is shown in Figures 1 and 2. The material is C50 pure concrete, whose mass density, thermal conductivity, elastic modulus, minor Poisson's ratio and secant coefficients of thermal expansion are 2500 kg/m\({}^{3}\), 1.28 W/(m.K), 3.45 \(\times\) 10\({}^{10}\) Pa, 0.2 and 1 \(\times\) 10\({}^{-5}\)/K, respectively. In calculating the temperature field, no boundary condition is applied; when calculating the thermal stress field, the bottom of the 0-block is fixed in all directions. Secondly, thermal stress field analysis is performed assuming the temperature changes vertically from 0 \({}^{\circ}\)C at the bottom of the box girder to \(-\)12.5 \({}^{\circ}\)C at the top of the box girder according to Ref. [17]. The calculated temperature field and the calculated longitudinal thermal stress field on the 0-block are shown in Figures 5 and 6, respectively. As can be seen, the temperature changes smoothly in the vertical direction on the concrete structure in Figure 5. According to Figure 6, the maximum longitudinal thermal stress (3.43 MPa) is found at the bottom of the box girder, which is greater than the tensile strength of C50 concrete (2.65 MPa). This indicates that the stress induced by the uneven temperature is significant for Huangsha Harbor Bridge under construction. Multiple thermal stress field analyses are undertaken based on one-factor design and the temperature variation range stipulated in Ref. [17], and the stress at the position of the realistic upper measurement point is extracted for each case, which is listed in Table 1. Then, using data listed in Table 1, a linear regression model (Equation (1)) describing the relationship between the temperature discrepancy (in \({}^{\circ}\)C) and the induced stress at upper measurement point (in Pa) is established for use according to the RS method. The significant test results for the established regression model are shown in Table 2. Table 2 has three important results to show that Equation (1) is valid: (1) F = 1.780\(\times\) 10\({}^{8}\) > F\({}_{0.05}\), \(p\) < 0.0001. This shows that the regression model is significant, and the lack of fit term is insignificant. (2) R\({}^{2}\) = 1. This proves that the model is accurate, and the experimental error is small. (3) Adequate precision equals 23,532.462. This provesthat the signal is sufficient and the fitting cure is valid over the entire design space. Further, the temperature discrepancy against the calculated stress curve is plotted in Figure 7 to further show the rationality of Equation (1). From Figure 7, a linear relationship exists between the temperature discrepancy and the stress. \[\text{Stress}=+1.54620-4174.98819\times\text{Temperature discrepancy} \tag{1}\]  
Finally, by feeding the temperature discrepancy measured between the top and the bottom of the measurement cross-section in each construction stage in real time to Equation (1), the stress due to the uneven temperature on the structure was calculated for each construction stage and used to compensate for the deviation in all measured data.