# Random Field-Based Time-Dependent Reliability Analyses of a PSC Box-Grider Bridge
## 5 Time-Dependent Reliability Analysis of the PSC Box-Girder Bridge
### Statistical Characteristics of Random Parameters  
The probabilistic models of the structural parameters contain random variables and random fields in this time-dependent reliability analysis. The uncertainties of shrinkage and creep models controlling prestressing stress and ambient humidity are considered as random variables, which are shown in Table 2. The coefficients of \(\psi_{1}\) and \(\psi_{2}\) are uncertainty factors of the creep and shrinkage models, respectively. For the CEB-FIP90 model, they approximately follow the normal distribution with the mean value of 1, and the coefficients of variation (COVs) are 0.339 for creep and 0.451 for shrinkage, respectively [42].The density of concrete (DC) is regarded as a random variable, which obeys a normal distribution, with a mean value 25.5 kN/m\({}^{3}\) and a COV of 0.046 [43]. The control prestressing stress (\(\sigma_{\text{con}}\)) is a normally distributed variable with a mean value of 1395 MPa, a COV of 0.04, and a truncated upper bound at 1488 MPa [44]. According to the tests, the concrete strength (f\({}_{\text{cm28}}\)) followed a normal distribution with a mean value of 63.9 MPa and a COV of 0.089 [45]. The distribution of annual relative humidity (RH) at the bridge site was obtained from the statistical results of monitoring data provided by the China Meteorological Administration. The initial loading age (t\({}_{0}\)) was assumed to follow the uniform distribution with a mean value of 7 days and a COV of 0.19, which ranged from 5 to 9 days during the construction of this bridge. The elastic modulus, secondary dead load, and live load are simulated as 2D random fields, and the statistical properties are summarized in Table 3. Similar to the FE model of the structure, the accuracy of discreteness is higher as the mesh density increases in the random field. The difference lies in that the mesh density of FE model is determined by the stress gradient, while for the random field, it mainly depends on the correlation length [22]. When the correlation length is longer, the correlation between two points in the distribution space will be stronger, and fewer discrete elements will be required to achieve the same accuracy. Given that the determination of the correlation length requires more measured data, and the paper mainly discusses the reliability estimation method of a PSC box-girder bridge based on random field theory, the longitudinal and transverse correlation lengths are both assumed to be 40 m. To balance the computational accuracy and efficiency, the size of the random field element is generally 1/3 to 1/2 of the correlation length. Meanwhile, considering the convenience of grid discretization, the element size of the random field for the elastic modulus at the top plate of the box-girder is set as 7.3 m \(\times\) 20 m, and the transverse direction of the random field is symmetrically meshed along the central line of the bridge. The number of discretized elements is 32, as shown in Figure 12. The meshes of the random fields of the elastic modulus at the bottom plate or the webs are similar to those at the top plate, which are both discretized into 32 elements. The meshes of the random fields for the secondary dead load and live load are the same as those of the elastic modulus random field at the top plate, and the random fields of the structure are divided into 160 random variables. It should be noted that the grid number of the random fields should be consistent with the discrete random variables, whose properties are endued to the corresponding elements of the structural FE model.