# Random Field-Based Time-Dependent Reliability Analyses of a PSC Box-Grider Bridge
## 5 Time-Dependent Reliability Analysis of the PSC Box-Girder Bridge
### Finite Element Model  
In order to maintain a balance between computational efficiency and accuracy, the composite degenerated shell element in the FE program _DIANA_[35] is adopted to simulate the mechanical behaviors of a thin-walled box-girder structure, as shown in Figure 12. This shell element is easily capable of modeling non-prestressed reinforcement and prestressed tendons. Non-prestressed reinforcement is modeled by a steel grid, which is embedded in the shell element, see Figure 12a. The reinforcement ratios in two perpendicular directions are represented by two equivalent thicknesses, \(t_{\text{eq}}\) (the area of cross section per unit length) of the grid, see Figure 12b. The parameter z identifies the location of each reinforcement grid, which determines the distance from this grid to the shell midsurface, as shown in Figure 12c. The area of the steel grid is divided into several segments, as shown in Figure 12d, which add stiffness to the shell element. Prestressed tendons can be established easily using an automatic tendon generation scheme, which defines tendons by several key position points in shell elements and shape functions (i.e., straight, quadratic, or cubic curve), as shown in Figure 12e. _DIANA_ automatically searches the shell elements for the intersections of the tendons with its boundaries. These intersections serve as location points, which divide the tendons into segments. For each tendon segment, the numerical integration is carried out separately. The embedded tendons add stiffness to the FE model and their strains are calculated from the displacement field of the surrounding shell elements based on deformation coordination. Figure 11: Profile of the Jinghang Canal Bridge: (**a**) Cantilever construction; (**b**) completed elevation; (**c**) segment division. Meanwhile, the shell element can be further integrated with some time-dependent constitutive models, to simulate the time-varying responses of PSC box-girder bridges, such as creep and shrinkage and steel relaxation. At present, dozens of creep and shrinkage models have been proposed, including the commonly used CEB-FIP model [36], the B3 model [37], the EC2 model [38], the ACI model [39], and the GL2000 model [40]. According to the research results by [PERSON] [41], it is concluded that the CEB-FIP90, B3, and GL2000 models can predict creep deformation well. In addition, the CEB-FIP90 and GL2000 models require fewer input parameters than other creep models, and in this study, the CEB-FIP 90 model is used to describe the concrete creep and shrinkage. As one important source of long-term prestress loss, the steel relaxation is modeled by the relaxation function and the generalized Maxwell model [12]. Moreover, the shell element can automatically consider the effect of shear lag on the long-term deflection of PSC box-girder bridges. The CEB-FIP 90 and steel relaxation models are integrated into the FE program _DIANA_, and users can adjust the parameters according to their needs. In this way, the influences of concrete creep and shrinkage, steel relaxation, and shear lag on the long-term deformation of PSC box-girder bridges can be computed automatically through FE analyses. In view of the symmetry of structure and load, only half of the bridge was modeled using _DIANA_ software to reduce the computation cost, as shown in Figure 13. The aforementioned composite degenerated shell element was used to simulate the thin-walled box-girder, and there were 1768 elements in total. The distributed reinforcements were simulated by using a series of steel grids, with two layers of steel mesh for each slab (i.e., top plates, bottom plates, and webs) and the prestressed tendons were established using the automatic tendon generation scheme [35]. Moreover, the unique phase analysis in the _DIANA_ program was used to consider the influence of the cantilever construction process on the bridge.