## 1 Introduction  
Modern transportation systems, such as high-speed train systems, demand for long-span bridges are increasing. The longer span length and faster operating speed make the system very sensitive to deflections of super and sub-structures [1]. Accordingly, accurate prediction of the time-dependent deflections of these structures (especially slender bridges) has become imperative. Long-span prestressed concrete (PSC) bridges often suffer excessive deflections during their service lives. This excessive deflection is mainly owing to the creep and shrinkage of the concrete. Current design codes provide practical methods for predicting the long-term deflections of concrete bridges, but they often underestimate the results [2; 3; 4; 5]. One cause of such underestimation is that the codes overlook the effects of the nonuniform shrinkage caused by an uneven moisture distribution along the section of the girder. The uniform shrinkage only induces an axial shortening, while the non-uniform shrinkage can induce a curvature. Thus, the nonuniform shrinkage strain can lead to significant additional deflections in PSC bridges [2; 3; 4], especially in large box girder bridges. However, in the design of a PSC bridge, the effects of nonuniform shrinkage are usually ignored. A number of studies have been attempted to investigate this nonuniform shrinkage with varying degrees of accuracy and complexity. [PERSON] et al. [2] assessed the effects of nonuniform shrinkage independently from other creep-related analyses and verified that the thickness of a girder has a significant effect on the long-term deflection. [PERSON] et al. [3] performed a parallel multi-scale thermo-hygral analysis that coupled moisture migration with the structural response to stress and strain. In their approach, the actual deflections of bridges were found to be between two analysis results performed while assuming extreme wet and dry exposure conditions. In addition, they pointed out that the distribution of the relative humidity (RH) within the cross-section affects the creep deformation until it reaches an equilibrium state with the external RH. The rate of drying might vary according to the size of the structure even if the geometric shape is the same, resulting in different rates of creep. [PERSON] et al. [4] proposed an algorithm for a more realistic creep analysis with a rate type creep formulation based on a Kelvin chain model to capture the effects of humidity and temperature variations. They also considered changes in the degree of hydration according to humidity and temperature, bond-slip behaviors, and cyclic creep. Furthermore, [PERSON] et al. [4] discussed that the estimation of long-term prestress loss is generally insufficient for creep-sensitive structures. As the prestress loss due to steel relaxation may significantly depend on the strain variation in concrete, it needs to be calculated as a part of structural analysis together with creep and shrinkage. They noted that current design practices significantly underestimate the effects of these factors on multi-decade creep in large-span PSC bridges. In PSC bridges with partial prestressing, where tensile stress and cracking are allowed under service load, flexural cracking may be another reason for the errors in calculations of the deflection (in addition to the nonuniform shrinkage). Cracks cause stiffness degradations and promote deflection at the mid-span. The creep and shrinkage of the concrete will further increase the deflection and promote the propagation of cracks and the emergence of additional cracks [5]. Current design codes also provide a way to consider the effects of cracking, so called the effective moment of inertia approach [6; 7]. This approach uses a reduced moment of inertia to calculate the deflection by ignoring regions subjected to higher stress than the modulus of rupture of the concrete after onset of cracking. That is, this approach considers cracking as a one-time event. However, actual cracking is a more rigorous and progressive phenomenon.