## 3 Determination Method of a Reasonable Completion State
### Prediction Method of Concrete Shrinkage and Creep  
The concrete shrinkage and creep effects during construction and operation periods were predicted using the CEB-FIP 90 model [28] and age-adjusted effective modulus method [29]. The relationship between the initial instantaneous elastic deformation and final concrete creep deformation is approximately linear when the stress of concrete does not exceed 40% of the ultimate strength. The superposition principle can be used to calculate the strain caused by the stress applied step-by-step [30]. The relationship between the strain increment and stress increment during the time interval of (\(t_{i}\), \(t_{i-1}\)) caused by concrete shrinkage and creep is shown in Equation (1).  
\[\begin{array}{ll}\Delta\varepsilon_{CS}(t_{i},t_{i-1})=&\frac{\Delta \varepsilon_{C}(t_{i},t_{i-1})}{E(t_{i-1})}[(1+\chi(t_{i},t_{i-1})\phi(t_{i},t_ {i-1})]\\ &+\sum\limits_{j=1}^{+1}\frac{\Delta\sigma(t_{j})}{E(t_{j})}[\phi(t_{i},t_{j}) -\phi(t_{i-1},t_{j})]+\Delta\varepsilon_{S}(t_{i},t_{i-1})\end{array} \tag{1}\]  
where \(C\) and \(S\) represent the creep and shrinkage effects, respectively. \(\Delta\sigma C(t_{i},t_{i-1})\) is the stress increment caused by creep. \(\Delta\varepsilon S(t_{i},t_{i-1})\) is the strain increment caused by shrinkage. \(E\) is the elastic modulus of concrete. \(\chi(t_{i},t_{i-1})\), \(\phi(t_{i},t_{i-1})\), and \(R(t_{i},t_{i-1})\) are the aging coefficient, the creep coefficient, and the relaxation coefficient, respectively, which are detailed as follows.  
\[\chi(t_{i},t_{i-1})=\frac{E(t_{i-1})}{E(t_{i-1})-R(t_{i},t_{i-1})}-\frac{1}{ \phi(t_{i},t_{i-1})} \tag{2}\]  
\[R(t_{i},t_{i-1})=\frac{1}{J(t_{i},t_{i-1})}\Bigg{[}1+\frac{c_{1}\alpha(t_{i},t _{i-1})(t_{i},t_{i-1})}{10J(t_{i},t_{i}-1)}\Bigg{]}^{-10} \tag{3}\]  
\[c_{1}=0.0119\ln t_{i-1}+0.08,\;\alpha(t_{i},t_{i-1})=\frac{J(t_{i-1}+\zeta,t_{ i-1})}{J(t_{i},t_{i}-\zeta)}-1,\;\zeta=\frac{t_{i}-t_{i-1}}{2} \tag{4}\]  
The age-adjusted effective modulus \(E_{C}\)\"(\(t_{i},t_{i-1}\)) can be defined as Equation (5).  
\[E_{C}^{\prime\prime}(t_{i},t_{i-1})=\frac{E(t_{i-1})}{1+\chi(t_{i},t_{i-1}) \phi(t_{i},t_{i-1})} \tag{5}\]  
The step-wise analysis method of concrete shrinkage and creep effect is shown in Figure 6. First, the elastic modulus of concrete during each time interval of (\(t_{i}\), \(t_{i-1}\)) was replaced by the age-adjusted effective modulus \(E_{C}\)\"(\(t_{i}\), \(t_{i-1}\)). The structural responses caused by concrete creep were obtained by the sequential calculation and gradual accumulation. Moreover, the initial strains applied on the prestressed tendons during each time interval (\(t_{i+1}\), \(t_{i}\)) were updated to consider the pre-stress relaxation, which was based on the pre-stresses calculation results of the former time interval (\(t_{i}\), \(t_{i-1}\)). The changes in structural stress and geometry during the construction period caused by concrete shrinkage and creep were considered in the bridge completion state, which was based on the drafted construction time from the cast of concrete to the completion state for each girder segment. In addition, the concrete shrinkage effects were calculated using the analogous means adopted in the temperature effect calculation based on the shrinkage strain of each step.  
Figure 6: Schematic diagram of the stepwise analysis of concrete shrinkage and creep effect.