## 2 Refined Numerical Modelling Approach
### Performance Degradation Simulation Approach
#### 2.5.2 Concrete Creep  
As a time-varying material, the creep development of concrete will cause its mechanical properties to change with service time. It is generally believed that the creep strain develops most rapidly at the initial stage of bridge construction, and then as the service time of the bridge increases, the concrete hydration gradually becomes sufficient and the creep development tends to become stable. The age-adjusted effective modulus (AAEM) method proposed by [PERSON] [25] and modified by [PERSON] [26] was used within this study to consider the long-term performance of Bridge A as affected by creep. The effective modulus \(E_{\mathit{eff}}\) can be obtained from:  
\[E_{\mathit{eff}}=\frac{E(t_{0})}{1+\chi(t,t_{0})\varphi(t,t_{0})} \tag{1}\]  
\[\chi(t,t_{0})=\frac{1}{1-e^{-\varphi(t,t_{0})}}-\frac{1}{\varphi(t,t_{0})} \tag{2}\]  
where \(E(t_{0})\) denotes the tangential elastic modulus of concrete at initial loading time \(t_{0}\); \(\varphi(t,t_{0})\) is the dimensionless creep coefficient, which indicates the ratio of creep strain to the elastic strain of concrete from the initial time \(t_{0}\) to the current time \(t\); and \(\chi(t,t_{0})\) is the aging coefficient of concrete, and its relationship with the creep coefficient is shown in Formula (2). The calculation method of creep coefficient \(\varphi(t,t_{0})\) given in the current Chinese Industry Specifications [27] was adopted to take into account the creep effect. The related expressions are listed within Equations (3)-(5):  
\[\varphi(t,t_{0})=\varphi_{n}(t,t_{0})\bigg{[}\frac{(t-t_{0})/t_{1}}{\beta+(t- t_{0})/t_{1}}\bigg{]}^{0.3} \tag{3}\]  
\[\beta=\min\bigg{\{}150\Big{[}1+(1.2\text{RH})^{18}\Big{]}\frac{h}{h_{0}}+250, 1500\bigg{\}} \tag{4}\]  
\[\varphi_{n}(t,t_{0})=\frac{5.93\bigg{(}1+\frac{1-RH}{0.46(h/h_{0})^{\frac{1}{ 3}}}\bigg{)}}{\bigg{(}\frac{f_{\mathit{rank}}}{f_{\mathit{conv}}}+1\bigg{)}^{0. 5}\bigg{[}0.1+\Big{(}\frac{t_{0}}{t_{1}}\bigg{)}^{0.2}\bigg{]}} \tag{5}\]  
where \(\varphi_{n}(t,t_{0})\) is the nominal creep coefficient and can be obtained with Equation (5); \(h\) represents the equivalent thickness (mm), which can be expressed as \(2A_{c}\)/\(C_{c}\), where \(A_{c}\) is the area of the concrete section (mm\({}^{2}\)) and \(C_{c}\) is the circumference of the concrete section in contact with the atmosphere (mm); \(f_{\mathit{conv},k}\) is the standard value of compressive strength of  
Figure 7: Schematic diagram of the three-dimensional finite element model for box beam cracks. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Locations**} & **Length of the Crack** & **Width of the Crack** & **Diameter of Drilling** & **Depth of Drilling** \\  & **(mm)** & **(mm)** & **Core (mm)** & **Core (mm)** \\ \hline Web of No.15 box beams & 3170 & 0.26 & 50 & 130, 160 \\ Web of No.16\({}^{\prime}\) box beams & 2800 & 0.20 & 70 & 180 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Crack information near the quarter span of Bridge A.  
the concrete cube (MPa); \(h_{0}=100\) mm, \(t_{1}=1\) d and \(f_{cm0}=10\) MPa are reference thickness, reference time and reference strength, correspondingly; and RH is the abbreviation of the relative humidity of the environment. Within this study, the starting time \(t_{0}\) was taken as 7 days. Due to differences in the construction completion time of different segments, the creep times of each segment were also different. However, since the creep development process is slow and gradual, especially the period from the completion of the whole bridge construction (around 400 days) to the late operation stage (10,000 days), the difference in the creep process caused by the difference in construction age is quite limited. Therefore, for the whole-bridge model analysis, the above age differences were negligible. It can be concluded that different creep strains occur within different segmental box beams with unequal equivalent thicknesses.