## 2 Long-Term Deflection Calculation in Existing Design Codes  
In current design practice, the elastic deflection of a concrete girder owing to various types of mechanical loads (such as self-weight, prestressing force, and other imposed loads) is calculated by considering the bridge superstructure as a 1D beam. Subsequently, the long-term deflection is calculated by multiplying the elastic deflection by coefficients representing the effects of creep and shrinkage, as suggested by design codes such as those from the American Association of State Highway and Transportation Officials (AASHTO) [7], ACI [10], Precast/Prestressed Concrete Institute (PCI) [12], International Federation for Structural Concrete [13], and Eurocode [14].  
For example, per the multiplier method in the PCI bridge design manual [12], the elastic deflection owing to the self-weight (\(\delta_{sw}\)), prestressing force (\(\delta_{pi}\)), and other permanent loads in service (\(\delta_{sl}\)) is initially calculated using the following formula:  
\[\delta_{e}=\ K\frac{ML^{2}}{E_{c}I_{g}} \tag{1}\]  
In the above, \(\delta_{e}\) is the maximum instantaneous elastic deflection along the beam, \(M\) is the applied moment (can be the result of either concentrated or distributed load), \(L\) is the length of a beam, \(E_{c}\) is the modulus of elasticity of the concrete, \(I_{g}\) is the gross 2 nd moment of the area, and \(K\) is a factor representing the support fixity and loading condition. Then, for a bridge with composite topping, the time-dependent deflections at important time stages are calculated as follows.  
1. Net erection deflection before permanent service load (\(\delta_{ne}\)) \[\delta_{ne}=\ 1.8\ \delta_{pi}\ +\ 1.85\ \delta_{sw}\] (2)
2. Net final deflection without permanent service load (\(\delta_{nf}\)) \[\delta_{nf}=\ 2.2\ \delta_{pi}\ +2.4\ \delta_{sw}\] (3)
3. Net final deflection with permanent service load (\(\delta_{nfs}\)) \[\delta_{nfs}=\ 2.2\ \delta_{pi}\ +2.4\ \delta_{sw}+\ 3\ \delta_{sl}\] (4)  
The multiplier coefficients in the above equations are determined by considering the creep and shrinkage of the concrete and may vary according to the creep and shrinkage models of the design code.  
Nevertheless, all the coefficients are calculated based on the same approach, i.e., a 1D simplification neglecting the nonuniform shrinkage effect.  
Another example is fib Model Code 2010 (MC2010) [13], which recommends the following simplified equation to calculate the long-term deflection (\(\delta_{t}\))  
\[\delta_{t}\ =(1+C_{c})\delta_{e} \tag{5}\]  
where \(\delta_{e}\) is instantaneous deflection and \(C_{c}\) is the creep coefficient.  
To account for the effects of cracking, the effective moment of inertia approach is common among design codes [6; 7]. In this approach, the gross area moment of inertia (\(I_{g}\)) is replaced with the effective area moment of inertia (\(I_{c}\)), i.e., the average between the cracked and gross area moments of inertia upon the initiation of crack. However, the effective moment of inertia approach does not consider crack growth or the development of multiple cracks with creep.  
In summary, as discussed in the introduction, the current design practice does not consider nonuniform shrinkage, progressive cracking, or the shear lag effect.