## 3 Modeling
### Moisture Transport Analysis  
The migration of moisture from a higher concentration zone to a lower concentration zone can be described using [PERSON]'s law of diffusion [2; 17; 18]. The equation for moisture transport can be written in a matrix form, analogous to the heat balance equation, as follows [2; 17; 19].  
\[[I]\frac{dh}{dt}+[D]h=F \tag{6}\]  
Here, \([I]\) is the identity matrix, \(F\) is the net moisture load vector (evaporation), \(h\) is the nodal RH vector, and [D] is the moisture diffusion coefficient matrix. The moisture diffusion coefficient of the concrete can be calculated using Equation (7), according to the fib MC2010 [13]. It should be noted that self-desiccation was not considered in this analysis.  
\[D(h)=D_{1}\Bigg{[}\alpha+\frac{(1-\alpha)}{1+\left(\frac{1-h}{1-h_{c}}\right) ^{n}}\Bigg{]} \tag{7}\]  
In the above, \(D_{1}\) represents the diffusion coefficient when \(h=1.0\), \(\alpha\) is a parameter representing the ratio \(D_{0}/D_{1}\), \(D_{0}\) is the lower limit of the diffusion coefficient for \(h=0.0\), \(D_{1}=D_{1,0}/f_{ck}\) (where \(f_{ck}\) is the characteristic compressive strength of the concrete (MPa) and \(D_{1,0}=1\times 10^{-8}\) (m\({}^{2}\)/s)), \(h_{c}\) denotesthe RH at \(D(h)~{}=~{}0.5\)D\({}_{1}\), and \(n\) is an empirical constant. According to MC2010 [13], it is assumed that \(\alpha\) = 0.05, \(h_{c}\) = 0.8, and \(n\) = 15.  
Drying shrinkage is proportional to the loss of water in concrete, and thus, the following relation has been suggested [18]:  
\[\varepsilon_{NUS}~{}=~{}\beta_{sh}(1-\Delta h) \tag{8}\]  
Here, \(\varepsilon_{NUS}\) is the nonuniform shrinkage strain, \(\Delta h\) is the difference in the RH, and \(\beta_{sh}\) is a hydro-shrinkage constant. It is suggested in the literature that \(\beta_{sh}\) lies in the range of \(1\times 10^{-3}\)-\(3\times 10^{-3}\)[20; 21; 20]. [PERSON] [22] used fitting to shrinkage test data to show that \(\beta_{sh}\) increases with an increase in RH in the RH range of 30 to 90%. In addition, \(\beta_{sl}\) can drop below \(1\times 10^{-3}\) for RH values of 60% or lower.  
DIANA FEA [15; 16] provides the general potential flow analysis, in which the potential flow is driven by the potential (temperature or concentration) gradient, as follows:  
\[q~{}=~{}-k\
abla\varphi \tag{9}\]  
Here, \(\
abla\varphi\) is the potential gradient, \(q\) is the specific flux vector, and \(k\) is the conductivity or diffusivity. Accordingly, a moisture transport analysis can be performed as a transient heat flow analysis by replacing the temperature and heat flow with the humidity and moisture flow, respectively. Then, the diffusion coefficient and hydro-shrinkage constant can be represented by the conductivity and thermal expansion coefficient, respectively. The boundary conditions are defined as a convection boundary condition as follows [19]:  
\[q_{c}\cdot n=-k_{c}(\varphi_{B}-\varphi_{E}) \tag{10}\]  
In the above, the evaporation at the surface of concrete can be represented by the convective discharge \(q_{c}\), the atmospheric RH by the environmental potential \(\varphi_{E}\), and the humidity at the surface of concrete by the boundary potential \(\varphi_{B}\). \(n\) is the normal vector to the surface. Parameter \(k_{c}\) is the conduction coefficient at the boundary for the heat transfer analysis but represents the evaporation rate in the moisture analysis. The evaporation rate mainly depends on moisture gradient, water cement ratio, wind speed and ambient temperature. Various researchers reported scattered empirical values in the range of \(3\times 10^{-9}-1\times 10^{-7}\) m/s [2; 16].  
In the potential flow analysis in DIANA FEA, quadratic hexahedron elements with 20 nodes are used for the concrete. Both the prestressing steel and non-prestressing reinforcing bars are modeled as embedded bar elements and assumed to be in a perfect bond with the surrounding concrete for the sake of simplicity.