# Analytical Study on the Load-Bearing Performance of RC Beams Subjected to ASR Expansion
## 4 Load-Carrying Capacity of RC Beams with ASR Deterioration
### Modeling of ASR Expansion
#### 4.3.1 Moisture Movement-Crack Coupling  
This study aims to develop a more complex degradation mechanism by considering that cracks in concrete affect not only the mechanical properties but also the mobility of materials in concrete. In other words, the cracked areas are considered more susceptible to moisture penetration, which is reflected in the mesoscale model. By changing the water penetration rate according to the cracks, we attempted to couple the moisture movement analysis with the stress analysis. Specifically, the cracks were considered in terms of two parameters, width and direction, and the effect of the cracks was reflected in the moisture inflow by changing the moisture diffusion coefficient based on these parameters. First, for the moisture movement performance of concrete without cracks, the moisture permeability \(K=50\) g/mm\({}^{3}\cdot\)hr\(\cdot\)mmHg and the moisture capacity \(C=0.1\) g/mm\({}^{3}\cdot\)mmHg, which were determined inverse analytically from the experiments in a previous study [37], were used to discretize Equation (9), which is portrayed as a one-dimensional diffusion equation and is solved by discretizing. \[C\frac{\partial P}{\partial t}=K\frac{\partial^{2}P}{\partial x^{2}} \tag{9}\]  
Next, the moisture migration performance at the crack initiation zone, i.e., the coupled moisture migration-cracking method, is described. This model evaluates the crack width as the strain after the tensile strength. In other words, the point at which cracking occurs is set to zero at the tensile strength, and the crack width is considered to increase as the strain increases from that point. As shown in Figure 22, the model defines this strain as the crack strain, which is calculated in each direction. Equations for crack strain are shown in Equations (10)-(12),  
\[\varepsilon_{cr\_x}=\varepsilon_{p\_x}-\varepsilon_{0} \tag{10}\]  
\[\varepsilon_{cr\_y}=\varepsilon_{p\_y}-\varepsilon_{0} \tag{11}\]  
\[\varepsilon_{cr\_z}=\varepsilon_{p\_z}-\varepsilon_{0} \tag{12}\]  
where \(\varepsilon_{cr\_x}\), \(\varepsilon_{cr\_y}\), and \(\varepsilon_{cr\_z}\) are the crack strain in the x, y, and z directions, respectively; \(\varepsilon_{p\_x}\), \(\varepsilon_{p\_y}\), and \(\varepsilon_{p\_z}\) are the principal strains in the x, y, and z directions, respectively; and \(\varepsilon_{0}\) is the crack strain. In this model, moisture movement at the cracks is assumed to be based on Hagen-Poiseuille flow. Therefore, the x-directional moisture movement velocity \(u_{xy}\) is associated with y-directional tensile failure and the x-directional moisture movement velocity \(u_{xz}\) is associated with z-directional tensile failure, as shown in Figure 23. The following Equations (13) and (14) can be expressed,  
\[u_{xy}=\frac{1}{2\rho v}\frac{\partial P}{\partial x}\left(y^{2}-\frac{{B_{y}} ^{2}}{4}\right) \tag{13}\]  
\[u_{xz}=\frac{1}{2\rho v}\frac{\partial P}{\partial x}\left(z^{2}-\frac{{B_{z}} ^{2}}{4}\right) \tag{14}\]  
where \(P\) is the water vapor pressure (MPa), \(\rho\) is the density of water vapor, \(v\) is the kinematic viscosity, \(B_{y}\) is the crack width in the y-direction, and \(B_{z}\) is the crack width in the z-direction. The area divisions of Equations (13) and (14), respectively, give the flow rate of moisture moving through the cracks. Equations (15) and (16) are shown below.