# 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
\[Q_{xy}=\int\limits_{-\frac{B_{y}}{2}}^{\frac{B_{y}}{2}}u_{xy}dA=-\frac{L}{12\rho \
u}{B_{y}}^{3}\frac{\partial P}{\partial x} \tag{15}\]  
\[Q_{xz}=\int\limits_{-\frac{B_{z}}{2}}^{\frac{B_{z}}{2}}u_{xz}dA=-\frac{L}{12 \rho\
u}{B_{z}}^{3}\frac{\partial P}{\partial x} \tag{16}\]  
Also, considering the averaged moisture flux \(q\) at the cracked section,  
\[q_{xy}=\frac{Q_{xy}}{B_{y}L}=-\frac{{B_{y}}^{2}}{12\rho\
u}\frac{\partial P}{ \partial x} \tag{17}\]  
\[\varepsilon_{cr\_x}=\varepsilon_{p\_x}-\varepsilon_{0} \tag{10}\]0  
Thus, the average moisture flux in the x direction is  
\[\varepsilon_{cr\_x}=\varepsilon_{p\_x}-\varepsilon_{0} \tag{10}\]1  
\[\varepsilon_{cr\_x}=\varepsilon_{p\_x}-\varepsilon_{0} \tag{10}\]2 Cracking and permeability concept. \[\varepsilon_{cr\_x}=\varepsilon_{p\_x}-\varepsilon_{0} \tag{10}\]3 Stress–strain relationship in the uniaxial conversion of rebar. Similarly, the average fluxes in the y- and z-directions are  
\[\varepsilon_{cr\_x}=\varepsilon_{p\_x}-\varepsilon_{0} \tag{10}\]4  
\[\varepsilon_{cr\_x}=\varepsilon_{p\_x}-\varepsilon_{0} \tag{10}\]5  
The matrix notation of Equations (19)-(21), based on the crack strain described above, can be expressed as follows:  
\[ \begin{split}\lambda_{x}&=\frac{{B_{y}}^{2}+{B_{z }}^{2}}{12\rho\
u}=\frac{{l_{0}}^{2}}{12\rho\
u}({\varepsilon_{cr,y}}^{2}+{ \varepsilon_{cr,z}}^{2})\\ \lambda_{y}&=\frac{{B_{x}}^{2}+{B_{z}}^{2}}{12\rho \
u}=\frac{{l_{0}}^{2}}{12\rho\
u}({\varepsilon_{cr,z}}^{2}+{\varepsilon_{cr,z}}^ {2})\\ \lambda_{z}&=\frac{{B_{x}}^{2}+{B_{y}}^{2}}{12\rho \
u}=\frac{{l_{0}}^{2}}{12\rho\
u}({\varepsilon_{cr,z}}^{2}+{\varepsilon_{cr,z}}^ {2})\\ (B_{x}&=l_{0}{\varepsilon_{cr}}\quad B_{y}=l_{0}{ \varepsilon_{cr}}_{y}\quad B_{z}=l_{0}{\varepsilon_{cr}}_{z})\\ \begin{bmatrix}q_{x}\\ q_{y}\\ q_{z}\end{bmatrix}&=\begin{bmatrix}-\lambda_{x}&0&0\\ 0&-\lambda_{y}&0\\ 0&0&-\lambda_{z}\end{bmatrix}\begin{bmatrix}\frac{\partial P}{\partial x}\\ \frac{\partial P}{\partial y}\\ \frac{\partial P}{\partial z}\end{bmatrix}\end{split}  \tag{22}\]  
Here, Equations (22) and (23) are equivalent to [PERSON]'s law, which describes diffusion in proportion to the energy (concentration) gradient, thus defining the moisture diffusion coefficient \(\lambda\) and allowing moisture movement to be described as a diffusion equation. Next, the direction of the crack is calculated from the direction of the principal stress and is calculated as the angle of rotation from the overall coordinate. In the analysis, the effect of the crack direction is reflected in the moisture movement by multiplying the crack direction coordinate transformation matrix \(\mathbf{r}\), shown in Equation (24), by the moisture transfer coefficient vector \(\lambda\):  
\[ \begin{split}\mathbf{r}=\begin{bmatrix}cos\psi&-sin\psi&0\\ sin\psi&cos\psi&0\\ 0&0&1\end{bmatrix}\begin{bmatrix}cos\theta&0&sin\theta\\ 0&1&0\\ -sin\theta&0&cos\theta\end{bmatrix}\begin{bmatrix}1&0&0\\ 0&cos\varphi&-sin\varphi\\ 0&sin\varphi&cos\varphi\end{bmatrix}\\ =\begin{bmatrix}cos\psi cos\theta&cos\psi sin\theta\text{sin} -cos\varphi&\psi&sin\psi sin\theta+cos\psi cos\theta\text{cos}\theta\\ cos\theta sin\varphi&cos\varphi+sin\psi sin\theta\text{sin}\varphi&cos\varphi \text{sin}\theta-cos\psi\text{sin}\theta\\ -sin\theta&cos\theta sin\varphi&cos\theta cos\varphi\end{bmatrix}\end{split}  \tag{24}\]  
Therefore, considering the moisture diffusion coefficient \(\mathbf{K_{cr}}\), which takes into account the effect of crack width and direction, and the moisture movement through the concrete pores before crack initiation as mentioned above, the governing equation for moisture movement in concrete in this study can be expressed as Equation (25):  
\[ \begin{split} C\frac{\partial P}{\partial t}=K_{x}\frac{\partial^ {2}P}{\partial x^{2}}+K_{y}\frac{\partial^{2}P}{\partial y^{2}}+K_{z}\frac{ \partial^{2}P}{\partial z^{2}}\\ K=\begin{bmatrix}K_{x}\\ K_{y}\\ K_{z}\end{bmatrix}=K_{0}+\mathbf{K_{cr}}=\begin{bmatrix}K_{x0}\\ K_{y0}\\ K_{z0}\end{bmatrix}+\begin{bmatrix}K_{x-cr}\\ K_{y-cr}\\ K_{z-cr}\end{bmatrix}\end{split}