# 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
#### 4.3.2 Modeling of ASR Aggregate Expansion  
Because this study aims to understand analytically the change in mechanical behavior of concrete due to ASR, ASR is considered as aggregate expansion and is represented as a mechanical model based on a three-dimensional mesoscale. In the following, we describe how aggregate expansion is represented in the actual analysis. In this study, ASR is considered as aggregate expansion, and the expansion is given as an increase in the volumetric strain of the aggregate. In general, it is more realistic to treat ASR gel in the analysis as an actual porous material that can hold stress, and such a model is also considered in the existing research [38,39]. It is said that there are two types of expansion models for ASR gel in the analysis: a reaction rim model and a gel pocket model [24]. In this study, in order to model the expansion state of the reaction rim model, the above-mentioned expansion-giving method is adopted. The model was developed considering water and alkali content as the strain parameters. The model was developed by referring to papers by [PERSON] et al. [40] and [PERSON] et al. [41]. Figure 24 shows a schematic diagram of ASR expansion.  
In this study, ASR is considered as a two-step reaction. In the first step, the reactive silica of the aggregate reacts with alkali to form an alkali-silica gel. Gel formation is assumed to be isotropic, as shown in the figure, and the amount of gel formed is given by Equation (26),  
\[T_{pr}=\int_{0}^{t}k_{gel}\times C\times A\times dt \tag{26}\]  
where \(T_{pr}\) is the amount of ASR gel produced (-), \(A\) is the unit reaction coarse aggregate volume (g/L), \(dt\) is the time increment (T), \(k_{gel}\) is the coefficient of the expansibility of the gel (-), and \(C\) is the alkaline ion concentration (mol/L). At this point, the gel is assumed not to affect the deformation of the concrete, and no aggregate expansion is assumed to have occurred. In the second stage, the gel generated in the first stage absorbs water and causes volumetric expansion strain in the aggregate. The volumetric expansion strain is given by Equation (27),  
\[\varepsilon_{V}=f(C,\Delta R)=k_{gel}\times\left(\frac{E_{gel}}{E_{agg}} \right)\times T_{pr}(C)\times\Delta R \tag{27}\]  
where \(\Delta R\) is the change in relative moisture content of the reactive aggregate (-), \(k_{gel}\) is the coefficient on the expansibility of the gel (-), \(E_{gel}\) is the Young's modulus of the gel, and \(E_{agg}\) is the Young's modulus of the reactive coarse aggregate. As shown above, the aggregate volume expansion strain could be expressed as a function of the alkali and moisture content. In the present paper, the objective is to evaluate the effect of moisture on ASR at the three-dimensional mesoscale and the relationship between aggregate expansion, cracking properties, and load-bearing capacity:  
\[\alpha=k_{gel}\times\left(\frac{E_{gel}}{E_{agg}}\right)\times T_{pr}(C) \tag{28}\]  
And then, the volume expansion strain \(\varepsilon_{V}\) is  
\[\varepsilon_{V}=f(C,\Delta R)=\alpha\Delta R \tag{29}\]  
In other words, the ASR phenomenon assumes that aggregate expansion occurs as a result of the action of moisture only on concrete that has already produced a certain amount of alkali-silica gel. In this case, \(\alpha\) is a function of the alkali content and is a constant  
Figure 24: Cracking and permeability concept.  
that determines the expansion based on the alkali content because it is a coefficient for the change in relative humidity.