# Construction of a Dimensional Damage Model of Reinforced Concrete Columns under Explosion Loading
## 2 \(\Pi\) Theorem of Dimensional Analysis
### Cylinder Damage Model Based on the \(\Pi\) Theorem  
Figure 1 shows the damage model of reinforced concrete support columns under blast loading, including reinforced concrete columns (longitudinal bars, stirrups and concrete), spherical explosives, rigid walls and air domains. The symbolic interpretation of the physical parameters in the damage model of reinforced concrete columns is shown in Table 1.  
The basic physical parameters involved in the damage model of reinforced concrete columns are mass \(M\), length \(L\) and time \(T\); this paper selects the 'MLT' dimensional analysis system to analyze the damage model. If it is determined that the parameters in Table 1 are the influencing factors of cylinder damage and the degree of disturbance in the column is used as the basis for evaluating the degree of cylinder damage, then the physical problem can be described as follows according to the principle of dimensional analysis:  
\[\xi=f(m,r,L,H,a,b,\rho_{TNT},\rho,\sigma,E,I) \tag{1}\]  
The bending stiffness of the cylinder is characterized by \(EI\), which is the combination of Young's modulus \(E\) and the cross-section moment of inertia \(I\). At the same time, the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Unit** & **Explanation** \\ \hline \(m\) & kg & Explosive equivalent \\ \(r\) & m & Radius of spherical explosive \\ \(L\) & m & The minimum distance from the detonation point of \\  & & explosive to the surface of reinforced concrete column \\ \(\rho_{TNT}\) & kg/m\({}^{3}\) & Explosive density \\ \(\rho\) & kg/m\({}^{3}\) & Column density \\ \(H\) & m & Column height \\ \(a\) & m & The section (square) side length of the column \\ \(\sigma\) & Pa & The yield strength of the column \\ \(E\) & Pa & [PERSON]’s modulus of column \\ \(I\) & m\({}^{4}\) & Sectional moment of inertia of the column \\ \(b\) & m\({}^{2}\) & On the cross-section of the column, the total \\  & & cross-sectional area of the longitudinal reinforcement \\ \(\xi\) & m & Disturbance in column \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Interpretation table of physical parameters in damage model. Figure 1: Damage diagram of a reinforced concrete column. density of spherical explosive \(\rho_{TNT}\) can be expressed by explosive equivalent m and radius \(r\). Then, the above equation can be simplified to:  
\[\xi=f(m,r,L,H,a,b,\rho,\sigma,EI) \tag{2}\]  
Equation (2) has a total of 10 change parameters, including 1 dependent variable parameter and 9 independent variable parameters, and its dimensional power exponent is shown in Table 2.  
The density \(\rho\), column height \(H\) and yield strength \(\sigma\) of concrete columns are selected as reference physical quantities, and Table 2 is sorted and reconfigured. Table 3 can be obtained as follows. According to the properties of \(\Pi\) theorem and dimensional analysis, six dimensionless independent variables and one dimensionless dependent variable can be obtained, see Formula (3). In this formula, dimensionless independent variables are represented by \(\pi_{i}\), and the dimensionless dependent variable is represented by \(\pi\). \[\left\{ \begin{array}{l}\pi_{1}=\frac{m}{\rho H^{3}}\\ \pi_{2}=\frac{r}{H}\\ \pi_{3}=\frac{r}{H}\\ \pi_{4}=\frac{a}{H}\\ \pi_{5}=\frac{b}{H^{2}}\\ \pi_{6}=\frac{EI}{\sigma H^{4}}\\ \pi=\frac{c}{H}\end{array} \right. \tag{3}\]  
Therefore, Formula (3) can be written as a dimensionless function expression:  
\[\frac{\xi}{H}=f(\frac{m}{\rho H^{3}},\frac{r}{H^{4}},\frac{L}{H^{4}},\frac{b}{ H^{2}},\frac{EI}{\sigma H^{4}}) \tag{4}\] \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline  & \(\rho\) & \(H\) & \(\sigma\) & \(m\) & \(r\) & \(L\) & \(a\) & \(b\) & \(EI\) & \(\xi\) \\ \hline \(\rho\) & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ \(H\) & 0 & 1 & 0 & 3 & 1 & 1 & 1 & 2 & 4 & 3 \\ \(\sigma\) & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 \\ \hline \end{tabular}
\end{table} 
Table 4: Dimension power exponent table of physical quantities (after transformation).