# Black Loaded Columns--State of the Art Review
## 3 Numerical Modeling
In the simulations, they obtained a higher residual bearing capacity of the column when the explosive was placed at the height of 1.5 m from the bottom than when it was placed at the bottom. [PERSON] et al. [7], for calculation of impulse variations per unit length along the height of the column, were using the Bridge Explosive Loading (BEL) [90] program. BEL also considers the reflected pressure of the blast wave on the surface of the superstructure and on the ground. [PERSON] et al. [47] studied the behavior of four types of column cross-sections in MSC.Dytran software on blast load: single-cell hollow steel section, multi-cell hollow steel section, single-cell hollow composite column, and multi-cell composite column. Compared to steel columns, the composites showed negligible deflection on blast load. The best stress distribution in the element was achieved with multi-cell composite columns, which is also visible by the displacements shown in Figure 17.  
[PERSON] et al. [49] analyzed three reinforced concrete columns with the same dimensions, material strengths, and reinforcement ratios but subjected to blast loads of different scaled distances. They wanted to find the failure probability using the computer code CARLER, which is verified with Monte Carlo simulations. They defined four damage levels (\(D\)) that depend on the ratio of residual axial load carrying capacity (\(N_{\text{residual}}\)) when the column is damaged and the axial load of the undamaged column (\(N_{0}\)), shown in Table 5. Through numerical simulations, they concluded that neglecting some of the material properties of the column has minimal impact on the probability of failure, while the random changes in the blast loading have a much greater role. [PERSON] and [PERSON] [51] emphasized the spalling of side-cover concrete because, in previous works, only the spalling of concrete off the back of reinforced concrete columns was mentioned. The aim of their research was to make and validate a numerical model with respect to the experiment explained in [38] and justified the cross-sectional response mechanisms that cause loss of side-cover concrete. For numerical simulations, they used the LS-Dyna program and the Karagozian and Case concrete (KCC) material model. Numerical simulations in LS-Dyna showed that the shape of the column cross-section has a large influence on the resulting impulse. The authors developed expressions for calculating column shape factors for circular and square columns. The expressions are used when the R / D ratio is less than 4.5 because they provide sufficiently conservative loads but less than those experimentally determined on the walls [52]. [PERSON] [10] performed numerical simulations in LS-Dyna of RC columns and columns retrofitted with fiber-reinforced plastic (FRP). FRP increases the resistance of the RC columns to the blast load. For numerical modeling, the choice of concrete material model is very important, and the analysis was performed with four different concrete  
Figure 17: Time history of displacements for 4 column types [47]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Level of Damage**[49] & \(D\)**=\(1-\frac{N_{\text{residual}}}{N_{0}}\) & **Damage Limit States**[57] \\ \hline Low damage & 0–0.2 & Low damage \\ Medium damage & 0.2–0.5 & Repairable damage \\ High damage & 0.5–0.8 & Repairable damage \\ Collapse & 0.8–1.0 & Collapse \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Damage levels of RC column in terms of axial load capacity. models (KC, Winfrith, Continuous Smooth Cap, and RHT model). The best results were obtained with the KC model. [PERSON] et al. [53] performed a parametric numerical analysis in Abaqus to see which of the six varied parameters (section ratio, compressive strength of concrete, column height and thickness, charge radius, and ratio between standoff distance to the charge radius) had the greatest impact on column damage.