# Dynamic Behavior Analysis of I-Shaped RC Beams under Combined Blast and Impact Loads
## 2 Finite Element Analysis of the I-Shaped RC Beam under the Blast Load
### Material Model  
In order to describe the nonlinear characteristics of the material, the material behavior of concrete was modeled using the MAT_CSCM_CONCRETE Keywords: The model included hardening, damage, and strain rate effects of the material under the blast loading. The dynamic enhancement factor (DIF) was used to reflect the enhancement effect of the material strength, and the concrete compressive strength enhancement factor (CDIF) [25] is:  
\[CDIF=\frac{f_{cd}}{f_{cs}}= \begin{cases}\left(\frac{\dot{\epsilon}_{dd}}{ \dot{\epsilon}_{cs}}\right)^{1.026\alpha}for\;\;\dot{\epsilon}_{cd}\leq 30 \text{s}^{-1}\\ \gamma\left(\frac{\dot{\epsilon}_{dd}}{\dot{\epsilon}_{cs}}\right)^{1/3}for\; \;\dot{\epsilon}_{cd}>30\text{s}^{-1}\end{cases}  \tag{2}\]  
where \(f_{cd}\) is the dynamic compressive strength under the strain rate \(\dot{\epsilon}_{cd}\); \(f_{cs}\) is the static compressive strength under the strain rate \(\dot{\epsilon}_{cs}\); the interval of the dynamic strain rate \(\dot{\epsilon}_{cd}\) is \(\left[3\times 10^{-5}\text{s}^{-1},3\times 10^{-2}\text{s}^{-1}\right]\), and the static strain rate \(\dot{\epsilon}_{cs}\) is \(3\times 10^{-5}\text{s}^{-1}\); \(f_{c0}=10\) MPa; \(\alpha=1/\left(5+9f_{cs}/f_{c0}\right)\), \(log\gamma=6.156\alpha-2\). The tensile strength enhancement factor (TDF) of the concrete is [26]:  
\[TDF=\frac{f_{td}}{f_{ts}}= \begin{cases}\left(\frac{\dot{\epsilon}_{td}}{ \dot{\epsilon}_{hs}}\right)^{\delta}for\;\;\dot{\epsilon}_{td}\leq 1\text{s}^{-1}\\ \beta\left(\frac{\dot{\epsilon}_{dd}}{\dot{\epsilon}_{hs}}\right)^{1/3}for\; \;\dot{\epsilon}_{td}>1\text{s}^{-1}\end{cases}  \tag{3}\]  
where \(f_{td}\) is the dynamic compressive strength at the strain rate \(\dot{\epsilon}_{td}\); \(f_{ts}\) is the static compressive strength at the strain rate \(\dot{\epsilon}_{ts}\); the interval of the dynamic strain rate \(\dot{\epsilon}_{td}\) is  
Figure 1: Geometric model of I-shaped RC beam; (**a**) experimental model, and (**b**) numerical model. \(\left[10^{-6}\text{s}^{-1},\,160\text{s}^{-1}\right]\), the static strain rate \(\dot{\epsilon}_{\text{cs}}\) is \(10^{-6}\text{s}^{-1}\); \(\delta=1/(1+8f_{cs}/f_{0})\), \(f_{\text{c0}}=10\) MPa; \(log\beta=6\delta-2\). In order to accurately describe the mechanical characteristics of the reinforcement, the keyword MAT_PLASTIC_KINEMATIC was used for simulation [26]. At the same time, the Cowper-Symonds model, which can consider both the strain rate effect and the dynamic strengthening effect, was used for analysis. The yield stress values of the transverse and longitudinal reinforcement are 320.3 MPa and 410.6 MPa, respectively. When the maximum principal strain exceeds 0.14, the reinforcement element quits working. The model strain rate enhancement factor was calculated as [27]:  
\[DIFs=1+\left(\frac{\dot{\epsilon}}{C}\right)^{1/P} \tag{4}\]  
where \(\dot{\epsilon}\) is the strain rate of the reinforcement, \(C\) and \(P\) are the strain rate parameters of the Cowper-Symonds model. CONSTRAINED_LAGRANGE_IN_SOLID was used for coupling between reinforcement and concrete elements. According to [24], frictionless sliding between steel and concrete elements was considered, allowing uniform meshing over the whole structure. In addition, the keyword MAT_VISCOELASTIC was used to simulate the material of the rubber bearing. RDX, as a high explosive, has an internal energy almost 1.5 times that of trinitrotoluene (TNT), and a density of about 1890 kg/m\({}^{3}\). For numerical analysis, TNT is a commonly used material in explicit material libraries. Therefore, in order to facilitate the calculation and quantitative comparison of the explosive performance of the structure under different explosion scenarios, the TNT equivalent coefficient was used to convert the RDX weight into the TNT equivalent weight. Use the keyword LOAD_BLAST_ENHANCED to simulate blast loads, where the default explosive is TNT. Set a blast surface on the structure with the LOAD_BLAST_SEGMENT_SET command to simulate the interaction between the blast shock wave and the structure.