## 2 Development of Finite Element Model of Axially Loaded RC Columns for Impact Load
### Constitutive Models of the Materials  
The material properties of transverse and longitudinal steel bars were modeled using MAT-24 (piecewise linear plasticity model) from the material library of LS-DYNA [33]. This material model is very popular and widely used for modeling steel materials under dynamic or rate-dependent loads. It takes into account the material's isotropic plasticity and kinematic hardening properties. The user-defined stress-strain parameters can also be used to define the stress-strain curve of the steel material. This material model allows us to incorporate the strain rate effect on the steel material. The following equations (Equations (1) and (2)) define the relationship between the tensile and compressive dynamic increase factor (_DIF_) of steel and the strain rate [34]:  
\[DIF=\left(\frac{\dot{\epsilon}}{10^{-4}}\right)^{\alpha} \tag{1}\]  
\[\alpha=0.074-\frac{0.04f_{y}}{414} \tag{2}\]  
where \(f_{y}\) and \(\dot{\epsilon}\) are the steel yield strength (MPa) and strain rate, respectively. The steel material of impactor, roller support, and end plates were modeled using MAT-3 (Plastic kinematic) from the material library.  
Modeling concrete materials in FE analysis is always a challenge. It is important to select an appropriate concrete material model in order to achieve a realistic result. The LS-Dyna material modeling library includes a large number of constitutive models for concrete [33]. MAT-159 (continuous surface cap model) and MAT-72 RII (concrete damage model) have the ability to generate the required parameters using only the unconfined cylinder compressive strength and the maximum aggregate size of the concrete. Other concrete material models, such as MAT-84/85 and MAT-272, can also be used by providing some parameters that can be determined manually using compressive strength. All of the material models discussed here have their own set of benefits and drawbacks. There is no single concrete material model that can be applied to all impact problems and produce the  
Figure 3: Numerical model of the column under impact load.  
best results in terms of damage, deflections, and impact forces. In some cases, the material model MAT-72 RIII produces good results; however, when the impact velocity is increased, it appears that shear deformation is prominent around the projectile rather than global beam deformation [35; 36]. Nevertheless, this material model gives very excellent results with low-velocity impact (i.e., low impact energy) by adjusting some parameters. To adjust the different parameters in this model needs very expert knowledge on the constitutive material properties of concrete [37].  
The material model MAT-159 is appropriate for simulating the impact problems of RC beams and columns. However, with the default fit values, it shows some concrete hardening [38]. For this reason, some of the cases underestimate the deflection of the beam. However, with the suggested adjustment on compressive fracture energy, tensile fracture energy, and shear fracture energy [38], this problem can be overcome. Additionally, this material model can define the damage profile in the RC members more reasonably than other concrete of the material models. Thus, this material model has been used by many researchers to model reinforced concrete structures for dynamic analysis [26; 35; 36]. In light of the above discussion, MAT-159 was used to simulate the concrete material of the column under impact load in this research.  
MAT-54 (ENHANCED_COMPOSITE_DAMAGE) specified with Chang-Chang failure criteria was used to model the AFRP material. This material model was specially developed for arbitrary orthotropic materials. To define FRP materials, this material model is widely used in many studies [17; 39]. The properties of the AFRP are given in Table 1. The remaining input parameters are taken from [39].