## 4 Predicting Response of Reinforced Concrete Columns
### Finite Element Modeling  
The 2D nonlinear finite element analysis program, VecTor2 [28] was used to predict the response of columns experiencing a range of failure modes and levels of ductility. The program employs a smeared, rotating crack model which is based on the Modified Compression Field Theory and the Disturbed Stress Field Model [29] and hence is capable of predicting shear failures and shear distress.  
The cracked concrete is treated as an orthotropic material with cracks distributed throughout the elements. The cracks can freely rotate to remain parallel with the principal compressive stresses in the material. Shear slip along the crack is accounted for, allowing the principal stresses to lag in their rotation relative to the principal strains.  
The default material models were used in all the finite element predictions, except for tension stiffening, which was accounted for using the [PERSON] 1982 model [2]. The effects of concrete confinement are accounted for by increasing the concrete strength and  
Figure 4: Shear strength predictions of rectangular, circular, and octagonal columns using the AASHTO General Procedure: (**a**) With reduction in \(V_{c}\) for low axial load levels; (**b**) Without reduction in \(V_{c}\).  
associated strains according to [PERSON] [30] and [PERSON] et al. [31]. The hysteretic response of the concrete is modelled with plastic strain offsets and nonlinear unloading [32]. The reinforcing steel model includes the effect of yielding, strain hardening, and bar rupture. The hysteretic response of the reinforcement includes the Bauschinger Effect [33]. In the finite element analysis, bar buckling was modeled using the Akkaya [34] model that was based on the model developed by [PERSON] and [PERSON] [35]. This bar buckling model accounts for the spacing of the transverse reinforcement, the diameter of the longitudinal reinforcing bars, and the stiffness of the transverse reinforcement that is controlling the dilation of the column in the plastic hinge region. The analysis considered spalling of the concrete cover by eliminating boundary elements based on their principal compressive strains, crack widths, and crack inclinations. Discrete truss elements were used to model the longitudinal reinforcement while smeared reinforcement ratios were used to model the in-plane and out-of-plane transverse reinforcement.  
To model circular columns in VecTor2, the cross-section must be discretized into layers and the smeared reinforcement ratios determined as shown in Figure 5. The width and thickness of these elements vary according to their location in the cross-section. The layers were selected such that their boundaries occurred at the levels of longitudinal reinforcement. For each of these layers, the in-plane (\(\rho_{x}\)) and out-of-plane (\(\rho_{z}\)) smeared reinforcement ratios must be determined. The in-plane component resists the shear, while the out-of-plane component is used to determine the confinement effects on the concrete. These reinforcement ratios are dependent on the cross-sectional area of the hoop/spiral, \(A_{b}\), and the angle, \(\alpha\), and are given by Equations (7) and (8). For spirally reinforced columns, the pitch is typically small, such that the inclination of the spiral can be ignored [6].  
\[\rho_{x}=\frac{2A_{b}\cos\alpha}{ws} \tag{7}\]  
\[\rho_{z}=\frac{2A_{b}\sin\alpha}{ts} \tag{8}\]  
The following sections provide details and response predictions of columns that displayed different failure modes. These predictions using nonlinear finite element analysis are compared with the experimental responses. In making the response predictions using nonlinear finite element analysis, the target displacements in the analysis were chosen to match, as closely as possible, the positive and negative peak displacements achieved in the reversed-cyclic loading tests.