## 4 Predicting Response of Reinforced Concrete Columns
### Square Columns Tested by [PERSON] et al. 1996  
[PERSON] et al. [15] tested a series of square columns under reversed-cyclic loading to investigate the behavior of columns with different levels of axial load, varying amounts of longitudinal reinforcement, different tie details, and large tie spacings typical of pre-1970's construction. Figure 10 shows typical cross-sections and the finite element model for three columns with different failure modes that were selected for analysis. Each column was subjected to a constant axial compression with the reversed-cyclic loading consisting of 3 cycles at displacements corresponding to increments of the calculated yield displacement. The columns were loaded in double curvature and had heavily reinforced end regions (see Figure 10). To simulate the effect of 90\({}^{\circ}\) end anchorages on the ties, the stress in the ties was limited to the yield stress without strain hardening in the finite element analysis. The finite element analysis predicts that buckling of the longitudinal reinforcement occurred as was reported by the researchers.  
Figure 8: Example of finite element model and typical cross-section of columns tested by [PERSON] et al. [22].  
The finite element response predictions for the [PERSON] et al. (1993) columns analyzed are compared with the experimental results in Figure 11. The three columns had a shear span-to-depth ratio of 3.22 and shear reinforcement consisting of No. 3 square and diamond-shaped ties at a spacing of 457 mm or 305 mm (see Figure 10). The axial load ratios (\(P/f_{c}^{\prime}A_{S}\)) were 0.073 for column 2 CLH18, and 0.262, for columns 3 CMD12 and 2 CMH18.  
Column 2 CLH18 achieved a ductility level of 4.2 and experienced a predominately flexural failure with yielding and buckling of the longitudinal reinforcement occurring at displacements of 15 mm and 46 mm, respectively. The finite element model gives reasonable predictions of the response of the longitudinal reinforcement with yielding and buckling occurring at displacements of 17 mm and 36 mm, respectively. The concentrated inelastic rotations reported within the bottom 102 mm of the column are also present in the finite element model (see Figure 11a). The influence of buckling can be seen in Figure 11. The predicted strength was 257 kN compared with the experimental strength of 241 kN.  
Column 3 CMD12 exhibited a flexure-shear failure with yielding of the longitudinal reinforcement occurring first, at a displacement of 30 mm followed by transverse reinforcement yielding at 46 mm. All ties except one indicated yielding prior to failure. The response is predicted well by the finite element model which indicates a drop in lateral load at a displacement target of 32 mm followed by yielding of the longitudinal reinforcement in the following cycle (see Figure 11b). The predicted strength was 367 kN compared with the experimental strength of 356 kN.  
Figure 9: Nonlinear finite element analysis predictions of shear-deflection responses and deflected shapes at failure (20\(\times\) magnification) for columns tested by [PERSON] et al. (2019): (**a**) Column 3; (**b**) Column 5.  
Column 2 CMH18 experienced a brittle shear failure during the last cycle at a displacement of 30 mm. The failure occurred due to the formation of an inclined crack and yielding of the tie crossing the crack. Longitudinal reinforcement at the column base had yielded just prior to failure but flexural hanging had not occurred. The finite element analysis gives reasonable predictions of the column shear-displacement response, and correctly predicted that a brittle shear failure would occur (see Figure 11c). The analysis also predicts that the longitudinal reinforcement just reaches yield at a displacement of 30 mm. The predicted strength was 301 kN compared with the experimental strength of 306 kN.  
Figure 10: Example of finite element model and typical cross-sections of columns tested by [PERSON] et al. [15].