## 4 Predicting Response of Reinforced Concrete Columns
### Circular Columns Tested by [PERSON] et al. 1993  
[PERSON] et al. [22] tested a series of circular columns under reversed-cyclic loading. The two columns for which predictions were made vary in their axial load level and grade of longitudinal reinforcement. Figure 8 shows a typical cross-section and the finite element model for one of these columns. The constant axial load was applied first. The reversed-cyclic loading consisted of 5 cycles at a load level corresponding to half the flexural strength followed by 3 cycles at each target displacement level. The columns were loaded in double curvature with lateral displacements applied on the top heavily reinforced end region, and the columns were supported by a heavily reinforced concrete end region anchored to the laboratory floor (see Figure 8). The hoops were anchored with 305 mm lap splices and hence to simulate this effect, strain hardening of the transverse reinforcement was neglected because of the uncertainty of developing stresses higher than the yield stress.  
The finite element response predictions for the [PERSON] et al. [22] columns are compared with the experimental results in Figure 9. The columns had a shear span-to-depth ratio of 2.0 and shear reinforcement consisting of No. 2 hoops at a spacing of 127 mm. The constant axial load ratios were 0.18, and 0.057 for Columns 3, and 5, respectively. The columns had different grades of longitudinal steel with Column 3 having Grade 40 (275 MPa) and Column 5 having Grade 60 (400 MPa).  
Column 3 experienced a limited displacement ductility of 2 with yielding of the longitudinal reinforcement followed by a shear failure (see Figure 9a). First yielding of the longitudinal reinforcement was predicted by the finite element model at a displacement of 10.0 mm and occurred during testing at a displacement of 9.4 mm. Yield strains in the transverse reinforcement were also reasonably well predicted and occurred at a displacement of 12.0 mm compared with a displacement between 11.9 mm (ductility 1.0) and 17.8 mm (ductility 1.5) during testing. At failure, the predicted crack pattern shows the same steeply inclined flexure-shear cracks and splitting cracks in the central core as observed in the test. The predicted strength was 720 kN compared with the experimental strength of 746 kN.  
Figure 9b shows the finite element predictions and experimental response of Column 5 that experienced a brittle shear failure with large shear cracks and fracture of hoops. This column had a higher moment capacity due to the higher yield strength of the longitudinal reinforcement and was predicted to have a shear failure. The sudden drop in strength and stiffness which occurred at a displacement of 18.5 mm was initiated by the formation of a wide diagonal shear crack from the top of the column. Yielding of the longitudinal reinforcement was predicted at a displacement of 15.0 mm which closely matched the 13.0 mm displacement at which yielding was observed during testing. Due to the very brittle shear failure, the finite element analysis failed to converge beyond the peak load. The predicted strength was 625 kN compared with the experimental strength of 599 kN.