## 4 Predicting Response of Reinforced Concrete Columns
### Circular Columns Tested by [PERSON] et al. 1989  
[PERSON] et al. [6; 7] tested a large number of circular columns under reversed-cyclic loading. Figure 6 shows a typical cross-section and the finite element model for one of these columns. The various colors in the mesh indicate varying concrete thicknesses and amounts of  
Figure 5: Discretization of circular column and components of hoop/spiral reinforcement. smeared reinforcement (\(\rho_{x}\) and \(\rho_{z}\)). The constant axial load was applied first and then imposed displacements were applied in the direction of shear. The reversed-cyclic loading consisted of 5 cycles at each displacement level. The columns were loaded in single curvature with lateral displacements applied at the top of the columns and were supported by a heavily reinforced concrete beam that spanned between supports at its ends (see Figure 6). It was important to include the deformations of this beam in the model to realistically predict the displacements at the top of the column. The predicted responses are compared with the experimental results in Figure 7. In addition, the shear span-to-depth ratio, \(a/D\), the axial load ratio, \(P/f_{c}^{\prime}A_{g}\), and the amount of transverse reinforcement are shown in the figure. Column 9 had an axial load level of 0.2, a shear span-to-depth ratio of 2.5, and No. 2 spirals at a pitch of 30 mm. It displayed a very ductile response achieving a displacement ductility ratio of about 8 with little shear distress. The finite element analysis also predicted a ductile response with significant yielding of the flexural reinforcement (see Figure 7a). The precited response agrees well with the experimental response for this ductile column. The longitudinal reinforcement was predicted to yield at a displacement of 7.7 mm. For this ductile column, yielding of the transverse reinforcement occurred at an applied shear of about 350 kN during testing compared with a predicted shear of 314 kN. Much of the damage to the column was concentrated in the hinge at the column base, as predicted by the finite element analysis. The predicted strength was 382 kN compared with the test result of 393 kN. Column 13 had a reduced axial load level of 0.1, \(a/D\) of 2.0, and had the same transverse reinforcement as Column 9. Both the experimental response and the predicted response indicated flexural yielding followed by shear distress and then shear failure (see Figure 7b). The column was only able to reach a ductility ratio of about 4 (moderate ductility) before failing in shear. Yielding of the longitudinal reinforcement was predicted by the finite element analysis to occur at a displacement of 5.8 mm. The predicted strength and ductility at maximum load compare well with the experimental results. The predicted strength was 418 kN compared with the test result of 440 kN. The nonlinear finite element analysis was not able to converge after the brittle shear failure had occurred and hence result in conservative predictions after failure occurred. Figure 6: Example of finite element model and typical cross-section of columns tested by [PERSON] et al. [67]. Column 17 had an axial load level of 0.1 and the same shear span and half the amount of transverse reinforcement as Column 9. This column behaved in a similar manner to Column 13 but had a reduced displacement ductility ratio of 3 (limited ductility) that was limited by shear failure (see Figure 7c). Yielding of the longitudinal reinforcement was  
Figure 7: Nonlinear finite element analysis predictions of shear-deflection responses and deflected shapes at failure (20\(\times\) magnification) for columns tested by [PERSON] et al. [6,7]: **(a)** No. 9 (ductile flexure failure); **(b)** No. 13 (shear failure with moderate ductility); **(c)** No. 17 (shear failure with limited ductility); **(d)** No. 16 (brittle shear failure). predicted to occur at a displacement of 7.9 mm. It is noted that the transverse reinforcement experienced high strains during testing that were also predicted by the finite element analysis.