C) Numerical vs. Experimental Column Response

Now that the shear strength of each column has been calculated, the numerical response history based on the implemented hysteretic law can be obtained. Some additional parameters must first be calculated from the shear strength equation components. The first thing that needs to be calculated is the cracking shear force, Vcr. This was given by equation IV.18 as:

e

cr

f c A

V = 3 . 5 '

With the calculation of the cracking shear force, the remaining values for the hysteretic law can be determined. These are the pinching parameter in the y direction, py, the initial shear stiffness, GAe , the pinching parameter in the x direction, px, which is a function of the shear ductility, µshr ,and the damage factors, d1 and d2. These parameters were developed in chapter IV and are detailed in equations IV.30 to IV.35. The resulting values for columns R-3 and R-5 are shown in table V-6.

Table V-6: Hysteretic Input Values – Column R-3 and R-5

R-3 R-5

With the parameters for the shear hysteretic law developed, the numerical analysis can be performed on both columns for comparison with the experimental data. The loading procedure used in the numerical analysis will be a little different than that used in the experimental procedure because of the limitations of the FEAP proportional loading command. The program does not allow a change from displacement control to force control in the same analysis. Further, because the degradation of shear resistance depends on the damage parameters and the pinching parameters, and these depend on the maximum previous shear strain, there is no degradation for repeated loading cycles with the same ductility value. So, only one cycle will be performed at each ductility level. The loading history used in the numerical analysis is shown in figure V-7.

Figure V-7: Loading Procedure for Numerical Analysis

The results of the Numerical Analysis for columns R-3 and R-5 are shown in figure V-8 and V-9 respectively. It can be seen from these figures that the numerical response from the hysteretic law implemented matches the experimental results well in capturing the initial stiffness and the maximum shear resistance of the columns, however there are large discrepancies between the experimental and numerical results once the peak lateral force has been exceeded. Namely, the hysteretic law implemented is not capable of capturing the softening effect (from the opening of large shear diagonal cracks) apparent in the experimental data. Also, the combination of increased pinching and reduced shear resistance once the peak lateral force has been exceeded is missed by the numerical analysis. While the overall shape of the response is satisfactory, some important response aspects are neglected. The question, however, arises as to whether the law can be used accurately for the Pushover Analysis.

µ = 1

∆ µ = 1.5 µ = 2

Figure V-8: Numerical and Experimental Results – Column R-3

Figure V-9: Numerical and Experimental Results – Column R-5 -150

-100 -50 0 50 100 150 200

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Top Displacement (in)

Lateral Force (kips)

Experimental Numerical

-200 -150 -100 -50 0 50 100 150 200

-1.5 -1 -0.5 0 0.5 1 1.5

Top Displacement (in)

Lateral Force (kips)

Experimental Numerical

To determine how well the law implemented applies to members subject to monotonically increased loads such as those defined by the Pushover Analysis, this type of loading will be applied to both column R-3 and R-5 for comparison to the experimental result envelopes. The resulting plots are shown in figure V-10 and V-11 for columns R-3 and R-5 respectively.

Figure V-10: Monotonically Increasing Loads and Experimental Results – Column R-3

Figure V-11: Monotonically Increasing Loads and Experimental Results – Column R-5 -150

-100 -50 0 50 100 150 200

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Top Displacement (in)

Lateral Force (kips)

Experimental Numerical

-200 -150 -100 -50 0 50 100 150 200

-1.5 -1 -0.5 0 0.5 1 1.5 2

Top Displacement (in)

Lateral Force (kips)

Experimental Numerical

From figures V-10 and V-11, it can be seen that while the hysteretic law implemented misses some important information in comparison with the experimental results for cyclic loads, it is accurate enough for the Pushover Analysis as it captures the initial stiffness of the structure and the maximum resisting shear that was calculated above with the underestimation shown in table V-5. However, one thing to note is that the displacement controlled loading shown in the figures could have increased forever without the section ever losing its shear resistance. This is a definite draw back of the hysteretic law, however since the shear ductility was calculated above, the engineer has a relation to obtain the maximum displacement the structure would be able to undergo.

Mentioned in chapter IV was the fact that the shear deformations are uncoupled with the axial – flexural deformations on the section level, but are coupled on the element level. Since the moment and shear responses are linked through equilibrium on the element level, the maximum strength available in the section will depend on the minimum of the individual components, i.e. axial – flexural or shear. This is illustrated by plotting the Moment – Curvature response and the Shear Force – Shear Strain response for column R-3 at different sections. For the numerical analysis, both columns were divided into five Guass – Lobatto integration points as shown in figure V-12. Using the cyclic load analysis results for column R-3, it is seen in figure V-14 that the Moment – Curvature response of the sections one and two vary slightly corresponding to the linear moment distribution along the element. From figure V-13 it is seen that the Shear Force – Shear Strain response of sections one and two are equal corresponding to constant shear along the element. Further it is seen that the shear response of the element controls the total response as a clear yielding point is recognized while the Moment – Curvature response remains basically elastic. This corresponds to the column failing in shear which agrees with the experimental data. So, while the axial – flexural and shear responses are uncoupled at the section level, coupling (through equilibrium) at the element level ensures that the element response will be controlled by the minimum of the individual components. This corresponds to reality in that the maximum strength of a member will depend on its weakest component.

Figure V-12: Section Locations of Columns for Numerical Analysis

Figure V-13: Shear Force vs. Shear Strain Column R-3, Sections 1 and 2

Figure V-14: Moment vs. Curvature Column R-3, Sections 1 and 2 -150