SB1 Rectangularity Study

4.3.1.1 Investigated Specimens

The first column rectangularity investigation was a hypothetical extension of the testing program by Adetifa and Polak (2005). The hypothetical program included slab SB1, tested by Adetifa and Polak (2005), and five hypothetical specimens supported on increasingly rectangular columns with constant critical perimeter lengths according to ACI 318M-14. The column dimensions, column aspect ratios and ratio of the minimum column dimension to the effective slab depth (cmin/d) are summarized in Table 4-2. Other than the column dimensions and longitudinal column reinforcement, the slabs were identical to SB1.

Table 4-2: Summary of Column Sizes Considered in SB1 Rectangularity Study Slab cmin (mm) cmax (mm) β (cmax/cmin) cmin/d

SB1 (Control) 150 150 1.0 1.7 C1 125 175 1.4 1.4 C2 100 200 2.0 1.1 C3 75 225 3.0 0.8 C4 50 250 5.0 0.6 C5 25 275 11.0 0.3

4.3.1.2 Finite Element Model

The ABAQUS model used to analyze the six slabs was based on the calibrated finite element model by Genikomsou (2015) with one minor change. To reduce computational time in some of the analyses, Genikomsou used a bilinear compressive stress-strain relationship for the concrete instead of the complete Hognestad Parabola, since the punching capacity of slabs is known to be primarily related to the tensile strength of concrete. In this thesis, the full Hognestad parabola introduced in Section 4.2 is used, which resulted in a slightly higher capacity and deflection at failure compared to the results from the Genikomsou model. A comparison of the predicted load-deflection response using the bilinear stress-strain relationship used by Genikomsou (2015) and the Hognestad parabola is provided in Figure 4-5. As expected, the predicted response for both models correlated well with the experimental results, though the initial stiffness predicted by the FEM is much higher than that observed experimentally. The discrepancy in predicted stiffness likely occurs because the FEM does not account for cracking due to temperature, shrinkage and specimen transportation in the laboratory.

Figure 4-5: Comparison of FEA Results of SB1 With Different Concrete Stress-Strain Curves

4.3.1.3 Finite Element Analysis Results

The predicted load-displacement response for the six slabs is provided in Figure 4-6. Increasing the column rectangularity was found to have a minimal impact on the predicted punching capacity and stiffness of the slab-column connection. As will be discussed in Section 5.1.1, this was expected since the cmin/d ratio decreased as the column rectangularity was increased. According to Model Code 2010, the impact of column rectangularity is very small when the cmin/d ratio is small.

0 50 100 150 200 250 300 0 5 10 15 20 L o a d ( k N ) Displacement (mm)

Figure 4-6: Load-Displacement Response Predicted by ABAQUS

A comparison of the predicted crack patterns for slabs SB1 (β=1), C2 (β=2), C4 (β=5) and C5 (β=11), visualized through the contours of maximum principal plastic strain, are shown in Figure 4-7. As expected, the crack pattern for slab SB1, which was loaded through a square column, is

approximately uniform in both orthogonal directions. The crack patterns for the other analyzed slabs, except for slab C5, which has a column aspect ratio of 11, are also approximately uniform. For slab C5, the cracks perpendicular to the long side of the column are slightly longer than those

perpendicular to the short side of the column. The lack of non-uniformity in the crack patterns supports the conclusion that column rectangularity does not have a large effect on punching shear behaviour when the cmin/d ratio is small.

0 50 100 150 200 250 0 5 10 15 20 L o a d (k N ) Displacement (mm) Control C1 - 175x125mm C2 - 200x100mm C3 - 225x75mm C4 - 250x50mm C5 - 275x25mm

Figure 4-7: Tension Surface Crack Patterns Predicted by ABAQUS

A comparison of the punching capacity predicted by the FEA, ACI 318M-14, Eurocode 2 (2004) (EC2) and Model Code 2010 (MC 2010) LoA I is provided in Table 4-3. The simplest level of approximation was used for MC 2010 since the primary objective of this study was to confirm that ABAQUS was properly predicting the impact of column rectangularity for the modelled slabs. As such, the overall trend predicted by MC 2010 is more important than accurate punching capacity estimates. Comparing the predicted capacities, it was found that ACI 318M-14 predicts a much more significant impact of column rectangularity compared to the other two design codes and the finite

SB1 – β = 1 C2 – β = 2

element results. Both Model Code 2010 and EC2 (2004) predict a constant capacity as the column rectangularity was increased since the critical perimeter length was constant for the investigated slab- column connections. Only for slab-column connection C5, which had a rectangularity of 11, did Model Code 2010 predict a slightly lower capacity (112kN vs 114kN). Model Code 2010 predicted a slightly lower capacity for column C5 because the maximum column dimension of 275mm is 5mm greater than 3 times the average effective slab depth (90mm), and so the effective critical perimeter length is lower than the total critical perimeter length for this connection only. It should be noted that a constant flexural reinforcing ratio of 1.2%, which is the average reinforcing ratio of SB1, was assumed in the EC2 (2004) calculations, instead of calculating the reinforcing ratio in each direction over a slab width equal to the column width plus three times the effective slab depth on each side of the column. This assumption simplified the EC2 calculations and removed the influence of slight changes in the reinforcing ratio on the final results. The use of a constant reinforcing ratio resulted in an approximately 4% increase in the predicted capacities for both the control and C4 specimens and a less than 1% increase in capacity for the remaining four specimens.

Table 4-3: Comparison of Punching Capacity Predicted by Codes and FEA

Predicted Punching Shear Capacity (kN)

Slab Column Aspect _{Ratio (β)} FEA ACI 318M-14 EC2 (2004) MC 2010 (LoA I) Control 1 230.8 189.1 207.8 114.1 C1 1.4 233.7 189.1 207.8 114.1 C2 2 233.6 189.1 207.8 114.1 C3 3 229.2 162.4 207.8 114.1 C4 5 229.7 136.4 207.8 114.1 C5 11 219.2 115.1 207.8 112.8

A comparison of the normalized nominal shear stress capacity per unit length along the ACI 318M- 14 critical perimeter, 𝑣_{𝑛𝑜𝑟𝑚}, calculated according to equation 4.3 is provided in Figure 4-8.

𝑣_{𝑛𝑜𝑟𝑚}= 𝑉

where 𝑉 is the punching shear capacity predicted by the FEA or the design code (N), 𝑏_{𝑜,𝐴𝐶𝐼} is the length of the critical perimeter according to ACI 318M-14 (2 × (𝑐_𝑚𝑎𝑥+ 𝑑) + 2 × (𝑐_𝑚𝑖𝑛+ 𝑑), mm), 𝑑 is the average effective slab depth (mm) and 𝑓_𝑐′ _{is the concrete compressive strength (MPa)}

Figure 4-8: Comparison of Nominal Shear Capacity Along the ACI 318 Critical Perimeter Predicted by the FEA and Various Design Codes

Again, the ACI 318 provisions predicted a much more substantial impact of column rectangularity compared to the other design codes and the FEA. As previously discussed, MC 2010 LoA I is very conservative (Muttoni & Fernández Ruiz, 2012) so the nominal stresses predicted using this level of approximation are very low, but the minimal change in nominal stress as the column rectangularity is increased is the important factor, and not the actual magnitude. The EC2 (2004) provisions predict a nominal shear stress close to that predicted by the FEA which validates the predicted behaviour.