## 3 New Shear Resistance Formula for the Y-Type Perfobond Rib Shear Connector
### Regression Analysis for the New Shear Resistance Formula
#### 3.2.5 New Shear Resistance Formula  
The two coefficients, \(\beta_{1}\) and \(\beta_{2}\), for the two terms in the new shear resistance formula (Equation (2)) represent the contributions related to rebar and rib, and it is found that the contributions are about 45% and 55% of a total of shear resistance, respectively. The optimal values of \(\beta_{1}\) and \(\beta_{2}\) are searched with the fixed contribution ratio between two terms as well as the index values of alpha selected through the repeated regression fit tests. The following new nominal shear resistance formula for a multiple Y-type perfobond rib shear connector, Equation (3), is proposed:  
\[Q_{n}=n^{0.67}\left(970d_{r}f_{yr}{}^{0.2}+4240\sqrt{f_{y}}\left(\frac{t}{10} \right)\left(\frac{w}{80}\right)^{0.95}\left(\frac{h}{100}\right)^{0.45}\right) f_{ck}{}^{0.3} \tag{3}\]  
where, the \(Q_{n}\) (N) represents the nominal shear resistance of multiple Y-type perfobond rib shear connectors, \(n\) is the individual number of ribs and transverse rebars, \(d_{r}\) (mm) is the diameter of a transverse rebar, \(f_{yr}\) (MPa) is the transverse rebar's yield strength, \(f_{y}\) (MPa) is the steel plate (Y-rib) yield strength, \(t\) (mm) is the steel plate (Y-rib) thickness, \(w\) (mm) is the Y-rib width, \(h\) (mm) is the Y-rib height, and \(f_{ck}\) (MPa) is the concrete strength. To calculate the nominal resistance, the nominal values need to be input for all design variables in Equation (3). In order to verify the accuracy of the new formula, Figure 5 compares the shear resistances from experimental tests with the estimation results (Equation (3)). The datasets are grouped into four parts, such as 2-rib (27 data: Figure 5a), 4-rib (54 data: Figure 5b), 6-rib (3 data: Figure 5c), and total (84 data: Figure 5d). The estimation resistance is calculated with the nominal values for all design variables except the concrete compressive strength, which is adopted from the cylinder tests for the test specimens. As summarized in Tables 16 and 17, the new shear resistance formula (Equation (3)) provides better estimations than the current formula (Equation (1)). The estimations range from 0.794 to 1.064, which is better than the current estimation ranging from 0.777 to 1.189 in Table 6. The COV is reduced from 0.109 (current formula) to 0.074 (proposed formula). Especially for 4-rib specimens, the estimation errors range from 0.895 to 1.064 with a mean of 1.000 (COV of 0.033) in the proposed formula, whereas the current formula ranges from 0.862 to 1.189 with a mean of 1.034 (COV of 0.067). All the ultimate shear resistances in Figure 5 and Table 17 are the resistances of one-pair sets of n-rib specimens. The proposed formula generally overestimates the resistances for the specimens with 2 ribs. However, the estimation errors are improved from 0.777-0.902 with mean of 0.844 to 0.794-0.914 with mean of 0.865. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**Group 1**} & \multicolumn{2}{c}{**Group 2**} & \multicolumn{2}{c}{**Ratio**} & \multicolumn{1}{c}{**Index (G = logF/logE)**} & \multicolumn{1}{c}{**Test No.**} \\ \hline \(n\)(A) & \(P_{u}\)**(B)** & \(n\)**(C)** & \(P_{u}\)**(D)** & \(n\) **(E = C/A)** & \(P_{u}\)**(F = D/B)** & \multicolumn{1}{c}{} & \\ \hline
2 & 1185.4 & 4 & 1805.7 & 2.000 & 1.523 & 0.607 & 2R/4R-FEA \\
4 & 1805.7 & 6 & 2352.0 & 1.500 & 1.303 & 0.652 & 4R/6R-FEA \\
6 & 2352.0 & 8 & 2884.4 & 1.333 & 1.226 & 0.709 & 6R/8R-FEA \\
8 & 2884.4 & 10 & 3411.0 & 1.250 & 1.183 & 0.751 & 8R/10R-FEA \\
2 & 1185.4 & 6 & 2352.0 & 3.000 & 1.984 & 0.624 & 2R/6R-FEA \\
2 & 1185.4 & 8 & 2884.4 & 4.000 & 2.433 & 0.641 & 2R/8R-FEA \\
2 & 1185.4 & 10 & 3411.0 & 5.000 & 2.878 & 0.657 & 2R/10R-FEA \\
4 & 1805.7 & 8 & 2884.4 & 2.000 & 1.597 & 0.676 & 4R/8R-FEA \\
4 & 1805.7 & 10 & 3411.0 & 2.500 & 1.889 & 0.694 & 4R/10R-FEA \\
6 & 2352.0 & 10 & 3411.0 & 1.667 & 1.450 & 0.728 & 6R/10R-FEA \\ Average & & & & & & 0.674 & \\ \hline \hline \end{tabular}
\end{table} 
Table 15: Index value for the variable of \(n\).