## 6 Experimental Results and Discussion
### Effective Stiffness  
The effective stiffness of a structural member is calculated as the slope of the secant line to the point of first yield of the main longitudinal reinforcement. For a cantilever member, the normalized value of the overall effective stiffness can be computed from the load-deformation experimental data as follows:  
\[\frac{\text{EI}_{\text{total,eff}}}{\text{E}_{\text{c}}\text{I}_{\text{g}}}= \frac{\text{L}_{\text{s}}{}^{3}}{3\text{E}_{\text{c}}\text{I}_{\text{g}}} \left(\frac{\text{F}_{\text{y}0}}{\Delta_{\text{y}0}}\right) \tag{3}\]  
where \(\text{I}_{\text{g}}\) is the gross inertia of the beam's cross-section (the effect of the longitudinal reinforcement on the transformed section is not accounted for), \(\text{E}_{\text{c}}\) is the young modulus of concrete, \(\text{L}_{\text{s}}\) is the shear span, and \(\text{F}_{\text{y}0}\) and \(\Delta_{\text{y}0}\) are the load and displacement values associated with first yield of the main longitudinal reinforcement, respectively. Analogously, the normalized values of the effective flexural and shear stiffnesses can be computed from measured flexural and shear displacement components at first yield (\(\text{A}_{\text{f,y}0}\) and \(\Delta_{\text{s,y}0}\)) as follows:  
\[\frac{\text{EI}_{\text{flex,eff}}}{\text{E}_{\text{c}}\text{I}_{\text{g}}}= \frac{\text{L}_{\text{s}}{}^{3}}{3\text{E}_{\text{c}}\text{I}_{\text{g}}} \left(\frac{\text{F}_{\text{y}0}}{\Delta_{\text{f,y}0}}\right) \tag{4}\]  
\[\frac{\text{G}_{\text{c}}\text{A}_{\text{eff}}}{\text{E}_{\text{c}}\text{A}_ {\text{w}}}=\frac{\frac{\text{F}}{\text{Y}_{\text{s}}}}{\text{E}_{\text{c}} \text{A}_{\text{w}}}=\frac{\text{L}_{\text{s}}}{\text{E}_{\text{c}}\text{A}_ {\text{w}}}\left(\frac{\text{F}_{\text{y}0}}{\Delta_{\text{s,y}0}}\right) \tag{5}\]  
where \(\text{A}_{\text{w}}\) is the gross web area and \(\text{G}_{\text{c}}\) is the shear modulus of concrete, with \(\text{G}_{\text{c}}\approx 0.4\) E\({}_{\text{c}}\),  
Normalized values of the total, flexural, and shear secant stiffnesses at first yield for specimens S1A, S1B, and S5 are summarized in Table 10. Compared to the prescriptive effective rigidity values for RC MRF beams in Table 10-5 of ASCE 41-13, where \(\text{EI}_{\text{flex,eff}}=0.30\text{E}_{\text{c}}\text{I}_{\text{g}}\) and \(\text{GA}_{\text{eff}}=0.40\text{E}_{\text{c}}\text{A}_{\text{w}}\), the elastic experimental responses of all three specimens are found to be stiffer in flexure (i.e., >0.30) but much more flexible in shear (i.e., <0.40). Specifically, the experimentally derived flexural effective stiffness values in downward and upward directions are 20% and 17% higher than the ASCE 41 recommended value for S1A, respectively, 13% and 60% higher for S1B, and 17% and 23% higher for S5. On the other hand, the experimental shear effective stiffness values are between 63% and 96% lower than the ASCE 41 value.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Total Stiffness**} & \multicolumn{2}{c}{**Flexural Stiffness**} & \multicolumn{2}{c}{**Shear Stiffness**} \\ \cline{2-7} Specim & \multicolumn{2}{c}{\(\text{EI}_{\text{total,eff}}/\text{E}_{\text{c}}\text{I}_{\text{g}}\)} & \multicolumn{2}{c}{\(\text{EI}_{\text{flex,eff}}/\text{E}_{\text{c}}\text{I}_{\text{g}}\)} & \multicolumn{2}{c}{\(\text{GA}_{\text{eff}}/\text{E}_{\text{c}}\text{A}_{\text{w}}\)} \\ \cline{2-7}  & **Down (+)** & **Up ( –)** & **Down (+)** & **Up ( –)** & **Down (+)** & **Up ( –)** \\ \hline S1A & 0.14 & 0.14 & 0.36 & 0.35 & 0.018 & 0.018 \\ S1B & 0.15 & 0.18 & 0.34 & 0.48 & 0.018 & 0.026 \\ S5 & 0.18 & 0.2 & 0.35 & 0.37 & 0.07 & 0.15 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Experimental effective stiffness values.