# Evaluation of Shear Effect on Deflection of RC Beams
## 4 Discussions
### Deflection  
Figure 7 compares the experimental and analytical results of the load-deflection relationships. It is noted that the analytical results were evaluated up to the yield value, while the experimental results expand to the whole measured range. The analytical results for the deflection in Figure 7 were obtained by substituting the effective moment of inertia obtained from Equation (4) into the following equation:  
\[\Delta_{ana.}=\frac{M_{a}}{24E_{c}I_{e,ana.}}(3l^{2}-4a_{\gamma}^{2}) \tag{5}\]  
The experimental deflections were grouped into two categories: (1) deflection measured directly from the LVDT at the mid-span of the specimen and (2) deflection obtained using the experimental curvature at the mid-span of the specimens. The experimental  
Figure 7: Comparison of experimental and analytical results: (**a**) B3-2.5a; (**b**) B3-2.5b; (**c**) B3-2.5c; (**d**) B6-2.5a; (**e**) B6-2.5b; (**f**) B6-2.5c; (**g**) B6-4.0a; (**h**) B6-4.0b; (**i**) B6-4.0c. curvature \(\varphi_{exp. }\) was obtained from the values of the strain gauges attached to the tension reinforcing bar and the extreme compression fiber of concrete as follows:  
\[\varphi_{exp.}=\frac{1}{R}=\frac{\epsilon_{c}+\epsilon_{s}}{d} \tag{6}\]  
where \(R\) is the radius of curvature, \(\epsilon_{c}\) is the concrete strain of the extreme compression fiber, and \(\epsilon_{s}\) is the strain of the tension reinforcing bar. The curvature obtained from Equation (6) can be substituted into the following equation to obtain the change in the effective moment of inertia for the entire loading history:  
\[I_{e,\varphi}=\frac{M_{a}}{E_{c}\varphi_{exp.}} \tag{7}\]  
The deflection can be obtained by substituting the effective moment of inertia obtained from Equation (7) into the following equation:  
\[\Delta_{exp,G}=\frac{M_{a}}{24E_{c}I_{e,\varphi}}(3l^{2}-4a_{v}^{2}) \tag{8}\]  
Substituting Equation (7) into Equation (8), the equation for \(\Delta_{exp,G}\) is only dependent on the curvature \(\varphi_{exp.}\). Since the experimental curvature was obtained using the value of strain gauges at mid-span of the specimens, Equation (8) reflects only the flexural deformation. Regarding the deflection at the first yielding of the tension reinforcement, Table 3 shows the experimental deflections \(\Delta_{y,L}\) and \(\Delta_{y,G}\) obtained from LVDT and Equation (8), and the analytical deflection \(\Delta_{y,ana. }\) calculated using ACI 318-19 formula. It is noted that \(\Delta_{y,G}\) is the deflection due to flexural deformation, and \(\Delta_{y,L}\) is the deflection due to combined bending and shear deformations. As shown in Table 3, \(\Delta_{y,L}/\Delta_{y,G}\) was approximately 1.5 for the B3-2.5 series and B6-2.5 series specimens with a small shear span-to-depth ratio, which means that the deflection due to shear deformation is approximately 50% of the deflection due to flexural deformation. As shown in Figure 7, the analytical result obtained from ACI equation had similar behavior to the load-deflection relationship obtained from the strain gauges attached to the pure bending region. In addition, from Table 3, it can be confirmed that the ratio of experimental and analytical results for deflection \(\Delta_{y,G}/\Delta_{y,ana. }\) are in agreement, with an average of 0.98 and a COV of 5.08% for all specimens. However, the analytical results obtained using ACI equation significantly differed from the load-deflection relationship measured directly from the LVDT. When the shear stress acting on the shear span of specimens increased, that is, the lower the shear span ratio or the higher the tension reinforcement ratio, the difference between the experimental and analytical results also increased. This is because the deflection measured directly by the LVDT is not only due to flexure, but also due to shear deformation. For example, as shown in Table 3, for the B6-4.0 series specimens, \(\Delta_{y,L}/\Delta_{y,ana.