# Nonlinear Finite Element Analysis Formulation for Shear in Reinforced Concrete Beams
## 9 Results
### Comparison with Other Shear Models
The material properties are: The compressive concrete strength is 28.1 MPa, and the yield strength of the longitudinal bar is 620 MPa. The shear span ratio (_a_/_d_) is 3.36, and the beam was subjected to a central-point loading. The beam failed at the shear force of 54.7 kN by diagonal tension cracking. Figure 11 shows the applied shear force versus the mid-span deflection curves obtained by using the four different shear models. The force-based model, including the proposed model and the constant shear flow model, terminated the analysis around the ultimate shear strength of the beam observed in the actual test. In contrast, the predefined shear strain models, including the parabolic shear strain model and the constant shear strain model, led to a shear strength more than twice as much as the actual shear strength; the analysis with the latter continued beyond the end point shown in the figure. The predefined shear strain models gave much stiffer response than the force-based models. For example, at the shear force of 54.7 kN, the constant shear strain model gave the mid-span deflection value of half as much as that obtained by the proposed model. To explain these discrepancies, the local behaviors were compared at the critical section (the effective depth away from the support) at the shear force of 54.7 kN, as shown in Figures 12-14. Figure 12 shows that the moment curvature values were analogous to the previous findings. The proposed model showed the largest curvature--more than twice as much as that of the constant shear strain model--and the smallest neutral surface depth. This suggests that the different shear model significantly affects the flexural behavior for the beam having a non-rectangular section. Figure 13 shows the shear strain distributions over the cross section at the same level of load. The proposed model described a parabolic shear distribution at the uncracked region (located top one-third portion), and dramatically increasing strains towards the bottom. Although a large discrepancy existed in the shear strain distributions, the contribution to the mid-span deflection of this seemed negligible. For example, the maximum shear strain estimated by the proposed model and the constant shear flow model were approximately 0.00135 and 0.00091, respectively, while the mid-span deflections  
Figure 10: Applied load versus mid-span deflection (OA-3 specimen). were 2.54 mm and 2.79 mm, respectively. Their difference in maximum shear strain is 67% and that in mid-span deflection is 9%. Figure 14 shows the shear stress distributions. The proposed model gave a similar shape of the shear stress distributions to that of the shear strain distribution; a parabolic shape in the uncracked region, and increasing stress towards the bottom. It is interesting to see the distributions obtained from the constant shear flow were similar to those from the proposed model. Together with the previous observations, this suggests that the constant shear flow model is a good approximation for the shear behavior of a reinforced concrete beam. In contrast, with the predefined shear strain models, a large portion of the shear stress residues are expected in the flange area. This may have reduced the shear stress in the web leading to the stiffer response. Figure 11: Placas and Regan Beam (T1 specimen). Figure 12: Longitudinal strain distribution over the cross section (at critical section, \(V=54.7\) kN).