# Nonlinear Finite Element Analysis Formulation for Shear in Reinforced Concrete Beams
## 9 Results
### Comparison with Other Shear Models  
1. Rectangular beam: The analysis results obtained for specimen OA-3 were compared with those by other shear models, such as (1) constant, or (2) parabolic shear strain model, and (3) constant shear flow model. The constant and parabolic shear strain models assumed the predefined shear strain functions over the cross-section first and then estimated the corresponding shear stresses. The advantage of the shear strain models was that the calculation procedure was simple, but those models can result in redundant constraints. On the other hand, the constant shear flow model, in which the shear force acting on the section considered is given first to obtain the shear strain of the section, can provide a more accurate shear response than the shear strain models. The constant and parabolic shear strain distributions were obtained by using the following relations for \(\psi(\zeta_{k})\) in Equation (6):  
\[\psi(\zeta_{k})=1 \tag{52a}\] \[\psi(\zeta_{k})=\frac{3}{2}\bigg{[}1-\bigg{(}\frac{\zeta_{k}}{h/2}\bigg{)}^{ 2}\bigg{]} \tag{52b}\]  
The constant shear flow model by using the secant stiffness method was discussed by [PERSON] and [PERSON] ([PERSON] and [PERSON], 2007). The shear stresses were determined by dividing a given section shear force by the height of each fiber (or layer). In the proposed model, the constant shear flow model was applied by replacing Equation (2) with the following relation:  
\[\varphi(\zeta)=\frac{V^{Trial}}{b_{w}(\zeta)} \tag{53}\]  
Figure 10 shows the comparison of the load-deflection relations obtained by using four different shear models. All four models showed similar initial stiffness up to the first cracking point, where the curves started to diverge. The parabolic and constant shear strain models, which had predefined shear strain distribution shape, showed a very similar response. In comparison with these models, the proposed model and constant shear flow models, which were a force-based method, showed a relatively softer behavior. This discrepancy between the constant/parabolic shear strain methods and the force-based methods comes from the fact that shear strains over the cross-section were assumed as the predefined shapes, such as a constant or parabolic, in the constant/parabolic shear strain methods, while they are not in the force-based method, because the predefined shear strains inevitably create redundant constraints in the deformability of the cross-section. In the force-based method, the shear stresses over the cross-section are calculated from equilibrium, and afterwards, the corresponding shear strains were determined, which does not cause such redundant constraints. The curve obtained by using the proposed model was closely related to the actual behavior with regards to both stiffness and strength. It is clear from the figure that the proposed model predicted the shear strength most  
accurately, provided that the termination point caused by the vertical equilibrium instability defined the shear failure point. 2. T-beam: The advantage that the proposed model possesses is that it describes the shear strain distributions over the cross section very accurately, and, therefore, the state of the section as well. This can best be illustrated by using T-beams, the section of which has larger shear strength than a rectangular section with the same depth and height. For this problem, the predefined shear strain methods, such as constant or parabolic shear strain method, would overestimate the shear strength by drawing more shear stress in the flange area, and keeping the web area stressed in lower intensity. A T-beam (beam no. : T2) was selected from the extensive test conducted by [PERSON] and [PERSON] [27]. The sectional dimensions are: The web width is 152 mm; the total depth is 305 mm; the depth to the center of the steel is 272 mm; \(A_{s}/b^{\prime}d=1.46\%\); the flange breadth is 76 mm; and the flange width is 610 mm.