# Prediction of Deflection of Shear-Critical RC Beams Using Compatibility-Aided Truss Model
## 3 Shear Analytical Model for Calculating Shear Deflection
### Governing Equations  
As shown in Figure 2, it can be assumed that a shear critical element exists in the web of an RC beam, and the stress states can be expressed as shown in Figure 3. Based on the assumption that reinforcing bars transmit only axial force, the stress equilibrium equations can be derived by considering the equilibrium between the external and internal forces shown in Table 1. Furthermore, using the transformation formula for strain under the assumption that the strains of the concrete and rebar are the same, the strain compatibility equations of the shear critical RC element can be obtained, as shown in Table 1.  
In a beam without an axial force, the initial shear crack occurs in a 45-degree direction. As the load increases after the initial crack occurs, the principal compressive stress direction, which was the same as the initial crack direction, rotates in the \(l\)-direction with more reinforcement. In particular, since the difference between the steel ratio in the \(l\)- and \(t\)-directions is generally large for RC beams, the actual experimental results can be overestimated if the rotations of the principal stress and strain are not considered. The TATM calculates the stress and strain in the \(m\)-\(n\) coordinate system from the principal stress and strain as follows using coordinate transformations, as shown in Table 1. The constitutive laws of materials used in shear analytical model TATM are listed in Table 2.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Conditions** & **Equations** \\ \hline Stress equilibrium & \(\sigma_{l}=\sigma_{l}^{c}+\rho_{l}f_{l}=[\sigma_{m}^{c}\cos^{2}\alpha+\sigma_{ n}^{c}\sin^{2}\alpha+2\tau_{mn}^{c}\sin\alpha\cos\alpha]+\rho_{l}f_{l}\) \\  & \(\sigma_{t}=\sigma_{t}^{c}+\rho_{l}f_{l}=[\sigma_{m}^{c}\sin^{2}\alpha+\sigma_{ n}^{c}\cos^{2}\alpha-2\tau_{mn}^{c}\sin\alpha\cos\alpha]+\rho_{l}f_{l}\) \\ equations & \(\tau_{l}=\tau_{l}^{c}=[(-\sigma_{m}^{c}+\sigma_{n}^{c})\sin\alpha\cos\alpha+ \tau_{mn}^{c}(\cos^{2}\alpha-\sin^{2}\alpha)]\) \\  & \(\sigma_{m}^{c}=\sigma_{\sigma}^{2}\cos^{2}\beta+\sigma_{1}^{c}\sin^{2}\beta\) \\  & \(\sigma_{n}^{c}=\sigma_{\sigma}^{2}\sin^{2}\beta+\sigma_{1}^{c}\cos^{2}\beta\) \\  & \(\rho_{l}=\rho_{lt}-\rho_{lf}=\frac{\lambda_{m}}{\mathrm{bd}}-\frac{V(\alpha-d _{m})}{\gamma_{ll}\beta\cdot bd}\) \\ Strain compatibility & \(\varepsilon_{l}=\varepsilon_{m}\cos^{2}\alpha+\varepsilon_{n}\sin^{2}\alpha+ \gamma_{mn}\sin\alpha\cos\alpha\) \\   equations & \(\varepsilon_{t}=\varepsilon_{m}\sin^{2}\alpha+\varepsilon_{n}\cos^{2}\alpha- \gamma_{mn}\sin\alpha\cos\alpha\) \\  & \(\gamma_{lt}=2(-\varepsilon_{m}+\varepsilon_{n})\sin\alpha\cos\alpha+\gamma_{mn} (\cos^{2}\alpha-\sin^{2}\alpha)\) \\  & \(\varepsilon_{m}=\varepsilon_{2}\cos^{2}\beta+\varepsilon_{1}\sin^{2}\beta\) \\  & \(\varepsilon_{n}=\varepsilon_{2}\sin^{2}\beta+\varepsilon_{1}\cos^{2}\beta\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Equilibrium and compatibility equations used in shear analytical model [16].  
Figure 3: Stress states and coordinates of shear critical element.  
Figure 2: Shear critical element in a RC beam.