## 2 Shear Analytical Model for RC Elements with High-Strength Concrete  
### Equilibrium Conditions  
Figure 1a shows a typical simply supported RC beam subjected to a concentrated load. It is assumed that the RC beam resists the external shear force with a truss mechanism consisting of a concrete compression strut and longitudinal and transverse steel bars. The stress state of a shear critical RC element located in the shear critical section separated by \(d_{es}\) from the maximum moment position can be expressed as shown in Figure 1b. In this study, the effective shear depth, \(d_{es}\), is taken as 0.9d [17]. Figure 2 shows the stress state and coordinates of the RC element. The \(l\)- and \(t\)-directions are the coordinate systems on which external loads are applied, and the \(m\)- and \(n\)-directions are the coordinate systems related to the crack direction. Furthermore, the 1- and 2-directions represent the principal stress coordinate system. As shown in Figure 2, the equilibrium of the RC element can be expressed as follows, based on the assumption that the reinforcing bars transmit only axial stress.  
\[\sigma_{l}=\sigma_{l}^{c}+\rho_{l}f_{l} \tag{1}\]  
\[\sigma_{t}=\sigma_{t}^{c}+\rho_{t}f_{t} \tag{2}\]  
\[\tau_{lt}=\tau_{lt}^{c} \tag{3}\]where \(\sigma_{l}\) and \(\sigma_{t}\) are the applied stresses acting in the \(l\)- and \(t\)-directions, respectively, \(\tau_{lt}\) is the applied shear stress in the \(l\)-\(t\) coordinate system, \(\sigma_{l}^{c}\) and \(\sigma_{t}^{c}\) are the concrete resistances in the \(l\)- and \(t\)-directions, respectively, \(\rho_{l}\) and \(\rho_{t}\) are the steel ratios in the \(l\)- and \(t\)-axes, respectively, \(f_{l}\) and \(f_{t}\) are the steel stresses in the \(l\)- and \(t\)-axes, respectively, \(\tau_{lt}^{c}\) is the shear resistance of concrete in the \(l\)-\(t\) coordinate system.  
The resistance of concrete can be expressed as follows:  
\[\sigma_{l}^{c}=\sigma_{m}^{c}\cos^{2}\alpha+\sigma_{n}^{c}\sin^{2}\alpha+2\tau _{nm}^{c}\sin\alpha\cos\alpha \tag{4}\]  
\[\sigma_{t}^{c}=\sigma_{m}^{c}\sin^{2}\alpha+\sigma_{n}^{c}\cos^{2}\alpha-2\tau _{nm}^{c}\sin\alpha\cos\alpha \tag{5}\]  
\[\tau_{lt}^{c}=(-\sigma_{m}^{c}+\sigma_{n}^{c})\sin\alpha\cos\alpha+\tau_{nm}^ {c}(\cos^{2}\alpha-\sin^{2}\alpha) \tag{6}\]  
where \(\sigma_{m}^{c}\) and \(\sigma_{n}^{c}\) are the normal stresses of concrete in the \(m\)- and \(n\)-axes, respectively, \(\tau_{nm}^{c}\) is the shear stress developed along concrete diagonal cracks, \(\alpha\) is the initial crack angle due to applied external load.  
After the initial crack is caused by an external load, the directions of the principal stresses changes as the load increases, depending on the state of the materials. As shown in Figure 3a, the proposed model transforms the stresses in the principal stress directions to the stresses in the \(m\)- and \(n\)-directions using the transformation angle, \(\beta\), as follows.  
\[\sigma_{m}^{c}=\sigma_{2}^{c}\cos^{2}\beta+\sigma_{1}^{c}\sin^{2}\beta \tag{7}\]  
\[\sigma_{n}^{c}=\sigma_{2}^{c}\sin^{2}\beta+\sigma_{1}^{c}\cos^{2}\beta \tag{8}\]  
Figure 1: Shear critical element and coordinates of a RC beam: (**a**) RC beam; (**b**) shear critical RC element.  
Figure 2: Stress state of shear critical element.  
where \(\sigma_{1}^{c}\) and \(\sigma_{2}^{c}\) are the principal tensile and compressive stresses of concrete, respectively, and \(\beta\) is the difference in the angles in the crack and principal compression directions.