## 2 Shear Analytical Model for RC Elements with High-Strength Concrete
### Constitutive Laws  
This study used the principal compressive stress-strain relationship, as shown in Figure 4a, proposed by [PERSON] and [PERSON] [19] for high-strength concrete as follows.  
\[\sigma_{2}^{c}=-f_{p}\left[\frac{n(\varepsilon_{2}/\varepsilon_{p})}{\left(n- 1\right)+\left(\varepsilon_{2}/\varepsilon_{p}\right)^{n\!k}}\right] \tag{14}\]  
\[n=0.8+f_{c}^{{}^{\prime}}/17 \tag{15}\]  
where \(f_{p}\) is the peak stress of concrete (=\(\
u f_{c}^{{}^{\prime}}\)), \(f_{c}^{{}^{\prime}}\) is the uniaxial strength of cylinder concrete (positive), \(\
u\) is the softening coefficient of cracked concrete, \(\varepsilon_{p}\) is the strain at peak stress (=\(\
u\varepsilon_{0}\)), \(\varepsilon_{0}\) is the strain at peak stress of uniaxial cylinder concrete (negative), \(n\) is the curve fitting factor, and \(k=1\) for the ascending branch and \(k=0.67+f_{c}^{{}^{\prime}}/62\) for the  
Figure 3: [PERSON]’s stress and strain circles: (**a**) [PERSON]’s stress circle; (**b**) [PERSON]’s strain circle.  
descending branch. The web of the RC beam undergoes compression softening because it is in a state of biaxial stress that is simultaneously subjected to compressive and tensile stresses. In this study, the softening coefficient was applied to both the stress and strain, as shown in Equation (14).  
The softening coefficient, the strain at the peak stress of the cylinder concrete, and the elastic modulus of the concrete used in this study are as follows [19]:  
\[v=\frac{1}{\left(0.8-0.34\frac{\varepsilon_{1}}{\varepsilon_{0}}\right)\left( 0.9+0.0045f_{c}^{{}^{\prime}}\right)}\leq 1.0 \tag{16}\]  
\[\varepsilon_{0}=\left(\frac{f_{c}^{{}^{\prime}}}{E_{c}}\right)\frac{n}{n-1} \tag{17}\]  
\[E_{c}=3,320\sqrt{f_{c}^{{}^{\prime}}(MPa)}+6,900 \tag{18}\]  
The concrete tensile stress-strain relationship proposed by [PERSON] and [PERSON] [12] are used as follows, as shown in Figure 4b.  
\[\sigma_{1}^{c}=E_{c}\varepsilon_{1}\ (\text{for}\ \varepsilon_{1}\leq \varepsilon_{cr}) \tag{19}\]  
\[\sigma_{1}^{c}=\frac{f_{cr}}{1+\sqrt{500\varepsilon_{1}}}\ (\text{for}\ \varepsilon_{1}> \varepsilon_{cr}) \tag{20}\]  
where \(\varepsilon_{cr}\) is the cracking strain of the concrete (=\(f_{cr}/E_{c}\)) and \(f_{cr}\) is the tensile strength of the concrete (=\(0.33\sqrt{f_{c}^{{}^{\prime}}(MPa)}\)).  
The truss model based on the fixed angle has the advantage of being able to directly consider the shear transfer owing to the aggregate interlock. Unlike normal-strength concrete, high-strength concrete has smoother crack surfaces and penetrates the aggregate rather than bypassing it. The [PERSON]'s stress and strain circles shown in Figure 3 can be used to derive the following equation.  
\[\sin 2\beta=\frac{2\tau_{mn}^{c}}{\sigma_{1}^{c}-\sigma_{2}^{c}}=\frac{\gamma_ {mn}}{\varepsilon_{1}-\varepsilon_{2}} \tag{21}\]  
Based on Equation (21), \(\tau_{mn}^{c}-\gamma_{mn}\) relationship is summarized as follows.  
\[\tau_{mn}^{c}=\frac{\gamma_{mn}}{2(\varepsilon_{1}-\varepsilon_{2})}(\sigma_ {1}^{c}-\sigma_{2}^{c}) \tag{22}\]  
Figure 4: Constitutive relationships of used materials: (**a**) concrete in compression; (**b**) concrete in tension; (**c**) rebar in tension.  
For the stress-strain relationship of reinforcing bars, the following equation proposed by [PERSON] and [PERSON] [20] was used, but was limited to not exceed the yield strength of steel bars, as shown in Figure 4c.  
\[f_{s}=E_{s}\varepsilon_{s}\left(\text{for }\varepsilon_{s}\leq\varepsilon_{sy}^{{}^{ \prime}}\right) \tag{23}\]  
\[f_{s}=f_{y}\big{[}(0.91-2B)+(0.02+0.25B)\varepsilon_{s}/\varepsilon_{sy}\big{]} \left(\text{for }\varepsilon_{s}>\varepsilon_{sy}^{{}^{\prime}}\right) \tag{24}\]  
where \(\varepsilon_{sy}^{{}^{\prime}}=\varepsilon_{sy}(0.93-2B)\), \(B=(1/\rho_{s})(f_{\alpha}/f_{y})^{1.5}\).