# Triaxial Compression Performance Research of Steel Slag Concrete on the Unified Strength Theory
## 3 Test Results and Discussion
### Failure Criterion  
The unified strength theory established by Professor \"[PERSON]\" [27] can be applied to concrete under the triaxial compression. Its expression is as follows:  
\[(1+b)(3+\beta)\frac{\rho}{\sqrt{2}}\cos\theta+\sqrt{3}(1-\beta)(1-b)\frac{\rho}{ \sqrt{2}}\sin\theta+f_{1}(\xi)=0\ \ \ \ 0^{0}\leq\theta\ \ \leq\theta_{b}, \tag{1a}\]  
\[(3+\beta-2b\beta)\frac{rho}{\sqrt{2}}\cos\theta+\sqrt{3}(1-\beta+2b)\frac{\rho} {\sqrt{2}}\sin\theta+f_{2}(\xi)=0\ \ \ \ \ \theta_{b}\leq\theta\ \ \leq 60^{0}, \tag{1b}\]  
The relationship between the cylindrical coordinates (\(\xi\),\(\rho\),\(\theta\)) and the primary stresses (\(\sigma_{1}\),\(\sigma_{2}\),\(\sigma_{3}\))are shown below:  
\[\xi=\frac{1}{\sqrt{3}}(\sigma_{1}+\sigma_{2}+\sigma_{3}).\ \ \ \ \rho=\frac{1}{\sqrt{3}}\sqrt{ \left(\sigma_{1}-\sigma_{2}\right)^{2}+\left(\sigma_{2}-\sigma_{3}\right)^{2 }+\left(\sigma_{3}-\sigma_{1}\right)^{2}}\cos\theta=\frac{2\sigma_{1}-\sigma _{3}-\sigma_{1}}{\sqrt{\phi}\rho}\]  
\(f_{1}(\xi)\),\(f_{2}(\xi)\) are the quadratic function of the hydrostatic stress axis, and the expressions be calculated from:  
\[f_{1}(\xi)=-2\sqrt{3}C+2[\beta(1+b)+A_{1}]\xi+\frac{2}{\sqrt{3}}B_{1}\xi^{2}, \tag{2a}\]  
\[f_{2}(\xi)=-2\sqrt{3}C+2[\beta(1+b)+A_{2}]\xi+\frac{2}{\sqrt{3}}B_{2}\xi^{2}, \tag{2b}\]  
The tension meridian and the compress meridian expression as follows:  
\[\rho_{t}=m_{0}+m_{1}\xi+m_{2}\xi^{2}\ \ \ \ \ \theta=0^{0}\ \, \tag{3a}\]  
Figure 6: Variation of peak strain with stress ratio under triaxial compression: (a) \(\sigma_{1}:\ \sigma_{3}=0.05\):1; (b) \(\sigma_{1}:\ \sigma_{3}=0.1\):1. \[\rho_{c}=n_{0}+n_{1}\xi+n_{2}\xi^{2}\ \ \ \ \ \ \theta=60^{0}\ \, \tag{3b}\]  
The parameters \(m_{0}\),\(m_{1}\),\(m_{2}\),\(n_{0}\),\(n_{1}\), and \(n_{2}\) in Equation (3a,b) can be determined by the test data and the characteristics of the destruction envelope surface. When the hydrostatic pressure increases gradually, the partial plane is approximately circular, and \(m_{2}=n_{2}\). In the triaxial experiments, the triaxial tensile stress point (\(\sigma_{1}=\sigma_{2}=\sigma_{3}=f_{ttt}>0\)) is difficult to test, we used the point (0,\(\xi_{0}\)) to replace it. In the cylindrical coordinates (\(\xi\),\(\rho\),\(\theta\)), the boundary conditions are uniaxial compressive stress (\(\sqrt{6}\alpha/3,\sqrt{6}/3,60^{\prime}\)), uniaxial tensile stress (\(\sqrt{3}\alpha/3,\sqrt{6}\alpha/3,0\)), biaxial compressive stress (\(-2\sqrt{3}\alpha/3,\sqrt{6}\alpha/3,0\)), a point in the tensile meridian (\(\xi\),\(\rho\),\(\theta\)), and a point in the compression meridian (0,\(\xi_{0}\),60\({}^{0}\)). By simplifying the boundary conditions, the coefficients of Equation (3a,b) can be expressed by Equation (4a-f). \[m_{2}=\frac{3\Big{\{}\alpha(\sqrt{3}\alpha-3\xi)(\sqrt{2}\xi-\rho)+\pi[3\sqrt {2}\alpha^{2}-2\sqrt{3}\alpha(2\sqrt{2}\xi+\rho)+3\xi(\sqrt{2}\xi+\rho)]\Big{\}} }{(2\pi+\alpha)(\alpha-\sqrt{3}\xi)[2\pi(\sqrt{3}\alpha-3\xi)+3\xi(\alpha- \sqrt{3}\xi)]\Big{\}}, \tag{4a}\] \[m_{1}=\frac{(2\pi-\alpha)}{\sqrt{3}}m_{2}+\frac{3\alpha}{\sqrt{2}(2\pi+\alpha)}- \frac{\sqrt{2}}{2},\] (4b) \[(3+\beta-2b\beta)\frac{rho}{\sqrt{2}}\cos\theta+\sqrt{3}(1-\beta+2b)\frac{\rho} {\sqrt{2}}\sin\theta+f_{2}(\xi)=0\ \ \ \ \ \theta_{b}\leq\theta\ \ \leq 60^{0}, \tag{1b}\]0 (4c) \[(3+\beta-2b\beta)\frac{rho}{\sqrt{2}}\cos\theta+\sqrt{3}(1-\beta+2b)\frac{\rho} {\sqrt{2}}\sin\theta+f_{2}(\xi)=0\ \ \ \ \ \theta_{b}\leq\theta\ \ \leq 60^{0}, \tag{1b}\]1 (4d) \[(3+\beta-2b\beta)\frac{rho}{\sqrt{2}}\cos\theta+\sqrt{3}(1-\beta+2b)\frac{\rho} {\sqrt{2}}\sin\theta+f_{2}(\xi)=0\ \ \ \ \ \theta_{b}\leq\theta\ \ \leq 60^{0}, \tag{1b}\]2 (4e) \[(3+\beta-2b\beta)\frac{rho}{\sqrt{2}}\cos\theta+\sqrt{3}(1-\beta+2b)\frac{\rho} {\sqrt{2}}\sin\theta+f_{2}(\xi)=0\ \ \ \ \ \theta_{b}\leq\theta\ \ \leq 60^{0}, \tag{1b}\]3  
where \(\alpha=f_{t}/f_{c}=0.1\), \(\alpha=f_{cc}/f_{c}\), \(f_{t}\) represents tensile strength, \(f_{c}\) represents compressive strength, and \(f_{cc}\) represents biaxial compressive strength.