# Experimental Study on Strength of Polypropylene Fiber Reinforced Cemented Silt Soil
## 3 Analysis of Unconfined Compressive Strength and Unconsolidated Underained Test Result
### Stress-strain Curve Fitting of Fiber Cement Soil with Different Curve Equations
#### 3.4.2 Fitting Curve of Fiber Cement Soil under Confining Pressure  
In 1963, through the stress-strain curve of the triaxial test, [PERSON] found that the hyperbolic curve can be used as the fitting curve of the stress-strain relationship of soil. The Kondner hyperbolic curve is as follows:  
\[\sigma_{1}-\sigma_{3}=\frac{\varepsilon}{\mathrm{a}\varepsilon+\mathrm{b}} \tag{6}\]  
where a and b are constants, \(\sigma_{1}\) is major principal stress, \(\sigma_{3}\) is minor principal stress and \(\varepsilon\) is axial strain.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & **Fiber Content** & **Fiber Content** & **Fiber Content** & **Fiber Content** \\  & **0\%** & **0.1\%** & **0.2\%** & **0.4\%** \\ \hline a\({}_{3}\) & 1 & 1 & 1 & 1 \\ b\({}_{1}\) & 2.5460 & 2.3190 & 2.2811 & 1.9494 \\ Adjusted R square & 0.7517 & 0.9602 & 0.9627 & 0.9714 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Coefficient table of constitutive equation of soil sample under uniaxial compression.  
Figure 8: The measured value and curve fitting diagram of cement soil with different fiber content under uniaxial compression. (**a**) fiber content 0%; (**b**) fiber content 0.1%; (**c**) fiber content 0.2%; (**d**) fiber content 0.4%.  
In this paper, the stress-strain curve under confining pressure was fitted by using the Kondner hyperbolic curve. The triaxial test results were adjusted according to the Kondner hyperbolic curve as follows:  
\[\frac{\varepsilon}{\sigma_{1}-\sigma_{3}}=\mathrm{a}\varepsilon+\mathrm{b} \tag{7}\]  
That is, \(\varepsilon/(\sigma_{1}-\sigma_{3})\) and \(\varepsilon\) were linear. In the triaxial test, \(\mathrm{d}\sigma_{2}=\mathrm{d}\sigma_{3}=0\); therefore the tangent modulus was  
\[\mathrm{E_{t}}=\frac{\mathrm{d}(\sigma_{1}-\sigma_{3})}{\mathrm{d}\varepsilon }=\frac{\mathrm{b}}{\left(\mathrm{a}\varepsilon+\mathrm{b}\right)^{2}} \tag{8}\]  
If \(\varepsilon=0\), \(\mathrm{E_{t}}=\mathrm{E_{0}}\), then \(\mathrm{E_{0}}=1/\mathrm{b}\), \(\mathrm{E_{0}}\) was the initial tangent modulus.  
In Figure 5b-d were the stress-strain curves under confining pressure (the confining pressures were 100, 200, and 300 kPa) and had the following characteristics:  
1. The curve was hyperbolic, and the slope increased with the increase in the confining pressure;
2. In the hardening stage before reaching the peak, the stress velocity gradually slowed down after reaching the peak.  
The fitting equation under confining pressure can be established by a hyperbolic function. Bases on the Kondner hyperbolic curve, a new expression was established, where a, b, and c were constants.  
\[\sigma_{1}-\sigma_{3}=\frac{\varepsilon}{\mathrm{a}\varepsilon^{2}+\mathrm{b} \varepsilon+\mathrm{c}}=\frac{1}{\mathrm{a}\varepsilon+\mathrm{b}+\mathrm{c}/\varepsilon} \tag{9}\]  
Equation (9) indicates that the curve was a parabola. Through curve fitting, the specific parameters were shown in Table 8, curve fitting of fiber cement soil under confining pressure was shown in Figure 9. The value of the adjusted R square was closest to the value of 1, indicating that the fitting effect of the fitting curve was better.