## 4 Analysis of Test Results
### Parameter Determination Based on the Duncan-Chang Model  
The stress-strain curves of the fiber-reinforced soil the triaxial tests possessed strain-hardening characteristics, and the relationship of \((\sigma_{1}-\sigma_{3})\) with \(\varepsilon\) could be fitted by a hyperbolic curve. The stress-strain relationship of the fiber-reinforced soil could be mathematically expressed by the Duncan-Chang model, a nonlinear elastic model [34]. The triaxial test results show that the relationship of \((\sigma_{1}-\sigma_{3})\) with \(\varepsilon\) in fiber-reinforced soil could be expressed as  
\[\sigma_{1}-\sigma_{3}=\frac{\varepsilon}{a+be} \tag{2}\]  
An approximately linear relationship between \(\sigma_{1}-\sigma_{3}\) and \(\varepsilon\) can be obtained, as shown in Figure 10:  
\[\frac{\varepsilon}{\sigma_{1}-\sigma_{3}}=a+be \tag{3}\]  
where \((\sigma_{1}-\sigma_{3})\) is the stress, \(\varepsilon\) is the strain, and \(a\) and \(b\) are the linear intercept and slope. Figure 10: Diagram of the relationship of \(\varepsilon/(\sigma_{1}-\sigma_{3})\) and \(\varepsilon\)  
Figure 9: Relationship between the failure stress and confining pressure of the fiber-reinforced soil. For triaxial tests, \(\mathrm{d}\sigma_{2}=\mathrm{d}\sigma_{3}=0\), and the tangential modulus is  
\[E_{t}=\frac{\mathrm{d}(\sigma_{1}-\sigma_{3})}{\mathrm{d}\varepsilon}=\frac{a}{ \left(a+b\varepsilon\right)^{2}} \tag{4}\]  
At the starting point of the test, \(\varepsilon=0\), \(E_{t}=E_{i}\), and the initial tangent modulus \(E_{i}\) is  
\[E_{i}=\frac{1}{a} \tag{5}\]  
where \(a\) represents the reciprocal of \(E_{i}\) during the test. In Equation (2), if \(\varepsilon\!\rightarrow\!\infty\), then  
\[\left(\sigma_{1}-\sigma_{3}\right)_{\mathrm{ult}}=\frac{1}{b} \tag{6}\]  
where \(b\) represents the reciprocal of the ultimate deviator stress (\(\sigma_{1}\!-\!\sigma_{3}\))\({}_{\mathrm{ult}}\) corresponding to the asymptote of the hyperbola. From the triaxial test data, coefficients \(a\) and \(b\) were found by curve fitting, and then the values of \(E_{i}\) and (\(\sigma_{1}\!-\!\sigma_{3}\))\({}_{\mathrm{ult}}\), the Duncan-Chang model parameters, were determined from Equations (5) and (6) (Table 5). Comparing the soil reinforced with the optimal fiber content of 0.2% to pure fill and fill with a concentration of 0.3%, the \(E_{i}\) was 17% and 8% higher, and (\(\sigma_{1}\)-\(\sigma_{3}\))ult was 31% and 13% higher, respectively, which indicates that fiber concentration has a greater effect on the strength of soil than on its stiffness. Effect of fiber length  
Figures 13 and 14 show that as the fiber length increased, both \(E_{i}\) and (\(\sigma_{1}\)-\(\sigma_{3}\))ult, the fiber-reinforced soil model parameters, first increased and then decreased, approximating a Gaussian function, with a peak value point near the length of 12 mm. Figure 11: Relationship between \(E_{i}\) and the fiber content of the fiber-reinforced soil. Figure 12: Relationship between \((\sigma_{1}-\sigma_{3})_{\text{ult}}\) and the fiber content of the fiber-reinforced soil. Figure 10:0 Relationship between \(E_{i}\) and the fiber length of the fiber-reinforced soil. Comparing the soil reinforced with the optimal fiber length of 12 mm to the pure fill and fill with a fiber length of 18 mm, the \(E_{i}\) was 14% and 7% higher, and \((\sigma_{1}-\sigma_{3})_{\text{ult}}\) was 30% and 12% higher, respectively, which indicates that fiber length has a greater effect on the strength of soil than on its stiffness. Figures 15 and 16 show that, as the confining pressure increased, the \(E_{i}\) and \((\sigma_{1}-\sigma_{3})_{\text{ult}}\), the model parameters of the fiber-reinforced soil, increased, approximating a power function, with power indices of 0.57 and 0.69, respectively. These findings indicate that confining pressure has a more significant effect on the strength of soil than on its stiffness. Figure 10:1 Relationship between \((\sigma_{1}-\sigma_{3})_{\text{ult}}\) and the confining pressure of the fiber-reinforced soil.