## 3 Strength Test
### Establishing Fatigue Equations Based on the Weibull Distribution of Interval Parameters
#### 4.3.3 Establishing the Interval Fatigue Equation under the Weibull Distribution  
The fatigue characteristic of cement-treated aggregate base materials is an important index to characterize the pavement base performance. The fatigue test method is related to several factors; therefore, the fatigue equation for cement-treated aggregate base materials is extremely complicated. However, the conventional fatigue equation was expressed as point numerical. In fact, because of the experimental equipment and human factors, the fatigue equation for cement-treated aggregate base materials typically exhibits a certain error. The interval parameter was used instead of the uncertainty parameter for engineering analysis, which showed a better agreement to the actual characteristics of the project. Hence, it is a point numerical, indicating that the fatigue equation does not satisfy the actual engineering requirements. Therefore, the interval analysis method is introduced, which could not only present all the possible real values present in the operation results but also exert significance for practical applications. From the standardized fatigue equation in Table 9, parameters a and b with the cement content of 3-5% comprised three groups, with the maximum and minimum. According to the interval theory, parameters a and b can be regarded as one interval variable. Parameter a values for the fatigue equation standardized for cement-treated aggregate base materials at a curing time of 28 days were 9.5423, 10.4689, and 10.7642, while parameter b values were 10.3342, 11.4541, and 11.7034. Parameter a values for the standardized fatigue equation for cement-treated aggregate base materials at a curing time of 90 days were 9.2115, 9.8136, and 9.5517. The corresponding parameter b values were 10.4075, 10.8184, and 10.8485. The minimum and maximum values for parameters a and b were estimated. According to Formula (10) and (11), the intervals of a and b were as follows:  
\[\left\{ \begin{array}{l}a=[9.2,\ 10.8\ ]\\ b=[10.3,\ 11.8]\end{array} \right. \tag{12}\]According to the interval values of a and b obtained from Formula (12), the standardized interval parameters for the fatigue equation under different stress states, different curing times, and cement content of 3-5% at a guaranteed rate of 95% can be obtained:  
\[lgN_{i}=[10.3,11.8\ ]-[9.2,10.8](\sigma/S_{\mathrm{f}}) \tag{13}\]  
Let us assume parameters a and b in the standardized fatigue equation with a cement content of 4% and 5% at a curing time of 28 days and 90 days. The minimum and maximum values for parameters a and b were calculated, and the intervals of a and b were still in accordance with Formulas (10) and (11):  
\[\left\{ \begin{array}{l}a=[9.5,10.8\ ]\\ b=[10.8,11.8]\end{array} \right. \tag{14}\]  
From the parameter value obtained by Formula (14), the standardization interval parameters from the fatigue equation under different stress states, different curing times, and cement content of 4-5% at a guarantee rate of 95% were obtained:  
\[lgN_{i}=[10.8,11.8\ ]-[9.5,10.8](\sigma/S_{\mathrm{f}}) \tag{15}\]  
When performing interval analysis calculations, it is imperative to consider the length of the interval. The longer the interval, the higher the probability that the sample data was within the interval. The lower the accuracy, the higher the permissible error. By comparing Equations (13) and (15), the lengths of the interval of parameter a in Equations (13) and (15) were 1.6 and 1.3, respectively. On the other hand, the lengths of parameter b in Equations (13) and (15) were 1.5 and 0.9, respectively. The interval length of Equation (13) was greater than that of Equation (15), indicating that the life interval allowed by Equation (13) is greater than that of Equation (15), and the error allowed by Equation (13) is also higher. If the interval length was extremely long, the estimated fatigue life interval would be extremely large, and the actual engineering would exert marginal significance.