# Effect of Drying-Wetting Cycle and Vibration on Strength Properties of Granite Residual Soil
## 3 Test Results and Analysis
### Analysis of Direct Shear Test Results
#### 3.1.3 Modification of the Shear Strength Formula  
The above analysis indicates that both the DW cycle and vibration have a significant influence on the cohesion and internal friction angle of GRS. Therefore, it is necessary to modify the shear strength formula.  
Figure 9 shows the relationship between the cohesion degradation index \(\xi_{t,n}\) and the number of DW cycles. The relationship for unvibrated and vibrated specimens is different. When there is no vibration, \(\xi_{t,n}\) is approximately exponentially related to the DW cycle. However, when there is vibration, \(\xi_{t,n}\) is approximately linear with the number of DW cycles.  
For the purposes of nondimensionalization and applicability, let \(v\) equal \(t/45\), and by substituting Equation (1) into the fitting formula, the modified formulas for cohesion under different vibration times are obtained, as shown in Equation (2).  
\[c_{v,n}=\begin{cases}c_{0,0}(1.23-0.23e^{n/5.13}),v=0,0\leq n\leq 5\\ c_{0,0}(0.75-0.03n-0.13v),\frac{1}{3}\leq v\leq 1,0\leq n\leq 5\end{cases} \tag{2}\]  
where \(v\) is the normalized vibration time, \(v=t/45\), where \(t\) is the vibration time and 45 is the maximum vibration time; \(c_{v,n}\) is the cohesion under nth DW cycles and the normalized vibration time \(v\); and \(c_{0,0}\) is the initial cohesion (without DW cycles and vibration).  
Figure 10 presents the relationship between the normalized internal friction angle and the number of DW cycles. The internal friction angle ratio is approximately linear with the number of DW cycles, and the correction formula for the internal friction angle is shown in Equation (3).  
\[\varphi_{n}=\begin{cases}(0.073n+1)\varphi_{0},v=0,0\leq n\leq 5\\ \beta(0.073n+1)\varphi_{0},\frac{1}{3}\leq v\leq 1,0\leq n\leq 5\end{cases} \tag{3}\]  
where \(\varphi_{n}\) and \(\varphi_{0}\) are the internal friction angle for the \(n\)th and initial friction angle, respectively; \(\beta\) is the correction factor, taking 1.1 for \(0\leq n\leq 2\) and 1.03 for \(3\leq n\leq 5\).  
Figure 9: Cohesion degradation index vs. DW cycles.  
So, the fitting formula for shear strength can be modified as follows:  
\[\tau_{v,n}=\left\{\begin{aligned} \sigma\tan[(0.073n+1)\varphi_{0}]+c_{0,0}(1.23 -0.23e^{n/5.13}),& v=0,0\leq n\leq 5\\ \sigma\tan[\beta(0.073n+1)\varphi_{0}]+c_{0,0}(0.75-0.03n-0.13v), \frac{1}{3}\leq v\leq 1,0\leq n\leq 5\end{aligned}\right. \tag{4}\]  
where \(\tau_{v,n}\) is the shear strength under \(n\)th DW cycles and the normalized vibration time \(v\); \(\sigma\) is the normal stress.  
Figure 11 shows a comparison of shear strength calculated using Equation (4) and the measured values for all DW cycle numbers and vibration times. The diagonal line is the contour line, where the measured values are equal to the predicted values. The data errors are all within 10%, demonstrating that the modified shear strength formula has high prediction accuracy.  
Figure 11: Comparison of measured and calculated values of shear strength.  
Figure 10: Internal friction angle ratio vs. DW cycle.