# Research on Shear Behavior of Sand-Structure Interface Based on Monotonic and Cyclic Tests
## 4 Cyclic Shear Tests
### Analysis and Discussion
#### 4.2.2 Cyclic Loading and Unloading Shear Stiffness  
Comparing the initial loading stiffness in the cyclic test with the initial loading stiffness in the monotonic shear test above, the corresponding shear stiffness data are basically the same. Additionally, in the cyclic test, the initial loading phase curves of different stress level amplitudes almost overlapped on one curve. This also reflects the reliability of this monotonic shear test and cycle test. Due to the dense distribution of the loading and unloading curves in the cyclic test image, the shear stiffness of the cyclic loading and unloading curves is better reflected by the respective secant stiffness. The cyclic load and shear deformation incremental model shown in Figure 10 can be used to consider the change of the interface cyclic load path to construct a calculation method for the shear stiffness of the cyclic loading and unloading curve. The shear stiffness of the Nth loading and unloading secants are:  
\[K_{IN}=\frac{\tau_{max}-\tau_{min}}{u_{slNmax}-u_{sNmin}} \tag{4}\]  
\[K_{uN}=\frac{\tau_{max}-\tau_{min}}{u_{slNmax}-u_{sNmin}} \tag{5}\]  
Figure 9: Influence of different stress level amplitudes on residual deformation: (a) \(R_{n1}\). (b) \(R_{n3}\). where \(K_{IN}\) is the secant shear stiffness of the loaded branch corresponding to the Nth cycle and \(K_{uN}\) is the secant shear stiffness of the unloaded branch corresponding to the Nth cycle. \(\tau_{max}\) and \(\tau_{min}\) are the maximum and minimum interface shear stresses, respectively. \(u_{slNmax}\) and \(u_{suNmax}\) are the cumulative shear displacements of the maximum loading and unloading branches during the Nth cycle, respectively. \(u_{sNmin}\) is the cumulative shear displacement of the minimum loading and unloading support during the Nth cycle. According to formulas (4) and (5), the variation law of the secant shear stiffness of the interface cyclic loading and unloading branches with different relative roughness (\(R_{n1}\), \(R_{n3}\)) under different normal stresses can be calculated. The secant shear stiffness of the loose sand loading curve on the interface of different relative roughness gradually increases with the increase in the number of cycles (Figure 11a). Additionally, the growth amplitude of its loading shear stiffness gradually decreases and finally the situation tends to be stable. For example, when \(\tau_{max}\) is 78.3 kPa, \(K_{I1}=156.35\) kPa\(\cdot\)mm\({}^{-1}\), \(K_{I2}=176.67\) kPa\(\cdot\)mm\({}^{-1}\), \(K_{I9}=228.28\) kPa\(\cdot\)mm\({}^{-1}\), and \(K_{I10}=229.62\) kPa\(\cdot\)mm\({}^{-1}\), it can be seen that the loading shear stiffness of the 10 th cycle is 30.21% higher than that of the first cycle. \(K_{I2}\) to \(K_{I1}\) is about 13% higher, and \(K_{I10}\) to \(K_{I9}\) is only 0.59% higher. It is also found that the loading stiffness of the same relative density sand increases with the increase in the normal stress. However, the growth rate of its loading stiffness decreases. The main reason is that the greater the normal stress, the greater the initial shear stiffness. Moreover, the loading shear stiffness of sand under a rough interface is generally greater than that of the relatively smooth interface. It can be seen from Figure 11b that the loading stiffness change trend of dense sand is basically the same as that of loose sand. The difference is that the loading shear stiffness of dense sand increases more slowly. If the normal stress and the interface roughness are the same, the loading stiffness of dense sand is generally higher than that of loose sand. This shows that both loose sand and dense sand show a cyclic hardening tendency during the cyclic shearing process. This is consistent with the change rule of residual deformation mentioned above. The loading stiffness increases with the increase in the number of cycles, and the unloading stiffness is almost unchanged, resulting in a continuous increase in residual deformation.