## 3 Results and Discussions
### Residual Strength  
The shear stress reaches a steady value after the peak strength point in the direct shear tests of dense sand, which is known as residual strength. As illustrated in Figure 3, fiber strips obviously increased the residual strength of dense sand in this study. The residual strength of sand specimens was selected as the shear stress at the displacement of 4 mm (i.e., the maximum shear displacements of some specimens). The ratios of residual to peak strength were calculated and are plotted in Figure 5. The residual strength is  
Figure 4: Shear strength envelope of reinforced sand with fiber contents of (**a**) 0.04%, (**b**) 0.2%, and (**c**) 0.4%.  
a valuable parameter for evaluating the engineering performance of soil because high residual strength helps to avoid the continuous failure of soil after a short period of high load. This, in turn, is helpful in the serviceability design of structures.  
As shown in Figure 5, the ratios of residual to peak strength had a positive correlation with fiber content. Similar to the change rule of peak strength, the fiber strips initially orientated at 60\({}^{\circ}\) had the most obvious effect on the increase in residual strength, and the fibers initially orientated at 90\({}^{\circ}\) had relatively weaker but still obvious effects. Note that the fiber strips initially orientated at 120\({}^{\circ}\) slightly increased the residual strength of dense sand, although it did not help to increase the peak strength. This indicated that the increase in residual strength may not only be related to stress mobilization in fiber strips, as mentioned above.  
The softening of dense sand is closely related to its volume dilation characteristic. When the stress ratio reaches a characteristic state stress ratio, the volume of dense sand changes from contraction to dilation during the shearing process, and then softening occurs. The increase in the residual strength led by the introduction of fiber strips may be attributed to the restraint effect of fibers on sand dilation. The volume change of sand in direct shear tests can be observed from the vertical displacement of the top surface of the specimen. The top vertical displacement of the sand specimens with fibers initially orientated at 60\({}^{\circ}\) under normal pressures of 400 and 800 kPa are presented in Figure 6. The positive values denote downward displacement and volume contraction, and vice versa. Typical dense sand usually exhibits volume contraction and then dilation during the shearing process. As illustrated in Figure 6, with the increase in fiber content, the volume contraction of reinforced sand was enhanced, and the dilation was inhibited. This reflected the restriction of fiber strips on the movement among sand particles, which was caused  
Figure 5: The ratio of residual to peak strength of sand specimens under normal pressures of **(a)** 100, **(b)** 200, **(c)** 400, and **(d)** 800 kPa.  
by the interaction between fibers and the sand matrix. Under the effects of deformation restriction, the softening behaviors of reinforced sand specimens were limited, and then the residual strength increased.