# Test Granular Materials Failure Using Bi-Directional Simple Shear Apparatus: A Review
## 4 Bi-Directional Simple Shear Tests with Complex Loading Stages
It should be noted that the plain rubber membrane-enclosed specimens were directly placed in the chamber of the apparatus, without rings or a reinforced membrane placed outside the soil specimen, and the undrained condition was achieved by closing the drainage and maintaining the constant height of a specimen. As a result, the horizontal (radial) strain was not zero, and there were non-uniformity issues in the vertical direction of the specimen, like  
Figure 5: The maximum normalized shear stresses and total shear stresses versus \(\theta\) (\(\tau_{hc}/\sigma^{\prime}_{vc}=0.2\)) (Reprinted from [55]). the triaxial samples or hollow cylinder samples. Single amplitudes of 3% and 7.5% were defined as failure criteria. The results in these studies show that when the two stresses are in the same direction, the liquefaction resistance usually increases with the increasing consolidation shear stress. In contrast, when the cyclic shear stress is applied perpendicular to the consolidation shear stress, the liquefaction resistance decreases with the increasing consolidation shear stress. The liquefaction resistance (with failure criteria of 3% or 7.5% single amplitude shear strain at the 10 th or 30 th cycle) of the perpendicular cyclic loading is 5% to 30% less than that in the parallel cyclic loading for tests with a consolidation shear stress ratio (\(\frac{\tau_{5}}{\sigma_{v}}\), is consolidation shear stress and is vertical stress) of 0.2 to 0.3, and relative densities of 35%, 45%, and 55%. [PERSON] and [PERSON] [38] stated that the reduction of cyclic shear resistance in perpendicular tests is due to the difference of the principal stress rotation and stress reversal, which have a profound influence on the excess pore water generation and shear strain. Using the same apparatus as [PERSON] and [PERSON] [38], [PERSON] [37] conducted a comprehensive study of multidirectional simple shear tests on a saturated medium dense to dense sand, under multidirectional stress paths. The initial driving shear stress was considered as a core parameter in this study. As shown in Figure 7, three types of cyclic loading were tested, namely: (a) one-directional cyclic loading; (b) loadings with oval/circular stress paths; and (c) loadings with Figure 8 stress paths. To create different levels of stress reversal, different levels of initial driving shear stress were considered by [PERSON] [37], as shown in Figure 8. However, only 0\({}^{\circ}\) and 90\({}^{\circ}\) were used as the direction of the initial driving shear stress, and the cyclic shear stress was not considered in the consolidation stage. The study used the stress-controlled method, and a double amplitude of 6% was defined as the triggering of liquefaction. Figure 6: Two dynamic loading conditions considered in the tests (Reprinted from [38]). The results show that the shear stress and shear strain responses of sand under multidirectional loadings are far more complex than those in the uni-directional simple shear tests. The shear strain in the bi-directional simple shear tests can be much larger than the estimated shear strain base on the uni-directional simple shear tests. Specifically, at the low values of the initial static driving stress, increasing the cyclic shear stress increases the cyclic and permanent shear strains. At the high values of the initial static driving stress, increasing the cyclic shear stress decreases the cyclic and permanent shear strains. The cyclic shear strain and permanent shear strain are defined in Figure 9. In addition, increasing the cyclic shear stress reduces the number of cycles at the limiting pore water pressure and the strain potential after reaching the limiting pore water pressure. The limiting pore water pressure is defined as the maximum pore water pressure at a stable state (a constant value), as shown in Figure 10.  
In most tests conducted by [PERSON] [37], a large strain occurs before reaching 100% excess pore water pressure (full-liquefaction).