Civil and Building Engineering.

The changes in stress conditions in a sand specimen during shearing may be understood by plotting the states of stress on a Mohr-Coulomb diagram. During Stage 1, a hydrostatic pressure is applied and the specimen is allowed to consolidate completely. The three principal stresses are all equal to 3, and Mohr’s circle plots as a single point in Figure 3.37a. In Stage 2a (Figure 3.37b) a stress difference is applied representing an intermediate state in loading where the applied stress difference is not large enough to cause failure. Shear stresses exist on all inclined planes through the specimen. In Stage 2b (Figure 3.37b), the axial stress has been increased to its maximum value and the soil specimen fails.

If such tests are performed on a series of identical specimens under various cell pressures and all failure circles are plotted on a single diagram, a single
line can be drawn from the origin that is tangent to each circle. Such a series of circles is shown in Figure 3.38. Specimens of sand do not fail until some part of the stress circle touches this line. Therefore, this line is called a failure envelope. By analogy with the direct shear tests and, more exactly, by consideration of the stresses on the planes of failure, the slope of this line represents the obliquity of the resultant stress on the failure plane. As before, the slope of this line is called the angle of internal friction of the sand.

The relationships between stresses, strains, and the angle of internal friction vary from one sand to another and with the density of an individual sand. At the levels of confining stress usually encountered in foundation engineering, angular sands tend to have higher angles of internal friction than sands with rounded grains. Dense sands have higher angles of internal friction than loose sands.

A classic study of the shearing behavior of fine sands was reported by Bjerrum et al. (1961). The study concerned both the drained and undrained behavior of a fine sand that is widely found in Norwegian fjords. It was found that when a dense sand is sheared under drained conditions, it usually undergoes a small decrease in volume at small strains, but the denseness of the packing prevents significant volume decreases. As the sand is strained further, the grains in the zone of failure must roll up and over one another. As a
result, the sand expands with further strain. The relationship between volumetric and axial strain for different densities of sand is shown in the lower half of Figure 3.39. Loose sands undergo a decrease in volume throughout the shear test.

The expansion of dense sand with increasing strain has a weakening effect and the specimens fail at relatively low strains, as seen in Figure 3.40. The densification of initially loose specimens with increasing strain makes such sands strain-hardening materials, and failure occurs at larger strains.

The relationship between angle of internal friction and initial porosity for fine quartz sand is shown in Figure 3.41. For this sand, the angle of internal
Figure 3.39 Typical volume change versus axial strain curves for fine sand (from Bjerrum et al., 1961).

friction decreases gradually until the initial porosity is about 46 or 47% and then decreases rapidly. Sand with such high initial porosities is not common, but the occasional catastrophic landslides in deposits of loose sand suggest that low angles of internal friction can occur in nature.

The data for porosity shown in Figure 3.41 were converted to void ratios and redrawn in Figure 3.42. Also drawn in this figure are vertical lines denoting the ranges of relative density for the sand that was tested. The maximum and minimum void ratios for relative density were evaluated on dry samples of sand and the shear tests were performed on saturated sand. For these data, the angle of friction is found to drop when the void ratio is larger than the maximum value measured on dry sand for relative density evaluations. These extremely low friction angles were possible because the test
Figure 3.40 Typical axial stress difference versus axial strain curves for fine sand (from Bjerrum et al., 1961).

specimens were formed by deposition in water. Thus, the loosest sand specimens were normally consolidated and never subjected to the higher stresses than can exist in dry sand. The lesson to be learned from this study is that the range of friction angles that may exist for saturated sands in nature can
be both larger and smaller than the values measured in the laboratory if the sand is permitted to dry.

The application of shearing deformation results in densification of loose sands and expansion of dense sands. Available data suggest that, for a given sand and given confining pressure, the density at high strains is the same for all specimens. Analysis of the stress-strain curves suggests that loose sands get stronger with strain (because of an increase in density) and approach a given strength. The data suggest that the strength of a dense specimen of the same sand peaks at low strains where interlocking of grains is maximum and then decreases and approaches the same limiting level of strength as for the loose sands. The strength of dense sands decreases toward the limiting level and the strength of loose sands increases toward the limiting level. The strength of a soil at large strains, where neither volume nor strength is changing with strain, is termed the ultimate strength.