52 Ground improvement by deep vibratory methods
below that achieved by vibro compaction in the field, may result in relative
densities D r in excess of 100%.
It is important to understand that all mechanical parameters describing the behavior of sand, such as stiffness and strength or permeability,
are directly related to its relative density. The modulus of elasticity of the
quartz grain itself, for example, is about 1000 times higher than that of
sand, even at its densest stage.
All forces acting on the soil such as gravity or additional loading are
transferred by grain contacts. The contact points themselves are able
to transfer normal and shear forces, the latter depending on the surface
characteristics of the grains. If an idealized sand material is considered to
consist of spherical grains of equal diameter, the loosest arrangement is
hexahedral whereby each sphere has six contact points with its neighboring spheres. In the densest stage, the spheres form an arrangement which is
characterized by 12 contact points with their neighboring spheres, allowing
considerably higher stresses per unit volume to be transferred with much
less deformations.
As already mentioned, the stiffness of a sand deposit depends on the prevailing stress level and therefore rises generally with depth and the relative
density as shown in Figure 3.11.
Beyond a certain threshold, all additional stresses acting on a granular
soil are accompanied by a rearrangement of its grains and a nonreversible (plastic) deformation. Unloading of the soil leads to a stress relief and
Vertical stress σ z (kN/m 2 )
D = 100%
80%
50%
20%
Modulus (MN/m 2
)
0
10
20
100
50
0
D =
n max − n
n max − n min
0.06 mm < d < 4 mm
Figure 3.11 Relationship between density D, vertical stress σ z , and constrained modulus
of sand. (After Smoltczyk, H.-U., Wasser und Boden, 9, 1966.)
below that achieved by vibro compaction in the field, may result in relative
densities D r in excess of 100%.
It is important to understand that all mechanical parameters describing the behavior of sand, such as stiffness and strength or permeability,
are directly related to its relative density. The modulus of elasticity of the
quartz grain itself, for example, is about 1000 times higher than that of
sand, even at its densest stage.
All forces acting on the soil such as gravity or additional loading are
transferred by grain contacts. The contact points themselves are able
to transfer normal and shear forces, the latter depending on the surface
characteristics of the grains. If an idealized sand material is considered to
consist of spherical grains of equal diameter, the loosest arrangement is
hexahedral whereby each sphere has six contact points with its neighboring spheres. In the densest stage, the spheres form an arrangement which is
characterized by 12 contact points with their neighboring spheres, allowing
considerably higher stresses per unit volume to be transferred with much
less deformations.
As already mentioned, the stiffness of a sand deposit depends on the prevailing stress level and therefore rises generally with depth and the relative
density as shown in Figure 3.11.
Beyond a certain threshold, all additional stresses acting on a granular
soil are accompanied by a rearrangement of its grains and a nonreversible (plastic) deformation. Unloading of the soil leads to a stress relief and
Vertical stress σ z (kN/m 2 )
D = 100%
80%
50%
20%
Modulus (MN/m 2
)
0
10
20
100
50
0
D =
n max − n
n max − n min
0.06 mm < d < 4 mm
Figure 3.11 Relationship between density D, vertical stress σ z , and constrained modulus
of sand. (After Smoltczyk, H.-U., Wasser und Boden, 9, 1966.)