Environmental Engineering Reference
In-Depth Information
pressure, (2) soil is isotropic, nonlinear, and elastic,
(3) strains are small, (4) pore-water is incompressible, and
(5) the effects of air diffusing through water, air dissolving
in the water, and the movement of water vapor are negligi-
ble. Tensor notation is used for the presentation of the theory
of one-dimensional heave.
14.6.1 Strain-Displacement and Compatibility
Relations
Let us consider a three-dimensional field with x, y, and z as
the rectangular Cartesian coordinates (i.e., x and z for hori-
zontal directions and y for vertical direction), with u i being
components of the displacement vector (i.e., comprised of
u , v , and w for the x -, y - and z -directions, respectively).
The components of strain tensor for the soil structure, ε ij ,
are written in terms of displacements as follows:
∂u i
∂x j +
Figure 14.52 Ratio of total heave for partial excavation and
backfilling to total heave for wetting of entire active depth.
∂u j
∂x i
1
2
ε ij =
(14.45)
The normal strains can be designated as ε x , ε y , and ε z for
the x -, y -, and z -directions, respectively. For infinitesimal
deformations, volumetric strain ε v is the sum of the normal
strain components:
and swelling pressures (provided the variables are constants)
and show the influence of excavating and backfilling on
computed total heave.
Figure 14.52 shows, for example, that if 50% of the active
depth were removed, the total heave would still be 86% of
that anticipated for the case of no excavation. However, if
the excavated portion is backfilled with a nonexpansive soil,
the total heave would be only 15% of that anticipated if no
material were excavated. In other words, partial excavation
of the expansive soil depth does not significantly reduce the
total heave unless there is backfilling with an inert soil.
These examples illustrate the type of plots that can readily
be generated using the closed-form equations for predicting
total heave. Other possible boundary conditions could also
be assumed. The plots presented are of assistance in making
engineering decisions when placing structures on expansive
soils. Possible remedial measures for heave reduction by
flooding, excavation, and backfilling can be studied by com-
paring the presented plots.
∂u i
∂x i =
∂u
∂x +
∂ν
∂y +
w
∂z
ε v =
=
ε x +
ε y +
ε z
(14.46)
The constitutive volume change behavior of the unsatu-
rated soil is described in terms of two independent stress
state variables: net normal stress σ
u a and matric suc-
tion u a
u w . The total volume change of an unsaturated
soil element must be equal to the sum of volume changes
associated with each phase and the continuity requirement
reduces to
V v
V 0
V w
V 0
V a
V 0
=
+
(14.47)
where:
V 0 =
initial overall volume of an unsaturated soil ele-
ment,
V v =
volume of soil voids,
14.6 SWELLING THEORY FORMULATED
IN TERMS OF INCREMENTAL ELASTICITY
PARAMETERS
V w =
volume of water in the element, and
V a
=
volume of air in the element.
The mechanics of volume change in unsaturated, expan-
sive soils under isothermal condition involves two primary
processes: water flow and stress-deformation. However, the
desire is to present each of the physical processes as inde-
pendent phenomenon. Therefore, Chapter 14 presents only
the stress-deformation theory and its application to expan-
sive soils problems. Coupled behavior (i.e., seepage and
volume change) is presented in Chapter 16.
The governing partial differential equations for stress-
deformation are derived based on the following assumptions:
(1) the air phase is continuous and remains at atmospheric
Two constitutive relationships are required to describe the
volume-mass changes associated with an unsaturated soil:
one for the soil structure (in terms of void ratio or volu-
metric strain) and another for the water phase (in terms of
degree of saturation or water content). The constitutive rela-
tionships for soil structure and water phase are presented
using indicial notation for the formulation of the partial dif-
ferential equations for the stress-deformation processes. The
stress state of an unsaturated soil can be written in terms of
the net normal stress tensor σ ij
u a δ ij and matric suction
tensor (u a
u w ij .
 
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