Civil Engineering Reference
In-Depth Information
2.3 Plane strain and axial symmetry
In general we should consider stresses and strains, or forces and displacements, in
three dimensions, but then the algebra becomes quite complicated and it is difficult to
represent general states on flat paper. There are, however, two cases for which only
two axes are required and these are illustrated in Fig. 2.2.
Figure 2.2(a) shows plane strain where the strains in one direction are zero and
the stresses and strains are vertical (
ε
h
). (It would be best
to use v as the subscript for vertical stress and strain but we will need to keep the
subscript v for volumes and volumetric strains.) This corresponds to conditions in the
ground below a long structure, such as an embankment or wall or a strip foundation.
Figure 2.2(b) shows axial symmetry where the radial stresses and strains (
σ
z
,
ε
z
) or horizontal (
σ
h
,
σ
r
,
ε
r
) are
equal and the other stresses and strains (
a
) are axial. This corresponds to conditions
in the ground below a circular foundation or a circular excavation. Throughout this
topic I will consider only plane strain and axial symmetry and I will use the axes z, h
(vertical and horizontal) for plane strain and the axes a, r (axial and radial) for axial
symmetry.
σ
a
,
ε
2.4 Rigid body mechanics
When soils fail they often develop distinct slip surfaces; on a geological scale these
appear as faults. Slip surfaces divide soil into blocks and the strains within each
block may be neglected compared with the relative movements between blocks, so
the principles of rigid body mechanics are applicable for failure of slopes and founda-
tions. To demonstrate this examine how sandcastles and claycastles fail in unconfined
compression tests.
Equilibrium is examined by resolution of forces in two directions (together with
moments about one axis) and this is done most simply by construction of a polygon
of forces: if the polygon of forces closes then the system of forces is in equilibrium.
Figure 2.3(a) shows a set of forces acting on a triangular block. We will see later that
this represents the conditions in soil behind a retaining wall at the point of failure.
Figure 2.3(b) shows the corresponding polygon of forces where each line is in the same
direction as the corresponding force and the length is proportional to the magnitude
of the force. The forces are in equilibrium because the polygon of forces is closed.
Figure 2.2
Common states of stress.
