Civil Engineering Reference
In-Depth Information
2.5 Analysis of stress
Within a loaded body the stresses generally vary from point to point so, for example,
the stresses below the edge and centre of a foundation are different. At any point the
stresses are different on different planes and it is necessary to relate the stresses on the
different planes.
The simplest form of analysis is through the Mohr circle construction which is
covered in courses on strength of materials. The only difference for soil mechanics is
that the sign convention is changed so that compressive stresses and counter-clockwise
shear stresses are positive.
Figure 2.5(a) shows principal stresses
σ
h
on the faces of an element of soil
and Fig. 2.5(b) shows the corresponding Mohr circles of stress. The pole P of the Mohr
circle is defined so that a line from P to
σ
z
and
σ
z
gives the direction of the plane on which
σ
z
acts. In Fig. 2.5(a) there is an element rotated to an angle
θ
as shown and the stresses
(
m
) on the faces of this element are at N and M in Fig. 2.5(b). From
the geometry of the Mohr circle the angle 2
τ
n
,
σ
n
and
τ
m
,
σ
subtended at the centre by the point
representing the major principle plane and the point N is twice the angle between the
planes on which these stresses act. From the geometry of the figure,
θ
τ
=
τ
m
. Using
n
Fig. 2.5(b) it is possible to calculate
h
or vice versa,
and in order to construct the Mohr circle it is necessary to know the stresses on two
(preferably orthogonal) planes.
τ
n
,
σ
n
and
τ
m
,
σ
m
from
σ
z
and
σ
2.6 Analysis of strain
Analysis of strains at a point using the Mohr circle of strain is similar to that for stress,
but there are a few points to note about strains. Firstly, while it is possible to talk about
a state of stress with respect to zero stress (taken as atmospheric pressure), there is no
absolute zero for strain so we have to talk about changes, or increments, of strain.
These may be small increments (denoted by
)
and generally they occur as a result of corresponding large or small increments of stress.
δε
) or large increments (denoted by
ε
Figure 2.5
Analysis of stress using a Mohr circle.
