Chapter -6
Stress and Strain Path Representation
Introduction
The stress path
▪ Is a graphical representation of stresses in a stress space.
▪ Is a line that connects a series of points each of which represents a
successive stress state experienced by a soil specimen during the
progress of a test.
▪ Represents the successive states of stress in a test specimen of soil
during loading or unloading.
▪ Is plotted as deviatoric stress (q) versus mean effective stress (p).
The response, stability, and failure of soils depend on stress paths.
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Stress and Strain Path Representation
The stress path
▪ A series of Mohr circles can be drawn to represent the successive
states of stress, but it is difficult to represent several circles in one
diagram.
▪ The results of triaxial tests can be represented by diagrams called
Stress paths.
Successive Mohr circles
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Stress path for constant 3
Stress and Strain Path Representation
General about stress paths
▪ It is the locus of the stress point.
locus; is a set of all points (commonly, a line, a line segment, a
curve, or a surface), whose location satisfies or is determined by
one or more specified conditions.
▪ This locus called the stress path, is plotted on what we call a p-q
diagram. Note that both p and q could be defined in terms of either
total stresses or effective stresses.
▪ It represents the state of stress and successive state of stress.
▪ The stress path needs to be a straight line.
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Stress and Strain Path Representation
Why do we need this stress and strain path??
▪ An element of soil will experience changes in its state of stress or
strain as the lab test progresses or is loaded in service.
▪ Sometimes it is convenient to represent that state of stress by a stress
point, which has the coordinates (1 - 3)/2 and (1 + 3)/2.
▪ In geotechnical engineering, we assume 1 and 3 act on vertical and
horizontal planes.
▪ So, the coordinates of the stress point are (v - h)/2 & (1 + 3)/2, or
simply q and p, respectively.
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Stress and Strain Path Representation
Why do we need this stress and strain path??
▪ Mohr circle may be employed to analyze the state of stress/strain at a
particular instant during loading.
▪ We may, however, need to trace the history of the changes in the state
of stress or strain.
▪ The behavior of a perfectly elastic material (loaded or unloaded
within an elastic range) is independent of the route taken during the
loading process. Only depends on the initial and final stress states.
▪ This is not true with soils!
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Stress path on p and q plot
▪ Both q and p could, of course, be defined in terms of the principal
stresses.
▪ By convention, q is considered positive when v > h; otherwise, it
is negative.
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Stress path on p and q plot
▪ A simple case to illustrate stress paths using the triaxial test in which
3 remains fixed as we increase 1 (45o stress path line).
▪ The initial conditions are v = h an equal-all-around or hydrostatic
state of stress.
Fig. 8.16
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Stress path on p and q plot
Fig. 8.17
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Stress path on p and q plot
Prove!
Verify that stress paths A, B, and C of hydrostatic and A and D of nonhydrostatic compression are correct as shown.
Solution:
The initial conditions for all stress paths in hydrostatic compression are
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Stress path on p and q plot
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Stress path on p and q plot
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Stress path on p and q plot
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Stress ratio (K)
▪ The stress ratio is defined as the ratio of the second invariant of the
deviatoric stress tensor to the first invariant of the stress tensor.
▪ It is often convenient to consider stress ratios.
▪ The lateral stress ratio K is the ratio of the effective horizontal stress
to the effective vertical stress.
▪ The coefficient of lateral earth pressure at rest (Ko)for conditions of
no lateral strain, or the assumed condition in the ground.
▪ Finally, we can define a ratio Kf for the stress ratio at failure:
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Stress ratio (K) and Stress Path
▪ Usually, Kf is defined in terms of effective stresses, but it could also
be expressed in terms of total stresses.
▪ Constant stress ratios appear as straight lines on a p-q diagram.
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Stress ratio (K) and Stress Path
▪ These lines could also be stress paths for initial conditions of v = h
= 0 with loadings of K equal to a constant (that is, constant v /h ).
▪ Other initial conditions are, of course, possible, such as those shown
in Figs. 8.16 and 8.17.
▪ Note that:where,  is the slope of the line of constant K when K < Kf. At failure,
the slope of the Kf line is indicated by the symbol .
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Stress paths for shear strength testing
▪ As a reminder, stress paths are a convenient representation of
multiple Mohr circle states for a soil element undergoing any
number of loading and/or unloading processes.
▪ Both q and p could be defined in terms of the principal stresses, and
by convention, q is considered positive when v > h; otherwise, it is
negative.
▪ We are often interested in conditions of failure, and it is useful to
know the relationship between the Kf line and the Mohr-Coulomb
failure envelope.
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Stress paths for shear strength testing
▪ To establish the slopes of the two lines and their intercepts, several
Mohr circles and stress paths, determined over a range of stresses,
were used.
▪ The equation of the Kf line is:-
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Stress paths for shear strength testing
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Stress paths for shear strength testing
Relationship between the K, line and the Mohr-Coulomb failure envelope
The equation of the Mohr-Coulomb failure envelope is:
From the geometries of the two circles, it can be shown that
and
So, from a p-q diagram, the shear-strength parameters  and c may
readily be computed.
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Reading assignment
Stress path based on Total stress path(TSP), effective stress
path(ESP) for CD, CU and UU tests…
Assignment 6
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