of p0c, the points representing the values of v and p

0

at failure would lie on or close to a curve S00S00 of similar shape to the consolidation
curve.
The data represented in Figures 4.20(a) and (b) can be combined in a three-

dimensional plot with coordinates q0, p0 and v, as shown in Figure 4.20(c). On this
plot the line OS0 and the curve S00S00 combine as the single curve SS. The curve SS is
known as the critical-state line, points on this line representing combinations of q0, p0

and v at which shear failure and subsequent yielding at constant effective stresses
occur. In Figures 4.20(a) and (b), OS0 and S00S00 are the projections of the critical-state
line on the q0–p0 and v–p0 planes, respectively. The stress paths for a consolidated–
undrained test (CA) and a drained test (CB), both consolidated to the same pressure
p0c, are also shown in Figure 4.20(c). The stress path for the consolidated–undrained
test lies on a plane CKLM parallel to the q0–p0 plane, the value of v being constant
throughout the undrained part of the test. The stress path for the drained test lies on a
plane normal to the q0–p0 plane and inclined at a slope of 3:1 to the direction of the q0

axis. Both stress paths start at point C on the normal consolidation curve NN which lies
on the v–p0 plane.
The stress paths for a series of consolidated–undrained and drained tests on speci-

mens each consolidated to different values of p0c would all lie on a curved surface,
spanning between the normal consolidation curve NN and the critical-state line SS,
called the state boundary surface. It is impossible for a specimen to reach a state
represented by a point beyond this surface.
The stress paths for a consolidated–undrained test and a drained triaxial test (D0E0

and D0F0, respectively) on specimens of a heavily overconsolidated clay are shown in
Figure 4.21(a). The stress paths start from a point D0 on the expansion (or recompres-
sion) curve for the clay. The consolidated–undrained specimen reaches failure at point
E0 on the line U0H0, above the projection (OS0) of the critical-state line. If the test were
continued after failure, the stress path would be expected to continue along U0H0 and
to approach point H0 on the critical-state line. However, the higher the overconsolida-
tion ratio the higher the strain required to reach the critical state. The deformation of
the consolidated–undrained triaxial specimen would become non-uniform at high
strains and it is unlikely that the specimen as a whole would reach the critical state.
The drained specimen reaches failure at point F0 also on the line U0H0. After failure,
the stresses decrease along the same stress path, approaching the critical-state line at
point X0. However, heavily overconsolidated specimens increase in volume (and hence
soften) prior to and after failure in a drained test. Narrow zones adjacent to the failure
planes become weaker than the remainder of the clay and the specimen as a whole does
not reach the critical state. The corresponding relationships between v and p0 are

120 Shear strength

Figure 4.21 Critical-state concept: overconsolidated clays.

represented by the lines DE00 and DF00, respectively, in Figure 4.21(b); these lines
approach but do not reach the critical-state line (S00S00) at points H00 and X00, respectively.
The volume of the undrained specimen remains constant during shearing but that of
the drained specimen, after decreasing initially, increases up to and beyond failure.
The line U0H0 is the projection of the state boundary surface, known as the Hvorslev

surface, for heavily overconsolidated clays. However, it is assumed that the soil cannot
withstand tensile effective stresses, i.e. the effective minor principal stress (�03) cannot
be less than zero. A line (OU0) through the origin at a slope of 3:1 (q0/p0 ¼ 3 for �03 ¼ 0
in Equations 4.14 and 4.15) is therefore a limit to the state boundary. On the q0–p0–v
plot, shown in Figure 4.21(c), this line becomes a plane lying between the line TT
(referred to as the ‘no tension’ cut-off ) and the v axis. Thus, the state boundary surface
for heavily overconsolidated clays lies between TT and the critical-state line SS. In
Figure 4.21(c) the undrained stress path (DE) lies on a plane RHUV parallel to the
q0–p0 plane. The drained stress path (DF) lies on a plane WXYD0 normal to the
q0–p0 plane and inclined at a slope of 3:1 to the direction of the q0 axis.
Also shown in Figures 4.21(a) and (c) are the stress paths for consolidated–undrained

and drained tests (G0H0 and G0J0, respectively) on lightly overconsolidated specimens
of the same clay, starting at the same value of specific volume as the heavily over-
consolidated specimens. The initial point on the stress paths (G0) is on the expansion
(or recompression) curve to the right of the projection S00S00 of the critical-state line in
Figure 4.21(b). In both tests, failure is reached at points on or close to the critical-state
line. During a drained test, a lightly overconsolidated specimen decreases in volume
and hardens; no decrease in stresses therefore occurs after failure. As a result the
deformation of the specimen is relatively uniform and the critical state is likely to be
reached.
The section of the complete state boundary surface, for normally consolidated and

overconsolidated clays, on a plane of constant specific volume is RHU in Figure
4.21(c). The shape of the section will be similar on all planes of constant specific
volume. A single section (TSN) can therefore be drawn with respect to coordinate axes
q0/p0e and p

0/p0e, as shown in Figure 4.22, where p
0
e is the value of p

0 at the intersection of
a given plane of constant specific volume with the normal consolidation curve. In
Figure 4.22, point N is on the normal consolidation line, S is on the critical-state line
and T is on the ‘no tension’ cut-off. A specimen whose state is represented by a point

Figure 4.22 Section of the complete state boundary surface.

122 Shear strength

lying between N and the vertical through S is said to be wet of critical (i.e. its water
content is higher than that of clay at the critical state, at the same value of p0). A
specimen whose state is represented by a point lying between the origin and the vertical
through S is said to be dry of critical.
To summarize, the state boundary surface joins the lines NN, SS and TT in Figure

4.21(c) and marks the limit to all possible combinations of stresses q0 and p0 and
specific volume v. The plane between TT and the v axis is the boundary for no tension
failure. The critical-state line SS defines all possible states of ultimate failure, i.e. of
continuing strain at constant volume under constant stresses. In the case of normally
consolidated clays the stress paths for both drained and undrained tests lie entirely on
the state boundary surface, failure being reached at a point on the critical-state line; the
state of the clay remains wet of critical. In the case of overconsolidated clays, the stress
paths prior to failure for both drained and undrained tests lie inside the state boundary
surface. A distinction must be made between heavily overconsolidated and lightly
overconsolidated clays. Heavily overconsolidated clays reach failure at a point on the
state boundary surface on the dry side of the critical-state line; subsequently the stress
path moves along the state boundary surface but is unlikely to reach the critical-state
line. Lightly overconsolidated clays remain wet of critical and reach failure on the
critical-state line.
The characteristics of both loose and dense sands during shearing under drained

conditions are broadly similar to those for overconsolidated clays, failure occurring on
the state boundary surface on the dry side of the critical-state line.
The equation of the projection of the critical-state line (OS0 in Figure 4.20(a)) on the

q0–p0