Yang, J. (2002). Géotechnique 52, No.

10, 757–760

TECHNICAL NOTE

Non-uniqueness of flow liquefaction line for loose sand

J. YA N G

KEYWORDS: failure; laboratory tests; liquefaction; sands steady-state concept and in many respects follows the princi-
ples of critical-state soil mechanics (Schofield & Wroth,
1968). The key point of this method is that the collapse line
INTRODUCTION is unique: the parameters describing its position (the slope
It has been consistently observed in undrained triaxial tests and the intercept on the vertical axis) in the stress path
(e.g. Casagrande, 1971; Castro et al., 1982; Vaid & Chern, space can be converted into parameters analogous to Mohr–
1985; Ishihara, 1993) that, subjected to monotonic loading, Coulomb failure parameters, and may therefore be used in
very loose sand exhibits a peak strength at a small shear conventional limit equilibrium stability analysis.
strain and then collapses to flow rapidly to large strains at The collapse line concept has been recognised by some
low effective confining pressure and low strength, as illu- researchers. For example, Ishihara (1993) showed the exis-
strated in Fig. 1. Furthermore, there exists an ultimate state tence of such a unique collapse line in the stress path space
of shear failure at which the sand flows continuously under for loose Toyoura sand. On the other hand, Vaid & Chern
constant stress and constant volume. The ultimate state, (1985), Lade (1993) and some recent experimental investiga-
termed as steady state (Poulos, 1981), is essentially the same tions have assumed that the locus of peak points in the
as the well-known critical state (Roscoe et al., 1958). The effective stress paths is a straight line, the flow liquefaction
type of behaviour described above is now recognised as flow line, that passes through the origin rather than through the
liquefaction, which may produce the most devastating effects steady state, as schematically shown in Fig. 2(b). The flow
of all the liquefaction-related phenomena. Flow liquefaction liquefaction surface or line, which also assumes the unique-
failures are characterised by a sudden loss of strength and a ness of the line in stress path space, seems to receive more
rapid development of large deformation. The collapse of recognition (Kramer, 1996).
Lower San Fernando Dam (Seed et al., 1975) is a typical In this study a new interpretation, a flow liquefaction line
example of such failures. varying with the state of soil rather than unique in the stress
Owing to its dramatic effects and its complex nature, path space, is proposed, based on careful examination of
considerable efforts have been made to understand and experimental data. The flow liquefaction line is defined here
characterise flow liquefaction. Sladen et al. (1985) proposed as a line that connects the peak point in any one single
the concept of a collapse surface based on some triaxial test stress path with the stress origin. Within the framework of
results. In this concept, for a series of specimens initially critical-state soil mechanics, a dependence of the slope of
consolidated at the same void ratio at different confining the flow liquefaction line upon a state parameter that simul-
pressures, the locus of peak points in the effective stress taneously accounts for the stress level and soil density is
paths is a straight line that projects linearly through the suggested.
steady-state point in q  p9 space (see Fig. 2(a)), where
p9 ¼ (
19 þ 2
39 )=3 is mean effective stress and q ¼
1 
3
is deviator stress in a triaxial setting. In q  p9  e space, EXPERIMENTAL EVIDENCE
where e is void ratio, these points form a space that passes The triaxial test data analysed are from Castro et al.
through the steady- or critical-state line. The collapse sur- (1982), Sladen et al. (1985) and Ishihara (1993). In Table 1
face or line concept is fundamentally an extension of the the index properties of the sands tested (Banding sand No. 6,

Steady/critical-state line
q

Peak point
Deviator stress, q

Steady state
Steady state Flow failure

Mean effective stress, p′ Axial strain, ε1

Fig. 1. Flow liquefaction of loose sand in undrained monotonic loading tests

Manuscript received 1 October 2002; revised manuscript accepted 28 October 2002.

Discussion on this paper closes 1 June 2003, for further details see p. ii.

Geotechnical Institute, Technical University of Berlin, Germany.

757

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 758 YANG

q′ ⫽ σ′1 ⫺ σ 3′ M
Steady/critical-state line
1
4 Varying flow
liquefaction line
Collapse line
3
Steady state

q/p′ss
2

A B p′ ⫽ (σ′1 ⫹ 2σ 3′ )/3 0
(a) 0 2 4 6 8
p ′/p′ss
(a)
M
q′ ⫽ σ′1 ⫺ σ 3′

Steady/critical-state line
1
12
Flow liquefaction Varying flow
line 10
liquefaction line
Steady state 8

q/p′ss
6

4
A B p′ ⫽ (σ′1 ⫹ 2σ 3′ )/3 2
(b)
0
0 4 8 12 16 20
p ′/p ′ss
Void ratio, e

(b)
Steady state
Fig. 3. Test results for (a) Leighton Buzzard sand and (b)
Nerlerk sand in stress path space (data from Sladen et al., 1985)
A B

The evidence of a moving flow liquefaction line and the

Steady/critical-state line similar tendency of the variation of the slope of the line with
confining pressure are also clearly observed in Figs 4 and 5
for Banding sand and Toyoura sand respectively.
p′ ⫽ (σ′1 ⫹ 2σ 3′ )/3
(c)
STATE-DEPENDENT FLOW LIQUEFACTION LINE
Fig. 2. Schematic diagram of: (a) collapse line concept; (b) flow A proper evaluation of the behaviour of the varying flow
liquefaction line concept liquefaction line with different initial states of soil is of
value. The nature of the steady- or critical-state line implies
the limited applicability of absolute measures of density,
Leighton Buzzard sand, Nerlerk sand and Toyoura sand) are such as void ratio, for characterising a potentially liquefiable
summarised. Fig. 3 shows the test results for the Leighton soil. The behaviour of a cohesionless soil should be more
Buzzard sand and Nerlerk sand in the stress path space. Both closely related to the proximity of its initial state to the
the deviator stress and the mean effective stress were normal- critical-state line. With the critical state as a basis, a state
ised by the mean stress at the steady/critical state at the same parameter (Been & Jefferies, 1985) can be defined as
void ratio, p9ss (Sladen et al., 1985). It is evident that the ł ¼ e  ec (1)
peak points in the stress paths do not lie on a single line
through the origin, but rather, the flow liquefaction line where e is the void ratio at the initial state and ec is the
varies with the stress level. It is found that the slope of the void ratio at the critical state under the same mean effective
flow liquefaction line increases with decreasing confining stress, as illustrated in Fig. 6. The state parameter is a
pressure, and the critical-state line provides an upper bound. measure of how far the material state is from the critical

Table 1. Index properties of sands tested

Sand D50 : mm Cu emax emin Fc : % e Reference
Banding No. 6 0·157 1·70 0·82 0·52 – Loose state Castro et al.
(1982)
Leighton Buzzard 0·86 1·16 0·75 0·58 0 Loose state Sladen et al.
(1985)
Nerlerk 0·28 2·0 0·94 0·62 2 Loose state Sladen et al.
(1985)
Toyoura 0·17 1·7 0·977 0·597 0 0·908 Ishihara (1993)
Note: D50 ¼ mean grain size; Cu ¼ uniformity coefficient; Fc : % ¼ fines content; e ¼ void ratio

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 FLOW LIQUEFACTION LINE FOR LOOSE SAND 759
20

Void ratio, e
Initial state: loose

Undrained
16 Varying flow
liquefaction line
Ψ ⫽ (e ⫺ ec) ⬎ 0
12
q/p′ss

Critical state
4

Critical-state line
0
0 4 8 12 16 20 24 28
p′/p ′ss
p′ss Mean effective stress (log)

Fig. 4. Test results for Banding sand in stress path space (data Fig. 6. Definition of state parameter
from Castro et al., 1982; also see Sladen et al., 1985)

8 state in terms of density. When ł is positive the soil is in a

loose state that is susceptible to liquefaction.
Varying flow For laboratory tests, the state parameter ł at the initial
liquefaction line state can be conveniently estimated provided the critical-state
6
line is determined in the e  p9 plane. It is worth mention-
ing here that it is simply a matter of mathematical conve-
q/p′ss

nience that the critical-state line is usually assumed as being

4
linear in a semi-log form. Alternative representations of the
line on a different scale may be made to better fit experi-
mental data: this is the case when a wide range of stresses is
2 considered (e.g. Verdugo & Ishihara, 1996).
Figure 7 shows the stress ratio, q= p9, at the peak points
in the stress paths as a function of the state parameter for
0
2 4 6 8 10 12 14
the Leighton Buzzard sand (Fig. 3(a)) and Toyoura sand
p ′/p ′ss (Fig. 5). It is clear that a correlation exists between the two
variables. For the data analysed, an exponential function as
Fig. 5. Test results for Toyoura sand in stress path space (data suggested below can describe the relationship between q= p9
from Ishihara, 1993) and ł reasonably well for both sands:

1.2 1.2

0.9 0.9
Stress ratio at peak point, q/p′

Stress ratio at peak point, q/p′

0.6 0.6

q/p ′ ⫽ 0.8 M exp(AΨ)

M ⫽ 1.19; A ⫽ ⫺5.2 q/p′ ⫽ 0.8 M exp(AΨ)
M ⫽ 1.24; A ⫽ ⫺4.0
0.3 0.3

0 0
0 0.03 0.06 0.09 0.12 ⫺0.02 0.04 0.1 0.16 0.22
State parameter, Ψ State parameter, Ψ
(a) (b)

Fig. 7. Relationship between peak stress ratio and state parameter: (a) Leighton Buzzard sand (data from Sladen et al.,
1985); (b) Toyoura sand (data from Ishihara, 1993)

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 760 YANG
 
q liquefaction line and the state parameter that accounts for
¼ 0  8M exp (Ał) (2)
p9 peak both stress level and the density of soil. The present work
clarifies the confusion and contradiction related to the
in which M is the stress ratio q= p9 at critical state and A is collapse line concept and the flow liquefaction line concept.
a parameter that is less than zero. For Leighton Buzzard In particular, it provides a useful framework for conceptual
sand the critical stress ratio M is 1·19 (Sladen et al., 1985), understanding of the complicated behaviour of liquefiable
and A is calibrated as 5·2. For Toyoura sand M is 1·24 soil both before and after liquefaction. Obviously, more
(Ishihara, 1993), and an appropriate A is found to be 4·0. experimental data of high quality are needed to further
Note that the relationship in equation (2) is established validate this new interpretation and to improve the relation-
based on limited experimental data, and an alternative fit ship suggested.
might be possible. Provided more data of quality are avail-
able, the relationship could be improved. For this reason, a
more general relationship may take the following form: ACKNOWLEDGEMENTS
  The financial support provided by the Alexander von
q M Humboldt Foundation via its prestigious Fellowship pro-
¼ exp (Ał) (3)
p9 peak B gramme is gratefully acknowledged. The author wishes to
thank Professor S. Savidis of the Technical University of
where B is an additional parameter that is expected to vary Berlin for his support.
within a narrow range. As the critical-state line serves as the
upper bound for the flow liquefaction line (see Figs 3–5),
the value of B is required to satisfy the condition B > 1:0. NOTATION
The state parameter ł has been related to CPT resistance A, B parameters used in the relationship in equation (3)
and other in situ test results (Been, 1998), and therefore the e void ratio
relationship established in equation (3) is of interest in ec critical void ratio
regard to its potential applications in engineering practice. M critical stress ratio
Based on equation (3), furthermore, a friction angle FLL for p9 mean effective stress
p9ss mean effective stress at critical state
characterising the slope of the flow liquefaction line (FLL)
q deviator stress
can be introduced as FLL friction angle for characterising the flow liquefaction line
 
M ł state parameter
3 exp (Ał)
sin FLL ¼ B  (4)
M REFERENCES
6þ exp (Ał)
B Been, K. (1998). The critical state line and its application to soil
liquefaction. In Physics and mechanics of soil liquefaction, pp.
It can be readily shown that the flow liquefaction angle 195–204. Rotterdam: Balkema.
decreases with increasing state parameter: that is, the looser Been, K. & Jefferies, M. G. (1985). A state parameter for sands.
the sand the smaller the flow liquefaction angle. The flow Géotechnique 35, No. 2, 99–112.
liquefaction line separates the liquefaction process into Casagrande, A. (1971). On liquefaction phenomena. Géotechnique
stable and unstable states in the stress path space. As a 21, No. 3, 197–202.
Castro, G., Enos, J. L., France, J. W. & Poulos, S. J. (1982).
result, if the stress conditions in an element of soil reach
Liquefaction induced by cyclic loading, Report No. NSF/CEE-
this line, flow liquefaction is to be triggered and the shear 82018. Washington, DC: National Science Foundation.
resistance will be reduced rapidly to the critical-state Ishihara, K. (1993). Liquefaction and flow failure during earth-
strength. quakes. Géotechnique 43, No. 3, 351–415.
Kramer, S. L. (1996). Geotechnical earthquake engineering. Engle-
wood Cliffs, NJ: Prentice Hall.
CLOSING REMARKS Lade, P. V. (1993). Initiation of static instability in the submarine
Owing to the complex nature and devastating effects of Nerlerk berm. Can. Geotech. J. 30, No. 6, 895–904.
the flow liquefaction of loose sand, its proper characterisa- Poulos, S. J. (1981). The steady state of deformation. J. Geotech.
tion has been, and continues to be, a very challenging topic. Engng Div., ASCE 107, No. 5, 553–562.
Roscoe, K. H., Schofield, A. N. & Wroth, C. P. (1958). On the
The collapse line (Sladen et al., 1985; Ishihara, 1993) and yielding of soils. Géotechnique 8, No. 1, 22–52.
flow liquefaction line (Vaid & Chern, 1985; Lade, 1993) are Schofield, A. N. & Wroth, C. P. (1968). Critical state soil mech-
two approaches that have been in widespread use. Both anics. London: McGraw-Hill.
approaches assume that the locus of peak points in the Seed, H. B., Lee, K. L., Idriss, I. M. & Makdisi, F. I. (1975). The
effective stress paths is a unique line that passes through slides in the San Fernando Dams during the earthquake of
either the steady-state point or the origin in stress path February 9, 1971. J. Geotech. Engng Div., ASCE 101, No. 7,
space. The slope of the collapse or flow liquefaction line is 651–688.
hence treated as a material constant irrespective of the state Sladen, J. A., D’Hollander, R. D. & Krahn, J. (1985). The liquefac-
of soil. tion of sands, a collapse surface approach. Can. Geotech. J. 22,
No. 4, 564–578.
In this study a new concept has been proposed that states
Vaid, Y. P. & Chern, J. C. (1985). Cyclic and monotonic undrained
that the flow liquefaction line is not unique but rather varies response of saturated sands. In Advances in the Art of Testing
with the state of soil. Experimental evidence for different Soils under Cyclic Conditions, pp. 120–147. American Society
sands was shown to clearly support this interpretation. With- of Civil Engineers.
in the framework of critical-state soil mechanics, an explicit Verdugo, R. & Ishihara, K. (1996). The steady state of sandy soils.
relationship was suggested between the slope of the flow Soils Found. 36, No. 2, 81–91.

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