The Effective Principal Stress Domain and Stress Locus of Soft Soil
Jeffery C. I. Chang
C & M Hi-Tech Engineering Co., Ltd., Taipei, Taiwan
ABSTRACT: This paper quote the following concepts proposed in the relative and time dependent soil theory
(Chang, 2001, pers. comm.) and the time dependent excessive pore water pressure (Chang & Wu, 2001, pers.
comm.): 1. The time dependent state is the condition that the effective principal stress ratio does not reach a
stable condition, i.e. Kt ﹤1 - sinφ’. 2. For soils in steady state, once the internal stress changed with the
amount of +Δσv, the soil skeleton will deform and slide. At this time, the time dependent excessive pore water pressure and the relative time dependent excess effective stress are rapidly induced in a saturated soft soil.
That is Δσv = Δσ’vt + Δut. 3. For soil in the time dependent condition, there exists an anisotropic time dependent effective principal stress. Such effective stress is the initial principal stress of the next change in the
external loading. 4. When the external loading keeps unchanged, the effective principal stress will continuously adjust to a static and steady condition. The absolutely static condition is that all the soil particles and internal stresses become static and stable. 5. The mechanical behaviors of soil is dependent on the ratio of the anisotropic time dependent principal stress, Kt. By determining the location of the initial stress in the effective
principal stress domain associated with appropriate soil classification, we can identify the type of soft soil. For
a known initial internal stress and stress characteristics, following the stress locus we can understand the mechanical behaviors including shearing failure, and primary and secondary consolidation of the soft soil under a
given loading condition.
1 PREAMBLE
Taipei soft soil (Wu, 1988; Lin and Huang, 1988;
Chen and Shieh, 1996) is an alluvium deposit. Based
on the initial stress condition can be considered as a
normally consolidated soft soil. From lots of practical experiences and field data, the author found that
the behaviors of such soil condition could not be explained with a static and absolute concept in an logical way. The author found that soil behaviors are relative and time dependent, it can be explained with
the time dependent excessive pore water pressure.
By using the following two new concepts associated
with the effective principal stress domain, we can
determine the stress path of the effective principal
stress to explain the shear failure, primary consolidation, secondary consolidation and passive failure of
the soft soil.
1. A normally consolidated soil consists of alluvium
soil, the internal stresses are in an equilibrium
condition. When the external loading is applied,
the effective principal stresses between soil particles will continuously adjust and reach to a new
stable condition. During the process of adjustment, the soil skeleton is in a time dependent condition (not static condition).
2. The mechanical behaviors of soils are depended
on the ratio of the time dependent effective principal stresses.
With these two new concepts associated with the
effective principal stress domain, we can then determine the stress path of the effective principal
stress to explain the shearing failure, primary consolidation, secondary consolidation and passive failure
of the soft soil.
2 TIME DEPENDENT SOIL BEHAVIORS
For soil in a stable condition, change of the external
loading will cause the positions and internal stresses
change relatively and then reach a new equilibrium
position. Based on the rate of deformation, the apparent mechanical behaviors of soil can be divided
into flow, shearing failure, primary consolidation,
secondary consolidation and elastic deformation.
Present instrumentation technique can be used to
measure the extent of change and to demonstrate that
the soil is under a time dependent change condition.
When soil in a complete stable state, the accumulated amount of change in each time period is the total change caused by external loading. That is
ΣD = ΣD(S) + ΣD(MC) + ΣD(SC)
= ∫Δυ(S)dt +∫Δυ(MC)dt +∫Δυ(SC)dt (1)
Soil start to move once external loading is applied. The movements could be in form of shearing
slide (S), the relatively time dependent movement of
primary consolidation (MC), and the relatively static
movement having secondary consolidation (SC).
Once the external loading keeps unchanged, the internal stresses between soil particles become stable
and the rate of movements become slow. As shown
in Figure 1, it reaches a relatively static, static-like or
absolutely static conditions that can not be measured
or felt. At this time, the accumulated change from
each time period will present another apparently stable result. For instance, “There is a 30cm settlement
which causes structure cracks of this structure which
was built twenty years ago.” Such description explains that when the loading of the structure was applied to soil, there was not immediate shear failure,
but resulted in 30cm settlement which was caused by
primary and secondary consolidation after twenty
years. It demonstrates that during the past twenty
years, the soil particles kept changing with only a
small rate of change in υave = 4.75×10-8 cm/sec per
day. The rate of change is large at the beginning and
small later. At this time, the soil condition is still
under change at a small rate. As long as the time of
consolidation is long enough, it will reach an absolutely static condition permanently. Even though it
reaches to the static-like condition, it still changes
slightly. The accumulated amount of change at later
stage no longer influence the structure. At this time,
the time dependent principal stress is close to the
static principal stress, i.e. the apparent total stress.
Figure 1. The time dependent and static relationship of soil.
3 RELATIONSHIP OF MECHANICAL
CHARACTERISTIC AND ENVIRONMENT
3.1 Mechanical characteristics of dry loose sand
Taking samples of Taipei soft soil, when dried in laboratory and tested, the following mechanical characteristics are observed:
1. Regardless SM, ML or CL types of soil, the angles of repose are about 30°±4°, in case of randomly accumulated in dry loose condition.
2. The effective friction angles obtained from DS,
CIU and CID tests in remolded loose soil samples
are about 30°±4°.
3. The cohesion c is close to 0 for dry, remolded
and normally consolidated soil. SinceΔut = 0, the
shear stressτ = σvtanφ = σ’vtanφ’. At this moment, σv = σ’v and φ = φ’.
4. Slowly increasing loading on horizontally
heaped loose soils could induce large settlement
without immediate shearing failure, but instantly
increasing loading or impact could result in shearing failure.
5. For any dry loose soils horizontally heaped to
stable state, the internal cemented stresses at any
depth can be represented as follows:
σv0 = σ’v = σv = γdz
(2)
and when φ’ ≒ 30°,
σh0 = σ’h = σh = (1-sinφ’) σv0 ≒0.5γdz
(3)
where σv0 and σh0 = vertical and horizontal internal
stresses in the final stable condition at this stage,
they are also the initial effective principal stresses at
the next change stage; γd = dry unit weight of soil;
σ = apparent total stress; and σ’ = effective stress.
Any dry soil in stable condition, i.e. after complete
consolidation, φ equals to φ’ and σ = σ’.
3.2 Mechanical characteristics of saturated loose
sand
The mechanical characteristics of saturated loose
sand is basically the same as dry soil, the only factor
influenced is the excessive pore water that can not
dissipated instantly. Such excessive pore water pressure results from the instant deformation of soil particles or applying loading at a very short time. It affects the ratio of the effective principal stresses. In
this case, a small loading can cause shearing failure.
When the rate of shearing deformation keeps increasing, the final minimum total friction angle is
the total friction angle φ generally presented soil
mechanics textbook.
In saturated loose sand, the effective unit weight
of soil isγsat – γw, if ground water level is located
at ground surface. For normally consolidated soil,
the anisotropic effective principal stress can be represented as:
σv0 = σ’v = σv = (γsat –γw)z
(4)
and when φ’ ≒ 30°,
σh0 = σ’h = σh = (1-sinφ’) σv0 ≒ 0.5(γsat –γw)z
(5)
For soil in such stress condition, the apparent
shear stress τv during the horizontal shearing failure
is equal to σv0tanφ. If the excessive pore water pressureΔ ut exists in the shearing plane of saturated
soil, the time dependent shear stress becomes:
τvt = (σv0 - Δut)tanφ’ = σ’vttanφ’ = σv0tanφ (6)
where Δut = the excessive pore water pressure generated at the shearing plane; σ’vt = the time dependent vertical effective principal stress between soil
particles along the shearing plane, it is affected by
the excessive pore water pressure.
3.3 The time dependent effective principal stress
(Chang, 2001a, pers. comm.)
For any stable state, Δut = 0, σv0 = σ’vt = constant,
and σh0 = K0(NC)σv0. In such stable condition, applications of external loading causes increment of internal stress +Δσvi. The soils will then be under a
consolidation or shearing failure process in which
the behavior is time dependent. Assume that it is in a
consolidation process, the stresses at this time period
are defined as follows:
Vertical pressure (Pvi) = vertical total loading per
unit area at time stage i, i.e. the apparent total pressure,
vertical total stress (σvi) = vertical apparent total
stress transmitted by soil particles at time stage i,
vertical time dependently effective stress (σ’vt) =
vertical effective internal stress transmitted by soil
particles at time t,
increment of internal stress (Δσvi) = the change
of stress at time stage i, it is predictable,
increment of time dependent effective stress (Δ
σ’vi) = Δσvi – Δut, it is also called as increment of
the excess effective principal stress,
vertical initial effective stress (σ’v(i-1)) = the time
dependent effective principal stress at time stage i-1,
it is the initial effective stress at time stage i, i.e.
σ’v0. When σ’v(i-1) becomes stable, σv(i-1) = σ’v(i-1)
=σ’v0, σh(i-1) = σ’h(i-1) =σ’h0.
time dependent excessive pore water pressure (Δ
ut) = the saturated pore water pressure that exceeds
static pore water pressure and can not dissipate instantly generated by the change of Δσvi. It can dissipate to zero.
static pore water pressure (u0) = pore water pressure resulted from ground water level.
The author proposes the time dependent effective
principal stress and the time dependent excessive
pore water pressure. They are different from the static concept provided in the present soil mechanics
theory. From the concept presented in Figure 2, we
get,
σ’vi = σ’v(i-1) +Δσvi
= σ’v(i-1) + (Δσ’vt + Δut) = σ’vt + Δut
(7)
Pvi = σvi + u0 = (σ’vt + Δut) + Δu0
(8)
In the above equations, σvi is vertical total stress
transmitted by soil particles and is the maximum total apparent stress. Δσ’vt and Δut are generated by
the change of external loading at time stage i. Both
Δσ’vt and Δu0 take loading of Δσvt. WhenΔu0
reaches to maximum, it will dissipate and such dissipated pressure transfers to effective stress between
soil particles. Since Δ σ’vt is a continuously increased variable, when σ’vt = σ’v(i-1) + Δσ’vt, (σ’vt is
also a variable, not a constant). Only in the condition
that Δut = 0, σ’vt will become constant. When Δut
= 0 or Δut ≒ 0, σvi = σ’vt, Pvi = σvi + u0 = σ’vt + u0,
that is the condition of the time dependent effective
stress keeps unchanged and equals to static effective
principal stress and apparent total stress.
σ vi=σ v(i-1)+△σ v
=σ v(i-1)+(△σ vt+△ut)
=(σ v(i-1)+△σ vt)+△ut
=σ vt+△ut
When △ut=0
σ vi=σ vt=σ vi
uo
Pvi=σ vi+ uo=(σ vt+△ ut)+ uo
When △ut=0
Pvi=σ vi+ uo=σ vt+ uo=σ vi+ uo
△ut=time dependent excess pore water pressure
σ vt=time dependent effective principal stress
in vertical direction
Figure 2.
Time
dependent
effec-
tive principal stress.
4 JUDGMENT OF SOFT SOIL FROM
MECHANICAL POINT OF VIEW
4.1 A subjective point of view on soft soil
1. The soil that will cause shearing failure and large
settlement.
2. With clay soil with SPT-N value less than 4 and
sandy soil with SPT-N value less than 10 (Inata,
1980) for soil with ground water table located at
or near the ground surface.
3. Saturated alluvium deposits for which w ≧ 36%,
e0 ≧ 1.0 and LL ≒ w.
4. Geotechnical engineers think the soft deposit is
soft clay or soft silt deposit. Such soft deposit is
distributed over coast, marsh, lake or riverbank.
Soft deposit is a normally alluvium with SPT-N
value less than 4
4.2 Form of soft soil and its initial internal stress
Taipei soft deposit can be divided into the following
two types. They are as follows:
1. Normally consolidated soil (NC soil): The primary and secondary consolidation are completed,
and Δut = 0, σvi = σvt = constant and σhi = σht =
K0(NC)σvi = constant.
2. Under consolidated soil (UNC soil): After the last
filling or loading, the soil is undergoing the primary consolidation process, i.e. Δut ≠ 0, σvt and
σht are still variables or in a secondary consolidation process. In this case, Δut ≒ 0, σvi = σvt =
constant, but σ’ht ＜ (1 – sinφ’)σ’vt ≠ σvi.
A normally consolidated soft soil is an alluvial
deposit. It is in a static condition which the cemented
consolidation has been completed. Many articles
(Lambe, 1969; Das, 1998; Craig, 1992) shows that
the internal stresses of such soil are located on the
K0(NC) line of p-q diagram. The effective principal
stress of this soil at any given depth is shown in Figure 3.
z1
σh
m
When Ko(NC)= σ v
=1-sinφ
sat-1
zi
σ vz=γm z1+(γsat -1)(zi -z1)
=σ vz
σ hz=σ vz(1-sinφ)
=σ hz
σ vz
σ hz
4.3 The influence of external loading on dry soft
soil
For normally consolidated soft soil in dry condition,
change of the external loading should induce the adjustment of internal stresses and change of voids.
Since there is no pore water in dry soil, change the
external loading will not induce any time dependent
excessive pore water pressure, i.e. Δ ut = 0. As
shown in Figure 4, at the moment of the external
loading (+ΔPt) changes, the increment of the vertical effective stress at any given depth (Δσv) can be
calculated from the theory of elasticity, and then
plotted in p-q diagram. The process of the change in
internal effective stress is shown in point 1 to point 6
of Figure 4. If point 6 does not exceed Ka, i.e. Kt ﹥
K0(NC), the internal stress will not be in a permanently stable condition, since the horizontal effective
principal stress is too small. Therefore the horizontal
stresses between soil particles will continuously adjust. It is followed by the sliding movements of soil
particles, the settlement induced by secondary consolidation in greater depth, and then position readjusting to top soil skeletons. Such adjustment continuous until the ratio of effective principal stress
reaches to a permanent stable condition K0(NC). The
process of change of internal stresses at this stage is
represented in point 6 to point 7 of Figure 4. Normally, the secondary consolidation will adjust back
and forth. The time to complete secondary consolidation is much longer than primary consolidation.
From practical experiences, in 10m thick sandy soil,
it takes about 6 to 12 months. In clayey soil, the time
to complete secondary consolidation and reach a
static state could take more than 30 to 60 years.
Figure 3. The initial effective principal stress of normally consolidated soil.
σvf = σv0 = σ’v0
= γmz1 + (γsat - γw)(z – z1)
(9)
σhf = σh0 = σ’h0 = K0σ’v0 = (1 – sinφ’)×
[ γmz1 + (γsat - γw)(z – z1)]
(10)
Kt = σ’ht/σ’vt
= time dependent effective principal stress
(11)
Pvi = σvi + u0 = (σ’vt + Δut) + Δu0
(8)
where σvf is vertical principal stress at depth z during
completion of the (i–1)th cemented consolidation. It
is also the vertical principal stress σv0 at the beginning of the ith cemented consolidation. When normal
consolidation condition is attained, σv0 = σ’v0.
From the data shown in the articles (Lambe,
1969; Das, 1998; Craig, 1992), for normally consolidated soft deposit, the ratio of time dependent principal stress is located in the zone between K0(NC) line
and Ka line of p-q diagram, i.e. K0(NC) ﹤ Kt ﹤Ka.
Figure 4. The influence of time dependent excessive pore water
pressure parameter on the stress paths of effective principal
stress in p-q diagram.
4.4 The influence of external loading on saturated
soft soil
A saturated soft soil is generally referred to an alluvial deposit with high ground water level. When the
effective principal stresses between soil particles
reach to static state of normally consolidated condi-
tion, the stress path will be located at K0(NC) line in
p-q diagram.
In saturated soft soil, the initial effective principal
stresses at the same depth are same for all soil particles, as shown in point 1 of Figure 4. Assume that
there are three types of soil, clay, silt and sand with
Skempton A pore pressure parameters ACL = 1.0,
AML = 0.5, and ASM = 0.0, respectively. At the instant of applying the same loading Δσv on these
soils, the induced time dependent excessive pore water pressures will be totally different. For soil A
(clay), ACL equals to 1.0, at the instant of applying
Δσv, Δut ≒ Δσv. For soil B (silt), AML equals to
0.5, at the instant of applying Δσv, Δut = 0.5Δσv.
For soil C (sand), ASM equals to 0.0, at the instant of
applying Δσv, Δut = 0.0.
At the time of applying Δσv, the stress paths of
the effective principal stresses will move from point
1 to point 2, point 1 to point 3, and point 1 to point 4
for soil A, soil B and soil C, respectively. During the
process of primary consolidation, they are from point
2 to point 4, point 3 to point 4 and point 4 to point 4,
for soil A, soil B and soil C, respectively. In secondary consolidation stage, all of them move from point
4 to point 5.
For the three types of soil shown in Figure 4, the
magnitude of parameter A is proportional to the inducedΔut. For large value of parameter A, the stress
path will turn to left as loading applying. When parameter A is equal to 1 or 0, the lengths of the vectors in p-q diagram will be the same, but in a 90° difference.
The major discrepancy of applying the same loading on saturated soft soil and dry soft soil is that the
time dependent excessive pore water pressure will be
generated at the instant of load application. This
makes the stress path in p-q diagram turning a direction of 90°. Such excessive pore water pressure
brings soil to primary consolidation condition. For
secondary consolidation, the stress changes of the
saturated and dry soft soil are the same, but saturated
soft clay or silt need longer consolidated time to
reach the final settlement.
5 STRESS DOMAIN OF SOFT SOIL
5.1 The initial internal stress of soft soil and its
stress domain
At any depth of a soil, when it reaches to a state of
normal consolidation condition, the initial internal
stress can be calculated from Equations 9 and 10.
Normally consolidated soil can be considered as
normally consolidated soft soil. We can describe soil
exactly with classification of physical properties,
mechanically related parameter A and consolidated
condition. For example, normally consolidated soft
clay with A = 0.9 or UNC clay with A = 1 and degree of consolidation U = 50%. In a more precise
way, the above descriptions explain the initial internal stress of soft soil and its relationship with the
time dependent excessive pore water pressure. For
normally consolidated soft soil, the stress locus of all
the effective principal stresses at any depth are distributed along K0(NC) line in p-q diagram. For soft
soil under consolidation, the stress locus is located at
zone between Ka line and K0(NC) line. It can be considered as a very soft soil. The ratio of time dependent effective principal stresses can be used to cover
all the stress ratios from the time of change to static
state. Figure 5 shows that when Kt = K0(NC), the soil
is NC soft soil and when K0(NC) ﹤ Kt ﹤Ka, the soil
is UNC soft soil.
+q
K a -line
A≒0,+△σ v
K o(NC) -line
A≒0.5,+△σ v
A≒1,+△σ v
O 1.5-line
A≒0,+△σ h
A≒1,+△σ h
A≒0.5,+△σ h
p
1 -line
K o(NC)
-q
K p-line
Figure 5. The effective principal stress domain and stress locus
of soft soil.
5.2 The stress path of soft soil
Large building on soft soil is the major geotechnical
engineering problem. Shearing failure of the building in soft ground caused by insufficient bearing capacity is not often emphasis. However large settlement caused by long term consolidation is quite
often which it makes foundation crack and supperstructure deform. The time dependent effective principal stress path shown in Figure 5 can explain the
behaviors of heave failure (shearing failure), piping
or large lateral deflection in deep excavation in soft
soil deposit. The stress paths shown in Figure 5 are
explained as follows:
1. Point 1 → point 2: For normally consolidated
clay with A = 1, when the applied loading is
greater than σ’v/6, shearing failure will occur immediately. Lateral unloading in the excavated side
of deep excavation in sand or clay deposit, i.e.
applying a –Δσ’h loading, will make the stress
path to Ka line (active failure failure).
2. Point 1 → point 3 → point 4 → point 5: For normally consolidated clay with A = 1, it is located at
point 3 when vertical loading is applied. No
shearing failure but settlements caused by primary
consolidation (point 3 to point 4) and secondary
consolidation (point 4 to point 5) occurs.
3. Point 1 → point 6: For normally consolidated silt
with A = 0.5, large vertical loading causes instant
shearing failure.
4. Point 1 → point 7 → point 8 → point 9: For normally consolidated silt with A = 0.5, it is located
at point 7 when vertical loading is applied. No
shearing failure but settlements caused by primary
consolidation (point 7 to point 8) and secondary
consolidation (point 8 to point 9) occurs.
5. Point 1 → point 10: For normally consolidated
sand with A = 0, large vertical loading causes instant shearing failure.
6. Point 1 → point 11 → point 12: For normally
consolidated sandy gravel with A = 0, applying a
large vertical loading instantly will not cause
shearing failure. But it induces instant primary
consolidation settlement (point 1 to point 11) and
secondary consolidation settlement (point 11 to
point 12).
In Figure 5, the initial effective principal stress of
the normally consolidated soft soil is in point 1.
When a visible passive failure or an invisible settlement occurs in this case, the time dependent effective principal stress should locate in the zone bounded by point 1, 2, 10 and 13. When it reaches to a
long-term static state, the effective principal stress
should locate on K0(NC) line.
The initial stress condition of the above normally
consolidated clay or silt is located at point 1 of Figure 5. When any visible passive failure or heave
happen, the final effective principal stresses will be
in the zone bounded by point 15, 16, 17 and 18.
6 CONCLUSION
The use of time dependent effective principal stress
and its stress ratio can exactly define the normally
consolidated soft soil and very soft soil. By incorporating Skempton parameter A and the description of
soil classification, one can precisely determine the
mechanical characteristics of soft soil. For any given
degree of consolidation U or thickness of overburden
and its history, we can then estimate its internal
stress condition.
Form p-q diagram, we know that the stress path
of normally consolidated soft soil should be located
on K0(NC) line and very soft soil between Ka line and
K0(NC) line. In this way, we can find the initial position of the internal stress of soft soil (σ’v0, σ’h0), variations of internal stresses (±σv, ±σh) caused by external loading and maximum time dependent
excessive pore water pressure induced in the instant
of internal stress change. We can then plot these variables in p-q diagram to clearly identify its stress
path for all the processes of mechanical behaviors.
With the effective principal stress domain and its
stress paths, we can predetermine the behaviors of
soft soil, safely avoid the causes of failures.
7 PREFERENCES
Chang, Jeffery C. I. 2001. The Relative and Time Dependent
Soil Theory. Third International Conference on Soft Soil
Engineering: accepted for publications.
Chang, Jeffery C. I. and Wu, Jason Y. 2001. The Effect of Unsteady Excess Pore Water Pressure on Soil Stability. Third
International Conference on Soft Soil Engineering: accepted for publications..
Wu, Wei Teh. 1988. Geotechnical Engineering of Soils in Relation to Horizontal Zoning in Taipei Basin. SinoGeotechnics, No. 22: 5-27.
Lin, Hsiuan and Shieh, Bai-Jung. 1988. The Undrained Shear
Strength of Cohesive Soils in Taipei Basin. SinoGeotechnics, No. 22: 75-79.
Inata, Baiho. 1980. Soil Mechanical of Soft Deposit (in Japanese). Kajima.
Lambe, T. W. and Whitman R. V. 1969. Soil Mechanics. John
Wiley & Sons
Das, B. M. 1998. Principles of Geotechnical Engineering.
Fourth Edition, Thomson Information Publishing Group.
Craig R. F. 1992. Soil Mechanics. Fifth Edition, Chapman &
Hall.