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Computers and Geotechnics 104 (2018) 54–68
Contents lists available at ScienceDirect
Computers and Geotechnics
journal homepage: www.elsevier.com/locate/compgeo
Research Paper
Predicting deformation of multipropped excavations in soft clay with a
modified mobilizable strength design (MMSD) method
T
⁎
L.Z. Wanga,b, Y.J. Liub, Y. Hongb, , S.M. Liuc
a
City College, Key Laboratory of Offshore Geotechnics and Material of Zhejiang Province, Zhejiang University, China
Key Laboratory of Offshore Geotechnics and Material of Zhejiang Province, College of Civil Engineering and Architecture, Zhejiang University, China
c
PowerChina Huadong Engineering Corporation, China
b
A R T I C LE I N FO
A B S T R A C T
Keywords:
Modified mobilizable strength method
Multipropped excavation
Deformation
Soft clay
Field validation
Design charts
Although serviceability (i.e., deformation) governs the performance of deep excavations in urban areas with soft
clay deposits, the common design methods are only able to check the potential of base stability, rather than the
deformation. This limitation has led to the development of an analytical approach entitled the mobilizable
strength design (MSD) method, which can predict both serviceability and safety in a single step of calculation.
The modification of the MSD method (i.e., the MMSD method) was recently made by the authors, by implementing a more realistic plastic deformation mechanism into the calculation. The objectives of this study are
(a) to validate the MMSD method by comparing its predicted deformation with eight well-documented case
histories in Shanghai and Hangzhou soft clay and (b) to quantify the influence of some potentially important
factors on the deformation of deep excavations in Shanghai and Hangzhou. The accuracy of the MMSD predictions for the maximum lateral wall displacement (δh-max) have been found to fall within 35% of the field data
in most of the cases. This accuracy is more satisfactory than the accuracy (i.e., 120%) of the MSD predictions for
the corresponding case histories, suggesting the effectiveness of the modified plastic deformation mechanism
considered in the MMSD calculation. Based on the verified MMSD method, preliminary design charts that predict
δh-max of narrow excavations (specifically for a metro station) considering different geometries and strength
mobilization characteristics have been developed.
1. Introduction
Th unsatisfactory performance of deep excavations in urban areas
with soft clay deposits is usually attributed to excessive deformation,
which could impose damage on the fragile neighboring infrastructures.
In contrast, the conventional limit analysis methods based on the plastic
limit equilibrium analysis [61,2,62,21] only check against collapse,
which is associated with large deformations far exceeding the limits
permitted in practice. This conventional limit analysis has necessitated
the development of methods for excavation-induced deformation in the
past decades.
The commonly adopted methods for predicting excavation-induced
deformation involve the semi-empirical approach that generalizes
trends from similar types of case histories or model tests
[52,9,50,37,44,23,64,31,59,60,10,22,8,33],
numerical
analysis
[54,55,46,28,41,45,18,53,57]
and
probabilistic
methods
[39,51,7,74,13]. One major deficiency of the semi-empirical approach
is that the expansion of the database may even lead to contradictions
⁎
[44], rather than a more generalized trend in deformation, because the
semi-empirical approach cannot inherently account for the non-linear
soil behavior and construction sequences. These case-specific conditions can be inherently considered in a numerical analysis. Satisfactory
numerical predictions for excavation-induced deformation could be
achieved with the use of sophisticated constitutive models considering
the non-linearity of soil stiffness at small strains, such as the Brick
model [55], MIT-E3 model [15], hardening soil small-strain model [1],
and the hypoplastic models with the intergranular strain concept [42].
Most of these sophisticated constituted models contain quite a few
parameters, which are sometimes not easy to calibrate [53,18]. The
successive use of numerical analysis also requires the designer having a
solid base in soil mechanics and finite elements.
In view of the over-simplification of the semi-empirical approach
and the relative complication of the numerical analysis, Osman and
Bolton [49] proposed a relatively simple analytical approach (mobilizable strength method, or MSD) for predicting both deformation and
stability of multipropped wide excavations (B > 2 2 (l−h), where
Corresponding author.
E-mail addresses: wanglz@zju.edu.cn (L.Z. Wang), yajing_liu@zju.edu.cn (Y.J. Liu), yi_hong@zju.edu.cn (Y. Hong), liu_sm@ecidi.com (S.M. Liu).
https://doi.org/10.1016/j.compgeo.2018.07.018
Received 31 January 2018; Received in revised form 8 July 2018; Accepted 16 July 2018
0266-352X/ © 2018 Elsevier Ltd. All rights reserved.
Computers and Geotechnics 104 (2018) 54–68
L.Z. Wang et al.
B = excavation width, l = wavelength of the deformation and h = excavation depth below the lowest prop) in clay. The MSD method can
consider the case-specific construction sequences and non-linear soil
behavior without the use of a constitutive model by assuming a plastic
deformation mechanism and a working stress field around a multipropped excavation. Based on the equilibrium at any given excavation
stage, mobilized average shear strength can be deduced and converted
to mobilized average shear strain through the measured represented
stress-strain relationship of the in situ soil. The incremental deformation can then be calculated based on the mobilized average strain and
the assumed plastic deformation mechanism (details to be given later).
The applicability of Osman and Bolton’s [49] MSD method was later
extended for predicting the deformation of narrow excavations
(B < 2 2 (l−h)) by Lam and Bolton [26].
The authors [65,36] recently proposed a modified MSD method
(i.e., MMSD method) for both wide and narrow excavations, by implementing a more realistic ground deformation mechanism into the
MSD methods [49,26]. Although the MMSD method has been shown to
reasonably predict the base instability in three case histories [65], the
predictive capability of the MMSD method for excavation-induced deformation has not been examined.
This study aims to validate the MMSD method by comparing its
predicted deformations with seven well-documented case histories in
Shanghai soft clay and one case history in Hangzhou soft clay and the
corresponding predictions made by the MSD method. Parametric studies using the verified MMSD method are also performed to quantify
the influence of excavation geometries and strength mobilization
characteristics on the deformation of narrow excavations (specifically
for metro stations) in Shanghai. Preliminary design charts considering
these factors are then developed with the aim of assisting decisionmaking prior to any detailed analysis in the future design.
Δδ h
Δδ h
Δδ h
Δδ h
(a)
Δδ h
2. A modified mobilizable strength design (MMSD) method
Δδ h
2.1. Modified incremental wall displacement profile
Δδ h
In the MSD method, the incremental profile of the ground displacement below the lowest prop is idealized to be a cosine function
(see Fig. 1), as proposed by O’Rourke [48]. The cosine function is formulated by Eq. (1):
2πy ⎤
⎞
Δδh = Δδh-max ⎡1−cos ⎛
⎝ l ⎠⎦
⎣
(b)
(1)
Fig. 2. Plastic ground deformation mechanism in the MSD method: (a) wide
excavation [49]; (b) narrow excavation [26].
where Δδh-max and Δδh denote the maximum incremental displacement
and the incremental displacement at any distance y underneath the
lowest prop, respectively. The parameter l represents the full wavelength of the deformation profile, which is correlated to the length of
the wall below the lowest prop (s), as shown in Fig. 1, as follows:
l = αs
(2)
The parameter α in Eq. (2) depends on the fixity condition at the
Fig. 1. Incremental displacement profile for a multipropped excavation assumed in the MSD [49] and MMSD [65] methods.
55
Computers and Geotechnics 104 (2018) 54–68
L.Z. Wang et al.
wall toe. It is bounded by a lower limit of 1 for a short wall embedded in
stiff clay [49] and an upper limit of 2 for a short wall embedded in very
soft clay [14]. For simplicity, the value of α was usually taken as a
constant during the entire excavation process [49,5]. In this study, the α
was optimized after each increment of calculation to satisfy the principle of minimum energy dissipation, as suggested by Wang and Long
[65].
After re-visiting the typical deformation profile of multipropped
excavations in clay, Wang and Long [65] found that the incremental
profile of lateral wall displacement can be more realistically represented by an exponential family distribution. The following exponential family function (Eq. (3)) is, therefore, adopted by the authors
to replace the cosine function:
Δδ h
Δδ h
Δδ h
Δδh = Δδh-max
4y
1 8y 2
exp ⎛ − 2 ⎞
l
⎝2 l ⎠
⎜
⎟
(3)
The newly proposed distribution in Eq. (3) forms the basis for
modifying the ground deformation mechanism of the MSD method (to
be presented in the following section).
(a)
2.2. Modified ground deformation mechanism
Fig. 2(a) shows the plastic ground deformation mechanism assumed
in the MSD method for a multipropped wide excavation [49]. In the
mechanism, the incremental profile of the soil deformation also follows
the cosine function defined in Eq. (1), to ensure the deformation
compatibility between the soil and the wall. The plastic deformation
mechanism consists of four zones of distributed shear, namely, a rectangular zone (ABCD) on the retained side above the lowest prop, a
circular fan zone (CDE) centered at the lowest prop, a circular fan zone
(FEH) centered at the interception between the wall and the excavation
bottom, and a triangular zone (FHI) under the excavation bottom. Each
dotted line with arrows, which passes across the four zones of distributed shear, represents an equal-deformation line. By doing so, the
soil within the four zones deforms continuously and compatibly, with
no occurrence of sliding between each pair of the neighboring zones.
Lam and Bolton [26] further extended the applicability of Osman and
Bolton’s [49] MSD method for wide excavations to the scenario of
narrow excavations by replacing the mechanism in the passive zones
(FEH and FHI, see Fig. 2(a)) with a rectangular one (FEHI, see
Fig. 2(b)).
Based on the aforementioned framework, the authors [65,36]
modified the ground deformation mechanisms by implementing a more
realistic incremental deformation profile (see Eq. (3)). Fig. 3(a) and (b)
illustrates the modified ground deformation mechanisms for wide and
narrow multipropped excavations in clay, respectively.
Δδ h
Δδ h
Δδ h
(b)
Fig. 3. Plastic ground deformation mechanism in the MMSD method: (a) wide
excavation [65]; (b) narrow excavation [36].
Table 1
Comparison of engineering shear strain increment of a wide excavation calculated by the MMSD and MSD methods.
Zone
ABDC
Engineering shear strain increment calculated by MMSD method for a wide
excavation
δγ = Δδh-max
4
l
(1− x ) exp ⎛⎝ −
16 2
l2
1
2
Engineering shear strain increment calculated by MSD method for a wide excavation
π
l
8x 2
⎞
l2
δγ = −Δδh-max sin
⎠
x = distance from the wall
CDE
δγ = Δδh-max
64r 2
1 8(r + h)2
⎞
exp ⎛ −
2
l3
l2
π
l
δγ = −Δδh-max sin
⎝
⎠
r = radial distance from the center of the circular arc (D)
EFH
FIH
64
−Δδh-max 3 (r
l
1 8(r + h)
h)2 exp ⎛ −
l2
⎝2
4
+
(1−
16
(r
l2
)
1
8(r + h)2
⎞
+ h)2 exp ⎛ −
l2
⎝2
⎠
r = distance along FH between point F and the intersection with a plastic flow
line
δγ = −Δδh-max
l3
2πr
l
+
Δδh-max
2r
(1−cos )
2πr
l
r = radial distance from the center of the circular arc (D)
2
⎞
⎠
r = radial distance from the center of the circular arc (F); and h = distance
between the lowest support and the excavation level
δγ =
2πx
l
x = distance from the wall
56
π
l
δγ = −Δδh-max sin
2π (r + h)
l
+
Δδh-max
2r
(1−cos
2π (r + h)
l
)
r = radial distance from the center of the circular arc (F); and h = distance between
the lowest support and the excavation level
π
l
δγ = −Δδh-max sin
2π (r + h)
l
r = distance along FH between point F and the intersection with a plastic flow line
Computers and Geotechnics 104 (2018) 54–68
L.Z. Wang et al.
Table 2
Comparison of engineering shear strain increment of a narrow excavation calculated by the MMSD and MSD methods.
Zone
ABDC
Engineering shear strain increment calculated by MMSD method for a narrow excavation
δγ = Δδh-max
4
l
(1− x ) exp ⎛⎝ −
16 2
l2
1
2
π
l
8x 2
⎞
l2
δγ = −Δδh-max sin
⎠
δγ =
1 8r 2
64
Δδh-max 3 r 2 exp ⎛ − 2 ⎞
2
l
l
δγ =
16y 2
4
π 2l
1 8y 2
Δδh-max ⎧⎛ 3 ⎛1− 2 ⎞− 2 ⎞ exp ⎛ − 2 ⎞
2
l
4B
l
⎨ l
π
l
δγ = −Δδh-max sin
⎝
⎠
r = radial distance from the center of the circular arc (D)
FIEH
⎟
2πr
l
+
Δδh-max
2r
(1−cos )
2πr
l
r = radial distance from the center of the circular arc (D)
2
2
π 2le−7.5 ⎫
πx
cos
B
4B2 ⎬
( )
+
⎝
⎠
⎠
⎠
⎩⎝ ⎝
⎭
x = distance from the wall; y = vertical distance from the interception between the wall and
the excavation bottom (F)
⎜
2πx
l
x = distance from the wall
x = distance from the wall
CDE
Engineering shear strain increment calculated by MSD method for a
narrow excavation
DSS: direct simple shear
PSA: plain strain active
PSP: plain strain passive
(
)
2π y
2πy
π l
πl
π
πx
δγ = Δδh-max ⎡ 2 + 2 +
− sin ⎤ cos
l ⎦
B
4B l
4B2 l
⎣ 4B
x = distance from the wall; y = vertical distance from the midpoint of the
FE
( )
normalized maximum incremental lateral wall displacement Δδh-max / l .
It is therefore anticipated that the shear strain increment (δγ) in the
solution of the MMSD method differs from that in the MSD method due
to the modified ground deformation mechanism. Table 1 compares the
equations of the incremental shear strains that are compatible with the
deformation mechanisms defined in the MSD and MMSD methods for a
wide excavation. The shear strain increments in the MSD and MMSD
methods for a narrow excavation are compared in Table 2.
2.4. Shear mode and strength mobilization within the mechanism
The incremental mobilization of shear strain (as formulated in
Tables 1 and 2 is accompanied by mobilization of shear strength, which
depends on the mode of shearing. Fig. 4 illustrates the shearing modes
(indicated by the orientation of the major principal stress) of the soil
within the assumed plastic mechanism in the MMSD method for wide
excavations. The soil at different locations around the wide excavation
is subjected to three distinct shearing modes, i.e., direct simple shear
(DSS), plain-strain active (PSA) and plain-strain passive (PSP).
To account for the degree of strength mobilization at each excavation stage, a parameter β is defined as follows:
Fig. 4. Shear modes (directions of major principal stress) compatible with the
plastic deformation mechanism in the MSD and MMSD methods for a wide
excavation.
β = su_ mob/ su
(4)
where su_mob and su denote the mobilized and the undrained shear
strength of soil. The value of β is deduced based on the principle of
energy conservation, as given in the following section.
2.3. Modified distribution of shear strain
2.5. Energy conservation
Since the soil deformation mechanism is compatible with the lateral
wall displacement, the incremental shear strain (due to excavation) in
each zone of the proposed mechanism should be proportional to the
Based on the principle of energy conservation, the virtual loss of
potential energy resulting from the incremental ground settlement (Δδ v )
Table 3
Comparison of incremental vertical displacement of a wide excavation calculated by the MMSD and MSD methods.
Zone
ABCD
Incremental vertical displacement calculated by MMSD method for a wide excavation
Δδ v = Δδh-max
4x
1 8x 2
exp ⎛ − 2 ⎞
l
⎝2 l ⎠
Δδ v =
Δδ v = Δδh-max
4r cosw
1 8r 2
exp( − 2 )
2
l
l
Δδ v =
Δδ v = −Δδh-max
4(r + h)
1 8(r + h)2
⎞ sinw
exp ⎛ −
l
l2
⎝2
⎠
Δδ v = −Δδh-max
Δδh-max cosw
2πr
⎡1−cos
2
l
⎣
Δδ v = −
( )
(
Δδh-max sinw
⎡1−cos 2π (r + h)
2
l
⎣
) ⎤⎦
r = radial distance from the center of the circular arc (F)
r = radial distance from the center of the circular arc (F)
FIH
( ) ⎤⎦
⎤
⎦
r = radial distance from the center of the circular arc (D)
r = radial distance from the center of the circular arc (D)
EFH
Δδh-max
2πx
⎡1−cos
l
2
⎣
x = distance from the wall
x = distance from the wall
CDE
Incremental vertical displacement calculated by MSD method for a wide
excavation
4(r + h)
1 8(r + h)2
⎞ sin π
exp ⎛ −
l
l2
⎝2
⎠ 4
Δδ v = −
(
Δδh-max
⎡1−cos 2π (r + h)
l
2
⎣
) ⎤⎦ sin
π
4
r = radial distance from the center of the circular arc (F)
r = radial distance from the center of the circular arc (F)
57
Computers and Geotechnics 104 (2018) 54–68
L.Z. Wang et al.
Table 4
Comparison of incremental vertical displacement of a narrow excavation calculated by the MMSD and MSD methods.
Zone
ABCD
Incremental vertical displacement calculated by MMSD method for a narrow excavation
Δδ v = Δδh-max
4x
1 8x 2
exp ⎛ − 2 ⎞
l
⎝2 l ⎠
Δδ v = Δδh-max
4r cosw
1 8r 2
exp ⎛ − 2 ⎞
l
⎝2 l ⎠
Δδ v = −
Δδh-max
2πx
⎡1−cos
2
l
⎣
Δδ v =
Δδh-max cosw
2πr
⎡1−cos
2
l
⎣
( )
( )
⎤
⎦
r = radial distance from the center of the circular arc (D)
r = radial distance from the center of the circular arc (D)
FIEH
Δδ v =
⎤
⎦
x = distance from the wall
x = distance from the wall
CDE
Incremental vertical displacement calculated by MSD method for a narrow
excavation
πlΔδh-max ⎡
πx
1 8y 2
exp ⎛ − 2 ⎞−e−7.5⎤ sin
B
4B
⎢
⎥
⎝2 l ⎠
( )
Δδ v = −
⎣
⎦
x = distance from the wall; y = vertical distance from the interception between the wall
and the excavation bottom (F)
lΔδh-max
⎡π
4B
⎣
+
2πy
l
+ sin
( ) ⎤⎦ sin ( )
2πy
l
πx
B
x = distance from the wall; y = vertical distance from the midpoint of the FE
Beijing
Shanghai
(a)
(b)
Fig. 5. (a) The location of Shanghai in the People’s Republic of China; (b) the locations of the seven case histories and soil sampling for the validation proposed in
Shanghai.
58
Computers and Geotechnics 104 (2018) 54–68
L.Z. Wang et al.
Fig. 6. A simplified soil profile and geotechnical parameters in Shanghai [5].
is assumed to be balanced by the virtual plastic work done in shearing
the soil to a certain degree of strength mobilization (β):
∫v γsat Δδv dv = ∫v βsu δγdv
(5)
where γsat and v denote the saturated weight of soil per unit volume and
the whole volume of the deformed zones, respectively. Numerically, β is
calculated using Eq. (6) derived from the principle of virtual work, i.e.,
virtual loss of potential energy (∫v γsat Δδ v dv ) balancing the virtual
plastic work in distributed shearing (∫v βsu δγdv ) .
β=
∫v γsat Δδv dv
∫v su δγdv
(6)
In Eq. (6), the incremental exponential shear strains (δγ ) are calculated based on the equations summarized in Tables 1 and 2 for wide
and narrow excavation, respectively. Additionally, the incremental
vertical displacement (Δδ v ) of wide excavation and narrow excavation
in Eq. (6) is calculated based on the equations summarized in Tables 3
and 4, respectively. By substituting the equations in Tables 1–4 into Eq.
(6), the β value for wide and narrow excavations can be calculated
numerically using Eqs. (7) and (8), respectively.
(
) exp ( − ) dxdy
exp ( −
) rdrdw
exp ( −
) rdrdw
(1− ) exp ( − ) rdrdw⎞⎠
exp ( − ) dxdy
exp ( − ) coswdrdw
exp ( −
) sinwdrdw
exp ( −
) sin dr ⎞⎠
H−h
l
4
16x
β = ⎛∫0
∫0 su l 1− l2
⎝
π
l
π
l
+ ∫0 2 ∫0 su
64r 2
l3
+ ∫0 4 ∫0 −su
π
l
/ ⎛∫0
⎝
4
∫
π
l
4r 2
l
l
1
2
− ∫0 γsat
4r (r + h)
l
2
1
2
⎜
⎝
0
π
4x
1 8x 2
exp ⎛ − 2 ⎞ dxdy + ∫0 2
0 sat l
⎝2 l ⎠
1
πl
∫hl ∫0B γsat 4B ⎡⎢exp ⎛ 2 −
⎣
⎝
∫0l γsat
π 2le−7.5 ⎤
πx
cos
B
4B2 ⎥
⎦
( )
⎞
dxdy ⎟/
⎠
4r 2
1 8r 2
exp ⎛ − 2 ⎞ coswdrdw−
l
⎝2 l ⎠
8y 2 ⎞ −7.5⎤
πx
−e
sin
dxdy ⎟⎞
B
⎥
l2 ⎠
⎦
( )
⎠
(8)
The maximum incremental wall deflection is calculated by substituting the equations in Tables 1 and 2 into Eq. (9).
8(r + h)2
l2
π
4
⎟
⎛⎜∫ H − h ∫l γ
8(r + h)2
l2
8r 2
l2
8(r + h)2
l2
2
2
16y
4
1 8y
π l
∫0H − h ∫0l su ⎡⎢ ⎛ 3 ⎛1− 2 ⎞− 2 ⎞ exp ⎛ 2 − 2 ⎞ +
l ⎠ 4B ⎠
l ⎠
⎝
⎣⎝ l ⎝
8(r + h)2
l2
8x 2
l2
1
2
1
2
π
4
16x 2
1 8x 2
64r 2
1 8r 2
H−h l
β = ⎜⎛∫0
∫0 su l ⎛1− 2 ⎞ exp ⎛ 2 − 2 ⎞ dxdy + ∫0 2 ∫0l −su 3 exp ⎛ 2 − 2 ⎞ rdrdw+
l ⎠
l ⎠
l
l ⎠
⎝
⎝
⎝
⎝
8(r + h)2
l2
1
2
1
2
l
4r (r + h)
γ
l
0 sat
− ∫0 ∫
8x 2
l2
16(r + h)2
l2
l
4x
γ
0 sat l
+ ∫0 2 ∫0 γsat
π
4
1
2
1
2
64(r + h)2
l3
+ ∫0 4 ∫0 −su 3
l
H−h
2
Fig. 7. Undrained soil strength profile adopted for MMSD and MSD calculations
(a) Shanghai soft clay; (b) Hangzhou soft clay. (See above-mentioned references
for further information.)
δγmob (β ) =
(7)
∫v δγ (Δδh-max ) dv
∫v dv
(9)
where the mobilized shear strain increment (δγmob ) can readily be deduced based on the known β value and a given stress-strain curve. The
59
Computers and Geotechnics 104 (2018) 54–68
L.Z. Wang et al.
Table 5
Geometrical properties of the eight case histories analyzed in this study.
Station
Thickness of diaphragm wall
(mm)
Wall depth
(m)
Maximumexcavationdepth (m)
Excavation length on plane
(m)
Excavation width on plane
(m)
Location
Yishan Station
South Xizan Station
South Pudong Station
Pudongdadao Station
Pudian Station
Yanchang Station
Gubei Station
West Wenyi Station
600
800
600
600
600
600
600
800
28.0
38.0
27.0
27.0
26.5
27.0
26
38.5
15.5
20.6
17.3
16.0
16.5
15.2
14.5
15.0
335.0
169.0
196.0
200.0
194.0
226.0
149.5
151.0
17.3
22.8
20.8
18.5
20.4
18.1
17.5
24.1
Shanghai
Shanghai
Shanghai
Shanghai
Shanghai
Shanghai
Shanghai
Hangzhou
the four construction sites is thought to be representative for the seven
case histories.
As shown in Fig. 7(a), the vane shear strength measured within a
depth of 30 m below the ground surface can be fitted by the following
linear function:
only unknown (Δδh-max ), which is a variable of δγ (see Tables 1 and 2,
can then be solved using Eq. (9).
3. Validation against case histories in soft clay
The validation of the MMSD method is examined by comparing its
predictions with the measured deformations from eight well-documented multipropped excavations in Shanghai soft clay [68,47] and
Hangzhou soft clay [67]. The predictive capability of the MSD method
for the eight case histories is also evaluated. This section reviews the
ground conditions and the eight selected case histories in Shanghai and
Hangzhou and then validates the MMSD method against the case histories.
15 + 1.7Z
Z ⩽ 15m
su = ⎧
⎨
⎩− 49.5 + 6.0Z 15m < Z ⩽ 30m
(10)
where Z represents the depth below the ground surface.
Considering the absence of field data concerning the strength of the
layer below the depth of 30 m (i.e., silty clay with sand), the corresponding su profile was deduced with a theoretical undrained strength
ratio (su/ σ ′v ) of the layer. According to Wang et al. [66], the su/ σ ′v
value for clay has an effective friction angle ϕ′ of 30°–35° (measured
from the triaxial compression test) falling in the range between 0.240
and 0.298. Wang’s [69] triaxial compression test on the silty clay with
the sand layer suggests an effective friction angle ϕ′ of 35°. The su/ σ ′v
value of the layer is, therefore, taken as 0.298. Since the excavations
analyzed in this study were carried out under the plane strain condition,
a strength anisotropy factor of 1.25 [24,25] was applied to convert the
value of su/ σ ′v under the triaxial condition (i.e., 0.298) to that under
the plane strain condition (i.e., 0.373). Correspondingly, the su profile
of the layer at the depth between 30 and 50 m was semi-empirically
determined by the following equation, given that the average soil unit
effective weight is 8.5 kN/m3:
3.1. Geotechnical profiles and soil properties of Shanghai and Hangzhou
soft clay
Shanghai is located on the west bank of the East China Sea (see
Fig. 6(a)), while the seven selected case histories are positioned around
the Pearl 2 circle line of Shanghai (see Fig. 5(b)). The top alluvial sediments in Shanghai, which were deposited during the Quaternary
Period, are present to a depth up to 400 m [27]. The soil profiles in
Shanghai are found to be relatively uniform [27,17] and can be divided
into many layers.
Fig. 6 illustrates a simplified soil profile for Shanghai, which was
generalized by Bolton et al. [5] based on the borehole information of
249 sites in Shanghai (as documented by Xu [72]. Within the depth
below 50 m, which is of engineering significance to the seven excavations analyzed herein (excavation depth from 14.5 to 20.6 m), the
ground consists of six layers, i.e., a top fill layer underlain with five
fine-grained layers. The ground water table is located at approximately
1 m below the ground surface [47]. Therefore, the great majority of the
six soil strata are fully saturated with water, except the fill within the
top 1 m. The top two layers (i.e., fill and silty clay with sand) are
slightly over-consolidated due to the desiccation in the stress history
[29], while the underlying four layers are normally consolidated [5].
There may be an over-consolidated layer at depths between 25 and
30 m below the ground surface, as implied by the profile of cone penetration resistance (qc) of a specific site near the underground railway
line 4 in Shanghai [73,56].
Fig. 7(a) shows the in situ su profiles, which were measured by the
vane shear apparatus at four typical construction sites in Shanghai.
According to Bjerrum [4], the ratio of vane shear strength su_vane to
triaxial undrained compressive strength su ranges from 0.95 to 1.05,
which gives a mean value of 1.0 (i.e., su_vane = su) for clay with a
plasticity index (Ip) of 10–24. Since the Ip values of the soil in the case
histories analyzed in this study fall within the range between 10 and 24,
the assumption of su_vane = su is adopted in the MMSD and MSD calculations reported herein. The locations of these vane shear measurements, as shown in Fig. 5(b), are in close vicinity to the seven case
histories analyzed in this study. Considering the reasonable uniformity
of Shanghai soil profiles [27], the vane shear strength measured from
Su = 37.5 + 3.1Z
(11)
The vertical permeability of the five fine-grained layers (from the
second to the sixth layers below the ground surface) is in the range
between 2.4 × 10−9 and 7.7 × 10−9 m/s, while the horizontal permeability is in the range between 1.2 × 10−8 and 7.6 × 10−8 m/s [19],
which leads to approximately undrained behavior of the conventional
excavations in Shanghai [47].
Hangzhou is located in the east of China, where the Qiantang River
flows. The selected case history in Hangzhou is situated in the West
Wenyi Road, where the soil profile near the ground surface consists of
six layers. The top layer is 1.5 m thick fill, whereas the second layer is a
2.5–3.0 m thick silt clay. Beneath these layers is a 25–32 m thick soft silt
layer, which has a high void ratio and low shear strength. The saturated
permeability of this layer is of the order of 10−9 m/s. The fourth and
fifth layers are gravel silt clay and gravel containing mica, oyster shells,
and other organic material, whereas the sixth layer is weathering
breccia. The ground water table in this case history is found to be approximately 1.5 m below the ground surface.
Fig. 7(b) shows the in situ shear strength profile measured by a vane
shear in the case history in Hangzhou. This profile can best be fitted by
the following equation, given that the average soil unit effective weight
is 8.5 kN/m3:
su = 3.1Z
60
(12)
Computers and Geotechnics 104 (2018) 54–68
L.Z. Wang et al.
600 km2, within which the ground conditions are similar. The length (L)
and width (W) of the seven station excavations in Shanghai, which are
rectangular-shaped on the plane, fall in the ranges of 169–335 m and
17–23 m, respectively. The aspect ratio of excavation in Hangzhou is
approximately 6.2, which is slightly lower than that in Shanghai (i.e.,
L/W = 7.4–19.4). Due to the large aspect ratios (i.e., L/W = 6.20–19.4)
of the eight-station excavations, the ground deformations near the
middle span along the longitudinal axis of each excavation were found
to exhibit a plane-strain behavior [68,17].
The eight excavations were all retained by concrete diaphragm
walls and multi-levels of prop, due to the large excavation depths
(15 ∼ 23 m) and the thick soft clay deposit. The height between each
level of the excavation typically ranged between 3 and 4 m. After each
level of excavation, a level of prop was installed at a height of approximate 0.3 m above the excavation bottom. For ground improvement purposes, compaction grouting was installed inside four of the
eight excavations (i.e., Yishan Station, South Xizan Station, Gubei
Station and West Wenyi Station).
The excavation rate of the eight case histories varies from 0.33 to
0.61 m/day. The previous field measurements suggest that at an excavation rate of 0.4 m/day in soft clay, the excess pore water pressures
on both sides of the wall merely dissipated during the excavation [16].
According to the coupled-consolidation finite element analysis of typical excavations in soft clay [16], an excavation rate of 0.3 m/day still
results in the undrained soil response, implying that all the case histories reported herein were likely to be excavated under the undrained
condition.
Considering that the eight excavations were carried out under the
undrained condition and deformed in a plane-strain manner, they are
identified as ideal case histories for validating the MMSD method.
Despite the grouting installed inside four of the eight excavations, the
beneficial effect of this grouting on the ground deformation was not
evident [68]. The presence of the compaction grouting was, therefore,
not considered in the MMSD and MSD predictions.
3.3. Input parameters and calculation procedure of the MMSD method
The input soil parameters include mainly su profiles (as illustrated in
Fig. 8 and quantified by Eqs. (10)–(12)) and strength mobilization
curves of Shanghai and Hangzhou clay.
Fig. 8(a) shows relationships between the degree of mobilized undrained shear strength (β = su_mob/su) and the mobilized shear strain of
Shanghai clay. These data resulted from Li’s [34] undrained triaxial
tests and from the authors’ direct simple shear test on intact K0-consolidated clay sample [67]. The soil samples tested by Li [34] and in
this study were cored from the clay layer at a depth of 8 and 14.5 m in
the construction sites for Longhua Station and Guiqiao Station, respectively. Considering the nearness of the soil sampling locations to
the seven selected case histories and the uniformity of the Shanghai soil
profile [27], the measured strength mobilization curves in Fig. 8(a) are
likely to be representative for analyzing the seven case histories. The
quality of these soil samples was evaluated using the criterion of Lunne
et al. [38], which suggested the sample used in the triaxial tests [34]
and the direct simple shear test (this study) are of rate 2 (good to fair).
Despite the differing depths and locations of the clay sample tested,
the three measured strength mobilization curves exhibit very similar
trends, confirming the reasonable uniformity of Shanghai clay. As anticipated, the strength mobilization curve resulting from the direct
simple shear test is bounded by that from the triaxial compression and
triaixal extension tests. The measured strength mobilization curves
were fitted by a power function, which was shown to broadly fit the
data of 19 types of clay [63]. As far as the Shanghai clay is concerned,
however, the power function generally underestimates the undrained
strength of the clay and overestimates the initial stiffness of the clay.
The overestimation of the initial stiffness by the power function may be
due to the use of fair to good quality samples. Stress-strain curves from
Fig. 8. (a) Normalized stress-strain relationship of intact K0-consolidated
Shanghai clay samples obtained from undrained shearing test; (b) comparisons
of the strength mobilization curve between Shanghai clay and soft clays
worldwide; (c) normalized stress-strain relationship of intact K0-consolidated
Hangzhou clay samples obtained from undrained shearing test. (See abovementioned references for further information.)
3.2. Construction details and geometrical properties of the case histories
Table 5 summarizes construction details and geometrical properties
of the eight multipropped excavations analyzed in Shanghai and
Hangzhou, which were all for metro stations. As shown in Fig. 5(b), the
seven stations in Shanghai are in a plane area of approximately
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L.Z. Wang et al.
Fig. 9. Comparisons between the measured and calculated (by the MMSD and MSD methods) lateral wall displacement in the seven case histories: (a) Yishan Station;
(b) South Xizan Station; (c) South Pudong Station; (d) Pudongdadao Station; (e) Pudian Station; (f) Yanchang Station; (g) Gubei Station.
the geotechnical parameters are unavoidably uncertain, which can be,
rationally, treated under a probabilistic framework [6,32].
Fig. 8(b) compares the strength mobilization curve of Shanghai clay
with that of eight types of clays worldwide. The stiffness of Shanghai
clay under relatively small strains (shear strain < 1%) is higher than
the stiffness of most of the clays in other areas around the world, partly
explaining why the observed excavation-induced ground deformations
in Shanghai clay are relatively small compared with similar case histories worldwide [29]. This observation also suggests that it could be
non-conservative to extrapolate the calculation chart developed for
predicting deformation of excavation in Shanghai (presented in latter
part of this study) to similar case histories in softer soil around the
world.
Fig. 8(c) shows the relationships between the degree of mobilized
high quality samples will be required for better calibration of the
strength mobilization curve in the future. Alternatively, the strength
mobilization curves of Shanghai clay were found to be better fitted by a
rational function, as follows:
β=
γ
⎧ a + γ − (1 + a) γ 2 (γ ⩽ b)
⎨1
(b < γ ⩽ 1)
⎩
(13)
where a and b are fitting parameters. The best fitting parameters for a
and b are 0.0035 and 0.06, respectively, which are adopted as the input
of the MMSD and MSD predictions for the seven case histories in
Shanghai. The other two variants of “a” used in the parametric study
(i.e., a = 0.002 and 0.005) represent the upper and lower bound
parameters that can fit the test data of strength mobilization curves for
Shanghai clay, as shown in Fig. 8(a). With a limited number of test data,
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Computers and Geotechnics 104 (2018) 54–68
L.Z. Wang et al.
(e)
(g)
(f)
Fig. 9. (continued)
undrained shear strength (β = su_mob/su) and the mobilized shear strain
of the Hangzhou clay. These data were obtained from the undrained
triaxial compression tests performed by the authors on two intact K0consolidated clay samples. The two soil samples tested in this study
were cored from the construction sites of West Wenyi Station at depths
of 9 and 19 m below the ground surface. The strength mobilization
curves of the Hangzhou clay are refitted by a rational function (see Eq.
(13)).
(b) Read the corresponding mobilized shear strain (δγmob) from the
strength mobilization curve of a representative intact sample, using
Eq. (13).
(c) Deduce the maximum incremental wall movement Δδh-max by substituting the δγ in Tables 1 or 2 and δγmob into Eq. (9).
(d) Calculate the incremental displacement profile, based on Eq. (3).
(e) Obtain the cumulative deformation profile by accumulating the
incremental movement profiles within the proposed mechanism.
3.3.1. Calculation procedure for the MMSD method
Based on the fitted su profile (Eqs. (10)–(12)) and strength mobilization curve (Eq. (13)) described above, the following calculation
procedure is programmed in MATLAB to calculate the excavation-induced lateral wall displacement in the eight selected case histories:
3.4. Validation of the MMSD method against the case histories
Fig. 9(a)–(g) compare the measured and predicted lateral wall displacement from the seven selected excavations in Shanghai at two typical stages, namely, one intermediate stage and the final excavation
stage. In the figure, H and He denote the excavation depth at the intermediate stage and at the final stage, respectively. The predictions in
the figure are made based on the MMSD and MSD methods, which differ
only in the form of incremental displacement profiles (see Figs. 2 and 3.
(a) Calculate the average mobilized strength (β = su_mob/su) within the
proposed mechanism based on equilibrium and energy conservation after each excavation stage according to Eq. (6).
63
Computers and Geotechnics 104 (2018) 54–68
L.Z. Wang et al.
Fig. 10. Comparison between the measured and calculated (by the MMSD and
MSD methods) lateral wall displacement in West Wenyi Station in Hangzhou
soft clay.
In both methods, the kinematic mechanism for a narrow excavation
(B < 2 2 (l−h), see Figs. 2(b) and 3(b)) is considered. Calculated deformation wall profiles by both MMSD and MSD methods are similar to
the measured data in each case history, with some certain extent of
overestimation in the maximum lateral wall displacement. The overestimation is probably associated with the use of strength and stiffness
parameters obtained from only fair to good quality samples. As suggested by Osman and Bolton [49], high-quality samples will be required
to better calibrate the stress-strain curves for improving the MMSD
prediction in the future. Comparatively, the MMSD method yields a
better prediction than the MSD method.
Fig. 10 compares the performance of the MSD and MMSD methods
for predicting the lateral wall displacement (δh) of the excavation in
Hangzhou. Compared to the MSD method, the MMSD yields a better
prediction for δh at the two typical stages. Qualitatively, the MMSD
method overestimates the measured δh at the last stage of excavation by
10%, which is smaller than the overestimation by the MSD method (i.e.,
15%).
To quantify the accuracy of the MMSD method, the measured and
predicted maximum lateral displacement (using the MMSD method)
after each excavation is extracted from Figs. 9 and 10 and presented in
Fig. 11(a) to make a direct comparison. As illustrated, 85% of the
MMSD predictions (i.e., 7 out of the 8 excavations) fall within 35% of
the corresponding measurements. Considering the ignorance of the
bending elastic energy stored in the wall in the current MMSD method
[26], its predictive capability is considered fairly satisfactory.
The accuracy of the MSD method for the same excavations is examined in Fig. 11(b), which compares the MSD predictions with the
measured data. The comparison suggests that the results predicted by
MSD fall within 120% of the field observations. This error band given
by the MSD method for predicting the maximum lateral wall displacement (i.e., 120%) is obviously larger than the error band of the
MMSD predictions, i.e., 35% (see Fig. 11(a)). In other words, the accuracy of the MMSD method for predicting the maximum lateral wall
displacement has been substantially improved by implementing a more
realistic incremental deformation profile than that assumed in the MSD
method.
Fig. 12(a) and (b) examine the error bands of the MMSD and MSD
Fig. 11. Accuracy of the calculated maximum lateral wall displacement by: (a)
MMSD and (b) MSD methods.
methods in predicting the depth of the maximum lateral wall displacement, respectively. As anticipated, the MMSD method (error
band = 30%) yields a better prediction in the depth of the maximum
lateral wall displacement than the MSD method (error band = 60%),
due to the acceptance of a more realistic incremental wall deformation
profile.
4. Parametric study and design charts of narrow excavations (for
metro station) in Shanghai soft clay
Having verified the MMSD method against eight well-documented
multipropped excavations, a series of parametric studies is performed to
quantify the effects of three potentially important variables on the deformation induced by narrow excavations specifically for metro stations
in Shanghai soft clay. The three variables include excavation width (B),
embedment depth of the wall (D) under the base and the height of the
lowest prop above the excavation bottom (h). These variables are
normalized by the final excavation depth (He).
By examining the geometrical properties of the 23 metro station
excavations out of more than 300 Shanghai case histories documented
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L.Z. Wang et al.
by Liu et al. [29] and Wang et al. [64], the dimensionless groups B/He,
D/He and h/He in this parametric study are determined to be in the
ranges of 0.50–1.50, 0.60–1.20 and 0.235–0.365, respectively. In each
analysis, the value of He is kept as a constant, i.e., 15 m. In each analysis, the undrained shear strength profile (see Eqs. (10) and (11)) and
the strength mobilization curve (see Eq. (13)) are identical to those
adopted in the analyses of the seven case histories in Shanghai.
Fig. 13 shows the calculated δh-max due to multipropped excavations
with different geometrical properties in Shanghai soft clay, which has
fitting parameters a of 0.0035 and b of 0.06 (see Fig. 9(a)). It is anticipated that for each given D/He and h/He, the value of δh-max reduces
as the excavation becomes narrower (i.e., smaller B/He), with a maximum percentage reduction of 72.0%. The figure also shows that for
each given combination of B/He and h/He, the value of δh-max declines
as the wall penetrations go more into the ground (i.e., larger D/He),
with a maximum percentage difference of 67.7%. This finding is attributed to a larger D/He resulting in a longer wavelength, which increases the integral domain for the soil in shearing to resist the potential
energy loss and therefore a declined mobilization of shear strain. Although a similar trend in δh-max is also observed for excavations with a
decreasing h/He at a given combination of B/He and D/He, the variation
in h/He leads to a much smaller percentage reduction (i.e.,
0.2%∼9.8%) than those caused by changing B/He and D/He (i.e.,
72.0% and 67.7%, respectively) because the h/He in the range of engineering interest (0.235–0.365) can only alter the wavelength by up to
14.8% (i.e., D/He = 0.6, α = 1), leading to a very limited change (up to
9.8%) in the mobilization of shear strain.
Based on the two most influential dimensionless forms (i.e., B/He
and D/He), as identified above, a more detailed parametric study was
performed to develop preliminary design charts, with the aim of assisting decision-making prior to any detailed analysis in design. In the
detailed parametric study, different rates of strength mobilization with
strain are considered by adopting three different groups of fitting
parameters (as defined in Eq. (13)), namely, a = 0.0020, b = 0.045;
a = 0.0035, b = 0.06; and a = 0.0050, b = 0.09. In the meantime, the
strength profiles of the three soils are identical (Fig. 7(a)). For the soil
with each given strength mobilization factor, 121 analyses considering
different combinations of D/He and B/He are performed. Fig. 14(a)–(c)
shows the design charts for estimating the normalized maximum lateral
wall displacement (δh-max/He) of excavation in clay with three different
groups of fitting parameters, namely, a = 0.0020, b = 0.045;
a = 0.0035, b = 0.06; and a = 0.0050, b = 0.09. In each chart, the
deformation control criteria for deep excavations with different protective grades (i.e., I, II and III) in Shanghai [30] are also marked to
gauge the environmental effects due to the wall deformation.
The three figures show that the lateral wall displacement can be
reduced by either decreasing the excavation width (B/He) or increasing
the embedment depth of the wall (D/He), as anticipated. A comparison
between the three figures also suggests that for a given geometrical
property of an excavation (i.e., B/He and D/He), smaller lateral wall
displacement results in a relatively stiffer ground (i.e., with a smaller
value of parameter a). Fig. 14(a) and (b) shows that within the geometrical properties of interest, the resulted maximum lateral wall displacements fall within the Grade III deformation control criterion (δhmax/He < 0.7%) for excavations in the soil with the fitting parameter a
of 0.0020 and 0.0035. When the parameter a is larger than 0.0050 (see
Fig. 14(c)), however, the maximum lateral wall displacement can exceed the Grade III deformation control criterion (δh-max/He > 0.7%).
Despite the influence of parameter a (governing the strength mobilization rate) and b on the wall deformation, the factor of safety (FOS)
against basal heave should not be altered provided that an identical
strength profile is adopted. Fig. 15 shows the design chart for estimating the FOS of the cases analyzed in this parametric study. Within
the range of geometries considered, the FOS falls within the range between 1.3 and 2.2, implying that all the excavations considered in this
parametric study have sufficient resistance against base heave failure,
Fig. 12. Accuracy of the calculated depth of the maximum lateral wall displacement by: (a) MMSD and (b) MSD methods.
Fig. 13. Influence of normalized excavation width, wall embedded depth and
height of the lowest prop on the maximum lateral wall displacement.
65
Computers and Geotechnics 104 (2018) 54–68
L.Z. Wang et al.
(a)
(b)
(c)
Note: Grade I, Grade II and Grade III are based on the deformation control criteria for deep excavations in Shanghai (Liu and Hou, 1997).
Fig. 14. Design charts for the normalized maximum lateral wall displacement (δh-max/He) of Shanghai excavations considering different geometries and strength
mobilization factor a (as defined in Eq. (13)): (a) a = 0.0020, b = 0.045 (b) a = 0.0035, b = 0.060 (c) a = 0.0050, b = 0.090.
eight carefully selected and well-documented deep excavations in
Shanghai and Hangzhou soft clay. A comparison was also made between the calculated results of both MMSD and MSD methods for the
seven case histories.
The accuracy of the MMSD predictions for the maximum lateral wall
displacement (δh-max ) falls within 35% of the measurements in most of
the eight case histories (i.e., 7 out of the 8 excavations). This accuracy is
obviously more satisfactory than the accuracy (i.e., 120%) of the MSD
method for the corresponding case histories. Additionally, the MMSD
yields a slightly better prediction for the depth where the δh-max occurs
than the MSD method. These comparisons have demonstrated the improvement made by the MMSD method, due to the implementation of a
more realistic plastic deformation mechanism in the latter.
With the validated MMSD method, a series of parametric studies is
carried out for developing calculation charts to predict the values of
δh-max and the factor of safety (FOS) against basal heave for typical
metro station excavations (narrow excavations) in Shanghai soft clay.
Various excavation geometries and strength mobilization characteristics of soil are considered in the 363 runs of the parametric study.
By considering the excavation geometries of common engineering
practice in Shanghai, the parametric study reveals that the most effective way to reduce the excavation-induced deformation is to either
narrow the excavation or increase the wall embedded depth. The results
of the parametric study also confirm that there is no one-to-one correlation between the deformation and FOS. For example, at a given FOS
of 1.5, the induced δh-max can vary between 0.14% and 0.60% of the
final excavation depth, due to excavations in soils with different
strength mobilization characteristics but having the same strength,
practically implying that the existing semi-empirical methods that relate the values of δh-max to the FOS may be treated with caution.
Fig. 15. Design chart for factor of safety (FOS) against basal heave considering
different excavation geometries.
although excessive deformation (see Fig. 14(c)) can occur. A comparison between Figs. 14 and 15 suggests that there is no one-to-one correlation between deformation and FOS. To be more specific, the value
of δh-max/He can vary between 0.14% and 0.60% at a given FOS of 1.5.
In other words, semi-empirical approaches (such as the chart of Mana
and Clough [40], which correlate deformation to FOS, may, therefore,
be treated with caution, highlighting the usefulness of the mobilization
strength methods that are capable of predicting both deformation and
FOS in a single step of calculation.
Acknowledgements
5. Summary and conclusions
The work presented in this paper is supported by National Key
Research and Development Program (Grant No. 2016YFC0800204),
National Natural Science Foundation of China (51338009), Zhejiang
Provincial Key Research and Development Program (2018C03031),
Zhejiang Provincial Natural Science Foundation of China (Y17E090016,
LQ18E080004), Shanghai Science and Technology Development Funds
(16QB1403400).
Despite the recent development of the modified mobilizable
strength design (MMSD) method by the authors, its predictive capability for excavation-induced deformation in undrained soft clay has
not been evaluated. The performance of the MMSD method in comparison with the earlier versions of MSD methods remains unknown.
This paper presents a detailed validation of the MMSD method against
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L.Z. Wang et al.
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