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Basal stability analysis of braced excavations with embedded walls in undrained clay using the upper bound theorem
Tunnelling and Underground Space Technology 79 (2018) 231–241
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Basal stability analysis of braced excavations with embedded walls in undrained clay using the upper bound theorem
T
⁎
Maosong Huang , Zhen Tang, Juyun Yuan Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University, Shanghai 200092, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Basal stability Limit analysis Wall embedment Anisotropic shear strength
The embedded wall has beneficial effects on the basal stability of braced excavation in clay. Conventional basal stability analysis methods cannot evaluate the lateral resistance afforded by the wall reasonably. This paper presents a new failure mechanism to evaluate the basal stability of excavations with embedded walls in undrained clay based the upper bound theorem. The proposed mechanism consists of a rigid block and three shear zones, and the horizontal reinforcing effects of wall penetration blow the excavation base are considered. The embedded wall is regarded as an elastic beam and deforms consistently with the velocities of adjacent soil in the shear zone. The elastic strain energy stored in the wall is incorporated in the upper bound calculation to increase the stability of excavation. The proposed mechanism has also been extended to the basal stability of excavations in anisotropic and non-homogeneous clay. The applicability of the proposed mechanism was validated by comparison with the results of numerical limit analysis and FEM, as well as five field cases near or at failure. Two failure field cases in anisotropic clay were also studied using the proposed mechanism. The results showed that the proposed method is capable to yield reasonable estimation by using both isotropic and anisotropic shear strength.
1. Introduction For deep excavations in soft clay, the factor of safety against base heave failure plays a key role in assessing the amount of ground movement generated by the plastic deformation of clay around the excavation. The retaining walls used to penetrate the excavation base with certain depth to prevent the base heave failure and limit the ground movements. For calculations of basal stability, it is desirable to consider the effects of wall embedment on the basal stability. The study of the basal stability of excavations in soft clay has been investigated by several authors in literature. Most authors have performed numerical or analytical approaches. The most widely used numerical approach for basal stability analysis may the finite element method (FEM). The FEM provides a comprehensive framework to evaluate multiple facets that affect the basal stability and thus results in an accurate estimation. The non-linear elastoplastic finite element analysis conducted by Hashash and Whittle (1996) showed the wall embedded depth and support conditions can significantly affect the failure models and the stability of excavation base. Goh (1990) and Faheem et al. (2003) adopted the elastoplastic FEM with shear strength reduction technique to study the basal stability
⁎
of excavations. Parametric study showed that the basal stability of excavations increases with the embedded depth and the stiffness of wall. Ukritchon et al. (2003) used the numerical limit analysis to study the stability the braced excavations, whose results agreed well with those by Goh (1990) and Faheem et al. (2003). The results of numerical analysis had shown the close dependency of the basal stability on wall embedded depth and stiffness. However, the numerical approaches have not been widely used in practice engineering due to the difficulties in selecting suitable constitutive models and the time-consuming convergence process. Most analytical approaches for calculating stability are through limit equilibrium analysis or upper bound limit analysis. Fig. 1(a) shows two failure mechanisms of limit equilibrium analysis, as presented by Theoretical (1943) and Bjerrum and Eide (1956). These two methods assumed that the failure of the excavation base is analogous to the failure of a wide footing located at the excavation base. This assumption provided a convenient way to calculate the basal stability factor by introducing the bearing capacity expressions. Obviously, these two methods neglect the effects of wall penetration. The modified versions of those two methods to account for the embedded depth of wall were proposed by Terzaghi (1943) and Eide et al. (1972), as shown in
Corresponding author at: Department of Geotechnical Engineering, Tongji University, 1239 Siping Road, Shanghai 200092, China. E-mail address: [email protected] (M. Huang).
https://doi.org/10.1016/j.tust.2018.05.014 Received 31 December 2016; Received in revised form 8 January 2018; Accepted 14 May 2018 0886-7798/ © 2018 Elsevier Ltd. All rights reserved.
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B B/2
B
2
Q
Supports
H
he
H
įm
D
m
2
įm įm
Prandtl zone 45°
(a) Failure mechanism
Terzaghi(1943)
Bjerrum & Eide(1956)
m
1 1 cos 2
x l
(a)
B B/2
B
l
2
wall
H
įm soil
(b) Deformation of wall and adjacent soil Fig. 3. Failure mechanism proposed by O'Rourke (1993).
D
Adhesion fs=Įcu 45°
Fig. 1(b). The two modified mechanisms do not consider the lateral resistance afforded by the wall. Therefore, these two modifications can only be applied in the case of fully rigid walls, which is unrealistic in practical engineering. The slip circle method (Hsieh et al., 2008) is widely used in practical engineering, which defines the ratio of resisting moment to the driving moment of the lowest support as the factor of safety for basal stability, as shown in Fig. 2. The resisting moment afforded by the wall is also not incorporated in the slip circle method. Moreover, the slip circle method deduces a constant stability factor with different wall embedded depth in the case of homogeneous clay. The analytical approaches based on upper bound theory are also widely used, such as Su et al. (1998), Liao and Su (2012), Chang (2000), Faheem et al. (2003) and Huang et al. (2011). These approaches either neglected the wall embedment or regarded the wall as fully rigid, and thus had the same limitations with those from the limit equilibrium analysis. O'Rourke (1993) proposed a distinctive analytical approach based on conservation of energy, as shown in Fig. 3. The failure mechanism consists of a Prandtl zone and an embedded wall, the wall increases the stability due to the elastic energy stored in flexure. As seen in Fig. 3, the wall deforms inconsistently with the adjacent soil, thus the velocity boundary conditions of soil-wall interface are not satisfied. Actually, this approach did not necessarily provide a rigorous basal stability estimation of excavation with embedded wall. In the framework of upper bound approach and Tresca’s yield criterion, the velocity fields should satisfy the incompressibility condition and any imposed velocity boundary condition. Ingenuity is required in upper bound approach from the need to devise velocity fields satisfying kinematic admissible conditions and yet producing a sufficiently low value of the upper bound of the stability factor. Most failure mechanisms of basal stability analysis presented in literature consist of rigid blocks and uniform shear zones with either straight or circle velocity fields. Among these failure mechanisms, some consist of a uniform
Terzaghi(1943) Eide et al (1972)
(b) Fig. 1. Conventional basal stability mechanisms: (a) without consideration of wall embedded; (b) with consideration of wall embedded.
Lowest support
H
he X
D
Fig. 2. Failure mechanism of the slip circle method.
232
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B
shear zone just located at the excavation base, which involve an embedded wall. These kinds of mechanisms neglected the velocity boundary conditions along the soil-wall interface (e.g. Su et al., 1998; Liao and Su, 2012; O'Rourke, 1993). Others consist of a uniform shear zone located at the toe of the wall and consequently implied an unrealistic rigid wall (e.g. Faheem et al., 2003; Huang et al., 2011), especially for long walls. Therefore, both kinds of failure mechanisms are unable to consider the horizontal deformation nor the lateral resistance afforded by the wall. In this paper, two distinct failure mechanisms are first presented in detail to explain the limitations of using uniform shear zones in conventional failure mechanisms. Then a modified failure mechanism consisting of a rigid block and three nonuniform shear zones is proposed. Within the modified mechanism, the deformations of the wall are compatible with the adjacent soil, without generating any unwanted gaps or overlaps between the wall and the adjacent soil. The elastic strain energy produced by the wall is incorporated in the upper bound calculation. Thereafter, the modified mechanism is extended to the basal stability analysis of excavations in anisotropic clay. Least upper bound solutions are obtained by optimizing the geometry of the kinematic mechanism and the velocity fields with respect to the geometric variables. The applicability of the modified mechanism is validated by the results of parameter studies of numerical limit analysis and FEM, as well as several field cases.
H
E ș
v0
v0 R
K' K
v0
v0
(a) M1 mechanism
B Rigid block
A'
C'
Shear zone Struts
H N'
The upper bound theorem shows that collapse will occur under the smallest values of the surface tractions for which it is possible to find a kinematically admissible velocity field. Under the condition of Tresca’s yield criterion, Drucker et al. (1952) demonstrated that the upper bound solution for the surface traction T = {Tx , Ty, Tz } can be calculated from the following equation:
v0 M'
2v0
v0
D
F'
cu |Δv| dS
v0 G' G
D
2.1. Upper bound theorem
D
C
Struts F
2. Failure mechanism and upper-bound solution
∫S T T vdS = ∫V 2cu |ε ̇|max dV + ∫S
A
Rigid block Shear zone
G'
E' ș
v0 R
v0
v0 (b) M2 mechanism
Fig. 4. Two distinct failure mechanisms.
(1)
where v is the kinematically admissible velocity field; |ε ̇|max is the absolutely largest principal component of the plastic strain rate; Δv is the velocity jump across any discontinuity; S is the surface that bounds the body or the assemblage of the bodies; V is the volume of the assemblage of the bodies; and SD is the surface of all discontinuity.
theorem, the factors of safety for the basal stability obtained from two mechanisms are expressed as follows. M1 mechanism
FsM1 =
2.2. Two distinct failure mechanisms
cu H + cu πR + cu πR γsat HR
(2)
M2 mechanism
Two distinct failure mechanisms are shown in Fig. 4, referred to as the M1 and M2 mechanisms. Both mechanisms consist of rigid blocks and uniform shear zones. As seen, the uniform shear zone in M1 mechanism is located at the excavation base and involves an embedded wall. The uniform shear zone in M2 mechanism is located at the toe of the wall. A key feature of uniform shear zone is that the velocity at any point maintains the same value and is perpendicular to the corresponding radial line. In M1 mechanism, the wall should be “cut off” into two separated parts to satisfy the velocity boundary conditions along the soil-wall interface, which is obviously unrealistic. Thus, M1 mechanism is actually not kinematically admissible. In M2 mechanism, the wall is located between two rigid blocks with vertical velocities. The kinematically admissible conditions are obviously satisfied. However, the M2 mechanism implies that, the wall is rigid enough to prevent the soil behind the wall move toward the excavation. In the case of flexible walls, the M2 mechanism may apparently overrate the basal stability. Notice that the two mechanisms assumed a rough soil-wall interface, and the wall moves vertically with the rigid block outside the excavation. Therefore, the relative velocity on velocity discontinuity M′G′ should equal to 2v0 in the M2 mechanism. Bases on upper bound
FsM1 =
cu (H + D) + cu πR + cu πR + cu D + 2cu D γsat HR
(3)
Here, cu is the undrained shear strength of clay; γsat is the unite weight of clay; H and B are the depth and width of excavation; D is the embedded depth of wall; and R is the radial of the fan-shaped shear zone. The factors of safety in Eq. (2) and Eq. (3) reach the least value when R = B. Therefore, the dimensionless stability numbers obtained from the two mechanisms are expressed as follows. M1 mechanism
NM1 = 2π +
H B
(4)
M2 mechanism
NM2 = 2π +
H D +4 B B
(5)
As seen in Eq. (4) and Eq. (5), the stability numbers obtained from the both mechanisms are irrelevant with the wall stiffness. This conclusion is significantly discrepant with that from numerical analysis (e.g. Ukritchon et al., 2003). The discrepancy actually due to the 233
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a
Rigid block
b
Shear zone
Hp
vz
H
v0
Lowest support
he e o
f
ș
vș
R
vrș
vr (r , θ) =
(8)
−v0 (R + m) m sin θ R2−m2sin2 θ
(m cos θ +
vrθ =
θ0 =
The modified kinematic mechanism shown in Fig. 5 is considered. It consists of a rigid translational block abcf and three shear zones fcd, fdd′ and ee′fd′. As shown in Fig. 5, he is the length of the lowest support to the base of the excavation. Three geometric parameters are required to determine the mechanism. These are the length Hp of the rigid block abcf, the radius R of slip fan fcd and horizontal distance m from the center of slip fan ocd to the retaining wall. Due to the existence of horizontal struts, it is assumed that the wall can only deform blow the lowest struts, and the length Hp of the rigid block abcf is limited to Hhe ≤ Hp ≤ H + D. Hp = H + D corresponding to the case of a fully rigid wall. The function of arc cd could be expressed in two different ways, as shown in Fig. 6. In slip fan fcd, function of arc cd is expressed as
vθ2 + vr2 =
v0 (R + m) R R2−m2sin2 θ
(m cos θ +
R2−m2sin2 θ )
(10)
π 1 ∂R (θ) ⎤ π −m sin θ ⎤ + arctan ⎡ = + arctan ⎡ ⎢ R2−m2sin2 θ ⎥ ⎢ 2 2 ⎣ R (θ) ∂θ ⎥ ⎦ ⎦ ⎣
vx = vrθ cos(θ0−θ) = −
vz = vrθ sin(θ0−θ) =
v0 (R + m) sin θ R2−m2sin2 θ
(12a)
v0 (R + m)( R2−m2sin2 θ cos θ−msin2 θ) R2−m2sin2 θ (m cos θ +
R2−m2sin2 θ )
(12b)
In the polar coordinates, whose origin defined at o, component θ z can be represented as θ = arctan m , as shown in Fig. 7. Therefore, the velocity components vx and vz at the surface fhd can be rewritten as
In slip fan ocd, function of arc cd is expressed as
R2−m2sin2 θ
(11)
In this study, the wall is assumed cannot be extended of compressed, therefore the wall deforms compatible with the velocity component at x direction of the resultant velocity at the surface fhd. In the local rectangular coordinates (x, z), whose origin is defined at the point f, the velocity components at x direction and z direction at the surface fhd can be written as
(6a)
R (δ ) = R
(6b)
Eq. (6a) and Eq. (6b) determine the geometric shape and velocity fields of slip fun fcd. The velocity fields in shear zone fcd are devised according to the geometry parameters of slip fan ocd, which will be
wall
m
f
ș
c
į
(9)
The resultant velocity vrθ inclines at an angle θ0 to the radial R(θ), as shown in Fig. 7, and θ0 can be deduced as
2.3. Modified kinematic mechanism and velocity fields
f
R2−m2sin2 θ )
Eq. (7) and Eq. (9) show that at the surface fc (θ = 0), the circular velocity vθ = v0 and the radial velocity vr = 0, so there is no relative velocity on fc. The resultant velocity of slip fan fcd can be obtained by combing Eq. (7) and Eq. (9), which is expressed as follows
limitations of uniform shear zones, which can hardly be compatible with the feasible deformation of the embedded wall. To solve this issue, a modified kinematic mechanism is presented in next section. The velocity fields are devised to satisfy both the incompressible condition and the velocity boundary conditions along the soil-wall interface.
o
(7)
Substituting Eq. (7) into Eq. (8), the radial velocity of slip fan fcd can be expressed as
Fig. 5. Modified kinematic mechanism.
R (θ) = m cos θ +
R2−m2sin2 θ
∂vr v 1 ∂vθ + r + =0 ∂r r r ∂θ
vr
w(z) h
v0 (R + m) m cos θ +
For undrained conditions, there is no volumetric change in soil, which is represented as
c
ȕ į
vȕ
vȡȕ
vθ (r , θ) =
e'
m
d' D
illustrated in detail in next section. For undrained conditions, the relative velocities are parallel to the respective discontinuities. The value of relative velocity on discontinuity bc is defined as v0. The velocities in slip fan fcd are expressed in polar coordinates (θ, r), by taking the point o as the origin, the following circular velocity of slip fan fcd will be selected:
wall-soil interface
vrș
R(ș)
R(ș)
vz h
d Fig. 6. Geometry of arc cd.
Fig. 7. Relative velocities along the soil-wall interface. 234
x
soil
ș0
vx R
c
į
R
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the wall is also incorporated in total internal energy calculation. Rate of work done by internal stress along velocity discontinuity bc
max
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
0.2 0.4 z/(D+he) 0.6
-0.6
-0.8
-1.0
Ebc = v0 m/R=0.3 m/R=0.5 m/R=1 m/R=1.5 m/R=2
∫0
H − he
cu dz
(18)
Rate of work done by internal stress along velocity discontinuity cd
Ecd =
∫0
R arctan m
cu vrθ R (θ) dθ
(19)
Rate of work done by internal stress in the radial shear zone fcd 0.8
R=D+he
Efcd =
Fig. 8. Normalized deflection curves of wall.
vx =
vz =
∫0
R arctan m
v0 (R + m)(m
R2 (m2
+
(13b)
The rate of work done by internal stress in shear zone fdd′ and ee′fd′ can be divided into two cases. Case 1 Hp ≤ H Rate of work done by internal stress in the radial shear zone fdd′
−v0 (R + m) z R2 (m2 + z 2)−m2z 2
(14)
Efdd′ =
Note that the deflections of wall increase from zero to a maximum value on the toe of wall according to Eq. (14). By comparison with the M1 and M2 mechanism, the devised velocity fields in the modified mechanism are more flexible and compatible with the potential deformations of wall. Fig. 8 plots the different normalized wall deflection curves with different m/R ratios (assumed that the radius of the slip fan fcd equals the value of he + D). As seen, the wall exhibits firstly a deep inward movement for m/R values less than unit and then a kick-out movement for m/R values larger than unit, which are different from the empirical constant cosine curve proposed by O'Rourke (1993). The velocity fields in slip fan fdd′ should also satisfy the boundary condition along the soil-wall interface. Thus in the polar coordinate, whose origin is defined at point f, the circular velocity of slip fan fdd′ is selected as
vβ (ρ , β ) =
∫π2
v (R + m) vρβ = 0 R
R
̇ ′|max ρdρdβ ∫(H−H ) 2cu |εfdd p
(22)
v0 (R + m) ρ2 (R2−m2) 3
2(R2 (m2 + ρ2 )−m2ρ2 )
2
(23)
Rate of work done by internal stress along velocity discontinuity dd′
Edd′ =
∫π2
π + arccos ⎡ (H − Hp ) ⎤ ⎢ ⎥ 2 R ⎣ ⎦
cu
v0 (R + m) Rdβ R
(24)
For the case Hp ≤ H the shear zone ee′fd′ no longer exist, therefore the rate of work done by internal stress is zero. Case 2 Hp > H Rate of work done by internal stress in the radial shear zone fdd′
Efdd′ =
(15)
π
∫π ∫0
R
̇ ′|max ρdρdβ 2cu |εfdd
(25)
2
Rate of work done by internal stress along velocity discontinuity dd′
Edd′ =
∫π
π
cu
2
v0 (R + m) Rdβ R
(26)
Rate of work done by internal stress in the shear zone ee′fd′ (16)
Eee′fd′ =
It can be seen from Eq. (15) that the circular velocity of the slip fan fdd′ is independent with β. Therefore, by taking the point f as the origin, the vertical velocity vz in the shear zone ee′fd′ can be deduced by setting ρ = x in Eq. (15), which is expressed as
R
Hp
∫0 ∫H
2cu |εeė ′fd′|max dzdx
(27)
where |εeė ′fd′|max is expressed as
|εeė ′fd′|max =
v0 (R + m) x R2 (m2 + x 2)−m2x 2
(21)
sin β
̇ ′|max = |εfdd
v0 (R + m) ρ R2 (m2 + ρ2 )−m2ρ2
π + arccos ⎡ (H − Hp ) ⎤ ⎢ ⎥ 2 R ⎣ ⎦
2
̇ ′|max is expressed as where |εfdd
Due to the incompressible condition, the radius velocity of slip fan fdd′ is selected as vρ (ρ , β ) = 0 . Setting ρ = R in Eq. (15), the velocity along velocity discontinuity dd′ can be expressed as
vz (x , z ) =
v0 (R + m) R2 2r (R2−m2sin2 θ)3
|ε ̇|max =
According to Eq. (13a), the deformation of the wall is selected as
w (z ) =
(20)
(13a)
z 2)−m2z 2 −mz 2)
R2 (m2 + z 2)−m2z 2 + m2 R2 (m2 + z 2)−m2z 2
̇ |max rdrdθ 2cu |εfcd
cos θ
̇ |max equals to the absolutely largest principal component where |εfcd of the plastic strain rate in shear zone fcd and expressed as follows
−v0 (R + m) z R2 (m2 + z 2)−m2z 2
R (θ)
∫m
(17)
v0 m2R2 (R + m) 2(R2 (m2
3
+ ρ2 )−m2ρ2 )
2
(28)
Rate of work done by internal stress along velocity discontinuity d′e
The horizontal velocity vx in the shear zone ee′fd′ is selected as vx = 0 according the incompressible condition.
Edd′ =
2.4. Upper-bound calculation
Hp
∫H
cu
v0 (R + m) dz R
(29)
It is assumed that the wall is on the verge of yield, and the plastic deformation of the wall is avoided in this study. It can be found from Eq. (14) that the rotation of the wall w′(z) ≠ 0 at the point f (z = 0), so the portion of the wall at the point f can be simulated as an “elastic hinge”. The elastic strain energy stored in the wall can be expressed as follows (after O'Rourke (1993))
According to the upper bound theory, any variation in the maximum shearing stress, cu, between the bodies in the assemblage, within the volume of the bodies, and along discontinuities, must be took into account in the evaluation of plastic work. The elastic strain energy of 235
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M. Huang et al.
EI 2
Ue =
∫0
D + he
(w′ ′ (z ))2dz +
1 My w′ (0) 2
B
(30)
My = EI (w′ ′ (z ))max
Ue =
2(w′ ′ (z ))max
∫0
1 (w′ (z )) dz + My w′ (0) 2 ′
D + he
∫0
Egh =
∫h
Ehd =
∫D+h
D + he
e
R
e
D
cu v0 dz
(34)
cu vz dz
H
γvz dxdz
Wγ = v0 γRH
(36)
(37)
(38)
cu (z ) = cu0 + ηz
(39)
cu (z , ζ ) = (cu0 + ηz )[k + (1−k )cos2 ζ ]
(43)
(44)
3. Verification of the modified mechanism 3.1. Excavation cases with different wall strength
(40) The modified mechanism is first used to analyze excavation cases with embedded walls in isotropic and homogeneous clay for an aspect ratio H/B = 0.375 and he = 0. The excavation is assumed be well braced above the excavated grade, and the wall penetrates the base of the excavation with a depth D. The strength of wall can be expressed by the relative strength parameter, Mp/(cuD2) (Ukritchon et al., 2003); Mp is the plastic bending moment of wall, and Mp = 1.5My, for rectangular wall cross section. The upper-bound results are compared with the results of numerical limit analysis presented by Ukritchon et al. (2003). Fig. 10 shows variation of the undrained stability number Nc = γsatH/cu with relative wall strength parameters for two different wall embedded depth D/H = 2/3, and D/H = 2. As seen, the results of modified
B2 + (H −Hp )2 , Hp ⩽ H R ⩽ B, Hp > H
(42)
in which cu(z) is the undrained shear strength at any given depth z, cu0 is the undrained shear strength at the ground surface, and η = dcu/dz is the rate of shear strength increase with depth. η = 0 in Eq. (43), according to uniform shear strength. By combining Eq. (43) and Eq. (42), the anisotropic and nonhomogeneous undrained shear strength of soil can be expressed as
so that the velocity fields kinematically admissible. Another limitation applied is
⎧R ⩽ ⎨ ⎩
cu(į-45º)
where k is anisotropic ratio of undrained shear strength (k = cue/cuc), where cuc is the undrained shear strength obtained from CK0UC test, and cue is undrained shear strength obtained from CK0UE test), and ζ is the inclination of major principle stress with vertical direction, for undrained condition, the values of ζ are show in Fig. 9. Due to one dimensional deposition and subsequent K0 consolidation, the undrained shear strength of clay is not uniform with depths, this study assumed the undrained shear strength of clay increases linearly with depth according to:
Eq. (39) can be solved for any given combination of m and R. The combination of these parameters that minimizes the solution defines the critical kinematic mechanism and gives the least upper-bound solution. The minimization can be performed with Optimization Tool Pattern Search in Matlab. The following limitations are applied to the solution
R ⩾ D + he , m > 0 and H + D ⩾ Hp ⩾ H −he
ı1
cu (ζ ) = cuc [k + (1−k )cos2 ζ ]
The factor of safety in this study is defined as the ratio of the sum of total work done by internal stress and elastic strain energy of the wall to the sum of work done by external force
∑ E + Ue ∑W
c
į ȕ
values of undrained shear strength change with the rotation of the largest major principle stress should be expressed in a mathematical form. The mathematical expressions to simulate the anisotropic undrained shear strength could be found in many literatures such as, Casagrande and Carrilo (Skempton, 1951), Hill (1998), and Davis and Christian (1970). This study selects the function presented by Casagrande and Carrilo as follows:
The total rate of work done by external forces q is written as
Wq = v0 qR
f
Fig. 9. Directions of major principle stresses.
Case 2 Hp > H
Fs =
ș
ı1
cu(ȕ-45º)
(35)
R − z2
o
e'
h
The total rate of work done by the soil unit weight γ is also divided into two cases. Case 1 Hp ≤ H
p
cu(45º) d'
m
(32)
(33)
∫H ∫0
e
2
cu (vz + v0 ) dz
Wγ = v0 γRHp +
Lowest support
he
The modified mechanism assumes the wall is fully rough, so the adhesion along the soil-wall interface equates cu. On the verge of failure, the wall moves vertically downward with the rigid block abcf. Rate of work done by internal stress along the soil-wall interface fh (right side), gh (left side) and velocity discontinuity hd
Efh =
cu(45º)
H
(31)
D + he
b
Shear zone
Combining Eq. (31) and Eq. (30), the elastic strain energy of the wall can be determined by
My
a
Rigid block
where E and I are the elastic modulus and moment of inertia per unite length of wall. The first term on the right of Eq. (30) denotes the elastic strain energy of the wall blow the point f, and the last term on the right of Eq. (30) denotes the strain energy associated with elastic rotation of the wall at the point f. The yield moment My, can be equated to:
(41)
This ensures that the failure mechanism does not extend beyond the excavation. 2.5. Anisotropic and non-homogeneous undrained strength In order to consider the anisotropic undrained shear strength, the 236
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13
13
UB(Ukritchon et al., 2003) D/H=0.67 UB(Ukritchon et al., 2003) D/H=2 Modified mechanism D/H=0.67 Modified mechanism D/H=2 M1 mechanism
12 11
11
N
9
M2 mechanism D/H=2 M2 mechanism D/H=0.67
8
9 8
7
7
6
6
5 0.01
0.1
1 Mp/(cuD2)
10
5 0.01
100
mechanism show the stability number increases with wall embedment depth and relative strength. Meanwhile, the results of modified mechanism have excellent agreement with the upper-bound results from numerical limit analysis for a wide range of relative strength parameters; while slightly overestimate the stability for relatively rigid walls (Mp/(cuD2) > 6). Also shown in Fig. 10, the M1 mechanism fails to estimate the increase in basal stability associated with wall embedment and relative strength. The M2 mechanism takes into account for the increase in basal stability associated with wall embedment, however, it fails to estimate the effects of wall relative strength. The M2 mechanism actually corresponds to the case of a fully rigid wall. Even through the results of modified mechanism are larger than those from numerical limit analysis for Mp/cuD2 > 6, they are still lower than the results of M2 mechanism, which corresponds to the case of a rigid wall. The comparisons clearly show the advantages of the modified mechanism. Among the analytical approaches, a distinct one was proposed by O'Rourke (1993), which had been mentioned previously. As seen in Fig. 3, the wall deforms consistently with an assumed cosine function, therefore the elastic strain energy stored in the wall was incorporated. While the soil just adjacent to the wall has constant value of velocities that are perpendicular to the wall. In fact, the mechanism implied a kinematically inadmissible velocity field. The stability number proposed by O’Rourke is given as
1 Mp/(cuD2)
10
100
12
Mp/cuD2=6
11
Mp/cuD2=1
10
Mp/cuD2=0.1
Ukritchon(1998) UB Modified mechanism
N
9 8 7 6 5 0.0
0.3
0.6
H/B
0.9
1.2
1.5
Fig. 12. Effects of excavation aspect ratio on stability of braced excavations with wall embedment.
3.3. Numerical excavation cases in anisotropic and non-homogeneous clay The MIT-E3 model which describes the anisotropic strength behavior of soil was used by Hashash and Whittle (1996) for the nonlinear finite element analysis of excavation in Boston blue clay. The excavation models investigated here are long braced excavations with a wide of 40 m, retained by 0.9 m thick concrete diaphragm walls, the braced systems are assumed incompressible and spaced at equal vertical intervals, he = 2.5 m. Four numerical cases were studied with different wall lengths (L = 12.5 m, 20 m, 40 m, 60 m). Failure in the finite element analysis was defined when the numerical convergence is not achieved within a specified excavation step. For normally consolidated Boston blue clay, the normalized peak undrained shear strengths cuc/σ′v and cue/σ′v are 0.335 and 0.155, respectively; total unite weight is 18kN/m3; and vertical effective consolidation pressure σ′v (kPa) is equal to 8.19z + 24.5. The yield moment for a 0.9 m-thick medium to heavily reinforced concrete diaphragm wall is selected as My = 0.67 MN m/m. The anisotropic undrained strength criteria is used in the analysis of the modified mechanism to investigate the failure depths Hf. The results of finite element method and those calculated from the modified mechanism are plotted in Fig. 13. As seen, the results of the proposed method are in excellent agreement with the results of finite element analysis for short walls (L = 12.5 m and L = 20 m), while slightly larger than the results of finite element method for long walls (L = 40 m and L = 60 m).
π 2My D (D + he ) cu
0.1
Fig. 11. Comparison with the method of O'Rourke (1993).
Fig. 10. Comparison with the upper bound results of numerical limit analysis.
N = Nc1 + x
Modified mechanism D/H=2 Modified mechanism D/H=0.67 O'Rourke (1993)
10
N
10
12
(45)
where Nc1 is the bearing capacity factor proposed by Skempton (1951); x = 1/8, 9/32, 1/2 for free, slide and fixed wall condition. Fig. 11 shows the stability number obtained from Eq. (45) and the proposed method. As seen, Eq. (45) fails to predict the increase of stability due to the increase of wall embedment ratios. Furthermore, the results of Eq. (45) are conservative for relative flexible walls and overrated the stability for relative rigid walls. 3.2. Excavation cases with different aspect ratios For further examination of the proposed method, excavation cases with wall embedment ratio, D/H = 0.67, relative wall strength Mp/ cuD2 = 0.1, 1 and 6, and depth to width ratio, H/B = 0.2, 0.25, 0.33, 0.5, 1 and 1.5 are selected. Fig. 12 shows the stability numbers calculated by the proposed method and the numerical limit analysis (calculated by the empirical equation presented by Ukritchon et al. (2003)). As seen, the results presented in this paper are actually very close to that from numerical limit analysis. 237
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60 50
Failure depth.Hf (m)
braces moved upward as a result of large inward deformation of the wall. As a consequence, the wall was only supported by the upper level of struts. The cumulative horizontal displacement of the wall exceed 400 mm according to the measurement results. Although the excavation was backfilled to prevent the massive failure, it is reasonable to suppose the excavation had underwent failure. Using the depth H and the length from the remaining level of struts to the bottom of the sheets, the Fs calculated by the proposed method is 0.98, which is consistent with the field observation. Another failure excavation case was a lagging excavation in soft clay in Washington DC. The piles deformed as much as 450 mm into the excavation. Using the proposed method, a Fs = 0.95 is calculated, which is closer to the value that is supposed to be attained in a critical state. For the Mexico City and Vaterland 1 excavations, the factor of safety obtained from the proposed method are 1.057 and 1.03, respectively, which correspond with the actual excavation performance, where no failure was observed.
Present study FEM (Hashash,1992)
40 30 20 10 0
0
10
20 30 40 Wall length. L(m)
50
60
Fig. 13. Comparison with finite element method of failure depth of excavation cases in normally consolidated clay in Boston.
4.2. Taipei rebar broadway case of Hsieh et al. (2008)
4. Comparison with field excavation cases
Two failure cases of excavations in Taipei were presented by Hsieh et al. (2008), which are employed to further validate the proposed method. The first case is the Taipei Rebar Broadway case, which is about a deep rectangular internally braced excavation site, the length and width of the excavation is 100 m and 25.8 m, respectively. The final excavation depth was 13.45 m. Fig. 14 plots the subsoil profile, comprising four alternating layers of clayey and sandy soils. The top 8.7 m was the backfill level, and the main soil layer was thick soft silty clay with depth ranging from 10.7 m to 44.7 m. The water table was located at GL-2.8m. The undrained shear strength of the soil was obtained from the results of triaxial K0-consolidated undrained compression tests (CK0UAC), extension tests (CK0U-AE), and direct simple shear tests (CK0UDSS), as shown in Fig. 15. Soil samples used for the above tests were taken at 200 m far from the construction site. The diaphragm wall used to retain the excavation was 0.7 m in thickness and 24 m in depth. The plastic bending moment of the wall, Mp, is taken as 975 kN m/m . The strut layers were installed at depths of 1.0 m, 3.55 m, 6.85 m, and 10.15 m from the ground level. Fig. 13 shows the construction sequence for this case. After completion of last stage (GL-13.45m), the basal heave failure occurred in the construction site, causing collapse of the braced system and a 132 × 40 m subsided area. Fig. 16 shows the relationship between the factor of safety Fs with the wall embedded depth D using the anisotropic undrained shear strength. The critical factor of safety Fs drived from the proposed method is 1.046. In this case, the failure happened after the final stage
4.1. Field excavation cases of O'Rourke (1993) In this section, five field excavation cases presented by O'Rourke (1993) are adopted to validate the proposed method, all those cases are known to have been at failure or near failure. The basic geometry, soil properties and wall characteristics of the excavations are listed in Table 1. For comparison, the factor of safety calculated from two conventional methods (Terzaghi, 1943; Bjerrum and Eide, 1956) are listed next to the factor of safety calculated by the proposed method. As mentioned previously, these two conventional methods do not consider the effects of wall embedment blow the excavation base. The factor of safety based on the Terzaghi approach and Bjerrum-Eide approach are less than one in all the cases, even for the cases where there is no failure occurred. In contrast, the proposed method predicts Fs ≥ 1, for stable cases (Mexico city, Davidson Ave. 1, Vaterland 1), and does predict Fs < 1, for failure cases (Davidson Ave. 2 and Washington DC). The Davidson Ave. 1 excavation refers to the general conditions surveyed along the braced cut. Although there is no visible failure observed, the measured maximum wall deflection varied from 100 mm to 250 mm, which is fairly high for a 9.1 m deep cut. The excavation performance is consistent with the Fs = 1.0 obtained from the proposed method, but inconsistent with the Fs = 0.865 and 0.72 obtained from the conventional methods. At the location of the Davidson Ave. 2, a slope failure inside the excavation exposed a portion of wall to an unsupported depth H = 7.3 m. The second level of wales rotated and the corresponding
Table 1 Validation of the modified mechanism with field excavation cases at or near failure. Parameter
H (m) B (m) D (m) My (kN m/m) cu (kPa/m2) γsat (kN/m3) Fs Terzaghi (1943) Bjerrum and Eide (1956) Present study Observed behavior δwall (cm) δwall/H (%)
Excavations Mexico City
Davidson Ave.1
Davidson Ave. 2
Washington DC
Vaterland 1
9 7 5 455.7 12.2 12 0.865 0.72 1.057 No Failure 10–15 cm 1.1–1.7%
9.1 7.6 4.6 426.7 19.2 16.3 0.844 0.77 1.0 No Failure 10–25 cm 1.1–2.7%
7.3 7.6 6.4 426.7 19.2 16.3 0.88 0.86 0.98 Failure
9.1 21.3 6.7 416.3 23 17 0.939 0.81 0.955 Failure
11 11 5.3 553.5 23.5 18.6 0.818 0.76 1.03 No Failure 10–13 cm 1.1–1.4%
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GL 0.0m
GL -1.0m SM
GL -3.55m
2.5
GL -2.8m
2.0
GL -4.5m
GL -6.85m GL -8.7m
SM
GL -10.7m
1.5 Fs
GL -10.15m
ML
1.0
GL -13.45m
0.0
Fig. 14. Subsoil profiles and construction sequence of Taipei Rebar Broadway Case.
50
cu(kPa)
100
150
CK0U-AE
Depth(m)
CK0U-DSS (AC+AE+DSS)/3*0.817
20 Used in analysis (CK0U-AC values) 25
10
15 D(m)
20
25
30
4.3. Taipei Shi-Pai case of Hsieh et al. (2008) The Taipei Shi-Pai case was a rectangular excavation with 12.3 m width and 45 m length. The soil profiles and construction sequence of this excavation are shown in Fig. 18. The depth of diaphragm ranges from 13.8 to 17 m and is 15.4 m on average with a thickness of 0.5 m. The plastic bending moment, Mp, of the wall is taken as 570 kN m/m. The basal heave failure occurs after the completion of final stage (GL9.3m), with the collapse of the bracing system and adjacent buildings. The soil conditions were reinvestigated at a site 500 m from the
Used in analysis (CK0U-AE values)
30 Fig. 15. Variation of undrained shear strength with depth for different test types in Taipei Rebar Broadway Case (modified from the original figure by Hsieh et al. (2008).
2.5
of the excavation was completed and its stability should have been in a critical state. If the influence of surcharge on the ground surface is taken into consideration, the factor of safety Fs should be even lower. Under such a condition, the factor of safety Fs derived from the proposed method is consistent with field observation. The calculations considering the anisotropic shear strength may relatively complicated for engineering practice, therefore, two isotropic undrained shear strength are considered in this study. Istropic strength I1 – the undrained shear strength are taken as that from the CK0U-DSS test results. Istropic strength I2 – following the suggestion by Ladd and Foott (Ladd and Foott, 1974), the undrained shear strength was taken as an average result of CK0U-AC, CK0U-AE, and CK0U-DSS tests (Do et al., 2016), while it is necessary to consider the strain rate effect by multiplying a correction factor (μt), which was summarized by Kulhawy and Mayne (1990) as follows:
μt = [1−0.1log10 (t f /tlaboratory )]
5
where tf = time to get failure of soil in the field; and tlaboratory = time to get failure of soil in the laboratory. It was assumed that tf was 2 weeks for all failure cases as a rough estimate, and tlaboratory was 5h for the consolidated undrained test with a strain rate of 1%/h, which was considered as the standard reference test. Hence, μt = 0.817. Fig. 17 shows the result obtained from the proposed method with the consideration of isotropic shear strength. The critical Fs obtained from the calculation using isotropic strength I1 is 1.27. It is obvious that the results of isotropic strength I1 greatly overestimate the stability of the excavation. By using the isotropic strength I2, the critical Fs obtained from the proposed method is 1.022, which is consistent with the one (Fs = 1.0) from FEM (Do et al., 2016) and closer to the critical state.
CK0U-AC 15
0
Fig. 16. Relation between factor of safety and wall penetration depth for anisotropic strength analysis in Taipei Rebar Broadway Case.
GL -24m
10
D=10.55m
0.5
CL
0
1.046
Istropic strength I1 Istropic strength I2
2.0
Fs
1.5 1.335 1.022
1.0 0.5 D=10.55m 0.0
0
5
10
15
20
25
30
D(m) Fig. 17. Relation between factor of safety and wall penetration depth for isotropic strength analysis in Taipei Rebar Broadway Case.
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Fill
GL -1.4m GL -3.7m
GL 0.0m GL -1.5m
2.5 2.0
ML
1.5 Fs
GL -5.5m
GL -6m
0.997
1.0 0.95
CL
GL -9.3m
GL -8.5m
0.5 0.0
CL
D=6.1m
0
5
D=7.7m
10
15 D(m)
20
25
30
Fig. 20. Relationship between safety factor and wall penetration depth for anisotropic strength analysis in Taipei Shi-Pai Case.
2.5
Istropic strength I1 Istropic strength I2
Fig. 18. Subsoil profiles and construction sequence of Taipei Shi-Pai Case.
10
20
(AC+AE+DSS) /3*0.817
5
2.0 50
60
70
1.5
CK0U-AE CK0U-AC
0.5
Depth(m)
D=6.1m 0.0
15
25 30
1.03
1.0 0.981
CK0U-DSS
10
20
1.326
1.266
Fs
0 0
cu(kPa) 30 40
0
5
D=7.7m 10
15 D(m)
20
25
30
Fig. 21. Relationship between safety factor and wall penetration depth for isotropic strength analysis in Taipei Shi-Pai Case.
Used in analysis (CK0U-AE values)
Fig. 21 shows the relationship of Fs and D calculated from the proposed method using the isotropic shear strength I1 and I2. As this is a failure case, it is obvious the Fs calculated by using isotropic shear strength I1, is apparently overestimated. By using the isotropic shear strength I2, the critical Fs obtained from the proposed method are 0.981, for wall embedded depth D is 6.1 m, and 1.03, for wall embedded depth is 7.7 m, respectively, which are close to one (Fs = 1.0, for D = 6.1 m) from FEM. Therefor, with the consideration of isotropic shear strength I2, the Fs values obtained from the proposed method is more closer to the value that is supposed to be attained in a critical state.
Used in analysis (CK0U-AC values)
Fig. 19. Variation of undrained shear strength with depth for different test types in Taipei Shi-Pai Case (modified from the original figure by Hsieh et al. (2008).
excavation and a series of undrained tests were conducted. The undrained shear strength of the soil was obtained from the results of triaxial K0-consolidated undrained compression (CK0U-AC) tests, extension (CK0U-AE) tests, and direct simple shear (CK0U-DSS) tests, as shown in Fig. 19. Fig. 20 shows the relationship of Fs and D calculated from the proposed method with the consideration of anisotropic undrained shear strength. Two depth of wall penetration are selected to examine the stability of base at final excavation stage. The critical Fs values is 0.95, for wall embedded depth D is 6.1 m, and 0.997, for wall embedded depth D is 7.7 m, respectively. The basal heave failure occurs after the final stage, so the critical Fs of the excavation should be closer to 1.0. Therefor, the Fs derived from the proposed method if consistent with the failure conditions observed.
5. Conclusions Based on the upper bound theory and with the usage of nonuniform shear zone, this paper presents a kinematic approach to estimate the basal stability of excavations with wall embedment in soft clay, with the consideration of wall strength and embedded depth. The anisotropic and non-homogeneous undrained shear strength are also incorporated in the proposed mechanism. The wall deforms consistently with the adjacent soil in the mechanism. The influence of the wall stiffness on the stability is taken as the elastic strain energy added to the total internal work. The least upper-bound solutions are obtained through the 240
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optimization of the kinematic mechanism. The following conclusions can be drawn on the basis of the work presented herein:
for Distinguished Young Scholars of China (Grant No. 50825803). These supports are gratefully acknowledged.
1. For excavations with wall embedment in isotropic and homogeneous clay, the proposed method is capable to evaluate the effect of wall stiffness on the basal stability by comparison with the upper bound results of numerical limit analysis. The factor of safety obtained from the proposed method can reasonably describe the undrained stability of excavation base for five field cases of O'Rourke (1993), which had been at or near failure. 2. The results of numerical excavation cases in anisotropic and nonhomogeneous clay indicate that the predicted failure depth obtained by the proposed method were consistent with those obtained by the finite element analysis. 3. Two failure excavation cases of Hsieh et al. (2008) in anisotropic and non-homogeneous clay were also studied to verify the validity of the proposed method. It was found that the proposed method gave reasonable estimate of basal stability by using the anisotropic undrained shear strength. The proposed method overestimated the basal stability by using the results of CK0U-DSS tests, and yielded reasonable estimation by using the average result of CK0U-AC, CK0U-AE, and CK0U-DSS tests with a multiplicative of a correction factor μt for strain rate effect.
References Bjerrum, L., Eide, O., 1956. Stability of strutted excavations in clay. Geotechnique 6 (1), 32–47. Chang, M.F., 2000. Basal stability analysis of braced cuts in clay. J. Geotech. Geoenviron. Eng. 126 (3), 276–279. Davis, E.H., Christian, J.T., 1970. Bearing capacity of anisotropic cohesive soil. J. Soil Mech. Found. Div., ASCE 97, 753–769. Do, T.N., Ou, C.Y., Chen, R.P., 2016. A study of failure mechanisms of deep excavations in soft clay using the finite element method. Comput. Geotech. 73, 153–163. Drucker, D., Prager, W., Greenberg, H., 1952. Extended limit design theorems for continuous media. Q. Appl. Math. 381–389. Eide, O., Aas, G., Josang, T., 1972. Special application of cast-in-place walls for tunnels in soft clay in Oslo. In: Proc. 5th European Conf on Soil Mechanics and Foundation Engineering, Madrid, Spain. pp. 485–498. Faheem, H., Cai, F., Ugai, K., Hagiwara, T., 2003. Two-dimensional base stability of excavations in soft soils using FEM. Comput. Geotech. 30 (2), 141–163. Goh, A.T.C., 1990. Assessment of basal stability for braced excavation systems using the finite element method. Comput. Geotech. 10 (4), 325–338. Hashash, Y.M.A., Whittle, A.J., 1996. Ground movement prediction for deep excavations in soft clay. J. Geotech. Eng. 122 (6), 474–486. Hill, R., 1998. The Mathematical Theory of Plasticity. Clarendon Press, New York. Hsieh, P.G., Ou, C.Y., Liu, H.T., 2008. Basal heave analysis of excavations with consideration of anisotropic undrained strength of clay. Can. Geotech. J. 45 (6), 788–799. Huang, M.S., Yu, S.B., Qin, H.L., 2011. Upper bound method for basal stability analysis of braced excavations in K0-consolidated clays. China Civ. Eng. J. 44 (3), 101–108 (in Chinese). Kulhawy, F.H., Mayne, P.W., 1990. Manual on estimating soil properties for foundation design. Rep. EL-6800, Electric Power Research Institute, Palo Alto, CA. Ladd, C.C., Foott, R.F., 1974. New design procedure for stability of soft clays. J. Geotech. Eng. Div., ASCE 100 (7), 763–786. Liao, H.J., Su, S.F., 2012. Base stability of grout pile-reinforced excavations in soft clay. J. Geotech. Geoenviron. Eng. 138 (2), 184–192. O'Rourke, T.D., 1993. Base stability and ground movement prediction for excavations in soft clay. In: Retaining Structures. Thomas Telford, London. pp. 131–139. Skempton, A.W., 1951. The bearing capacity of clays. In: Proc. Building Research Congress, Thomas Telford, London. pp. 180–189. Su, S., Liao, H., Lin, Y., 1998. Base stability of deep excavation in anisotropic soft clay. J. Geotech. Geoenviron. Eng. 124 (9), 809–819. Terzaghi, K., 1943. Theoretical Soil Mechanics. Wiley Online Library. Ukritchon, B., Whittle, A.J., Sloan, S.W., 2003. Undrained stability of braced excavations in clay. J. Geotech. Geoenviron. Eng. 129 (8), 738–755.
Finally, is should be pointed out that, the deformations of the wall in the proposed mechanism are dependent on the assumed velocity fields and are not completely in conformity with engineering practice. The proposed mechanism still yields rigorous upper bound solutions of the safety of factors, which are important in the preliminary design of excavations in undrained clay. Although the suitable of the proposed mechanism had been verified by parameter studies and some field cases, its applications to excavations not supported by high stiffness walls still needs further study. Acknowledgements This work was financially supported by the National Key R&D Program of China (Grant No. 2016YFC0800200) and the National Fund
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Basal stability analysis of braced excavations with embedded walls in undrained clay using the upper bound theorem
T
⁎
Maosong Huang , Zhen Tang, Juyun Yuan Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University, Shanghai 200092, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Basal stability Limit analysis Wall embedment Anisotropic shear strength
The embedded wall has beneficial effects on the basal stability of braced excavation in clay. Conventional basal stability analysis methods cannot evaluate the lateral resistance afforded by the wall reasonably. This paper presents a new failure mechanism to evaluate the basal stability of excavations with embedded walls in undrained clay based the upper bound theorem. The proposed mechanism consists of a rigid block and three shear zones, and the horizontal reinforcing effects of wall penetration blow the excavation base are considered. The embedded wall is regarded as an elastic beam and deforms consistently with the velocities of adjacent soil in the shear zone. The elastic strain energy stored in the wall is incorporated in the upper bound calculation to increase the stability of excavation. The proposed mechanism has also been extended to the basal stability of excavations in anisotropic and non-homogeneous clay. The applicability of the proposed mechanism was validated by comparison with the results of numerical limit analysis and FEM, as well as five field cases near or at failure. Two failure field cases in anisotropic clay were also studied using the proposed mechanism. The results showed that the proposed method is capable to yield reasonable estimation by using both isotropic and anisotropic shear strength.
1. Introduction For deep excavations in soft clay, the factor of safety against base heave failure plays a key role in assessing the amount of ground movement generated by the plastic deformation of clay around the excavation. The retaining walls used to penetrate the excavation base with certain depth to prevent the base heave failure and limit the ground movements. For calculations of basal stability, it is desirable to consider the effects of wall embedment on the basal stability. The study of the basal stability of excavations in soft clay has been investigated by several authors in literature. Most authors have performed numerical or analytical approaches. The most widely used numerical approach for basal stability analysis may the finite element method (FEM). The FEM provides a comprehensive framework to evaluate multiple facets that affect the basal stability and thus results in an accurate estimation. The non-linear elastoplastic finite element analysis conducted by Hashash and Whittle (1996) showed the wall embedded depth and support conditions can significantly affect the failure models and the stability of excavation base. Goh (1990) and Faheem et al. (2003) adopted the elastoplastic FEM with shear strength reduction technique to study the basal stability
⁎
of excavations. Parametric study showed that the basal stability of excavations increases with the embedded depth and the stiffness of wall. Ukritchon et al. (2003) used the numerical limit analysis to study the stability the braced excavations, whose results agreed well with those by Goh (1990) and Faheem et al. (2003). The results of numerical analysis had shown the close dependency of the basal stability on wall embedded depth and stiffness. However, the numerical approaches have not been widely used in practice engineering due to the difficulties in selecting suitable constitutive models and the time-consuming convergence process. Most analytical approaches for calculating stability are through limit equilibrium analysis or upper bound limit analysis. Fig. 1(a) shows two failure mechanisms of limit equilibrium analysis, as presented by Theoretical (1943) and Bjerrum and Eide (1956). These two methods assumed that the failure of the excavation base is analogous to the failure of a wide footing located at the excavation base. This assumption provided a convenient way to calculate the basal stability factor by introducing the bearing capacity expressions. Obviously, these two methods neglect the effects of wall penetration. The modified versions of those two methods to account for the embedded depth of wall were proposed by Terzaghi (1943) and Eide et al. (1972), as shown in
Corresponding author at: Department of Geotechnical Engineering, Tongji University, 1239 Siping Road, Shanghai 200092, China. E-mail address: [email protected] (M. Huang).
https://doi.org/10.1016/j.tust.2018.05.014 Received 31 December 2016; Received in revised form 8 January 2018; Accepted 14 May 2018 0886-7798/ © 2018 Elsevier Ltd. All rights reserved.
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B B/2
B
2
Q
Supports
H
he
H
įm
D
m
2
įm įm
Prandtl zone 45°
(a) Failure mechanism
Terzaghi(1943)
Bjerrum & Eide(1956)
m
1 1 cos 2
x l
(a)
B B/2
B
l
2
wall
H
įm soil
(b) Deformation of wall and adjacent soil Fig. 3. Failure mechanism proposed by O'Rourke (1993).
D
Adhesion fs=Įcu 45°
Fig. 1(b). The two modified mechanisms do not consider the lateral resistance afforded by the wall. Therefore, these two modifications can only be applied in the case of fully rigid walls, which is unrealistic in practical engineering. The slip circle method (Hsieh et al., 2008) is widely used in practical engineering, which defines the ratio of resisting moment to the driving moment of the lowest support as the factor of safety for basal stability, as shown in Fig. 2. The resisting moment afforded by the wall is also not incorporated in the slip circle method. Moreover, the slip circle method deduces a constant stability factor with different wall embedded depth in the case of homogeneous clay. The analytical approaches based on upper bound theory are also widely used, such as Su et al. (1998), Liao and Su (2012), Chang (2000), Faheem et al. (2003) and Huang et al. (2011). These approaches either neglected the wall embedment or regarded the wall as fully rigid, and thus had the same limitations with those from the limit equilibrium analysis. O'Rourke (1993) proposed a distinctive analytical approach based on conservation of energy, as shown in Fig. 3. The failure mechanism consists of a Prandtl zone and an embedded wall, the wall increases the stability due to the elastic energy stored in flexure. As seen in Fig. 3, the wall deforms inconsistently with the adjacent soil, thus the velocity boundary conditions of soil-wall interface are not satisfied. Actually, this approach did not necessarily provide a rigorous basal stability estimation of excavation with embedded wall. In the framework of upper bound approach and Tresca’s yield criterion, the velocity fields should satisfy the incompressibility condition and any imposed velocity boundary condition. Ingenuity is required in upper bound approach from the need to devise velocity fields satisfying kinematic admissible conditions and yet producing a sufficiently low value of the upper bound of the stability factor. Most failure mechanisms of basal stability analysis presented in literature consist of rigid blocks and uniform shear zones with either straight or circle velocity fields. Among these failure mechanisms, some consist of a uniform
Terzaghi(1943) Eide et al (1972)
(b) Fig. 1. Conventional basal stability mechanisms: (a) without consideration of wall embedded; (b) with consideration of wall embedded.
Lowest support
H
he X
D
Fig. 2. Failure mechanism of the slip circle method.
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B
shear zone just located at the excavation base, which involve an embedded wall. These kinds of mechanisms neglected the velocity boundary conditions along the soil-wall interface (e.g. Su et al., 1998; Liao and Su, 2012; O'Rourke, 1993). Others consist of a uniform shear zone located at the toe of the wall and consequently implied an unrealistic rigid wall (e.g. Faheem et al., 2003; Huang et al., 2011), especially for long walls. Therefore, both kinds of failure mechanisms are unable to consider the horizontal deformation nor the lateral resistance afforded by the wall. In this paper, two distinct failure mechanisms are first presented in detail to explain the limitations of using uniform shear zones in conventional failure mechanisms. Then a modified failure mechanism consisting of a rigid block and three nonuniform shear zones is proposed. Within the modified mechanism, the deformations of the wall are compatible with the adjacent soil, without generating any unwanted gaps or overlaps between the wall and the adjacent soil. The elastic strain energy produced by the wall is incorporated in the upper bound calculation. Thereafter, the modified mechanism is extended to the basal stability analysis of excavations in anisotropic clay. Least upper bound solutions are obtained by optimizing the geometry of the kinematic mechanism and the velocity fields with respect to the geometric variables. The applicability of the modified mechanism is validated by the results of parameter studies of numerical limit analysis and FEM, as well as several field cases.
H
E ș
v0
v0 R
K' K
v0
v0
(a) M1 mechanism
B Rigid block
A'
C'
Shear zone Struts
H N'
The upper bound theorem shows that collapse will occur under the smallest values of the surface tractions for which it is possible to find a kinematically admissible velocity field. Under the condition of Tresca’s yield criterion, Drucker et al. (1952) demonstrated that the upper bound solution for the surface traction T = {Tx , Ty, Tz } can be calculated from the following equation:
v0 M'
2v0
v0
D
F'
cu |Δv| dS
v0 G' G
D
2.1. Upper bound theorem
D
C
Struts F
2. Failure mechanism and upper-bound solution
∫S T T vdS = ∫V 2cu |ε ̇|max dV + ∫S
A
Rigid block Shear zone
G'
E' ș
v0 R
v0
v0 (b) M2 mechanism
Fig. 4. Two distinct failure mechanisms.
(1)
where v is the kinematically admissible velocity field; |ε ̇|max is the absolutely largest principal component of the plastic strain rate; Δv is the velocity jump across any discontinuity; S is the surface that bounds the body or the assemblage of the bodies; V is the volume of the assemblage of the bodies; and SD is the surface of all discontinuity.
theorem, the factors of safety for the basal stability obtained from two mechanisms are expressed as follows. M1 mechanism
FsM1 =
2.2. Two distinct failure mechanisms
cu H + cu πR + cu πR γsat HR
(2)
M2 mechanism
Two distinct failure mechanisms are shown in Fig. 4, referred to as the M1 and M2 mechanisms. Both mechanisms consist of rigid blocks and uniform shear zones. As seen, the uniform shear zone in M1 mechanism is located at the excavation base and involves an embedded wall. The uniform shear zone in M2 mechanism is located at the toe of the wall. A key feature of uniform shear zone is that the velocity at any point maintains the same value and is perpendicular to the corresponding radial line. In M1 mechanism, the wall should be “cut off” into two separated parts to satisfy the velocity boundary conditions along the soil-wall interface, which is obviously unrealistic. Thus, M1 mechanism is actually not kinematically admissible. In M2 mechanism, the wall is located between two rigid blocks with vertical velocities. The kinematically admissible conditions are obviously satisfied. However, the M2 mechanism implies that, the wall is rigid enough to prevent the soil behind the wall move toward the excavation. In the case of flexible walls, the M2 mechanism may apparently overrate the basal stability. Notice that the two mechanisms assumed a rough soil-wall interface, and the wall moves vertically with the rigid block outside the excavation. Therefore, the relative velocity on velocity discontinuity M′G′ should equal to 2v0 in the M2 mechanism. Bases on upper bound
FsM1 =
cu (H + D) + cu πR + cu πR + cu D + 2cu D γsat HR
(3)
Here, cu is the undrained shear strength of clay; γsat is the unite weight of clay; H and B are the depth and width of excavation; D is the embedded depth of wall; and R is the radial of the fan-shaped shear zone. The factors of safety in Eq. (2) and Eq. (3) reach the least value when R = B. Therefore, the dimensionless stability numbers obtained from the two mechanisms are expressed as follows. M1 mechanism
NM1 = 2π +
H B
(4)
M2 mechanism
NM2 = 2π +
H D +4 B B
(5)
As seen in Eq. (4) and Eq. (5), the stability numbers obtained from the both mechanisms are irrelevant with the wall stiffness. This conclusion is significantly discrepant with that from numerical analysis (e.g. Ukritchon et al., 2003). The discrepancy actually due to the 233
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a
Rigid block
b
Shear zone
Hp
vz
H
v0
Lowest support
he e o
f
ș
vș
R
vrș
vr (r , θ) =
(8)
−v0 (R + m) m sin θ R2−m2sin2 θ
(m cos θ +
vrθ =
θ0 =
The modified kinematic mechanism shown in Fig. 5 is considered. It consists of a rigid translational block abcf and three shear zones fcd, fdd′ and ee′fd′. As shown in Fig. 5, he is the length of the lowest support to the base of the excavation. Three geometric parameters are required to determine the mechanism. These are the length Hp of the rigid block abcf, the radius R of slip fan fcd and horizontal distance m from the center of slip fan ocd to the retaining wall. Due to the existence of horizontal struts, it is assumed that the wall can only deform blow the lowest struts, and the length Hp of the rigid block abcf is limited to Hhe ≤ Hp ≤ H + D. Hp = H + D corresponding to the case of a fully rigid wall. The function of arc cd could be expressed in two different ways, as shown in Fig. 6. In slip fan fcd, function of arc cd is expressed as
vθ2 + vr2 =
v0 (R + m) R R2−m2sin2 θ
(m cos θ +
R2−m2sin2 θ )
(10)
π 1 ∂R (θ) ⎤ π −m sin θ ⎤ + arctan ⎡ = + arctan ⎡ ⎢ R2−m2sin2 θ ⎥ ⎢ 2 2 ⎣ R (θ) ∂θ ⎥ ⎦ ⎦ ⎣
vx = vrθ cos(θ0−θ) = −
vz = vrθ sin(θ0−θ) =
v0 (R + m) sin θ R2−m2sin2 θ
(12a)
v0 (R + m)( R2−m2sin2 θ cos θ−msin2 θ) R2−m2sin2 θ (m cos θ +
R2−m2sin2 θ )
(12b)
In the polar coordinates, whose origin defined at o, component θ z can be represented as θ = arctan m , as shown in Fig. 7. Therefore, the velocity components vx and vz at the surface fhd can be rewritten as
In slip fan ocd, function of arc cd is expressed as
R2−m2sin2 θ
(11)
In this study, the wall is assumed cannot be extended of compressed, therefore the wall deforms compatible with the velocity component at x direction of the resultant velocity at the surface fhd. In the local rectangular coordinates (x, z), whose origin is defined at the point f, the velocity components at x direction and z direction at the surface fhd can be written as
(6a)
R (δ ) = R
(6b)
Eq. (6a) and Eq. (6b) determine the geometric shape and velocity fields of slip fun fcd. The velocity fields in shear zone fcd are devised according to the geometry parameters of slip fan ocd, which will be
wall
m
f
ș
c
į
(9)
The resultant velocity vrθ inclines at an angle θ0 to the radial R(θ), as shown in Fig. 7, and θ0 can be deduced as
2.3. Modified kinematic mechanism and velocity fields
f
R2−m2sin2 θ )
Eq. (7) and Eq. (9) show that at the surface fc (θ = 0), the circular velocity vθ = v0 and the radial velocity vr = 0, so there is no relative velocity on fc. The resultant velocity of slip fan fcd can be obtained by combing Eq. (7) and Eq. (9), which is expressed as follows
limitations of uniform shear zones, which can hardly be compatible with the feasible deformation of the embedded wall. To solve this issue, a modified kinematic mechanism is presented in next section. The velocity fields are devised to satisfy both the incompressible condition and the velocity boundary conditions along the soil-wall interface.
o
(7)
Substituting Eq. (7) into Eq. (8), the radial velocity of slip fan fcd can be expressed as
Fig. 5. Modified kinematic mechanism.
R (θ) = m cos θ +
R2−m2sin2 θ
∂vr v 1 ∂vθ + r + =0 ∂r r r ∂θ
vr
w(z) h
v0 (R + m) m cos θ +
For undrained conditions, there is no volumetric change in soil, which is represented as
c
ȕ į
vȕ
vȡȕ
vθ (r , θ) =
e'
m
d' D
illustrated in detail in next section. For undrained conditions, the relative velocities are parallel to the respective discontinuities. The value of relative velocity on discontinuity bc is defined as v0. The velocities in slip fan fcd are expressed in polar coordinates (θ, r), by taking the point o as the origin, the following circular velocity of slip fan fcd will be selected:
wall-soil interface
vrș
R(ș)
R(ș)
vz h
d Fig. 6. Geometry of arc cd.
Fig. 7. Relative velocities along the soil-wall interface. 234
x
soil
ș0
vx R
c
į
R
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the wall is also incorporated in total internal energy calculation. Rate of work done by internal stress along velocity discontinuity bc
max
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
0.2 0.4 z/(D+he) 0.6
-0.6
-0.8
-1.0
Ebc = v0 m/R=0.3 m/R=0.5 m/R=1 m/R=1.5 m/R=2
∫0
H − he
cu dz
(18)
Rate of work done by internal stress along velocity discontinuity cd
Ecd =
∫0
R arctan m
cu vrθ R (θ) dθ
(19)
Rate of work done by internal stress in the radial shear zone fcd 0.8
R=D+he
Efcd =
Fig. 8. Normalized deflection curves of wall.
vx =
vz =
∫0
R arctan m
v0 (R + m)(m
R2 (m2
+
(13b)
The rate of work done by internal stress in shear zone fdd′ and ee′fd′ can be divided into two cases. Case 1 Hp ≤ H Rate of work done by internal stress in the radial shear zone fdd′
−v0 (R + m) z R2 (m2 + z 2)−m2z 2
(14)
Efdd′ =
Note that the deflections of wall increase from zero to a maximum value on the toe of wall according to Eq. (14). By comparison with the M1 and M2 mechanism, the devised velocity fields in the modified mechanism are more flexible and compatible with the potential deformations of wall. Fig. 8 plots the different normalized wall deflection curves with different m/R ratios (assumed that the radius of the slip fan fcd equals the value of he + D). As seen, the wall exhibits firstly a deep inward movement for m/R values less than unit and then a kick-out movement for m/R values larger than unit, which are different from the empirical constant cosine curve proposed by O'Rourke (1993). The velocity fields in slip fan fdd′ should also satisfy the boundary condition along the soil-wall interface. Thus in the polar coordinate, whose origin is defined at point f, the circular velocity of slip fan fdd′ is selected as
vβ (ρ , β ) =
∫π2
v (R + m) vρβ = 0 R
R
̇ ′|max ρdρdβ ∫(H−H ) 2cu |εfdd p
(22)
v0 (R + m) ρ2 (R2−m2) 3
2(R2 (m2 + ρ2 )−m2ρ2 )
2
(23)
Rate of work done by internal stress along velocity discontinuity dd′
Edd′ =
∫π2
π + arccos ⎡ (H − Hp ) ⎤ ⎢ ⎥ 2 R ⎣ ⎦
cu
v0 (R + m) Rdβ R
(24)
For the case Hp ≤ H the shear zone ee′fd′ no longer exist, therefore the rate of work done by internal stress is zero. Case 2 Hp > H Rate of work done by internal stress in the radial shear zone fdd′
Efdd′ =
(15)
π
∫π ∫0
R
̇ ′|max ρdρdβ 2cu |εfdd
(25)
2
Rate of work done by internal stress along velocity discontinuity dd′
Edd′ =
∫π
π
cu
2
v0 (R + m) Rdβ R
(26)
Rate of work done by internal stress in the shear zone ee′fd′ (16)
Eee′fd′ =
It can be seen from Eq. (15) that the circular velocity of the slip fan fdd′ is independent with β. Therefore, by taking the point f as the origin, the vertical velocity vz in the shear zone ee′fd′ can be deduced by setting ρ = x in Eq. (15), which is expressed as
R
Hp
∫0 ∫H
2cu |εeė ′fd′|max dzdx
(27)
where |εeė ′fd′|max is expressed as
|εeė ′fd′|max =
v0 (R + m) x R2 (m2 + x 2)−m2x 2
(21)
sin β
̇ ′|max = |εfdd
v0 (R + m) ρ R2 (m2 + ρ2 )−m2ρ2
π + arccos ⎡ (H − Hp ) ⎤ ⎢ ⎥ 2 R ⎣ ⎦
2
̇ ′|max is expressed as where |εfdd
Due to the incompressible condition, the radius velocity of slip fan fdd′ is selected as vρ (ρ , β ) = 0 . Setting ρ = R in Eq. (15), the velocity along velocity discontinuity dd′ can be expressed as
vz (x , z ) =
v0 (R + m) R2 2r (R2−m2sin2 θ)3
|ε ̇|max =
According to Eq. (13a), the deformation of the wall is selected as
w (z ) =
(20)
(13a)
z 2)−m2z 2 −mz 2)
R2 (m2 + z 2)−m2z 2 + m2 R2 (m2 + z 2)−m2z 2
̇ |max rdrdθ 2cu |εfcd
cos θ
̇ |max equals to the absolutely largest principal component where |εfcd of the plastic strain rate in shear zone fcd and expressed as follows
−v0 (R + m) z R2 (m2 + z 2)−m2z 2
R (θ)
∫m
(17)
v0 m2R2 (R + m) 2(R2 (m2
3
+ ρ2 )−m2ρ2 )
2
(28)
Rate of work done by internal stress along velocity discontinuity d′e
The horizontal velocity vx in the shear zone ee′fd′ is selected as vx = 0 according the incompressible condition.
Edd′ =
2.4. Upper-bound calculation
Hp
∫H
cu
v0 (R + m) dz R
(29)
It is assumed that the wall is on the verge of yield, and the plastic deformation of the wall is avoided in this study. It can be found from Eq. (14) that the rotation of the wall w′(z) ≠ 0 at the point f (z = 0), so the portion of the wall at the point f can be simulated as an “elastic hinge”. The elastic strain energy stored in the wall can be expressed as follows (after O'Rourke (1993))
According to the upper bound theory, any variation in the maximum shearing stress, cu, between the bodies in the assemblage, within the volume of the bodies, and along discontinuities, must be took into account in the evaluation of plastic work. The elastic strain energy of 235
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EI 2
Ue =
∫0
D + he
(w′ ′ (z ))2dz +
1 My w′ (0) 2
B
(30)
My = EI (w′ ′ (z ))max
Ue =
2(w′ ′ (z ))max
∫0
1 (w′ (z )) dz + My w′ (0) 2 ′
D + he
∫0
Egh =
∫h
Ehd =
∫D+h
D + he
e
R
e
D
cu v0 dz
(34)
cu vz dz
H
γvz dxdz
Wγ = v0 γRH
(36)
(37)
(38)
cu (z ) = cu0 + ηz
(39)
cu (z , ζ ) = (cu0 + ηz )[k + (1−k )cos2 ζ ]
(43)
(44)
3. Verification of the modified mechanism 3.1. Excavation cases with different wall strength
(40) The modified mechanism is first used to analyze excavation cases with embedded walls in isotropic and homogeneous clay for an aspect ratio H/B = 0.375 and he = 0. The excavation is assumed be well braced above the excavated grade, and the wall penetrates the base of the excavation with a depth D. The strength of wall can be expressed by the relative strength parameter, Mp/(cuD2) (Ukritchon et al., 2003); Mp is the plastic bending moment of wall, and Mp = 1.5My, for rectangular wall cross section. The upper-bound results are compared with the results of numerical limit analysis presented by Ukritchon et al. (2003). Fig. 10 shows variation of the undrained stability number Nc = γsatH/cu with relative wall strength parameters for two different wall embedded depth D/H = 2/3, and D/H = 2. As seen, the results of modified
B2 + (H −Hp )2 , Hp ⩽ H R ⩽ B, Hp > H
(42)
in which cu(z) is the undrained shear strength at any given depth z, cu0 is the undrained shear strength at the ground surface, and η = dcu/dz is the rate of shear strength increase with depth. η = 0 in Eq. (43), according to uniform shear strength. By combining Eq. (43) and Eq. (42), the anisotropic and nonhomogeneous undrained shear strength of soil can be expressed as
so that the velocity fields kinematically admissible. Another limitation applied is
⎧R ⩽ ⎨ ⎩
cu(į-45º)
where k is anisotropic ratio of undrained shear strength (k = cue/cuc), where cuc is the undrained shear strength obtained from CK0UC test, and cue is undrained shear strength obtained from CK0UE test), and ζ is the inclination of major principle stress with vertical direction, for undrained condition, the values of ζ are show in Fig. 9. Due to one dimensional deposition and subsequent K0 consolidation, the undrained shear strength of clay is not uniform with depths, this study assumed the undrained shear strength of clay increases linearly with depth according to:
Eq. (39) can be solved for any given combination of m and R. The combination of these parameters that minimizes the solution defines the critical kinematic mechanism and gives the least upper-bound solution. The minimization can be performed with Optimization Tool Pattern Search in Matlab. The following limitations are applied to the solution
R ⩾ D + he , m > 0 and H + D ⩾ Hp ⩾ H −he
ı1
cu (ζ ) = cuc [k + (1−k )cos2 ζ ]
The factor of safety in this study is defined as the ratio of the sum of total work done by internal stress and elastic strain energy of the wall to the sum of work done by external force
∑ E + Ue ∑W
c
į ȕ
values of undrained shear strength change with the rotation of the largest major principle stress should be expressed in a mathematical form. The mathematical expressions to simulate the anisotropic undrained shear strength could be found in many literatures such as, Casagrande and Carrilo (Skempton, 1951), Hill (1998), and Davis and Christian (1970). This study selects the function presented by Casagrande and Carrilo as follows:
The total rate of work done by external forces q is written as
Wq = v0 qR
f
Fig. 9. Directions of major principle stresses.
Case 2 Hp > H
Fs =
ș
ı1
cu(ȕ-45º)
(35)
R − z2
o
e'
h
The total rate of work done by the soil unit weight γ is also divided into two cases. Case 1 Hp ≤ H
p
cu(45º) d'
m
(32)
(33)
∫H ∫0
e
2
cu (vz + v0 ) dz
Wγ = v0 γRHp +
Lowest support
he
The modified mechanism assumes the wall is fully rough, so the adhesion along the soil-wall interface equates cu. On the verge of failure, the wall moves vertically downward with the rigid block abcf. Rate of work done by internal stress along the soil-wall interface fh (right side), gh (left side) and velocity discontinuity hd
Efh =
cu(45º)
H
(31)
D + he
b
Shear zone
Combining Eq. (31) and Eq. (30), the elastic strain energy of the wall can be determined by
My
a
Rigid block
where E and I are the elastic modulus and moment of inertia per unite length of wall. The first term on the right of Eq. (30) denotes the elastic strain energy of the wall blow the point f, and the last term on the right of Eq. (30) denotes the strain energy associated with elastic rotation of the wall at the point f. The yield moment My, can be equated to:
(41)
This ensures that the failure mechanism does not extend beyond the excavation. 2.5. Anisotropic and non-homogeneous undrained strength In order to consider the anisotropic undrained shear strength, the 236
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13
13
UB(Ukritchon et al., 2003) D/H=0.67 UB(Ukritchon et al., 2003) D/H=2 Modified mechanism D/H=0.67 Modified mechanism D/H=2 M1 mechanism
12 11
11
N
9
M2 mechanism D/H=2 M2 mechanism D/H=0.67
8
9 8
7
7
6
6
5 0.01
0.1
1 Mp/(cuD2)
10
5 0.01
100
mechanism show the stability number increases with wall embedment depth and relative strength. Meanwhile, the results of modified mechanism have excellent agreement with the upper-bound results from numerical limit analysis for a wide range of relative strength parameters; while slightly overestimate the stability for relatively rigid walls (Mp/(cuD2) > 6). Also shown in Fig. 10, the M1 mechanism fails to estimate the increase in basal stability associated with wall embedment and relative strength. The M2 mechanism takes into account for the increase in basal stability associated with wall embedment, however, it fails to estimate the effects of wall relative strength. The M2 mechanism actually corresponds to the case of a fully rigid wall. Even through the results of modified mechanism are larger than those from numerical limit analysis for Mp/cuD2 > 6, they are still lower than the results of M2 mechanism, which corresponds to the case of a rigid wall. The comparisons clearly show the advantages of the modified mechanism. Among the analytical approaches, a distinct one was proposed by O'Rourke (1993), which had been mentioned previously. As seen in Fig. 3, the wall deforms consistently with an assumed cosine function, therefore the elastic strain energy stored in the wall was incorporated. While the soil just adjacent to the wall has constant value of velocities that are perpendicular to the wall. In fact, the mechanism implied a kinematically inadmissible velocity field. The stability number proposed by O’Rourke is given as
1 Mp/(cuD2)
10
100
12
Mp/cuD2=6
11
Mp/cuD2=1
10
Mp/cuD2=0.1
Ukritchon(1998) UB Modified mechanism
N
9 8 7 6 5 0.0
0.3
0.6
H/B
0.9
1.2
1.5
Fig. 12. Effects of excavation aspect ratio on stability of braced excavations with wall embedment.
3.3. Numerical excavation cases in anisotropic and non-homogeneous clay The MIT-E3 model which describes the anisotropic strength behavior of soil was used by Hashash and Whittle (1996) for the nonlinear finite element analysis of excavation in Boston blue clay. The excavation models investigated here are long braced excavations with a wide of 40 m, retained by 0.9 m thick concrete diaphragm walls, the braced systems are assumed incompressible and spaced at equal vertical intervals, he = 2.5 m. Four numerical cases were studied with different wall lengths (L = 12.5 m, 20 m, 40 m, 60 m). Failure in the finite element analysis was defined when the numerical convergence is not achieved within a specified excavation step. For normally consolidated Boston blue clay, the normalized peak undrained shear strengths cuc/σ′v and cue/σ′v are 0.335 and 0.155, respectively; total unite weight is 18kN/m3; and vertical effective consolidation pressure σ′v (kPa) is equal to 8.19z + 24.5. The yield moment for a 0.9 m-thick medium to heavily reinforced concrete diaphragm wall is selected as My = 0.67 MN m/m. The anisotropic undrained strength criteria is used in the analysis of the modified mechanism to investigate the failure depths Hf. The results of finite element method and those calculated from the modified mechanism are plotted in Fig. 13. As seen, the results of the proposed method are in excellent agreement with the results of finite element analysis for short walls (L = 12.5 m and L = 20 m), while slightly larger than the results of finite element method for long walls (L = 40 m and L = 60 m).
π 2My D (D + he ) cu
0.1
Fig. 11. Comparison with the method of O'Rourke (1993).
Fig. 10. Comparison with the upper bound results of numerical limit analysis.
N = Nc1 + x
Modified mechanism D/H=2 Modified mechanism D/H=0.67 O'Rourke (1993)
10
N
10
12
(45)
where Nc1 is the bearing capacity factor proposed by Skempton (1951); x = 1/8, 9/32, 1/2 for free, slide and fixed wall condition. Fig. 11 shows the stability number obtained from Eq. (45) and the proposed method. As seen, Eq. (45) fails to predict the increase of stability due to the increase of wall embedment ratios. Furthermore, the results of Eq. (45) are conservative for relative flexible walls and overrated the stability for relative rigid walls. 3.2. Excavation cases with different aspect ratios For further examination of the proposed method, excavation cases with wall embedment ratio, D/H = 0.67, relative wall strength Mp/ cuD2 = 0.1, 1 and 6, and depth to width ratio, H/B = 0.2, 0.25, 0.33, 0.5, 1 and 1.5 are selected. Fig. 12 shows the stability numbers calculated by the proposed method and the numerical limit analysis (calculated by the empirical equation presented by Ukritchon et al. (2003)). As seen, the results presented in this paper are actually very close to that from numerical limit analysis. 237
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60 50
Failure depth.Hf (m)
braces moved upward as a result of large inward deformation of the wall. As a consequence, the wall was only supported by the upper level of struts. The cumulative horizontal displacement of the wall exceed 400 mm according to the measurement results. Although the excavation was backfilled to prevent the massive failure, it is reasonable to suppose the excavation had underwent failure. Using the depth H and the length from the remaining level of struts to the bottom of the sheets, the Fs calculated by the proposed method is 0.98, which is consistent with the field observation. Another failure excavation case was a lagging excavation in soft clay in Washington DC. The piles deformed as much as 450 mm into the excavation. Using the proposed method, a Fs = 0.95 is calculated, which is closer to the value that is supposed to be attained in a critical state. For the Mexico City and Vaterland 1 excavations, the factor of safety obtained from the proposed method are 1.057 and 1.03, respectively, which correspond with the actual excavation performance, where no failure was observed.
Present study FEM (Hashash,1992)
40 30 20 10 0
0
10
20 30 40 Wall length. L(m)
50
60
Fig. 13. Comparison with finite element method of failure depth of excavation cases in normally consolidated clay in Boston.
4.2. Taipei rebar broadway case of Hsieh et al. (2008)
4. Comparison with field excavation cases
Two failure cases of excavations in Taipei were presented by Hsieh et al. (2008), which are employed to further validate the proposed method. The first case is the Taipei Rebar Broadway case, which is about a deep rectangular internally braced excavation site, the length and width of the excavation is 100 m and 25.8 m, respectively. The final excavation depth was 13.45 m. Fig. 14 plots the subsoil profile, comprising four alternating layers of clayey and sandy soils. The top 8.7 m was the backfill level, and the main soil layer was thick soft silty clay with depth ranging from 10.7 m to 44.7 m. The water table was located at GL-2.8m. The undrained shear strength of the soil was obtained from the results of triaxial K0-consolidated undrained compression tests (CK0UAC), extension tests (CK0U-AE), and direct simple shear tests (CK0UDSS), as shown in Fig. 15. Soil samples used for the above tests were taken at 200 m far from the construction site. The diaphragm wall used to retain the excavation was 0.7 m in thickness and 24 m in depth. The plastic bending moment of the wall, Mp, is taken as 975 kN m/m . The strut layers were installed at depths of 1.0 m, 3.55 m, 6.85 m, and 10.15 m from the ground level. Fig. 13 shows the construction sequence for this case. After completion of last stage (GL-13.45m), the basal heave failure occurred in the construction site, causing collapse of the braced system and a 132 × 40 m subsided area. Fig. 16 shows the relationship between the factor of safety Fs with the wall embedded depth D using the anisotropic undrained shear strength. The critical factor of safety Fs drived from the proposed method is 1.046. In this case, the failure happened after the final stage
4.1. Field excavation cases of O'Rourke (1993) In this section, five field excavation cases presented by O'Rourke (1993) are adopted to validate the proposed method, all those cases are known to have been at failure or near failure. The basic geometry, soil properties and wall characteristics of the excavations are listed in Table 1. For comparison, the factor of safety calculated from two conventional methods (Terzaghi, 1943; Bjerrum and Eide, 1956) are listed next to the factor of safety calculated by the proposed method. As mentioned previously, these two conventional methods do not consider the effects of wall embedment blow the excavation base. The factor of safety based on the Terzaghi approach and Bjerrum-Eide approach are less than one in all the cases, even for the cases where there is no failure occurred. In contrast, the proposed method predicts Fs ≥ 1, for stable cases (Mexico city, Davidson Ave. 1, Vaterland 1), and does predict Fs < 1, for failure cases (Davidson Ave. 2 and Washington DC). The Davidson Ave. 1 excavation refers to the general conditions surveyed along the braced cut. Although there is no visible failure observed, the measured maximum wall deflection varied from 100 mm to 250 mm, which is fairly high for a 9.1 m deep cut. The excavation performance is consistent with the Fs = 1.0 obtained from the proposed method, but inconsistent with the Fs = 0.865 and 0.72 obtained from the conventional methods. At the location of the Davidson Ave. 2, a slope failure inside the excavation exposed a portion of wall to an unsupported depth H = 7.3 m. The second level of wales rotated and the corresponding
Table 1 Validation of the modified mechanism with field excavation cases at or near failure. Parameter
H (m) B (m) D (m) My (kN m/m) cu (kPa/m2) γsat (kN/m3) Fs Terzaghi (1943) Bjerrum and Eide (1956) Present study Observed behavior δwall (cm) δwall/H (%)
Excavations Mexico City
Davidson Ave.1
Davidson Ave. 2
Washington DC
Vaterland 1
9 7 5 455.7 12.2 12 0.865 0.72 1.057 No Failure 10–15 cm 1.1–1.7%
9.1 7.6 4.6 426.7 19.2 16.3 0.844 0.77 1.0 No Failure 10–25 cm 1.1–2.7%
7.3 7.6 6.4 426.7 19.2 16.3 0.88 0.86 0.98 Failure
9.1 21.3 6.7 416.3 23 17 0.939 0.81 0.955 Failure
11 11 5.3 553.5 23.5 18.6 0.818 0.76 1.03 No Failure 10–13 cm 1.1–1.4%
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GL 0.0m
GL -1.0m SM
GL -3.55m
2.5
GL -2.8m
2.0
GL -4.5m
GL -6.85m GL -8.7m
SM
GL -10.7m
1.5 Fs
GL -10.15m
ML
1.0
GL -13.45m
0.0
Fig. 14. Subsoil profiles and construction sequence of Taipei Rebar Broadway Case.
50
cu(kPa)
100
150
CK0U-AE
Depth(m)
CK0U-DSS (AC+AE+DSS)/3*0.817
20 Used in analysis (CK0U-AC values) 25
10
15 D(m)
20
25
30
4.3. Taipei Shi-Pai case of Hsieh et al. (2008) The Taipei Shi-Pai case was a rectangular excavation with 12.3 m width and 45 m length. The soil profiles and construction sequence of this excavation are shown in Fig. 18. The depth of diaphragm ranges from 13.8 to 17 m and is 15.4 m on average with a thickness of 0.5 m. The plastic bending moment, Mp, of the wall is taken as 570 kN m/m. The basal heave failure occurs after the completion of final stage (GL9.3m), with the collapse of the bracing system and adjacent buildings. The soil conditions were reinvestigated at a site 500 m from the
Used in analysis (CK0U-AE values)
30 Fig. 15. Variation of undrained shear strength with depth for different test types in Taipei Rebar Broadway Case (modified from the original figure by Hsieh et al. (2008).
2.5
of the excavation was completed and its stability should have been in a critical state. If the influence of surcharge on the ground surface is taken into consideration, the factor of safety Fs should be even lower. Under such a condition, the factor of safety Fs derived from the proposed method is consistent with field observation. The calculations considering the anisotropic shear strength may relatively complicated for engineering practice, therefore, two isotropic undrained shear strength are considered in this study. Istropic strength I1 – the undrained shear strength are taken as that from the CK0U-DSS test results. Istropic strength I2 – following the suggestion by Ladd and Foott (Ladd and Foott, 1974), the undrained shear strength was taken as an average result of CK0U-AC, CK0U-AE, and CK0U-DSS tests (Do et al., 2016), while it is necessary to consider the strain rate effect by multiplying a correction factor (μt), which was summarized by Kulhawy and Mayne (1990) as follows:
μt = [1−0.1log10 (t f /tlaboratory )]
5
where tf = time to get failure of soil in the field; and tlaboratory = time to get failure of soil in the laboratory. It was assumed that tf was 2 weeks for all failure cases as a rough estimate, and tlaboratory was 5h for the consolidated undrained test with a strain rate of 1%/h, which was considered as the standard reference test. Hence, μt = 0.817. Fig. 17 shows the result obtained from the proposed method with the consideration of isotropic shear strength. The critical Fs obtained from the calculation using isotropic strength I1 is 1.27. It is obvious that the results of isotropic strength I1 greatly overestimate the stability of the excavation. By using the isotropic strength I2, the critical Fs obtained from the proposed method is 1.022, which is consistent with the one (Fs = 1.0) from FEM (Do et al., 2016) and closer to the critical state.
CK0U-AC 15
0
Fig. 16. Relation between factor of safety and wall penetration depth for anisotropic strength analysis in Taipei Rebar Broadway Case.
GL -24m
10
D=10.55m
0.5
CL
0
1.046
Istropic strength I1 Istropic strength I2
2.0
Fs
1.5 1.335 1.022
1.0 0.5 D=10.55m 0.0
0
5
10
15
20
25
30
D(m) Fig. 17. Relation between factor of safety and wall penetration depth for isotropic strength analysis in Taipei Rebar Broadway Case.
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Fill
GL -1.4m GL -3.7m
GL 0.0m GL -1.5m
2.5 2.0
ML
1.5 Fs
GL -5.5m
GL -6m
0.997
1.0 0.95
CL
GL -9.3m
GL -8.5m
0.5 0.0
CL
D=6.1m
0
5
D=7.7m
10
15 D(m)
20
25
30
Fig. 20. Relationship between safety factor and wall penetration depth for anisotropic strength analysis in Taipei Shi-Pai Case.
2.5
Istropic strength I1 Istropic strength I2
Fig. 18. Subsoil profiles and construction sequence of Taipei Shi-Pai Case.
10
20
(AC+AE+DSS) /3*0.817
5
2.0 50
60
70
1.5
CK0U-AE CK0U-AC
0.5
Depth(m)
D=6.1m 0.0
15
25 30
1.03
1.0 0.981
CK0U-DSS
10
20
1.326
1.266
Fs
0 0
cu(kPa) 30 40
0
5
D=7.7m 10
15 D(m)
20
25
30
Fig. 21. Relationship between safety factor and wall penetration depth for isotropic strength analysis in Taipei Shi-Pai Case.
Used in analysis (CK0U-AE values)
Fig. 21 shows the relationship of Fs and D calculated from the proposed method using the isotropic shear strength I1 and I2. As this is a failure case, it is obvious the Fs calculated by using isotropic shear strength I1, is apparently overestimated. By using the isotropic shear strength I2, the critical Fs obtained from the proposed method are 0.981, for wall embedded depth D is 6.1 m, and 1.03, for wall embedded depth is 7.7 m, respectively, which are close to one (Fs = 1.0, for D = 6.1 m) from FEM. Therefor, with the consideration of isotropic shear strength I2, the Fs values obtained from the proposed method is more closer to the value that is supposed to be attained in a critical state.
Used in analysis (CK0U-AC values)
Fig. 19. Variation of undrained shear strength with depth for different test types in Taipei Shi-Pai Case (modified from the original figure by Hsieh et al. (2008).
excavation and a series of undrained tests were conducted. The undrained shear strength of the soil was obtained from the results of triaxial K0-consolidated undrained compression (CK0U-AC) tests, extension (CK0U-AE) tests, and direct simple shear (CK0U-DSS) tests, as shown in Fig. 19. Fig. 20 shows the relationship of Fs and D calculated from the proposed method with the consideration of anisotropic undrained shear strength. Two depth of wall penetration are selected to examine the stability of base at final excavation stage. The critical Fs values is 0.95, for wall embedded depth D is 6.1 m, and 0.997, for wall embedded depth D is 7.7 m, respectively. The basal heave failure occurs after the final stage, so the critical Fs of the excavation should be closer to 1.0. Therefor, the Fs derived from the proposed method if consistent with the failure conditions observed.
5. Conclusions Based on the upper bound theory and with the usage of nonuniform shear zone, this paper presents a kinematic approach to estimate the basal stability of excavations with wall embedment in soft clay, with the consideration of wall strength and embedded depth. The anisotropic and non-homogeneous undrained shear strength are also incorporated in the proposed mechanism. The wall deforms consistently with the adjacent soil in the mechanism. The influence of the wall stiffness on the stability is taken as the elastic strain energy added to the total internal work. The least upper-bound solutions are obtained through the 240
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optimization of the kinematic mechanism. The following conclusions can be drawn on the basis of the work presented herein:
for Distinguished Young Scholars of China (Grant No. 50825803). These supports are gratefully acknowledged.
1. For excavations with wall embedment in isotropic and homogeneous clay, the proposed method is capable to evaluate the effect of wall stiffness on the basal stability by comparison with the upper bound results of numerical limit analysis. The factor of safety obtained from the proposed method can reasonably describe the undrained stability of excavation base for five field cases of O'Rourke (1993), which had been at or near failure. 2. The results of numerical excavation cases in anisotropic and nonhomogeneous clay indicate that the predicted failure depth obtained by the proposed method were consistent with those obtained by the finite element analysis. 3. Two failure excavation cases of Hsieh et al. (2008) in anisotropic and non-homogeneous clay were also studied to verify the validity of the proposed method. It was found that the proposed method gave reasonable estimate of basal stability by using the anisotropic undrained shear strength. The proposed method overestimated the basal stability by using the results of CK0U-DSS tests, and yielded reasonable estimation by using the average result of CK0U-AC, CK0U-AE, and CK0U-DSS tests with a multiplicative of a correction factor μt for strain rate effect.
References Bjerrum, L., Eide, O., 1956. Stability of strutted excavations in clay. Geotechnique 6 (1), 32–47. Chang, M.F., 2000. Basal stability analysis of braced cuts in clay. J. Geotech. Geoenviron. Eng. 126 (3), 276–279. Davis, E.H., Christian, J.T., 1970. Bearing capacity of anisotropic cohesive soil. J. Soil Mech. Found. Div., ASCE 97, 753–769. Do, T.N., Ou, C.Y., Chen, R.P., 2016. A study of failure mechanisms of deep excavations in soft clay using the finite element method. Comput. Geotech. 73, 153–163. Drucker, D., Prager, W., Greenberg, H., 1952. Extended limit design theorems for continuous media. Q. Appl. Math. 381–389. Eide, O., Aas, G., Josang, T., 1972. Special application of cast-in-place walls for tunnels in soft clay in Oslo. In: Proc. 5th European Conf on Soil Mechanics and Foundation Engineering, Madrid, Spain. pp. 485–498. Faheem, H., Cai, F., Ugai, K., Hagiwara, T., 2003. Two-dimensional base stability of excavations in soft soils using FEM. Comput. Geotech. 30 (2), 141–163. Goh, A.T.C., 1990. Assessment of basal stability for braced excavation systems using the finite element method. Comput. Geotech. 10 (4), 325–338. Hashash, Y.M.A., Whittle, A.J., 1996. Ground movement prediction for deep excavations in soft clay. J. Geotech. Eng. 122 (6), 474–486. Hill, R., 1998. The Mathematical Theory of Plasticity. Clarendon Press, New York. Hsieh, P.G., Ou, C.Y., Liu, H.T., 2008. Basal heave analysis of excavations with consideration of anisotropic undrained strength of clay. Can. Geotech. J. 45 (6), 788–799. Huang, M.S., Yu, S.B., Qin, H.L., 2011. Upper bound method for basal stability analysis of braced excavations in K0-consolidated clays. China Civ. Eng. J. 44 (3), 101–108 (in Chinese). Kulhawy, F.H., Mayne, P.W., 1990. Manual on estimating soil properties for foundation design. Rep. EL-6800, Electric Power Research Institute, Palo Alto, CA. Ladd, C.C., Foott, R.F., 1974. New design procedure for stability of soft clays. J. Geotech. Eng. Div., ASCE 100 (7), 763–786. Liao, H.J., Su, S.F., 2012. Base stability of grout pile-reinforced excavations in soft clay. J. Geotech. Geoenviron. Eng. 138 (2), 184–192. O'Rourke, T.D., 1993. Base stability and ground movement prediction for excavations in soft clay. In: Retaining Structures. Thomas Telford, London. pp. 131–139. Skempton, A.W., 1951. The bearing capacity of clays. In: Proc. Building Research Congress, Thomas Telford, London. pp. 180–189. Su, S., Liao, H., Lin, Y., 1998. Base stability of deep excavation in anisotropic soft clay. J. Geotech. Geoenviron. Eng. 124 (9), 809–819. Terzaghi, K., 1943. Theoretical Soil Mechanics. Wiley Online Library. Ukritchon, B., Whittle, A.J., Sloan, S.W., 2003. Undrained stability of braced excavations in clay. J. Geotech. Geoenviron. Eng. 129 (8), 738–755.
Finally, is should be pointed out that, the deformations of the wall in the proposed mechanism are dependent on the assumed velocity fields and are not completely in conformity with engineering practice. The proposed mechanism still yields rigorous upper bound solutions of the safety of factors, which are important in the preliminary design of excavations in undrained clay. Although the suitable of the proposed mechanism had been verified by parameter studies and some field cases, its applications to excavations not supported by high stiffness walls still needs further study. Acknowledgements This work was financially supported by the National Key R&D Program of China (Grant No. 2016YFC0800200) and the National Fund
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