Modeling of an anchored diaphragm wall
A. Szavits-Nossan & M.S. Kovačević
University of Zagreb, Faculty of Civil Engineering, Zagreb, Croatia
Vlasta Szavits-Nossan
University of Zagreb, Faculty of Geotechnical Engineering, Varaždin, Croatia
ABSTRACT: The paper describes a case history and numerical modeling of an anchored diaphragm wall designed to secure an 18 m deep excavation pit for the construction of an underground garage and mall in downtown Zagreb. The numerical analysis performed by FLAC included detailed modeling of staged excavation
and prestressing of ground anchors. The soil behavior, ranging from large stiffness at very small strains to low
stiffness at large strains, was accounted for by a new elasto-plastic model, which was incorporated into FLAC
by the FISH programming language. The new model allows for adjustments to arbitrary shear stiffness reductions, depending on shear strains during loading. The required soil parameters were determined from filed and
laboratory tests, including triaxial tests with measurements of small strains and field geophysical tests. The
wall deflections, observed during different construction stages, compared very well with the predicted behavior.
1 INTRODUCTION
The finite element and finite difference methods are
powerful tools for a rational analysis of anchored
soil retaining structures. However, these methods often overpredict ground movements and wall displacements, when simple elastic models are used,
with stiffness parameters determined in conventional
laboratory tests. Thus, the limit analysis and the
Winkler spring model are still predominantly used in
practice.
Only recently, it became apparent that conventional laboratory tests overpredict strains, due to the
strong non-linearity of soil behavior in the small to
medium strain range (e.g. Jardine et al. 1986, Burland 1989, and Simpson 1992). The paper presents
the results of a numerical analysis intended to overcome this problem.
The analyzed anchored diaphragm wall is a cast
in place reinforced concrete structure, constructed in
the downtown area of Zagreb in Croatia. Existing
buildings sensitive to possible ground movements
closely surrounded the construction site. Ground settlements and anchored wall displacements were
monitored during the excavation and wall construction.
The numerical analysis was conducted by FLAC
(Version 3.22), using the new elasto-plastic soil
model described in the paper. The FISH programming language incorporated the new constitutive relationship into FLAC. The main features of this
model are, (a), accommodation of any given relationship between shear stiffness and shear strain including the strong non-linearity of soil behavior in
the range from small to large strains, and (b), proper
modeling of the increased soil stiffness upon unloading. The second feature is important, because it can
cover the increase in stiffness of the soil-structure
system during the ground anchor prestressing.
The selection of soil parameters, required for the
numerical analysis, was based on laboratory and
field test results. In situ measurements of shear wave
velocity by seismic down-hole tests, was an important source of data. Several investigators recently
proposed that the shear stiffness determined from
shear wave velocity could represent static shear
stiffness at very small strains (e.g. Tatsuoka &
Shibuya 1991, Tatsuoka & Kohata 1995). The shear
strength of fine-grained soils was determined in laboratory direct shear, and triaxial UU and CIU tests.
Field standard penetration tests were used for
strength characteristics of gravely soils. The triaxial
tests were performed with and without local strain
measurements. A limited number of triaxial tests
with small strain measurements provided data for
modeling the strong non-linearity of soil behavior in
the small to medium strain range.
The calculated displacements of the anchored diaphragm wall were compared to observed wall displacements during different construction stages.
depth, m
2 SOIL PROFILE AND WALL LAYOUT
0
The soil profile at the construction site is shown in
Figure 1. It was obtained from standard laboratory
and field tests. In addition, Figure 1 shows the
measured and modeled shear wave velocities.
The average strength parameters used in the numerical analysis are listed in Table 1, along with the
analysis type, for each soil layer. The referent shear
modulus and the modulus exponent, also presented
in Table 1, are variables used in the described new
soil model. Their values were determined from the
measured shear wave velocity.
Table 1. Design soil parameters (soil profile from Figure 1) and
related
analysis types.
___________________________________________________
Soil
_____ c'
' c_____
G
m
u
0
_____ 
___
______
__ Analysis
layer
kN/m3 kN/m2 deg. kN/m2 MN/m2 ___________________________________________________
Fill
19
5
30 -60
0.5 drained
Firm clay 19
15
22 -170
0.5 drained
Gravel
21
5
38 -170
0.5 drained
Stiff
clay
20
--150
320
0.0 undrained
___________________________________________________
 = unit weight; c' = cohesion; ' = friction angle; cu = undrained strength; G0 = referent shear modulus for mean effective stress s0 = 100 kN/m2; m = modulus exponent.
Fraction
content (%)
0 20 40 60 80 100 0
w (%)
20
40
-2
80 0
200
±0
elevation 119.6 m
fill
firm clay
prestressed
anchor
-10
gravel
diaphragm
w all
-15
stiff clay
-18
grouted
length
free
length
The cross section of the anchored diaphragm wall is
shown in Figure 2. The wall is a 22 m high and
0.8 m thick structure. The Young modulus of concrete was assumed 3 x 107 kN/m2. Commercial
ground anchors of high strength steel, with a nominal Young modulus of 1.95 x 108 kN/m2, were used.
The excavation and ground anchor prestressing proceeded in stages, as indicated in Table 2.
400
N (SPT)
600 0
20
40
60
vs (m/s)
80 0
200 400 600
0
5
Depth (m)
10
15
20
25
30
fill
firm clay
gravel
stiff clay
fraction of clay+silt
fraction of clay+silt+sand
plastic limit
natural water content
liquid limit
Figure 1. Soil profile at the site of the anchored diaphragm wall.
-23
Figure 2. Cross section of the cast in place diaphragm wall with
three rows of prestressed ground anchors.
qu (kPa)
60
0.8
unconfined compressive strength
SPT blow count
shear wave velocity (down-hole)
shear wave velocity (model)
Table
2. Construction stages and anchor characteristics.
___________________________________________________
Stage Description
z_ L
GL
A
F
_
__ 
___ a_
___
__0
m m
m deg. m
cm2 kN
___________________________________________________
1
Excavation
-5
2
1st anchor row -4 27 7
20
2.6 6.6 420
3
Excavation
-11
4
2nd anchor row -10 20 7
12
2.6 9.7 400
5
Excavation
-16
6
3rd anchor row -15 17 7
15
2.6 9.7 400
7
Excavation
-18
___________________________________________________
z = depth; L = anchor length; GL = anchor grouted length;
 = anchor inclination; a = horizontal distance between anchors; A = anchor cross section area; F0 = anchor prestressing
force.
3 THE SOIL MODEL
3.1 Shear stiffness
The new soil model was developed for the plane
strain condition. Plane stress invariants t and s', defined as
c
t  dij dij 2
h,
12
dij  ij
s ij , s21  ii
(1)
were used, where t = maximum shear stress; d = deviator stress tensor; ' = effective stress tensor;
s' = mean effective stress; and  = the twodimensional unit tensor (indices may take values 1
and 2). The effective stress analysis was performed
by introducing the effective stresses as
 ij ij uij
(2)
where  = total stress tensor; and u = pore water
pressure. The standard continuum mechanics sign
convention is used, as in FLAC.
Three features characterize the shear stiffness of
the new soil model: The elastic shear modulus, the
shear strength, and a target shear modulus function.
The elastic shear modulus Ge defines the maximum
tangential shear stiffness of the soil at a given stress
state, and will be covered in the next section.
The Mohr-Coulomb law given by
t f c
s sin 
(3)
governs the shear strength, where tf = t at failure;
c' = the cohesion intercept; and ' = the friction angle.
The target shear modulus function rt is a normalized function with a normalized argument. It is given
by
G G e rt 
x
, x r
(4)
where G = secant shear modulus for monotonic
loading, defined as
G t 
 = the maximum shear strain, defined as
(5)
c h,
 2eij eij
12
1
eij 
vij , v 
ij 
ii
2
(6)
e = deviator strain tensor;  = strain tensor; and
v = volumetric strain.
The referent shear strain is defined as
r t f G e
(7)
The target backbone function t/tf = bt(x) is related
to the shear modulus function rt by
b t 
x rt 
x x
(8)
Two assumptions are made on the form of the
target backbone function, i.e. two limits are imposed. The first limit is the elastic limit el, below
which the target backbone function is equal to x
(rt = 1). The second limit is the failure shear strain f,
beyond which the target backbone function is equal
to unity (rt = 1/x). Within these two limits, the target
backbone function bt can assume any form. It is assumed unique for a given soil. The uniqueness of
this function complies with the experimental evidence provided by Anderson & Richart (1976), and
Richart (1977, 1978, and 1982). Puzrin  Burland
(1996) discussed specific forms of backbone functions for different soils. A normalized backbone
function for Toyoura sand was developed by
Szavits-Nossan  Kovačević (1994).
3.2 Elastic strains
The proposed soil model incorporates a plane strain
case of the simple nonlinear elastic constitutive
equation proposed by Vermeer (1978, 1984),
 ij2G e ije
(9)
where e = elastic strain tensor; and the elastic shear
modulus Ge is given as a function of the stress invariant I, or the elastic strain invariant I
b g G bI v g
G e G0 I  s0
m (1
m)
m
0
 0
(10)
with
c
I    ij
 ij 2
h,
12
d i
e e
I  2
ij ij
12
(11)
The soil parameter G0 is the referent elastic shear
modulus, defined for a referent stress state given by
t = 0 and s' = s0; v0 is the referent volumetric strain
given by v0 = s0/G0, and m is the shear modulus exponent.
Equation (9) may also be written as
sG e v e , t G ee
(12)
where ve and e are the volumetric strain and the
maximum elastic shear strain, respectively.
It can be shown that this elastic constitutive equation has a strain energy potential U given by
2
m
U 21
Ge I

m
(13)
1.2
The effective stress is the gradient of this potential.
The tangential form of equation (9) may be written as
e
11
e
22
e
12
L
2G E
M

2 G E
M
M
N 0
e
e
t
1
t
1
e

2 t2 G e E 2t
t
1
2G
e
E 2t
0
O
L
O
P
M
P
P
M
P
 P
M

1P
N
Q Q

0  11
0  
22
0.8
(14)
12
z = y/x
L
 O

M
P 1
 P

M
M
 P

N
Q2G
1
1.0
0.4
where
Eit
t2
m
(2G ) (1 R ) Ai , R  1 
2
e
1t
2
E2t
E1t
,
2
A1 1 
1 m R ,
it
mR Ai ,
A2 
1 mR 2
0.2
m+1
(15)
0.1
m+1
y
(16)
where (m)e, (m)p = elastic and plastic strain tensor
for the m-th element, respectively. The elastic range
of each element is bounded by a limit strain (m)l, so
that
d
i
( m) e ( m) e 1 2 ( m)
ij
ij  l
(17)
The m-th limit shear strain defines a cylindrical
yield surface in the elastic strain space of the m-th
element. The axis of this cylinder is the zero shear
strain line ((m)e = 0). Plastic strains may develop
when elastic strains reach the yield surface. The following flow rule is obtained by assuming normality
( m)
 ijp ( m) -2

l 2
( m) e
ekl
 kl

( m) e
eij
(18)
where <> denote Maculay brackets (x  x for x 
0; and x  0 for x  0). Because of the cylindrical
yield surface, and the associated flow rule, the model does not develop plastic volumetric strains. The
elastic strain rates are then calculated as
( m)
( m)  p
 ije 
 ij 

ij
(19)
The kinematic hardening is introduced by assuming that the elastic strain is a weighted average of element elastic strains
n
ije  ( m) w
m1
( m) e
ij
n
0.8
In the proposed new soil model, the kinematic hardening follows the concept of Iwan's array of parallel
elasto-plastic elements (Iwan 1967). The described
model consists of n parallel elements undergoing the
same strain history. If the element strains are decomposed into elastic and plastic components, it follows that
e  2
10.0
0.6
3.3 Kinematic hardening
( m)
1.0
x
1.0
It can be seen that equation (9) is isotropic with a
zero Poisson's ratio, whereas its tangential form is
orthotropic with a nonzero Poisson's ratio.
ije ( m) ijp ij , m 1,..., n
n
0.0
'1 and '2 are the principal effective stresses.
( m)
0.6
(20)
target
0.4
model
0.2
elastic
m
1
0.0
0
10
20
30
x
40
50
60
Figure 3. A target and a model shear modulus function, and the
corresponding target and model backbone functions (z = normalized shear modulus; y = normalized shear stress; x = normalized shear strain).
where (m)w = weight of the m-th element. It is readily
verified that this model has a piece wise linear backbone function of the form
n
y b m ( x )  ( m) w
c  h
( m)
e
r
(21)
m1
Weights are determined from a given target
backbone function bt. Requiring that the model
backbone function bm coincides with the target
backbone function bt at n points, for which  = (m)l,
the following expressions for weights are obtained
(provided the limit shear strains (m)l form an increasing order)
(1)
(1)
( n)
(n
1)
y (2) y 
y
y
y
w  (1) ( 2) (1) , ( n) w  ( n) ( n 1)
x
x x
x x
( m)
( m
1)
( m
1)
( m)
y

y
y

y
( m)
w  ( m) ( m1) ( m1) ( m)
x x
x x
(1)
(22)
where (m)y = bt((m)x); and (m)x = (m)l/r. It is to be noted that the weights are not constant. They are functions of the referent strain r, which is, in turn, a
function of the current stress state.
Figure 3 shows a target and a model backbone
function, and the corresponding modulus reduction
functions. The model backbone function is plotted
for a set of five element limit shear strains and a giv-
en referent strain. It is obvious that a better approximation of the target backbone function is obtained
for a greater number of elements. It is to be noted
that, for a given number of elements, a better approximation of the target backbone function is obtained by a convenient selection of element limit
shear strains.
If the first and the last limit shear strain satisfy
the following inequalities
(1)
l el ,
( n)
l f
(23)
where el and f were defined in Section 3.1, it can
be shown that
n
 (m) w 1
(24)
m 1
The strain tensor of the model can then be decomposed into an elastic and a plastic component, so that
n
ije ij ijp , ijp  ( m) w
( m) p
ij
(25)
m 1
and the stress tensor can then be calculated using the
elastic constitutive equation (9).
This new model satisfies the Masing rules for
regular cyclic loading (Masing 1926, Pyke 1979),
expressed by
b
g
y y0 2b m ( x x0 ) 2
(26)
where (x0, y0) are the coordinates of the unloading
(or reloading) point on the stress-strain hysteresis.
Furthermore, the new model naturally extends the
Masing rules to irregular cyclic loading, without resorting to any arbitrary rules (Pyke 1979). Due to
constant element limit shear strains, the apparent
yield surface of the model is characterized by the
constant shear strain surface, which complies with
experimental evidence for sands (Tatsuoka  Ishihara 1974).
ed, ten variables remain to be stored for each of four
triangular subzones of the FLAC quadrilateral zone,
giving a total of forty variables. These forty variables were stored using variables declared by the
fprop statement.
Any target backbone function may be given as
input to FLAC by a table of (x, y) pairs. The table
function was used for this purpose.
3.5 Material functions and parameters
Four material parameters and one material function
are used in the constitutive equation: the referent
shear modulus G0, the modulus exponent m, the cohesion c, the friction angle , and the target backbone function bt(x). The material parameters for the
analysis of the diaphragm wall are listed in Table 1.
The design shear wave velocities (vs = (Ge/)1/2,  =
soil density), calculated from these material parameters, are shown in Figure 1.
The same backbone function, shown in Figure 4,
was used for all soil layers. This function was developed from extensive studies of numerous laboratory
data on Toyoura sand (Szavits-Nossan & Kovačević
1994). Figure 4 also shows typical data from triaxial
CIU tests, with local strain measurements, performed on the stiff clay from the site of the diaphragm wall. CIU triaxial tests were performed in
the hydraulic triaxial apparatus from GDS, UK. Hall
effect transducers (Clayton & Khatrush 1986) were
used to measure local vertical and lateral strains on
soil specimens.
Despite the fact that the backbone function was
developed for a specific sand, it approximates the
stiffness of the stiff clay fairly good.
The constitutive model uses element limit shear
strains, for which the model backbone function
matches the target backbone function (see Section
3.3). For the five element model used in the present-
3.4 Implementation of the model into the FLAC
code
Normalized shear modulus
The constitutive equation described in previous sections was implemented into FLAC by the use of the
FISH language. Version 3.22 of the code was used at
the design stage, but the results presented in this paper are from reruns by Version 3.4. Although the
constitutive equation may use any number of parallel
or overlaying elements, the code was developed for
five elements only. It is the authors' opinion that five
elements may model any backbone function with
sufficient accuracy for practical purposes.
Being developed for plane strain only, the fiveelement model has to store ten deviator elastic strain
components, two for each local element, and one
elastic volumetric component. Since FLAC automatically stores the global stress components, from
which the elastic volumetric strain may be calculat-
1.5
1.5
Local strain measurement:
loading
unloading
reloading
second unloading
second reloading
External strain measurement
Target backbone
1.0
0.5
0.0
0.01
1.0
0.5
0.0
0.1
1
10
Normalized shear strain
100
Figure 4. The design (target) shear modulus function and typical results from CIU triaxial tests with local strain measurements on the stiff clay at the diaphragm wall site.
4 CONSTRUCTION STAGES: PREDICTIONS
AND OBSERVATIONS
All construction stages indicated in Table 2 were
modeled by FLAC. A rectangular mesh of 28 horizontal x 35 vertical zones (56 m x 35 m) was used
for the spatial discretization. The initial vertical
stresses were calculated using the soil unit weights.
The initial horizontal stresses were assumed equal to
vertical stresses. The drained conditions were assumed for the upper three soil layers, as indicated in
Table 1. The drained state was modeled by fixed
values of pore water pressure, zero bulk modulus of
water, and strength parameters related to effective
stresses. It was assumed that the undrained state is
more appropriate for the bottom stiff clay layer. The
undrained condition was modeled by a stiff bulk
modulus of water, and strength parameters related to
total stress. Hydrostatic water pressures were assumed in the gravel layer.
Predictions by the numerical model were made at
the design stage, i.e. before the construction of the
anchored diaphragm wall. Wall movements were
measured during the excavation of the construction
pit, and compared to predicted values. Surveying
methods and several inclinometers installed in the
concrete wall were used for the measurement.
At the end of Stage 7, the maximum horizontal
displacement at the top of the wall, measured by
surveying methods, was about 1 cm, which is less
than the corresponding predicted displacement of
about 2.5 cm. Taking into consideration the low accuracy of the used surveying method, and the uncertainty of soil parameters, this prediction is considered satisfactory.
As for the displacements along the whole wall
length, it has to be noted that it was not possible to
rectify reliably the inclinometer measurements, because there was no fixed point either on the top, or
on the bottom of the wall, to refer them to. Therefore, only relative displacements of the wall with respect to its top can be compared to predicted values.
Figure 5 shows computed horizontal displacements
of the wall at the end of stages 2, 4 and 7. It also
shows horizontal displacements of the wall measured by inclinometers. These values were rectified
by adding a constant displacement, in order to have
the computed and measured values coincide at the
wall top. The resulting curves show a good agreement of predicted and measured values. The bending
moments were also well predicted.
Figures 6 to 9 show some typical FLAC output.
Figure 8 is interesting in showing the lines of equal
maximum shear strains. Shear strains are about 0.4%
close to the wall, and then they rapidly decrease by
an order of magnitude, or more, with the distance
from the wall, indicating the corresponding increase
in shear stiffness of the soil.
5 CONCLUSIONS
From the presented case history and numerical analysis of an anchored diaphragm wall, with the emphasis on constitutive modeling and the determination of soil parameters, the following conclusions
are proposed:
 The shear wave velocity determined in field tests,
which is a ready available and reliable, but not
commonly required parameter in soil-structure interaction projects, can be used to determine the
static soil stiffness at very small strains; this is
one of the few soil parameters required by the
proposed new soil model.
 The new kinematic hardening constitutive equation is capable of modeling any measured relationship between the soil shear stiffness and shear
strain in the range from very small up to the failure strains; this is very important in soil-structure
interaction analyses; good predictions of the behavior of an anchored diaphragm wall were obtained when this relationship was properly chosen.
 The new soil model requires a normalized shearstress-shear-strain backbone function, which results from special laboratory tests on undisturbed
soil specimens; the presented analysis has shown
0
Stage 2
5
Depth (m)
ed analysis, the following element limit strains were
found appropriate: (m)l = 10-5, 10-4, 10-3, 10-2, and
10-1.
10
Stage 4
15
Stage 7
20
predicted
measured
25
0
1
2
Displacement (cm)
3
Figure 5. Predicted horizontal displacements of the anchored
wall after completion of stages 2, 4 and 7, and displacements
measured by inclinometers, rectified to coincide with the computed displacements at the top of the wall.
that
of wall displacements can
JOBgood
TITLE : predictions
Phase VII
even be obtained by using a normalized backbone
FLAC (Version
function
from the3.40)
literature.
 The new soil model was incorporated into the
FLAC code,
and used to solve a complicated
LEGEND
practical problem of a non-linear soil-structure
in(*10^1)
teraction in the design stage; the observed behavior during the construction stage compared very
well with the predictions.
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H: 1.500E-02
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0.000
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JOB TITLE : Phase VII
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Figure 6. Vertical displacements at the end of Stage 7.
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step
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Figure 7. Horizontal
displacements at the end of Stage 7.
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Contour interval= 5.00E-04
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C
I IH
GF
E
D
C
0.000
-1.000
Figure 8. Maximum shear strains at the end of Stage 7.
-2.000
3.000
LEGEND
21-Mar-99 17:26
step
8967
-3.911E+01 <x< 2.311E+01
-2.561E+01 <y< 3.661E+01
2.000
Boundary plot
0
1.000
1E 1
Axial Force on
Structure
Max. Value
# 1 (Cable) -2.048E+02
# 2 (Cable) -2.516E+02
# 3 (Cable) -2.300E+02
# 4 (Beam ) -4.178E+02
0.000
-1.000
Figure 9. Beending moments and anchor forces at the end of Stage 7 (maximum bending moment in the diaphragm wall:
418kNm/m)
-2.000
University of Zagreb
REFERENCES
Faculty of Civil Engineering
-3.000
-2.000
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