A Qexibility number for the
displacement controlled design of
multi propped retaining walls
by TI Addenbrooke, Imperial College, University
of London. Cooling Prize winner 1994.
Introduction
To support excavations in urban areas where construction
space is limited and excavation depths large (deep building
substructures, or cut and cover transportation tunnels) it is
common practice to use vertical retaining walls supported
internally by multi level propping. Construction is usually
carried out in close proximity to existing roads, buildings and
services.
The analysis of such an excavation should therefore examine:
(1) The possibility of collapse of the retaining structure, either
as a result of excessive prop loads or failure of the soil at the
bottom of the excavation.
(2) The predicted movements caused by excavation and
dewatering, particularly with regard to their effects on nearby
structures (Clayton & Milititsky, 1986).
(3) The displaced shape of the wall, as large bending moments
may be induced (Potts & Fourie, 1985).
This paper considers points (2) and (3), movements and
induced bending moments. It is rare for a multi propped
excavation to fail due to structural problems. More commonly
'failure'f a support system is because of unacceptable
movements (Clough et al, 1989). Buildings sited adjacent to
deep cuts are generally less tolerant of the subsequent
excavation —induced differential settlements than similar
structures settling under their own weight (Boscardin &
Cording, 1989).
Potts & Day (1990) state that both experimental work (eg
Rowe, 1952) and more recent numerical work (eg Potts &
Fourie, 1985) indicate that under the same operating
conditions stiffer walls attract larger bending moments than
more flexible walls. The stresses imposed by the soil are &ee to
redistribute through a more flexible structure thus reducing the
structural forces imposed on the wall. This in itself is beneficial,
but occurs at the expense of larger wall and soil movements.
They observe that there is therefore a compromise between
reduced bending moments and increased movements as the
flexibility of the wall increases.
If greater movements cannot be tolerated, then more props
are required if a more flexible wall is to be employed. The
engineer needs a framework within which bending moment
reduction can still be considered, balanced against the required
increase in the number of propping levels. This paper
introduces a new flexibility number for multi propped retaining
wall design, the 'displacement flexibility'. The displacement
flexibility number permits the engineer to confidently consider
cost and construction variations within a displacement
controlled design &amework. Its applicability is justified using
finite element predictions of movements for various specific,
but representative support systems.
Two sets of analyses are presented. Set A models different
retaining wall systems supporting an excavation in dry soil,
while Set B models systems supporting a rapid excavation
followed by consolidation in a water bearing soil.
Review
Potts & Fourie (1985) presented the results of numerical
predictions for a propped retaining wall. It was demonstrated
that by increasing the flexibility of a wall in stiff clay, the
induced bending moment can be reduced significantly. It may
be possible therefore to provide a range of design solutions
&om a stiff to a flexible wall, noting that while a more flexible
wall may be cheaper, it will allow greater soil and wall
movements.
Potts & Day (1990) went on to investigate the complex
interaction between bending moment, soil and wall
movements, and cost, as the flexibility of the wall changes. By
considering three cases (Bell Common tunnel, George Green
tunnel, and the House of Commons car park) they showed that
a five fold reduction in maximum wall bending moment could
be achieved by using a sheet pile wall in place of a lm concrete
section wall. It was concluded that if the increased movements
associated with more flexible walls can be tolerated, or
reduced by extra propping, such walls provide a viable
alternative.
Flexibility nuxnbers
There are a number of existing ways of representing a retaining
wall's stiflness —material flexibility, Rowe's flexibility number
(Rowe, 1952) for single prop retaining walls, and system
flexibility (Clough et al, 1989) for multi propped retaining
walls.
~Material flexibility.
The flexibility of the material &om which a wall is composed is
commonly defined by:
ln
(EI)
The natural logarithm of EI, the Young's modulus, multiplied
by the second moment of area of its section.
~Wall flexibility.
The flexibility of a wall can be defined by the logarithm of
Rowe's flexibility number, log p:
log (
—)
where H is the wall height. Appendix A shows the theoretical
derivation of the flexibility number. It was devised by Rowe
(1952) to define a single propped model wall which
deformed to the same deflected shape as the prototype it
represented.
~ System flexibility.
Clough et al (1989) introduced the idea of system
flexibility, p,:
(EI
y„h4
)
where h is the average vertical prop spacing of a multi propped
support system and y„ is the bulk unit weight of water. This
was related to wall displacements, and hence ground
movements for a given factor of safety against basal heave and a
given support system.
~Displacement flexibility.
A logical extension to Rowe's flexibility number is to derive a
number, the constancy of which defines support systems with
the same absolute displacement (refer to Appendix B for
derivation). The 'displacement flexibility', A, of a support
system is defined thus:
A=(-h'I )
GROUND ENGINEERING
~
SEPTEMBER ~ 1994
Analysis
in clay
of a multi propped excavation
zom
oem
The finite element program ICFEP (Imperial College finite
~
——L
CA
element program) was used to perform different analyses of a
20m and 23m wall in a clay material with a large factor of
safety against base heave. This is representative of typical deep
excavations in London.
~Details of analyses
The retaining wall modelled was 20m in height, supporting a
40m wide excavation of depth 16m. One analysis was carried
out with a 23m wall, an increase in embedment depth of 75%.
Various elastic wall types were analyzed and are listed in
Table 1. Five different rigid prop spacings (h) were employed,
corresponding to the use of 2, 3, 4, 5, and 10 props. The props
were evenly spaced over the 16m excavation depth with the first
prop at the original ground surface on each occasion, and no
prop at the final reduced level of excavation.
The soil was assumed to be linearly elastic —perfectly plastic
with a Mohr-Coulomb yield surface. A non associated
condition in which the plastic potential is defined by an angle
of dilation, v, was adopted. The soil properties were as follows:
c'
Strength parameters
toom
xmv
r/Lvv
CAvv
//IXP
I
ov
+
KOav
0
25'
=
Angle of dilation
Poisson's ratio
Young's modulus
MOP
AT ONOUND EUNPACE
EXCAVATE TO NEXT MOPPINO LEVEL
12.5',=02
E'
6000z kN/m'here
z is the depth below the ground surface.
= 20kN/m'
= 1.0 x 10-"m/s
Ko = 1.0
Bulk unit weight
7
Permeability
Initial efFective stress ratio
Table 1: Support systems.
CONTINUE TO ~INAL NEDUCED LEVEL
(Potts & Day, 1990)
E
In (EO
Prep
spso919 (GN/msi
1m Concrete
8m
Description
28
Diaphragm
4m
High Mod. 1BXN
Frodingham
Frodingham
'Soft'all
28
28
sheet 5.33m 210
4N Sheet
1N sheet
14.70
14.70
14.70
'l2;85
210
11,29,
5.33m 210
11.29
11,29
10.19
6.65
8m
4m
210
3.2m
210
1.6m
210
iog p
log
a
Rsl'e
-1.18 60.27 -1.87 1
-1.18 305.66 -2.75 2
-1.18 964.31 -3.37 3
-0.29 39,38 -1.86 4
-0.39 5
0.30 1.99
0.30 10.11 -1.27 6
0.30 31.86 -1.89 7
-1.89 8
0.78 25.89
-1.87 9
2.32 12.02
~Finite element simulation
42
One of the three finite element meshes used for this study is
shown in Figure 1. Symmetry has been assumed, and so the
mesh represents only one half of the excavation, with A-B being
the centre line. The soil was modelled using eight noded
isoparametric plane strain quadrilateral elements. The wall was
modelled using three noded isoparametric plane strain beam
elements.
Construction was simulated by removal of the elements in
front of the wall row by row to a depth of -16m. Three
different meshes were required to ensure that all the desired
prop positions matched with the top of a row of elements to be
excavated. The initial displacement boundary conditions
GROUND ENGINEERING
SEPTEMBER ~ 1994
Figure 1 (TOP): Typical 300 element mesh.
Figure 2 (BOTTOM): Example construction sequence.
applied were the same for all three meshes and are shown on
Figure .1. No displacements were permitted along the bottom
boundary, and only vertical displacements were permitted
along the two side boundaries. The boundaries were remote to
the excavation.
The following stages of construction and unloading were
modelled (see Figure 2):
ANALYSES SET A:
The soil was dry. Systems 1-9 (Table 1) were modelled.
(1) Wall installation.
A wished in place wall was adopted. The assumption therefore
was that the initial state of stress in the ground and the soil
strength were unaffected by wall installation.
(2) Supported excavation.
A prop was simulated at the ground surface by the specification
of zero horizontal displacement of the beam element node at
that point as excavation proceeded. When the next propping
position coincided with the reduced level of excavation a prop
was again simulated by the prevention of horizontal
displacement at that point on further excavation. This process
continued until the reduced level reached -16m.
ANALYSES SET B:
A water table at 2m below the ground surface was modelled,
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Set 8 afler cowolidation
Figure 3a (TOP): Maximum lateral wall displacement
and horizontal ground surface movement (as a % of
excavation depth) against ln (EI) and (EVE+4) for Set A
on completion of excavation.
FIGURE 3b (BOTTOM) 8 Maximum lateral wall
displacement (as a % of excavation depth) against log A.
Figure 4 (TOP): Maximum horizontal ground
surface movement (as a % of excavation depth) against
log A.
Figure 5 (BOTTOM): Maximum vertical ground
surface movement (as a % of excavation depth) against
log A.
with an initial hydrostatic pore water pressure distribution. The
boundary representing the line of symmetry was a no flow
boundary, the other boundaries were fully drained. Systems 1,
3, 4, 5, and 7 were modelled.
(1) Wall installation.
As for Set A.
flexibilities, and a complete log cycle of system flexibilities.
These five systems do however have the same displacement
flexibility. Figures 3b and 4 plot the same displacements
against log A. The sign convention adopted for the horizontal
ground movements in Figure 4 is such that a positive
displacement is a movement towards the excavation, this results
in the same trend for all three situations. There is clearly no
trend between displacements and ln (EI). The system flexibility
and the displacement flexibility however both relate to
movements. The reduced scatter apparent irom using
displacement flexibility, A, and its theoretical justification
(Appendix B) suggest it to be the best parameter to plot
displacements against. The predictions from analysis Set B
show similar results.
~Wall movements
Figure 3b shows maximum lateral wall displacement plotted
against log A. The results fall onto curves between the two
extreme cases of a four prop diaphragm wall and a two prop
sheet pile wall. The smaller the displacement flexibility, the less
the maximum lateral wall displacement.
~ Ground movements
Figures 4, 5 and 6 consider the movement of the ground
surface behind the completed excavation. It is these movements
(2) Supported excavation.
As for Set A. The excavation was carried out over a simulated
period of four months with the excavated surface having a pore
water pressure of zero at all times.
(3) Consolidation.
The pore pressures were permitted to equilibrate over a period
of 90 years. At the end of this period steady state conditions
had been achieved.
Support system behaviour
The maximum lateral wall displacement and the maximum
horizontal ground surface movement (a negative displacement
is towards the excavation), on completion of excavation from
analysis Set A are plotted as percentages of the excavated depth
against both ln (EI) and (EV7+4) in Figure 3a. It is evident
that there is very little variation in displacement for support
systems 1, 4, 7, 8, and 9 which cover a wide range of material
GROUND ENGINEERING
'EPTEMBER
'994
Wall type
ifo of propi
Ref No
logs
(fable
I
-3.37
4
Oiaphragm
line ttyie
I)
d.
I
that cause damage in adjacent structures.
Figure 4 is a plot of maximum horizontal ground surface
movement as a percentage of excavation depth against log A.
Again the results fall onto curves between the two extreme
cases of a four prop diaphragm wall and a two prop sheet
pilewall, with systems 1, 4, 7, 8, and 9 showing very little
spread.
Figure 5 is a plot of maximum vertical ground surface
movement as a percentage of excavation depth against log t5,.
The excavations modelled in dry soil produce heave behind the
wall (plotted as positive displacement). The analyses indicate
that this heave is controlled by the unloading infront of the
wall, lifting the support system upwards. All the support
systems modelled gave rise to the same maximum heave
adjacent to the back of the wall. Excavation in a water bearing
soil followed by consolidation produces settlement profiles
behind the wall (as shown in Figure 6). Plotting the maximum
vertical ground surface movement as a percentage of excavation
depth against log A reveals that the lower the system's
displacement flexibility, the less the maximum ground surface
settlement.
Figure 6 shows the marked similarity in settlement profiles
behind support systems with the same displacement flexibility
(systems 1, 4, and 7), bounded by a stiffer system (3) and a
softer system (5).
~Prop forces and wall bending moments
Figure 7 is a plot of the sum of prop forces against log A for all
the support systems. A linear relationship is evident. The lower
the displacement flexibility, the greater the sum of the prop
forces.
The bending moments induced in a wall are dependent on
the deflected shape. Bending moments are not therefore
uniquely related to the displacement flexibility which is
representative of a support system's absolute displacement
potential. Figure 8 shows that for a given A value (analyses 1,
4, 7, 8, and 9), the use of a more flexible wall with a
corresponding increase in the number of propping levels
(analysis 9 compared to 1) reduces the maximum bending
moment.
The single analysis performed with a four propped 23m
sheet pile wall with the same prop spacing as the four propped
20m wall, representing an increase in embedment depth of
75% showed no significant change in the prop forces (0.96%),
the wall and ground movements (2.5%), or the maximum
bending moment (3.5%).The average prop spacing therefore
appears to be an acceptable characteristic dimension.
Engineering application
The displacement flexibility,
A, as defined here, where h is the
average prop spacing can be used as part of a displacement
controlled design.
44
The prescribing of a maximum allowable displacement as a
design parameter does not restrict the engineer in the selection
of a support system. By keeping A constant the bending
moment reducing benefits &om the use of a more flexible wall
material, lower EI, can be accurately balanced against the
construction requirement of more propping levels, smaller h
(Figure 8). Several different support systems can therefore be
considered in the knowledge that the maximum allowable
GROUND ENGINEERING
SEPTEMBER '994
4-
I
2.
~
-2-
0
I
0
0.6
0.5-
0.4Iet 0 after QNsk4bN
0.3C
5et 6
0.2-
5et 0 after excavation
C
E
0.15
0
-3.5
-3
-2.5
I
I
I
I
-2
-1.5
-I
45
0
Figure 6 (TOP): Settlement profiles —Set B after
excavation.
Figure 7 (BOTTOM) t Sum of prop loads (non
dixnensionalised) against log A.
Figure 8 (RIGHT) i Magnitude of the maximum bending
moment (non dimensionalised) against ln EI for support
systems with the same log A.
displacement will not be exceeded and the profile of ground
settlement behind the excavation will be unchanged.
Conclusions
It has been shown that existing flexibility numbers for retaining
walls do not provide a complete &amework for displacement
controlled design of deep excavations supported by multi
propped systems.
A new flexibility number, the displacement flexibility
h, (= h'/EI), has been defined to enable the engineer to consider
different support options which will meet the same
displacement criteria.
Support systems with the same displacement flexibility Lec, on
completion of excavation, give rise to the same maximum
lateral wall displacement, the same ground surface response
(vertical and horizontal displacements), and the same sum of
prop loads.
0.02
1
O.O16
0.012
l5
Set
17
0.004
E
A
O.OO8
8
9
~
I
—-1—
I
~
Therefore the requirement
1011 12131415
5 6 7 8 9
where z is the depth below the original ground surface level.
In EI
0.02
If we
J
0.016
t
I
L
1
0.012
have,
Z
(H—
)modeJ
J
and,
P
=
Z
(—
E
0.008
'4
0.004
o —-I- —s. —-I ——I- —s—-I —~
Set B after excavation
(IPr.H.vt)
E.I
~
I
0
5 6 7 8 9
101112131415
sos
p =
In EI
E
(FPc.H.vt)~„~
E.I
—) = constant
(E.I
Hd
of p are
likely to yield the same
deflected shape.
0.08
L
0.06
P
0.04
4
1
The displacement flexibility nuxnber.
I
17 -I- Os-—
0.02
B
Appendix
Set B after consolidation
R
~
Walls with the same value
0.1
m Vi
pmrorypr
then,
17
psororysse
—
(M.
—
(M
modd
IQ
is,
(M.z)
(Mz)~,
E.I
E.I
0
~
For walls to have similar displacements;
(3 ) sysrem I = (3 ) sysrem 2
0
5 6 7 8 9 1011 12131415
In El
This can be expressed as:
(f ~dz),~rm(f ~dz),
2
Now,
(f
References
Boscardin, MD & Cording, EJ (1989). 'Building response to excavation—
induced settlement'. Journal of Geotechnical Engineering, ASCE, 115, No 1, 1-
21.
Clayton, CRI & Milititsky, J (1986). 'Earth pressures and earth retaining
structures'. Surrey University Press, London.
Clough, GW, Smith, EM & Sweeney, BP (1989). 'Movement control of
excavation support systems by iterative design'. Foundation engineering, current
principles and practices, Pmc ASCE, Vo12, 869-884.
Potts, DM 8t Fourie, AB (1985). 'The effect of wall stiffness on the behaviour of
a propped retaining wall'. Geotechnique, 35, 347-352.
Potts, DM 8t Day, RA (1990). 'Use of sheet pile retaining walls for deep
excavations in stiff clay'. Proceedings, ICE, Part I, 88, 899-927.
Rowe, P (1952). 'Anchored sheet pile walls'. Proceedings, ICE, I, 27-70.
where M is the bending moment which is assumed
proportional to (z'), and EI is the material flexibility.
If we
Rowe's flexibility number (Rowe, 1952).
Rowe derived this number to enable him to perform scaled
model retaining wall experiments, the deflected shape of which
were similar to the prototype, ie
(dg)modd —(
as for Rowe's derivation
have the same requirements
Appendix A), with D as depth,
( D )prororype
( D)modd
(see
'q
and,
M
M
(~)~m
(~)
f ~~
Then we have,
(
Appendix A
(f Mz«)
dz)
~d
.vi
dD
)o,,
m
(
f ~~D
.vl
dD
),
Integrated over the average prop spacing, h (the characteristic
dimension of a multi propped retaining system), this leads to a
solution of the form
1
2
hs
EI
g~)roororJvse
Thus the 'displacement
'
1
2
h'
EI
flexibility's defined:
where y is in the horizontal plane and z in the vertical.
—
')
From structural analysis,
=( E.I
(@)
dz
Walls with the same value of 1S are likely to have the same
magnitudes of displacements.
GROUND ENGINEERING
SEPTEMBER 1994
45