Delattre - A century of desing methods for retaining walls

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33
 
A century of design methods for retaining
walls – The French point of view
 
I. Calculation-based approaches –
conventional and subgrade reaction methods
 
Luc DELATTRE
 
Laboratoire Central des Ponts et Chaussées
 
Introduction
 
A very wide variety of design methods for retaining walls are in current usage. Technical docu-
mentation, whether textbooks or documents intended for engineers, contains methods that date
from the beginning of the century as well as methods developed from the 1970s, completely empir-
ical methods and methods that are based solely on theoretical models, methods which claim to take
account of the in-service behaviour of structures while also being described as limit equilibrium
methods and methods that actually take account of the behaviour of structures at failure, although
this last group is small.
However, the situation is not chaotic. It is simply the result of repeated attempts to deal with one of
the most complex types of geotechnical structure, which is not only supported by the soil, as is the
case with foundations, but also loaded by the soil.
After a survey of the state of the art as it existed at the start of the twentieth century, this paper uses
the bibliography in order to describe the development of design methods while explaining how
each new generation of method provided responses to the general problem of retaining walls and
by situating, when relevant, the problem in the engineering context of its day.
This account will be presented in three parts. The first, i.e. this paper, deals with the theoretical
approaches to the design of retaining walls, which have dominated in France from the start of the
twentieth century until the present, and the so-called “classical” design methods which are based
on them, and the subgrade reaction method. The second part will deal with the empirical approach
to the design of retaining walls which was mainly developed in the English-speaking world from
the 1930s. Lastly, the third part will deal with the contribution which finite element method has
made, from the 1970s, to the design of retaining walls.
 
The question of the design of retaining walls
 
For retaining walls, the domination of masonry gravity walls until the beginning of the twentieth
century led researchers, from the predecessors of Coulomb to Boussinesq, to concentrate on the
active earth pressure exerted on structures of this type (Delattre, 1999). In this area, the success of
eighteenth and nineteenth century engineers is undeniable, as their work still provides the basis for
the design of structures in France and elsewhere in the world. 
The responses that were given at this time (Coulomb’s method and the theories of Rankine and
Boussinesq in particular) provide a satisfactory method of dealing with the problem: gravity walls
are rigid structures whose kinetics generally involve overturning of the structure under the action
of sustained earth pressure, the latter being thus decompressed laterally and brought to active fail-
ure (Fig. 1). 
At the start of the century, the development of flexible structures that are supported by the soil and
subject to deforma:tion (Delattre, 2000), added major new elements to the question of the soil-
structure interaction. Passive earth pressure was now considered in addition to active earth pres-
sure, in view of the soil’s response to the embedded portion of the structure. The kinematics of
 
34
 
rotation at the top of the wall in the
case of an excavated structure were
added to those of rotation at the base
which apply in the case of walls that
retain fill. Finally, new forms of inter-
action with flexible retaining walls
were added to those that were known
in the case of rigid gravity walls
(Fig. 2). 
The successive identification of these
new aspects of the soil-retaining wall
interaction and the attempts to provide
solutions (on the basis of the general
techniques available at the time) each
constitute stages in the development
of design methods for retaining walls.
These developments took place in four directions (Fig. 3): initially, the approach seems to contin-
ued the tradition of research into retaining walls that started in the eighteenth century. This work
made considerable use of existing theories of active earth pressure, in addition to statics and the
strength of materials. This work led to the developed of so-called “classical” methods;
 
➢ 
 
this first approach was joined fairly soon by the approach based on the concept of the modulus
or coefficient of subgrade reaction, the basis of which was developed during the nineteenth century.
This approach remained relatively undeveloped during the first half of the twentieth century as it
required integration capabilities which only really became available to engineers with the advent of
computing in the 1960s;
 
➢ 
 
very soon these theoretical approaches encountered limits as regards the representation of phys-
ical phenomena and an empirical approach developed which was to remain present throughout the
twentieth century and play a role in the development of a frame of reference for the observed behav-
iour of structures which is in use to this day;
 
➢ 
 
later, retaining walls, like the other types of geotechnical structures, were to benefit from
advances in the fields of mechanics and numerical design methods with the application, from the
1970s, of the finite element method.
This article covers work in the first two of the above directions, and furthermore analysis is
restricted to developments related to the modelling of structures under in-service loading condi-
tions*. Three major stages in the development of these methods have been identified. The first was
the consideration of a flexible retaining wall with the introduction of modelling of the reaction of
the soil to the embedded portion of the structure. Associated with this were developments with
respect to the computation of passive earth pressure and the method used to analyze the
equilibrium of the structure known as the free earth support method. The second stage involved
consideration of the flexibility of the embedded portion of the retaining wall. This was the main
issue for research in the field during the thirty years between 1930 and 1960 and generated a large
number of propositions. The third stage was the comprehensive consideration of the relative
flexibility of the wall and the soil, made possible by the subgrade reaction method. 
 
From active earth pressure to passive earth pressure – the free earth support 
method 
 
The most simple method for designing anchored retaining walls, and probably the first to be devel-
oped, is the free earth support method. This assumes that the retaining wall is displaced in a rigid
manner under the effect of active earth pressure and mobilizes both passive earth pressure along its
embedded portion and stresses on the support in its upper portion (Fig. 4). This method made it
 
* In modern terminology these are known as serviceability limit state analysis methods.
Overturning of wall
Lateral decompression and
settlement of supported soil
 
 Fig. 1 - Kinetics of a retaining wall and the supported soil.
 
35
1
1
2
3
4 1
2
3
4
1
2
3
4
1
2
3
1
2
3
1
4
4
2
3
Kinematics of a rigid retaining wall and the surrounding soil
Kinematics of an excavated retaining wall with evenly-spaced bracing installed during excavation
Kinematics of a flexible retaining wall that is anchored at the top and embedded in the soil
Displacement of wall
Lateral decompression and settlement of supported soil
Lateral decompression and raising of soil in front of wall
Support (tieback, bracing)
Displacement of wall:
bracing of the wall (installed from the top downwards,
as excavation progresses) leads to increasing displacement
of the wall with depth in the supported zone.
Lateral decompression of the soil increasing with depth
Lateral compression and raisingof the soil in front of the wall
Low lateral decompression of the supported soil in the vicinity of the support
High lateral decompression of the supported soil beneath the support
Reversal of the lateral compression-decompression regime
at the base of the embedded portion of the wall
Fig. 2 - Soil-retaining wall interactions.
 
36
Blum (1931)
Tschebotarioff (1948)
Butée simple
Rowe (1952)
1Winkler (1867)
Baumann (1935)
Rifaat (1935)
Terzaghi (1955)
Rowe (1955)
Ménard et al. (1964)
Haliburton (1968)
Balay (1985)
Simon (1995)Schmitt (1995)
Monnet (1994)
2
Clough et
Woodward (1967)
Bjerrum et al. (1972)
3
Peck (1943)
Peck (1969)
Terzaghi (1936)
Clough et
O'Rourke (1990)
4
Brinch-Hansen (1953)
5
Fig. 3 - Major directions and principal stages in the development of design methods for retaining walls.
The left side of the diagram shows serviceability limit state analysis methods (from right to left (1): classical methods, 
(2): subgrade reaction method, (3): finite element method (*) and (4): empirical methods) and the right side of the diagram 
(5) shows ultimate limit state analysis methods.
(*) The finite element method has undergone much development since the 1970s, so much so that all the major references could not be 
included here. Only the first references concerning the application of the finite element method to retaining structures have been shown.
F
N T N
T
1
2
1 : pa = Ka cosδa . σ'v
2 : pp = 1/Fb . Kp cosδp . σ'v
T: support reaction (tieback, bracing)
F: resultant of pressure
Fig. 4 - Kinematics of an anchored rigid retaining wall and the resulting pressure 
distributions (case of a homogeneous frictional soil).
 
37
 
possible to make immediate use of the results concerning active earth pressure that were provided
by retaining wall theory but soon encountered limits due to the inadequacy of the knowledge at the
time concerning passive earth pressure.
Thus while Vauban (Kerisel, 1993) or at a later date Poncelet (1840) already used passive earth
pressure to prevent partially embedded gravity walls built in the clay of the North of France from
sliding on their base, passive earth pressure did not, generally, play a decisive role in the stability
of gravity retaining walls. Coulomb (1776) mentioned the concept in his essay, but failed to
describe to what use it might be put. Later, Boussinesq only solved his equations for active earth
pressure. However, in the case of embedded retaining walls, the role of passive earth pressure is of
prime importance and the concept has been the subject of numerous developments. Extension of
Coulomb’s method
In the first justifications of the equilibrium of embedded retaining walls, calculation of the passive
earth resistance was based on Coulomb’s method: it was simply an extension to the calculation of
passive earth pressure of the active earth pressure calculation method in use at the time, indeed
Coulomb himself had envisaged an extension of this type. However, this calculation method was
very soon found to be wanting, from both the experimental and theoretical standpoints.
On the basis of experimental observations and for the sake of simplicity, Coulomb had limited
application of the “rules of Maximi and Minimi” for active earth pressure to flat rupture surfaces.
Debate concerning this fundamental hypothesis, particularly at the end of the nineteenth century,
demonstrated that it was acceptable in the case of most of the active earth pressure problems faced
by engineers at the time. It is still widely accepted a century later.
However, the hypothesis of flat rupture surfaces is open to question when passive earth pressure is
considered. It has thus been shown that the use of broken lines, arcs of circles, logarithmic spirals,
combinations of flat surfaces and logarithmic spirals, or combinations of flat surfaces and arcs of
circles in order to model rupture surfaces provides lower “Minimi” for passive resistance than
those obtained with flat surfaces. Of such methods, that developed by Ohde (1938), which com-
bines flat surfaces and logarithmic spirals, has become the standard “comprehensive” method
(Fig. 5). 
 
The Boussinesq-Caquot method
 
The application of Boussinesq’s work (1882) on the equilibrium of soil masses behind retaining
walls was, initially, mainly concerned with the problem of active earth pressure. Two principal
reasons can be suggested for this. First, as has already been stated, engineers at the time were more
concerned by active earth pressure than passive earth pressure; second, Boussinesq’s equations
were only solved, and this in an approximate manner, for certain active earth pressure configura-
tions.
O
M
τ
σ
θ
π 
−
 ϕ
2
π 
−
 ϕ
4 2
π 
−
 ϕ
4 2
Fig. 5 - Modelling of the rupture surface (passive case) with a combination of flat surfaces 
and logarithmic spirals (after Terzaghi, 1943).
 
38
 
The value of this method became apparent gradually, as the problem of passive earth pressure
gained importance and the shortcomings of Coulomb’s method became clearer.
However, the difficulty of solving the equations that resulted from this method stood in the way of
its implementation. Thus, during the first half of the twentieth century it was not possible to apply
Boussinesq’s method to passive earth pressure, as Boussinesq’s equations were not solved, for pas-
sive earth pressure, until Caquot (1934).
The publication of tables of active and passive coefficients calculated using this method (Caquot
and Kerisel, 1948) and its extension to the case of cohesive soils (Theorem of corresponding states,
Caquot, 1934) and to the case of loaded retained soil masses (passive and active earth pressures in
a surcharged non-cohesive weightless medium. L’herminier and Absi, 1962a, 1962b, 1965, 1969)
made this the standard method for calculating passive and active earth pressures, at least in France
(Kerisel and Absi, 1990).
 
The question of the fixity of the embedded portion
 
Blum’s method for the design of anchored retaining walls*
 
The rather basic free earth support method was shown to be inadequate as soon as a connection was
made between the loading exerted by the soil and the deformation of the retaining structure, with
reference to its real deformability, as it was by Blum (1931)**.
In the case of a retaining wall anchored by a row of tiebacks near its top, Blum analyzed, for differ-
ent depths of embedment, the distributions of the pressures acting on the structure, the bending
moments and the horizontal deflection of the structure. His analysis was essentially qualitative and
based on the interdependence of the distributions (the deflection of the retaining wall has points of
inflexion where the moment is zero, as does the plot of moments where the pressure is zero and the
mobilized pressure is related to the lateral deflection of the wall). This analysis allowed Blum to
observe that those structures with a short embedded depth will simply be supported by the soil and
that increasing the embedment depth mobilizes fixity in the soil (Fig. 6). 
Among all the possible configurations, Blum considered that the best compromise with regard to
the fixity of the retaining wall is obtained for embedment such that the tangent to the deflected wall
at its toe passes through the anchorage point. Greater embedment depths do not lead to a significant
increase in the fixity of the wall, while smaller embedment depths result in a reduction in the fixity
moment.
The problem thus posed can be solved graphically, but the process is nevertheless relatively
long***. To simplify computation, Blum stated that in the case of an embedded retaining wall, the
point at which the bending moment is equal to zero is fairly close to the point at which the resultant
pressure is zero. He therefore proposed that for computation the bending moment should be consid-
ered to be zero at the point of zero pressure (the so-called “approximate loading” of the “equivalent
beam”,Fig. 7b). In view of the small difference that is observed between the position of the point
where the bending moment is zero and the point of zero resultant pressure, it is assumed that no
significant error is introduced into the estimation of the maximum bending moment and the support
reaction.
In addition, with the aim of simplifying calculation of the embedment depth, Blum proposed that
the distribution of resultant passive pressures acting on the fixed portion of the wall should be
modelled by a single force, applied to the wall’s axis of rotation (Fig. 7c). A comparison between
the embedment depth obtained with this “idealized loading” method that obtained with the
“approximate loading” method shows that the ratio between them depends simply on the mobiliza-
tion of the resultant of passive pressure acting behind the wall whose ratio to active pressure acting
 
* Blum proposed a similar development for the analysis of unanchored retaining walls.
** The fixed earth support method is attributed to Blum (1931), however, Baumann (1935) gives Lohmeyer (1930)
credit for a similar method.
*** The graphical solution method, known as the elastic line method, is now used in a numerical form.
 
39
A
A
A
A
A
A
A
A
C
Pressures Moments Wall deflection
Fig. 6 - Influence of the embedment depth on the pressures acting on the retaining wall, 
the bending moments to which it is subjected and the wall deflection (after Blum, 1931).
 
40
 
in front of the wall, is denoted by n* (Fig. 7d). This led Blum to propose the “idealized” load
solution increased by a factor of 20% as the design value for embedment depth. 
 
Tschebotarioff
 
A major criticism that can be levelled at Blum’s method is that it has no direct experimental basis.
In contrast, the work conducted by Tschebotarioff between 1941 and 1949 largely consisted of
tests conducted on reduced scale models of retaining walls (Tschebotarioff, 1948).
These tests mainly involved flexible retaining walls embedded to 30% of their total height in mod-
erately dense to dense sand and supported near the top. The principle findings relate to the distribu-
tion of lateral pressure applied to the wall by the supported earth and the distribution of pressures
in the embedded portion of the wall, at different degrees of backfilling and excavation of the retain-
ing wall (Fig. 8). 
In qualitative terms, for the embedment depths used in the experiments, Tschebotarioff’s results
confirm Blum’s hypotheses. Thus, fixity of the wall in the soil is achieved through large mobiliza-
tion of passive earth pressure in the upper part of the embedded portion of the wall and the toe of
 
* This coefficient in fact describes the relative flexibility of the retaining wall in the soil.
A
B0
B0 B0
E E
H
T T T0
t t0
t'F F
b
a
gwl gwr f.t ν.f.t
y
O O
C
a b c
h' = b-y
0 1 2 3 4 5 6 7 8 9 10
1
1,1
1,2
1,3
1,4
1,5
1,6
1,7
µ = t / t0
v = gwr / gwl
d
H: free height of the wall
T: embedment depth
t: "net" embedded depth
y: level of point where the moment is zero
b: level of point with zero resultant pressure
a: upper support reaction 
B0: shear stress at the point where the moment of zero
A: upper support reaction 
C: resultant of active pressure acting behind the wall
 and active pressure acting in front of the wall
gwl: net passive pressure at foot of screen
gwr: net resultant of active pressure acting behind
 the wall and active pressure acting in front of the wall
 divided by the net embedment depth
f: net passive pressure passive pressure at toe of wall
ν: net resultant of active pressure acting behind the wall
 and active pressure acting in front of the wall divided
 by the net passive pressure
Fig. 7 - Blum’s method for the design of anchored sheet piles.
a. Resultant distribution of earth pressures on an anchored and embedded retaining wall.
b. “Approximate” loading for the embedded portion.
c. “ Idealized” loading ”.
 
41
 
passive pressure acting behind the wall. The latter is always much smaller than suggested by
Blum’s diagrams, as its values are similar to the earth pressure at-rest.
Fixity produces a point where the bending moment is zero which is closer to the dredge line than
the level where the resultant pressure is zero (Blum’s hypothesis). This finding led Tschebotarioff
to propose a new method for the design of flexible retaining walls set in sand and anchored near the
top which was based on the principle (1) of an embedded portion equal to 30% of the total height
and (2) a hinge in the wall at the dredge line level (Fig. 9). 
 
Rowe
 
Tschebotarioff’s experimental results demonstrated, qualitatively, the phenomenon of fixity in the
case of flexible retaining walls embedded in moderately dense to dense sand.
By conducting a series of tests, which also used reduced-scale models with materials of different
relative densities (gravels, sands, ash, wood chips), Rowe (1952) attempted to analyze this phe-
nomenon in greater detail. His results confirmed that because of the mobilization of fixity in the
After fill
Fill Excavation
Redistribution of
pressures after
vibration of ground
Normal relaxation
of support
No relaxation
of the support
Fig. 8 - Pressure distributions measured after various construction procedures 
(after Tschebotarioff and Brown, 1948).
-M
+M
(1)(2) (3)
Ap Ap
Rb
H
(A) (B)
βH
αH = 0,7 H
(α − β)H
[γ'(α − β)H]
(γβHKA)
D = 0,43αH
H: total height of wall
γ: density of soil
KA: active pressure coefficient
Ap: upper support reaction
Rb: shear stress at the base of excavation
(1), (2), (3): plots of the moments for different embedment depths
Fig. 9 - Influence of the degree of fixity of the wall on the distribution of bending moments (A) and diagram showing 
the principle of Tschebotarioff’s method (B), which corresponds to the distribution (2) of bending moments.
 
42
 
soil, the stresses (maximum bending moment, support reaction) in flexible retaining walls are
lower than in a perfectly rigid wall (free earth support method). On the basis of detailed analysis of
his results, Rowe showed (Fig. 10) that for walls with an embedment depth equal to approximately
30% of their total height, the reduction in stresses depended mainly on the flexibility of the wall
(expressed by the parameter where H denotes the total height of the retaining wall, E is
its elasticity modulus and I is its inertia) and the relative density of the soil. Furthermore it was
almost unaffected by the other factors that applied during the tests (internal angle of friction and
density of the soil, loads acting on the ground surface, relative position of the support at the top of
the wall).
Rowe also showed that the stress reductions can exceed those calculated using Blum’s or Tsche-
botarioff’s methods. This led him to describe additional reductions in stresses due to the mobiliza-
tion of passive earth pressure above the anchorage height, the mobilization of a shearing force at
the base of the structure and the influence of the anchorage, the embedded portion of the wall and
the flexibility of the wall on the vertical stresses applied to it. 
ρ H
4
EI
-------=
-4 -3,5 -3 -2,5 -2
0
20
40
60
80
100
Loose ash
Sand
Gravel
Wood chips
Plots of average values for 250 tests
Log ρ
Points based on the results
from Tschebotarioff’s tests
 Dense Loose
Maximum moment
(percentage of the maximum value
computed using the free earth
support method
Fig. 10 - Rowe’s experimental 
results (1952).
 characterizes the flexibil-
ity of the wall.
On this diagram, ρ is expressed in foot5/
pound.inch2.
ρ
H
4
EI
-------=
qs
Pa
Pp
Ps + Pw
θH
βH
T
R
(1
/2
 -
 β
)H
(1
 -
 α
)H
(2
/3
 -
 β
)HαH
1/3(1 - α)H
Ts
H: total height of wall
T: stress in upper support
Pa: effective earth pressure
Ps: pressure due to loads acting on surface
Pw: hydraulic pressure resulting from water level difference
Pp: effective passive pressure
Ts: frictionat base of wall
R: resultant support force at toe of wall (R = Pp + Ts)
Fig. 11 - Loading diagram used by Rowe for the free earth support calculation.
 
43
 
From this Rowe (1952) derived a new design method for embedded retaining walls. This method
begins with a free earth support design calculation, which is modified to take account of the mobi-
lization of a shearing force at the base of the wall (Fig. 11). The stresses thus calculated are then
corrected using design charts obtained from tests (Fig. 10) to take account of the flexibility of the
wall and the density of the soil. 
 
The subgrade reaction method
 
A solution to the problem of fixity of the embedded portion
 
The forerunners
 
Rowe’s design method for embedded retaining walls (1952) was a major advance over Blum’s
method (1931) as it made direct use of the two factors which govern the fixity of the wall, namely
its flexibility and the stiffness of the soil (which is related to its density). However, these factors are
introduced during the final stages of design of the wall as correction factors, after a calculation in
which neither the flexibility of the wall nor the stiffness of the soil are fundamentally present.
The flexibility of the wall and the stiffness of the soil only became really central to the modelling
of the behaviour of the embedded portion of retaining walls with the introduction of the subgrade
reaction method. This method, which can be applied to all interactions between a solid and a mass
of soil, lays down that the opposing reaction of the soil to the solid consists of a distribution of
pressures along the interaction surface, whose intensity p at a point is expressed by an equation of
the form p = k
 
y
 
, where y denotes the displacement of the interaction surface at the point in question
and k is a coefficient of subgrade reaction (Winkler, 1867).
The first civil engineering applications of the subgrade reaction method were Zimmermann’s cal-
culations (1888) of the stresses in railway sleepers*. Subsequent development of the method
involved foundations and rafts. It was introduced, very soon, into the field of retaining walls by
Rifaat (1935) and Baumann (1935)**. 
Development of this method was nevertheless hampered by practical implementation problems.
Before the advent of computing, this was necessarily analytical and many different approaches
were proposed (Rifaat, 1935; Blum, 1951; Richart, 1957). These methods were, however, not suf-
ficiently direct to be readily applied in standard engineering without computer technology. 
 
Rowe
 
The first genuinely practical results with regard to use of the subgrade reaction method for the
design of retaining structures are due to Rowe (1955), who published design charts for reducing the
stresses computed by the free earth support method for a wide range of retaining walls (Fig. 12).
In order to draw these design charts, Rowe (1955) undertook a theoretical study of the influence of
the relative flexibility of an anchored retaining wall on the stresses applied to the structure when
this is installed in a cohesionless medium. The study was conducted by calculation using the coef-
ficient of subgrade reaction, employed analytically, on the basis of the following hypotheses:
 
➢ 
 
the pressure exerted by the earth on the unembedded portion of the wall is the active pressure;
 
➢ 
 
the reaction of the earth mobilized at a point on the front surface of the wall is expressed by the
formula:
where
 
➢ 
 
m is a soil stiffness coefficient which does not depend on the dimensions of the structure,
 
➢ 
 
D is the embedment depth of the wall,
 
➢ 
 
z is the depth of the point in question,
 
➢ 
 
y is the displacement of this point,
 
* Terzaghi (1955), p. 298.
** Terzaghi (1955), p. 314.
p m 
z
D
----y=
 
44
 
which implies that the soil’s coefficient of subgrade reaction is a linear function of
depth under the dredge depth.
Furthermore, Rowe imposes no limits on the stresses applied to the soil at the front surface of the
retaining wall: the “elastoplastic” behaviour model, which implies that the soil pressure acting on
the wall will be between the active and passive earth pressures, was introduced at a later date (Hal-
iburton, 1968).
Rowe expressed the stresses cal-
culated in this manner as a func-
tion of the stresses calculated
using the free earth support
method, of the flexibility of the
retaining wall and dimensionless
parameters 
and (Fig. 12): 
 
➢ 
 
the relative flexibility of the
retaining wall is defined by the
product of the stiffness of the soil
m and the flexibility of the
retaining wall;
 
➢ 
 
 a nd
 characterize respectively the percentage of the height of the wall that is embedded, the
position of the anchorage, and the relative intensity of the uniform loads acting on the ground sur-
face.
On the basis of his results, Rowe argued that the parameters that played a decisive role in the
behaviour of the retaining wall were its relative flexibility and the ratio 
 
α
 
 between its free
height and its total height, the role of other factors being secondary.
He therefore proposed a two stage calculation method, similar to the purely experimentally-based
method proposed in 1952:
 
➢ 
 
calculation of the stresses in the retaining wall using the free earth support method;
reduction in these stresses using design charts based on the subgrade reaction method (Fig. 12).
 
From the fixity of embedment to the soil-wall and support-wall interaction
 
The role of computing
 
The formation of equations for the subgrade reaction method resulted in a fourth order differential
equation. Solving this was for a long time a major problem that hindered application of the method
for retaining wall design.
The appearance and development of the computer and computer processing in the 1960s facilitated
numerical integration of the equations and radically changed the nature of the problem. Firstly,
practical use of the subgrade reaction method was facilitated, and secondly, it became possible to
solve problems that were more complex than those considered previously.
kh m 
z
D
----=
0 1 2 3 4
0
20
40
60
80
100
Maximum moment in wall
Percentage of maximum value computed using the free earth support method
Log mρ
q
0
0,2
b
0
0,1
0,2
0,1
0,2
Fig. 12 - Rowe’s reductions in bending moments (1955).
On this diagram, ρ is expressed in foot5/pound.inch2 and m is expressed in pounds/foot3. Log mρ 
can be converted into a quantity with no units by adding 2.2 to the values read off from the x-axis.
α
h
H
---- ,= β
ht
H
----=
q0
q
γH
-------=
H4
EI
-------
α
h
H
---- ,= β
ht
H
----=
q0
q
γH
-------=
m 
H4
EI
-------
 
45
 
The soil-wall and support-wall interaction
 
Until the 1960s, research concerned the interaction of the embedded portion of the retaining wall
with the soil, the problem tackled being how best to take account of its fixity when estimating the
stresses in the structure. The development of new techniques for integrating the equilibrium equa-
tion for the retaining walls allowed the subgrade reaction method to be applied to other aspects of
the problem. Thus, Turabi and Balla (1968) supplemented subgrade reaction modelling of the fix-
ity of the embedded portion of the retaining wall by modelling the action of the supported soil and
the supports. The active earth pressure acting on the retaining wall thus depended on its deflection,
being equal to the earth pressure at-rest when there is no deflection of the wall and falling in a
linear manner when the retaining wall moves away from the supported soil.
This modelling, however, had one serious shortcoming with regard to taking account of active and
passive failure. The reason for this is that, like previous attempts at modelling the interaction of the
embedded portion of the retaining wall, the pressure of the soil acting on the wall is simply mod-
elled by means ofa linear function of the displacement of the wall but is not limited by the pres-
sures corresponding to passive and active failure. It can therefore take on values that are higher
than that of the passive earth pressure or lower than the active earth pressure.
This approach could, if necessary, remain acceptable for the interaction of the embedded portion of
the wall, in view of the fact that, under the service conditions that apply for an analysis using the
subgrade reaction method, the structure only mobilizes the passive earth pressure over a limited
amount of the embedment depth*. However, this approach may rapidly become unacceptable for
modelling the stresses in the soil that is supported by the wall, as active failure is very soon
reached, even under service conditions.
Haliburton (1968, Fig. 13) was responsible for the introduction of a non-linear soil response model
which included thresholds that correspond to active and passive failure. The proposed model is also
able to deal with elastic supports or imposed deflections. Haliburton thus demonstrated new poten-
tial applications of the subgrade reaction method to the design of retaining walls. In particular, this
method differs from classical methods in that it makes it possible to consider various boundary
conditions and the position of supports at a number of different levels. Furthermore, it is able to
 
* With regard to the interaction of the embedded portion in sand, it should be noted that the fact that coefficients of sub-
grade reaction are considered to increase with depth means that low values of passive earth pressure, in the upper part of
the soil, go together with low coefficient of subgrade reaction values. The extent to which the passive pressures can be
exceeded is thereforerelatively limited. Rowe (1955) noted, however, that when modelling highly flexible sheet pile
walls embedded in very stiff sand, such an approach can lead to zones where the passive stresses are considerably
exceeded, which leads to an overestimation of the fixity of the wall.
y
y
P
Pa
Pp
Kh
a) distribution of soil reactions on the wall
b) plot of the reaction of the soil
on the wall at a given point
Pa: soil pressure at active failure
Pp: soil pressure at passive failure
Kh: coefficient of subgrade reaction
Fig. 13 - Interaction model used for the subgrade reaction method.
 
46
 
analyze the stress distributions obtained for various configurations of the structure, allowing the
engineer to experiment with different embedment depths and different positions of support.The
standard method in retaining structure engineering
The possibilities of the subgrade reaction method, which were clearly apparent in the work of Hal-
iburton (1968), were quickly put to use for the design of real structures. Thus, from the early 1970s,
Boudier et 
 
al.
 
 (1970), then Fages and Bouyat (1971a; 1971b) and Rossignol and Genin (1973)
developed dedicated software programs for the design of retaining walls using the subgrade reac-
tion method.
It is also important to note that these developments were primarily concerned with the design of
diaphragm walls, which were frequently anchored by active tiebacks that prevented the develop-
ment of active failure states. Thus, this design method was applied in the context of new construc-
tion techniques (Delattre, 2000) rather used to replace prevailing methods for existing techniques.
Wider application of the technique during the 1970s led to the development of new application
software, including DENEBOLA (Balay et 
 
al.
 
, 1982). It also led to the design hypotheses undergo-
ing some formalization, in particular the rules concerning the values of the coefficient of subgrade
reaction (Balay, 1985).
 
Choice of the values for design parameters
 
Application of the subgrade reaction method to retaining structures differs from application to
foundations because the soil provides both the loading and surrounding medium, while generally in
the case of foundations loading is independent of the soil.
In order to implement the method, two problems must be dealt with separately. Firstly, for each
stage of construction, the loading applied to the retaining wall must be calculated, with the assump-
tion of zero deflection. The retaining wall’s equilibrium position must then be found, with refer-
ence to the mobilization of subgrade reactions which are described by subgrade reaction coeffi-
cients and thresholds that correspond to the active and passive failure states.
In practice, the question of the loading applied to the retaining wall brings in the concepts of soil
compression and decompression coefficients, while the problem of subgrade reaction curves is
simply a question of subgrade reaction coefficients, with passive and active earth pressure thresh-
olds making use of previous developments for classical methods.
 
Coefficient of lateral decompression or compression of the soil
 
The principal loading of retaining walls is applied by the soil. Starting from the initial equilibrium
condition that applies before the wall is installed, this loading has two components. One direct
component comes from the stresses applied to the wall by the fill, or (in the opposite case) the
stresses which are removed as a result of excavation. An indirect component of loading is transmit-
ted to the retaining wall by the soil that lies below the fill or the excavation.
Little debate surrounds the assessment of the direct component of this loading. In the case of fill,
with zero deflection of the retaining wall, the stresses are evaluated on the basis of the coefficient
of earth pressure at-rest
 
 
 
K
 
0
 
, while in the case of excavation, they are defined 
 
a priori
 
.
With regard to the stresses that are transmitted by the foundation soil, in the case of loading applied
by fill Balay and Harfouche (1983) also proposed to evaluate the loading transmitted by the soil
beneath the retaining wall on the basis of its coefficient of earth pressure at rest K
 
0
 
. 
In the case of a reduction in loading caused by excavation, Balay and Harfouche (1983) proposed
two alternatives. The first, referred to as “irreversible” considers that the horizontal stress remains
unchanged so long as it remains less than the passive stress. The second, referred to as “reversible”
considers that unloading takes place in accordance with the slope K
 
0
 
 of first loading (as long as the
stress does not become lower than the passive stress). 
 
47
 
Monnet (1994) proposed, in the case of unloading, that the reduction in horizontal stress should be
calculated as a fraction of the reduction in vertical stress:
where K
 
d
 
 denotes a coefficient of soil decompression.
On the basis of an analysis of the results of other authors’ work on the behaviour of soil undergoing
decompression under oedometric conditions, Monnet proposed the following equation for K
 
d
 
 
It is noteworthy that this expression leads to a linear stress path for unloading, the slope of which is
fairly close to that described by Mayne and Kulhawy (1982) for second loading. 
 
Determination of coefficients of subgrade reaction
 
To design a retaining wall with classical design techniques it is necessary to select a stress diagram
for the structure. This obliges the engineer to consider the deformations to which the soil will be
subjected with reference to the type of structure that is planned (rigidity of the retaining structure
and the supports, nature of the soil and envisaged construction procedure). It is on the basis of these
factors that the engineer will be able to make the hypothesis that a given part of the retained soil
mass remains close to the at-rest state, while another undergoes decompression and therefore
approaches an active state.
In contrast to classical methods, the subgrade reaction method means that such questions do not
necessarily have to be answered. The hypotheses do not relate to deformations(which are com-
puted) but to the distribution of stiffness. It is this which is included in the calculation in order to
determine equilibrium.
Analysis of the design hypotheses should therefore deal with the distribution of the coefficients of
subgrade reaction acting on the wall, the stiffness of supports and the stiffness of the retaining wall.
 
Terzaghi
 
The two most important contributions to the evaluation of the coefficient of subgrade reaction were
made by Terzaghi (1955) and Ménard et al. (1964).
Discussion concerning experimental studies of the behaviour of structures that are embedded in the
soil (Rifaat, 1935; Loos and Breth, 1949), and concerning studies dealing with geotechnical struc-
tures of other types, supplemented by theoretical considerations regarding the coefficient of sub-
grade reaction concept, led Terzaghi (1955) to formulate general rules for deciding on coefficient
of subgrade reaction values for use in calculations. For the design of retaining walls, Terzaghi pro-
posed use of a coefficient that increases linearly with depth for structures embedded in sand and a
constant coefficient of subgrade reaction in the case of stiff clays. Furthermore, he demonstrated
that the coefficient of subgrade reaction falls the greater the surface area of soil that is subjected to
stress and the lower the stiffness of the soil.
This analysis led Terzaghi to express the coefficient of subgrade reaction by the following equa-
tion:
in the case of sand and
in the case of stiff clay.
Where z denotes the depth considered, D is the “characteristic length” which depends on the
embedment depth of the structure and the manner in which it functions, lh is a constant that charac-
∆σh Kd∆σν=
Kd
1
2
--- 1 ϕsin–( ) 3 3 ϕsin–( ).=
kh lh
z
D
----=
kh kh1
1
D
----=
48
terizes sands, on the basis of their density and the presence of a water table, and kh1 is a constant
that characterizes clays on the basis of their consistency.
Ménard
Ménard’s contribution applied the theory developed by Ménard and Rousseau (1962)* for calculat-
ing the settlement of shallow foundations on the basis of elastic theory and empirical adjustments.
Transposing the results obtained to the opposing reaction of the soil led Ménard to express the
coefficient of subgrade reaction with the following equation (Ménard et Rousseau, 1962; Ménard
et al., 1964):
where
➢ EM denotes the pressuremeter modulus of the soil, 
➢ a is the “characteristic length” which depends on the embedment depth of the structure and its
mode of operation,
➢ and α is a rheological coefficient that depends on the nature of the soil.
General application of this method in the 1970s led to some formalization of design assumptions,
in particular the rules for deciding on coefficient of subgrade reaction values (Balay, 1985). With
regard to the coefficients of subgrade reaction for the embedded portion of the retaining wall, these
recommendations used the proposals made by Ménard et al. (1964), with a correction for cases
where the embedded depth of the structure exceeded its height above the ground. In addition,
Ménard’s proposals were extended to the upper part of retaining structures (i.e. the reaction of
supported earth) with specific provisions for the reaction of the soil affected by the prestressing of
tiebacks. These provisions were derived from finite element analysis of the reaction of an elastic
block to a structure that is loaded at certain points and are based on observations of several actual
retaining structures (Gigan, 1984).
A topic that is still much debated
The seminal work of Terzaghi (1955) or Ménard and Rousseau (1962) began by applying the sub-
grade reaction method to the simplest structures (shallow foundations) and then proposed exten-
sions first to laterally loaded deep foundations then to retaining walls. In practice, the proposals
made for shallow foundations no longer generate much in the way of discussion. This is not true,
however, in the case of retaining walls for which the issue of the distribution of coefficients of
subgrade reaction is still the subject of much debate.
Recent proposals have followed two main directions. The first (Simon, 1995) consists of adopting
a more flexible approach than Balay for estimating the characteristic lengths a for application in
Ménard’s formulae. The mass of earth that is loaded by the retaining wall is divided into as many
parts as its operating mode requires, the compressed zones and decompressed zones needing to be
clearly identified when applying Ménard’s formulae. 
The second direction (Schmitt, 1984, 1995, 1998) is based on the nonlinear nature of the response
of the soil to the retaining wall. Thus, the coefficients of subgrade reaction proposed by Ménard
would be an acceptable approximation when the wall undergoes considerable displacement, but
would seem to underestimate the real reaction of the subgrade in the case of smaller deformations.
This analysis, which is supplemented by additional analysis that, like Simon’s, deals with the value
of the characterisitc length a taking into account, in particular, the flexural stiffness of the retaining
wall, led Schmitt to proposed considerably higher coefficients of subgrade reaction than those
derived from Ménard’s research.
* See also Cassan (1978, tome 2), p. 65.
kh
EM
αa
2
------ 0,133 9a( )α+
-----------------------------------------=
49
Chadeisson’s alternative
Chadeisson’s alternative* (in Monnet, 1994) consisted of establishing the value of the coefficient
of subgrade reaction with reference to the shear strength of the soil, characterized by the cohesion
and the internal angle of friction. This proposal, which takes the form of a design chart, in principle
has no basis apart from experience. Subsequently, some justification was provided by Monnet
(1994), who also proposed developments to the method, while Londez et al. (1997) gave an exam-
ple of the use of Chadeisson’s design chart on a real structure. 
Hybrid methods
The limits of the subgrade reaction method have led some authors to propose hybrid methods in
which the reaction of the soil on the retaining wall is computed as construction work progresses by
considering that the soil mass behaves in an elastic manner. The reaction of such a mass to dis-
placement of the retaining wall can therefore be based on developments of elasticity theory (Vaziri
and Troughton, 1992; Vaziri, 1995, using Mindlin’s, equations and additional developments; Papin
et al., 1992, in Potts, 1992; Creed and O’Brien, 1991, using numerical methods), this reaction
being, of course, limited by the usual active and passive thresholds.
Conclusions
The approach that consisted of calculating the in-service equilibrium of retaining walls was applied
throughout the twentieth century to all the developments of the technique and all the different
forms it took, from rigid to flexible retaining walls, whether of cantilever design or anchored by a
single row of passive tiebacks or later by several rows of tiebacks.
From the beginning of the twentieth century until the 1970s, this technique was centred on the
stresses to which structures are subjected. The issue of the deformation of the structure and the
adjacent soil was therefore not tackled, at least explicitly.
Research in this area was essentially concerned with two aspects of the soil-retaining wall interac-
tion:
➢ firstly, it continued the work on passive and active failure that had been done in the eighteenth
and nineteenth centuries. Consequently, extensions of the methods of Coulomb and Boussinesq
were made available to engineers enabling them to analyze both active and passive earth pressure;
➢ secondly, they attempted to provide an answer to the question of the embedment of the wall in
the soil, which is of fundamental importance for determining the stresses to which the wall as a
whole is subjected. The answer came gradually, and resulted in a whole set of calculation methods
each of which attemptedto embrace a wider perspective than those used previously. As a result,
these methods do not share the same domain of application, which can be quite limited in the case
of the most basic methods.
However, this approach was relatively uninterested in the interaction between the retaining wall
and the supported soil. The hypothesis that was applied until the early 1970s was that of active
failure, with the resultant actions on the wall assessed using the Coulomb, Rankine or Boussinesq
methods or their extensions. Thus, if we consider the different forms of interaction between the soil
and the retaining wall, we have to conclude that design methods failed to provide a firmly-based
solution to the problem of arching in the case of flexible retaining walls and the question of the
dependency of active earth pressure on the general kinematics of the retaining wall.
General application of the subgrade reaction method from the 1970s put an end to previous approx-
imations concerning the fixity of the retaining wall in the soil by proposing a solution that took
account of the properties of both the soil and the retaining wall. In addition, it made it possible to
consider new types of interaction between the retaining wall and the supported soil in addition to
the straightforward active earth pressure approach that had dominated previously. As a conse-
* Initially this involved rules developed during the 1970s by the firm of contractors Solétanche, with reference to struc-
tures that were built by the company. They were published by Monnet (1994).
50
quence of the development of active tiebacks, loading of the supported portion of the wall could
depend on soil states which are intermediate between active and passive failure. 
The general application of the subgrade reaction method should not, however, make us forget that
it fails to deal with the soil-retaining wall interaction in a completely satisfactory manner. Thus,
like the classical methods which preceded it, it is not able to take account of arching in the vicinity
of supports or the general kinematics of the retaining wall and provides only a very approximate
estimate of the deformations to which the structure is subjected. Furthermore, the evaluation of the
parameters used by the method (in particular the coefficient of subgrade reaction and the coeffi-
cient of decompression) remains a genuine difficulty.
These aspects, which are inadequately dealt with or ignored by classical methods and the subgrade
reaction method, explain the position of empirical and semi-empirical methods. They also allow us
to glimpse the benefits that might accrue from an application of finite element methods to retaining
walls, as these are able to take into account of more aspects of the soil-structure interaction. These
two topics will be covered by two further articles for publication in this journal.
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