A Century of Computation Methods for Designing Retaining Walls II

BULLETIN DES LABORATOIRES DES PONTS ET CHAUSSÉES - 244-245 - MAY-JUNE-JULY-AUGUST 2003 - REF. 4457 - PP. 31-51 31

A century of computation methods fordesigning retaining walls

II* – Empirical and semi-empiricalapproaches

Luc DELATTRESonja MARTEN

Laboratoire Central des Ponts et Chaussées

Introduction

Although gravity retaining wall systems have been adapted to retaining fills, excavation retainingwalls are the most frequently used type of retaining wall and have experienced continued develop-ment throughout the 20th century. The development and diversification of the technique relating toretaining walls, sheet piles, diaphragm walls, Lutecian type interpile sheeting, parisian interpilesheeting and cut-off walls, as well as the technique of using supports such as struts with pretensionedanchors have made it possible to build more and more subsurface walls: transport infrastructure,various networks, building bases, parking areas, etc. [Delattre, 2000].

This development and application of technologies resulted in a major development of computationmethods. In fact, this form of structure requires the diversification of the computation schemas ofearth pressure because their kinematic characteristics are different from those of gravity walls, themost popular kind of retaining walls to date.

* This is the follow-up article of an article dedicated to traditional methods and the coefficient method of reaction[DELATTRE, 2002]. It will be completed by an article about the application of the finite elements method on retain-ing walls since the 1970s.

BULLETIN DES LABORATOIRES DES PONTS ET CHAUSSÉES - 244-245 - MAY-JUNE-JULY-AUGUST 2003 - REF. 4457 - PP. 31-5132

Gravity walls are rigid and their movements combine lateral displacement and warping when thesupported soil experiences pressure. As a result the soil is decompressed laterally and is brought toa state of limited equilibrium of the thrust [Delattre, 2002]. The earth pressure computation methodsassocated with this kinematic characteristic of the wall were proposed by Coulomb [1776] andBoussinesq [1882] and have undergone many experimental validations [e.g. Darwin, 1883; Feld,1923].

Contrary to this kinematic characteristic of warping in rigid gravity walls that are used to retain fills,flexible retaining walls have more complex kinematic characteristics. Two main factors are respon-sible for this diversification in the walls [Delattre, 2002]:

Their relative flexibility leads to a variable deflection of the structure, as well as a redistributionof the thrust between the areas where supports exist and the the areas between supports;

The ways in which the retaining walls are implemented provoke a kinematic reaction of the entirewall that moves from a “top rotation” (warping) into a “base rotation”.

Original tools needed to be developed for ascertaining these new forms of interaction between thewall and the supported soil. The problem/task nevertheless proved to be difficult. In practice, it wasnecessary to wait for the appearance of the finite element method so that the different forms of soil-structure interaction concerned could be included into the details. In the mean time, empirical andsemi-empirical solutions were proposed by engineers. These solutions, which were initially devel-oped for calculating the stress in structural element, were otherwise to be used to assess the defor-mation of the structure and the supported soil block. The development of these empirical and semi-empirical methods as well as that which concerns stress assessment and distortion of structures isthe focus of this article.

Flexible walls and arching

Experimental observations

Christiani’s observations

The analysis of the collapse of the most commonly used equilibrium on the local level on homoge-neous soil supported by a restraining wall, allowed Boussinesq [1882], in his continuation of theworks of Rankine [1857], to offer an expression of stress distribution on a retaining wall. For gran-ulated material, this distribution is triangular with the stress moving in a linear manner and inaccordance with the depth. This work completed that of Coulomb, which only helped to determinethe intensity of the resultant thrust. The triangular distribution was only a theory.

The literature recognizes Christiani [in Brinch Hansen, 1953], a Danish engineer, as being the firstperson at the beginning of the 20th century to disprove this theory of pressure distribution behindflexible walls anchored at the top.

His research, which was empirical for the most part, proceeded from an analysis of the dimensioningof old structures made with timber sheet piling and dimensioned in an empirical manner. This anal-ysis helped him to show that in the areas between supports, the pressure that was applied to thesewalls were clearly lower than that inferred in the works of Coulomb or Boussinesq. Christianiinferred from this observation that the earth pressure was redistributed from the areas between sup-ports to the areas where the supports were.

This redistribution was lastly [e.g. Terzaghi, 1943a, pp.66] credited to “arching” effects in the soil:the difference in wall stiffness between the areas corresponding to supports and the intermediateareas cause a transfer of stress through shear mobilization in the soil towards the stiffest areas.


The study of earth pressure on flexible walls

The experimental validation of the empirical schema adopted by Christiani and his successors of theDanish school was continued by Stroyer [1935]. He started a series of tests on reduced models usinga device made specially for studying earth pressure distributions on flexible walls. In this device, therigid wall pivoting at the base is replaced by a wall that is jointed at the top and at the base and that isalso sufficiently flexible to undergo a deflection in its middle portion. This device allowed Stroyer toobserve that the stress in the middle part of the wall that was submitted to deflection, reduced as thedeflection increased. This meant that the wall was flexible. He otherwise observed that this stressreduction in the middle part of the wall was accompanied by a redistribution of stress towards thefixed points at the top and base of the wall and that this did not lead to reduction of the resultant stresson the wall.

Tschebotarioff and Brown [1948], then Rowe [1952, 1961] and Masrouri [1986] continued this firstexperimental study by equally conducting studies on reduced flexible models. The works of Tsche-botarioff helped to underline the fact that the redistribution of stress by arching appeared only onwalls sheeted from top to bottom, anchored rigidly at the top (to a platform, according to the circum-stances), then dredging (Fig. 1) and that such redistributions of stress should not be considered aswalls filled or anchored with tension rods that are relatively flexible.

The experiments of Rowe [1961] and Masrouri [Masrouri, 1986; Masrouri and Kastner, 1991]involved walls sheeted from top to bottom, then excavated with struts installed in the advancement.Similar to the experiments of Stroyer, Rowe’s experiments (Fig. 2), carried out on a flexible walland rigid struts, clearly showed a redistribution of stress on the wall, consisting of a reduction ofstress between the supports and a concentration of stress on the supports.

For her part, Masrouri concentrated on studying the influence of strut rigidity and the intensity ofinitial prestressing on stress distributions on a semi-flexible wall. She showed that, for a flexiblesupport that was not prestressed, the stress distribution behind the wall corresponded to the distri-bution calculated using the theories of thrust while the displacement in the base of the wall remainedlimited, but that this was no longer true the moment the prestressing and rigidity of the strut rose orwhen displacement in the base of the wall increased. She actually showed that both the increase instiffness of the support and the increase of initial prestressing caused the pressures applied to thewall in the support area to increase while the areas between the supports could be relieved. In thesame way she showed that if excavation is pursued in order to generate strong movements in thebase, stress concentrations will increase to the right of the support.

Similar work carried out this time on structures anchored with tension rods, enabled Masrouri toobtain analogue results.

Softclay

After filling

Filling Dragage

Thrust redistribution

after soilvibration

Normal relaxation ofthe support

Pas de relaxationde l'appui

Fig. 1 - Stress distribution stemming from different construction procedures [according to Tschebotarioff and Brown, 1948].


An empirical model of the phenomenon

Attempts were made, particularly inspired from computation methods for silos using the method ofattaching the soil onto the wall [Terzaghi, 1936b; Handy, 1985; Harrop-Williams, 1989], to take intoaccount anchorage effects during the computation of stress distribution. These attempts were notalways developed, and today, there only exists one semi-empirical method that integrates the effectsof anchorage into structure calculations.Research carried out in Denmark stemming from the researches of Christiani gave rise to the socalled “Danish” rules of dimensioning for retaining walls. These rules that are applied to sheet pilewalls anchored at the top, incorporate the redistribution of stress towards points in the wall that areless susceptible to displacement at the anchorage point (Fig. 3).

Excavation retaining walls and the kinematic characteristic of walls

Stress on excavation retaining wall supports

Since the beginning of the 20th century, the development of underground transport networks hasbeen one of the main driving forces behind the study of excavation retaining wall behaviour. In fact,constructing underground lines for the tube, the urban environmental stress associated to the geom-etry of the excavation, have often caused engineers to use the cut and cover technique with lateralsupport for the walls strutted on the advancement. The measurement system for these structures,motivated by the sensitive characteristic of the urban environment, provides rapid measurementsrelating to their behaviour.

Such structures are relatively flexible allowing for deflection between supports. They also have akinematic characteristic that causes the deflection to normally increase with depth. The flexibilityis inherent to the technology used producing naturally flexible forms: sheet piles or H-sections inthe case of interpile sheetings. The kinematic characteristic is the result of accumulated deflections

After excavation

Fig. 2 - Stress distribution taken on a reduced flexible wall model with struts rigidly installed in the advancement of

the excavation [Rowe, 1961].


experienced by the retaining wall at each level, crossed with the number of excavation phases andthen the depth, before it’s blocked by a support.

The measurements taken on these structures have revealed pressures on retaining walls that do notobey the triangular law of distribution predicted in the theories of Rankine or Boussinesq. Contraryto what was expected, the earth pressures were stronger in the middle part of the wall than in thebottom part.

Kinematic characteristic of structures and stress distribution

The kinematic analysis of strutted excavations, complemented by a series of experiments effectedon a semi-high wall submitted to movements of different nature, helped Terzaghi [1934, 1936a] tolink earth pressure distribution on retaining walls with the general kinematic characteristic of thestructure (Fig. 4).

Hence, gravity walls loaded by soil-filling have a tendency to turn with respect to their base, causinga strong lateral decompression of the soil at the surface of the supported soil. This lateral decom-pression decreases with depth to the zero level at the base of the wall. For Terzaghi, this kinematiccharacteristic enables a relatively homogeneous deformation in the prism of the supported soil ofeach structure, which is brought to a state of thrust equilibrium.

For the structures that move by translation, which is a kinematic characteristic similar to thatobserved for supported excavations or for structures that pivot at a point at the top, lateral decom-pression of the soil behind the wall is not homogeneous and has greater values at the base of the wallthan at the top. For Terzaghi, this schema of structural deformation leads to non-triangular stress

The structure is dimensioned using thesingle direction thrust bearing method,with the thrust being calculated using anangle of wall-soil friction equal to ϕ′/2.Firstly, the thrust on the structure is calcu-lated using the Coulomb method by stip-ulating a wall-soil friction equal to zero.The diagram of the thrust that is obtainedis then modified by constructing a parab-ola that reduces the pressure of the coeffi-cient q at the centre of the two supportsanchored from above and the applicationpoint of the bearing resultant and increas-ing from 1.5 q to the top anchorage level.To find q:

where pm is the average value of stressdistribution calculated with the Coulombmethod. H is the charge applied above theanchorage level, equal to the height of thesoil above the anchorage level, estab-lished while taking into account the sub-merged weight of the soil. L is the dis-tance between the two support points thatsupport the wall and k is a coefficientapproximately equal to 0.8.The pressure distribution that is calcu-lated helps to ascertain the bendingmoment of the wall and the anchorageforce. The depth given to the wall can beobtained by multiplying the depth neces-sary for limiting equilibrium by .

q

1,5q

A A

B

2/3 D

1/3 D

1/2 L

1/2 L

d √2

Fig. 3 - Danish computation method for walls anchored at the top [in Brinch-Hansen, 1953].

q k10H + 2L10H + 3L-----------------------Pm=

2


Fig. 4 - Dependence of the diagram of the thrusts appplied by the soil to the kinematics of the wall [according to Terzaghi, in Ohde, 1938].

Ever since, many authors have reconsidered, using experimental study results, the influence of the kine-matic characteristic of a structure on thrust distribution. The obtained results mainly concern thrust distri-bution applied to a wall for the three wall kinematics characteristics: translation, top rotation and base rota-tion.Concerning the transaltion movement, Sherif et al. [1982], as well as Fang et al. [1997], have demonstratedthat thrust distribution during movement of the wall stays noticeably triangular. This result is qualified byFang and Ishibashi [1986], for whom the thrust distribution deviates slightly from the triangular distribu-tion, the thrusts being slightly stronger than expected with this distribution in the upper section of the walland slightly lower in the bottom part of the wall.The case of walls pivoting at the top have been studied by James and Lord [1972]. The study revisited themeasuring devices used in the case of wing wall abutments [James and Bransby, 1970]. Although the devicedid not enable them to measure the thrust applied by the soil on the wall, owing to the fact that the thrustcells used were dimensioned to measure stress on the wing wall abutments, it enabled them to gain accessto the soil block deformations. They also observed that the deformations were located following a band forwhich the curve resembled the arc of a circle. This band of deformation started at the base of the wall andprogressed, throughout the displacement of the wall, towards the surface of the soil block. James and Lordotherwise observed that the volume of the soil, while slipping along this band of deformation is all theweaker bacause the sand is dense: the deformation band develops all the more behind the wall as the sandis loose.The thrust distribution applied by the soil to the wall for this structural kinematic characteristic was studied,always on reduced models, by Fang and Ishibashi [1986]. They observed that while the wall is rotating,moving from the "at rest" state, the stresses applied at the top of the wall have a tendency to decrease in thelower part. They also highlighted the stress distributions for which, in the upper quadrant of the wall, thestress is significantly higher at the earth pressures at rest state, while in the middle section, this stressapproaches the theoretical thrust and that in the bottom quadrant, this pressure decreases until it reachesvalues close to zero at the foot of the wall.They otherwise showed that the initial density state of the sand noticeably influenced the thrust distribution.In this way, the thrust distribution for loose sand, while being similar to the distribution described above,is not far removed from a triangular distribution. For greater initial densities, the thrust distribution isclearly more removed from a triangular distribution with the stress being concentrated at the top increasingthe density state of the material. Fang and Ishibashi explain this result by using the notion of anchorageefects created at the edges of the fixed point at the top of the wall.

For structures pivoting at the base, they finally observed that during rotation of the wall, the thrustsdecreased less rapidly in the bottom section of the wall than at the top. In this way, they showed that thewall movement led to the develoment of an equilibrium state limit in the top part of the soil block (approx.the upper half), while in the bottom part, the soil remained at an intermediate state between the thrust stateand the rest state.

Base rotation Translation Top rotation


distribution on the wall. In fact, when the movement of the upper part of the wall is insufficient atthis level to authorize the lateral decompression of the soil mass involved in the mobilisation of thethrust, an “anchorage effect” appears resulting in the increase in thrusts in the top part of the walland a reduction at the bottom part.

Support dimensioning for supported excavations - Theoretical computation methodsInitially, and in keeping with the traditional approach of active earth pressure studies, the issue ofthrust on walls submitted to different kinematic conditions of rotation at the base were addressed bythe computation. The initial contribution of Terzaghi [1936b] inspired a dimensioning method forsilos while postulating a priori a ratio value for the vertical and horizontal stresses on the supportedheight and on the friction of the wall. These theories enabled vertical stress variations in the structureto be determined, taking into account discharging from friction on the wall and the inference ofthrust distributions on the wall that approach the values observed during the experiment.The issue was taken up again by Ohde [1938, in Brinch-Hansen, 1953] who proposed that for ascreen pivoting at its summit, as in Coulomb’s method, the equilibrium state for a prism of soilbehind a sliding wall along the length of a circular surface (Fig. 5) should be considered. In justify-ing his choice of a circular form from the surface of collapse, he said that this form respected thekinematic condition imposed by the wall. The problem can then be resolved by calculating thestresses along the slide-bed plane by integrating the equation of Kötter [1903] and by writing thestatic equilibrium of the soil prism.The problem raised by Ohde was taken up again by Terzaghi [1943a] who had similar arguments,stating that the breaking plane was no longer parametered by an arc of circle, but by a logarithmicspiral arc. The solution to the problem is obtained by writing the result of the moments of stressapplied to the prism when sliding at the centre of rotation by means of a hypothesis concerning theposition of the gravitational centre of the thrusts on the wall. This theory stems from the experimen-tal results then available. For sand, the gravitational centre of thrust is taken mid-way along theheight of the retaining wall while for clay, this position depends on the relationship c/γh, where ccorresponds to soil cohesion, γ its unit weight and h, the clear height of the retaining wall.

Empirical computation methods for supported excavation supports The approach using the computation of stress distribution on walls that rotate in relation to theheight of the wall [Terzaghi, 1936b; Ohde, 1938; Terzaghi, 1943a] has never really been put intopractice and quickly encountered competition from a new approach where stress distributions to beintroduced in the computation for excavation retaining walls were to be taken directly from meas-urements on the structure. This approach was mainly developed in Germany and the Unites States of America during the sec-ond half of the 20th century using observations made on work carried out on underground railwaysin many metropolitan cities in addition to conducting experiments on reduced models whenevernecessary. The research carried out by these two schools had a partly common experimental basebut was developed further by the Germans.


The German school

Laboratory tests carried out by Press [1942] or Lehmann [1942] have demonstrated, like the tests ofTerzaghi [1934], that earth pressures largely depend on the wall kinematics. Lehmann focusedmainly on the issue of stress distribution at the moment when the soil decompresses in the lower partof the soil block, which is the case for top rotation. For these tests carried out on sand, he used aclosed box with sheets of glass on the sides in order to observe the behaviour of the soil block andthe wall. The wall, which was 100cm high and 98cm wide, was divided into four rigid parts attachedwith hinges. In doing so, he could simulate top rotation as well as a succession of earth movementsby turning one joint after the other. By measuring the stress in the springs used as struts, he wasalways able to trace an average line of stress distribution according to wall displacement, the centreof rotation, wall friction and “parasite” effects (if they were identified, like soil-glass friction)(Fig. 6a). From these tests, Lehmann formulated a semi-empirical method to calculate thruststrength using Coulomb’s method (his measurement results never exceeded this value) and to redis-tribute it using a line that encompasses all the results (Fig. 6c).

In a second series of articles, Ohde [1948, 1949, 1950, 1951, 1952] revisited his ideas of the end ofthe 1930s [Ohde, 1938], and qualitative tests that demonstrate the different fracture planes accord-ing to the kinematic characteristic of the wall. He presented strength distribution figures in agree-ment with Lehmann’s results indicated above.

a) Analysis diagram.

b) Distribution of resulting pressure.

Fig. 5 - Stress computation for the kinematic characteristic of rotaton for a wall at the top [Ohde, 1938

Ql

Qw

Q

he'

Ea

Eaw

R

G

ϕ1

δ

Q : Soil reaction alongthe fracture plane

Ea : Thrust on the wall

G : Soil block weight


The recent computation methods used in Germany were based on these first studies and additionalstudies done between 1960 and 1970. Briske [1958] proposed an initial synthesis of strength redis-tribution according to the type of retaining wall and the number of struts used for non-coherent soil(rectangular or trapezoidal redistribution). Subsequently, Müller-Haude and Scheibner [1965],Heeb et al. [1966], Briske and Pirlet [1968], Breth and Wanoschek [1969] and Petersen and Schmidt[1971] describe the work and measurements taken during the construction of underground railwaysin Berlin, Stuttgart, Cologne, Frankfurt am Main and Hamburg.

For strutted interpile sheetings built on the coherent soil of Stuttgart, Heeb et al. [1966] propose atriangular distribution with most of the struts mid-way up the wall. This idea was later revisited byBreth and Wanoschek [1969] for the more rigid walls of bored piles in the clay of Frankfurt. Briske[1971] stressed the importance of where the first strut was situated, the effect of time, as well as thedepth attained before the following struts are put in place. In Stuttgart (2 layers of struts) as in Frank-furt am Main (5 layers), the excavation below the second to last strut was relatively deep comparedto the spacing between the layers. Consequently, after installation, there was a large amount of stresson the bottom struts and this even increased further with time because of creep.

Since 1970, tension rods have been used more frequently and this led to a new series of laboratorytests. Schmitt and Breth [1975] first performed tests with one layer of tension rods. One year later,a second publication announced the results of tests using three layers of tension rods [Breth andWolff, 1976]. Some time afterwards, Briske [1980] analysed the observation results in situ of testscarried out on anchored walls.

Tests carried out on reduced models demonstrated that the amount and distribution of tension rodsdid not have a large influence on the earth pressures on the wall although the pressures decreasedwhen the tension rods got longer. Upon comparing strutted and anchored walls, Breth and Wolff[1976] discovered that the strength concentration around the supports was less strong for anchoredwalls. In their opinion, the strength distribution behind strutted walls depended mainly on the wayin which the excavation was dug while the advancing effect of excavation was less visible in thecase of tension rods. They explained this observation to be the result of elasticity and the more sym-metric vertical spacing of the anchorage system.

a) Strut stress E0 to E4 for a displacement of 14 mm.b) Test results with a rotation of level sections of the wall (max. = 12 mm).

c) Proposals of thrust redistribution.Fig. 6 - Lehmann’s tests on reduced models [1942].

a b c

Ei : Stress in the strutsei : Distributed pressuree’ : Relative pressure e’ = E / (γ . b . ha)ha : Height of zone of influence for a strutb : Width of zone of Influence (b = 1 m)

γ : Soil unit weighth : Free height of wallF : Earth pressures according to CoulombeL : Maximum trapezoidal pressure according to Lehmann


These works were revisited in different works of synthesis of which the most exhaustive one wasthe three volume work of Weissenbach [1975], who presented his own tests and accumulated theGerman know-how of the time. His work was also the base for national recommendations, amongwhich was the EAB which related to excavation retaining walls (“Empfehlungen des ArbeitskreisesBaugruben”; Fig. 7) and the EAU concerning works on river banks (“Empfehlungen des Arbeitskre-ises Ufereinfassungen”), as well as local ones (e.g. “Stadtbahn-Richtlinien” in Frankfurt am Main).

The American school

Terzaghi [1941] was responsible for the alternative proposal stating that structures dimensioneddirectly at the base of a distribution enveloped the experimentally ascertained pressures thus remov-ing the computation phase. A first trapezoid diagram applicable to excavations dug in sand wasestablished on the basis of the stresses measured by Spilker [1937] in the struts of the supportedexcavations in underground railway supported excavations of Berlin. This diagram was used subse-quently to enrich the results of measurements carried out on the marly soil of Munich by Klenner[1941]. A similar diagram, applicable to excavations dug in plastic clay was proposed by Peck[1943] based on experiments carried out during the building of the Chicago underground railwayand the theories formulated by Terzaghi [1943b].

The EAB, the recommendations of the work group on “excavation retaining walls”, is the result of workcarried out by several German researchers under the direction of J. Schmidbauer and A. Weissenbach. Thepublication of these technical rules started in 1970 and continued for a few years. They were published inthe “Die Bautechnik” journal and gathered since 1980 in a work published by the editor Ernst & Sohn. Theyare well established in present day Germany and even considered to have legislative value.

The recommendations address the following issues:

computation basis (determining stressing soil characteristics),thrust intensity and distribution and the rules for justifying bearing capacity,computation particularities for interpile sheeting, sheet pile walls, site-mixed concrete walls (bored pile

walls and diaphragm walls) and anchored excavations,security factor proposal,excavations of different shapes (round, rectangular, oval),excavations situated beside other constructions,excavations dug under ground water,excavations dug in soft rock,element dimensionings (shoring, sheet piles, struts, etc.),experimental measurements and the observational method.

The thrust is calculated with the traditional rules given in the DIN 4085 standard. We normally apply theCoulomb theory to a flat slide bed-plane or for ϕ > 30 degrees, it is advised to use the Caquot-Kérisel rules.The wall-soil frictional angle selection, the taking into account of a waterflow or even the computation ofearth pressures in non-flat conditions (before the vertical elements of composite walls) are alwaysaddressed by this standard. If it is probable that the expected retaining wall movements do not reach themovements necessary for activating the thrust limit state (e.g. δh/H ≥ 0.1% for a lateral displacement), theDIN standard refers to the EAB recommendations and obliges that we take an “increased thrust” intoaccount (erhöhter aktiver Erddruck). The EAB considers that the movements are limited if after installationthe supports are submitted to a prestress equivalent to more than 30% of the determined value for the laststage of excavation (for sheet-pile walls and walls made from site-mixed concrete). The EAB security sys-tem is a global system that only decreases the resistance (wing wall abutment or wall material and supports)with a security factor but it doesn’t increase the thrusts.

The most popular places outside of Germany deal with earth pressure redistribution according to the typeof retaining wall and the support conditions. The following figures show the thrust redistribution for wallsmade from site-mixed concrete, meaning diaphragm walls or bored pile walls.


hk

hk hkeho

eh

eh

ehu

eho

eho

ehoZe

ZeZe

ehoeho

eho

eho

eho

eho

ehu

ehu

ehuehu ehu

ehu

eho

ehu

H

H

HH H

H H

H HH'

H'

H' H' H'

H' H'

H' H'

U

U

U U U

U U

U U

Support at hk ≤ 0,1H Support at 0,1H < hk ≤ 0,2H Support at à 0,2H < hk ≤ 0,3H

EB 70-1. Redistribution schemas for site-mixed concrete walls with 1 support

Support at 1/4H and 3/4 H Top supports Bottom supports

EB 70-2. Redistribution schemas for site-mixed concrete walls with 2 supports

3 supports 4 supports 5 support

EB 70-3. Redistribution schemas for site-mixed concrete walls with 3 or more supports

These initial documents were modified by taking into account new experimental data coming fromvarious underground railway sites: Munich [Klenner, 1941], New-York [White and Prentis, 1940]and Cologne [Briske and Pirlet, 1968; Fig. 8] on sand; Tokyo, Osaka, Oslo [Kjaernsli, 1958] on softclay; Oslo [Di Biagio et Bjerrum, 1957] and London [Golder, 1948] on stiff clay. Stemming fromseveral works of synthesis [Terzaghi and Peck, 1967; Tschebotarioff, 1973, in particular], these suc-cessive changes facilitated the availability of diagrams that benefited from significant experimentalvalidation (Fig. 9). The transposition of the proposed rules of strut dimensioning relating to bored and prestressed ten-sion rods, developed in the 1960s, was addressed by Peck [1972]. The elements available allowedhim to establish that the apparent pressure diagrams published in 1969 could lead to overdimension-ing in the tension rods.

Deformation stress

Structure deformation

The theoretical computation methods for retaining wall structures have long been unable to predictthe deformations to expect in service situations and the engineer had to contend with orders of mag-

Fig. 7 - Principle of the redistribution of thrusts given by the EAB rules.


Fig. 8 - Measurements of stress for the underground railway at Cologne [Briske

and Pirlet, 1968].

SandSoft and moderately

soft clatStiff and cracked

clay

Struts

H

0,25 H 0,25 H

0,5 H

0,25 H

0,75 H

0,65KA γH 1,0KA γH 0,2 γH à 0,4 γH

KA = tan2 (45 - Ø/2)

m = 1,0 without exception

KA = 1 - m4CuγH

Fig. 9 - Diagram of thrust to consider for dimensioning excavation shoring, according to Terzaghi and Peck [1967]. γ is the unit wieght of the supported soil and m an empirical coefficient of reduction

of the undrained cohesion taking the value of 1 except where the excavation involves"genuinely" and normally consolidated clay and is characterized by the index γH/cu > 4.

The coefficient m can then take a value equal to 0.4.

0 10 20 30 40 50 60 70 80 90 100Constructionphase C1

Constructionphase C2

Constructionphase E

Stress in the struts (Mp)

Bea

m a

xis

Average values fora distance of 2mbetween the strutsA e = 4m

B e = 2m

(C)

A e = 4m

B e = 2m

C e = 2m

D e = 2m

A

B

C prestress

prestress

17,75 Mp

31,14 Mp

26,42 Mp

31,42 Mp

47,90 Mp

28,00 Mp

30,80 Mp

68,80 Mp

49,32 Mp

3,80

4,50

13

5,70

3,80

4,50

1

Σ A

....D

= 1

76,9

2 M

p

1 Mp = 10 kN


nitude derived from the behavioural observation of real structures. This empirical approachremained undeveloped for a long time.

Peck [1969] was responsible for the first detailed approach concerning structure deformation. Fol-lowing the principles used for the stress analysis of strutting on excavation retaining walls, and onthe base compilation of measurement results, Peck established diagrams of compactions resultingfrom excavations (Fig. 10). Like the distinctions made for calculating the stress in struts, he noteddifferent configurations in the structures linked to the nature of the soil.

This initial proposal by Peck was the focus of various improvement exercises.

Excavation stability index

In his analysis of deformations, Peck [1969] recognized the importance of conditions relative to thebottom of the excavation, particularly for clay, and for the conditions deriving from the presence ofa soft layer under the bottomgrade of the excavation.

D’Appolonia [1971] pointed out that the different situations encountered could no longer be identi-fied by using a simple qualitative method but by using the coefficient of security with respect to thecollapse of the bottom of the excavation (Fig. 11).

This correlation between the stability index of the structure and the observed deformations was speci-fied by Mana and Clough [1981 (Fig. 12)] on experimental and theoretical bases. For their study, theseauthors made a more profound selection of experimental data, excluding notably the deformationsobtained in exceptional situations (especially the faulty design and building of the structures) as well asduring the initial phases where the structure is simply embedded. On this basis, including the parametriccalculations obtained by the finite elements method, they noted that, while the relation between the crit-ical height and the height of the excavation remain higher than 1.5, the lateral deformations of the struc-ture remain low, in the order of 0.5% of the height of the structure. However, in excavations for whichthe depth approaches that of the critical depth (for the relation Hc/H < 1.5), the lateral deformations ofthe structure will be high. With regards to compactions of supported soil, Mana and Clough [1981]observed that their amplitude is generally made up of between 50% and 100% of the amplitude of thelateral displacements of retaining walls.

Zone I Excavation in sand and soft and stiffclay.

Zone II Excavations in very soft clay, with theclay surface having an extension limitunder the bottom grade of the excava-tion or being limited by a stiffer layerof clay.

Zone III Excavations in very soft clay, with thesoft clay layer being very thick under-neath the bottomgrade of the excava-tion.

Fig. 10 - Compaction due to excavations [Peck, 1969}.


The possible depth of an excavation is limited by the phenomenon of bottom heave, which occurs when thevertical stress differences on either side of the retaining wall become higher than that which the soil canmobilize, considering its shear strength. The first studies on the phenomenon of bottom heave were conducted by Terzaghi [1943a]. For excavationsdug in sand, Terzaghi showed, by using results relating to the bearing capacity of superficial foundations,that the security coefficient with respect to the bottom heave is independent from the excavation depth (itonly depends on the frictional angle of the sand) and is always much higher individually when the fric-tional angle is higher than 30 degrees and in the absence of unfavourable water circulation. For excava-tions dug in clay, a similar analysis based on the superficial foundations theory enabled Terzaghi [1943a]to show that in undrained conditions the depth of an excavation, where the length is greater than the width,is limited to the value:Hc = Nccu/(γ − 2cu/B )where cu is the undrained cohesion of the clay, Nc is the bearing factor applicable to the perfectly roughsuperficial foundations (Nc = 5,7 for Terzaghi), γ the unit weight of the clay and B is the width of the exca-vation.Later developments of this analysis method for the stability of the bottom of the excavation are linked tothe progress made in computation methods for the bearing capacity of superficial foundations [Tschebota-rioff, 1951, in Bjerrum and Eide, 1956, pp. 34], taking into account the shape of the excavation [Bjerrumand Eide, 1956], the anistropy of the clay [Clough et Hansen, 1981] or the resistance brought by the retain-ing wall in its driven section [O’Rourke, 1992] for taking into account the role played by the driven sectionof the wall in the analysis of the stability of the bottom of the excavation.

Fig. 11 - Development of the analysis method for the bottom of the excavation and illustration of the method proposed by Bjerrum and Eide [1956].

2

p

B

D

F = Nc

cu

γD + p

D : Excavation depthB : Excavation widthL : Excavation lengthp : Extra load

0 1 2 3 4 5 63

4

5

6

7

8

9Nc

Nc Rectangular = (0,84 + 0,16 B/L) Nc Square

Circular or square excavation B/L=1

Infinitely longn ny gexcavation B/L=0a

D/B

6,2

5,1

cu : Undrained clay cohesionγ : Unit weight of clayγNc : Bearing factorF : Coefficient of security

Fig. 12 - Correlation between the stability index of the excavation

bottom and observed deformations [Mana and Clough, 1981].


Deformation linked to excavations

The different research described above allowed Clough and O’Rourke [1990] to establish experi-mental diagrams relative to the behaviour of excavation retaining walls.

These diagrams were established for sand, stiff clay and soft clay. They concerned short term defor-mations caused by excavation, with the exception of excavations induced by other activities of thesite and deformations due to long term soil behaviour (deformations linked to the consolidation offine soils or creeping).

Maximum deformation values

For all the studies produced, the same convention was adopted to express the deformations of thestructure and the supported soil block. In this way, the deformation of the retaining wall is expressedin relation to lateral displacement at the depth of excavation, whereas the deformation at the surfaceis expressed in relation to the compaction observed at the depth of the excavation.

For excavations in sand, stiff clay and residual soil, Clough and O’Rourke show that the maximumdeformations of the retaining wall are generally lower than 0.5% and that their average value is0.2%. The maximum compaction of the supported soil is lower than 0.5% and, on average, equal to0.15% of the depth of the excavation.

For deformations linked to excavations in soft clay, Clough and O’Rourke propose to estimate themaximum deformation of the retaining wall according to the security coefficient of the excavationconcerning the bottom heave and an estimation of the rigidity of the retaining structure (Fig. 13).The maximum vertical deformations for the supported soil (compaction) are equal to the maximumdeformations of the retaining wall in the horizontal direction.

These results were complemented by Ou et al. [1993] who pointed out that the maximum deforma-tion of the retaining wall took place at the bottomgrade of the excavation. They otherwise qualifiedthe indications concerning the maximum compaction. In fact, Ou et al. [1993] propose taking avalue between a half and two thirds of the maximum retaining wall deformation (Mana and Clough,1981, proposed a value between half and the total maximum deformation of the retaining wall).

Fig. 13 - Deformations linked to excavations in soft clay, according to Clough and O’Rourke [1990].

0 30 50 70 100 300 500 700 1000 30000

0,5

1

1,5

2

2,5

3

0,91

1,1

1,423

Coefficient of securityvis-à-vis the retaining wall

Max

. lat

. dis

pl. o

f the

wal

l/ E

xcav

atio

n de

pth

(%)

(EI) / (γwh4) Increasing rigidity

Incr

easin

g sta

bility

Sheet pilesh = 3,5m

Diaphragm walle = 1m - h = 3,5m


They were also complemented by Carder [1995], on the basis of experiments conducted on structuresbuilt on stiff clay. For such works, Carder proposed that we keep the maximum deformation valuesof the retaining wall varying between 0.125 and 0.4%, following the stiffness of the support system.He also proposed to keep the maximum deformation values of compaction between 0.1 et 0.2% thatare in good agreement with the 0.15% value observed by Burland et al. [1979].

The results were also complemented by Muramatsu and Abe [1996], together with Long [2001].Muramatsu and Abe mainly dealt with circular rectangular or square wells. They pointed out that thedeformation of supported soil is significantly lower in circular wells than in square or rectangularones. Also, for circular excavations, the maximum deformation for the retaining wall does not exceed0.1% of the height of the excavation.

On the basis of analysis on a set of new cases for structures with high safety in relation to bottomheave, Long proposed lower overall values than those suggested by Clough and O’Rourke for thedeformation of retaining walls as well as for the compaction of the supported soil. The maximumvalues observed by Long [2001] were close to the values considered by Clough and O’Rourke[1990] as average values. He otherwise observed that for the cases studied, the influence of retainingwall rigidity on the measured deformations was low. For structures with lower safety with respectto bottom heave, the deformations observed by Long do, however, lie within the range delimited byMana and Clough [1981]. Long introduced the case of retaining walls driven into stiff soil and sup-porting a soft material. He noted that if the case he analysed agreed with the proposals of Cloughand O’Rourke [1990] and Clough et al. [1989] when the stiff soil reached the bottomgrade of theexcavation, these proposals tended to underestimate the deformation when soft soil was to be foundunderneath the excavation bottomgrade. For self-supporting structures, Long observed deforma-tions in the retaining wall that were relatively independent from their rigidity and for which the max-imum values reached 0.5%.

Compaction profiles

In establishing compaction profiles, Clough and O’Rourke [1990] qualified two profiles of defor-mation in retaining walls. For structures that are not supported at the top, the elastic curve of the wallis that of a structure simply embedded at one end; it decreases with the depth and reaches zero at thedigging point. The compaction profile associated with this kind of elastic curve is approximately tri-angular. For structures that are maintained at the top before excavation, the most important part ofthe curve is seen under the support points and compaction is maximum at a certain distance behindthe retaining wall.

These two schemas allow us to distinguish the different categories of structures. The compactionprofile that is associated with excavations in sand and stiff clay is triangular, with the maximumcompaction being produced close to the retaining wall and gradually decreasing with the distancefrom the retaining wall. It reaches zero at a distance of twice the depth of excavation for sand andthree times the depth of excavation for stiff clay (Fig. 14). For Carder [1995], the distance at whichthe compaction is sensitive can be brought to four times the height of the excavation for stiff clay.

The compaction profile associated with excavations in soft clays combine the two schemas of defor-mation and so takes on a trapezoidal shape. Clough and O’Rourke [1990] propose a profile estima-tion for maximum compaction. This involves considering a constant compaction at its maximumvalue at a distance of three quarters the depth of the excavation that then gradually decreases to reacha zero value at a distance of approximately twice the depth of the excavation (Fig. 15).

More complex profiles were proposed by Ou et al. [1993] and Hsieh and Ou [1998]. The triangularprofile proposed by Clough and O’Rourke for sand become convex. In the case of clay, the profileproposed by Clough and O’Rourke changes to have the maximum compaction appear at a retain-ment distance equal to half of the supported height, the compaction of the immediate neighbouringretainment wall being reduced to 50% of the maximum compaction. Furthermore, these profiles arecomplemented by a secondary zone extending to four times the supported height where the compac-tions are low.


Perspectives and limits of semi-empirical methodsThe development of digital methods from in the 1960s marked a downtime in the development ofempirical and semi-empirical methods. Their interest in fact appeared less clear, since a theoreticalanalysis would be able, using the behaviour of the soil and structure elements defined locally, to pre-dict the entire behaviour and notably the phenomenons associated to the kinematic behaviour of thestructure.

These digital methods are however limited in predicting the behavior of structures, some of whichis listed below:

the transfer of properties of a material measured locally on a laboratory tested sample, to thebehaviour of a soil block or a structural assembly remains difficult, taking into account the sensitiveinfluence of the many heterogeneous factors that affect the soil block (e.g. thin sand layers ororganic materials in a clay layer, real conditions support of struts on a diaphragm wall);

the real conditions of structure implementation are not accessible (e.g. case of inserting sheetpiles in the soil or the construction of diaphragm walls);

the bi-dimensional models adopted do not take into account the three dimensional characteristicsof many aspects of building the structure: e.g three dimensional characteristics of the structure butalso its manner of construction.Considering these difficulties, the global approaches to structure behavior represented by the empir-ical approaches have found a new place alongside local approaches that use numerical tools. In par-ticular, observational methods can be a suitable framework for the joint use of the models for struc-tural behaviour prediction, typically digital tools, and global criteria for behaviour assessment basedon the empirical knowledge of their behaviour.

In this context, a new development of these empirical approaches has been observed as evidence ofthe work described in the chapter “Deformation stress”. Beyond these works, one can rightfullyassume that these methods still have a potential for development. In this way, the parameters usedfor “explaining” the behavior observed in structures can be complemented in such a way as toreduce the dispersion of highlighted correlations [see Masuda, 1996], which is still strong.

Fig. 14 - Compaction profiles observed behind the retaining walls of excavations in

sand, according to Clough and O’Rourke [1990].

Fig. 15 - Compaction profiles observed behind the retaining walls of excavations in

soft clay, according to Clough and O’Rourke [1990].


However, these methods are still limited in their capacity to address all of the structural configura-tions that can be encountered since by nature they represent particular cases that have served as thebasis for their elaboration. Their generalization remains thus a significant problem.

ConclusionThe bibliographic analysis shows that although the empirical approach to the behaviour of excava-tion retaining walls is not widely recognized in France, contrary to the developments achieved inthe domain of foundations, it has greatly developed in Germany and the Unites States of America.The research centres of these countries have at their disposal a large amount of well balanced infor-mation on the behaviour of structures concerning the stress they undergo as well as their deforma-tions. This information has given rise to computational methods for structures that are commonlyused today in the engineering practices of these countries.These methods were initially proposed as a temporary solution to the development of theoreticapproaches to the behaviour of retaining walls for excavations, taking into account the complexforms of soil-structure interactions. These methods remained without any alternative until theappearance of the finite elements method in the 1960s. Nevertheless, the appearance of the finiteelements method did not condemn these methods. One can even observe that in collaboration withthe important development of the building of sub-surface structures, interest in these empiricalmethods has grown since the 1980s. Their centre of interest has nevertheless moved away from theinitial attempts to predict stresses, to the deformations that are the focus of most present day devel-opments, because controlling deformations is one of the main issues of excavation in urban areas.In fact, these methods complement digital methods to the extent that they provide a set of “refer-ence” behaviors that is useful at all stages of the project. In the study phase, they enable us to quan-tify the main phenomena and set limits. So they are useful for assessing a solution that is calculated,notably for deformations that are still poorly approached computation tools. In the execution stage,they enable us to set the thresholds for normally expected behaviour and are therefore a useful toolthat can be used on sensitive sites. They obviously lie within the domain of observational methods.In this context, the methods must be developed and fine-tuned on the basis of observed behavior.Their development justifies the more systematic use of instrumentation for the retaining walls oflarge excavations than in the past.

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