THE STABILITY OF EARTH DAMS

A Thesis Submitted by

Alan W. Bishop, M.A.(Cantab.) , A.M.I..C.E.

For the Dogma of Doctor of Philosophy

in the University of London

Imperial College May, 1952

The Stsbility of r3rth Dams

A Thesis sulzmit;;ed by

Alan 1.7. Bishop, TI.A.(0antab.),

For the Decree of 'Occtor of Philosophy

in the UniversIty of London.

The central problem in earth dam design Is that of shear

failure, either in the dam alone, or including the foundation

strata. A clear understandinc, of the factors involved is made

possible by do principle of efreetiye stress, and advances in

laboratory technique now enable shear strength rmrametcrs

measured in terms of effective stress to be used-as the basis

of design. The principle uncertainty in large dam design lies

in predicting the pore-pressure at each stage of its construction

and use.

Both elastic end limit elet;ign. methods of analysis are

discussed. A Relaxation solution is given for the elastic stress

distribution in a symmetrical triangular dam. Two classes of

problem for which limit design is used are distinguished, those

in which the pore-pressure is a function of the state of streams

in the coil ( i.e. excess pore-pressures set up durini; construction,

and during draw-down in fills which are not free d-eaining), o.nd

those in which thepore-preseuro is determined by the per:Ilcability and boundary conditions of the various zones of the fill (i.e. 0 t c a Ely seepage9 and draw-down la' free-draining materials).

Particular attention is given to the design of dar;as having

a pad ilea clay core9 and new methods are developed for

the lateral thrust of the clay core, and for estimating the gain

in strength of a sort foundation layer with -partial consoT'idation.

CONTENTS

Page

1. Introduction 1

2. Types of stability Analysis 6

3. Elastic Methods of Stability Analysis 16

4. Limit Design Methods of Stability Analysis 38

5. The Mechanics of the Circular Arc Analysis 43

6. Application to Specitic Design Problems:

(a) Stability of a Dhm in which Excess Pore-

pressures are set up during Construction 60)

7. (b) Stability of a Dam in which Excess Pore-

Chapter

pressures are set up due to Steady Seepage 88

8. cl Stability of. a Dam in which Excess Pore-

9.

10.

it.

pressures are set up due to Rapid Draw-down

Stability Problems associated with Puddled

94

Clay Cores 119

Resistance to Failure in a Thin Soft Layer

at the Base of the Dam 146.

Conclusion 160

Acknowledgements 163,

Appendix 1 164

Bibliography .168

CHAPTER I

Introduction

Earth embankments for retaining water may be classed

among the earliest large scale civil engineering works. Yet

.superficially they would appear to have undergone the least

development of any type of structure. Earth dams about

.60 feet in height were successfully constructed in India over

hundred years ago, and, though safe height is not the

:.,only criterion of development, it is interesting to note that

i)A4.'ghest earth dam constructed in this country during

the 19th century was the Yarrow dam, of 103 feet. The

tallest earth dam at present under construction in Britain is

130 feet in height (Daer Valley) , while in the United States

the Anderson Ranch dam, constructed over the period 1941-47/

has a maximum height of about 350 feet above ground level.

• There are probably few dams, however, for which stability

has been tie only factor controlling the height chosen.

Economic factors, the size of catchment area, or biological

actors controlling the maximum permissible depth of water

are of primary.importance; and though they will not be

considered in detail in this thesis it is of interest to

note that on a site underlain by a thick clay stratum the

volume of, bank fill required per unit length will be roughly

proportional to the (height)3, which places a severe

limitation on the economic height.

In most branches of engineering advances have depended

on two main factors: the development of design methods and

the improvement of the mdchanical properties of the materials

available or the introdubtion of new materials. In the case

of earth dams, the principal construction materials are of

nocessityAhose which lid within a reasonable haulage distance

of the site. The possible improvement of their mechanical

properties by compaction has been affected by the technical

developments of the industrial revolution to an important,

but rather limited extent. The modification of their mecha-

nical properties by the upe of chemical or bitumenous stabi-

lisers or cement is almost unknown in present earth dam

practice(x). Concrete and steel sheet-piling have been

used to a limited extent in the construction of cut-offs and

.core-walls„ but have not led to any major change in design.

On the other hand, important developments have occurred

in our understanding of the mechanical properties of the

traditional, materials and in our ability to measure them

quantitatively. Those have led to methods of analyols and

design which form the principal advance in this branch of

civil engineering. Though superficially no dramatic change

(x) The injection of cement grout is sometimes used as a

remedial measure in cases of leakage (see for example the discussion by J. Noel Wood on the paper by Bishop, 1946). Experiments have also been made on the admixture of bentonite to a sandy fill to decrease its permeability and increase the plasticity (Wetter 1948).

in profile has resulted, a rational basis is now available

for ensuring the most economical disposition of the construct-

ion materials and for calculating the factor of safety of

the structure with some 6onfidence.

Those methods must, however,-be considered to supplement

rattier than supersede the experience and judgement of the

practising engineer, The variability of some foundation

strata and the difficulty of predicting the effect of variable

weather conditions on the compaction and excess pore-pressures

in cohesive fills loave considerable scope for experience

and judgement, especially if based on well-conceived field measurements.

Much, in fact, could be gained from a more explicit and

active cooperation between the engineer and the scientist.

Confirmation of the validity of design methods in the labora-

tory is generally impossible in the case of problems where

body forces due to gravity predominate. The price the engi-

neer must pay for design methods in which he can have confi-

dence is a much greater willingness to take the necessary

field measurements of shear strength, pore-water pressure,

settlement etc., even on structures about whose stability

there is little doubt; Eind the price will generally prove

tote small compared witki the cost of the structure. The

research worker, on the ether hand, must give greater thought

to suggesting tests, design methods and specifications which

are practicable in terms of the soils and climatic conditions met with by the engineert

If the. need for work in this field is not at once

obvious, it should be noted that at least three major earth

dam failures have occurred in this country in the past twenty

years and at least six in the (x), involving expenses

running into many millions of pounds.

Vrom the data presented by Justin and others, it will

be seen that many of the earlier failures of earth dams were

not primarily due to shear failure, but followed over-top-

'ping due to inadequate spillway capacity. The importance

of this danger is now sufficiently realised, and as it is

outside the field of soil mechanics it will not be discussed

here.

Similarly, a number of the earlier failures were due to

ocalised piping or internal erosion caused by seepage paths

()riming at the junction of the fill with the outlet conduit

.or with bed rock. The importance of eliminating any such

is now fully recognised, and the outlet conduit is

now generally either constructed with collars or baffles

whore: . it passes through the impervious core, or driven in

tunnel through the impervious foundation stratum beneath the

darn

(x)Thoso are tabulated by Justin, Creager and Hinds (1945).

The typo of failure which has, however, persisted even

in recent years, and must therefore bo considered the hardest

to design against, is shear failure, either in the foundation

strata 'or in the bank fill itself, generally taking place

during construction. Recent examples in this country are

Muirhoad Reservoir, whore the shear failure occurred in tho

bank fill (Banks, 1948); and Chingforcl No. 2 Reservoir,

where the weakness of the soft underlying stratum was . the

primary cause of failure (Cooling and Golder, 1942).

Justin, Creager and Hinds (1945) quote five failures of

this type in the U.S.A. within the last twenty years,

including both hydraulic fill and rolled earth core dams,

..ranging from 52 to 225 feet in height. It is interesting

note that only one of these was caused by rapid drawdovm,

:Eilthoiigh in conventional design it is against this condition

that bank is usually found to have its lowest factor of (x) safety .

It is with a consideration of the problem of shear

failure that the subsequent chapters will be mainly concerned.

A brief history of design methods will be included, but no

attempt will be made to ()over the whole history of earth

dam construction, which is a subject in itself.

( x) Mayer reported in 1936 a number of clrawdovm failures which had occurred in France in the preceding fifty years, in dams constructed before quantitative design methods were in use.

CHAPTER 2

TYPES OP STABILITY ANALYSIS

The stability of an earth dam, as of any structure

in soil mechanics and, in fact, in civil engineering in

general, may be examined by methods of analysis falling

under two main headings. Both types of analysis must be

considered necessary to a full understanding of the problem,

though this is overlooked in much contemporary literature.

The first type is based on the principle of calculating

the stress distribution under the various conditions of

loading to which the dam will be subjected, and comparing

the stresses with the allowable strength of the soil. It

is thus in line with the classical procedure in structures,

though for practical reasons which will be discussed later,

its use is not very general in soil mechanics. As elastic

theory is usually used to ca l culate the stresses, the methods

are loosely termed elastic methods though in principle the

assumptions of a linear stress strain relationship and of

reversibility are unnecessary, and aro made only to avoid

difficulties in computation.

The second type is based on the principle of limit design.

If the load on a structure is gradually increased, or, in

the case of a dam, when the stresses are mainly duo to gravity

body forces, the strength is imagined to be progressively

decreased, zones will appear in which the material is over-

stressod. Failure will not occur until those zones have

extended to form a continuous path within which a continuous

slip surface could form. Tho analyses are concerned only

with the state of stress in the dam when this condition hap

boon reached. The actual procedure from this point varies

considerably, but basically it consists of finding by trial

and error the slip surface for which the disturbing and resto-

ring forces are in equilibrium for the smallest decrease ip

strength, i.e. the slip surface which would be most likely to

form. For simplicity the problem is considered to be two

dimensional, and the slip surface is assumed to be a plane or

a cylinder or a combination of both. More elaborate curves

such as log.spirals and cycloids have been used, but any

advantage claimed for them as a closer approximation to the

:actual curve is outwoighod by the mathematical difficulties

involved. It is in 'any case necessary to keep a sense of

perspective in this. respect by remembering that this procedure

is essentially an approximation as in fact the distribution

Stress along the slip surface is statically indeterminate

end introduces a small erlror into the calculation of the resto-

rinEforces (as indicated in Chapter 5).

might have appeared that this indeterminacy was some

fault inherent in the tsiicost method of analysis used in

conjunction with a slip surface, as Kotter's equation:—

don di — 2 afi tan 0 . dr7 =Tsin (r — cos 0. di

(whoro az = normal stress on failure plane

1 = distance along failure plane

r-1( = angle botytoon dl and horizontal)

is quoted by Torzaghi'(1q4.3) as permitting the distribution

of 01.1 to bo calculated from the shape of the sliding surfaco;

and Jaky (1936) and °tide (1938) are instanced as having used

it as the basis for stability and earth pressure calculations

respectively.. However, Coonen (1948) has shown that Kotter's

equation cannot in general be applied to any arbitrarily chosen

failure surface, independently of the state of stress in the

adjacent material, and , that Jaky's analysis is invalid.

Though Coonen does not discuss Ohdo's work it appears to

suffer from a similar defect.

An exact solution to tho problem is therefore impossible,

unless preceded by a step by step calculation of stress

distribution passing from the 'elastic' to 'plastic' states,

and it is doubtful whether our present knowledge of soil

properties in relation to compacted fill would justify this

procedure. It reintroduces the difficulties associated

with the first type of method for a probable gain in accuracy

of loss than about 10 per cent in a typical case.(x)

In practice, the choice of the shape and location of the

approximate surfaces to be tried is guided by the relative

strengths of the various sections of a zoned bank and its

foundation strata and by the records of actual slips.

A subdivision of thiS type of analysis which- is of

theoretical interest, tholigh of rather limited practical

application, is formed by those rases in which the stress

conditions in the limiting case are such that failure occurs

simultaneously over a considerable zone. Within this. zone

conventional plastic theory can be used. The bearing .

capacity of a thin clay stratum and the'lateral thrust of a

clay puddle core-wall are two cases which are relevant to

earth dam design•. In both cases the boundaries of the

plastic zone are determined by a discontinuity in the material,

along which the limiting stresses can be readily estimated.

In general, however, neither the boundary of the plastic

zone nor the stress conditions there can be determined

without a great deal of cbmputation, and, when foundswould

not, correspond to the few cases for which a solution by

plastic theory exists.

(x) A similar position exists in relation to theories of bea-ring capacity, though the approximation involved in a slip • surface analysis becomes less accurate as the angle through which the surface turns i increased. Here the Mott-Gibson analysis represents the elastic-plastic approach and • Moyerhoft8 analyses the limit design approximation.

- 10 -

Before examining in detail the assumptions and scope

of the two typos of analysis, it would be as well to consider

the criteria by which the practical engineer must judge them.

These may be summarised as:-

(1) simplicity

(2) reliability in practice

(3) small errors as compared with the most

rigorous analysis a'railable.

Simplicity is of groat importance if a method is to be

of general use to the engibeering profession. It should lie

within the ability of the average design engineer to use it,

and to appreciate the factors involved. Where some simpli-

fication of the mathematical stages of the analysis can be •

made, the elimination of possible sources of error may

actually increase its reliability. Simplification of the

physical properties of the material, however, should be done

with great care, and only where necessary to make the analysis

practicable. Ono of the factors in this simplifiCation is

-' - the need to keep testing proceduiie within the scope of the

normal laboratory.

Reliability in practice may be indicated in several

'different ways. Thu first, that structures designed by a.

given method have proved stable, provides satisfying, though

in fact .only negative evidence. It is usually invokLId to

— 11 —

justify semi-empirical rules, for which often no quantitative

cheek is possible, and it gives no indication of the gross

lack in economy which accompanies a high factor of safety.

If a new departure in magnitude or geological conditions is

made, this criterion alorie is of little use.

The second way in which reliability may be indicated is

that the stresses measured in the structure correspond to

those predicted by the design method. This check cannot

be made in the case of the limit design method unless the

factor of safety is equal to one, and the assumed condition

of incipient failure is in fact realised. In the case of

'elasti c' methods it is the only quantitative chock possible,

This may prove to be one of the limitations of their use in

practice, as the measurement of the state of stress within a

mass of soil presents great difficulty both experimentally

and in the interpretation of the results. Measurement of

the pore-water pressure, which is easier and more reliable,

is likely to prove the most important check in this case.

The third way is by checking in cases whereo the factor

of safety is known to be unity. This is probably the most

impressive test of all, and accounts for the great interest

shown in failures by specialists in this field of civil

engineering. This interest is due in part to the difficulty

of carrying out model tests in the case of gravity structures

- 12 -

of cohesive soil. Thu gravity stresses in a scale model

are so reduced( that a failure cannot be reproduced without

decreasing the strength of the soil to a point at which the

results cease to be convincing. • Even in practice, as the

site investigation, selection of samples etc., are rude after

the failure, when its loCation and often the shape of the

slip surface is revealed, some of the difficulties facing

...the designer are avoided. Nevertheless, the evidence of

YthiEr,Sort presented in support of the 0 = 0 analysis, whioh

is one of the typos of, limit design (Skempton and Golder,

1948, and Cadling and Odenstad, 1950) forms perhaps the

most convincing evidence we have of the validity of any of

the methods of analysis used in soil mechanics, or, in fact,

in civil engineering in general.

Elastic methods cannot be checked in this way at all

satisfactorily, as it is generally accepted that a factor of

safety of loss than unity against local overstressing will

only load to excessive deformation, which is very difficult

to assess quantitatively. Guthlac.Wilson t s results (1950) ,

in relation to the bearing capacity of screw piles, illus-

trate this difficulty.

The magnitude of the errors involved in a given design

(4Tests lava been carried ort'in a centrifuge to overcome thi6 difficulty but have obvious limitations.

- 13 -

method as compared with the most rigorous analysis possible

may seem rather academic to the practical engineer if there

is experimental evidence available that the approximate

methods are reliable in practice. It will be seen, however,

that in the case of large earth dams the evidence on the

reliability on the desig0 methods used is largely negative,

and therefore inconclusive. If judged by their successes,

Indian practice of the period 800 - 1600 A.D. (Rao, 1951),

elastic methods (Middiubrooks, 1936 and 1948, and limit

design (Terzaghi, 1936, Daehn and Hilf, 1951) are equally

reliable. A more critical examination of the methods is

therefore necessary.

The errors involved may enter at several different

stagesg-

(a) The soil. c nditions in the test may not correspond

to those in the foundation strata or in the bank duo

to unrepresentative or poor sampling; or to un-

reliable compaction technique or weather conditions.

(b) The stress conditions in the test may differ from

those in the daM, viz. the results of standard

triaxial tests 'ztre usually used in the solution of

plane strain prdblems.(x)

Hansen and Gibson (194.9) discuss some related difficulties.

— 14 —

• (c) The failure orituria and stress strain characteris-

tics are simplified to make the calculations pgla.

goable.

(d) The analysis is Simplifedloy the use of simple

slip surfaces, the neglect of some indeterminate

forces or the approximation of boundary conditions.

An analysis which in fact eliminates these orrors has

not yet been achieved. An estimate, to within reasonably

,close limits, of their magnitude in any particular case, is

the :limit of our present work.

1n -brief, it may be said that there are two main ways

of. approaching the stability analysis of an earth dam:-

By calculating the stress distribution within the

dam and its foundation and comparing this with the

strength of the soil.

(2)• By examining the conditions of equilibrium when

incipiqatfaildro is postulated, and comparing the

strength necessary to maintain equilirium with that

of the actual soil.

Approximations are inherent in both approaches which it

difficult to eliminate altogether; • a clearar understanding

of them, howover, might serve to lessen the burden of applied

mathematics thrust on the designer,

Convincing practical confirmation of the second group

- 151tr.

(of limit design methods) can only be achieved if the dam is

designed to have a factor vf safvoy cf unity and just fails.

Except in cases where failure due to an unpredicted change in

loading conditions, or initial poor design, presents this

opportunity to the investigator, he must generally remain

content with measuring stresses and pore-water pressures,

which can in fact sore as a direct check only in the case of

elastic or stress distributi.on methods;, though the results

may serve to narrow the margin of uncertainty in the use of

the limit design methods.

Neither method can be ignored by the investigators

though, as will be seen from the following sections, limit

design methods are much easier to use from the designer's

point of view.

CHAPTER 3 •

ELASTIC METHODS OF STABILITY ANALYSIS

The most widely used application of the theory of

elasticity in soil mechanics has been to the design of

foundations when considered from the point of view of settle-

ment.

This is discussed in detail by Terzaghi (1943), who

also considers the fundamental assumptions involved.

Its application to stability analysis is used to a

limited extent, and was first discussed in detail by

Jurgenson (1934). In both cases' the basis of the method

is the stress distribution in a semi-infinite elastic con-

tinuum under a syStem of surface loads, obtained either by

integrating Boussinesq's solution for a point load (1885),

or from Carothers' solutions for various systems of continuous

loading for which Jurgenson tabulates numerical solutions

(Carothers, 1924 etc.)

Jurgenson considers the case of an earth dam whose cross

section can be approximated to a symmetrical triangle, and

uses Carothers' solution for a triangular strip loading.

The weakness of this approach is that it ignores the hori-

zontal shear stresses transmitted from the dam to its foun-

dation at ground level, and gives no information about the

stresses within the dam itself. He observes that complete

failure will not rosUlt from local overstressing, unless

- 16 -

progressive failure occurs in the soil, but suggests that in

practice it may be desirable to design so as to avoid local

overstress (i.e. the maximum stress calculated on elastic

theory should not exceed the shear strength of the soil).

Middlebrooks (1936) Was aware of this weakness and

suggested a method by which the dam was divided into a series

of horizontal layers, and the stresses below each layer

calculated as though it yore a vertical load applied to the

surface of a semi-:i.nfinitS elastic continuum. This is open

to the obvious criticism that the condition of zero shear

and normal stress will not be satisfied on the surfaces of

the dam, but Bennett (1951) asserts that this error can be

neglected if the slopes are flatter than 3 to 1. While this

may be true of its effect on the values of the maximum stres-

ses, the stress pattern ai3thoseaces are approached will be

very misleading.

Parallel developments in the field of masonry dams

appear to have been rather. overlooked. The controversy in

this country on design methods, following the failure of the

Bouzey .dam in France in 113959 contains several contributions

of interest Which have been summarised by Pippard in his

Unwin Memorial Lecture (1949). Terzaghi (1943) quotes only

the solutions for a semi-infinite elastic wedge dbtainod by

Levy (1898) and Fillunge (1912). These lead to the result

- 17 -

that on a horizontal section of a dam the horizontal shearing

stress is a maximum at the outer edge, and the vertical stress

is uniform over the full width of the dam. Terzaghi com-

pares these stresses with those based on Rendulict s plastic

analysis for frictional soils (Terzaghi, 1945), and rejects

the elastic stress distribution as improbable. The effect

of continuity with the foundation on the stresses within

the dam itself had, howevdr, been taken into account by

Richardson in his analysis of the J5ssuan dam in 1908, using

elastic theory and a finite difference approximation.

Southwellts RelExation method has now made possible a more

accurate and extensive investigation of this type of problem,

and has been applied to the masonry dam by Zienkiewicz (1945).

Pippard draws attention to the good agreement between

Zienkiewiczts results and those obtained by Wilson and Gore

in 1908, using a rubber model; a method which has received

little attention .in soil mechanics.

The first application of Relaxation methods to this

aspect of earth daM desigp was made by the Author in 1943,

when, at the invitiation of Sir Richard Southwell, he spent

several months working with his research group on this

problem on behalf of the Metropolitan Water Board. The

results are presented here as they represent the latest

- 18 -

development in this field•

Several other approaches have been described which should

be mentioned although they do not fall, strictly within this

class.

Patrick (1948) has described a method in which the

elastic approximation duo to Middlobrooks is used as a baSis

for determining the stress distribution along a slip surface,

from whose equilibrium the factor of safety is derived.

Brahtz (1936) in a contribution to the 2nd Conference on

Large Dams, outlined a method in which the stress distribution

.was calculated assuming an arbitrary ratio (K, the compaction

factor)betweun the horizontal and vertical stress in any

element and using equilibrium equations which ignore strain

compatability. It appears to have neither the merits of

.simplicity nor accuracy, and has not boon generally used.

Glover and Cornwell (1941) proposed an analysis in

which the boundaries between elastic. and plastic zones were

arbitrarily chosen, and a stress distribution computed. This

suffers from the same limitations as that due to Brahtz, but

has recently been used as the basis for the estimation Of

pore-water pressures on rapid draw-down (Glover, Gibbs and

Dachn, 1948).

Assumptions implied in the Use of Elastic Theory

The idealised elastic solid may be defined as one in

- 19 -

which Hooke's law is obeyed fdr both positive and negativo

changes in stress, and which is homogeneous and isotropic in

respect of both the elastic modulus and Poisson's ratio

(which also is independent of the state of stress). Where

body forces due to gravity are considered, constant density

is assumed.

No engineering material completely fulfills these

requirements, and in the case of masonry and concrete dams,

when elastic analysis (including elastic model and photo-

elastic studies) is much used, very wide deviations are in

fact accepted (Zienkiewicz, 1945).

Probably the most serious departure in soil, as in con-

crete, is in relation to Hooke's law. 'Deviations from

linearity become large as the failure stress is approached

(Fig. 30) though it will be seen that under lower stresses

a linear law is a good first approximation in the case of

some undisturbed saturated soils, and in the case of well

compacted fills. Terzaghi (1943). stateb that where the

factor of safety exceeds 3 elastic theory is likely to give

results corresponding to. the actual stress distribution, bit

that the stresses in the soil in earth dams are usually well\

beyond the range of the approximate validity of Hooke's law \

(the factor of safety used in design is of the order of 1.5).

Ho suggests that this justifics the solution of those problems

3

4

ST

RE

NG

TH

. 4O

Ui

10 15 O/o STRAIN.

5 20 25

• , TYPICAL STRESS STRAIN CURVES FOR SOILS.

1. UNDISTURBED STIFF CLAY— WALTON — 2. UNDISTURBED SOFT CLAY— SHELLHAVEN — 3. :COMPACTED.BOULDER CLAY --DAER 4. COMPACTED MEDIUM SAND—BRASTED —

UNDRAIN ED TEST. UNDRAINED TEST. UNDRAINED TEST. DRAINED TEST.

- 20

by the theory of plasticity. While this is accepted in

principle by the common use of limit design, it offers no

reliable information about the distribution of stress in a

dam designed with a factor of safety of, say, 1.5, in which

plastic failure will be

A more logical approach

dams would be to obtain

and if appreciable zones

redistribute the excess

liMited to relatively small zones.

in the case of relatively homogeneous

a solution based ,on elastic theory,

of overstress were indicated, to

by a step-by-step use of the equations

of plastic equilibrium in the zones in which they were valid.

A more important objection is to be found in the fact

that the elastic modulus in general varies with the state of

stress. For any given soil the elastic modulus has been

found to be directly proportional to the strength, as a first

approximation. The ratio E/"C"(x)

depends somewhat on soil

type, state of compaction or consolidation history, but lies

within quite a narrow range of values for a particular soil

(Fig.3:2). Now the failure criterion of soils, with respect

to total stresses, may be approximately represented by the

Coulomb equation:-

s = c oh tan 0 ,. 3:1

taken, for soils, as the secant of the ultimate strength, and "c'

re generally, half the compression

( x ) E is Young's modulus, modulus at 50 per cent is the cohesion, or mo strength.

2 00r----- - -

IA AT 10 E

100— -

:50

10 20 30 40 50

lb OHESION C ----= I/2 COMPRESSION STRENGTH 4q.in.

FIG. 3:2. RATIO OF YOUNG1S MODULUS TO ULTIMATE

STRENGTH FOR VARIOUS SOILS— UNDRA IN E D TESTS.

C

0 SHELLHAVEN CLAY. 4.diam. UNDISTURBED AVERAGE

SAMPLES VALUE.

LONDON CLAY. 4'I diarn. UNDISTURBED SAMPLES AVERAGE VALUE.

COMPAC TED BOULDER CLAY. 4'.' darn. SAMPLES. IN TERPOLATED VALUES FOR VARIOUS MOISTURE CONTENTS (DAER).

COMPACTED LEAN SILTY CLAY 3% diam. SAMPLES. RESULTS FOR VARIOUS VALUES OF 0- (WALKER and HOLTZ 1951.). 3

the expression:-

sin cos 0 3:2 0-31 . (1.3 + - singl - sin'

- 21 -

where s denotes shear strength

o -denotes apparent cohesion

denotes angle of Shearing resistance

rfi denotes applied pressure normal to the shear surface.

If the results of a triaxial compression tesi are

expressed in the form of la Mohr envelope, it will be seen

that half the compression strengths(0-1 '13) is given by 2

Young's modulus is therefore given by the expression:-

E A (73 sin 0 +a cos 0 ) 1 - sin 0

3:3 - sin 0

where A is a constant.

It will be seen therefore that the use of elastic theory

with a constant value of E 'is only valid if 0 0 or if c is

very large compared with 73 tan 0. The first condition

' -rop're sent s the case of saturated soils (Bishop and Eldin,1950)

the second that of materials such as concrete in normal

gravity dams where o3 is relatively small compared with C.

The conditions under which 0 = 0 in soil have been discussed

in detail by the Author (Bishop and Eldin, 1950) ; the two

principal factors being the absence of drainage during the-

application of the stresses; and full saturation. Good

compaction technique seeks to avoid the latter owing to the

high pore-water pressures sot up. It is, however, in the

cases in which pore-pressures are likely to be high that

stress distribution studies aro important, and hero elastic

theory will approximate most closely to the actual conditions.

The fact that under an initial stressing the strains are

largoly.irroversible is a limitation which soil as an ideal

material shares with concreto.(x) it is probably an

essential characteristic of all granular solids in which part

of the strength is provided by intergranular friction, and

rules out the General use of the principle of superposition

if stress reversals are involved in the constructional

operations considered. Per the initial stresses in a gravity

structure, however, no major error should be involved.

The effects of anisotropy have only received serious

consideration so far in relation to normally consolidated

saturated soil (Hvorslev, 1936, Hansen and Gibson, 1949).

As this would not provide a suitable foundation for a largo

structure, and does not correspond to the condition of the

fill, little quantitative evidence exists. The error will,

(x) This is partly exhibited as long term creep, which in the case of concrete results in total strains which are pro-portional to stress, equivalent to a lower effective elastic modulus (Zienkiewicz, 19L1.5)

23

for the present, be considered small as compared with those

considered above.

All soils except those that are fully saturated undergo

an appreciable volume chahge during shear, which alters in

rate and usually in sign hs failure is approached. This

reinforces the suggestion that elastic theory is only valid

in the case of saturated soils, where Poissonts ratio will be

constant and equal to if. That this is a very close appro-

ximation is due to the high compressibility of the soil

structure compared with that of water, which controls the

bulk modulus of a saturated soil.

In relation to the present problem the particular diffi-

culties arise (a) of assuming the same elastic properties for

the dam and its foundation 'and (b) of assuming that if the

gravity field wore removed the dani and its foundation would

tree of residual stresses.

The cumulative effect of these approximations is difficult

to assess theoretically; and though substantial advances have

been made in obtaining field measurements of stress in the

last ten years (Pressure Distribution Studies on Soils -

Vicksburg, 1947, recent Swedish work etc.) errors of the order

of 4- 30 per cent must still be expected, and the data is very

incomplete, and largely of qualitative interest (Vicksburg

Report,. page 279).

-24.—

The present solution can, however, be considered as

at least an approximate basis for the discussion of factors

influenoed by the stress distribution.

A Relaxation Solution of the Stresses in an Earth Dam

after Constpuction.

The embankment problem reduced to its simplest form may

be 'considered as that of a symmetrical triangular strip made

continuous with the surface of a semi-infinite elastic con-

:tinuum of the same elastic properties, and subjected to a

uniform gravitational field. This permits the use of plane

strain equations, and the conditions of continuity and

elasticity can be satisfied by an .Airy stress function tVt whore V Liv = 0. Analytical solutions have boon obtained

representing certain states. of stress and boundary conditions,

but none approximate to the case under consideration.

As a numerical solution will correspond to a particular

geometrical shape, 3 to 1 slopes have been chosen as represen-

ting typical large dam practice. Fig. 3:3.

Basic Equations

If x and y represent the orthogonal coordinates of any

point and the gravitational pull acts in the direction of

the y aftis, the stress components at that point may be

expressed as:-

_ _ _ _ _ _ _ _ _ _ _

I d x

j>'. dx .dy r cr d x y X y

•4 • dx

Y V

dy , U ÷

KY

cY

y

1 d •

Y y

3: 4.

-25-

s2 v

3:4a

3:4b

3:4o

a d- + G(gY 6 x

= v dycy

6x6y

where ca, erS, , aky ard defined as in Fig. 5:4,I is the bulk

density of the soil, and V the Airy stress function, in

- terms of x and y.

These equations follow Airy (1862) and Richardson

(1909); the equilibrium of an element being automatically

satisfied by them, The compatability of strain equations

are satisfied if V4 V = 0, i.e. if

4 2 4 . 4 • + c..x2oy2 ) V = 0

3:5

For a unique solution a function V has to be found

which satisfies the stress conditions along the free surface

ABODE, and at an infinite distance within the elastic body.

This latter requirement is made more practicable for numeri-

cal Solution by the use of St.Venantls principle. Thus the

stresses at a - large distance from the dam will be the same

as those due to any statically equivalent load applied to

the surface of the semi-infinite elastic continuum. The

-26—

more nearly the arbitrarily chosen distribution of load

on the surface corresponds to that actually existing at

the base of the dam, the smaller will be the distance

required before the consequent stresses approximate to

the exact solution. A stress function representing those

stresses can be used to give values of V round this boun—

dary. Within this boundary the values of V are found by

Relaxation..

Stress functions giving the distribution of stress

within a weightless semi—infinite body are known for several

simple types of surface loading. In order to minimise

the area within which relaxation is required the stress

function for a triangular strip load is required. This

has been obtained ,by integration of the existing solution

for a uniform strip load. This derivation and its more

general form applicable to asymmetrical triangular or trape—

zoidal banks are given in appendix 1.

This stress function does not include a term for

stresses in the ground due to its own weight. As the state

of stress in natural ground depends largely on its previous

geological history, and as this solution is concerned

primarily with stresses consequent on the construcI

the dam, the ground will be assumed to be initially free

from shear stresses.

- 27 -

Hence, from equation 3:4., if y = 0 is taken as the

free surface, we have:-

a 2'\/* = cr3i eY

() 2V +1Y = crY = 61° Y ax

. -‘•.‘ 2 = °MY = °

3:6a

316b

3:6c

These equations are satisfied by the stress function(x)

V = j 3 3:7

The stress function for the stresses in the ground of

density beneath a triangular strip load of breadth. 2b and

having a peak value of ')/ H is 2.

V li-13 loge (x + b2 2

+ y2

2 Xtai x), .-67 (x2 + y2) 2 Y.

(3

b + 3 y2 (x + b) ) tan-1 x + b

(x o3 .1. 3 y2 (x - b) ) tan-1 x - y

b + 2102 y] -J

3.8 +. y3

(x) It should be noted that this is consistent with zero residual stresses on the removal - of the gravity field only if Poissont s ratio is. equal to 2 9 as by symmetry the lateral strains beneath the horizontal free surface are zero.

r. V = Vo ± (x xo ) 16—Y1 (y -

Lox. 0 2 xo) Y 2y0)

ei 0

- 28-

I.; remains, therefore, to determine the boundary values

of V along the free surfabe of the dam and the adjacent

ground. Consideration of the equilibrium of a surface ele-

ment (see Appendix i.) 31-lbws that in passing from xo, yo to

x, y along a straight section of the surface free from normal

pressure or tangential shear the changes in V satisfy the

following relationships:-

constant = 17, 6V! oy

I '6Yi0

f)v ,Sc,„1/71 6x 177

i L.) o 3:10

3:9

yo)

3:11

These equations enable any surface to be dealt with by

a step-by-step, method, the present case being particularly

simple.

Boundary values and gradients can thus be determined

around the whole of the section of the dam and its foundation

under consideration. Within this area arithmetical values of

V are then found by the Relaxation method of successive

approximations at a series of tmesh pointst so that the

- 29 - 4

equation v V = 0, expressed as a finite difference approxi-

mation, is satisfied throughout. For a mathematical justi-

fication of this method, and a discussion of the errors

involved in the use of a finite mesh, reference should be

made to papers by Southweil et al (1938 seq) and Zienkiewicz

(1945) •

The method may be understood by reference to Fig. 3:5.

If values of V are plotted 22 ordinates on a suction parallel

to the x axis, 23 a first approximation.

= V2 - Vi L J12 d 3 : 1 2

-S1

(.,V - V2 '6x(

J 23 d 3:13

and

v

a i r -' .__.. 1 6v __,. v! = V3 - 2V2 4-- V1 b xj .1...6x 23 r 12

d2 4

3 : 4

It follows directly that, if V has the values Vo, V1 etc.

as the intersections of• a square mesh as shown in Fig. 3:6

the stresses can be expressed as a finite difference approxi-

mation from equation 3:4. For the intersection 0, we have:-

V 2 Y

V2 V4 - 2Vo

3 1 3a

\ 0 /1. L' v VA V- - 2V w);c, Y71.1-. 'y' 4 ° Y' , "

d2 3:15b

Vo

V 41

V

V 6 V V6-

V, 0 1

V3

FIG, 3 5 .

FIG. 3.: 6.

- j0 — iv v , V V \ —' 5 — 6 -r 7 — 8

14. (12 I

3:15c

Similarly the equation \--1 14V 0 is written

20 (Vo ) E3(,V1 -1-V2+V3+V14) 2 (V5+V6+V7+V8) + V9+V1 0 +Vi +V,2. 0

3

Ecivations of the form of equation 3:16 can be written

for ovary point within the dam , and together with the boundary

conditions- , represent a set of simultaneous equations whose

solution gives the required. stress distribution.

Relaxation Method

As an accurate solution necessitates the use of a c.-iesh

involving a large number of intersections (c.170 in this c:_-.3e),

the corresponding number of simultaneous equations prs:-.J1ts

too formid.ablo a. problem for the use of ordinary methods.

If arbdtrary values are given to the stress function at

the points under consideration, equation 3:16 will in gen ral

not be satisfied, but the corresponding relationship can

be written:-

20(V0), 8(V1+V2+V3+V4) + 2(V5+V6+V7+V8) + V9+V1o+V1 --N12

3 : 1 7

The equation V 11. V 0 min only be satisfied when

Ro 0, and this condition can be reached by successively

- 31 -

a d j us t i n g the values of V. If unit change i s made i n the .

I

I

"The vaiue~.tlrl:I~~dual at any point can be' r e duce d to

zero by the appropriate adjustment of the vaiue of the st ~ ~~

~alue of Vo' ~he value of Ro will be changed by 20, and the

values ~f R at adjace~t points by smaller amounts as s ho m

in Fig. 3:7, Which is termed the relaxation pattern.

2 -8 21"-,-8 20: - 8-l ~ ·- · · - , :-

2 '

FI C, .3: 7.

.'\ "

.. function at that point, and is redistributed amon g t he

Ithe largest residual is a convergent pr oce ss , hi ch can,

'1:. _,.

,':, t"

~ , .. '

adjacent points as shown. The successive elimination of

however, be great ly speeded up i f gr oups of poin t s lines

- .or block s) hav i n g r e sidua l s of' predonin ent l y the 8 6..1 e s i z

are treat e d as units. The r e Laxat i on pat t e r n s r e qu :..l. e

for t he se can be simply derived frau I"':' r: .. 3: 7 b y S "

tion.

In orde r to en s ur e t e spe ci~ie d v~lue or t he ~t~e ~ ,

f unct i on an d it s gr a dien t al ong the bo u i c ar Le ,val ue 0

are calculate d f or three l ine s in dep t h 0~ t he bo undar y!

wi t hin :the .g.round , Alon g :'t he · r ee s ur-t'ace s use i s .ncc, . . .... -II

a l i ne of i magi nar y poin t s ,out s i de the sur f a ce S0 t la t

urface gz-ad.i.en t s whac h t he y define s at i s :;:'y t e vaL ue

gi ven by e ~ ation s 3~9, 1), 11 (Fi g . 3: 8 _

r MAX

-2.5027 7/268 67513,, 63760 60009 56259 5251/

9/9/4 8835 846384 • 6434 7

99_5/9 96007 92351 1 89123

.

779351. 74559 7/27/ 661071

85762 60461 76.?76 76160

2439/ 21736; 49447

21727

19444.

..1,11457

STRESSES FOR A DAM OF UNIT HEIGHT AND UNIT DENSITY. Icc

v x tic cry (=Sy 'MAX tan 2a a°

7575/ 750,-'' 7'25/ 6750/ 6375/ 6000/ .. ..,

g o 50 0 ; 0 8

•

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409 1 .0/5

oo .0,3 00 •,5 • 4S • 4S •

82527 78777 7502 7.268 675/3 , 6.3760 6000

J15 42.4-5'-

.251 .2.5.1 0.753 4 5,

-.004 • .003 • 0 004

47 0 1 •023. .076 , 072 3.24....- 7.::f.'... ?f,- ' -7.11t1.-• 7897.3• 75222 71466 677/1 63959 602/2 82723

-3.54 -7,32 -T.16

502 • SOS • .509 .506 ..g3; ..2EJ ;ST9 'Cl?

-244. 434.7 •

o; .79s

.0; 7

63265 79501 7,576/ 720 2 66264 64..1• 60787

-7;927 - .640 •

.7%. • 7 rn • 7 • •0201

042 . 04 47,

84283 80550 76811

- -44•2•

813509 648294 61115

1

02054 663.49, 84706

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I.

• 11021/, 4066/7 _103061

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i'l ' I

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0522 •••"591 1.130 -,47a ; I.524 1.337 0590

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444-.-- )4 f•i" /44•05 -• ,44.44-.0

1-790 0740 • 0.794 ,,,,,,,

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10.9.1 1•14.2 I•••.1 n ill 'say 2.0S• 2•097

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FIG.3:8. RELAXATION SOLUTION FOR A DAM OF TRIANGULAR CROSS SECTION.

-32-

The relaxation process is continued until the indivi-

dual residuals and their sum over any area is negligible,

and the calculated values of stress are not significantly

altered by further adjustment.

Discussion of Results..

The final values of the stress function are given in

Fig. 3:8. As the bank is symmetrical the detailed results

are presented for one slope only. In Fig. 3:8 the values

of the calculated stresses are given, numerical values being

shown for the vertical and horizontal normal stresses and

shear stress, the maximum shear stress and the inclination

of the planes on which it acts.

The interesing features of the results may be illustra-

ted in several ways.

(a) Distribution of Maximum Shear Stress.

Contours of the values of maximum shear stress through-

out the dam and its foundation are shown in Fig. 3:9; and

in Fig. 3:10 a comparison is made with.the values obtained

by Jurgenson (1934). While it is interesting to note that

the zone of maximum stress still lies below the level of the

base of the dam, the neglect of the base shear implied in

Jurgenson's solution leads to very serious error in the

strata lying immediately below the dam where the'greatest

uncertainty usually lies. For strata lying deeper than

iH (where H the height of the dam), Jurgenson's solution

3 9 t.c. SHEAR STRESS

ghl of the dam

FIG. 3:10. CONTOURS OF MAXIMUM SHEAR STRESS — A COMPARISON OF VALUES

TABULATED BY JURGENSON WITH RELAXATION SOLUTION .

VALUES OF 7 MAX RELAXATION SOLUTION ey- H JURGENSON 0934

-33-

involves an error of 20 per cent or less and may be

acceptable.(x)

A closer approximation in some respects is provided by

Middlebrooks' type of analysis. While this method involves

a great deal of computation to determine the complete stress

distribution, the values of maximum shear stress on the axis

of symmetry are easily obtained. In Fig. 3:11 the maximum

shear stress at the centre line is plotted against depth

below the crest of the dam for the different methods of

analysis. The values are obtained by a graphical integra-

tion corresponding to the use of an infinite number of hori-

zontal slices.

Comrarison with the results of the Relaxation solution

Midaleo:;,00:::.6 1: method gives a muoh more

realistic picture of the shear stresses it beneath

the dam than Jurgonson's method, it underestimates the shear

stresses in the bank by up to about 40 per cent. While

Bennett (1951) suggests that the errors involved in this

method are negligible for slopes flatter than 3 to 1, he

does not give any quantitative evidence. A method involving

a possible error of L.0 per cent. in the shear stresses cannot

be considered a reliable basis for design, though it may

(x)These comparisons are for banks whose slopes approximate to 3 to 1, though they will be qualitatively correct for other slopes.

RELAXATION

MIDDLEbR

-------. JURG

_...

---1

CREST LEVEL

DEPTH

GROUND LEVEL

F OUNDAT ION

2 H

SOLUTION

OOKS' METHOD

ENSON

T MAX /s" H

0

2

FIG. 3: II . VALUES OF MAXIMUM SHEAR STRESS

ON THE AX IS OF A TRIANGULAR DAM.

serve as an approximate guide in checking stress distribu- tiOns. Middlobrooks and Bertram (1948) suggest as a

typical case to which it is applicable that a strong .

embankment constructed on a foundation of soft clay, the

design criterion being local overstress in the soft clay.

Here the method due to Middlebrooks gives a maximum error

of 30 per cent on the low side; though redistribution of

stressiedue to the stronger bank might lead to shear stress

lower than those based on the assumption of a homogenous material throughout.

(b) Distribution of Stress at Base of Dam.

The distribution of horizontal shear is given in

Fig. 3:12. It is interesting to note that its maximum

value 0.083 H occurs almost exactly under the mid-point

of each slope, and that the distribution is approximately

parabolic except near the toe. The distribution of verti-

cal normal stress is given in Fig. 3:13, and this corres-

ponds closely to the vertical 'head' of soil above any given

point. • In this latter respect especially it agrees well

with the type of distribution..axpected by Terzaghi

(1943, p. 407 - 409) 'regardless of the value of the factor

of safety', although the solution he quoted was obtained

by Rendulic's method for plastic equilibrium in a sand fill.

It will be seen that the idea suggested by the use of

the semi-infinite elastic wedge solution, both that shear and

x y y H

3H 2H

TOE

FIG. 3 12. O IS TR IBU TI ON OF HORIZONTAL SHEAR ACROSS BASE

AND INTERMEDIATE HORIZONTAL SECT IONS OF DAM

,..-' .--------c--

.."-' ,•••'.

.---- .,-- ..---'

-1 .--''

----•

---- ..---

----- -----

0'2 ----

..-----

. 50 i

0'22 -itA

--- i

-.

.--

I. 0

H

0..5

0 3 H

TOE

FIG. 3 :13.

2 H

H

DISTRIBUTION OF VERICAL STRESS ACROSS BASE

AND INTERMEDIATE HORIZONTAL SECTIONS.

-35- normal stresses are distributed linearly over the cross

section, is misleading both at the base of the dam, and at

its mid height as well (Fig. 3:12). The method of analysis

used by Baumann (1942 - 1948) is based on linear distribu-

tions in an elastic triangular section and must therefore be

considered of very doubtful validity(c.f. Zienkiewicz,1945).

(c) Trajc6ories of Principal Shear Stress.

Two orthogonal sets of lines can be drawn representing

continuous planes on which the shear stress has its maximum

value at each point. One set of these trajectories is

shown in Fig. 3:14, and it will be seen that they correspond

to the type of slip surface generally observed in a relatively

homogeneous system. They do not in fact represent probable.

slip surfaces in detail, as the shear stress distribution

along them is not uniform (Fig. 3:15) and even if local

failure were to occur in the most highly stressed zone, the

average stress would be much below the strength. This

average is about 75 per.cent of the peak value for shallow

trajectories and 56 per cent for the deep trajectory which

shows the highest average shear stress (= 0.1375r).

It is interesting to compare these results with .the

slip surface obtained using the circular arc analysis, which

is alsci shown in Fig. 3:14. The average shear on the most

probable slip surface (assuming a cohesive soil of uniform

strength) is 0.1481(H. The average values are thus in close .

agreement. .

H

FIG. 3:14. TRAJECTORIES OF PRINCIPAL SHEAR STRESS AND SLIP CIRCLE FROM 5, 0 ANALYSIS.

/5 •

is

.25

. ,

. , .1f-'61

- (pi, - '--)1 i - '3'3

!i!EC'

.I.j 1 3 1 3 '3'

.103 ,p .0 Aci

/0 h .9 3 7 S S 3 2 ' O Pi3A-0.7:7e cr/or7g 1-rgyectoPy mecrsui-ed J2-o/77 4oper

FIG. 3:15. DISTRIBUTION OF SHEAR STRESS ALONG TRAJECTORIES OF

PRINCIPAL SHEAR STRESS.

-36-

It is also clear from the contours of maximum shear

that a bank designed just to avoid overstress would require

a strength of 0.258)(H. Its factor of safety against comp-

lete failure by the circular arc method would therefore be

0.258(̀H 1.75. Thus for banks with a factor of safety of 0.148 xEL less than 1.8 on limit design methods the stress distribution

from elastic theory can at best be an approximation (even

assuming ideal materials), as some local overstress is

bound to occur.

(d) Distribution of Major Principal Stress.

In order to make an estimate of excess pore-pressures

set up during construction it is theoretically necessary to

know both major and minor principal stresses, and these can

be calculated for any point in the dam and foundation strata

from the values tabulated in Fig. 3:8. There is at present,

however, insufficient experimental data on the pore-pressures

induced by various combinations of stress to justify this.

procedure in practice.c4Ourrent design methods imply that

the excess pore-pressure is primarily controlled by the

major principal stress and this is equated to the vertical

head of soil above the point under consideration (Bishop

1948, Hilf 1948, Lee 1948, Daehn and Hilf 1951). The.

approximations involved in the' first assumption will be

(x)The requirement of plane strain presents special experimental difficulties (Kjellman 1936, 1951).

120

115

110

105

100 95 90 I 90 1

FIG. 3 16. MAJOR PRINCIPAL STRESS AS PERCENTAGE OF y h .

-37-

discussed in Chapter 6, but it is appropriate hero to con-

sider the accuracy of the second.

In Fig. 3:16 the major principal stress is plotted as

a percentage of the vertical height of dam above the point,

multiplied by its density. It will be seen from the con-

tours that this method of estimating major principal stress

is _a good first approximation for much of the central zone

of the embankment, but would lead to errors near the toes.

As the pore-pressures are of greatest importance where the

stresses are high this is not a very serious objection.

CHAPTER 44.

LIMIT DESIGN METHODS OF STABILITY ANALYSIS

These methods represent the classical approach in soil

mechanics. Coulomb's work on earth pressure and on the

critical height of a vertical bank, published in 1776, was

based on the consideration of the statical equilibrium of a

wedge of soil above an assumed plane slip surface. Francais

(1820) extended this method to the case of a sloping bank.

It was realised that the plane slip surface was not a very

close approximation in the case of slopes and Collin (1846),

on evidence obtained by measuring actual slips in clay banks

and in the Cercey dam, used.a cycloid instead. This proved

mathematically difficult, and he was only able to present

an approximate method.

Collin was, however, the first engineer to carry out

shear tests on clay and measure the effect of variation in

water content on shear strength. This realistic approach

was not followed up, and little advande took place in methods of analysis for slopes or embankments until

Petterson suggested the circular arc approximation (c. 1916).

This alloWd aradLcal simplification of the applied mechanics

of the problem and is widely accepted. But it is only the

starting point of any analysis, and different treatments of

the mechanics of shear failure and the influence of pore—

pressures have led to procedures which differ not merely in

detail but in principle. 39,

_39-

The logarithmic spiral has received serious consideration

from a number of investigators (Rendulic 1935, 1940, Taylor

1937, Odenstad 1944) as it avoids the difficulty of making

an additional assumption about the distribution of normal

stress on the slip surface. But Taylor has shown that the

result given by such an analysis is not significantly diffe-

rent from that of the circular arc method, while the work of

computation is greatly increased.

It is necessary, hoWever, to consider other forms of

failure surface in the case of banks constructed on relatively

weak foundation strata, especially if the central core of the

dam is soft puddle clay, as is common in British practice.

Several alternative approaches are available, such as the

use of composite surraces of circular arcs or planes, or the

use of plastic theory. As this treatment is"csually limited

to conditions under which the 0 = 0 analysis is applicable

it will not be used to illustrate general principles, and

will be dealt with in a separate chapter.

Two problems must be faced at the present stage .of

development of the subject.. While it is necessary to dis-

cuss the mechanics' of stability analyses in terms of funda-

mental soil properties in order to examine their validity,

the measurement of these properties is outside the scope

of ordinary testing practice, especially in relation to the

less important jobs. 'Satisfactory approximate methods are

therefore of great practical importance. Can their

accuracy be assessed?

How should 'factor of safety' be defined, so as to

present a realistic picture to the engineer and yet avoid

inconsistencies where several alternative methods of analy-

sis can be used?

In any treatment of stability analysis in which the

pore-pressures along the failure surface are considered, two

classes of problem must be distinguished:-

(a) cases in which the pore-pressure is a function of

the state of stress in the soil at the time under

consideration;

(b) cases in which the pore-pressure is independent of

the state of stress in the soil.

Glass (a) includes the cases in which excess pore-

pressures are set up either in foundation strata or bank fill

during construction due to the low permeability of the soil,

and the case of rapid drawJwith a compressible fill. In

the first two cases the 0 = 0, or c, 0 analysis with respect

to total stresses may be used (depending on whether or not

the soil is saturated), and in .the case of rapid draw down

consolidated undrained tests are sometimes used. In all

three procedures the excess pore-pressure is not determined

explicitly, but is implied in the shear strength characteris-

tics used. •

- 41 -

These procedures cannot be considered as rigorous,

however, as the location of the failure surface is not

determined by the angle of shearing resistance, but by the

true angle of internal friction, which is not taken into

consideration. Field evidence (Skempton and Golder 19481

Ladling and Odenstad 1950)i indicate that, in spite of this

limitation, a reliable estimate of stability is given in the

limiting case (when the factor of safety = 1) for saturated

clays where Ou O.

A rigorous analysis, indeed, raises two major diffi-

culties. In a plane strain problem the pore-pressure due

to the application of stresses can only be determined if the

changes in major and minor principal stresses are known, or

alternatively the major principal stress and the state of

shear. In relation to limit design this will be so if the

limiting condition has in fact been reached, and if the

boundary conditions correspond to one of the known solutions

by plastic theory (unless a step by step elastic-plastic

solution is used). In design problems where the factor of

safety is greater than unity a wholly 'plastic' solution is

in any case inadmissible. In practice those difficulties

are avoided by assuming that, as far as its effect on pore-

pressure is concerned, the state of stress corresponds either

to no lateral yield (Bureau of Reclamation, Hilf 1948, 1951)

or to complete failure (Clarke 1948, Glynn 1948). Further;

the major principal stress (total) is taken to be equal to

the vertical head of soil above the point considered. It

will be seen from Fig. 3:16 that this latter approximation

. is within ± 15 per cent for most of the central zone of a

dam, where such pore-pressures are likely to be high.

Class (b) includes those cases in which the pore-pres-

sures are due to steady seepage and are determined by the

boundary conditions and permeability characteristics of the

dam; and the case in which partial or total submergence of

the bank causes pore-pressures with no flow. The case of

rapid draw down with an 'incompressible' fill is included in

this category by Terzaghi-(1948), Reinius (1948) and otherp,

but the implied neglect of the tendency of a compact fill of

sand or gravel to dilate under shear requires justification.

The following discussion of the mechanics of the circu-

lar arc analysis is presented in relation to the requirements

of these two classes of problem. It will be given first in

terms of total stresses, and then in terms of effective ,.

stresses. The particular problems will then be discusses.

in detail.

CHAPTER 5

The Mechanics of the Circular Arc Analysis.

x 0

/

/ RADIUS

(1) In relation to total stresses.

Consider the equilibrium of the mass of soil bounded

by the circular arc ABM, of radius R and centre at 0 (Fig.

5:1). In the case where no external forces act on the

surface of the dam, equilibrium must exist between the

weight of the soil above ABCD, and the resultant of the

total foces acting on ABCD. 141

If s is the available shear strength on the element

of surface BC (of length 1), the factor of safety F

may be defined as

F = s sm

where sm is the shear strength required for equilibrium.

Hence for a unit width of slip, we have

S = sm = . Ail, F

Since the line of action of all the normal forces

such as P passes through 0, they may be eliminated by

taking moments about 0, and a direct relationship bo-bw..-r,

the shear forces and the gravity forces on the soil mass

is obtained:-

4w.x S.R.

i .e . W x= zs 1R F

where Tr is the weight of the soil above ABCD, and x the horizontal distance of its centre of gravity

from 0. This may be written:-

. F R 2-s.1 5.4 7,7 3E

Now in general the shear strength of the soil can be

related to the total normal stress by Coulomb's equation:-

s = CA- o- n tan 0 .5:5

5-1.

5.2

5.3

- 45 -

Here cr

P and hence it follows that:-

R . 1(c 1 P tan 0) W 5:6

In other words, the factor of safety can be calculated

without knowing or assuming the distribution of normal stress

along the slip surface only if 0 = 0 with respect to total

stresses, when equation 5:6 may be written

F - _R 5:7 N -5c-

This is the basic equation of the 10 = 0' analysis, which

is supported by much field evidence in cases where F = 1.

Its use is limited to saturated soils, the shear strength

being measured under undrained conditions(x); and though

not rigorous, inthEtit is not concerned with fundamental

shear strength characteristics, it has the advantage of only

requiring the simplest testing procedure.

If 0 =4 0 with respect to total stresses, but the

conditions of construction are such no drainage can be assumed,

and conditions in an undrained test correspond to those in

the fill, then this analysis may be extended as below.

Consider the equilibrium of a vertical slice of the

bank above. BC, with respect to total stresses.

(x) Equation 5:7 then gives the factor of safety in the absence of further consolidation or softening.

Let En , En4.1 denote the resultants of the horizontal forces

on the sections n and n + 1 respectively, and Xn' Xn+1 denote the vertical shear forces.

Then, resolving in a direction normal to the surface

BC, we obtain an expression for P, i.e., P = ryi + x xn+1] cos dt..- [E - E I sin 5'8 n+1.1 where -y is the angle between the normal and the vertical.

Hence, from equation 5:6, the value of F is obtained:- F = R ,[c 1 + tan 0. Vticos 6-- tan 0 (En - En+1) sin (5,-

+ tan 0 (Xn Xn+i ) cos

5:9 Since the forces between two slices act in equal and

opposite directions on the adjacent slices, and. since there

are no external forces,

2 (En - En+1) = 0

and 1(Cn ;1+1 ) 0 5.10

So if tan 0 is a constant along ABCD and if (-Lis a constant,

equation 5:9_ reduces to

cos ck]

F R [cl W tan 0. 5.11 "r

In general, however, the arc does not reduce to a straight line and. therefore not a constant; and. in zoned fills 0 may have a significant variation along the arc.

A rigorous analysis would therefore require the evaluation

-47-

of En and. Xn on each section. It can be seen, however,

thqt they are statically indeterminate.

Resolving tangentially, the equilibrium of the slice

requires that:-

[W + Zn Xn+1] sin as+ [En - En+1] cos C_31/4

and from equation 5:5

S 11 (c]. + P tan 0)

i.e. S = el +. tan (111/ +

Xn Xn+i) copc-(En - En÷i )sin&

5.13 Eliminating S, we obtain

, cl tan 0. cosCL) (En -- En+1) = (W + - Xn+1)(sinot F

cos t an . sin c\L

5.14 Taking moments about the mid point of the base of the slice,

through which the resultant of the external forces may be

assumed to act,

(Xn + Xn -1 + = En • Yn En44 • Yn+1 .5:15

where yn and yn.o are the vertical heights of the lines of

action of En and En+1 recpectively above the mid-point of

the base BO.

Two courses are open to the engineer, either to

5.12

2

-48-

make same reasonable(x)assumption about En and Xn which satis-

fies equations 5:14 and 5:15 and use equations 5:9, or to

assume that the term

[tan 0 (En En+1) sin ck- tan 0 (Xn - X ) cos- n+1

5.16

may be neglected without serious loss of accuracy. The

work of Beichmann (1937)2 , Krey (1936)2 and Taylor (1948),

indicates that the loss in accuracy is not likely to exceed

10 per cent, and may be less. On the other hand, the

labour of calculating the E and X forces is such that very

few trial circles could be attemptbd, and failure to locate

the most dangerous circle can cause a very much larger error

than 10 per cent.

Thus, for practical purposes equation 5:11 ,may be used.

It should be noted that in using this equation, we are not

neglecting the forces between the slices, as is often stated.

That would be physically meaningless. All that is implied

is that their effect on the magnitude of the resultant

restoring moment may be neglected under certain circumstances.

If external forces are present, or if a two circle analysis

is used,this assumption must be re-examined.

(x) i.e. 141 1› chn + En tan 0 where h is the vertical height of the section, and tension avoided wherever possible.

Beichmann and Krey did not include the effect of pore-pressures or seepage forces in their analysis, and the validity of their conclusions is therefore limited to this particular case.

— 49 —

(2) In relation to effective Stresses.

So far the analysis has been considered

total stresses, and corresponds, in the case

soils, to the simplest laboratory technique.

To obtain the corresponding expressions

in terms of

of impervious

in terms of

effective stresses, the average pore-pressure along BC

may be taken as u.

Now the total normal stress on BC is

-- I o- n

Hence the effective normal stress is given by

5:17 If c' and 0' are the cohesion and angle of shearing

resistance of the soil with respect to effective stresses

s = o' - u) tan 0'

Taking moments for the equilibrium of the

ABCD we obtain

soil above

R

5.18

*11 (p- al) tan O1'] W

P is given by equation 5:8 in terms of the total forces

acting on 'the slice, and may be eliminated to give:-

1. • • = R 5.1 9

tan 0' EY cosci-L4H. tan 01 (En- En+i) sing "le

tan 0' (Xn Xn+i) cosai

5.20

CI L tan 0/2. yb h cos 0,(1 - u ih co

5:23

- 50 -

If we neglect the term

tan 0' (En - En+i) sin Cit. - tan 0' Xn+1) cos Cv-

5.21 and. put W =1. b. h. where h is the average height of the

slice, and ''the bulk density of the soil

and 1 = b 2 CO

then equation 5:20 may be written:-

F = v7R3a. I[.?1 1 4. tan 01 .,0

b h cosA- u.b 1]

5:22

Further, if L is the total length of the slip surface and

average values of c' and 01 are used, we have

F 177 3E

Attention is drawn to the term expressing the effect

of the pore-pressure. It should be noted that for uniform

soils equation 5:23 closely approaches the rigorous solution

as the arc approaches a plane slip surface. It is equiva-

lent to that used by the Bureau of Reclamation (Daehn and

Hilf, 1951) and agrees with Taylor's second interpretation

of the Fellenius assumption (Taylor, 1948). The pore

pressure term is consistent with that used by Terzaghi

(19L3, 1948) in his consideration of slopes and the. effect

of_ seepage on earth pressure and by Fellenius, 1936.

- 51 -

Earlier (Terzaghi 1936), he had introduced the effect of

pore-pressure due to seepage by reducing the weight of the

slice by an uplift equal to the pore-pressure, and resolving

the reduced weight normal to the slip surface to obtain the

effective normal stress. This leads to the expression

F R W

+ tan Or21b. h., cos CL(I u )

5:24

This method has been followed by Cedergren (1940), and Golder

and Ward (1950). It is difficult to justify it logically,

as in the limiting case of d-being constant it is obviously

incorrect, and the errors, which become considerable as Ct-

increases0 are not on the safe side(x) As an empirical

method it has the merit of being slightly simpler, and of

its error tending in some cases to cancel the conservative

approximation in equation 5:23. These factors are, however,

considered to weigh,lightly against the lack of generality

of the method.

( 3 ) The Effect of External Forces.

The most important external force to be considered in

the case of earth dams is water pressure on the face of the •

(x) This point is illustrated by Fig. 5:2, taken from the stability analysis for the Daer Reservoir Embankment, where the relationship of factor of safety and pore- pressure is plotted. Values of ud/N,h of from 40 to 50 per cent are typical both of construction pore-pres-sures and those set up by rapid draw-down. The use of equation 5:24 would lead to an over-estimate of the factor of safety by 10 to 20 per cent in these cases.

0 20 40 60 BO O/ o

100

3 .0

2' 5

2.0

FACTOR OF

SAFETY

1.5

1;0

O.5

. .

N

N._

\

N. N

N

'

------\\

X %N.

..\ FROM EQUATION 5:24

FROM EQUATION / 5:23

N \X \

s\X X

N

U, h

FIG. 5%2. INFLUENCE OF METHOD OF ANALYSIS

ON CALCULATED FACTOR OF SAFTEY.

DAER RESERVOIR— SLIP CIRCLE 4.

-52—

dam. Although it is usually appropriate to consider the

consequences of water pressure in terms of effective stresses

in the case of a saturated soil stressed under conditions

for which the 0 = 0 analysis is valid tho analysis may still

be made in terms of total stresses.

(a) The Effect of External Water Pressure in Terms of Tetal

Stresses.

As the conditions of drainage are such that the soil has

zero angle of shearing resistance with respect to total

stresses, the presence of water pressure on CDE (Fig. 5:3)

has no effect on the strength; and hence the restoring

moment due to shear strength

R r-

7 4,c1 5:25

The total disturbing moment is now W 7 (whore W is total weight of soil -in the section ABODE, and 1 the horizontal

distance of its centre of gravity fiiom 0)1 less the moment of

the water pressure on CDE about 0. Now if we imagine - a

section of water bounded by a free surface at BE and outlined

by BODE, and similarly take moments about 01 - the normal

forces on the a