FRY.

· TA

1

A Critical State

Soil Model

For Cyclic Loading

).P. CARTER

,.R. BOOKER

and

;.P. WROTH

. tl4956 ..

N0.6

2 ch Rep,ort No. CE6 'cember, 1979

TA I

. v tt cr f:Jb l�b

Ill �i Ill iilllllllllllllllll ��Ill �Ill� II 3 4067 03257 6166

CIVIL ENGINEERING RESEARCH REPORTS

This report is one of a continuing series of Research Reports published by the Department of Civil Engineering at the University of Queensland. This Department also publishes a continuing series of Bulletins. Lists of recently published titles in both of these series are provided inside the back cover of this report. Requests for copies of any of these documents should be addressed to the Departmental Secretary.

The interpretations and opinions expressed herein are solely those of the author(s). Considerable care has been taken to ensure the accuracy of the material presented. Nevertheless, responsibility for the use of this material rests with the user.

Department of Civil Engineering, University of Queensland, St Lucia, Q 4067, Australia, (Te1:(07) 377·3342, Telex UNIVQLD AA40315]

A CRITICAL STATE SOI L MODEL

FOR CYCLIC LOADING

by

J.P. Carter, BE, PhD, Syd, MIE Aust, AMICE,

Lecturer in Civil Engineering, University of Queensland,

J.R. Booker, BSc, PhD, Syd, Reader in Civil Engineering, University of Sydney,

and

C.P. Wroth, MA, PhD, Cantab, MICE, C Eng, Professor of Engineering Science, University of Oxford.

RESEARCH REPORT NO. CE 6 Department of Civil Engineering

University of Queensland

October, 1979

Synopsis

Recently, several sophisticated constitutive models have been proposed to predict the behaviour of soils under cyclic loading. In this paper the conoepts of the critical state soil mechanics have been used to develop a simple model which predicts many aspects of clays under repeated loading. The model employs the parameters that are usually associated with the Cam-clay family of models together with an additional parameter which characterises the cyclic behaviour. This parameter can conveniently be determined by performing cyclic triaxial tests under undrained conditions.

The behaviour of soils which are either, initially normally or initially overconsolidated is investigated for stress controlled and strain controlled loadings in the triaxial test. The results of this theoretical investigation show encouraging agreement with the results of laboratory tests on saturated clays, e.g. Taylor and Bacchus, 1969; Andersen, 1975, 1976.

m lU

c ··�

1 . INTRODUCTION

2. THEORETICAL DEVELOPMENT

Modified Cam Clay

CONTENTS

2.1

2.2 A Model for Cyclic Loading

1

2

2

8

3. PREDICTION OF THE BEHAVIOUR OF NORMALLY CONSOLIDATED CLAY 14

3.1

3.2

Stress Controlled Loading Strain Controlled Loading

4. PREDICTIONS OF THE BEHAVIOUR OF OVERCONSOLIDATED CLAY

4.1 The Effect of Initial OCR on Cyclic Behaviour

5. EXPERIMENTAL DETERMINATION OF THE MODEL PARAMETERS

6 . COMPARISON O F PREDICTIONS WITH EXPERIMENTAL RESULTS

6 .1

6 .2

Tests of Taylor and Bacchus

Tests on Drammen Clay

7. SUGGESTIONS F O R FUTURE RESEARCH

8. CONCLUSIONS

ACKNOWLEDGEMENTS

APPENDIX A NOMENCLATURE

APPENDIX B REFERENCES

15

20

32

34

39

41

41

45

48

49

50

5 1

52

1. INTRODUCTION

1.

A problem of considerable importance in geotechnical engineering

concerns the prediction of the behaviour of soils under repeated loading.

The necessity of understanding the response of soil under earthquake condit­

ions has long been appreciated, but more recently the problems of offshore

technology have accentuated the need for adequate descriptions of this

aspect of soil behaviour. Highway engineers have also been interested in

the response of soil and pavement materials to repeated loads of the type

caused by rolling vehicles, and testing of these materials under simulated

loading conditions has been carried out in the laboratory, e.g. Monismith

et al, 1975.

There exists a considerable body of data on the behaviour of sands under

cyclic loading conditions (e.g. Seed and Lee, 1966: Lee and Seed, 1967;

Pyke, 1973; Seed, 1979) and engineering theories have been developed for

particular classes of problems (e.g. Martin et al, 1970; Seed et al, 1975;

Seed and Booker, 1977; Rahman et al, 1977).

Recently, data for the behaviour of clays under cyclic loading have been

obtained, e.g. Seed et al, 1955; Taylor et al, 1965, 1969; Theirs and Seed,

1969; Sangrey et al, 1969; Wilson and Greenwood, 1974; Brown and Snaith, 1974;

Brown et al, 1975; Andersen, 1975, 1976; Lewin, 1978; van Eekelen and Potts,l978.

Although the conclusions of these examinations are not unanimous, several facts

emerge. The most important of these is that under undrained loading excess pore

pressures are generated and if cyclic loading is continued for a sufficiently

long time a failure or critical state condition may be reached.

A natural consequence of this interest in cyclic loading has been the

attempt to develop constitutive models to predict this type of behaviour

(Mroz et al, 1978, Prevost, 1977,1978). Generally, these models are complex,

involving nested yield surfaces and both kinematic and isotropic hardening,

and depend on the specification of a number of parameters. There seems to

2.

be no straightforward way of determining values for these parameters

directly and this places a severe limitation on the use of these models in

practical situations. A less complicated model , which is potentially

applicable to cyclic loading, has been suggested by Pender (1977,1978).

In this paper the concepts of critical state soil mechanics (Schofield

and Wroth, 1968), whose models sucessfully describe the behaviour of soil

under monotonic loading, have been extended to provide a description of the

response of clay to cyclic loading . This new model predicts the generation

of excess pore pressures and ultimate failure of the soil under repeated,

undrained loading conditions . It requires the specification of only one

addit ional soil parameter which can be conveniently determined from the

number of cycles to failure in an undrained stress controlled triaxial test .

2. THEORETICAL DEVELOPMENT

2.1 Modi fied Cam Clay

In order to clarify the presentation,some of the essential features of

critical state soil mechanics will be summarised. In particular, the theory

will be developed in terms of the modified Cam-clay soil model (Roscoe and

Burland, 1968) and attention will largely be restricted to triaxial conditions.

The extension to three dimensional conditions, using a von Mises failure

condition· is self-evident, and the extension to more general cases, e.g. the

Mohr-Coulomb failure condition is straightforward.

The state of effective stress of a soil specimen will be expressed in

terms of the stress invariants p' and q defined by

p' (1)

q (2)

3.

are the principal effective stress component s .

Under triaxial conditions, where it is assumed that 0 21 = o 31 , these

quantities reduce to

p' (3)

q (4a)

In the above description the subscripts l and 3 refer to the major and minor

principal stresses respectively (compression positive). When presenting

the results of triaxial tests it is often convenient to distinguish between

so called compression and extension. For this reason a stress difference

q*

is defined as

q* a' - o' z r (J - (J

z r (4b)

where oz' and or' are the axial and radial components of effective stress

respectively. The total stress or

is equal to the cell pressure in a

conventional triaxial test and Oz is equal to the total axial stress. Under

compression cond itions 01 = Oz , o3 = or

and *

q

while under extension conditions o1 = or

, 03

quantity.

is a positive quantity;

o and z q *

is a negative

The convenient measures of st rain for triaxial conditions are v ,

the volume strain, and E , a measure of octahedral shear strain, given by

v (5)

E (6)

where £1 and £3 are the major and minor principal strains, respectively.

Only saturated clays will be dealt with here and the symbol e is used as

usual to denote the voids ratio. With compressive strains taken :as positive the

incremental volume strain dv is related to the change in voids ratio de by

dv de

l+e (7)

4.

The modified Cam-clay model requires the spec ification of five parameters ,

values of which may be readily obtained from standard oedometer and triaxial

compression tests. These parameters are:-

>. the gradient of the normal consolidation line in e- �n p' space,

K the gradient of the swelling and recompression line in e - �n p'

space

e a value of voids ratio which locates the consolidation lines in cs

e - �n p' space, conveniently taken as the value of e at unit p'

on the critical state line,

M the value of the stres s ratio q/p' at the critical state condition;

M is related to �·. the angle of friction obtained in triaxial

compression tests, by

M = 6 sin � '

3- sin�'

G the elastic shear modulus .

(8)

For states of stress within the current yield surface the soil responds

elastically and the incremental effective stress-strain law may be written as

where the bulk modulus K is gi ven by

K (1 + e)p' K

and the shear modulus G is constant.

(9)

(10)

Yielding of the material occurs whenever the stresses· satisfy the following

criterion

0 (11)

where Pc' is a hardening parameter - analogous to a preconsolidation pressure

- which defines the non-zero intersection of the current ellipsoidal yield

q

Fig.l

5.

Critical state line

Elliptical yield surface

pi y

I (a)

Some aspects of the modified Cam-clay model for triaxial conditions

p'

p'

6 .

locus and t h e p' axis i n effective stress space - see Fig.l. Plastic flow

is determined by an associated flow rule and the permanent volume strain

dvP is related to the change in the hardening parameter Pc' as follows

dvP = (A- K)

1 + e (12)

Types of loading can be categorised in terms of a variable p� , defined as

p ' y

(13)

Equation (13) is also the locus of an ellipse in p'- q space which passes

through the current stress point and the origin, and is c entred on the p'

axis, i.e. it has the same shape as the yield locus - see Fig.2. This

variable p� is the (non-zero) value of p' at which the ellipse cuts the

p'-axis and is a convenient way of comparing the current stress state with

the current yield locus represented by p' c

The material is elastic whenever p�(P� and during the elastic deformation

dpc

Pc ' 0 (14)

The material behaves plastically whenever p; = pc' and three conditions can be

identified. These are (a) the material hardens whenever dp; = dpc' > o, (b) the

material softens whenever dpy

' = dpc' < o, and (c) 'neutral loading', w hen the

yield locus does not change while plastic behaviour occurs, dp; = dpc

' = 0.

Condition (a) requires p' > pc'/2, i .e . the material is said to be 'wet' of

critical, and (b) requires p' < pc

' /2, i.e. the material is said to be'dry1of critical

During plastic behaviour the yield locus changes according to the law

� p' y

(15)

The incremental stress-strain relation during yielding may be shown to be

(::) C12J ·

l dp ') c22 dq

where the compliance coefficients are given by

(16)

q

7.

.-·-�', /,...-- I ' . ...... ' 'I ',n .\ ' . '

\ p' y

• Curr�nt str�ss state

,...----...... Current yield surface

. ......- ·-..... C urr�nt "loading" surface

_, ..... --, New "loading" surfac�

I Elastic "loading"- Pc constant

D Elastic "unloading"- Pc d�cr�ases

Fig. 2 The yield surface and the "loading"

surface in p' - q space

p'

8.

(A- K) a K 1 en p' + (l +e) p' l+e

cl2 c21 (A- K) (1- a)

1 + e p'

(A- K) b 1 c22 p' +

3G 1 + e

and

M2 - n2 a M�

b 4T]2 �

T] the stress ratio q/p'

As would be expected, the relation (16) breaks down when the soil reaches

the critical state condition T] = M .

2.2 A Model for Cyclic Loading

The modified Cam-clay model has been shown to match well the observed

behaviour of insensitive clays subjected to monotonic loading for which the

stress level increases, and in particular, was used for the successful

prediction of the performance of the M.I.T. Trial Embankment (Wroth, 1977).

However, the predictions are not as satisfactory when the soil undergoes

repeated loading.

When saturated clay is unloaded and then reloaded it is found that

permanent strains occur earlier than predicted by the Cam-clay model. One

way of interpreting this real behaviour is to assume that the position and

perhaps the shape of the yield surface have been affected in some way by the

elastic unloading.

For the sake of simplicity in developing a new model it is assumed that

the form of the yield surface remains unchanged but that its size has been

reduced in an isotropic manner by the elastic unloading. This then can only

9.

mean that the hardening parameter Pc' has been reduced by this process.

In order to specify how this reduction occurs a relation is proposed between

the hardening parameter p' c and the loading parameter p' . In view of

y equation (14) it seems reasonable to postulate that when the material is

elastic (py

' < pc') and when dp; < 0, the following relation holds

(17a)

If 8 takes a value of unity, then the yield surface would shrink back in

such a way that the stress state always lay on it. It is to be expected that

the yield surface will recede only a fraction of this amount and the values

of 8 will tend to be quite small .

If, however , the material is elastic, but dpy

' � 0 , it is postulated

that the current yield surface is not changed , i.e.

0 (17b)

The distinction between these two types of behaviour is shown schematically

in Fig.2.

The mode of behaviour can be illustrated by considering a simple example

of isotropic effective stress change so that py

' is always equal to p'.

Suppose that a clay specimen is isotropically normally consolidated to a mean

effective stress of p' = Pc' = o0 and is subsequently allowed to swell

elastically by reducing the mean effective pressure to a value p' = a1

During the swelling, equation (17a) predicts that the value of pc' is reduced

to the value a1 8

p' = a (-l c a a0

If the specimen is then reconsolidated p' c remain unchanged and the material will behave elastically until

Cl 8 P' = Pc' = 0.0 Ccf) If loading is continued indefinitely the material will

0 deform plastically and thereafter Pc

' will be equal to p' . This means

that in a laboratory test, yielding of the soil will be observed during

will

10 .

reloading at a value of isotropic pressure which is smaller than the actual

preconsolidation pressure. Hence for this isotropic test the measured

overconsolidation ratio will be

OCR

a l-6

(....£) a,

This is less than the value given by the conventional definit ion , i.e.

OCR

(16)

(19)

The magnitude of this effect (usually quite small for one cycle) will depend

an the value of 6 , which can be thought of as an OCR degradation parameter.

For example, consider a specimen hav ing a value of 6 = 0.05, which is

initially normally consolidated under a mean effective pressure of a = 100 a

units. This mean effective pressure is then reduced to a1 = So units and

subsequently increased. Yielding will occur as soon as the pressure reaches

96.6 units again. When the sample has a mean effective stress of 50 units

the OCR given by the conventional definition is equal to 100 7 50 2

the value of OCR inferred from the behaviour on reloading is equal to

96.6 7 so 1.93.

In the modified Cam-clay model the positions of the normal consolidation

or "A l ines " in e-�np' space are assumed to be uniquely determined for

any clay by the value of the stress ratio n However, a consequence of

the new model is that these A lines "migrate11 with " el astic unloading .. and

so the position at any time is a function, not only of the current stress

ratio but also of the stress history. This feature can also be ill ustrated

by a simple example.

The behaviour of both models under repeated consolidation and swelling

at constant stress ratio n ' between the limits p' = p� and p' = p� ' is shown schematically in Figs.3 and 4. In modified Cam-clay (Fig.3) the

11.

L---------�x�l -----------------��--------�� loge p' p� p�

Fig.3 Repeated consolidation and swelling of modifed Cam-clay at constant stress ratio

12 •

• •

Dllnsification oftu n cycltZS

L----------x------------�----�------------------ log� p' p� P'o P'e

Fig.4 Repeated consolidation and swelling of

cyc l ic Cam-clay at constant stress ratio

13.

yield locus does not change during elastic swelling, and so after the

initial normal consolidation to p' = p� the materi al is always elastic

in this test. As a result the path in e - 9,n p' space varies continuously

along a "K line" between points Ai and Bi of Fig.3, i.e. the voids ratio

oscillates about some mean. In the new model a shrinkage of the yield

surface is predicted during each period of elastic swelling or "unloading",

as explained above. Hence, on reconsolidation in each cycle the material will

yield at some value of p' = p' less than p' ' iae. at points D2,D3, ... ,Dn D B in Fig.4. Movement from D. to B. down a "A line " implies some irreversible l l

or plastic volume change, so that the average voids ratio of the material is

reduced with each cycle of this type, i.e. the material becomes more dense.

As a result the normal consolidation or A line is seen to migrate. In

contrast to this, the A l ine of Fig.3 remains fixed. In both models the

A lines corresponding to different stress ratios will always remain parallel

to each other in e - 9-n p' space.

The amount of densification per cycle predicted by the new model will

depend on the value of the degradation parameter 6 . It is emphasised again

that for most soils 6 will be small so that the shift in any one cycle is

likely to be small, and thus laboratory specimens may require many such

cycles before they exhibit a measurable densification.

One of the most important features common to modified Cam-clay and the

new model is that concerned with the prediction of the undrained strength

under increasing deviator stress in a triaxial test. For both models this

strength is uniquely related to the current mean effective stress p' and

the hardening parameter Pd by

c u

where c is one half of the deviator stress at failure. It should be u

(20)

c u

14.

emphasised that the values of p' and pc

' occurring in eq. (20) are those

that exist in the sample before undrained testing proceeds. Given an initial

stress state modified Cam-clay pred icts a unique undra ined stress path in

p' -q space and a un ique failure point. However, for the new model this is

only true as long as the dev iator stres s q always increases. Thus cyclic

loading will have an influence not only on the effective stress path, but

also on the generation of excess pore pressure and the value of the undrained

strength.

It has been shown in this section that the modified ·cam-clay model and

the new soil model have many features in common. The criterion for yielding

is the same, the flow rule and the hardening law are the same, and the

incremental elastic and elastoplastic stress-strain relations are the same.

The only difference is the modification to the y ield surface associated with

"elastic unloading" (p; decreasing) . This slight modification has important

consequences to the repeated loading problem; some relevant to drained

conditions have already been discussed. Othe rs relevant to undrained condi-

tions are dealt with in the following sections .

3. PREDICTION OF THE BEHAVIOUR OF NORMALLY CONSOLIDATED CLAY

In order to illustrate the behaviour predicted by this model one set of

values for the conventional Cam-clay parameters has been selected. In this

and subsequent sections the following values have been adopted :

where c uo

;\. o.25

K 0.05

M 1.2 (¢

is the initial value of the undrained

15.

strength predicted by the modified Cam-clay model. The subs cript zero

indicates an initial value. For all calculation s in •"hich the soil i s

initially i n a normally consolidated state,the initial void s ratio i s taken

as e 0.6. 0 The effects of cyclic loading are most dramatic when the soil is loaded

in an undrained manner and so attention will be concentrated on undrained

triaxial condition s for both total stre s s and strain controlled loadings.

In all ca se s reported here the exce s s pore pressure u has been determined

from the effective stres s principle i.e. it is the difference between the

applied total stre ss and the effective stre ss. The latter ha s been calculated

using equation s (9) or (16), as appropriate, together with the constant

volume condition.

3.1 Stress Controlled Loading

The model presented in this paper involves the specification of the

additional degradation parameter 8 . In order to examine its effects,

calculations have been performed for the case of cyclic axial load at constant

cell pressure in the triaxial test. In each case loading is applied so that

the deviator stre s s *

q is varied continuously between limits of 0 and

qc ' i.e. one way compres sion loading where a 2: a with a con stant. z r r

Typical results for calculations with 6 = 0.1 and qc

= 0.75 2cuo

are

shown in Fig.S. The effective stres s path, plotted in p' - q space, is

shown in Fig.S(a). In the first half of the first cycle the yield surface

expands, i.e. the material work hardens, and the stress path is identical to

that predicted by modified Cam-clay. During the second half of the first

cycle the soil i s unloaded (q decreasing) and it responds elastically. A s

n o drainage occur s there i s no change i n p ' , however, the value of p' c

will have decreased according to equation (17a), i.e. the yield surface will

q 0 Cu

1·5 ;-

1·0 f-

0·5 -

0 1·0

Critical state N=12

r-....r---

I 1·5

'-1\.�

2·0 (a l

16.

q Cuo

N=l

1·0

0·5

p' Cuo

I I 2·5 3·0

o L__ _ __j __ __j __ __j __ � �-----L------L-----�E 0 0·05 0·10 0·15 1·0 1·5 2·0 2·5 3·0

Fig. 5

(c) (d)

Predictions for a one-way, stress controlled, undrained triaxial test: OCR � 1 , 6 � 0.1

17.

have contracted slightly. On reloading in the second cycle the material

behaves elastically until the stress point reaches the yield locus again

thereafter the material yields, the yield surface expands, further plastic

deformations occur, the stress state migrates toward the critical state

condition and additional excess pore pressure is generated. This sequence

is repeated at each additional load cycle and ultimately, if this process is

continued, a critical state condition is reached. In every cycle there is

yielding and associated permanent strains and, in particular, during any

cycle there is an increment of permanent volume strain. Because the defer-

mation occurs at constant volume there must be a corresponding elastic

volume increase and this implies a decrease in mean effective stress, i.e.

an increase in pore pressure. The accumulation of excess pore pressure

with each cycle is plotted against mean effective stress in Fig.S(c) and

against shear strain in Fig.S(d). The relation between deviator stress and

shear strain is also shown in Fig.S(b).

For this material, which has 6 = 0.1, failure occurs on the loading

portion (q increasing) of the 12th cycle. In general the number of cycles

to failure Nf will be dependent not only on the value of 6 but also on

the cyclic load level qc Results are presented in Fig.6 for a number of

values of 6 and a range of different load levels. It can be seen that for

a given material, i.e. a particular value of e ' the number of cycles to

failure increases as the amplitude of loading is decreased. For a given

amplitude of loading the number of cycles to failure decreases as e

increases. This is as expected since a larger value of 6 implies a greater

contraction of the yield surface with elastic 11unloading11, i.e. a greater

decrease in Pd . Consequently there are greater permanent volume strains

and greater excess pore pressures generated per cycle and thus the material

In modified Cam-clay the yield surface will have remained fixed during the unloading and elastic behaviour would be predicted for all subsequent cycles and there would be no further increase in pore pressure.

Fig. 6

lR.

Number of cycles to failure Nt

Variation of the number of cycles to failure with cyclic stress amplitude qc , in a on e - way , stress

controlled, undrained, triaxial test: OCR = 1

0 "

� u .Q .. 0 L

.c. .. 01 c "' L .. Ill

'0 "' £ 0·2 0 L

'0 c :::>

0

Fig.7

19.

0·2 0·4 0·6 0·8 Cyclt: ratio N/N1

1·0

Effect of cyclic stress amplitude � on the change

in undrained strength during a one-way, stress

controlled, triaxial test

20.

will reach critical state after less cycles.

The number of cycles to failure in this particular type of test is

independent of the elastic shear modulus G . The value o f this quantity

only affects the magn itude of the shear strains.

Another important feature predicted by this model is indicated in Fig.7

where the ratio of the undrained shear strength cu , measured immediately

after the "Nth" cycle , to the original undrained strength cuo , measured

before cycling, is pl otted against the ratio N/Nf . The results show a

continual reduction in the undrained shear strength for soils subjected to

repea ted increments of devia tor st. 2ss. Each of the curve s of Fig. 7 corres-

ponds to a di fferent amplitude of cyclic deviator stress and results for

materials with G in the range 0.001 $ 8 $ 0.1 appea r to lie on either

a unique curve or in a narrow region as shown. When the soil reaches

failure after Nf cycles the final undrained shear strength is equal to one

half of the amplitude qc of the cyclic deViator stress. This effect of a

reduction in strength after cycl ic loading with increasinq number of cycles

has been observed in tests on many clays (e.g. Taylor and Bacchus, 1969;

Anderson, 1975,1976; and Brown et al, 1975).

3.2 Strain Controlled Loading

Predictions have also been made using this model for samples which are

subjected to loading in which the axial strain is controlled and the cell

pressure is maintained constant. Both one way tests involving total compre-

ssive strain only, and two way tests involving both compression and extension

strains are considered.

Typical results of a two way cyclic test are shown in Fig.8 for which

e = 0.1 and the strain E: (= the axial strain in an undrained test) is

varied continuously in the range -s s E: $ E: where £ = o.ool. c c c

q* Cuo

I·

-4

-1·

u Cuo

0·8

0·6

0·4

0-2

-0·2

Fig.S

21.

q* Cuo

I·Ol

p' Cuo

r 2·9 -0·001

_, o] (a) (b)

-1·5

u Cuo

�· p' Cuo

2·7 2·9 -O·

(c l (d)

Predictions for a two-way, strain controlled,

undrained triaxial test - first 25 cycles only: OCR = l, B = 0.1, G = 200 Cuo

. . Q .. 0 L.

.s::. .. 01 c 1111 L. .. "'

"0 " .5 E

"0 c ::::>

0

22 .

G = 200 Cuo

OCR= I

2

Fig.9

5 10 Numb�r of cycl�s, N

Variation of undrained strength with number of cycles in a two-way, strain controlled triaxial test: , c = 0.001

q Cue

2 3.

G= 200Cue

OCR =I

0·5 1·0 1·5

Critical stat� lint:

q = Mp'

p' Cue

•

•

•

e = O· l

6 =0·03

6 = 0·01

Fig.lO Line of peaks in the effective stress path in a

two-way, strain controlled, undrained triaxial

test: rc

= 0.001

24.

In the interest of clarity results for only 25 cycles have been p lotted .

Fig.B(a) shows the effecti ve stress path during the test. When the str e ss

path is parallel to the deviator stress axis, i.e. p' constant, the soil

is responding elastically. It can be seen that y ield ing occurs in each

cyc le when q is largest in both compression and extension. In this type

of strain controlled test the stress path mi grates toward the critical

state condition, o sc illating between compression and extension, with the

mean effective stress gradually reduc ing to zero, i.e. the soil liquef ies .

This trend is also illustrated in Fig.lO where the line of peaks of the

stress path is also plotted. As this r educt io n in p' occurs the excess

pore pressure is gradually increased as shown in Fig.B(c) and B(d).

Cycling in this type of test also causes reduction in the undrained

shear strength and the strength ratio cu/c

uo is shown in Fig.9 aga inst

the number of cycles N, plotted as log10

(N +1). Results are given for

three d iff erent values of e ; 0.01, 0.03, 0.1 and it c an be seen that all

of those materials undergo a rapid reduction in shear strength i.e. they

tend to liquefy, as the number of cycl es N is increased. In Fig.lO all

of the materials follow the same line of peaks but movement to any given

point on the stress path occurs in fewer cycles as the value of 9 is

increased .

In Figs.ll =d 12 the results of one way cycling in the range 0 $

and cycling. in the and -Ec Ec

two way ranges -£ $ £ $ £ T $ £ $-c c 2

£

are

comp ared fur a particular value of e ; o.l . i.e. a g i ven materi al . The

$

strength ratio is plotted against the number of cycles in Fig . ll and it can

be seen that a given amount of damage, i.e. strength reduction, in both one

way and two way tests occurs in fewer cycles as the magnitude of the cyclic

strain Ec is increased. Perhaps the most interesting feature shown by

£

this figure is that the plots for one way testing in the range 0 $ £ $ 0.001

c

u 0

... 0 '-

"0 ., .5 0 '-� 0·2 :::)

OCR = I G = 20 0Cuo 8= 0·1

25.

2 way

---- I way

- -........ 0"-·o " OS

'\ \ \

\ \ \ \ \ \ I I I

2 5 10 20 50 100 200 500 1000 2000 Number of cycles N

Fig.ll Effect of cyclic strain amplitude on the undrained strength in one and two-way triaxial tests

2 6 .

and two way testing i n the range -0 . 0005 ,; E ,; 0 . 0005 ar e almost the same ,

i.e . the damage predicte d in a two way test is about the same as that in a

one way te st of th e same overa l l amplitude . Fig . 12 shows that the lines o f

peak for one way and two w a y testing a r e about the same i f the total strain

ampl itudes are equal .

The predic t ions of the soil response in a strain contro l l ed test are ,

unlike th e str e s s co ntro l l ed t es t , very much d epend ent on the value a s s igned to the

elastic shear modulus G . Thi s fac t i s observed when comparing the response

o f a soi l with G = 200 cue

, p lotted in Fig.B , w i th that of a soil with

G = 400 cuo , plotted in Fig . l 3 . The elastically sti ffer soil exhib its a

softening in its response, i. e . a reduction in the pe ak value of q , in

fewer cycle s than does the soi l with a l ower value of G - see Fi g . l 4 , and

tha rate o f increa se of average excess pore pressure is also greater fo r the

stiffer soil - compare Figs . l 3 ( c ) , ( d ) with Figs . S ( c ) , ( d ) . A given reduction

in undrained shea r strength also occurs in fewer cycle s if the soil i s

elasti c a l l y sti ffer. This effect i s illustrated in Fig . lS .

The result presented in F ig . l 3 also exhibit other features observed in

l aboratory tests on many clays . These include hyste r e s i s of the stress-strain

behaviour and the e ffects of c yc l ic loading on the apparent shear modulus

which i s defined in Fig . l6 . I t can b e seen i n Fi g . l 3 ( b ) that the apparent

shear modulus G decrea se s as the loading is repeated and it is noted that

the rate of decrease is greater as the amplitude of the shear strain incr eases.

Thi s feature has been observed in experiment s by many workers including Taylor

and Hughe s , 1 965 ; The i r s , 1966 ; The irs and Seed , 1 9 68 ; Taylor and Bacchus ,

1969 ; Harden and Drnevich , 19 70 ; Seed and Idri ss , 1 9 7 0 and Andersen , 19 7 5 .

q C uo

2 ·5

OCR = I

a = 2 00 C uo 9 = 0 · 1

2 7 .

Crit ical state line q = M p'

2 way

- - - I way

3 · 0

F ig . l 2 E f f ec t o f cycl ic stra i n amp l i tud e o n t h e l in e o f p ea k s i n t h e e f f e c t i v e s t r e s s pa th i n o n e and two -way u ndra in ed , tr i axial t ests

q * C uo

- 1 · 0

u C uo

- o.

28 .

( a )

p' �------�--------��+-�� C uo

2 · 5

( c l

.s: C uo

u cuo

F i g . 1 3 Pr ed i c t io ns for a two -way , strain c o n t ro l l ed , undr a i n ed tr i ax i a l t e s t - f ir s t 1 5 c yc l e s o n l y : OCR � l , 6 � O . l , G � 400 C ue

q C oo

O C R = I e = 0 · 1

0 · 5 1 · 0 1 · 5

29 .

Critica l stat� I intZ q = M p 1

G 4 00 - = C oo

2 00

100

2 · 0 2 · 5 3 · 0

Fig .l4 E f f e c t of the el a s t i c shear modulus on the l ine

of peaks i n the e f f ec t i v e str e s s path i n two­

way , undra i n ed t r i ax ial t ests

p ' C uo

0 :1

� u

.Q � 0 L..

� ... � c 61 L.. .. 1ft

� � c '6 .... "C c :l

f ·O

! i

o · 0

)0 .

= 100

OC R = ! (J = O· I

2 5 10 .20 N u m be r o f c y c le s N

Fig . l 5 Effect of the el a s t i c shear mod u l u s on th e undra i ned str ength i n a two-way , s tr a i n control l ed , tr i a x i a l t e s t : l c = 0 . 00 1

3 1 .

q

B

/

A p p o r<l nt sh.zor mo d u l u s = � )( s l o p tZ of O A

Fig . l 6 D e f i n i t io n o f t h e appa r e n t s h e a r mod u l u s

3 2 .

4 . P RE DICT IONS OF THE BEH AVIOUR OF OVE RCONSOLI DAT E D CLAY

The behaviour o f a n i n i t i a l ly ove r-consolidat ed sampl e when s ub j ec ted to

r e pea ted l oad i ng may be contrasted to t h a t o f a n i n i ti a l l y norma l l y c o n s o l i -

d a t e d s o i l . I n Fig . l 7 r es u l t s are pr e s en t ed for a m a te r i a l w i th 6 � 0 . 001

wh i c h h a s been i n i t i a l l y i so t ro p i c a l l y c o n s o l idated to an e f f e c t ive s t r e s s

o f 3 . 8 5 cue and has then be e n a l lowed t o swe l l t o a m e a n e f fective s tr e s s

equ a l to 0 . 96 1 Cue , s o t h a t t h e conve n t i ona l overc o n s o l i d a t ion r a t i o i s

e qu a l t o 4 . Th e so i l h a s then been s ub j ec ted t o a c o n t i n uo u s v a r i a t i o n o f

deviator s tr e s s be twe e n the l i m i t s 0 � q � qc whe r e q c � 1 . 9 c ue under

u n d r a i n ed cond i t i on s . A l l s tr e s s l eve l s quoted here have been expr e s sed a s

mu l ti p l e s o f the undra i n e d s t r e n g t h cue , which is the value a f t e r swe l l i ng

to an OCR of 4 but prior to cyc l i c lo ad i ng .

The i n i t i a l swe l l i n g a n d th e pe r iod when q de c rea s e s in e ach c yc l e

cons t i t u te e l a s t i c " u nloadi n g " a s de f i n e d above , i . e . p� d e c r e a s i ng . Dur i n g

e a c h o f t h e s e u n l o a d i n g e v e n t s t h e yie ld su r face c o n t r a c t s un t i l even t ua l l y

the s t r e s s po i n t r e a c h e s t h e y i e l d su r fac e . Th erea fter , th ere wi l l b e pe r iods

o f p l a s t i c l o a d i n g i n e a c h cyc l e . In th i s par t i cu l a r e x ampl e the f i r s t

pl a s t i c stra i n s w e r e obse rved i n the 5 1 st c yc l e . T h u s d u r i ng t h e f i r s t SO

c y c l e s t h e m a t e r i a l r e spo nds en t i r e l y e l a s t i c a l l y ; the r e a r e n o permane n t

s t r a i n s a n d the e x c e s s po r e p r e s s u r e o sc i l l a te s be twe en 0 and A f t e r

S i c yc l e s , perma n e n t s t r a i n s o c c u r a n d i n th i s p a r t i c u l a r c a s e the m a t e r i a l

d i l a t e s and p l a s t i c a l l y so f t e n s b e c ause t h e s t r e s s s t a t e i s on t h e " d ry "

s i d e o f c r i t i c a l . S in c e the d e forma t ion is o c c u r r i n g at con s ta n t vo lume the

inc r e a s e i n p l a st ic vo l ume s t r a i n mu s t be comp e n s ated by a decrease i n e l a s t i c

vo l um e strain , i . e . the s t r e s s s t a te m i g r a t e s t owa rd s c r i t i c a l s tate a nd the

pore p r e s s u r e d e c r e a s e s . In c ommo n wi th mod i f i ed Cam- c l a y the c yc l i c mod e l

pr e d i c ts a pe ak strength i n a s t r e s s d e f i n e d te s t under c e r ta i n c i r cumstanc e s ;

h e n c e fa i l u r e may oc cur e i t h e r b y t h e s t r es s s t a t e r e a c h i n g c r i t i c a l s t a t e

3 3 .

q " C uo N == I to 50 P�ok fai l ure C uo

u

2 · 0 - \

1 ·5 -

j 1 · 0 f-.

0 ·5 �

f---0 0 · 9

0· 8 r-Cuo

0 - 6 -·

0 - 4 -

0 - 2 f-- ,

0c 9 - 0 2 '--

- 0 -4 ___:

- 0 - 6 -

1 ·0 1 - 1 ( o )

I ·

I c )

N == 73

r-y....r--

I p '

-------, Cuo 0 1 · 2 1 · 3

C uo

11---

P I 0·2 C uo

1�2 1� 3 0

Fig . l 7 Pred i c t ions for a on e-way , stre s s concrol l ed , u nd r a i ned , t r i a x i a l t e s t : OCR = 4 , 8 : 0 . 00 1 , G ::: 200 cue

(

3 4 .

or by reaching t hi s pe ak undrained s t r en g t h , w hi c he v e r o c c u r s f i r s t . In

samp l e s wh i c h a re i n i t i a l l y hi g h l y overco n so l id a ted , suc h a s the one

con s id e red he re , peak fa i l u re is l i k e l y to occur . I n c o n t ra s t , so i l s w hi c h

are s l i g ht l y overco n so l i d ated , i . e . on t he " w e t " s i d e o f c r i t i c a l , wi l l ,

a f t e r su f f i ci ent c y c l e s , be have i n t he ma n n e r o f i n i t i a l ly no rmal l y conso l i ­

d a t ed so i l s .

It is a l so a fea ture o f t hi s mod e l t ha t a l l i n i t i a l l y overco n s o l i d a t ed

soil s wi l l eventu a l l y respond to c yc l i c load ing i n t he same ma n n e r as a n

i n i t i a l l y norma l l y consolid ated so i l , a s l o n g a s t h e d e v i a tor s t r e s s q

is never great e r t han M times p ' . To i l lu s trate t his feature

consider the pred ic t i on s o f Fi g . l 8 . T he i n i t i a l va lue o f convent ion a l OCR

i s 4 • t he i n i tial value of p ' i s p� = 0 . 9 6 cuo

a n d e has a v a l u e o f

0 . 1 . The ma t e r i a l i s o t herwi s e the s ame a s tha t fo r t he p revious ex am p l e

, g iven i n Fi g . l 7 . The cyc l i c d ev i a to r stress level i n t he present c a s e i s gi ve n by

qc = 0 . 5 8 cuo ' and he nce prior to fa i l u r e q wi l l a l ways be l e ss t han p ' t i me s

t he friction constant M , i . e . be l ow t he c r i t ic a l s t a t e l eve l . As s hown i n

F i g . l8 , t he f i r st 5 2 c y c l e s a r e e l a s t i c w hi l e d u r i n g c y c l e s 5 3 t o 64 e l a s t ic

a nd p l a s t i c be ha v iour is pred ic t e d . F a i lure oc c u r s d u r i n g cyc l e 64 w he n t he

soil sample comes into a c r it i c a l s t a t e condition wi t h q = Mp ' . I n Fig . 18 C c )

i t can b e s e en t hat , e ven i n t hi s c a s e o f a n i n i t i a l l y ove r-con so l id a t e d so i l ,

t he e x c e s s po r e pr e s sur e c a n gradua l l y bui l d up a s cyc l i c lo ad i n g i s co nt i nued .

Henc e in overconsolidated soi l s t he c y c l i c s t r e s s level has a s i gn i f ic a n t

e f f e c t on t he pred ic ted r e spo n s e .

4 . 1 T he E f f e c t o f I n i t i a l OC R on Cyc l i c B e havio u r

C a l c u lat i o n s ha ve be e n pe r formed for a n umb e r o f idea l soi l s w i t h

d i f f e r e n t values o f e b u t a l l having t he s a m e co nven t i o na l overco nso l ida t io n

rat io o f 4 . B e fo r e swe l l i n g eac h so i l w a s considered to be norma l ly con so l i -

d a ted wi t h a vo i d s r a t io e = e n c = 0 . 6 a t a me a n p r e s s u r e p ' = 2 . 9 0 2 c

uo ( n . c . )

.1 5 .

q � Crit i ca l stoVl N >= I to 52 q � C u o

0 · 6 r- I N = 65

� �� "�""'" ""� \.1\

C uo

0· 6

0- 4 - 0-4

0 · 2 r- 0 ·2

p ' I C u o

1 - 0 0 0·05 0 ·10 0· 15 ( a ) ( b }

u -0 · 6

0 · 4

0 · 2 0 ·2

P I

Q

L--------L--------L------L-L

C uo 0· 4 0 · 6 0 · 8

( c , 1 · 0 0 · 0 5 0 · 10 0· 1 5

( d l

f'.ig . 1 8 P r ed i c t io n f or a o n e-w.ay , s t r e s s co n t r ol l ed ,

u nd r a i n ed , t r i a x i a l t e s t : OCR = 4 , e = 0 . 1 , G = 200 cuo

( 0 ·20

3 6 .

whe r e cue

( n . c . ) is the v a l ue of the und r a i n ed s t r e n g t h of th e so i l i n

the n orma l l y c o n s o l i d a t ed c o nd i t io n . Al l sampl e s were s ub s e qu e n t l y a l l owed

to swe l l so that they had a vo i d s ratio e0 � 0 . 66 9 and a mean pr e s s u r e

p ' = o . 7 2 5 5 cu ( n . c . l , i . e . OCR 4 . During the e l a s t ic swe l l ing the va l u e

o f p; wi l l h a v e d e c r eased a n d thu s so w i l l p ' c The new va lue o f p ' c '

a nd hence c ue the undrained s trength in the overcon s o l i da t ion cond i t ion

will depend on the va l u e o f 6 . Th i s dependence i s s hown i n Table l for

the s everal va l u e s of e considered .

e p ' c /cu ( n . c . ) cu0/cu ( n . c . )

0 2 . 902 o . 7 5 8

0 . 001 2 . 8 98 0 . 757

0 . 003 2 . 890 0. 7 5 5

0 . 01 2 . 8 6 2 0 . 7 5 0

0 . 03 2 . 784 0 . 7 3 3

0 . 1 2 . 5 2 6 0 . 6 7 8

Table l . Var i a t ion o f pc' and strength wi th 8 a t OCR = 4

Fig . l 9 shows the pred ic tion of t h e number o f c yc l e s to fa i l ure Nf

in a one way s t r e s s contro l led t e s t p i o tted aga i n s t the m a g n i tude of the

app l i e d devia tor s tre s s qc

Curve s have b e e n p l o t t ed fo r t h r ee d i f fe r e n t

ma te r i a l s corr esponding t o 6 = 0 . 001 , 0 . 01 a n d 0 . 1 . Th e trend i s t h e same

a s t h a t for norma l l y co n s o l i d a ted s o i l s , i . e . the number o f cyc l e s to

fa i l u re i n c r e a s e s a s d e c r e a s e s a nd a s e decrease s . Broken c u r v e s

h a v e a l so b e e n p l o t t e d in Fig . l 9 for soi l s wi th OCR = l and the s ame va l u e s

o f 6 . A compa r i son of the three pa i r s of curves shows that the number o f

cyc l e s t o f a i l u r e i s a l so a fu n c t i o n o f the i n i t i a l ove rconso l id a t ion ratio

of any so i l . The mode l pred i c t s tha t overconso l i d a t e d so i l� f a i l sooner ,

q, 2cuo

.n .

---- OC R = 4 - - - - O C R = I

...... .......

e 0 = 0 · 669 } G = 200 C.uo ( I'I.C . l

e 0 = 0· 600 I

....... ....... ........ .......

' ........

0 · 001 0 ______ .L._ ______ L__ ______ L__ _____ _ _j

10 ro 2 ro3 10 4 ros

Fig . l 9

Nu mber o f cyc l�s to foilurcz N 1

E f f ec t o f ini tial OCR o n tn e number o f cyc) es to f a i l ur e i n a o n e-wa y , str ess control l ed , undrained �r iaxi�l tes t

z fll L. ::J

·-c -0 .. "' � u >-u -0 L � E ::J �

0 I

F' i g . 20

3 8 .

e c. o - o 1

' '

' .....

.......

2 C o nve n t iono I

....... .......

3

Q c -- = 0 · 7 5 .2 C u o

...... ...... ...._

O C R

- ...

4

E f f e c t o f i n i t i a l OCR on t h e number of c y c l e s to f a i l u r e i n a o n e- - w o y , s t. r e s s control l eC . u nd r a i n ed tY i a x i a l t es t

3 9 .

i . e . i n few e r c y c l e s , in repe a �ed load t e s ts than do n o rm a l l y cons o l i d a t e d

sampl e s o f t h e same so i l e . g . se e F i g . 20 . Th i s pred i c � ion i s i n a g reemen t

w i th the t r en d s shown in l a bo r a to r y te s t s on D•·arnrnen c l a y e . g . 1\nd er s e n ,

1975 .

5 . EJa>ERIMEN'I'AL OETE: RM I NAT I ON OF TH E MODEL PARAJ-IET£RS

For c a l cul a t io n s unde r fu l l y d r a i n e d o r undr ained cond i ti o n s th e mod e l

requ i r e s the s pec i- f i c a t i o n o f t h e f i v e ba s i c so i l pd r a m e t e r s A, K , M , G a nd e

ln add i tion , the i n i t ial s t a t e of e f fe c t i v e s tr e s s a n d t h e i n i t ia l vo i d s

ra tio e0

a r e r equ i r ed . Va l ues f o r A a n d · K m a y b e o b ta i ned d i r e c t l y

fr om the r e sul t s o f oedome t e r or d r a i ned t r i a x i a l te s t s ; bo th tests mus t

inc l ude un l oa d ing - r e l oa d i n g cyc l e s . Th e f r ictional p a rame t e r M i s

d i r ec t l y r e l a ted t o � · , s e e equ a t ion ( 8 ) , a nd i s conve n i e n t l y de t ermined

from good qua l i ty d r a i ned t r i ax i a l te s t s . Th e e l a s t i c cons t a n t G c a n b e

dete rm ined a s one th i rd o f the grad i e n t o f t h e d ev i a tor s t r e s s - a x i a l s t rain

c urve on an u n loading por t io n of an u n d r a i ned tr i ax i a l t e s t .

I n pr i n c i p l e , i t i s po s s i b l e to d e t ermine a va l u e for e from t h e

resu l t s o f o n e un load i n g - r e l oad i n g c y c l e i n a co ns o l ida t i o n tes t . B o wever ,

i n p r a c tice i t s e em s mor e r e a s o nab l e to base the e s tima te of e on the

r e s u l t s o f a l a r g e n umb e r of c y c l e s rather tha n o n one s i n g l e cyc l e . W h i l e

i t mo y be possible to i n te r p r e t t h e resu l ts o f m a n y cyc les o f conso l id a tion

i t is proba b l y more con v e n i en t to use the resul t s of u n d ra i n ed cycl i c t e s t s .

For i n s ta n c e i t is po s s i b l e to r ep l o t t h e res u l ts of F ig . 6 , which a r e f o r a

on e way , u n d r a i n ed , s t r e s s c on t rol led comp r e s s i o n te s t in t h e t r i a K i a l appara tus ,

in the form shown i n Fi g . 2 1 . For exampl e , in a t e s t i n wh ich t h e s tr e s s l eve l q c -2- - o . s c u e

has be e n r e pea ted , fa i l u re i s i n d ica ted f o r the pa r t i c u l a r

samp l e a fter 4 5 8 cyc l e s . U s i ng F i g . 2 1 , a s s h o wn , a v a l u e o f 8 = 0 . 001 could

f' i g . 2 1

4 0 .

X == 0 - 2 5 It:. = 0 - 05

M = 1 · 2 t 0 = 0 · 6 G = 2 00 c uo

oc� - 1 0./

N umbrr of c y cl e s t o fa i lur£ N ,

D e t el'm i na t i o n o f t h � d eg r ad a t 1on pa r am e- t e r 8 f rom

a o ne -wa y , s t r e s s co n t. ro l l ed , u nd r a i n ed tr i a x i a l t e s t

4 l .

be infexxed for ch i s soi l . Alchough i t i s con ven i e n � . i t is not nece s s a r y to u s e the numbe r o f

c yc l e s t o fa i l u� e a s a mea n s of d e t e rmi n i ng 8 . I t i s poss i bl e to c a l cu l a te a nd to plot a fam i t y of cu rves similar to those in Fig . 2 1 {or the numbe r o f

cyc l e s requ i r ed for the pe rman e n t a x ia l s t r a i n t o reach some c ho se n va l u e ,

3% for e xamp l e . �l t erna tive l y , the number of c yc l e s requ i red to genera te

a g iven exce s s pore pr e s sure in a s t r a i n con t ro l l ed te s t could be u sed to

deduce a va l u e for e . These t e s t s , to d e te rmine the Vb l U e of the addi tional d e g r ad a t ion pa r am e te r , a r e s t r a i g h tforward a nd e a si l y pe r formed :i n the lab-or a -

to ry . The physi c a l s i gni f i c a n c e which c a n be a t tac hed to this pa rame t e r i s

a n adva n t age w h i c h i s l ik e l y to be apprec ia ted b y e nq i ne e r s .

6 . COMPARISON OF PREDICTIONS W ITH E XPE R I MENTAL RESULTS

6 . 1 Tes t s o f Taylor and Bacchus

Taylor and Bacchus ( 1969 ) reported che resul �s of c yc l i c t r i a x i a l te s ts

in wh i c h one h u nd red s i n u soida l s t r a i n - co n t ro l l ed cycles were a ppl i ed to

a r t i f i c i a l l y prepa red satura�ed c l a y sampl e s . The sign i f i ca n t ef fec t on

norma l l y consol i da t ed c l ay was to reduce t h e mea n e f fec t i ve s t re ss p ' by

a n amoun t which d e pe n ded on the appl i ed s t r a i � ampl i tude . The resul ts of

on e of these tests i n �h i ch the i n i t i a l OCR 1 , e : 0 . 962 a nd 0

p� = 64 lbf/i n 2 , are pl ot ted in rig . 2 2 for the ca se whe r e t he s trai n wa s

va r i ed c o n t i nuou s l y in the range -0 . 00) � £ � 0 . 00) A l so shown on th i s

pl o t a r e some pr edi c t ions made u s in g t h e new mod e l . Va l u e s lor the mod e l

pa rame te r s and the source for e a c h a r e gi v en i n Table 2 . I t can be see n tha t

t h e pr ed i c t io n s in these stra i n - co n t ro l l ed te s t s a r e very depe nde n t on the value s e l ec ted for the e l a stic shear mod u l u s G . I n both pred i c t ions the

rate o f d e c r e a s e in p ' i s ove r pred i c ted i n the L a t t e r stages o f bo th tes t s

3 -Q

0 :::>

4 2 .

Me a s ur�d ( Taylor a n d Bacc h u s , 1969 )

- - - - Pr�d ic ted ( 9 = 0 · 03 G = 92 - 5 c u0l

- - - - - - - Pr�d icl�d ( 8 = 0 - 03 G = I B 5 c u0 l

-� ....... 0.. "' "' .., L .. "' Ill Ill .., r::: .., .!!: .. u .., ..... .... .., 1·0 r::: 0 .., E 'C "' "' 0 E L 0 z

0 0

2 - wa y c y c l i c

tr ia x ial lest. E.c = 0 · 003

OC R = I

' '-

\ \

'

\ \

'\ \

\

\ \

\ \

\

3 0 1 0 0 N u m be r o f c y c h r s . N

F i g . 2 2 Compa r i so n o f mod e l pr ed ic t io n s w i t h t e s t r e s u l t s o f

T a y l o r a nd B a c c h u s { 1 9 6 9 ) f o r a two - wa y , s t r a i n c o n t r o l l ed , u nd r a i n ed , t r i ax i a l t e s t

60

N .� 40 -..... -E. CT In ., til L.. ... In L. 20 0 ... . !:! > " Q

0

4 3 .

Meo !ured ( Ta y lo r and B a c c h u s I 1 Q 6 9 ) - - - - Pred i c tj!d G .::::; 92 · 5 C u o == 2 500 t,1 I i n 2 · - · - - P r cr d ict�d G o:::: i s s C uo = 50 00 l b f I i n 2

. :;:::;;; --//

. , ! , OC R = I

I f Fo r pr�d i c t io n s

j I · I I J .

I

X = 0 · 1 3 2 /tC = 0 · 02 1

M = 1 · 5 l o = 0 - 962 p� = 64 l b1 / in 2

0 · 05 0 · 1 0 Shea r strai n , €

F i g . 23 Compa r i son of model pr ed i c t ion w i � h t e s t r esul ts o f Taylor a nd Bacc h u s ( 1 969 ) for a n u nd r a i n ed ,

mo no�o n i c , t r iax ial compr e s s ion t e s t

Parameter

K

M

G

a

Va lue

0 . 1 3 2 0 . 02 1

1 . 5

5 000 lbf/in 2

2 5 00 lbf/in 2

0 . 03

4 4 .

Sour c e

Conso l i d a t i o n plo t , F i g . 2 o f Ta ylor a nd B a c c hu s

E f fective s t r e s s s ta t e a t f a i lure , Fig . l O o f Tay l o r

and Bacchus

Average s from unload ing c u rve s .

F i g . ? of Taylor a nd B a c c h u s

E s timated

Table 2 . Parame t e r s used in predictions o f

Taylor a n d B a cchus t e s t s .

a nd po s s i b l e r e asons for this behaviour are d i s c u s sed be low . Neverthe l e s s ,

t h e mode l predic t s the correc t trend i n th i s type o f cyc l i c test .

Te st resu l t s and pr edi c t io n s a r e shown in Fig . 2 3 f o r the case of a

monotonic t r i a xi a l compr e s s ion t e s t under undra i ned cond i t ion s . I t c a n be

seen , tha t a l though the pred ic t io n s for the u l t imate d e v i a tor s t re s s a r e very

accurat e , the pred i c t e d sh e a r s t r e s s - s tr a i n r e spo n s e s a r e both too s t i f f

pr io r t o f a i l u r e . The s e pred i c tions , wh ich a r e t h e same as wou l d be

prov ided by the mod i f i e d Cam-c l a y mod e l , do not s how enough p l a s t i c s h e a r

s t r a i n a n d in f a c t over pr e d i c t the p l a s t i c vo l ume s tr a i n . As a resu l t a

g iven drop i n p ' lor i nc re a s e i n u ) is pred i c t e d in a cyc l ic te s t i n

fewer c yc l e s t h a n i s ob s erved . Both s t a t i c a nd cyc l i c t e s t s sugge s t t h a t

t h e e l l ipt i c a l y i e l d l o c u s u s ed i n t h e mod e l ( w h i c h i s iden t i c a l to the

pl a s t ic po ten t i a l because o f an a s soc i a ted flow r u l e ) is not an ac c u r a t e

r e p r e s en t a t ion o f the ac tual behaviou r . Be tter pred i c t io n s might be obta i n ed ,

for th i s pa r t i c u l a r mater i a l i f some o th e r shape is used fo r a y i e l d locu s ;

one in which pla stic shear s tra i n s a r e g r e a t e r at l ower va l u e s o f d e v i a t o r

s t r e s s q , t h a n pr ed ic t ed )Oy t h e e l l ips e . A sh ape l i k e th e o r i g i n a l C a m - c l a y y i e l d

l o c u s m i g h t be b e t t er a s l o ng a s the s i ngu l a r i ty a t t h e i s o t r o p i c a x i s i s r emov ed .

6 . 2 Tes cs on Or Amme n C l ay

4 5 .

An extensive programme of cyc l ic te s t i n g has bee n c a r ri ed ov t on

Dr ammen c l ay and tne r e s u l t s nav e been brough t toge ther in a r e po r t

pub l ished by c h e Norweg i a n Geo t e c h n ica l I n s ti tu t e ( �dersen , 1 9 7 S ) . I nc l uded in ch is program were cyc l ic t r ia � i a l tes t s during � h i c h po r e

pressure mea suremen t s we re made . The r e su l ts o f one o f these t e s t g a r e

pl o t t ed in Fig . 2 4 f o r t h e c a se i n �h ich the i n i tia l OCR = l , e 0 = 1 , 09 2 and p; = 40 t/ro2 • In this t e s t the d ev i a tor stre s s was va r i ed con t i nuou s l y

i n compression i n the ra nge 0 f. q ::::: 0 . 8 cuo Al so shown i n th i s p l o t

a r e some pred ic t ions made u s ing t h e new mod e l . Va lues lor the model pa r ame ters a n d t h e source for e a c h a r e g iven i n 'fabl e ,l . In both pred i c -

t ion s the r a t e of i n c r ease i n exc e s s po r e pressu re is overpred i c ted i n t he

l a c te r stages of the t e s t , i . e . at l a rge r c yc l e numbe rs . This me ans that

the r a te of d e c r e a s e i n p' wi l l a l so be overpred icted a t l a rger cyc l e

numbe r s and t h i s i s t h e s a m e t rend as no ted i n t h e pred ic tions o f the t es t s

of Tay lor and Bacchu s . A g a i n a possibl e expl a na t ion for the d isc r epancy be tween pred ic ted and observed re s u l t s may be the cho ice of the yie ld surface used i n the mode l . I t i s proposed to i n ve s tigate th i s m a t t e r in fu tu re

r e s e a rch work .

Parameter

K

M

G

8

Va l ue

0. 3 4

0 . 07

1667 t /m 2 l = l l 7 c )

uo

0 . 1 0 . 01

Source

I n t e rpre ta t ion of t r i a � i a l

test d a ta made by v a n E e k el e n

a nd Pot t.s ll978)

From . s ta t ic t e s t . da ta

E s t im a t ed

Tab l e J . Parame t e r s u s ed i n pred ic t ions of tests on o r ammen c l ay

N E -...

Ill II\ t!l u )(

w

30

2 5

I S

1 0

s

46 .

C r i t ic a l S tet�

For predictions : ' \ X. = 0 · 34 I I

IC. = 0 · 07 I I M = 1 · 2 3 I G = 1 1 7 C uo I I � = 1 · 09 I p = 2 · B2 C uo I

O C R = I I I

G c I 2 C u0

= O · B I I

� I

/

I /

/ / M e o s ur�d

/ / ( o f tcz r Andersen ) ./ _.,/ _..,.

- � ... �- - -- _, - - - - P r cz d l c tczd 9 = 0·0 ! -· - · -- · - P r r d i c ted 9 = 0 · !

--�--------------_L ____ ____ 1 0 100 1000

N u mb e r of c y c le s N

Fig . 2 4 Com� r i son of model pr ed i c t ions w i �h test r esu l ts

for Dr amme n clay ( after Ander sen , 1 9 7 5) - o ne - way ,

s t r e s s control l ed , u ndr a i n ed , tr i a x i a l t es �

N e -... 1 5 cr ., lA bl L.. .... In I.. 1 0 0 .... _g > lilt 0

5

0

4 7 .

O C R ==

Measu red ( af t e r A n d � rse n l

- - - - Pred ic ted

F or pre d ict ion

X = 0 · 3 4 k. = 0 · 07

M = 1 · 2 3

G = 1667 t / m2

e o = 1 · 090 j

= 4 0 t / m 2 Po

0 · 02 5 0 - 0 5 0 S h e e r strain E

Fig . 2 5 Compa r i so n of model pr ed i c t ion w i t h t e s t r esu l ts

0 · 07 5

for Dr amm e n c l ay ( a f ter A nd er se n , 197 5) - u ndrai ned , mo no to n i c , t r iax i a l compr e s sion t es t

4 8 .

Fig . 2 5 s hows the compa r i son be twe e n ob s erved a nd pred i c t e d dev i a tor

s t r e s s - s t ra i n c u r v e s fo r a no rma l l y conso l i d a ted s amp l e o f Dramroe n c l ay

sub j ec te d to undr a ined , · mo no t o n i c , txiaxia l campr·e s s i o n . W i t h t he values of so i l

par amete r s given i n Tabl e 3 i t c a n b e seen tha t the predic t ion o f t h e

i n i ti a l s t i f f n e s s a ppe a r s to be ad equa te but the pr ed i c t ion o f the u l timate

strength i s too h i gh .

7 . SUGGESTIONS FOR FUTURE RESEARCH

The r e are several mat ters wh i c h r equ i re furth e r atten t ion . Same of the

mor e impo r t a n t a re : -

( a ) From the previous s e c t ion i t i s obvious that wore work mu st be

don e to d e t e rmine a c c u r a t e l y the shape of the yie l d surfac e . H i gh qua l i ty

testing i s requi red in a d d i t i o n to the theo r e t i c a l in terpr e t a t io n . It i s

apparent from the ca lcu l a t ions pre se n ted i n th i s pape r t h a t y i e l d su r faces

wh ich give r e a so n ab l e pr ed i c t i o n s for mo no t o n i c t e s t s may n o t be s a t i s fa c to r y

f a r pred i c t ions o f cyc l i c behaviou r . Any sma l l e r ror i n the s ta t i c t e s t

predict io n s i s l i k e l y t o become a s i gn i f i c a nt e r r o r a f.te r a l a r g e n u mb e r o f

load repe t i t ions . Al though the e l l i p t i c a l farm of t h e mod i f i e d Cam- c l ay

y i e l d l o c us h a s been used h e r e , it is a l so po s s i b l e to adopt a l t e r n a t ive

shape s .

(b ) Fur th e r compu t a t i o n s shou l d be performed to d e t e rmine t h e s i gn i ­

f i c a n c e o f t h e type o f " loadin g s u r f a c e " adopt ed i n t h e mod e l . S p e c i a l l y

d e s i g n e d l a bo r a tory t e s t s may prov i d e i n fo rma t ion th a t w i l l h e l p i n the

choice o f the best shape .

( c ) The pr e s e n t mode l a s sumes tha t the e l a s t i c shea r mod u l u s G for

the so i l is con s ta n t . Hou l sby ( 1 97 9 ) has sugges ted that more acc u r a te

pr ed ic t io n s might be obta i n e d i f t h e va l u e of G i s tak e n as a f u n c t i o n o f

49 .

pc' . so tha t �s t he preco nso lidation p� e s s u r e is increasM then the soil becomes e l a s � i ca l l y s c i e!er in $hear . Thi s sma l l a d J u s cm e n t i n the deta i l s o f che model wi l l enabl e h y s t�r e s i s t o b e more c los e l y predic ted .

( d ) when che po i n t s (a l to < c l have been r e so l ved i t wi l l chen be

possib le co ex tend the model co �ore gener a l stre s s cond i tions .

I t i s the i n �e n tion of the au thors to include an i nves t i ga t ion of these

above me n t ioned po i n t s in fu ture r e search .

9 . CONCLUS IONS

A so i l mod e l , ca pa b l e o f pred i cting ma ny o f the obser v ed fea tu r e s of the

behaviour of c l ay wh e n subj ected to r epea t ed load ing , has be e n presented . The m od e l po s s e s s e s m o s t o f the c ha rac t er is tics o f the form e r c r i t i c a l sta t e

mod el s but w i t h a s impl e , y e t impo r t a nt mod i f ic a t ion . Th i s i nvol v e s a

spec i f ied con trac t io n of t h e y i e l d su r fa c e as the so i l sampl e is u nloaded

{wi th th e d ef i n i t ion of u nload i ng a s g iv en above) . Wi th the �ntroduction o f th i s mod i f i c a t i o n a n add i t io n a l parameter mu s t a l so b e de f i ned . I t has

been s hown cha t a v a l u e for th i s pa rameter may be de ter m i n ed , in a s tra igh t ­

forward ma nner , from a labora tory t r i a x i a l test i nvol v i nq re�a ted , u nd r a i n ed

lo ad i ng .

Cal c u l a t ions have been mad e u s i ng chis mod el a nd th e r esu l t s hav e been presen t ed i n pa r ame t r ic form . Th e pr ed i c t ions exh ibi t mo ny o f the s ame

t r e nd s tha t have been observed in l a bo r a tory t e s t s i nvo l v i ng the r epea t ed

load i ng o f satu r a t ed c l ays . I n add i t io n , pred ic t ions have been made o f the

beha v iour o f two pa rticu lar clays and th e r e su l t s have bee n compar ed w i th

the actual tes t resul ts . R easonabl e agreement was fou nd betwe e n the mea su r ed

and pred ic ted b ehaviour .

5 0 .

As a r esul t of the parametric study a nd the pr ed ic t io n s of l a boratory

behav iour , some gugg e s t i o n s fo r futur e r esearch have been mad e . The mo st

important o f these is c o nc er n ed w i th the need for an accura t e d e t er m i na t io n

o f the yield sur fac e a nd p l a s t i c potentia l , f o r a n y p a r t i c u l a r c l ay , under

cond it i o n s of mono ton i c load i n g • . It is s ugge s ted t ha t t h e s hap e o f t hi s

sur fac e mu st b e known i n some d e t a i l b efore good qua l i ty pr ed i c t io ns c a n

be expec ted f o r t h e behaviour o f th e same s o i l u nd e r repeat ed l o a d i ng .

It should be empha s i s ed th a t t h e model d e s c r ibed in th i s pa per c annot

be expected to r eprodu c e accura tely a l l fea ture s of t h e behaviour o f a

r eal c l ay u nd er mono tonic and cyc l ic l o ad ing . I nd e ed , it is b e l i eved tha t

no mathema t ical model , tha t can be u s ed sens i b l y a nd eco nom ic a l l y for

d e s ign c a l c u l a t i o n s , i s l ikely to ach i eve this mod est a im . The ph ilo sophy

behind this work has been the need to d eve lop a s s impl e a fami l y of mod e l s

as po s s ible t ha t r eproduc e �Jal i ta t i v e l y th e ga l i ent f ea t ur e s of c yc l ic

behaviour of s o i l s , and t ha t are e�pr e s s ed in terms of s o i l parameters that

have phy s ical meaning a nd wh i c h can be e a s i l y m e a s ur ed i n c o nventional

l abo r a tory test s .

ACKNOWLEDGEMENTS

The �utho r s wish to acknowl edge that this work was produc e d on a S c i e n c e

Re s e arch Counc i l con trac t wi th t h e Un ive r s i ty o f Cambr idge , w h i c h p rovided

financi a l support for J . P . Carter as a re search a s s i s tant and s u ppor t for

J. R� Book e r a s a visito r .

The authors a re g rate fu l to C . M . S z a lw i n s k i for th e provi s i o n of comp u t e r

graph i cs r o u t i ne s .

A P P ENU I X A

e

e.c s

G

K

M

N

Nf OC R

p

p '

Pe:'

Pc�

Po'

Py'

q v

n e

0� Oz' a 1 ' , a 2 ', o l'

S L

NOHENCLA'l'UR.E:

Undra ined shear s t r e n9th

I n i t i a l va lue of und ra i n ed shea � s t r en9 th void s ra t io Re ference v a l ue of voids ratio

I n i t i a l va lue of voids ra tio Elas t ic shear modulus Elastic bulk modulus

S tress ra t i o at cr i t ica l sta t e

Number o f cycles

Cyc le n umber d u r ing wh ich failure occurs

Ove rcon sol ida t ion ratio Mea n to ta l stress M e a n e f f ec t i ve stress A mea s u r e of s i z e of the cu rren t y i e l d s u r face

I n i t i a l va l u e o f Pc '

I n i t i a l va l ue of mean e f f ect i v e s tr e s s

A measure o f s i ze o f th e current ' loadi n9 ' s u r face

A mea s u r e of d evia tor s t ress

Volume s tra in

A mea s ur e of dev iator s tr a in

Pr i n c ipal s t ra i n compon e n t s

S lope o f e l as t i c e v s Z n p ' l i ne

S lope of e lastopla s t i c e vs ln p ' l i ne

S t r e s s ra t io = qjp ' OCR degrad a t ion param e t e r

Ang l e o f ! r i c t i o n

Radial compon e n t o f total s t ress

A x ia l component of tota l s tr es s

Ra d ia l component o f e ff ective s tY e s s

�x i a l componen t o f ef fec tive s t r e s s

Pr i n c i pa l compone n t s o f e ffec t i ve s tr e s s

5 2 .

Al?PEND ! X B REFERENCES

1 . ANDERSE N , K . H . ( 1 9 75 ) , " Researc h pro j ec t , repeated load i n g on c l ay -

Summa ry and i n terpr e t a t ion of te s t r e s u l t s ' " , Norwe g i a n

Geo technica l Institute , Repo r t No . 7 4 0 3 7 - 9 , Oslo .

2 . ANDERSEN , K . H . ( 1 9 76 ) ' " Behaviour o f c l a y sub j e c ted to und r ained cyc l i c load ing" , Proc . BOS S Conf . , NGI , pp3 9 2-403 .

3 . BROWN' , S . F . , LASR I NE , A . K . F . and RYDE . A . F . L . ( 1 9 7 5 ) " Repe a t e d load triax i a l t e s t i n g o f a s i lty c l ay11 , Geo techn iqu e , Vo l - 2 5 , No . 1 , pp9 5- l l4 .

4 . BROWN . S . F . and SNAITH , M . S . ( 1 974 ) " The measurement of r ecover ab l e and

i r r ecoverab l e d e form a t ions i n repea ted loa d t r i a x i a l testing'; Geotechoique , Vo l . 2 4 , pp2 5 5 - 2 5 9 .

5 . EEKELEN , H . . I\ . M . van a nd ·POTI'S , D . M . ( l 9 7 8 ) " The behaviour o f Dr ammen c l a y

under cyc lic l oading " , Geotechnique , Vo l . 28 , No . 2 , ppl 7 3 - 196 .

6 . HARDI N , B . O . and DRNEVICH , V . P . ( 197 2 ) " Sh e a r modu l u s and dampi ng i n so i l s : Mea s u r ement a nd parameter e f fects " , J . So i l Mec h . and Found . Divn . , ASCE , Vol . 98 , No . SM6 , pp603 - 6 2 4 .

7 . ROULSBY , G . T . ( 1 9 7 9 ) Private Commun i c a t ion .

8 LEE , K . H . and SEE D , H . B . ( 1967 ) " Cyc l i c s t r e s s co nd i t ions causing l ique f a c t io n of sand" , J. S o i l Mech s . Found . Di vn . , ASCE , Vo l . 9 ) , NO . SMl , pp4 7 - 7 0 .

9 . LEWI N , P . I . I l 978 ) '" The de formation o f s o f t c lay under gene r a l i sed s t r e s s cond i tions ' " , Ph . D . Th e s i s , King ' s Co l l ege . Un ive rs i ty o f London .

10 . MARTIN G . R . , FINN , W . O . L . and SEE D , H . B . ( 1 9 7 4 ) ' " Fundamen t a l s o f l ique ­fact ion under cyc l i c load i ng11 , Univer s i ty o f Bri tish Columbia ,

Dept . of C i v i l Enginee r i n g , Soi l . Mec h . S e r i e s No . 2 3 .

1 1 . MONI SMITH , C . L . , OGAWA , N . and FR.EEME , C . R . ( 1 9 7 5 ) " Permane n t de formation chara c te r i s t i c s o f subgrade so i l s i n repe a ted lo ad i ng'" , Tra n s po r t a t i o n Research Record 5 3 7 , ppL - 1 7 .

1 2 . MROZ , Z . , NORRIS , V . A . and Z I E NKIEWI CZ , O . C . ( 1 979 ) " Appl i c a t ion of a n

a n i s o t r o p i c harde ning mod e l in the ana l ys i s o f e l a s to - p l a s t i c

de forma t i o n " , Ge o t e c h n i qu e , Vo l . 29 , No . 1 , ppl - 3 4 .

1 3 . PENDE R , M . J . ( 1 9 7 7 ) " Mode l l ing s o i l beha viour u nd e r cyc l i c l o ad i n g '" , Proc .

9 th I n t . Conf . Soi l Mec h . Found . Engng . , Tokyo , Vol . 2 ,

pp32 5 3 3 1 .

1 4 . PENOE R , M . J . ( l 9 7 8 ) " A mod e l for the behav i o u r of over-con s o l i d a t ed c l ay " ,

Geo techn ique , Vo l . 2 8 , No . 1 , ppl - 2 5 .

1 5 . PRtVOS T , J . R . ( 1 9 7 7 ) " Ma thema ti c a l mode l l i n g o f mono ton i c a nd cyc l i c

undrained c l ay behaviour " , I n t . J . Numer ical An a l yt i c a l Me thod s i n Ge ome c h a n ic s , Vo l . l , ppl 9S-21 6 .

5 3 .

1 6 . PREVOS T , J . H . ( 1 978 ) " P l a sti c i t y theo r y for so i l s t r e s g - s t. r a i n beha v i o u r " , J . Engng M e c ha n i c s Divn . , ASCE , Vo l . l 04 , No . EM3 , ppl l 7 7 - l l 9 4 .

1 7 . PYK.E , R . M . ( l 9 7 J ) " S e t t l emen t and l i q u e fa c t i on o f s a n d s unde r mul t i ­d i r ec t io n a l l o a d i ng " , Ph . D . T h e s i s , Un i ve r s i ty o f C a l i fo r n i a , B e r k e l e y .

1 9 . RAH MAN , M . S . , SEED , H . B . a nd BOOKE R , J . R . ( 1 9 7 7 ) " Po r e pres s u r e devel o p -

men t u n d e r o ffsho r e gravi ty s t ruc tu r e s " , J . Geotech . Engng . D i v n . , ASCE , Vo l . l Ol , No . GT1 2 , ppl 4 1 9- 1 4 J 6 .

1 9 . ROSCOE , K . H . a nd B U RLAND , J . 6 . ( 1 968 ) " On the g e ne r a l i s ed s t r e s s - s t r a i n

behavi o u r o f ' we t ' c l ay " , i n Eng i n e e r i n g P l a sti c i ty ,

(Ed . J . Heyroan and F . A . L e c k i e ) , C a mb r i dg e Un iversity P r e ss , ppS J S - E.o9 .

20 . SJ>.NGREY , D . l\ . , HENl<.EL , D . J . a n d ES R I G , M . I . ( 1 96 9 ) " Th e e f f e c t i ve s tr e s s respon s e o f a sa tu r a ted c l a y t o r e pea ted l oadi n g " , Cana d i a n G eo t e c hn i c a l Jou r n a l , Vol . 6 , No . ) , pp 2 4 l - 2 5 7 .

2 1 . SCHOPI ELD , A . N . a nd WROTH , C . P . ( l 96B ) C r i t i c a l S ta te S o i l M e c h a n i c s McGraw H i l l , Lond o n .

2 2 . S EE D , H . B . ( 1 9 7 9 ) N i ne te e n th Rank i n e Le c tu r e , to a ppe a r in Geo techn ique .

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2 S . SEED , H . B . a rtd I DRI S S , l . M . ( 1 9 7 0 1 " Soi l mod u l i and damping fac tors foJC d y n a m i c r e s ponse a na l ys i s " , E a r thqua k e Engi rteering Research Ce n t r e EERC , R e po r t No . E E RC 7 0- 1 0 , Un iversi ty o f Cal i forn i a , Berke l ey .

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No . SM6 , pplOS - 1 ] 4 .

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E a r thqua k e E n g i n ee r i n g R e s e a r c h Ce n t r e . R e po r t No . EERC 7 5 - 2 6 , Un iver s i ty of Ca l i fornia , B e r k e l ey .

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M e x ico , Vo l . l , pp4 0 l - 4 09 .

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3 2 . W I L S ON , N . E . a nd GREENWOO D , J . R . ( 1 974 1 " Po r e pre s s u r e s a nd s t r a i n s a f te r repeated loadi ng o f s a tu r a ted c l a y " , Canadi a n Geo tec hn i c a l � · Vo l . l l , No . 2 , pp2 6 9 - 2 7 7 .

3 J . WROTH . C . P . ( 1 9 7 7 ) " The pred i c t e d pe r f o rma n c e of a s o f t c l a y un d e r a

·-

r.r i al ernbanlune n t loading based on the Cam-cl ay rnod e l " . Ch . 6 o f F i n i te E l emen t s i n Geomech a n ic s , Ed . G . Gudehus , W i l ey , Londo <\ .

* * * * • • * * * *

E N G I N E E R I N G RESEA RCH REPORTS

CE N o .

.t '..J·· , Ti"tte --· Aut hor'( s ) Date

CURRE/JT REPORTS

2

3

4

6

f lood F r e q u e n c y Analysis ' Logis t i c Method for Incorpo r a t i n g Probab l e M a x imum f l oo d s

Ad j u s tment o f Ph r ea t i c Line i n Seepage A na l y s i s Sy F i ni te E l em en t M e thod

C r e e p S u c k l i n g O f R e i n fo r c e d Conc r e t e C o l umns

B u c k l i ng Prope r t i e s of Monos ymm e tr i c l - B eams

E l a s to - P l a s t i c �nal y s is of C a b l e Net S t r u c t ur e s

l\ Cri t i c a l S ta t e S o i l Hod e l fo r Cyc l i c Load i ng

Jl fU\ D 'l , D . K .

I S AACS , L . T .

8 £H A N , J . E . & O ' CONNO R , C .

K l T l PO RNC H � I , S . & TAAHA I R , N . S .

MEEK , .J . L . & 8 ROt I N , P . L . D .

CARTE R , J . l? . , BOOK£R , J . R . & W ROTH , C . P .

Feb r u a r y , 19 79

March , 19 79

�pr i l , 19 79

May , 1 9 7 9

Novembe r 1 1 9 79

December , 1 9 / 9

C U R R E NT C I V I L E N G I N E E R I N G B U L L ET I N S

4 Bri t tle Frac ture of S teel - Perform·

ance o f ND I B and SAA A I s truc tural

s teels: C. O 'Con n or ( 1 964)

5 Buc k ling in Steel Struc tures - 1 . The

use of a charac teris tic imperfe c t shape

and i ts aiJp/ica rion to the b uckling o f

an isola ted column : C. O 'Con n o r

( 1 965)

6 Buck ling in Steel Struc tures - 2. Th e

use of a chara c teriS Tic imperfect shap e

in the design of de termina te p lane

trusses agains t buckling in th e ir plane :

C. O 'Connor ( 1 965)

1 Wa ve Generated Curre n ts - Some

observa tions made in fixed b ed h y ·

draulic models : M R. Gourla y ( 1 965)

8 Brittle Frac ture of Sreel - 2 Th eore t ­

ical s tress dis tributions in a partially

yielded, non·uniform. iJOiycrystalline

ma terial: C. O 'Con .�or ( 1 966)

9 A nalysis b y Com(J u ter - Programmes

for frame and grid struc tures. J. L .

Mee k ( 1 96 1)

1 0 Force A nal ysis of Fix ed Supp ort Rigid

Frames : J. L . Meek and R. Owen

1 1 968)

I I Analvsis b y Co mp u te r A xisy· me tric solu tion o f elas ro-p lasric (Jro­b lems b y finite elem e n t m e th o ds :

J. L . Meek a n d G. Care y ( 1969)

1 2 Ground Wa ter H ydro logy: J. R. Wa tkins ( 1 969)

1 3 L a n d use (Jredic tion in transp o r ta tion plan ning: 5. Golding and K. B. Da vid· son ( 1969)

1 4

1 5

1 6

Fin i te Elemen t Meth o ds Two

dim ensional see(Jage wi th a free sur·

face: L . T. Isaacs ( 1 9 7 1 )

Transp ortation G ravi ty Models : A. T. C. Philbrick ( 1 9 7 1 )

Wa ve Clima te a t Mo ffa t Beach : M . R . Gourla y ( 1 9 73)

1 7. Quan tita tive Evalua ticm of Tra ffic

A ssignmen t Me th o ds : C. L u cas a n d K. B. Da vidson ( 1 9 74)

1 8 Planning and Eva lua tion of a High Speed Brisba n e - Gold Coast Rail L ink : K. B. Da vidson, e t a!. ( 1974)

19 Brisbane A irp orr Developmen t Flood· wa y Studies : C. J. A(Jelt ( 1 977)

20 Numbers of Engineering Graduates in

Queensland: C. O 'Conn or 1 1971)