constitutive laws

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Constitutive laws and numerical analysis for soilfoundations under static, transient or cyclic loads*

0. C. ZIEN KIEW ICZ

Department of Civil Engineering, Univ ersity College of Sw ansea, Sw ansea SA2 8PP, UK

In this paper we present the survey of research carried out over the past ten years at University

College of Swansea under the guidance of the author to determine a rational approach to the study

of foundation and other soil mechanics problems. The paper starts with a description of the need

for numerical approaches utilizing finite element or similar methodology and discusses various

constitutive models for static soil behaviour. Plasticity is adopted to describe the non-linear

characteristics of soil. A series of tests on ideally elasto/plastic, associative and non-associative,

models and on an extended critical state model show that with the latter it is possible to obtain

good predictions of the behaviour for drained and undrained behaviour of normally consolidated

materials and indeed to extend the results to over-consolidated situations. The remainder of thepaper concerns itself with the cyclic and transient load behaviour. Here the well known increase of

pore pressure under repeated loading has to be accounted for as this can either lead to liquefaction

or a very considerable weakening of the material. Two alternative approaches are proposed. In the

first a concept of an autogenous densification of the material is introduced to supplement the

original elasto/plastic models and this is shown to be effective in predicting liquefaction of sands.

An alternative model modifies the critical state using methods proposed by Mroz to describe

behaviour of clays more accurately. Finally, the paper deals with shakedown or ratchetting type

problems in which it is possible to obtain collapse without material deterioration merely by a

sufficient number of cyclic load repetitions. The numerical methods of dealing with such problems

are discussed.

INTRODUCTION

This paper gives a survey of research carried out in recent

years by the author’s group with the object of providing a

sound basis for computation of static, quasi-static and

dynamic responses of soil foundations of offshore struc-

tures. We shall therefore be concerned here more with the

philosophy of the approach than with the detail which can

be found in various publications cited.

With the development of efficient numerical procedures

for the solution’of boundary and initial value problems

and with the expanding power of the computer pro-

gressively less room remains for unquantified empiricism

and ‘trial and error’ approaches. Care must, however, betaken to use such methods intelligently with due regard to

geological and material uncertainties. We shall not dis-

cuss any details of numerical processes here and the

reader can refer to the author’s text’. It is assumed that

with a suitably formulated constitutive law, solutions can

be obtained to most problems of foundation and structure

interaction by methods known today.

The main objective of offshore foundation analysis

and design is to provide economical structures which will:

(1) not collapse on application of maximum anticipated

loads, (2) not collapse after period of exposure to transient

loads such as can be expected during storms and earth-

quakes, and (3) not sustain excessive deformation under

* A preliminary version of this paper was presented at the SecondConference ou the Behaviour of Offshore Structures August 1979 inLondon, Permission to publish has been kindly granted by the BritishHydromechanics Research Association and other sponsors of this

conference.

static or transient loads of magnitude lower than those

encountered under (2) or (3).

Limit analysis procedures, so useful in simple soil

mechanics studies, are limited generally to uniform homo-

geneous situations and to materials obeying certain

idealized assumptions. Further, only answers to problems

of category (1) can be provided by the methods of limit

analysis, all other studies requiring a fuller solution of the

properly posed boundary or initial value problems. For

this reason in what follows we shall assume that such

solutions need to be formulated, and that as a by-product

of analysis the limit values can also be obtained.

In offshore work the soil will always be in a saturated

state and this presents a considerable simplification of the

basic behaviour patterns. However, due to varying per-

iods of load, both drained and undrained extremes of

action will occur and we shall therefore seek to describe

the constitutive soil behaviour in terms of drained con-

ditions from which undrained or partly drained be-

haviour can be deduced. How far a constitutive re-

lationship needs to be specified and how to choose

between alternatives is a serious question to which the

answer is highly problem dependent. We shall thus

consider in turn the two extreme phases of behaviour and

suggest suitable models for each.

STATIC OR QUASI-STATIC BEHAVIOUR

This area is clearly the most studied one in soil mechanics

and many engineers will assert that safety against collapse

0141-1187/80/010023~9/%02.000 1980 CML Publications Appl. O cean Res. 1980, Vol. 2, No . 1 23



Con s t i t u t i ve l aws and numer i ca l ana ly s i s Jbr so i l f ounda t ions: O. C . Z i enk i ewic z

c a n b e r e a s o n a b l y a s s u r e d b y l i m i t c o m p u t a t i o n s w h i l e

v e r y s im p l if ie d o n e - d i m e n s i o n a l c o n s o l i d a t i o n t y p e s o -

l u t i o n s a r e a d e q u a t e f o r p r e d i c t io n o f d e f o r m a t f o n s . I n t h e

v i e w o f t h e a u t h o r t h i s is a c o n s i d e r a b l e o v e r s i m p l i f i c a ti o n

a n d p o t e n t i a l ly d a n g e r o u s w h e n l a rg e s t r u c tu r e s a r e

c o n s i d e r e d . W h i l e l i m i t l o a d c o m p u t a t i o n s a r e w e l l

d e v e l o p e d f o r t w o - d i m e n s i o n a l ' c o n s t a n t c o h e s i o n ' t y p e

m a t e r i a l s o b e y i n g p u r e l y c o h e s i v e t y p e b e h a v i o u r o f

a s s o c i a t e d p l a s t i c i t y , t h e r e a l s i t u a t i o n i s n o t s o r e a d i l y

t r e a t e d a n d s o m e o f t h e d i f fi c u lt ie s a r e c i t e d b e l o w : (1 )u n d e r u n d r a i n e d c o n d i t i o n s n o r m a l l y c o n s o l i d a t e d c l a y s

s h o w a n a p p a r e n t c o h e s i o n i n c r e a s i n g w i t h d e p t h f o r

w h i c h f e w s o l u t i o n s o f l im i t t y p e a r e a v a i l a b l e ; (2 ) i n

d r a i n e d c o n d i t i o n s w i t h v e r y s m a l l c o h e s i o n e x i s t i n g t h e

l i m i t l o a d c a l c u l a t i o n s a r e g e n e r a l l y n o t a p p l i c a b l e d u e t o

t h e n o n - a s s o c i a t i v i t y o f th e f l o w r u le a n d h e n c e o n l y i n

c e r t a i n c a s e s c a n r e a s o n a b l e p r e d i c t i o n s b e a c c e p t e d ; ( 3 )

o v e r c o n s o l i d a t e d b e h a v i o u r o f cl a y s c a n n o t r e a d i ly b e

t r e a t e d i n t e r m s o f t o t a l s t r e s s a n a l y s i s u s in g s i m p l e l im i t

t h e o r e m s ; ( 4) f o r c o m p l e x l o a d a n d n o n - h o m o g e n e o u s

m a t e r i a l s a s w e ll a s f o r t h r e e - d i m e n s i o n a l c o n d i t i o n s v e r y

f e w l i m i t s o l u t i o n s a r e a p p l i c a b l e a n d h e r e f u l l d e f o r -

m a t i o n s o l u t i o n s a r e d e s i r a b l e .W h e r e d e f o r m a t i o n s h a v e t o b e a s s e s s e d a n e e d f o r

s o m e s o l u t i o n s o f t h e b o u n d a r y v a l u e p r o b l e m s e x i s t a n d

h e r e f r e q u e n t l y t h e r e s o r t i s m a d e t o l i n e a r f in i t e e l e m e n t

a n a l y s i s w i t h s u i t a b l e a d j u s t e d m o d u l i . I t i s a c o n t e n t i o n

o f t h e a u t h o r t h a t i n t h e p r e s e n t s t a t e o f d e v e l o p m e n t i t is

e c o n o m i c a l l y f e as ib l e t o t r e a t t h e d e f o r m a t i o n a n d c o l-

l a p s e b e h a v i o u r i n a u n i f i e d w a y w i t h o u t e x c e s s iv e c o s t

a n d t h u s a v o i d i n g t h e d i f f i c u l t i e s m e n t i o n e d a b o v e . T h i s ,

h o w e v e r , r e q u i re s t h e d e v e l o p m e n t o f re a s o n a b l e c o n -

s t i t u t i v e m o d e l s a n d w e s h a l l n o w t u r n t o t h e i r

d e t e r m i n a t i o n .

Con s t i t u t i ve mode l s f or s ta t i c l oads

T h e p r o b l e m is o f c o n s id e r a b le i m p o r t a n c e a n d m u c h

w o r k h a s b e e n d o n e o v e r t h e y e a r s t o e n s u r e a d e q u a t e

m o d e l s f o r cl a y s a n d s a nd s . I f a p u r e l y m o n o t o n i c i n c r e a se

of s t ress oc curs a t a l l po in t s of the so i l it is c l ea r ly poss ib le

t o u s e a s s u m p t i o n s o f n o n - l i n e a r e l a s t i c it y in p r a c t i c e t o

d e s c r i b e t h e b e h a v i o u r a n d v a r i o u s s u c h m o d e l s h a v e

b e e n d e v e l o p e d . U n f o r t u n a t e l y i n m a n y r e a li st ic e n g i n e e r -

i n g s i t u a t i o n s s u c h m o d e l s f a il a s l o a d s a r e n o t i m p o s e d i n

a m o n o t o n i c s e q u e n c e (f o r i n s ta n c e t h e g r a v i t y l o a d s a n d

w a v e l o a d i n g i n o f f s h o r e s t r u c t u r e s f o l l o w s e q u e n t ia l l y ) . I f

n o n - m o n o t o n i c l o a d i n g e x i s t s t h e s t r e s s / s t r a i n r e l a t i o n -

s h i p m u s t d e f i ne ' lo a d i n g ' a n d ' u n l o a d i n g ' p h a s e s a n d f o r

t h i s r e a s o n t h e c h o i c e i s l i m i t e d t o p l a s t i c i t y , h y p o e l a s -

t ic i ty , a n d e n d o c h r o n i c m o d e l d e s c r ip t i o n s. F o r r e a s o n s

d i s c u s s e d e l s e w h e r e t h e a u t h o r h a s c h o s e n t o u s e

t h r o u g h o u t p l a st i c m o d e l s t o d e s c r i b e so i l b e h a v i o u r .

I n p l a s t i c i t y w e t h u s p o s t u l a t e a y i e l d s u r f a c e :

F ( o , e p ) = a ( 1 )

wh ere o i s the s t res s l eve l and e p i s a mea sur e of p las t i c

s t ra i n in g . T h e t o t a l s t r a i n i s n a t u r a l l y c o m p o s e d o f as u m m a t i o n o f e l as ti c a n d p l a s t ic c o m p o n e n t s :

e = e e + e p (2 )

A s i t i s p o s s i b l e t o d e s c r i b e f u l l y t h e b e h a v i o u r o f

s a t u r a t e d s o i l d e f i n i n g t h e l a w g o v e r n i n g t h e d r a i n e d o r

( s k e le t o n ) c o m p o n e n t w e s h al l b e o n l y c o n c e r n e d w i t h

s u c h ' e ff e c ti v e s t r e ss ' c o n d i t i o n s . I n o t h e r p u b l i c a t i o n s i t is

s h o w n h o w t h e k n o w l e d g e o f s u c h b a s ic c o n s t i t u t iv e l a w s

a l lo w s u n d r a i n e d , c o n s o l i d a t e d , o r d y n a m i c b e h a v i o u r t ob e r e a d i ly d e d u c e d 2 - 5.

W h e n c h o o s i n g t h e p l a s t i c i t y m o d e l t h e m o s t i m p o r -

t a n t f e a t u r e m u s t b e i ts a b i li t y t o r e p r o d u c e t h e c o l l a p s e o r

p e a k s t re s s s i t u a t i o n w h i c h is t h e b e s t k n o w n q u a n t i t y f o r

s o il s a n d w h i c h c o r r e s p o n d s t o a M o h r C o u l o m b s u r fa c e

i n t h e p r i n c i p a l s t r e s s s p a c e . W e s h a l l p r o p o s e t o m a k e a

s e l e c t io n f r o m a r a n g e o f th r e e b a s i c g e n e r a l p o s s i b i li t ie st o d e f i n e r e a s o n a b l y w e l l p l a s t i c s o i l b e h a v i o u r .

A. Idea l assoc ia t i ve e las to -p las t i c i t y . T h e M o h r

C o u l o m b s u r f a c e i s a s s u m e d t o a c t a s p l a s t i c y i e l d a n d

p o t e n t i a l s u r f a c e . T h i s t o g e t h e r w i t h i t s t r i a x i a l s t r e s s

sec t ion i s shown in F ig . l (a ) .

B. Idea l non-assoc ia t i ve , e las to -p las t i c it y . H e r e t h e m o -

d e l A i s e x t e n d e d b y d e f i n i n g a s e t o f p l a s t ic p o t e n t i a l

s u r fa c e s o f t h e s a m e t y p e b u t n o t p a r a l l e l to t h e M o h r

C o u l o m b y i e ld s u r f a c e Q( a) . W i t h t h e f l o w r u l e g i v e n a s:

dep = ~Q),,a" (3 )

t h i s a l l o w s a m o r e r e a l i s t i c d i l a t a n c y ( o r i n f a c t a z e r o

d i l a ta n c y ) t o b e i m p o s e d o n t h e m a t e r i a l d u r i n g y i e ld t h u s

a p p r o x i m a t i n g i n a b e t t e r w a y t h e t r u e b e h a v i o u r . T h i s

mode l i s shown in F ig . l (b) .

C. An assoc ia t i ve s t ra in harden ing p las t i c , c r it i ca l s ta t e

mode l . T h i s i s b a s e d o n t h e w e l l k n o w n c l a s s i c m o d e l

d e r i v e d b y R o s c o e a n d h i s c o l l a b o r a t o r s 5 - 7 . W e s h a l l u s e

\

02

o 3 / Y

! '

[ o l

B

Yie ld sur face :J Potential surface

Principal stress space

Yi eld ~ o I > 0 2 = 0 3

surft~c e / y . . _

<~O2=O3

Tr iax io l se ct ion OAB

F i g u r e l (a ) M o d e l A -- i dea l assoc ia t ed p las t i c i t y w i th

Mohr Coulomb y i e ld sur face .

2 4 A p p l . O c e a n R e s . 1980, Vol . 2, N o . 1



Cons t i tu t ive laws and numer ica l ana lys is for so i l foundat ions: O . C. Z ienk iew icz

Yield. surface

' r i n c i p a l s t r e s s_space

Potent ialsurface

Yie ld surface / Q=/O '~O, > o2 .03

Q Or.~.._"""".~ o l < o 2 = o 3

T r i o x i o [ s e c t i o n OAB

F i g u r e l (b) Model B - - idea l non-associa ted p las t ic i ty

wi th Mohr Coulomb y ie ld sur face .

h e r e a n e x t e n d e d f o r m f o r t h e g e n e r a l p r i n c i p a l s t r e s s

s p a c e i n t h e m a n n e r d o n e b y Z i e n k i e w i c z et al. s.

I n t h e c r i t i c a l s t a t e m o d e l t h e y i e l d s t a t e s u r f a c e

' h a r d e n s ' a s t h e v o l u m e t r i c p l a s t i c c o m p r e s s i v e s t r a i n

d e v e l o p s o r a s t h e m a t e r i a l ' d e n s i t i e s ' . W i t h t h e t o t a l

p l a s t ic s t r a i n b e i n g a s s o c i a t e d ( i .e . i n a d i r e c t i o n n o r m a l t o

t h e y i e l d s u r f a c e ) s u c h d e n s i f i c a t i o n o c c u r s o n t h e r i g h t

h a n d s i d e o f t h e e l l ip s e as s h o w n i n F i g . l ( c ) a n d t h e

h a r d e n i n g b e c o m e s z e r o o n t h e c r it ic a l s u r fa c e w h i c h i s

i d e n ti f ie d w i t h t h e M o h r C o u l o m b f a il u r e li ne . T h e c r i ti c a l

s u r f a c e i n f a c t d o e s n o t n e c e s s a r i l y c o r r e s p o n d t o t h e

m a x i m u m s tr e ss a t f ai l u re b u t t o t h a t a t w h i c h c o n t i n u i n g

d e f o r m a t i o n c a n e x is t. A s t r ai n s o f te n i n g r e g i o n a b o v e t h e

c r i t i c a l s t a t e s u r f a c e e x i st s b u t i s h o w e v e r h i g h l y u n s t a b l e

a n d a s w i ll b e s e en l a t e r i ts e x a c t d e f i n i t i o n a p p e a r s n o t tO

be es sent i a l .

Cho ice of 'reasonable' mode l

C o m p u t e r i n v e s t i g a t i o n s a r e n e c e s s a r y t o d e t e r m i n e a

' r e a s o n a b l e ' m o d e l w h i c h w o u l d g i v e p r e d i c t i o n s o f

c o l l a p se a n d d e f o r m a t i o n c o n s i s te n t w i t h t h o s e o b s e r v e d

o r p r e d i c t e d b y w e l l e s t a b li s h e d m e t h o d s a n d a l so d o t h is

w i t h o u t u n d u e c o m p l e x i t y . A la r g e s e ri es o f su c h c o m -

p u t a t i o n w a s i n f a c t u n d e r t a k e n 3 .4 , 8 a n d h e r e w e g i v e o n l y

a b a s i c s u m m a r y o f t h e f i n d in g s . T h e t h r e e s e t s o f

c o n d i t i o n s c o r r e s p o n d i n g t o f u ll y d r a i n e d , u n d r a i n e d a n d

u n d r a i n e d o v e r c o n s o l i d a t e d b e h a v i o u r w i ll b e c o n s i d e r e d

a n d i f t h e se e x t r e m e s a r e w e l l r e p r o d u c e d b y a m o d e l , i t

w i ll b e d e e m e d t o b e c a p a b l e o f r e s p o n d i n g s a t i s f a c t o r i l y

t o c o n s o l i d a t i n g b e h a v i o u r .

Drained test . I n F i g . 2 , w e s h o w t h e r e s u l t s o f f u l ly

d r a i n e d , n o r m a l l y c o n s o l i d a t e d , a n a l y s i s w i t h w e i g h t l e s s

Critical stateB

Y i e l d an d potentialsurface

P r i n c i p a l s t r e s s

T r i a x i a l s e c t i o n

A . ~ ~ . P I~p / O A B

Y i e l d a n d J' ~ °1> oz =o 3m t e n t i a l / / " I

F l a , e v ) ~

o1<02 0 '3Cri t ical state" J ~ ,,,~l",.uz-

F i g u r e l (c ) M ode l C - - s t ra in harden ing cr i t ica l s ta te

associa ted p las t ic i ty wi th Mohr Coulomb cr i t ica l s ta te

surface.

L o a d q ( I b / i n. }

0 100 200

l ° ' _. e - _ ,

"~~C 1'2- . _ q R=5~ bC '~ k"- " - - = 5 f t . -

, 6 - - f'o c8

o M e s h~ . 2 . 4 - -

ai Data • c = 10 I b / in 2

2 .s - - ~,= 2 0 o

E = 30 ,000 I b / in . 2

v = O . 3 o

Figure 2 . Axi - sy mm etr ic Jbo t ing (un iform load) dra ined

load-deformat ion behaviour for three mater ia l models . A ,Idea l associa ted p las t ic i ty (Mohr Coulomb) ; B, ideal non-

associa ted p las t ic i ty; C , extended cr i t ica l s ta te .

Appl . Ocean Res . 1980, Vol . 2, N o . 1 25



Co nsti tut iv e law s and num erical analysis fo r soil joundations." O. C. Zien kie wic z

0 2

0 4c

•~ " 0 6

i 0-8

gu 1 2

44 ~ 2.0

1- 6 ~ 4 0

6 0

Figure 3 .

A ~ e d f o oh n g p re s su r e P l L b / o n z )

t O 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0

% ~,~,

Bo

Co

P r e s s u r e P r i b / i n }

2 0 0 4 0 0 6 0 0 8 0 0

foun d I

Load dejbrmation characteris t ics (undrainedcondi t ions ) for p la te foo t ing .

Model A, elast ic ideal plast ic (Mohr Coulomb) associative

(dilatant) ; Model B, elast ic ideal plast ic (Mohr Coulomb)

non-associative (zero dilatancy); Model C, elast ic-strain

depend ent plastic; M od el D, elast ic ideal plast ic total s tress

analysis

s o i l f o r a t w o - d i m e n s i o n a l c i r c u l a r f o o t i n g . A s s e e n f r o m

t h e r e s u l t s t h e p e r f o r m a n c e o f a l l t h r e e m o d e l s i s v i r t u a l l y

i d e n t ic a l a n d i n d e e d t h i s is c o n f i r m e d b y o t h e r s i m i l a rs t u d i es s - 1 o

Undrained tests . R e s u l t s a r e s h o w n i n F i g . 3 i n d i c a t i n gt h a t f o r u n d r a i n e d s i t u a t i o n s , w h e r e a h i g h d e g r e e o f

r e s t r a in t i s p r o v i d e d b y t h e f l u id o n t h e d i l a t i o n o f th e s o i l

s k e l e t o n , q u i t e d i f f e r e n t a n s w e r s c a n b e o b t a i n e d . F o r

c o m p a r i s o n a t o t a l s t r e ss a n a l y s i s is a ls o i n c l u d e d i n t h e

r e s u l t s a n d w e n o w s e e t h a t m o d e l s B a n d C p r e d i c t

r e a s o n a b l y w e l l t h e b e h a v i o u r b u t t h e a s s o c i a t i v e i d e al l y

p l a s t i c m o d e l A d o e s n o t c o l l a p s e a t a l l d u e t o t h e

c o n t i n u o u s d e v e l o p m e n t o f n e g at i v e p o r e p r e s su r e s w i t h

t h e d i l a t i o n ( i n d e ed i t m i g h t b e c o n t e n d e d t h a t f i n a l ly

c o l l a p s e w o u l d o c c u r f o r s u c h a m a t e r i a l b y c a v i t a t i o n b u t

t h i s i s b e y o n d t h e a r g u m e n t s e x t e n d e d h e r e ) . T h e a b o v e

t e s ts i n d i c a t e t h a t f r o m a p r a c t i c a l p o i n t o f v i e w b o t h

m o d e l s B a n d C a r e v i a b l e a n d c a n b e u s e d i n a v a r ie t y o f

s i t u a t i o n s . T h e f i n a l t e s t s e r i e s , h o w e v e r , o n o v e r -

c o n s o l i d a t e d c l a y s i n d i c a t es t h a t h e r e o n e o f t h e m is

p r e f e r a b l e .

Overconsolidated tests . I t i s w e l l k n o w n t h a t o v e r c o n -

s o l i d a t e d c l a y s u n d e r u n d r a i n e d c o n d i t i o n s w il l e x h i b i t

m u c h h i g h e r f a i l u r e s t r e s s e s t h a n n o r m a l l y c o n s o l i d a t e d

m a t e r i a l s . T h i s w o u l d a p p e a r t o b e p r e d i c t e d o n l y w i t h

m a t e r i a l s o f t y p e C b u t c l e a r l y w o u l d n o t o c c u r i n e i t h e r A

o r B m o d e l s . H o w e v e r , in t h e l o a d i n g o f o v e r c o n s o l i d a t e d

s o il s a c o n s i d e r a b l e a m o u n t o f s t r a in s o f t e n in g m u s t o c c u r

w h e n t h e s t r e s s e s e x c e e d th e c r i t i c a l s t a t e a n d i t i s p o s s i b l e

t h a t t h e a c t u a l p e a k s o f s t re s s a r e o f li tt le i m p o r t a n c e i n

p r a c t i c a l s i tu a t i o n s . T h i s i s b o r n e o u t b y a s t u d y i n w h i ch

w e h a v e b e e n f o r t u n a t e t o h a v e r e s u lt s o f t e s ts c a r r ie d o u t

a t I m p e r i a l C o l le g e f o r a n o v e r c o n s o l i d a t e d c l a y m a t e r i a l

o n w h i c h a x i - s y m m e t r i c f o o t i n g s w e r e p la c e d . I n F i g . 4 w e

s h o w o n e s e t o f r e s u lt s o b t a i n e d b y e x p e r i m e n t a n d a

c o r r e s p o n d i n g c a l c u l a t i o n u s i n g t h e m o d e l s C a n d B

d i s c u s s e d p r e v i o u s l y . I t is s e e n t h a t t h e p r e d i c t i o n g i v e n b y

m o d e l C i s q u i t e r e m a r k a b l e c o n s i d e r i n g t h e r ig i d n a t u r e

o f t h e f o o t i n g a n d e x p e r i m e n t a l u n c e r t a i n t i e s , a n d t h a t

m o d e l B p r e d i c ts m u c h t o o l o w c o l la p s e v a l u e s.

T o t e s t th e i m p o r t a n c e o f t h e p o r t i o n o f t h e c r i ti c a l s t a te

m o d e l a b o v e t h e c r i t i c a l s t a t e l in e w e i n t r o d u c e h e r e a c a p

m o d e l w h e r e t h e l e f t h a n d s i d e s e c t i o n o f th e e l l i p s e o f

mo de l C ( v iz. F ig . l ( c )) is cu t o f f by the c r i t i ca l s ta t e l ine

n o w u s e d a s a p l a s t i c y i e l d s u r f a c e w i t h a n o n - a s s o c i a t i v e

f l o w r u l e . T h i s m o d e l i s s h o w n i n F i g . 5 a n d i n F i g . 6

p r e d i c t i o n s g i v e n b y t h i s m o d e l a r e s h o w n . I t i s i m -

m e d i a t e l y n o t i c e d t h a t , providin 9 an y dilat ion at al l occurs,

c o l l a p s e l o a d s s i m i l a r t o t h o s e p r e d i c t e d b y t h e s t a n d a r d

c r i t ic a l s t a t e m o d e l a r e o b t a i n a b l e . I f a f u l ly a s s o c i a t i v e

r u l e is t a k e n t h e p r e d i c t i o n i s e v e r y b i t a s g o o d a s t h a t

o b t a i n e d t a k i n g t h e f u l l e l li p s e. A ll th e t e s t s i n d i c a t e t h a t i f

a s i n g l e m o d e l i s s o u g h t , a f u ll e ll i p s e o r c a p t y p e c r i t i c a l

s t a t e m o d e l C i s p r o b a b l y t h e c h o i c e t h a t s h o u l d b e m a d e .

H o w e v e r , i t m u s t b e r e m e m b e r e d t h a t f o r a m a j o r i t y o f

s i t u a t i o n s p r e d i c t i o n s b y m o d e l B w i l l a l s o r e m a i n

a v a i l a b l e a n d g i v e s i m i l a r a n s w e r s . T h e c h o i c e w i l l o f t e n

b e m a d e o n c o m p u t a t i o n a l g r o u n d s in m a n y p r a c t ic a ls i t u a t i o n s .

3 0

Z

O 2 o

(3

~o

4

a , e o e o o o e o e ° ° ° l ° ° ° e ° ° ° ° e

• • ~ . . . . . . T an k

° /I I ~, i l

L . 1375 m m d _ jv÷ + ~ p ÷ + + ÷ + ÷ ÷ ' 1 - ÷ + ÷ + 4 - ÷ ÷ ÷ ÷ + ÷ ÷ + + + ÷ + +

0 I I 1

0 5 10 15 20

V e r t i c a l d i s p l a c e m e n t ( m m )

Figure 4. A foot in9 on overconso lidated clay (appro-x i m a t e o ver co n s o l i d a t i o n = 3 5 0 k N / m 2 / 5 0 k N / m 2. -

E xp er i m en t a l - - I m p er ia l C o l leg e ( M .C .R . M a r t i n s , 1977);

f in i te e lement - - U .C. Swansea (L . A . Winn icki and D . J .

N a y l o r 1978): (model C) ; . . . . , cr i tica l s ta te wi th M ohrCoulomb form; + + + +, idea l M ohr C oulomb p las t ic i ty

non-associated (model B) . E = 1 200 0 k N /m 2, (p = 30 ° , v

= 0 . 3 , H = 2 5

0

Po tential L E'p / ""e ur fa c e ---~ ~ g ,Evt)

F i g u r e 5 . M o d e l C C ca p t yp e m o d ~ ca t i o n o f m o d e l C

4" F iy . l (c)

20 Appl. Ocean Res. 1980, Vol. 2 , No. 1



Constitutive laws and numerical analysis for soil foundat ions: 0. C. Zienk iewi cz

Nan assoc. J, = O”

0

Certical displacement km1M

Figure 6. A footing on ov er-consolidat ed clay. A ssessment

of cap model of Fig. 5 wit h va rious degrees of association.

- - -, Experimental - Imperial College (M. C. R.

Martins, 1977); . . ., Finit e element - U.C. Sw ansea (L. A.

W innicki, 1978); Criti cal stat e w ith Moh r Coulomb cut-off:

E=12000 kN/m2, cp=30”, v=O.3, H=25. Non-

associated flow rule

Computations

Same remarks should be made with regard to the

numerical finite element processes used in the solution of

realistic problem utilizing above models. The choice of

model may to same extent be influenced by campu-

tatianal techniques used’. Thus if tangential methods are

used the critical state model is preferable to a non-

associative one as tangent matrix is then symmetric. On

the other hand, with initial stress techniques (or equiva-

lent viscaplastic processes) the ideally plastic model is

preferable as mare rapid convergence occurs and nan-

assaciativity is not important.In the context of two-dimensional plane strain or axi-

symmetric, static or dynamic, analysis, the solution

techniques are widely available at moderate cast with all

the approaches. In full three-dimensional situations,

however, we find that the cast is still large and refinements

of numerical methodology are proceeding. Here iterative

procedures will .ultimately became the basis and distin-

ctions between tangent and initial stress methods will

disappear.

In the intermediate case of axi-symmetric bodies sub-

ject to non-axi-symmetric loads such as occur frequently

in offshore platform analysis, an alternative to full three-

dimensional analysis exists using a Fourier type expan-sion in the circumferential direction. This reduces the cast

of full three-dimensional analysis considerably and details

of the process are described elsewhere. However, it should

be noted that for such methods an initial stress technique

is essential and hence preference far simple, non-strain

hardening models exists .3 In Fig. 7 same results of a nan-

linear analysis carried out far a three-dimensional loading

of a footing are shown.

CYCLIC AND TRANSI ENT LOADS

The offshore or onshore marine structure exposed to wave

loading or indeed any structure subject to dynamic shack

loading such as could be caused by earthquakes presents

special problems. Here tests indicate that in bath clays

and sands cyclic stress reversals progressively increase the

pare pressure and thus, for undrained conditions, will lead

towards a weakening of the material. None of the present

models discussed so far allows such weakening to occur

and the mechanism in which it happens will now have to

be considered. In the next section we shall introduce

possible models by which ‘densificatian’ of the soil

skeleton is introduced whilst preserving the previous

model characteristics. One simple and effective approach

is to consider that in addition to elastic and plastic strains

already recorded by the models discussed in the previous

sections a further purely volumetric, strain occurs which

densifies the material. This autogenous strain is a function

of some parameter which depends an shear stress levels

and the total straining path. Such a model has proved to

V

H-

EIn

f

IFigure 7(a). A three-dimensional, non- linear Fourier Scot ts

solut ion for a foot ing problem. Element mesh in r-z plane.

Footing: E =2.0 x lo6 kN/m’; v=O.3.

Soil: E = 1 O x lo4 kN/m2; v = 0.3.

fly =cahesion = 50 kN/m’; K, = 1.0; p = weight = 20

kN/m2

Limit load ratio

3D/2D =l-72

ttlement of the

Settlement of the edge

0.126 o-740

Figure 7(b). Behav iour of problem of Fig. 7(a) first

increasing vertical load, OA’ (or OA”), then the horizontal

load up to collapse.

Ap pl. O cean Res. 1980, Vol. 2, No . 1 27



Con s t i tu t i ve laws and numer ica l analys i s Jor so i l , [bundat ions :

0, , 4 '

Ouaywal

~ l t t e r t a t p r o p e r t t e l :

Q u l ~ r V l l l ; ¢ ~n l t d e r . d n • r t$111 t*loc l t

1 "l ~ t f I | l ~ d F O U D d l t l o l t

Y o ~ , = o a . z = 1 o 0 I ¢ ~ / = =

P o l s e o n ' e r l t l o u = 0 , 4

F r i e r : o n ~g l e ¢ - 4 0 °

t ) n t t / e i g h t = 1 2 ~ I f f / l 3

o _

o - - . X 1

I I I I0 20 40 60m

( 1 )

. : : fi0'- A . . . . . . .

°° ' I/b xAAAAA A Ao., v ,,v vvV W "

T i m e , s

( 2 )

7 2 .®©

A i~¢

. , t . . . .

4 = ec

10 tt ¢

131

O. C. Z ienk iewicz

b e e f f e c ti v e f o r s a n d s a n d h a s a l l o w e d l i q u e f a c t i o n e f f ec t s

t o b e p r e d i c t e d ~4. A n a l t e r n a t i v e m o d e l u s e s a m o d i f i -

c a t i o n o f i s o t r o p i c h a r d e n i n g s u r f a c e a s s o c i a t e d w i t h t h e

c r i ti c a l s ta t e m o d e l C i n a m a n n e r s u g g e s t e d b y c l a s si c a l

w o r k o f M r o z f o r m e t al s~ 5. T h i s m o d e l w i ll b e d e s c r i b e d i n

g e n e r a l t e r m s b u t s t i l l m u c h r e s e a r c h h a s t o b e d o n e t od e v e l o p i t t o f u ll a p p l i c a b i l i t y t 6 , 1 7 .

I n a d d i t i o n t o p r o b l e m s a s s o c ia t e d w i t h th e w e a k e n i n g

o f th e m a t e r i al d u r i n g r e p e a t e d c y c l ic o r d y n a m i c l o a d i n g ,

p r o g r e s s i v e d e f o r m a t i o n o r r a t c h e t t i n g c a n t a k e p l a c e

a f t er m a n y r e p e t i t io n s o f l o a d . T h i s p h e n o m e n o n k n o w n

t o s t r u c t u r a l e n g i n e e r s , is p e r h a p s n o v e l in s o i l m e c h a n i c s

a n d s o m e d i s c u s s io n o f th i s w i ll b e m a d e i n t h e l a st s e c t i o n .

Modi f i ed mater ia l mode l I - - dens i f i ca t ion concept

T h e w o r k o f S e e d , F i n n a n d o t h e r s ~ 8- 21 h a s i n d i c a t e d

t h a t u n d e r c y c l i c l o a d i n g d r y s o il s (s a n d s ) d e n s i fy w h i l e

s a t u r a t e d o n e s d e v e l o p c o n s i d e r a b le p o r e p r e s su r e . I f t h e

a m o u n t o f d e n s i f ic a t io n c a n b e r e l a te d i n s o m e m a n n e r t o

t h e s t r e s s a n d s t r a i n p a t h t h e m a t e r i a l u n d e r g o e s , b o t h

e ff e ct s c a n b e p r e d i c t e d b y t h e s a m e p h e n o m e n o n . C l e a r ly

w h e n t h e m a t e r i a ls a r e s a t u r a t e d a n d t h e s k e l e to n s h o w s a

t e n d e n c y t o c o n t r a c t i t w i l l t r a n s f e r a c o n s i d e r a b l e

p r o p o r t i o n o f it s m e a n t o t a l s tr e ss o n t o t h e f lu i d o r w a t e r

i n t h e p o r e s . T h e q u a n t i f i c a t i o n o f t h e s e e f f ec t s i s s i m p l e

a n d s t r a i g h t f o r w a r d i n t h e a n a l y s i s o n c e s u c h ' i n i t i a l '

s t r a i n c a n b e p r e d i c t e d . W r i t i n g t h u s t h a t t h e t o t a l s t r a i ncan be g iven as :

e = e e + e p + e a ( 4 )

w h e r e t h e t h r e e s t r a i n c o m p o n e n t s a r e t h o s e r e l a t e d t o

e l a st ic , p l a s ti c a n d d e n s i f i c a t i o n a c t i o n s , w e c o u l d u s e a n y

s u cc e ss f ul m o d e l o f st a ti c b e h a v i o u r a n d a u g m e n t t h is b y

d e v i s i n g a s u i t a b l e l a w f o r a u t o g e n o u s s t r a i n c a i n t h e

a n a l y s i s . Z i e n k i e w i c z et al. 14 s u g g e s t s u c h a l a w f o r a

p a r t i c u l a r s a n d a n d s h o w t h a t i t s e f fe c t s c a n b e t a k e n q u i t e

s i m p l y i n t o a c c o u n t i n t h e d y n a m i c n o n - l i n e a r c o m -

p u t a t i o n . I n F i g . 8 , w e s h o w a n a n a l y s i s o f a q u a y w a l l in

w h i c h s u c h d e n s i f i c a t i o n is n o w i n t r o d u c e d i n a d d i t i o n t o

e l a s t i c - p l a s t i c s t r a in o f m o d e l B d i s c u s s e d p r e v i o u s l y . T h e

f o r m u l a u s e d h e r e r e l a t e s a u t o g e n o u s s t r a i n t o ( a ) t h e r a t i o

o f t h e d e v i a t o r i c t o m e a n s t re s s e s a n d ( b ) t h e t o t a l l e n g t h

o f t h e s tr a i n p a t h t o b e m e a s u r e d i n a b s o l u t e t e r m s . T h u s

d d = m d e ~ m r = 0 , 1, 1, 0 , 0 , O)

A

de~ =f(~c)d~c f ( t¢) = - -I + B ~ c

d~c = g(# /a, . )d ( g(6/a , . ) = e x p (Ta/am)

( 5 )

d( = ~ /d~ud~ u (~u dev ia tor i c s t ra in)

F u r t h e r f o r m u l a e h a v e b e e n m o r e r e c e n t l y d e v e l o p e d 22

a n d w il l b e p u b l i s h e d s h o r t l y b u t a l t e r n a t i v e a p p r o a c h e sh a v e b e e n s u g g e st ed b y N e m a t - N a s s e r a n d S h o k o o h 23

w h i c h d e f i n e s u c c es s f u ll y t h e s a m e p h e n o m e n a .

I n o f f s h o re c o m p u t a t i o n s i t h a s b e e n q u i t e c u s t o m a r y

t o s i m p l y i n s e r t th e o b s e r v e d p o r e p r e s s u r e i n c r e a s e s in t o

Figure 8 . Non - l inear r esponse o f a quay w al l t o ear th-

quake . (1), Fini t e e l ement mesh and mater ia l proper t i e s ;

(2) , hor i zonta l d i sp lacement vs . t ime a t top o f quay wall ;

(3), de formed mesh (d i sp lacement x 100).

2 8 Appl . Ocean Res . 1980, Vol . 2, N o . 1



C o n s t i t u t i v e l a w s

t h e a n a l y si s o f a n u n d r a i n e d k i n d u s i n g t h e d r a i n e d s o il

p r o p e r t i e s a n d t h u s p r o g r e s s i v e l y w e a k e n i n g t h e m a t e ri a l .

T h i s i n s e r t i o n o f p o r e p r e s s u r e o b v i o u s l y w il l l e a d t o t h e

s a m e a n s w e r s p r o v i d i n g i t c a n b e r e l a t e d t o t h e s t r a i n p a t h

a n d s t r e s s l e v e l s a n d p e r h a p s a s s u c h p r e s e n t s a m o r e

d i r e c t a p p r o a c h . I f d r a i n a g e d o e s h o w e v e r o c c u r , it

a p p e a r s n e c e s s a r y t o u s e t h e d e n s i f i c a t i o n c o n c e p t d i -

r e c tl y . I n F i g . 9 w e s h o w h o w t h e p r o p o s e d d e n s i f i ca t i o n

f o r m u l a ta k e s i n t o a c c o u n t e x p e r i m e n t a l d a t a o b s e r v e d i n

a s a n d .

M o d i f i e d m a t e r i a l m o d e l I I - - i s o t r o p i c h a r d e n i n g c o n c e p t s

T h e m o d i f i e d m a t e r i a l m o d e l d i sc u s s ed i n t h e p r e c e d i n g

s e c t io n i s p e r h a p s t o o p r a g m a t i c a n d l a c ks t h e a e s t h e ti c

a p p e a l o f a s i n g l e m o d e l i n w h i c h t h e d e n s i f i c a t i o n e f f e c ts

c o u l d b e i n c o r p o r a t e d . T h e o r i g i n f o r t h e d e v e l o p m e n t o f

s u c h a m o d e l n o w a p p e a r s i n t w o s o u r c e s . I n t h e fi r st p l a c e

w e h a v e t h e o r i g i n a l w o r k o f M r o z ~5 w h e r e a s e r i e s o f

k i n e m a t i c a l l y h a r d e n i n g s u r f a c e s n e s t i n g w i t h i n e a c h

o t h e r w a s s u g g e s t e d t o d e s c r i b e m e t a l p l a s t i c i t y . I n t h e

s e c o n d o v e r l a y c o n c e p t s i n t r o d u c e d b y Z i e n k i e w i c z e t a l .

0 2

.0.1to

0.01

F i g u r e 9 .

s a n d )

s j ~ v experim ental data

' ' o '.1 1. 100

A u t o g e n o u s v o l u m e t r i c s t r a i n versus ~ : ( N . G . I .

a n d n u m e r i c a l a n a l y s is f o r s o i l f o u n d a t i o n s : O . C . Z i e n k i e w i c z

w e r e a p p l i e d t o s o il s 2 3. T h i s f i n al m o d e l i s d e s c r i b e d

q u a l i t a t i v e l y i n t h e m e a n s t re s s s e c t i o n o f t h e c r i t i c a l s ta t e

p l a s t i c i t y m o d e l i n F i g . 1 0 . I n t h i s m o d e l w e r e t a i n t h e

n o r m a l c r i t i c a l s t a t e c o n c e p t s b u t u s e t h e p r e v i o u s

p l a s t i c i t y s u r f a c e n o t a s a y i e l d c o n d i t i o n b u t a s a n

e n v e l o p e o f a l l y i e l d c o n d i t i o n s d e f i n e d b y e l l i p s e s o f

s m a l l e r s i z e ( u s u a l l y g i v e n a s a f r a c t i o n o f t h e e x t e r i o r

e l l ip s e si ze ) w h i c h a r e c a p a b l e o f m o v i n g i n t h e i n t e r i o r a s

t h e s t r e ss p o i n t m o v e s w i t h i n . B y im p o s i n g a s u i t a b l e f lo w

r u l e i t is e a s y t o a c h i e v e c o m p a c t i o n o r d e n s i f ic a t io n o f th e

m a t e r i a l b o t h f o r i n c r e a s i n g a n d d e c r e a s i n g s t r e s s m o -

t i o n s w i t h i n t h e o v e r a l l c r i t i c a l s t a t e e l l i p s e . T h i s m o d e l

h a s n o w b e e n c o n s i d e r a b l y r e f i n e d a n d i s w el l c a p a b l e o f

r e s p o n d i n g t o l o a d c y c l e s. I n F i g . 1 1 w e s h o w s o m e r e s u l t s

t f ik e n f r o m a r e c e n t p a p e r 17 i n w h i c h t h e d e n s i f i c a t i o n

w i t h i n c r e a s i n g c y c l e n u m b e r i s i n d i c a t e d . T h e m o d e l is

n o t o n l y c a p a b l e o f sh o w i n g t h i s d e n s i fi c a t io n b u t i m -

p r o v e s t h e g e n e r a l p e r f o r m a n c e o f t h e c r i ti c a l st a t e m o d e l s

i f r e f i n e m e n t is n e e d e d a n d a d a p t s t h i s m o d e l b e t t e r t o

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Figu r e 11 . C y c l i c load ing unde r d r a ine d t r iax ia l c ond i t ions . (a ) S h e a r s t r e s s vs . v o lume t r i c s t r a in ; (b ) v o lume t r i c s t r a in vs .

n u m b e r o f c y c le s ; (c ) s h e a r s t r e s s vs. s he ar s t r a in ; (d ) s h e a r s t r a i n ( a t e n d o f c y c le ) vs . n u m b e r o f c y c le s .

A p p l . O c e a n R e s . 1980, V o l . 2, N o . 1 2 9



C o n s t i t u t i v e l a w s a n d n u m e r i c a l a n a l y s is f o r s o i l. ] o u n d a t io n s : O . C . Z i e n k i e w i c z

Moment l : 200 k I, m D e" met,e 'i] : /.Z)0 WNm pc, melee

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F i g u r e 12 . C y c l i c l o a d i n g o f a J o o t i n 9 u n d e r u n d r a i n e d c o n d i t i o n s . (a ) F i n i t e e l e m e n t m e s h a n d s o i l p r o p e r t i e s ; (b )

d i s p l a c e m e n t s d u e t o v e r t i c a l l o a d in g f o l l o w e d b y a m o n o t o n i c m o m e n t ; (c ) d i s p l a c e m e n t d u e t o v e r t i c a l l o a d i n g j b l l o w e d b y a

c y c l i c m o m e n t ; (d ) v e r t i c a l c e n t r a l d is p l a c e m e n t d u e t o c y c l i c m o m e n t f o r M o h r C o u l o m b a n d s t r a i n - h a r d e n in 9 y i e l d

c ond i t i ons ; ( f) e xc e s s por e p r e s s u r e due t o m om e nt a f t e r 3 / 4 c y c l e ( k N / m 2 ) ; (g ) v a r i a t io n o f e x c e s s p o r e p r e s s u r e s d u e t o c y c l i c

m o m e n t .

o v e r c o n s o l i d a t e d c o n d i t i o n s . I n re f. 17 , a f u ll d e s c r i p t i o n

o f t h e p e r f o r m a n c e o f th e m o d e l i s g i v e n a n d i n t h e n e a r

f u t u r e w e e x p e c t t h a t t h i s m o d e l w i l l b e a b l e t o b e a d j u s t e d

s o a s t o g i v e a l l o b s e r v a b l e p r e s s u r e r i s e c h a r a c t e r i s ti c s .

S h a k e d o w n a n d r a t c h e t t i n 9 p r o b l e m s

W i t h a s t r u c t u r a l s y s t e m s u b j e c t t o l o a d s d u r i n g w h i c h

p l a s t i c d e f o r m a t i o n o c c u r s a n d i f f u r th e r , s o m e l o a d s a r es u b j e c t t o r e v e r s a l w h i l e o t h e r s r e m a i n c o n s t a n t , p r o -

g r e ss iv e p l a st i c d e f o r m a t i o n m a y d e v e l o p i n e a c h c y c l e

e v e n i f t h e t ot a l l o a d c o m b i n a t i o n i s w e l l b e l o w t h a t o f

c o l la p s e . T h i s i s a w e l l k n o w n p h e n o m e n o n o f r a t c h e t ti n g

a n d i n F i g . 1 2 w e s h o w a n e x a m p l e u s i n g a t ot a l s t r e ss

a n a l y s i s w h i c h s h o w s h o w a t l o a d s w e l l b e l o w t h o s e o f

c o l l a p s e c o n t i n u i n g d e f o r m a t i o n m a y o c c u r f o r a t y p i c a l

f o u n d a t i o n d u e t o r e ve r s a l s o f w a v e l o a d i n g u l t i m a t e l y

c a u s i n g u n s e r v ic e a b i l it y o r i n d e e d i n c r e m e n t a l c o l l a p s e .

I n t h e c a s e i l lu s t r a t e d a l a r ge n u m b e r o f r e - a n a l y s e s w e r e

c a r r i e d o u t . T h i s i s f e a s ib l e i n a s i m p l e c a s e b u t i s v e r yc o st l y. F o r p r a c t ic a l p u r p o s e s m o r e s o p h i s t i c a t e d a p -

p r o a c h e s w i l l h a v e t o b e d e v e l o p e d t o t e l l t h e e n g i n e e r

w h e t h e r s u c h p r o g r e s s i v e c o l l a p s e i s l i k e l y t o o c c u r . T h i s

3 0 A p p l . O c e a n R e s . 1 9 8 0 , V o l . 2 , N o . 1



C o n s t i t u t i v e l a w s

q u e s t i o n c a n n o t b e a n s w e r e d b y a n y o f th e l i m i t a n a l y s is

p r o c e d u r e s a n d w i ll o n ly b e o b t a i n e d b y r e s u lt s o f a

b o u n d a r y v a l u e a n al y s i s o r t he a p p l i c a t i o n o f sp e c i al

b o u n d i n g t h e o r e m s w h i c h a g a i n a r e a v a i l a b le o n l y f o r

s i m p l i f i e d c a se s . S t u d y i s c u r r e n t l y i n p r o g r e s s t o e x p l o r e

t h e a p p l i c a b i l i t y o f s u c h l i m i t t h e o r e m s a n d t o p r o v i d e

s i m p l i f i e d c a l c u l a t i o n s i n w h i c h t h e e f fe c ts o f c y c l i n g c a n

b e o b t a i n e d b y e x t r a p o l a t i o n . I n d e e d i f t h e c y c l i c s t r e s si n g

i s a s s o c i a t e d w i t h t h e d e t e r i o r a t i o n o f t h e m a t e r i a l a n d

p o r e p r e s s u r e d e v e l o p m e n t , t h is p r o c e d u r e a p p e a r s t o b e

t h e o n l y w a y i n w h i c h t r u e p r o g r e s s c a n b e a c h i e v e d a n d

t h e p r o b l e m r e m a i n s o n e o f t h e m o s t s e r io u s o n e s fo r

c y c l i c ly l o a d e d f o u n d a t i o n s . T h i s p a r t i c u l a r a r e a i s t h e

s u b j e c t o f m u c h r e s e a r c h a n d t h e a u t h o r b e l i e v e s t h a t h e r e

a t l e a s t n o s i m p l e a l t e r n a t i v e o r r u l e o f t h u m b c a n b e

a p p l i e d a t t h e p r e s e n t s t a ge o f k n o w l e d g e f o r i m p o r t a n t

s t r u c t u r e s .

C O N C L U S I O N S

A r e v i e w o f t h e r e s e a r c h f o r d e t e r m i n a t i o n o f t h e r e s p o n s e

o f s t r u c t u r e a n d t h e i r f o u n d a t i o n u n d e r v a r i o u s l o a d i n gc o n d i t i o n h a s b e e n g i v e n i n t h i s p a p e r . A s m e n t i o n e d

e a r l i e r t h e d e t a i l s o f t h e v a r i o u s p r o c e s s e s h a v e n o t b e e n

d i s c u s s e d d u e t o t h e i r a v a i l a b i l i t y e l s e w h e r e i n t h e l i t e r a -

t u r e b u t h o p e f u l l y t h e a r e a s o f i g n o r a n c e h a v e b e e n

h i g h l i g h t e d . T h e a u t h o r f ee ls s t r o n g l y t h a t t h e d e v e l o p -

m e n t o f n u m e r i c a l b o u n d a r y s o l u t i o n o f t h e s t r u ct u r e a n d

s o i l p r o b l e m s i s e s s e n t i a l i f p r o g r e s s i s t o b e m a d e a n d h a s

t o b e c o m b i n e d w i t h a d o c u m e n t e d a n d t h o u g h t o u t

e x p e r i m e n t a l p r o g r a m m e . T h e r e f in e d m e a s u r e m e n t s o f

s o i l c h a r a c t e r i s t i c s t h e m s e l v e s w i l l n o t h e l p t o p r e d i c t t h e

b e h a v i o u r o f f o u n d a t i o n s a n y m o r e t h a n r e f in e d n u m e r i -

c a l m e t h o d s c a n d o s o w i t h o u t t a k i n g t h e s e e x p e r i m e n t s

i n t o a c c o u n t. I t i s o n l y t h r o u g h c o m b i n e d p h y s i c a l a n dn u m e r i c a l e x p e r i m e n t s a n d s e n s i t i v i t y s t u d i e s th a t r u l e s

f o r t h e d e s ig n o f sa f e f o u n d a t i o n s o f m a r i n e s t r u c t u r e s c a n

b e o b t a i n e d .

ACKNOWLEDGEMENTS

T h e a u t h o r i s g r a te f u l t o D r . C . H u m p h e s o n , D r . D . J .

N a y l o r , M r . V . A . N o r r i s , D r . L . A . W i n n i c k i , D r . C . T .

C h a n g a n d P r o f e s s o r Z . M r o z f o r c o l l a b o r a t i o n a n d

e n t h u s i a st i c s u p p o r t a t v a r i o u s s t a g e s o f t h is w o r k , '

T h a n k s a r e a l s o d u e t o t h e S c i e nc e R e s e a r c h C o u n c i l f o r

t h e i r s u p p o r t o f t h e a b o v e .

REFERENCES

Zienkiewicz, O. C. The Finite Element Method, McGraw-Hill,London 1977Zienkiewicz, O. C., Hum pheson, C. and Lewis, R. W. A unifiedapproach to soil mechanics including plasticity and viscoplas-ticity, in Finite Elements in Geomechanics, (Ed. G. G udehus), Ch.4, London. 1977

a n d n u m e r i c a l a n a l y s i s f o r s o il f o u n d a t i o n s : O . C . Z i e n k i e w i c z

3 Zienkiewicz, O. C. and Humpheson, C. Viscoplasticity - - ageneralised model for description of soil behaviour, in FiniteElements in Geomechanics, (Eds . C. S. Desai and C. Christian)McGraw-Hill, London, 1977

4 Zienkiewicz, O. C., No rris, V. A., Winnicki, L. A., Na ylo r, D. J.and Lew is , R. W. A unified approach to the soil mechanicsproblems of offshore foundations Numerical M ethods in OffshoreEngineering, (Ed s. O. C. Z ienkiewicz, e t a l . ) John Wiley,Chichester, V ol. 12, 1978

5 Roscoe, K. H., Schofield, A. N. and Wro th, C. P. On the yieldingof soils, Geotechnique 1958, 8, 22

6 Roscoe, K. H. and Burland, J. B. On the generalized stress/strainbehaviour of 'wet ' c lay, Engineering Plasticity (Eds. J. Haym anand F. A. Lockead) Cambridge U niversity Press, London, 1968,pp. 535-609

8 Zienkiewicz, O. C., Hum pheson, C. and Lewis, R. W. Associatedand non-associated viscoplasticity and plasticity in soil me-chanics, Geotechnique 1975, 25, 671

9 Hum pheson, C. Fin ite element analysis of elasto/viscoplasticsoils, Ph.D. Thesis, University College of Swansea (1976)

10 Nay lor, D. J., No n linear finite element models for soils, Ph.D.Thesis, University College of Swansea (1975)

11 (a) Martins, M. C. R. Large model footing tests - - a firstinterpretation, Soil Mechanics section internal repor t, ImperialCollege, London, (December, 1977)(b) E1-Ghamrawy, M. K. An experimental s tudy of a re-sedimented low plasticity clay, Soil Mechanics section internal

report, Imperical College, London (Februa ry 1977)12 Winnicki, L. A., Na ylor , D. J. and Zienkiewicz, O. C. Finite

element analysis o f model .footing test, Department o f Civ i lEngineering, University College of Swansea, CR/323/78, 1978

13 W innicki, L. A. and Zienkiewicz, O. C. Plast ic or viscoplasticbehaviour of axi-symm etric bodies subject to non-symmetricloading, lnt . J . Num. Meth. Eng. (1979) in press

14 Zienkiewicz, O. C., Chang, C. T. and Hinton, E. No n linearseismic response and liquefaction, lnt I . Num . Analyt. Meth.Geomechanics 1978, 2, (4 ), 381

15 Mroz, Z. On the description of aniso trop ic work-hardening, ./.Mech. Phys. Solids, 1967, 15, 163

16 Mroz, Z., No rris, V. A. and Zienkiewicz, O. C. An aniso trop ichardening model for soils and its applic ation to cyclic loading,Int. J . Num. Analyt. Meth. Geomech. 1978, 2, 203

17 Mroz, Z., Norris , V. A. and Zienkiewicz, O. C. Application of ananisotropic hardening model in the analysis of elasto-plasticdeformation of soils, Geotechnique, (March 1979) in press

18 Lee, K. L. and Seed, H. B. Liquefaction of satur ated sands duringcyclic loading, J. Soil Mech . Found. D iv., Proc. Am. Soc. Cir. Eng.,1966. 92, 105

19 Silver, M. L. and Seed, H. B. Volum e changes in sands durin gcyclic loading, J. Soil Mech . Found. D iv., Proc. Am. Soc. Cir. Eng.1971, 97, 1171

20 Martin, G. R., Finn, W. D. L. and Seed, H. B. Fundam entals ofliquefaction under cyclic loading, J. G eotech. Eng. Div., Proc. Am.Soc. Cir. Eng. 1975, 101, 423

21 Finn, W. D. L., Lee, K. W. and M artin, G. R. An effective s tressmodel for liquefaction, J. Geotech. Eng. Div., Proc. Am. Soc. Cir.Eng. 1977, 103, 517

22 Chang, C. T. Nonlin ear response of earth dams and foundationsin earthquakes, Ph.D. Thesis, University College of Swansea,

(1979)23 Zienkiewicz, O. C., Norris, V. A. and Nayl or, D. J. Pla sticity and

viscoplasticity in soil m echanics with special reference to cyclicloading problems, Proc. Int. Conf. Finite Elements in NonlinearSolid and Structural Mechanics, Geilo, 1977, pp. 455 485, Vol. 2,Tapir Press , Norwegian Institute of Technology, Trondheim.

24 Nem at-Nasser, S. and Shokooh, A. Densification and liquefac-tion of sand in cyclic shearing, Cent. Am. Conf. Earthquake Eng.,El-Salvador, (January, 1978)

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