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