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University of Nigeria Research Publications
TAMUNOIYALA, Koko S.
Aut
hor
PG/M Engr./84/2489
Title
Analysis of Rectangular Plates on Elastic Foundations by the Method of Initial
Functions
Facu
lty
Engineering
Dep
artm
ent
Civil Engineering
Dat
e
July, 1986
Sign
atur
e
ANALYSIS OF RECTANGULAR PLATES ON ELASTIC FOUNDATIONS
BY THE METHOD OF INITIAL FUNCTIONS
TAMUNOIYALA S. KOKO
( ~ ~ / ~ . E n g r . / 8 4 / 2 4 8 9 )
A Thes i s
0 f
submi t ted t o the ~ e ~ a r i m e n ' t
C i v i l Engineering
I n P a r t i a l Fu l f i lment of t h e Requirements for t h e
Degree o f Master of Engineering
i n C i v i l Engineering
Univers i ty of Nigeria
Nsukka
J u l y 1986
Dedicated
To my brother
Anthony S . Koko
WLC a u t l m r wishes t o e x p r e s s h i s s i n c e r e g r a t i t u d e t o
D r . V.K. S c b a s t i a n f o r a n w t i c u l o u s s u p e r v i s i o n and f o r making
a v a i l a b l e r c l n v a n t l i t e r a t u r e .
The invnlunb1.e c o n t r i b u t i o n o f M r . l?.C .Nu O j i a k o whose
e a r l i e r work w a s frequently rc fc , r red t o ancl wlio also gave u s e f u l
a d v i c e is acknowledged. The c o n t r i b u t i o n s o f t h e a u t h o r ' s c o l l e a -
y u e s E.C. O g u c j i o f o r , M r . Mbajiorgu and O.A. Olukayode a t v a r i o u s
s t a g e s o f tlie p r o j e c t a r e a l s o a p p r e c i a t e d .
ML. V.O. Ekechukwu, M i s s R,O. Onuchukwu, M r s . C. Kalu
Mr , 13. Al\ancnw and Lhc F e d e r a l U n i v e r s i t y 01 Technology, O w e r r i ' s
Cornputcr C c r i L r c were ve ry h e l p f u l d u r i n g t h e p r o d u c t i o n of t h e
n u m e r i c a l r e s u l t s .
F i n a l l y , t h e a u t h o r wishes t o t h a n k Plrs E,C. Chinwuba f o r
t y p i n g t i i j s t h e s i s .
PAGE
ACKNOWLEDGEMENT
NOTATIONS
L I S T OF FIGURES
I t I S T OF TABLES
ABSTRACT
CHAPTER ONE: INTRODUCTION . ,
1.1 GENERAL BACKGROUND
1.2 O B J F C T I V E S AND SCOPE , .
CHAPTER TWO: LITERATURE REVIEW
2 - 1 THEORIES OF PLATE BENDING
2.1.1 C l a s s i f i c a t i o n of P l a t e s
2 . 1 . 2 . T h i n P l a t e T h e o r y
2.1.2.1 S i m p l i f y i n g a s s u m p t i o n s s G
2,1.2.2. S o l u t i o n s by t h e C l a s s i c a l theory
2.1,2.3 L i m i t a t i o n s of t h e C l a s s i c a l T h e o r y
2.1.3 H i g h e r O r d e r P l a t e T h e o r i e s .
2.2 PLATES ON ELASTIC FOUNDATIONS .
2.3 FOUNDATION MODELS
1 - 3 . 1 O n e - P a r a m e t e r ( W i n k l e r ) Foundation Model
2,3.1.1 C o n s i d e r a t i o n s of t h e F o u n d a t i o n M o d u l u s
2.3.1.2 L i m i t a t i o n s of t h e Winkler Foundation
2,3.2 T w o - P a r a m e t e r Foundation M o d e l s
2.3.2.1 F i l o n e n k o - B o r o d i c h F o u n d a t i o n
, 2.3.2.2 P a s t e r n a k Foundation
iii
vi
viii
X
xii
2 -3.2.3 G e n e r a l i s e d F o u n d a t i o n 17
2.3.2.4 H e t e n y i Foundation 18
2 -3 .2 .5 V l a s o v F o u n d a t i o n 18
2,3.2.5* n c i s s n e r F o u n d a t i o n 2 0
2 ,4 CI<NEIlAL RI31AT?!'.3 2 1
CHAPTER THREE : DKVE1,Ol3MCNT OF T I E GOVERNING EQUATIONS
3 . 1 L I!!? I D O F I N I T I A L FUNCTIONS ( M I F ) 2 3
I n t r o d u c t i o n
D e r i v a t i o n of t h e B a s i c E q u a t i o n s
A P P L I C A T I O N O F M I F TO PLATE ON WINKLEK FOUNDA- FOUNDATION
S y n u n e t r i c ( o r E x t e n s i o n ) P r o b l e m
A n t i - s y m m e t r i c ( o r B e n d i n g ) P r o b l e m
A l l r o u n d S i m p l y Supported P l a t e
L e v y T y p e S o l u t i o n
APP1,ICATION O F M I F TO PLATE ON TWO-PARAMETER
E L A S T I C FOUNDATIONS
S y m m e t r i c P a r t
A n t i - S y m m e t r i c Part
A l l r o u n d S i m p l y Supported Plate
L e v y T y p e S o l u t i o n for P la te on T w o - P a r a m e t e r
Foundat ion
A NEW FOUNDATION MODEL
S y m m e t r i c P a r t
A n t i - S y n ~ r n e t r i c P a r t
A p p l i c a t i o n t o A l l r o u n d S i m p l y Supported P l a t e
CHAPTER FOUR: NUMERICAL RESULTS
4 , 1 ALLROUND S I M P L Y SUPPOKTED PLATE ON WINKLER
FOUNDATION
4 , l . l P l a t e Subjected t o U n i f o r m L o a d
4.1,2 P l a t e Subjected t o P a r t i a l L o a d
4.2 UNIP'C)RI4LY LOADED ALLROUND SIMPLY SUPPORTED
PLATES ON VLASOV FOUNDATION
4 o 3 -. UNIF'ORJIT,Y L0N)ED ALLROUND SIMPLY SUPPORTED
PLA'I'E ON NEW FOUNDATION
CHAPTER F I V E : CONCLtlS I O N S
APPENL) I X A b
APPENDlX I3
APPENDIX C
REFEIZENC'!.:!:
NO'l'n'l'T ONS
b
cxd
D
E
f mn
F
G
h
k
1
m,n
P
q
qmn
r
UP'J ,w
U
v
w
X l Y l Z
S
- l e n g t h of p l a t e
- 3x3 ma t r i ce s
- width of p l a t e
- a r e a over which p a r t i a l l o a d a c t s
- F l e x u r a l r i g i d i t y o f p l a t e
- Young's modulus of p l a t e
- C o e f f i c i e n t s i n a u x i l l i a r y func t i on , F
- a u x i l l i a r y func t i on f o r ant isymmetr ic p a r t
- R i g i d i t y modulus o f p l a t e
- h a l f o f p l a t e t h i c k n e s s
- Winkler foundat ion modulus
- Second foundat ion parameter
- number of terms of double summation s e r i e s
- foundat ion r e a c t i o n
- a p p l i e d s u r f a c e l oad
- Four i e r c o e f f i c i e n t of a p p l i e d l oad
a - p a r t i a l d i f f e r e n t i a l w i th r e s p e c t t o z (-) a z
- displacement components i n x-, y-, z- d i r e c t i o n s
r e s p e c t i v e l y
- d i s t a n c e s a long x- ,y-, z- d i r e c t i o n s respec : t i v e ly
of Ca r t e s i an coo rd ina t e system
- p a r t i a l d e r i v a t i v e s wi th r e s p e c t t o x , y r e s p e c t i v e l y
- Po i s son ' s r a t i o
- C h a r a c t e r i s t i c r o o t (= -yZ )
- d i r e c t s t r e s s e s i n x-,y-,z- d i r e c t i o n s r e s p e c t i v e l y
- shear s t r e s s components
- C o e f f i c i e n t s i n a u x i l l i a r y func t ion , 4
- a u x i l l i a r y func t ion f o r symmetric p a r t
- shear func t ion
F i g 2.1
F i g 3.1
F i g 4 - 1
F i g 4.2
F i g 4 . 3
LIST O F FIGURES
D e f l e c t i o n o f f o u n d a t i o n s u r f a c e : ( a ) Winkler
f o u n d a t i o n ( b ) r e a l f o u n d a t i o n
( a ) Coord ina te System; ( b ) P l a t e on e l a s t i c
f o u n d a t i o n ; ( c ) Symmetric P a r t ; ( d ) anti-symme-
t r ic p a r t 34
Non-dimensional stress (0 /q) a c r o s s c e n t r e o f X
uni fo rmly l o a d e d s q u a r e p l a t e s on Winkler founda-
t i o n (k=54311.132 k ~ / r n ~ , fi=0.3) 62
Non-dimensional stress (0 /q ) a c r o s s c e n t r e o f X
uni fo rmly l o a d e d s q u a r e p l a t e on Winkler founda-
t i o n f o r v a r i o u s v a l u e s o f f o u n d a t i o n modulus
(;1/211 = 2 0 ) 6 3
Non-dimensional s h e a r stress (T /q) across edge X Z
o f un i fo rmly l o a d e d s q u a r e p l a t e s on Winkler
f o u n d a t i o n (k=54 3 1 1 . 1 3 2 k ~ / m ~ , y=0.3)
Non-dimensional s h e a r stress ( T /q) a c r o s s edge X Z
o f un i fo rmly loaded s q u a r e p l a t e on Winkler
f o u n d a t i o n f o r v a r i o u s v a l u e s o f f o u n d a t i o n
modulus (a/2h=20)
Non-dimensional t r a n s v e r s e d e f l e c t i o n a c r o s s
c e n t r e o f u n i f o r m l y loaded s q u a r e p l a t e s on
Winkle1 f o u n d a t i o n ( k = 5 4 3 1 1 . 1 3 2 k ~ / m ~ , p=0,3)
Non-dimensional d e f l e c t i o n a c r o s s c e n t r e o f
u n i f o r m l y loaded p l a t e on Winkler f o u n d a t i o n
f o r v a r i o u s v a l u e s o f f o u n d a t i o n modulus
(a /2h=20) 68
v i i i
PAGE
15
F i g 4.4
F i g 4.5
F i g 4 -6
F i g 4,7 Non-dimensional f o u n d a t i o n r e a c t i o n a l o n g
c e n t r e l i n e of un i fo rmly loaded s q u a r e p l a t e
r e s t i n g on Winkler f o u n d a t i o n s 69
F i g 4 .8 Non-dimensional s t r e s s (0 /q) a c r o s s c e n t r e o f X
p a r t i a l l y l o a d e d p l a t e s on Winkler founda t ion
F i g 4.9 Non-dimensional s h e a r s t r e s s (T /q) a c r o s s edge X z
of p a r t i a l l y l o a d e d s q u a r e p l a t e on Winkler
f o u n d a t i o n (k=54311. 132 k ~ / m ' , p=0.3) 74
F i g 4.10 Comparison on non-dimensional bending s t r e s s
(0 /q) a c r o s s c e n t r e o f s q u a r e p l a t e s on Winkler X
and Via:-,m:v f o u n d a t i o n s (a/2h=2.5 & 5 ) 7 9
F i g 4.11 Compa'rison o f non-dimensional bending s t r e s s
(ax/q) a c r o s s c e n t r e of s q u a r e p l a t e s on Wink]-er
and Vlasov f o u n d a t i o n s (a/2h=10 a 20)
F i g 4-12 Comparison of non-dimensional s h e a r s t r e s s
( ~ ~ = / q ) a c r o s s edge o f s q u a r e p l a t e s on Winkler
and Vlasov f o u n d a t i o n s 8 1
Table 3.1
Table 4.1
Table 4.2
Table 4.3
Table 4.4
Table 4.5
Table 4 -6
LIST O F TABLES
X
L i s t of d i f f e r e n t i a l ope ra to r s L uu I
Luv.. .L XX
Comparison of non-dimensional maximum
t r ansve r se d e f l e c t i o n s of uniformly loaded
square p l a t e s r e s t i n g on Winkler founda-
t i o n s (p=0.3) 59
Non-dimensional maximum s t r e s s e s i n uni-
formly loaded square p l a t e s r e s t i n g on
Winkler foundat ions (p=0.3) 60
Maximum foundation r eac t ion a t bottom of
uniformly loaded square p l a t e s r e s t i n g on
Winkler foundat ions (fi=O. 3) 6 1
Comparison of non-dimensional maximum
t r ansve r se d e f l e c t i o n of p a r t i a l l y loaded
square p l a t e s on Winkler foundat ion (fi=0.3,
k=54311.133 kN/m3 7 1
Non-dimensional maximum s t r e s s e s i n
p a r t i a l l y loaded square p l a t e s r e s t i n g on
Winkler foundation (p=0.3, k=543ll .I32
k ~ / m ~ , c=d) 7 2
Comparison of non-dimensional maximum
t r ansve r se d e f l e c t i o n of uniformly loaded
square p l a t e s r e s t i n g on Vlasov foundat ions
(fi=0.3) 7 7
PAGE
32
Table 4.7 Comparison of non-dimensional maximum s t r e s s e s
i n uniformly loaded square p l a t e s on Vlasov
foundat ions
Table 4.8 Comparison of non-dimensional maximum t r ans -
verse d e f l e c t i o n and s t r e s s e s i n uniformly
loaded square p l a t e s on t h e Winkler and new
foundat ions (p=0,3, k=543111.32kN/m3
ABSTRACT
A mixed method i n e l a s t i c i t y , f i r s t suggested by Vlasov i n 1957,
and a l s o c a l l e d method of i n i t i a l func t ions (MIF), i s used t o ana lyse
r ec t angu la r p l a t e s r e s t i n g on the Winkler and var ious two parameter
e l a s t i c foundat ions. A new foundation model i n which t h e Winkler founda-
r i o n r e a c t i o n i s modified by t h e inc lus ion of l a t e r a l displacements ,is
suggested and t h e method of i n i t i a l func t ions i s used t o ana lyse p l a t e s
r e s t i n g on t h i s foundation model.
Numerical s o l u t i o n s a r e presented f o r uniformly loaded a l l -
round simply supported square p l a t e of various l eng th t o th i ckness r a t i o s
r e s t i n g on t h e Winkler, Vlasov and new foundat ions and t h e r e s u l t s a r e
compared with those obta ined by t h e c l a s s i c a l and Reissner t h e o r i e s .
So lu t ions have a l s o been given f o r p a r t i a l l y loaded p l a t e s r e s t i n g on t h e
Winkler foundation.
I t is shown t h a t Navier and Levy type s o l u t i o n s a r e poss ib l e i n t h e
a n a l y s i s of p l a t e s on e l a s t i c foundat ions using t h e MIF. I t is a l s o seen
t h a t t h e MIF is more adequate than t h e c l a s s i c a l theory i n t h e a n a l y s i s
of t h i c k p l a t e s r e s t i n g on e l a s t i c foundat ions and i n gene ra l t h e MIF and
Reissner theory s o l u t i o n s a r e c lose . This work has a l s o revea led t h a t
" t h e MIF can-be convenient ly employed t o ana lyse p l a t e s on a foundation
model i n which t h e foundation r eac t ion i s defined i n terms of t r ansve r se
a s well a s l a t e r a l displacements.
*
x i i
CHAPTER ONE
INTRODUCTION
1.1 GENERAL BACKGROUND
The a n a l y s i s of va r ious s t r u c t u r a l forms r e s t i n g on e l a s t i c founda-
t i o n s and sub jec t ed t o va r ious t y p e s of load ing has wide a p p l i c a t i o n s i n
c i v i l , mechanical, marine and a e r o n a u t i c a l engineer ing. Areas of app l i ca -
t i o n i n c i v i l engineer ing inc lude f o o t i n g s under column loads , airway and
road pavements sub jec t ed t o wheel l oads , bur ied p i p e l i n e s and ra i lway
t r a c k s . The compression o f a l i q u i d i n a c y l i n d e r by a p i s t o n , t h e deve-
lopment of so l id -p rope l l an t r o c k e t motors, t h e use of s o f t f i l a m e n t s i n
aerospace s t r u c t u r e s and s h i p b u i l d i n g a c t i v i t i e s a r e some of t h e a r e a s
of a p p l i c a t i o n i n mechanical, a e r o n a u t i c a l and marine engineer ing .
I n t h e a n a l y s i s of p l a t e s r e s t i n g on e l a s t i c foundat ions , t h e
c l a s s i c a l theory f o r t h i n p l a t e f l e x u r e is modified by t h e i nc lu s ion of
t h e foundat ion r e a c t i o n i n t o t h e governing d i f f e r e n t i a l equa t ion . The
foundat ion r e a c t i o n included depends on t h e foundat ion model employed.
Usual ly , t h e one-parameter Winkler foundat ion model is used and here it
is assumed t h a t t h e foundat ion r e a c t i o n a t any p o i n t is p ropor t i ona l t o
t h e d e f l e c t i o n a t t h a t p o i n t . The phys i ca l i n t e r p r e t a t i o n of t h i s model
i s seen t o d e v i a t e much from t h e r e a l s i t u a t i o n and t h i s ha s l e d t o t h e
development of more r e a l i s t i c two-parameter models.
Due t o t h e approximations made i n i t s d e r i v a t i o n t h e c l a s s i c a l
t h e o r y r i s unsu i t ab l e f o r t h e a n a l y s i s of t h i c k p l a t e s and i n r e c e n t
y e a r s many r e s e a r c h e r s have proposed h ighe r o rde r t h e o r i e s f o r t h i c k
p l a t e s . To so lve t h e problem of t h i c k p l a t e s on e l a s t i c foundat ions ,
such h ighe r t h e o r i e s have e q u a l l y been modif ied by t h e i n c l u s i o n o f
t h e founda t ion r e a c t i o n i n t o t h e governing equa t i ons .
I n 1957, a mixed method i n e l a s t i c i t y , a l s o c a l l e d method o f
i n i t i a l f u n c t i o n s , was proposed by Vlasov f o r t h e a n a l y s i s o f t h i c k
p l a t e s . By t h i s method, t h e governing e q u a t i o n s a r e de r i ved from t h e
th ree-d imens iona l e l a s t i c i t y equa t i ons and t h e unknowns a r e expanded
i n MacLaurin s e r i e s i n t h e t h i c k n e s s coo rd ina t e so t h a t t h e s o l u t i o n
is ob t a ined i n t e rms of unknown i n i t i a l f u n c t i o n s . The method h a s been
a p p l i e d t o v a r i o u s problems i nc lud ing e l a s t o - s t a t i c and elasto-dynamic
a n a l y s e s o f t h i c k p l a t e s . However, t h e ex t ens ion o f t h e method t o t h e
a n a l y s i s o f p l a t e s r e s t i n g on e l a s t i c f ounda t i ons h a s n o t been r epo r t ed
i n t h e l i t e r a t u r e .
1.2 OBJECTIVES AND SCOPE
The o b j e c t i v e s o f t h i s work a r e ,
(i) t o use t h e method o f i n i t i a l f u n c t i o n s t o deve lop t h e
govern ing equa t i ons f o r t h e bending o f r e c t a n g u l a r p l a t e s
on e l a s t i c f ounda t i ons and t o produce numerical s o l u t i o n s
t o c e r t a i n problems
(ii) t o develope a new founda t i on model i n which t h e founda-
t i o n r e a c t i o n is d e f i n e d i n t e rms of t r a n s v e r s e a s wel l
a s l a t e r a l . d i sp l acemen t s and t o u se t h e method o f i n i -
, t i a l f u n c t i o n s t o ana ly se p l a t e s r e s t i n g on t h i s new
founda t ion .
CHAPTER TWO
LITERATURE REVIEW
2.1 THEORIES OF PLATE BENDING
2.1.1. C l a s s i f i c a t i o n of P l a t e s
The a n a l y s i s of var ious types of p l a t e s under d i f f e r e n t loading
and boundary cond i t ions has been widely repor ted i n t h e l i t e r a t u r e
and only a summary can be given i n t h i s review. P l a t e s are c l a s s i f i e d
according t o th i ckness , shape, boundary cond i t ions and p r o p e r t i e s of
t h e p l a t e ma te r i a l . The th i ckness c r i t e r i o n i s most commonly used and
p l a t e s a r e defined according t o how t h e th i ckness compares with t h e
l e a s t of t h e o the r dimensions of t h e p l a t e . On t h i s b a s i s t w o types
of p l a t e s d i s c e r n i b l e :
( i) t h i n p l a t e s
( ii) t h l c k p l a t e s
Depending on t h e asaumptions made, var ious p l a t e t h e o r i e s have
been developed and some of these a r e discussed below.
2.1.2 Thin P l a t e Theory
Thin p l a t e bending theory has been e s t a b l i s h e d s ince 1811 when
Lagrange obtained t h e d i f f e r e n t i a l equat ion governing t h e bending of a
t h i n p l a t e under t r ansve r se loading f ~ i m o s h e n k o and Woinowsky-Krieger,
IS'',^^\ . \ \ l l . l l < * t . . , . . I \ \ I ( $ 9 I l l l l l l < l 1 < ~ 1 1 I I :. I l l l ~ ~ l , l l , ~ : ~ : ~ I : : : ~ l l l . l l I ~ . L j l l l -
p.11 C ' L ~ w i l 11 i ( :. I .11 ~~r , I 1 ~1 1 1 l 1 1 . n : ; ~011:; (W,111g, 1353) . 'I'hin p l a t c s a r e
f u r t h e r classified i n t o two kinds by t h e e x t e n t t o which they d e f l e c t .
This g ives r i s e t o two t h e o r i e s namely,
(i) small d e f l e c t i o n theory
4
(ii) large def lec t ion theory,
Norris e t a1 (1976) and Timoshenko and Woinowsky-Krieger (1959)
have described the small-deflection theory a s being applicable t o t h i n
p l a t e s t h a t a re primari ly subjected t o t ransverse loading. The e f f e c t
of in-plane o r membrane forces is neglected and the ac tua l de f lec t ions
a r e small compared t o t he thickness of the p la te . On the other hand
the large-deflect ion theory appl ies when the e f f e c t of t he membrane
forces becomes important. The def lec t ions a re l a rge compared with the
p l a t e thickness but s t i l l small compared with the other dimensions and
the p l a t e edges tend t o move p a r a l l e l t o i t s i n i t i a l plane o orris e t
a1 1976; Williams and Aalami,1979). The large-deflect ion theory is
nonlinear i n character and d i f f i c u l t t o apply. The discussion w i l l be
r e s t r i c t e d t o the small-deflection theory.
2.1.2.1 Simplifying assumptions
In the small-deflection theory, the three-dimensional e l a s t i c i t y
problems a r e reduced t o two-dimensional ones by the imposition of the
following assumptions:
(i) the p l a t e is f l a t , of a small thickness and of homo-
geneous i so t rop ic material
(ii) p l a t e i s of uniform thickness but i f not uniform, the
r a t e of change of thickness is very small
( i i i l t he t ransverse def lec t ions a re small compared with the
' p l a t e thickness
(iv) loads and react ions a r e normal t o the plane of the
p l a t e
(v ) normals t o t h e middle plane before bending remain
normal t o t h e middle plane a f t e r bending
( v i ) t h e middle plane i s unstrained before and a f t e r
bending
( v i i ) t h e v e r t i c a l s t r e s s Gz is small compared with t h e
o the r s t r e s s components
( v i i i ) t h e p l a t e is not s t r e s s e d beyond t h e e l a s t i c l i m i t .
These c o n s t i t u t e t h e c l a s s i c a l theory (Wang, 1953; Timoshenko and
Woinowsky-Krieger, 1959; Bares, 1969; Roark and Young, 1975; Norris
e t a l , 1976 and Nwoji, 1985)
2.1.2.2 Solut ions by t h e c l a s s i c a l theory
The d i f f e r e n t i a l equation governing t h e t ransverse de f l ec t ion ,
w of a t h i n p l a t e i s given by
v 4 w = q / ~ . . . (2.1)
where v4 is t h e bi-harmonic operator which depends on t h e coordinate
system used, q i s t h e appl ied t ransverse load and D is t h e p l a t e
f l e x u r a l r i g i d i t y given by
where E i s Young's modulus of p l a t e , H is t h e p l a t e th ickness and
t h e Poisson ' s r a t i o .
The so lu t ion of equation (2.1) depends on t h e coordinate system
used. For axi-symmetric c i r c u l a r p l a t e s Polar coordinates a r e used
and t h e so lu t ion has been given by Timosheno and Woinowsky-Hrieger
(1559) , Roark and Young (1975) , and Lancaster and Mitchell (1980) . The
6
Cartesian coordinate system i s used f o r rectangular p l a t e s and by
t h i s system t h e Laplace operator is given by
where x, y a r e t h e two perpendicular axes. Thus i n Cartesian coordinates,
the governing d i f f e r e n t i a l equation (2.1) is given by
This equation has been solved f o r p l a t e s with various edge condit ions
subjected t o d i f f e r e n t types of loading. For al lround simply supported
p l a t e s a Navier type so lu t ion is employed, where t h e de f lec t ion and
load a r e expressed i n double trigonometric s e r i e s . Levy type so lu t ions
a r e a l s o ava i l ab le f o r p l a t e s with various edge condit ions (Wang, 1953;
Timoshenko and Woinowsky-Krieger , 1959; McFarland e t a l , 1972)
2.1.2.3 Limitat ions of the c l a s s i c a l theory
The bas ic d i f f e r e n t i a l equation of t h e c l a s s i c a l theory is of
four th order and hence requires four boundary condit ions f o r complete
solut ion. However, f o r a f r e e edge the re a r e too many ( s i x ) condit ions
a s pointed ou t by Kirchkoff i n 1850. This controversy was l a t e r resolved
by Kelvin and T a i t i n 1883 (Timoshenko and Woinowsky-Krieger, 1959).
Ho,wever, the approximations inherent in the der ivat ion of the
c l a s s i c a l theory r e s t r i c t i t s appl ica t ion t o very t h i n p l a t e s and it
cannot be applied with a reasonable degree of accuracy t o th ick p l a t e s
o r problems where shear o r l o c a l e f f e c t s such a s s t r e s s concent ra t ion
predominate. This has l e d t o t h e use of more r e f i n e d t h e o r i e s and i n
r e c e n t y e a r s many s i g n i f i c a n t c o n t r i b u t i o n s have been made i n t h e
f i e l d of a n a l y s i s of t h i c k p l a t e s .
2.1.3 Higher Order P l a t e Theories
he shear deformation theory o f Reissner (1945) i s by for t h e
most widely accepted c o n t r i b u t i o n t o the improvement of t h e c l a s s i c a l
t h i n p l a t e theory . By t h i s theory , it is assumed t h a t t h e bending
s t r e s s e s 0 0 and T vary l i n e a r l y a c r o s s t h e th i ckness of t he p l a t e X I Y XY
and t h a t a t an edge t h e displacements u and V a l s o vary l i n e a r l y a c r o s s
t h e p l a t e t h i ckness while t h e t r ansve r se d e f l e c t i o n i s cons tan t over
equat ions . However, Green (1949) has shown t h a t t he Reissner equat ions
can be ob ta ined d i r e c t l y from t h e t h r e e - dimensional e l a s t i c i t y equa-
t i o n s without t h e use of v a r i a t i o n a l techniques.
Seve ra l i n v e s t i g a t o ~ s have d iscussed t h e a p p l i c a t i o n of t h e
Reissner t heo ry t o var ious problems. Herrmann (1967) has included
t h e e f f e c t of t r a n s v e r s e shear deformation by employing a v a r i a t i o n a l
' p r i n c i p l e . -.The a n a l y s i s i s -appl ied t o simply supported c i r c u l a r p l a t e s
and c x c e l l e n t agreement e x i s t s with an e x a c t t h i c k p l a t e s o l u t i o n .
Carley and Langhaar (1968) have employed R e i s s n e r ' s theory t o s tudy
t h e behaviour of t r a n s v e r s e shear s t r e s s e s i n a l l round simply supported
p l a t e s sub jec t ed t o uniformly d i s t r i b u t e d loads . The study shows
genera l agreement between t h e Reissner and Kirchhoff t h e o r i e s i n r eg ions
away from t h e edges b u t a s h a r p disagreement a t t h e edges . Smith
(1968) h a s a p p l i e d R e i s s n e r ' s t h e o r y t o moderate ly t h i c k p l a t e s wi thout
cons ide r i ng deg ree s o f freedom a s s o c i a t e d w i th s h e a r deformat ions .
Aderheggen (1969) ha s used a complementary energy approach i n which t h e
d i sp lacement paramete rs a r e Lagragean m u l t i p l i e r s . Pryor e t a1 (1970)
have used a f i n i t e e lement which c o n s i d e r s deg ree s of freedom a s s o c i a t e d
w i th shea r deformat ions and t h r e e boundary c o n d i t i o n s on an edge. Deshmukh
and Robert (1974) employ t h e edge f u n c t i o n method t o s o l v e f l e x u r a l pro-
bleme of moderate ly t h i c k p l a t e s . The b a s i c e q u a t i o n s a r e t aken from t h e
Re i s sne r t h e o r y u s ing t h e two v a r i a b l e approach. Speare and :Kemp (1977) ,
and Vay i ad j i s and Baluch (1979) have g iven a one v a r i a b l e fo rmula t ion o f
t h e Re issner equa t i ons .
Donne11 (1954) h a s g iven a h ighe r o r d e r p l a t e i n which t h e s o l u t i o n
is i n t h e form o f an i n f i n i t e s e r i e s where t h e c l a s s i c a l t h i n p l a t e
t h e o r y is r ep re sen t ed by t h e f i r s t t e rm and a l l t h e o t h e r t e r m s a r e made
t o approach t h e three-dimensional s o l u t i o n . T h i s method h a s been a p p l i e d
t o s imply suppor ted r e c t a n g u l a r p l a t e s by Lee (1967) . P o n i a t o v s k i i (1962)
ha s used a Legendre polynomial expansion i n t h i c k n e s s c o o r d i n a t e t o o b t a i n
a system of two-dimensional e q u a t i o n s whi le Lur ' e (1964) u se s a g e n e r a l
power s e r i e s method o f s o l u t i o n o f t h e e l a s t i c i t y equa t i ons .
F r i e d r i c h s and D r e s s l e r ( 196 1 ) and Goldenweizer (1962) have g iven
a n approximate s o l u t i o n by t h e method o f asympto t ic i n t e g r a t i o n of t h e
gove rn ing equa t i ons . Bache and Hegemier (1974) have employed asympto t ic
expansion t o s o l v e e las to-dynamic p l a t e problems. J i a - r ang (1982) h a s
so lved t h e problem o f p l a t e s o f va ry ing t h i c k n e s s by t h e use of double
Four i e r t r an s fo rma t ions and two t r a n s c e n d e n t a l f u n c t i o n s .
S r i n i v a s e t a 1 (1969) have developed a three-dimensional so lu -
t i o n f o r r e c t a n g u l a r p l a t e s i n which t h e s o l u t i o n f o r d i sp lacements
a r e t aken i n t h e form o f double t r i gonome t r i c s e r i e s s a t i s f y i n g t h e
e q u a t i o n s o f e q u i l i b r i u m i n t e r m s o f d i sp lacements . S e b a s t i a n (1983)
h a s a l s o o b t a i n e d a s o l u t ~ ~ ; n f o r r e c t a n g u l a r p l a t e s based on t h e
Gala rk in v e c t o r s t r a i n f u n c t i o n approach of e l a s t i c i t y .
Vlasov (1957) , u s ing a mixed method i n e l a s t i c i t y has developed
a t h i c k p l a t e t heo ry t h e governing e q u a t i o n s o f which a r e ob t a ined
from t h e th ree-d imens iona l e l a s t i c i t y equa t i ons . The unknowns o f t h e
problem a r e expanded i n MacLaurin s e r i e s i n t h e t h i c k n e s s coo rd ina t e
and t h e s o l u t i o n i s o b t a i n e d i n terms o f unknown i n i t i a l f u n c t i o n s on
t h e r e f e r e n c e p l ane and fo l l owing Vlasov, t h e method is r e f e r e d t o as
t h e method o f i n i t i a l f u n c t i o n s (MIF). By p r o p e r l y t r u n c a t i n g t h e
s e r i e s approximate t h e o r i e s o f any d e s i r e d o r d e r are e a s i l y ob t a ined .
D a s and S e t l u r (1970) , Rao and Das (1977) , Bahar (1977) and Iyengar
and Roman (1980) have employed t h e method t o s o l v e e las todynamic p ro-
blems. Bahar (1972) and Oj iako (1985) have a p p l i e d t h e method t o
l a y e r e d systems whi le Iyengar e t a1 (1974, 1974) have used t h e same
method t o , a n a l y s e t h i c k r e c t a n g u l a r p l a t e s and.beams.
2.2 PLATES ON ELASTIC FOUNDATIONS
A survey of t h e l i t e r a t u r e shows t h a t i n t h e a n a l y s i s o f v a r i o u s *
s t r u c t u r a l forms r e s t i n g on e l a s t i c f ounda t i ons , t h e r e l e v a n t governing
e q u a t i o n s a r e o n l y modif ied t o t a k e t h e e f f e c t o f t h e founda t ion re-
a c t i o n p i n t o account . The work o f ~ e ' t e n ~ i (1946) is one o f t h e
10
formost i n the s tudy of s t r u c t u r a l forms on e l a s t i c Foundations.
H i s work cons iders various types of beams r e s t i n g on e l a s t i c foundations
and s e v e r a l techniques inc luding t h e use of s e r i e s so lu t ions (Hetenyi,
1977) have been employed t o solve t h e governing d i f f e r e n t i a l equat ions.
Hetenyi ' s work has been extended t o p l a t e s by seve ra l i nves t iga to r s .
McFarland e t a 1 ( 1972) , Timoshenko and Woinowsky-Krieger ( 1959) ,
S c o t t (1981) have d iscussed t h i n p l a t e s on e l a s t i c foundat ions. The
b a s i c d i f f e r e n t i a l equat ion of t h e c l a s s i c a l t h i n p l a t e theory i s
modified by adding t h e
rec tangular p l a t e s t h e
e f f e c t of t h e foundation r eac t ion p. Thus f o r
equat ion becomes
The form of t h e foundation r eac t ion p depends on t h e foundation model
empl~yed. This is d iscussed i n d e t a i l i n sec t ion 2.4. Navier and
Levy type so lu t ions a r e a v a i l a b l e f o r equat ion (2 .5 ) . Livesely (1953)
has given a mathematical t rea tment of e l a s t i c p l a t e s on e l a s t i c founda-
t i o n s under s t a t i c and dynamic loads. Sonoda and Kobayash (1980), have
i n l i n e with P i s t e r and Williams (1960) s tudied t h i n rec tangular p l a t e s
on Reissner type v isco e l a s t i c foundations. Brown e t a 1 (1977) have
a l s o s tud ied beam-plate systems r e s t i n g on e l a s t i c foundat ions. Many
con t r ibu t ions have a l s o been made i n t h e a n a l y s i s of t h i n c i r c u l a r
p l a t e s bn e l a s t i c foundations. The works of Krajcinovic (19761, Sonoda
6
e t a 1 (1978) , Gazetas (1982) and Kamal and Durvasula (1983) (Ire some
of t h e con t r ibu t ions i n t h i s a rea .
The a n a l y s i s of t h i c k p l a t e s on e l a s t i c foundat ions has been
made poss ib l e by t h e in t roduc t ion of t he foundation r e a c t i o n i n t o t h e
equat ions of t h e r e l e v a n t t h i c ~ p l a t e t h e o r i e s . Freder ick (1957) ,
using t h e b a s i c equa t ions of t h e Reissner theory ana lysed t h i c k p l a t e s
on e l a s t i c foundat ions by inc luding t h e e f f e c t of t h e foundat ion i n t o
t h e equa t ions . I t is shown t h a t s o l u t i o n s of t h e Levy and Navier t ypes
can be used. Svec (1976) and Vykuti l (1982) have used f i n i t e e lements
t o ana lyse t h i c k p l a t e s r e s t i n g on e l a s t i c foundat ions. Shear e f f e c t s
a r e considered and Re i s snc r ' s theory is b a s i c a l l y app l i ed . V o y i a d j i ~
and Baluch (1979) have used a one-variable formulat ion o f t h e Reissner
theory and employed a success ive approximation technique t o t h e a n a l y s i s
of t h i c k p l a t e s on e l a s t i c foundat ions.
2.3 FOUNDATION MODELS
A d e t a i l e d review of t h e va r ious foundat ion models a v a i l a b l e has
been made by Kerr (1964) and Zhaohua and Cook (1983). I t i s seen t h a t
t h e va r ious models can be c l a s s i f i e d broadly i n t o one-and two-parameter
foundakion models depending on t h e number of cons t an t s used i n d e f i n i n g
t h e r e a c t i o n of t h e foundat ion t o load. F l e t c h e r and Herrmann (1971)
have desc r ibed a three-parameter model b u t a s S c o t t (1981) p o i n t s o u t ,
it is very u n l i k e l y f o r a case t o be made f o r t h e use of a three-para-
meter model and hence it w i l l n o t be d i s cus sed f u r t h e r i n t h i s r e ~ i e w .
The va r ious foundat ion models w i l l now be d i scussed .
2.3.1 One-Parameter (Winkler) Foundation Model
The s imp le s t and most widely accepted r e p r e s e n t a t i o n of a con- b
t i nuous e l a s t i c foundat ion under va r ious s t r u c t u r e s was given by
Winkler i n 1867. H e assumed t h a t t h e r e a c t i o n o f t h e foundat ion a t
a given p o i n t is d i r e c t l y p ropor t i ona l t o t h e d e f l e c t i o n a t t h a t po in t .
12
Mathematically, t h i s means t h a t t h e r e a c t i o n p ( x , y ) and the d e f l e c t i o n
w(x ,y) bo th i n t h e t r ansve r se z -d i r ec t ion a r e r e l a t e d by t h e expression
p ( x , y ) = k w(x,yl . . . (2.6)
where k is t h e foundat ion parameter o r modulus. The phys i ca l meaning
is t h a t t h e foundat ion c o n s i s t s of c l o s e l y spaced, indepeddent l i n e a r
s p r i n g s and it is a one-parameter foundat ion s i n c e only one modulus is
used t o de f ine t h e behaviour of t h e foundat ion.
By s u b s t i t u t i n g equat ion (2.6) i n t o equat ion (2.5) w e ob t a in t h e
d i f f e r e n t i a l equat ion f o r t he bending of a r ec t angu la r p l a t e on t h e
one-parameter Winkler foundat ion:
Navier and Levy type s o l u t i o n s havc been given t o t h i s equat ion
(Timoshenko and Woinowsky-Krieger, 1957, McFarland e t a l , 19?2)
2.3.1.: C o n s i 6 e r a t i ~ n s of t h e foundat ion moduius
In any a n a l y s i s o r des ign of any s t r u c t u r a l form on t h e Winkler
foundat ion, t h e foundation modulus K must be known. Unfortunately, due
t o t h e n a t u r e of most foundat ion m a t e r i a l s which a r e n e i t h e r homogeneous
nor i s o t r o p i c ( s o i l s f o r example), t h e d e t e r ~ u n a t i o n of t h e parameter k
i s n o t an easy matter . I t has u n i t s of KIV/rn3 and what is mostly seen
i n t h e l i t e r a t u r e a r e ranges of va lues f o r var ious s o i l types (Timoshenko
and Woinowsky-Krieger, 1959, McFarland e t a l , 1972)
However, some experimental i n v e s t i g a t i o n s i n t o t h e value of k
have been c a r r i e d o u t by V e s i c (1961) and F le t che r and Herrmann (1971).
Vesic , working on beams r e s t i n g on i s o t r o p i c e l a s t i c s o l i d chose a r e l a -
t i o n between k and t h e m a t e r i a l p r o p e r t i e s a s
0 . 65E E. B~ 6 S
k = . . . (2 .8 ) 1 -p2 s 124"'
where E and p a r c t h c Young's msdulus and Po i s son ' s r a t io o f t h e S S
s&grzde, B is t h e width o f t h e beax and E , I a r c t h e Young's modulus,
moment o f i n e r t i a of t h e beam. I n o r d e r t o match v a l u e s o f v a r i a b l e s
between t h e Winkler and continuum s o l u t i o n s some m o d i f i c a t i o n s have
been made t o e q u a t i o n (2.8) a s r e p o r t e d by S c o t t (1981) . The modifica-
t i o n s made by Vesic i n o b t a i n i n g t h e subgrade r e a c t i o n f o r s l a b s o r
p l a t e s on Winkler founda t ion have a l s o been r e p o r t e d by S c o t t (1981).
The modif ied va lue k is g iven by 0
where H = p l a t e t h i c k n e s s and y i s P o i s s o n ' s r a t i o o f p l a t e . Thus
k depends on t h e r e l a t i v e p r o p e r t i e s of t h e s l a b and subgrade and 0
a l s o v a r i e s i n v e r s e l y a s t h e t h i c k n e s s o f t h e slab. F l e t c h e r and
Herrmann (1971), r e s t r i c t i n g t h e r a t i o o f subgrade Young's modulus,
E t o beam modulus E t o l e s s t han 0.01 ob t a ined c u r v e s f o r t h e selec- s
t i o n o f t h e founda t ion modulus. I t should b e no ted t h a t i n a l l t h e s e
i n v e s t i g a t i o n s , t h e founda t ion r e a c t i o n i s d e f i n e d on ly i n terms of
t r a n s v e r s e d e f l e c t i o n . The g r e a t e s t asset o f t h e Winkler model is
t h e s i h p l i c i t y o f i t s mathemat ical r e p r e s e n t a t i o n .
2.3.1.2 Limi ta t ions of t h e Winkler foundation
Fig 2 . l ( a ) shows the deformations of the Winkler foundation
su r face f o r a uniform load. I t is seen t h a t t h e displacements i n
t h e loaded a r e a a r e cons tant while o u t s i d e t h i s region t h e d isp lace-
ments a r e zero. This does not r ep resen t t h e a c t u a l foundation be-
haviour a s t h e displacement of t h e foundation sur face f o r most ma te r i a l s
is a s shown i n f i g 2.1 (b ) (Kerr, 1964 . Vesic (1961) has found t h a t t he Winkler model r ep resen t s t h e
behaviour of beams on s o i l s f a i r l y wel l b u t Jones and Xenophontos
(1377) have s t a t e d t h a t t h e model is less s a t i s f a c t o r y when app l i ed
t o p l a t e s . Due t o these s h o r t comings seve ra l foundat ion models
have been suggested, i n r ecen t yea r s , i n an at tempt t o g ive phys ica l ly
c l o s e r ep resen ta t ions . I n these models t h e foundation is descr ibed
by two p r o p e r t i e s and they a r e thus r e f e r r e d t o a s two-parameters
models.
2.3.2 Two-Parameter Foundation Models
I n o rde r t o desc r ibe models t h a t c l o s e l y r ep resen t r e a l founda-
t i o n s some i n v e s t i g a t o r s have assumed some kind of i n t e r a c t i o n between
t h e Winkler s p r i n g s while o t h e r s s t a r t i n g from continuum equat ions
have introduced s impl i fy ing assumptions with r e s p e c t t o displacements
and/or s t r e s s e s . The assumptions made have l e d t o many two-parameter
models which a r e now discussed . 6
2.3.2.1 Filonenko-Borodich foundation
This model, developed i n 1940, is due t o Filonenko-Borodich who
s t a r t e d from t h e Winkler model and assumed some degree of i n t e r a c t i o n
Fig. 2.1 Deflection of foundation surface (a) Winkler foundation
(b) real foundation
between t h e s p r i n g elements (Kerr , 1964). I t i s assumed t h a t t h e t o p
ends of t h e s p r i n g s a r e connected t o an e l a s t i c membrance which is
s t r e t c h e d by a c o n s t a n t t en s ion f i e l d T. By t h i s model, t h e r e l a t i o n -
s h i p between t h e foundat ion r e a c t i o n p ( x , y ) due t o a given l o a d and t h e
d e f l e c t i o n w(x,y) is given by
where k i s t h e Winkler foundat ion modulus and VZ is t h e Laplac ian opera-
t o r i n x and y. The e v a l u a t i o n of t h e second parameter T is n o t r e p o r t e d
i n t h e l i t e r a t u r e a v a i l a b l e t o t h i s Gtudy.
2 .3 .2 .2 Pas te rnak foundat ion
The Pas te rnak founda t ion model w a s developed i n 1954 and it assumes
t h e e x i s t e n c e o f s h e a r i n t e r a c t i o n s between t h e s p r i n g elements o f t h e
Winkler model. The t o p ends of t h e s p r i n g s a r e connected to a beam o r
p l a t e c o n s i s t i n g of incompress ib le v e r t i c a l e lements , which deform on ly
by t r a n s v e r s e shea r (Jones and Xenophon t o s , 1977) . To d e r i v e t h e load-
d e f l e c t i o n r e l a t i o n t h e v e r t i c a l equ i l i b r i um of an e lement of a shea r
l a y e r is cons ide red and assuming t h e founda t ion m a t e r i a l t o be homogeneous
and i s o t r o p i c , t h e foundat ion r e a c t i o n p ( x , y ) , due t o a g iven l oad , is
g iven byp-
p ( x , y ) = kw(x,y) - G v2w(x ,y) S
... (2.11)
where G i s t h e s h e a r modulus o f t h e foundat ion m a t e r i a l and i s given by 6
17
Thus G can be eva lua ted i f Young's modulus and Poisson ' s r a t i o of t h e s
foundat ion ( E ,ps) a r e known. The curves given by F le t che r and Herrmann S
(1971) can a l s o be used t o determine t h e parameters i n t h i s model.
Von Mises ha.s obta ined , f o r t h e two-dimensional case an expression
of t h e form given i n equat ion (2.11) by expanding Wieghardt's i n t e g r a l
r e l a t i o n f o r load and d e f l e c t i o n s (Kerr, 1964). This confirms equat ion
(2.11) a s a c o r r e c t approximation of t h e foundation response. This
argument, a s wel l a s t h e mechanical behaviour of t h e model and t h a t of
t h e r e a l medium has made Kerr t o s t a t e t h a t t he Pasternak model i s t h e
most n a t u r a l ex tens ion of t h e Winkler model of a l l t h e models t h a t
s t a r t e d wi th t h e Winkler foundation. The bending of p l a t e s r e s t i n g on
Pasternak-type foundat ions has been d iscussed i n d e t a i l by Kerr (1964)
2.3.2.3 General ised foundation
This model is a l s o an extens ion t o t h e Winkler hypothesis . An
a r b i t r a r y assumption is made t h a t a t each po in t where t h e foundation
r e a c t i o n i s propor t ional t o t h e d e f l e c t i o n w, t h e r e is a l s o a moment
m(x,y) which i s propor t ional t o t h e angle of r o t a t i o n . This a d d i t i o n a l
assumption can be expressed mathematically a s
where n is any d i r e c t i o n a t a po in t i n t h e p lane of t h e foundation
su r face and K is t h e p r o p o r t i o n a l i t y f a c t o r . By conver t ing m(x,y) 0
i n t o r a n equiva lent su r face load we ob ta in t h e foundation r e a c t i o n
p ( x , y ) due t o a given p res su re a s
p ( x , y ) = I w(x,y) - K v2w(x,y) 0
Although t h e genera l i sed foundation i s r e l a t i v e l y a r b i t r a r y , it
has been shown by Kerr (1964) t h a t t h e Pasternak foundation i s a p o s s i b l e
model f o r t h e gene ra l i s ed foundation, hence g iv ing it a phys ica l meaning,
2.3.2.4 Hetenyi foundation
Hetenyi has a l s o suggested a modif icat ion t o t h e Winkler model. H e
assumes i n t e r a c t i o n between t h e sp r ing elements by imbedding an e l a s t i c
beam i n t h e two-dimensional case and a p l a t e i n t h e three-dimensional case ,
The fondat iDnreact ion i s expressed as
where D i s t h e f l e x u r a l r i g i d i t y of t h e p l a t e
2.3.2.5 Vlasov foundation
Vasov s t a r t e d from a continuum p o i n t of view. He assumed t h e founda-
t i o n as a semi - in f in i t e e l a s t i c medium, imposed c e r t a i n r e s t r i c t i o n s on
t h e poss ib l e deformations of an e l a s t i c l aye r and used a v a r i a t i o n a l
method t o ob ta in a load-def lec t ion r e l a t i o n t h a t is given by
p (x ,y ) = kw(x,y) - 2 t V2w(x,y) . . . (2.16)
where k and 2 t a r e parameters i n terms of e l a s t i c cons tan t s and dimensions
of p l a t e and foundation i n v e s t i g a t o r s and is being used i n t h e design
of s t r u c t u r e s on s o i l foundations. I t has t h e advantages of being
derived-from a continuum approach and i s a l s o a simple coupled sp r ing
model. The model is, however, s t rong ly dependent on t h e assumed
form o f t h e v e r t i c a l deformation and a good choice must be made t o b
ob ta in good r e s u l t s . Jones and Xenophontos (1977) using a d i f f e r e n t
v a r i a t i o n a l p r i n c i p l e have obtained t h e Vlasov model and provided a
r igorous t h e o r e t i c a l b a s i s for t he form of t h e v c r t i c a l deformation
p r o f i l e . They have found a good agreement between t h e t h e o r e t i c a l l y
p r e d i c t e d and exper imenta l ly determined v e r t i c a l deformation p r o f i l e s .
The func t ion desc r ib ing t h e v e r t i c a l deformation p r o f i l e g(Z)
is s e l e c t e d t o f i t t h e boundary cond i t i ons namely, u n i t value a t ground
s u r f a c e and zero a t dep th H i n a f i n i t e l a y e r o r a t i n f i n i t y f o r a f
ha l f - space . For t h e f i n i t e l a y e r Vlasov assumes
where $ is an unknown cons t an t determining t h e v a r i a t i o n of t h e v e r t i c a l
d i sp lacements wi th depth. However, S c o t t ( 1981 ) has used t h e fol lowing
express ion :
- 1 f o r t h e ha l f space where V is a cons t an t with dimension L express ing
t h e r a t e a t which v e t t i c a l displacement decays with z and
f o r t h e f i n i t e l a y e r .
The parameters k and 2 t of t h i s foundat ion model a r e ob t a ined
r a t i o n a l l y . k c h a r a c t e r i s e s t h e compressive s t r a i n i n t h e foundat ion
and..is e q u i v a l e n t t o t h e Winkler modulus while t c h a r a c t e r i s e s t h e
shear ing s t r a i n i n t h e foundat ion. They a r e given f o r t h e h a l f space
20
where E and p a r e t h e Young's modulus and P o i s s o r . ' ~ r a t i o of t h e S s
f o m d s t i o n medium i n a plan€ s t r a i c problem. In q plane s t r e s s con-
d i t i o n they a r e rep laced by En and p where - 0
Taking t h e v a r i a t i o n of t h e v e r t i c a l d e f l e c t i o n i n t h e form given bj
equat ion (2.18) t h e i n t e g r a l s i n equat ions (2.20) and (2.21) a r e eva-
lua t ed and t h e foundat ion parameters a r e thus given by
f o r t h e p lane s t r a i n case and
f o r t h e plane s t r e s s c a s e ,
f o r both plane s t r a i n and plane s t r e s s except t h a t f o r plane s t r e s s ,
pc a r c rep laced by Eo, Eg,
pa (Sco t t , 1381; Jones and Xenophontos,
1977)
2.3.2.6 Reissner foundation
Reissner a l s o s t a r t e d from t h e equat ions of a continuum and
a.ssumed t h a t t h e in-plane s t r e s s e s throughout the foundation l aye r a r e
neg l ig ib ly smal l : = 0 = T = 0 and t h a t ho r i zon ta l displacements Ox Y XY
a t t o p and bottom of tlic foundation a r c zero. I l e obtained t h e r e l a t i o n
where
For a cons t an t o r l i n e a r l y varying load , a f t e r r ede f in ing c o n s t a n t s
( C =k and c = G 1 eqca t ion (2.26) f s i d e n t i c a l t o equat ion (2.11) 1 2 S
The assumptions t h a t t h e in-plane s t r e s s e s a r e n e g l i g i b l e l e a d s
t o t h e f a c t t h a t t h e shear s t r e s s e s T and T are independent of z X z YZ
and hence cons t an t throughout t h e depth of t h e foundat ion. This i s
u n r e a l i s t i c e s p e c i a l l y f o r t h i c k foundat ion l a y e r s . However, t h i s
de f i c i ency is of no s e r i o u s consequence s i n c e foundat ion models study
t h e response of t h e foundat ion t o loads and n o t t h e stresses wi th in
t h e foundat ion.
In gene ra l , t h e r eac t ion of a two-parameter foundat ion t o a
given load is given by
p (x ,y ) = k w(x,y) - kl VZw(x,y)
g iv ing t h e d i f f e r e n t i a l equat ion of a t h i n p l a t e r e s t i n g on a two-para-
meter foundat ion a s
2.4 GENERAL REMARKS
From t h e l i t e r a t u r e it i s seen t h a t t h e behaviour of t h i n and
t h i c k p l a t e s r e s t i n g on e l a s t i c foundat ions has been widely t r e a t e d .
T$e c l a s s i c a l t h i n p l a t e theory i s modified by inco rpora t ing t h e founda-
t i o n r e a c t i o n i n t h e a n a l y s i s of t h i n p l a t e s on e l a s t i c foundat ions. In
t h e case of t h i c k p l a t e s , Re i s sne r ' s o r o t h e r h igher o rde r t h e o r i e s a r e
s i m i l a r l y modified t o take t h e e f f e c t of t h e foundation i n t o considera-
t i o n . However, t h e l i t e r a l ru rc shows t h a t Vlasov' s ~ricthod of i n i t i a l
2 2
f u n t i o n s has not been a p p l i e d t o p l a t e s on e l a s t i c foundat ion.
It i s a l s o observed t h a t t h e Winkler foundat ion i s s t i l l widely
used t o r e p r e s e n t t h e behaviour o f e l a s t i c foundat ions under load
d e s p i t e i t s depa r tu re from r e a l i t y . However, many models have been
proposed i n o r d e r t o b r i n g t h e t h e o r e t i c a l formula t ions c l o s e t o
r e a l i t y . I n t h i s connection t h e model proposed by Pasternak has been
accepted a s a very n a t u r a l ex tens ion t o t h e Winkler model. The Vlasov
foundat ion model i s seen a s a r a t i o n a l model s i n c e it is de r ived from
a continuum p o i n t view and i t s v a l i d i t y has been confirmed by r igo rous
mathematical d e r i v a t i o n s .
CHAPTER THREE
DEVELOPMENT OF THE GOVERNING EQUATIONS
3.1 THE METHOD OF INITIAL FUNCTIONS (MIF)
3.1 .1 I n t r o d u c t i o n
The method of i n i t i a l f u n c t i o n s (MIF) , proposed by Vlasov i n
1957, is a mixed method i n e l a s t i c i t y . I n o b t a i n i n g t h e s o l u t i o n s ,
no p r e f e r e n c e i s g iven t o s t r e s s e s o r d i sp l acemen t s as i n t h e s t r e s s
o r d i sp lacement approach.
Vlasov h a s s t a r t e d wi th t h e three-dimensional e q u a t i o n s o f e l a s -
t i c i t y and e l i m i n a t e d t h e in-plane s t r e s s e s o ,o and -r t o reduce x Y XY
t h e nanber o f equa t i ons . The unknowns o f t h e problem a r e expanded i n
Mac-Laurin s e r i e s i n t h e t h i c k n e s s c o o r d i n a t e s o t h a t t h e s o l u t i o n is
ob t a ined i n t e rms of unknown i n i t i a l f u n c t i o n s on t h e r e f e r e n c e p l ane .
He obse rve s t h a t t h e i n f i n i t e s e r i e s i n t h e d i f f e r e n t i a l o p e r a t o r can
be w r i t t e n i n c l o s e d form. By p rope r t r u n c a t i o n o f t h e s e r i e s , approx i -
mate t h e o r i e s of any d e s i r e d o r d e r a r e ob t a ined . The fo rmula t ion of
t h e MIF is now p r e s e n t e d as g iven by Iyenqar e t a 1 (1974) , Rao and
Das (1977)
3.1.2 De r iva t i on of t h e Basic Equat ions
For a homogeneous, i s o t r o p i c and l i n e a r l y e l a s t i c s o l i d , t h e
equa t i ons o f e q u i l i b r i u m i n t h e absence of body f o r c e s a r e :
and the stress-displacement equations a re
where (5 , u y~ uZI T xy' ' xzl T~~ a re the s t r e s s components in the
X
respective direct ions , u,v,w are the displacements components in the
x ,y ,z direct ions , p is ~ o i s s o n ' s r a t i o and G the modulus of r i g id i ty .
Using the following symbols:
and eliminating the in-plane s t resses 0 0 T between equations x' y' xy
( 3 . 1 ) and ( 3 - 2 ) since 'they. do-not 'appear in the surface conditions a t the
top and bottom faces , the following equations a re obtained:
The in-plane stresses a r e given by
Vlasov has assumed t h e so lu t ions of equations (3.4) i n t h e
form of in£ i n i t e MacLaurin s e r i e s i n z as
where a l l the higher de r iva t ives a r e obtained from equations (3 .4 ) .
Rao and Das (1977) have used matrix no ta t ion t o solve t h e equations
(3.4) and 6ollowing t h e i r method equations (3.41 a r e r e c a s t i n matrix
hota t ion as
and
where
and 1'-
The d i f f e r e n t i a l operators a,B follow the ru les of algebra. Vlasw 's
suggested form of the general solutions of equations (3.4) can now be
put i n matrix form a s
2 7
where U o, Vo, ... X a r e t h e respect ive values of U, V, ... X on the 0
i n i t i a l plane Z=0 and a r e c a l l e d the i n i t i a l functions. In general ,
any plane Z = constant can be taken a s t h e i n i t i a l plane and t h e
above expressions can be modified appropriately.
The higher order de r iva t ives of t h e funct ions with respect t o
Z a r e obtained by repeated operat ion of r on equations (3.7). For
example, by opera t ing r once on equations (3.7) we obta in higher-
order even d e r i v a t i v e s i n t h e form
where [Cl = [A] [B l o r
and
*
I t is s e e n from e q u a t i o n s (3 .12) a n d ' (3 .13) t h a t m a t r i x [R] can b e
o b t a i n e d by t r a n s p o s i n g t h e e l e m e n t s o f m a t r i x [CI a b o u t t h e secondary
d i a g o n a l . Higher even o r d e r d e r i v a t i v e s can b e o b t a i n e d by r e p e a t e d
u s e o f e q u a t i o n s (3.11) w h i l e h i g h e r o r d e r odd d e r i v a t i v e s c a n be
o b t a i n e d by u s i n g e q u a t i o n s ( 3 . 7 ) and
1:7 - - LC' J
L z J
( 3 . 1 1 . Thus w e have
By g r o u p i n g cvcn and odd powers o f r , and also s u b s t i t u t i n g e q u a t i o n s
( 3 . 7 ) , (3 .11) and ( 3 . 1 4 ) i n e q u a t i o n s (3 .19) t h e r e s u l t i n g e x p r e s s i o n s
are
where I i s a 3x3 i d e n t i t 1 m a t r i x .
E q u a t i o n s ( 3 . 1 5 ) c o n t a i n f u n c t i o n s o f t h e m a t r i c c z [C] and [R]
a n d S y l v e s t e r ' s and Cay ley -Hami l ton ' s t heo rems , which depends on t h e
characteristic r o o t s o f t h e m a t r i c e s [C] and [ t i ] , h a v e been u s e d t o
s o l v e t h e e q u a t i o n s , A s m a t r i x [ R ] i s o b t a i n e d by t r a n s p o s i n g t h e
e l e m e n t s o f m a t r i x [C] a b o u t i t s secondary d i a g o n a l , it f o l l o w s t h a t
t h e two matrices have i d e n t i c a l c h a r a c t e r i s t i c r o o t s . The t h r e e
c h a r a c t e r i s t i c r o o t s o f t h e m a t r i c e s IC] and [ R ] are g i v e n by
= = < = c = - 2 3 Y . . . (3 .16a)
as shown i n Appendix A i n which
Applying Sy l v c s t s r ' s and Cay ley-Hamilton ' s theorems equations (3.15)
can be wr i t t en as
Z S i n yz = [ C o s y2.c - ,(------)C 1
1 ,.. Y 2
where
(See Appendix A )
The c l o s e d form e x p r e s s i o n s observed i n e q u a t i o n s (3.17) f o r two
d imens iona l c a s e have been o b t a i n e d d i r e c t l y by Bahar (1975) who
used a s t a t e space approach. From e q u a t i o n s (3.17) t h e s t r e s s e s and
d i s p l a c e m e n t s a t any p o i n t i n t h e body a r e e x p r e s s e d i n t e r m s o f
t h e i n i t i a l f u n c t i o n s as
where L L ..., L a r e l i n e a r d i f f e r e n t i a l o p e r a t o r s . They uu' uv' XX
a r e l i s t e d i n Tab le 3.1.
3.2 APPLICATION OF MIF TO PLATE ON WINKLER FOUNDATION
The a p p l i c a t i o n o f t h e MIF t o t h e a n a l y s i s o f t h i c k r e c t a n g u l a r b
p l a t e s h a s been d i s c u s s e d by Vlasov (1957) and Iyengar e t a 1 (1974) .
Proceeding on s i m i l a r . l i n c s t h e method i s now a p p l i e d t o t h e a n a l y s i s
o f p l a t e s r e s t i n g on a Winklcr foundclt ion
Fig 3 , l ( a ) shows t h e coo rd ina t e system adopted i n t h e a n a l y s i s
of a r ec t angu la r p l a t e of dimensions axbx 2h. The p l a t e is sub jec t ed
t o a su r f ace load of q kN/m2 a s shown i n F ig 3 . l ( b ) . The foundat ion
r e a c t i o n p (x ,y ) i s given by equat ion (2.6). Using t h e n o t a t i o n s of
equat ions (3.3') t h e foundat ion r e a c t i o n can be expressed a s
k ~ ( X I Y ) = F Wh(x1y) . . . (3.20)
where t h e t r a n s v e r s e d e f l e c t i o n , W component i s taken a t z = h. To h '
s impl i fy t h e d e r i v a t i o n , t h e problem is d iv ided i n t o symmetric and a n t i -
symmetric p a r t s as i n Fig 3.1 (c) and (dl and these w i l l be d iscussed
sepa ra t e ly .
3.2.1 Symmetric (o r Extension) Problem-
The middle p lane of t h e p l a t e i s taken a s t h e r e f e rence ( o r
i n i t i a l ) p lane and due t o symmetry Wo, Xo, Y a r e a l l zero. The a p p l i e d 0
loads a r e Z = ~ C ~ ~ ( X , ~ ) + ~ ( X , ~ ) } , Y = 0 and X = 0 on z = +h and by us ing
them i n p l a c e of t h e stress components Z,Y,X i n equat ions (3.19) t h e
fol lowing d i f f e r e n t i a l equat ions a r e obta ined f o r t h e s o l u t i o n of t h e
unknown i n i t i a l func t ions :
Fig, 3.1 (a) Coordinate
part , ( d l system, (b) Plate on elastic foundation (c) Symmetric
anti-symmetrlc part
The l a s t two of equat ions (3.21) are s a t i s f i e d i f a funct ion $'
is chosen such t h a t
- Vo - - (Lxu L yz - Lyu L x z h 1 $ ' . . . (3.22)
Bp expanding t h e o p e r a t o r s using Table 3.1 t h e f i r s t of equat ions
(3.21) becomes
2
2 Sinyh (yh + Sinyh Cosyh)$' = - fq (x ,y ) - tp (x ,y ) . . . (3.23) 1 -B
Pu t t ing
Y (- Sinyh)$ ' = $ 1 -B
equat ion (3.23) can be w r i t t e n as
y (yh + Sinyh Cosyh)$ = -fq (x,y) - tp (x ,y) . . . (3.25)
where 4 i s a new funct ion . S u b s t i t u t i n g f o r p ( x , y ) from equat ion
(3.201, t h e d i f f e r e n t i a l equat ion (3.25) becomes
K y ( y h + Sinyh Cosyh)$ + -
2G Wh(x,y) - - tq (x ,y ) . . . (3.26)
Now, by expanding equat ions (3.22) us ing Table 3.1 and s u b s t i t u t i n g
(3.241, t h e i n i t i a l func t ions becomes
a , uo = - [ ( I - 2 ~ ) - -
2 hcosyh] @
Y B S inyh
Vo = - [ ( I - 2 ~ ) - - hCosyh1 $ 2 Y
b
Using equat ions (3.19) t h e d e f l e c t i o n W a t t h e bottom of t h e p l a t e
(z=h) i s g'iven by
which on e x p a n s i o n and s i m p l i f i c a t i o n , u s i n g Tab le 3.1 and e q u a t i o n s
(3.271 a r e o b t a i n e d i n t e r m s o f t h e f u n c t i o n (i, a s
s u b s t i t u t i n g e q u a t i o n (3 .29) i n e q u a t i o n (3 .26 ) t h e g o v e r n i n g d i f f e r e n -
t i a l e q u a t i o n i s g i v e n by
[y2 h i -ys inyh Cos yh + X ~ i n ' y h ] @ = - $ q ( x , y ) . . . (3 .30)
where
3.2.2 Anti-Symmetric ( o r Bending) Problem
The midd le p l a n e (z=O) is s t i l l t a k e n as t h e i n i t i a l p l a n e and
as t h e l o a d i n g is a n t i - s y m m e t r i c , "0
Vo, Z a r e z e r o on t h i s p l a n e . 0
The a p p l i e d l o a d s are X = \r = 0 , Z = l : f q ( x , l r ) - p ( x , y ) ) on Z = th.
Using t h e s e i n e q u a t i o n s (3 .19 ) t h e f o l l o w j n g d i f f e r e n t i a l e q u a t i o n s
are o b t a i n e d :
The f i r s t two o f e q u a t i o n s ( 3 . 3 2 ) are s a t i s f i e d i f a f u n c t i o n F ' is
chosen s u c h thl:
This reduces t o
where
F = (Cosyh) F' . . . (3.36)
By expanding equat ions (3.33) using Table 3.1 and s u b s t i t u t i n g equation
(3.36) t h e i n i t i a l funct jons i n terms o f F a r e given by
xo = - a h Sinyh . F
1 -P
From equat ions (3.19) t h e d e f l e c t i o n W a t t he bottom of t h e p l a t e '
(z=h) is given by
Wh(x,y) = (L 1 W + (LwyIh Yo+(LWX X . . . (3.38) ww h o
Using Table 3.1 and equations (3.37) t h i s s i m p l i f i e s t o
wh(x1y) = ( c o s 2 y h ) ~ . . . (3.39)
Subs tu t ing equation (3.39) i n equation (3.35 ) t he governing d i f f e r e n t i a l
equat ion becomes
which i n expanded form i s given by . .
b
where is given by equation (3.31).
To ob ta in t h e so lu t ion of the problem i n Fig. 3 . l ( b ) t h e so lu t ions
f o r t h e symmetrical and anti-sy~nmetrical. p a r t s a r e sumnied up.
The funct ions 4 and F arc now cvaluatcd fo r problcnis w l t h various boundaqy
c o n d i t i o n s .
3 . 2 . 3 Allround Simply Supported P l a t e
The boundary cond i t i ons i n terms o f s t r e s s e s and displacements are:
Equations (3.30) and (3.40) a r e s a t i s f i e d by t a k i n g 4, F and t h e l oad
q (x ,y) i n t h e form
= 4 S i n a x S in Bny Inn m
F = 3 f m S i n amx S in Bny
= 2 q S i n a x s i n Bny mn m
where q i s t h e f o u r i e r c o e f f i c i e n t o f t h e l oad and mn
S u b s t i t u t i n g equa t ions (3.42) i n t o t h e d i f f e r e n t i a l equa t ions (3.30)
and (3.40) t h e c o e f f i c i e n t s 4 and f 3re ob ta ined ( s ee Appendix B ) a s rnn mn
Hence t h e f u n c t i o n s @ and F a r e given by
q S ina x s i n 6 4 = c c mn m n Y
m n 6 (sinh26h + 26h) + 2Xsinh2 6h
39 Now, using equat ions (3.19) t h e s t r e s s e s and displacements can be
obta ined by summing up f o r each quan t i ty , t h e r e s u l t s of t h e symmetric
and anti-symmetric p a r t s . They a r e given by
By using able 3.1 and equat ions (3.27) and (3.37) and s impl i fy ing , the
fol lowing express ions a r e obtained (see Appendix B)
. . . (3.47) Equat ions (3.45) a r e .now s u b s t i t u t e d i n t o equa t i ons ( 3 -47) t o o b t a i n
the exp re s s ions f o r t h e s t r e s s e s and d i sp lacements a t any p o i n t ( x , y , z )
(See Appendix B) t h u s
aln C&Z { ( 1 - 2 ~ ) ~ i . n h 6 h - 6 h ~ o s h S h ) + 6 z ~ i n h 6 z ~ i n h 6 h u = +C C --[ m n d 6 ( ~ i n h 2 d h + 26h) + 2Asinh2 8h
- ~ i n h d z {( 1-2p) ~osh-6h~in6h~+6z~oshbzCos h6h d (Sinh 26h-2831) + 2Xcosh2 611 l q mn Cosa m xSinBny
6 z ~ o s h 6 z ~ i n h 6 h - ~ i n h B z ( 2 (1-1) ~inh6h+8hCosh6h} w = '6 (Sinh 26h+26h) +2?&inh2 6h
6zs inh6z~osh6h-coshbz {2 ( l -~~ )cosh6h+bhs inh6h} - ] q S i n a xSinB y 6 (S inh 26h-26h) +2Xcosh2 dh mn n n
6z~osh6z~nh6h-6h~inh8zCosh6h = k i B n ' 6 (Sinh 26h+26h) +2hsinh2 6h
1 a2 k o s h 6 z (S inh6h-6h~oshdh) +6z~inh6z~in6h}+2pp~~osh6z~inh6h
m cr = C C -[-A
x m n 6 6 (Sinh 26h+26h) +2hsinh2 dh
... - - 6 (Sinh 26h-26h) + 2 k o s h 2 6h
I
XL S i n a xSin 6 y m n
Xqmn S i n a xSin6 y
m n
a ampn cosh6z { (1-21) ~inh6h-6h~osh6h}+6z~inh6z~inh6h = C C - - xy m n [ 6 (S inh 26h+26h) +2kSinhz 6h
3.2.4 Levy Type Solut lon
The a n a l y s i s of a p l a t e with two oppos i te edges simply supported
and the remaining edges assuming any boundary cond i t ions f o r t h e p l a t e
bending problem using t h e MIF has been given by Iyenger e t a 1 (1974)
and is now extended t o p l a t e on Winkler foundation. The expanded form
of t h e governing d i f f e r e n t i a l equat ion f o r t h e bending of a p l a t e
r e s t i n g on a ~ i n k l e r foundation has been given i n equat ion (3.40b) . The homogeneous p a r t is given by
where
The above d i f f e r e n t i a l equat ion can be f a c t o r i s e d i n t h e form
(v' V; "' v4 + k g v;" ... V;~)F = o
where v4 and v * ~ a r c biharrnonic ope ra to r s defined thus n n
V4= V4- 2 ( € I - q2 ) V2 +$ + r + 2~~ q2 (n=1,2,. . .) n n n n n
where t h e q u a n t i . 1 . j ~ ~ r and 1) a r e r c l a t c d t o 1 t h e r o o t s of t h e 11 I 1 11 '
equat ion
4 3
asX = E + i o n . . . (3.54) n n
and E* and n* are r e l a t e d t o A* t h e r o o t s o f t h e equa t i on n n n '
Since equa t i on (3.49) is l i n e a r i t s s o l u t i o n can be w r i t t e n a s
t h e sum o f t h e s o l u t i o n s o f
V ~ F = 0 , V ~ F = 0 , V L ' F ~ = 0 ( n = l r 2 , 3 , . . .) . . . (3.57) o n n
Choosing t h e s o l u t i o n s o f (3.57) i n t h e form o f
F =C f S i n ci x (n = 0 , 1 , 2 , ...I m
. . . (3.58) n m n
e q u a t i o n s (3.57) can be reduced t o a s e t o f o r d i n a r y d i f f e r e n t i a l
e q a a t i o n s and t h e s o l u t i o n of t h e homogeneous equa t i on (3.49) can
be w r i t t e n as
= C[f +C(f + k ' f i ) ] S i n a x F ( ~ , Y ) o n . . . (3.59) m
where
f =APCosha y+BOci ySinhci y+C;Sinha y+DOa yCosha y 0 m m m m m m m m
n n n -f = ~ " c o s h g yCost y+B Sinhg y S i n t y+C Coshg y S i n t y+D Sinhg y c o s t y
n m n n m n n m n n m n n
4 d =c4 +n4 + 2 ~ ~ l l ~ +2 ( c 2 -1l2 )a2 43
n n n n n n m r n
and t h e Am ... D ' s a r e i n t e g r a t i o n cons t an t s . m
I n o rde r ' t o determine t h e p a r t i c u l a r s o l u t i o n Fp a t heo ry of any
d e s i r e d o r d e r i s cons t ruc ted . For example, by assuming a f o u r t h o r d e r
theory w e have t h e fol lowing
This i s equ iva l en t t o t h e c l a s s i c a l t h i n p l a t e theory . S i x t h and h igher
order t h e o r i e s can be obta ined by in t roducing a shea r func t ion x such t h a t
( C o s y h ) ~ = 0 . . . (3.521
and proper ly t r u n c a t i n g t h e s e r i e s . For a s i x t h ordwtheory, t h e system
of equat ions c o n s i s t s of t h e f i r s t o f equa t ions (3.61) and t h e fol lowing:
2 (v2- x = 0 . . . (3,631
I n t h i s s tudy , it is intended only t o show t h a t a Levy type s o l u t i o n
is p o s s i b l e f o r t h e a n a l y s i s o f p l a t e s on e l a s t i c foundat ions us ing
method of i n i t i a l func t ions , Therefore t h e s i x t h order equa t ions of
(3.61) and (3.63) a r e used t o eva lua t e t h e p a r t i c u l a r s o l u t i o n F . The P
f i r s t of Equations (3.61) when reduced t o a d i f f e r e n t i a l equat ion i n
mid-plane d e f l e c t i o n (w i s 0
which on recas t ing becomes
~ D V ' W ~ -kH1~'wO + kw0 = ( I - H ' v ' ) ~
where
The p a r t i c u l a r so lu t ion (w ) p and t h e load a r e taken i n t h e form 0
( w ~ ) ~ = Za Sina x m m
q = Cqm~inamx
The f i r s t of equations (3.67) s a t i s f i e s equation (3.65) a s well a s
the boundary condi t ions (supports a r e assumed s imple) ,Gubst i tu t ing
equations (3.67) i n (3.65) gives
m = 1,3,5,.,.
where q is a uniformly d i s t r i b u t e d load. Thus
Using equations (3,3) t h i s can be wri t t en a s
where
Thus by e q u a t i o n (3.61
To s o l v e e q u a t i o n (3.63) it is f a c t o r i s e d i n t h e form
where t h e 2 ' s a r e t h e r o o t s o f Cos Rh = 0 . The s o l u t i o n o f (3.73)
i s w r i t t e n i n t h e form
~ ( x I Y ) = ~ x ~ I Y ) . *. (3,741
S Assuming x is o f the forwx = CE Cosa x w e o b t a i n
s m m in
where
d = R; + a;, s m
The complete s o l u t i o n oL c q u a t i o n 40) is g i v e n by t h e sum o f t h e
homogeneous and p a r t j . c u l a r solul ; ions ,
3.3 APPLICATION OF M I f i ' 'rO PLATE ON TWO-PARAMETI!R ELASTIC
, FOUNDATIONS
The problem o f p l a t e on a two-parameter founda t ion is a l s o d i v i d e d
i n t o symmetric and an t i - symm(? t r i c i,,irts a s shown i n F i g 3 , l . For a
two-parameter founds t i o n , the fot~ndat iorr r e a c t ion h a s been d e f i n e d i n
e q u a t i o n (2.28) a x
o r by u s i n g t h e n o t a t i o n s i n equat i -ons (3 ,3 )
3.3.1 Symmetric P a r t a . ,
The s t a t i c a l boundary c o n d i t i o n s a r e :
a t z = h , X = Y = O . . . (3.77)
1 z = - f q ( x , y ) - - 2 G { h ~ ~ ( x ~ y ) - k i y ~ ~ ~ ( x . y ) 1
Thus, p roceed ing on s i m i l a r l i n e s as i n s e c t i o n 3.2, t h e governing
d i f f e r e n t i a l equa t i on s i m p l i f i e s t o
Ly2 h + y h ~ i n y h ~ o s y h + y ~ i n ' yh-h lYi s i n 2 yhl @ = -3q (x , y ) .. (3.78) where i s a s g iven i n equa t i on (3.49 ) and
I n t h i s c a s e t.he s t a t i c a l boundary c o n d i t i o n s a r e :
Hence by proceeding i n t h e same l i n e s as i n s e c t i o n 3.2 t h e governing
d i f . £e r en t i a l equctt ion simplif ics t o
which i n expanded f o ~ m i.s !~i .ven by
3,3,3 Allround Simply Supportea P l a t e
A Navier type s o l u t i o n i s used i n t h i s ca se , By assuming @
and F i n t h e form of equa t ions (3.42) and s u b s t i t u t i n g i n t o t h e
d i f f e r e n t i a l equa t ions (3,78) and (3.81a) t h e c o e f f i c i e n t s $ and mn
fmn a r e obta ined a s
0, = SIln inn G (Sin h 26h + 26h) + 2 ( A + A1A2 s i nh26h
Thus @ and F a r e def ined a s
q S ina xSinB y mn m n @ = c c
m n 6 ;S inh 26h + 28h) + 2 ( h + A1d2)sinz8h
c c (1-p) .qmSin mxSin n Y F =
m n 6 ( s i n 26h-26h) + 2 ( A + A1G2 )Sinh2 6h
S u b s t i t u t i n g @ and F i n equa t ions (3.58) t h e express ions f o r t h e
s t r e s s e s and displacements a t any p o i n t (x ,y ,z ) a r e given by
. - Sinh6z {( 1-2p) cosh6h-6h~inhdh}+6z~osh6z~osh6h~ Cosa xSinBny
6 (Sinh 26h-26h) +2 (A+A162 )cosh2 6h mn m
a x =IzL[- mn6 6 ( s i n h 26ht26h) + 2 (h+XiS2 ) Sinh2 6h
Xqmn S i n a xSinBny
m
1 s2 { C O S ~ ~ Z (~in1161~-611~0shSl1) +6z~i.nh6z~ir1ll61~}+2p~l~~osh6~~inh&h
n a =.CC--[- Y mn6 6 ( S i n h 26h+26h) + ~ ( x + A 62 1 S i n h Z 6 h 1
3 3.4 Levy Type Solu t ion f o r P l a t e on Two-Parameter Foundation
The expanded form of t h e governing d i f f e r e n t i a l equat ion i s given
i n equat ion (3.8lb) . The homogeneous p a r t is given by
where
The d i f f e r e n t i a l equat ion (3.85) can be f a c t o r i s e d i n t h e form
v'v;. . .v;+k1 P I . . .v*'++k"v2 .v*' V * F = o n 1 . . . (3.87) n
snd . fo l lowing t h e proceedure i n s e c t i o n 3,2.4 t h e s o l u t i o n of t h e homo-
geneous equat ion (3.85) can be w r i t t e n a s
b
where
and a l l o t h e r t e r m s a s def ined before. Using a s i x t h o rde r theory
t h e fo l lowing system o f equa t ions a r e ob t a ined f o r t h e eva lua t ion of
t h e p a r t i c u l a r s o l u t i o n .
The f i r s t o f equa t ions (3.90) can be reduced to a d i f f e r e n t i a l equa t ion
i n t h e mid-plane d e f l e c t i o n a s
4 1 v W = - [ I - 0 8 D
h2 - ) 8' I x lq- {kwo-k lV2 wo 11 2 (1-P)
. . . (3.91)
which on r e c a s t i n g becomes
( ~ D + ~ , H ' ) V ~ W ~ - ( ~ ~ ~ + k ~ ) V ~ w ~ + k w ~ = ( l - ~ l V ~ ) q . . , (3.92)
where H' i s a s de f ined i n (3.66). By t a k i n g t h e p a r t i c u l a r s o l u t i o n
( W and t h e load i n t h e form o f (3,671 and s u b s t i t u t i n g equa t ions O P
(3.67) i n t o (3.91) we g e t
m = l , 3 , 5 , . .. and q is a uniformly d i s t r i b u t e d load. Thus
Uting t h e n o t a t i o n s of cqua t ions (3.3) t h i s reduces t o
where
Since by equat ions (3 ,61) , Wo = F it fol lows t h a t F = (W ) P O P
The s o l u t i o n f o r t he shear func t ion is a s given i n s e c t i o n 3 .2 .4
and t h i s completes t h e so lu t ion .
3 .4 A NEW 'FOUNDATIN MODEL
I n t h e Winkler and most o t h e r foundat ion models t h e foundation
r eac t ion is def ined i n terms of t h e t r ansve r se d e f l e c t i o n only.
However, t h e presence o f f a c t o r s such a s f r i c t i o n a t t h e p l a t e -
funct ion i n t e r f a c e sugges ts t h a t t h e foundation r e a c t i o n might a l s o
depend on l a t e r a l d e f l e c t i o n s . Thus a new foundat ion model is pro-
posed such t h a t t he r e a c t i o n of t h e Winkler foundation is 111odifJed
by t h e add i t ion of t h e l a t e r a l displacements. I t is suggested t h a t
t h e foundation r eac t ion p ( x , y ) i s given by
P(x ,y) = ~ ~ , Y ) + M ~ ~ C U ~ ( X , ~ ) + Gv(x ,y) ) . , , ( 3 -98)
which, using t h e n o t a t i o n s i n cquat ions (3.3) reduces t o
k 1 - w ( x , y ) + - C a ~ ( ~ , y ) + B ~ ( x , y ) p ( x , y ) = . . 0 (3.99 G
AS usua l , t l ~ c problem i s d iv ided i n t o sylnrnetric and anti-symmetric
p a r t s
3.4.1 Symmetric P a r t
S t a t i c a l equ i l i b r ium r e q u i r e s t h a t a t z = h , X = Y = 0
and . . . (3.100)
Proceeding as i n s e c t i o n 3.2, t h e governing d i f f e r e n t i a l equat ion reduces t o
where A is a s given by (3.31) and
3.4.2 Anti-Symmetric P a r t
The s t a t i c a l boundary cond i t i ons are at
z = h r X = Y = O
and
1 a = +q(x .y) - =[kw ( X , Y ) + ~ ~ { W ( X ~ Y ) + ~ ~ V ( X , Y ) 11 . . . (3.103)
and t h e governing d i f f e r e n t i a l equat ion i s t h u s obta ined (proceeding a s i n
s e c t i o n 3.2) a s
' which in.expanded form is given by
where c'= A /A 3 2
3.4.3 A p p l i c a t i o n t o Al l round Simply Supported P l a t e
For a n a l l r o u n d simgly suppor ted p l a t e we assume and F i n t h e form
(3.42) which s a t i s f y b o t h t h e boundary c o n d i t i o n s and t h e governing d i f f e r e n -
t i a l e q u a t i o n s . S u b s t i t u t i n g e q u a t i o n s (3.42) i n t o e q u a t i o n s (3.101) and
(3.104) we o b t a i n t h e c o e f f i c i e n t s ($ and fmn as mn
- - fmn 6 (S inh 26h-26h) + 2 k o s h 2 6h+6 (h2sinh26h+2A b%)
3
Hence t h e f u n c t i o n s + and F a r e g iven by
q S i n a xs inB y + =C C m 91 n m n --
6 (S inh 26h+26h) +2hsinh2 6h+ 6 (A2sinh26h-2X 6h) 3
(1-p)qm S i n a S i n 6 y F =C C a-
m , n m n 6 (Sinh26h-26h) + Z h ~ o s h ' 6 ~ + 6 (h2sinh26h+h36h?
S u b s t i t u t i n g f o r 6 and F i n e q u a t i o n s (3 .47) t h e e x p r e s s i o n s f o r t h e s t r e s s e s
and d i s p l a c e m e n t s a t any p o i n t ( x , y , z ) a r e g i v e n by
sizh6z {! 1 - 2 ~ ) ~ o s h 6 h - 6 h ~ i n h 6 h ) + 6 z ~ o s h 6 z ~ o s k , 6 h - -.-- --lq Sina xCosB y 6 (Sinh26h-2 6h) +2XCosh2 6h+6 (X2Sinh26h+2X 6h) mn
3 m n
6zCosh6z~inh6h-6h~inh6zCosh8h - - X = C C a t --.
m n m 6(~inh26h+26h)+2XSinh~6h+6(X~~inh28h-2A Ah). , 3
CHAPTER FOUR
NUMERICAL RESULTS
4 , l ALLROUND SIMPLY SUPPORTED PLATE ON WINKLER FOUNDATION
The e x p r e s s i o n s f o r t h e s t r e s s e s and d i sp lacements i n an a l l -
round s imply suppor ted p l a t e r e s t i n g on a Winkler founda t ion a r e g iven
by e q u a t i o n s (3.48). The problem is so lved f o r two c a s e s of load ing :
(i) uniform load o f i n t e n s i t y q on z = -h
(ii) p a r t i a l l oad o f i n t e n s i t y q on z = -h, o v e r an a r e a cxd
For c a s e (i) t h e F o u r i e r c o e f f i c i e n t q i n equa t i ons (3.48) i s g iven mn
by
and f o r c a s e (ii)
. For t h e sake of compansion, t h e fo l l owing paramete rs as used by
Voy iad j i s and Baluch (1979) have been used:
The va lue s o f founda t ion modulus, k used a r e
54311,132, 152933.4 and 543111.32 KN/mJ
R e s u l t s a r e p r e sen t ed f o r p l a t e s wi th l e n g t h t o t h i c k n e s s r a t i o s
(a /2h) o f 2.5,5,10,20 and 40. The maximum va lue s of each of t h e q u a n t i t i e s
t r a n s v e r s e d e f l e c t i o n , w; bending s t r e s s , 0 and shea r s t r e s s , T a r e X X z
c a l c u l a t e d and t h e i r v a r i a t i o n s a c r o s s t h e t h i c k n e s s have been s t u d i e d
by c v d u a t i ~ i c j q u m t i t ie:: ,it z/li - -1 -0, -0. ' ) , O,.') ,iricl 1 - 0 . 'I'Llc' fo t~ndd-
t i o n p r e s s u r e a t t h e bottom o f t h e p l a t e i s a l s o eva lua ted . A l l
r e s u l t s a r e p re sen t ed i n non-dimensionalised form. The computer pro-
gram developed f o r t h e s o l u t i o n is given i n Appendix C. Comparisons
a r e made with c l a s s i c a l and Reissner theory s o l u t i o n s .
4.1.1 P l a t e Subjec ted t o Uniform Load
I n Table 4.1 t h e maximum t r a n s v e r s e d e f l e c t i o n s ob t a ined by t h e MIF
a r e compared with r e s u l t s based on t h e c l a s s i c a l and Reissner t h e o r i e s .
The percentage e r r o r s i n t h e c l a s s i c a l theory s o l u t i o n s a r e a l s o given.
I t is seen t h a t f o r t h i c k p l a t e s t h c r c i s a marked d i f f c r c n c e between t h e
MIF and c l a s s i c a l theory s o l u t i o n s b u t an agreement of t h e s o l u t i o n s i s
observed f o r t h i n p l a t e s . The t a b l e a l s o shows t h a t t h e MIF and Reissner
theory s o l u t i o n s a r e c l o s e f o r bo th t h i c k and t h i n p l a t e s . For a given
p l a t e , t h e non-dimensional d e f l e c t i o n (Gw/2qh) decreases a s t h e value o f
t h e foundat ion modulus i nc reases . Th i s can be expla ined by t h e f a c t t h a t
a h igher k value means a s t i f f subgrade which o f f e r s more r e s i s t a n c e t o
deformation than a weaker subgrade wi th a smal l va lue o f k. Also, it is
noted from Table 4.1 t h a t f o r a given foundat ion, t h e d e f l e c t i o n i n c r e a s e s
a s t h e p l a t e becomes t h i n n e r . Table 4.2 shows a s i m i l a r t r e n d f o r t h e
maximum bending and shea r s t r e s s e s . However, a s shown i n Table 4.3, f o r
a p l a t e of given t h i c k n e s s , t h e maximum foundat ion r e a c t i o n , p i n c r e a s e s
wirth t h e foundat ion modulus while f o r a foundat ion of g iven modulus p
i nc reases a s t h e p l a t e g e t t h i n n e r .
The v a r i a t i o n s of t h e bending s t r e s s , 0 a c r o s s t h e c e n t r e o f va r ious X
p l a t e s a r e shown i n F igu re s 4.1 a n d 4 -2 . I t is seen t h < i t t h c v a r i a t i o n s o f
TABLE 4.1 Comparison o f non-dimensional rnaximum t r a n s v e r s e d e f l e c t i o n s
o f u n i f o r m l y l o a d e d s q u a r e p l d t e s r e s t i n g on Winkler Founda-
t i o n s (D = 0031
CLASSICAL
0.666
10.653
l69,3O9
2571.957
2921 1.822
0 ,666
10.632
166.783
2204.577
17363.096
0.666
10.562
158.455
1684.488
6854.031
REISSNER - 1.134
12.521
176.674
2598.245
29259.510
1.134
12.492
173 -894
2325.134
17373.251
1.132
12.393
164.803
1694.199
6848,808
-
"Percen tage E r r o r
Value by MIF - Value by Class ical t h e o r y *percentdge e r r o r =
Value by MIF
Table 4.2
-z&---
N9n-dimensional maxi~um s t r e s s e s i n uniformly loaded square p l a t e s r e s t i n g on Winkler foundat ions (~=0.3)
MIF ---
2.123
7.501
2b.810
107.977
294.995
C l a s s i c a l
TABLE 4,3 Maximcim fonndat ion r e a c t i o n a t bottom of uniformly loaded square p l a t e s r e s t i n g on Winkler foundat ions (p=0.3)
MIF
------ Classical
% a r e i n f luenced by p l a t e t h i cknes s . The v a r i a t i o n s t e n d t o d e v i a t e
from t h e l i n e a r assumption o f t h e c l a s s i c a l t h e o r y f o r t h i c k p l a t e s b u t
f o r t h i n p l a t e s , t h e assumption of t h e c l a s s i c a l t heo ry is d isp layed .
F i g 4.2 shows t h a t f o r a p l a t e o f given t h i c k n e s s , t h e v a r i a t i o n of U X
a c r o s s t h e t h i c k n e s s is t h e same no matter t h e va lue o f t h e founda t ion
modulus, a l though t h e maximum and i n t e rmed ia t e va lues a r e d i f f e r e n t f o r
d i f f e r e n t va lue s o f k. Msximum a always occu r s a t t h e c e n t r e o f t h e t o p X
s u r f a c e o f t h e p l a t e ( a t x = a /2 , y = b/2 and z =-h). Thickness a l s o
a f f e c t s t h e v a r i a t i o n of t h e shea r s t r e s s T as shown i n F igs . 4.3 and xz
4.4. The maximum va lue of T t ends t o s h i f t from t h e middle p l ane xz
towards t h e t o p of t h e p l a t e i n t h e c a s e o f t h i c k p l a t e s b u t t h e v a r i a t i o n
becomes symmetrical f o r t h i n p l a t e s . F ig 4 .5 shows t h a t f o r t h i c k p l a t e s
t h e v a r i a t i o n of t h e t r a n s v e r s e d e f l e c t i o n a c r o s s t h e t h i c k n e s s is non-
l i n e a r as a g a i n s t t h e assumption o f c o n s t a n t d e f l e c t i o n o f t h e c l a s s i c a l
and Reissner t h e o r i e s . I t is o n l y . # o r t h i n p l a t e s t h a t t h e d e f l e c t i o n
is f a i r l y c o n s t a n t th rough t h e t h i c k n e s s as shown i n F i g 4.6. Maximum
d e f l e c t i o n o c c u r s a t t h e middle p l ane excep t f o r t h e p l a t e wi th l e n g t h
t o t h i c k n e s s r a t i o o f 2.5 where t h e maximum i s a t t h e t o p su r f ace .
F i g 4.7 i n d i c a t e s t h a t t h e maximum founda t ion p r e s s u r e occu r s a t t h e
p l a t e c e n t r e .
The r e s u l t s p r e sen t ed have been c a l c u l a t e d t o c e r t a i n deg ree s o f
accuracy depending on t h e number of terms o f t h e s e r i e s employed. I t
i s found t h a t a good convergence is ach ieved by t a k i n g t h e fo l lowing
number of terms of t h e double s e r i e s :
f o r w: m=n=15
f o r p: m=n=15
f o r OX: m=n=2 1
f o r TXZ: m=n=9
Thus using t h e r e s u l t s corresponding t o k=162933.4 k ~ / m ~ and a/2h=2.5,
t h e e r r o r s i n t h e va lues accepted a r e 0.54%, 4.57% and 0.35% f o r w, 0 X
and T r e s p e c t i v e l y while t h e same r e s u l t s f o r a/2h = 40 a r e 0.002%, xz
0,14% and 4.52% respec t ive ly . Because o f t h e low r a t e of convergence
of t h e double s e r i e s it does n o t seem d e s i r a b l e t o ex tend t h e s e r i e s
beyond t h e number of terms ind ica t ed .
4.1.2 P l a t e Subjected t o P a r t i a l Load
Table 4.4 g i v e s a comparison of non-dimensional maximum mid-plane
d e f l e c t i o n f o r two va lues of c / a (c being t h e l eng th over which t h e
p a r t i a l load a c t s ) , It is seen a s be fo re t h a t f o r a given loading , t h e
MIF and c l a s s i c a l s o l u t i o n s a r e only c l o s e f o r t h i n p l a t e s while t h e
MIF and Reissner s o l u t i o n s a r e c l o s e f o r both t h i n and t h i c k p l a t e s .
Also, by r e f e r i n g t o Table 4.1 it is seen from Table 4.4 t h a t f o r a
given p l a t e t h e non-dimensional d e f l e c t i o n decreases wi th t h e loaded
a r e a (o r c / a ) .
The maximum s t r e s s e s f o r k=543ll . 132 k ~ / m ~ a r e given i n Table 4,5.
It .is observed t h a t maximum 0 and 'r decrease with c/a. However, X X Z
t h e decrease i s more apparent i n t he maximum shea r stress, 'r . Thi s X z
marked decrease i n t h e maximum 'r va lues might be due t o t h e f a c t t h a t X Z
tfle loaded a r e a becomes f a r t h e r from t h e p l a t e edge where maximum T xz
occurs , a s c / a decreases . A: A layed i n F ig 4.8 0 v a r i e s i n t h e manner X
descr ibed f o r uniformly loadcd p l a t e s but- i t is sccn thaL f o r a given
p l a t e the nnn- l inea r i t y i n O x i nc rcascs a s Chc loadcd area dc.c3rearw.s. F3949
Table 4.4 Comparison of non-dimensional maximum t r ansve r se d e f l e c t i o n of p a r t i a l l y loaded square p l a t e s on Winlkler foundat ion
MIF -- ;w/2qh
C l a s s i c a l .Percentage ---,
Error
Value by MIF-Value by C l a s s i c a l theory *Perct-ntage e r r o r = v a l u e by MIF
x 100
TABLE 4.5 Non-dinensional maximum stresses i n p a r t i a l l y loaded square p l a t e s r e s t i n g on Winkler foundation (b=0.3,
MIF C l a s s i c a l
MIF
---- Classical
Fig. 4.8 : Non-dimensional stress (crX/q) across centre of partially loaded plates Winkle, foundation ( K = 34311.132 K N / ~ ~ , f l = 0, a/2h ~ 2 . 5 )
does no t r e v e a l any s i g n i f i c a n t change i n t h e v a r i a t i o n of T a c r o s s X z
t h e th i ckness
4,2 UNIFORMLY LOAPED ALLROUND SIMPLY SUPPORTED PLATE ON VLASOV
FOUNDATION
I n equat ions (3.84) a r e given t h e express ion f o r t h e s t r e s s e s and
displacements i n an a l l round simply supported p l a t e r e s t i n g on t w o -
parameter foundat ions . The Vlasov model i s taken a s an example o f a two-
parameter foundat ion and t h e problem is so lved f o r t h e ca se of uniform
load only s i n c e t h e same t r e n d o f behaviour of t h e va r ious q u a n t i t i e s
a s d i scussed i n s e c t i o n 4.1 is expected f o r t h e ca se of p a r t i a l load.
I (
The Vlasov model is chosen a s t h e foundat ion parameters a r e obta ined
r a t i o n a l l y . The express ions f o r t h e foundat ion parameters k and 2 t a r e
given i n equat ions (2,23) and (2.25) r e spec t ive ly . From equat ion (2 -23) ,
p u t t i n g VB = 1.0, t h e Young's modulus E of t h e subgrade is given by S
Assuming D = 0.3, w e f i n d t h a t E = 1 .a86 k S S
.'. For k = 54311.132, Es = 1,486~54311.132 = 80690.8XN/m2
hence
S imi l a r ly f o r k = 162933,4 k N / m 3 / , 2 t ( = k = 48000 kN. The p r o p e r t i e s of 1
b
t h e p l a t e a r e t h e same a s t hose used i n s e c t i o n 4.1. The s t r e s s e s and
displacements a r e computed t o t h e same convergence l e v e l as d iscussed i n
76
wi th t h e va lues ob ta ined by t h e c l a s s i c a l theory i n Table 4.6. It
is seen t h a t , a s i n t h e ca se of p l a t e on Winkler foundation, f o r t h i c k
p l a t e s t h e va lues ob ta ined by t h e MIF are on ly c l o s e t o t h e c l a s s i c a l
t heo ry s o l u t i o n s f o r t h i n p l a t e s . A comparison with t h e app rop r i a t e
d a t a i n Table 4.1 shows t h a t f o r a given p l a t e t h e v e r t i c a l d e f l e c t i o n
o f t h e p l a t e on Vlasov foundat ion is sma l l e r than t h a t on t h e Winkler
foundat ion and t h i s discrepancy is more pronounced i n t h i n p l a t e s and
a l s o f o r b igger foundat ion modulii . Thus i f t h e Vlasov foundat ion ,
being de r ived from c o n t i n u i t y equa t ions is regarded a s a more r e a l i s t i c
model, it means t h a t t h e Winkler theory over e s t i m a t e s t h e d e f l e c t i o n
of t h i n p l a t e s r e s t i n g on e l a s t i c foundat ions. Table 4.7 shows t h e
same t r e n d f o r maximum s t r e s s e s , a l though t h e d i f f e r e n c e s a r e smal le r
i n t h e ca se o f T The s t r e s s e s a r e p l o t t e d i n F i g s 4.10 t o 4.12 t o XZ.
show how they compare with s t r e s s e s ob ta ined i n t h e c a s e o f p l a t e on
Winkler foundat ion.
A . 3 UNIFORMLY LOADED ALLROUND SIMPLY SUPPORTED PLATES ON NEW
FOUNDATION MODEL
The r e l e v a n t exp re s s ions f o r s t r e s s e s and displacements a r e given
i n equa t ions (3.107) . A dimensional a n a l y s i s o f equa t ion (3.98) r e v e a l s
-2 t h a t t h e second foundat ion modulus ki has u n i t s o f FL . Values o f k 1
a r e assumed a r b i t r a r i l y and t h e problem is so lved f o r p l a t e s having t h e
same p r o p e r t i e s a s those used i n s e c t i o n 4.1. S t r e s s e s and d isp lacements
a r g com2uted t o t h e same convergence l e v e l a s d i scussed i n s e c t i o n 4.1.1
us ing t h e app rop r i a t e program i n Appendix C
The maximum stresses and t r a n s v e r s e displacements for t h e K =200!,, 1
10,000, 5OC)OO arc. g i v e n irl 'l'clblc 4 . 8 w l ~ c ~ r e tllc curr-c.sporld~ng va lues due
7 7
TABLE 4.6 Comparison of non-dimensional maximum t r a n s v e r s e d e f l e c t i o n
of uniformly loaded square p l a t e s r e s t i n g on Vlasov founda-
t i o n ( p 0 . 3 )
Foundat ion parameters
MIF
1.139
12.516
169- 498
1956 -09
.006 1.02
1.132
12.335
154.013
1228.74
3879.91
C l a s s i c a l
0.666
10.596
162.381
e r r o r s
41.53
*Percentage e r r o r = Value by MIF - Value by C l a s s i c a l theory
Value by MIF x 100
TABLE 4.7 Comparison of non-dimensional maximum stresses in uniformly
loaded square plates on Vlasov foundations (p=0.3)
Foundat ion parameters
Classical
1.764
7 a016
26.794
78.141
89.205
O.!
1.C
Fig, 4.10 : Comparison of non-dimensional bending stress (uK/q) across centre of
square plateson Winkler and Vlasov foundations(a /2h = 2.5.5)
Fiq 4.12 Comparison of non-dimensional shear stress (xxz/q) across edge of CJ
square plates on Winkler and Vlasov foundotions F
82
TABLE 4.8 Comparison o f non-dimensional maximum t r a n s v e r s e d e f l e c t i o n
and s t r e s s e s i n uniformly loaded square p l a t e s on the Winkler
and new foundat ions (y=0.3, K=543111.32Wy/m3)
t o the Winkler theory a r e a l so incorporated fo r the sake of comparison.
I t is seen t h a t the second parameter does not introduce any s i gn i f i c an t
changes i n the s t r e s s e s and transverse de f lec t ion espec ia l ly f o r th ick
p la tes . For a given foundation, it is seen t h a t the s l i g h t change in-
troduced by the second parameter i s more i n t h in p l a t e s and fo r a given
p l a t e t he change i s more pronounced f o r higher values of K This shows 1
t h a t t he deviat ion might become large i f kl i s su f f i c i en t l y large . The
smallness of the contr ibut ion of K might be due t o the f a c t t h a t the 1
p l a t e is subjected t o only transverse loads. It is t o be expected t h a t
k might make a subs tan t ia l contribution i f the p l a t e i s subjected t o 1
l a t e r a l loads a s well.
The var ia t ions of the s t r e s s e s and transverse displacements a r e
similar t o those discussed i n sect ion 4.1.1 and since the values a re
almost a l i ke , they a r e not presented here a s the re levant t ab l e s and
f igures i n t h a t sect ion su f f i c i en t l y describe the var ia t ions of the
s t r e s s e s and displacements i n p l a t e s res t ing on the new foundation
model.
CHAPTER FIVE
CONCLUSIONS
I t h a s been shown t h a t t h e method o f i n i t i a l f u n c t i o n s (MIF) can
be used i n t h e a n a l y s i s o f p l a t e s r e s t i n g on one- and two-parameter e l a s t i c
f ounda t i ons , Numerical :;elutions of t h e Navier t ype have been p r e s e n t e d
f o r a l l r o u n d simply suppor ted p l a t e s and it h a s a l s o been shown t h a t a Levy
t ype s o l u t i o n is p o s s i b l e u s ing t h e MIF.
The numerical s o l u t i o n s o f p l a t e s r e s t i n g on t h e Winkler and Vlasov
founda t i ons r e v e a l t h a t f o r a g iven p l a t e r e s t i n g on a founda t ion o f given
p a r a m e t e r ( s ) , t h e t r a n s v e r s e d e f l e c t i o n , bending and s h e a r s t r e s s e s g e n e r a l l y
i n c r e a s e as t h e p l a t e g e t s t h i n n e r . The e f f e c t o f l a r g e r founda t ion modul i i
is t o reduce t h e d e f l e c t i o n and s t r e s s e s i n a g iven p l a t e , t h e e f f e c t be ing
more prominent i n t h i n p l a t e s . I t h a s been f u r t h e r demonstra ted t h a t t h e
d i f f e r e n c e s i n s t r e s s e s and d i sp lacements o b t a i n e d i n t h i c k p l a t e s on t h e
Winkler and Vlasov founda t i ons a r e small. However, f o r t h i n p l a t e s t h e s e
d i f f e r e n c e s a r e l a r g e and i f t h e Vlasov founda t i on , be ing d e r i v e d from a con-
tinuum p o i n t o f view, is regarded as be ing more c l o s e t o r e a l i t y t hen it can
be concluded t h a t t h e Winkler model ove r e s t i m a t e s t h e s t r e s s e s and d i s p l a c e -
ments i n t h i n p l a t e s .
A comparison of . t h e r e s u l t s o b t a i n e d by t h e MIF and t h e c l a s s i c a l
t h i n p l a t e t heo ry shows t h a t t h e c l a s s i c a l theory unde re s t ima t e s t h e stresses
and d i s ~ l a c e m e n t s i n t h j ck p l - ~ t e s r e s t i ~ l y on e l a s t i c f ounda t i ons . I t , however,
becomes more a c c u r a t e a s thc p l a t e bccolncs t h i n n e r . For example, t h e e r r o r
i n t r a n s v e r s e d e f l e c t i o n u s ing t h e c l a s s i c a l t heo ry f o r a p l a t e wi th l e n g t h
t o t h i c k n e s s r a t i o o f 2 . 5 i s about 4 4 % whi lc t h a t f o r a l c n g t h t o t h i c k n e s s
r a t i o of 40 i s j u s t 0.2%. Thus t h e use o f t h e c l a s s i c a l t heo ry f o r t h i c k
p l a t e s is n o t recommendcd. However, it i s i n t e r e s t i n g t o no t e t h a t t h e
MIF and Reissner theory s o l u t i o n s are c l o s e f o r a l l l eng th t o t h i c k n e s s
r a t i o s . The maximum d i f f e r c n c e s i n t r a n s v e r s e d e f l e c t i o n ob t a ined by t h e
two methods i s about 4.0%.
I t is seen t h a t MIF can be convenien t ly used t o ana lyse a p l a t e r e s t i n g
on a foundat ion which is desc r ibed i n t e r m s o f t r a n s v e r s e and l a t e r a l d i s -
placement so The i n t r o d u c t i o n o f l a t e r a l d e f l e c t i o n s i n t o t h e foundat ion
r e a c t i o n of t h e Winkler model does n o t in t roduce s u b s t a n t i a l changes i n t h e
s t r e s s e s and d isp lacements f o r t h e a r b i t r a r i l y chosen va lues of t h e second
parameter. Th i s i s probably due t o f a c t t h a t t h e p l a t e i s sub jec t ed t o
t r a n s v e r s e l oads on ly . I t i s l i k e l y t h a t s u b s t a n t i a l mod i f i ca t i ons i n s t r e s s e s
and d isp lacements w i l l be introduced i f t h e p l a t e is sub jec t ed t o l a t e r a l f o r c e s
a s wel l . However, an experimental i n v e s t i g a t i o n of t h e second parameter ought
t o be done t o confirm t h e exac t na tu re of t h e new foundat ion.
The MIF has t h e advantages t h a t t h e governing equa t ions a r e convenien t ly
expressed i n terms of a d i f f e r e n t i a l o p e r a t o r and t h a t approximate t h e o r i e s
o f any d e s i r e d o r d e r can be c r e a t e d by proper t r u n c a t i o n of t h e s e r i e s . How-
eve r , it reqwces enormous computcr t i m e t o produce good r e s u l t s and can only
be app l i ed t o problems wi th c e r t a i n boundary cond i t i ons .
APPENDIX A
DETERMINATION OF CITARACTERISTIC ROOTS OF MATRICES [C] AND [R] , AND APPLICATION OF SYLVESTER'S AND CAYLEY-HNULTON'S THEOREMS
I n o r d e r to e v a l u a t e t h e c h a r a c t e r i s t i c r o o t s o f t h e ma t r i c e s
[C] and [R] , t h e fo l lowing equa t i on i s used:
Expanding we have
On f u r t h e r expansion and s i n ! p l i f i c a t i o n t h i s r educes t o b
5' + 3y2c2 + -jy4~, + Y6 = 0
. . The c h a r a c t e r i s t i c r o o t s o f t h e m a t r i c e s [ C ] and [I?]
a r e
where Y2 = ct2 -c f3'
S y l v e s t e r ' s Theorem;
A s g iven by F raze r e t a 1 (1960) , S y l v e s t e r ' s theorem s t a t e s t h a t
i f n l a t e n t r o o t s o f a ma t r i x M are a l l d i s t r i n c t and P(M) i s any
polynomial o f M t hen
Thus f o r a t h i r d o r d e r ma t r i x P ( h ' d can be expressed as
.where el, t2 , c3 <ire the d i s t i n c t c h a r a c t e r i s t i c r o o t s o f M.
b.
In the case when a l l the character is t ic roots are equal
the confluent form of Sylvester's theorem is used:
In part icular , for a matrix of order m ,
where there are two equal la tent roots and
the expression becomes
one d is t inc t root,
. . . (6)
where 5 is the repeated root and 5 is the d is t inc t root, j i
~ ( 5 , ) .L
F(Ei) = .j. (5.1-M) and Q = 3 1 3 0
Cayley-Hamilton's Theorem:
The theorem s ta t e s tha t any square matrix M s a t i s f i e s i t s
character is t ic equation given by
f o r an nxn m a t r i x . b'or a t h i r d order m a t r i x . ,
we p u t n = 3 and o b t a i n
(S,I-M)' = 0
s u b s t i t u t i n g equa t i on (8) i n equa t i on ( 4 ) g i v e s
P ( M I = P (En) M1 -P ( C n ) M 2 + p ( 2 ) ( 6 n ) ~ 3 . . . (9b)
Also s u b s t i t u t i n g equa t i on ( 4 ) i n equa t i on ( 5 ) w e o b t a i n
p u t t i n g m = 2 we have
M' = CAI-2 ( ~ , I - M ) + (5 I-MI 2 n
From e q u a t i o n s (9b) and (11)
M = I 1
and M = ( C n 1 - m 2
3
Applying t h e s e t o C and R m a t r i c e s w e have
A ~ p l y i n g equation (10) t o the mztrices C and R , t h e following
expressions a r e obtained:
S u b s t i t u t i n g f o r C and R in equations (3.15) and c o l l e c t i n g like
terms t h e following s e t of equations a r e obtained:
P u t t i n g 6 = -y2 and s impl i fy ing , equat ions (15) reduce t o .
1 S i n z
Y Y ['o] ... (16)
9 2 APPENDIX B
DERIVATION OF EXPRESSIONS FOR STRESSES AND DISPLACEMENTS I N
TERMS OF THE FUNCTIONS (I AND F
From equat ions (3.46) each o f t h e q u a n t i t i e s is made up of two
p a r t s . For example t h e displacement U can be w r i t t e n a s
where
S u b s t i t u t i n g f o r t h e ope ra to r s , L u u p L u v ~ - - - from Table 3.1 and t h e
i n i t i a l func t ions U o, Vo, ... from equat ions (3.27) and (3.37) we have
1 6 S inyh - (-- . aBz ) . -[(I-2p)- - h Cosyh](I 2 ( 1 p) - Sinyz 2
Y Y
a z - - Sinyz. y [Sinyh+ hCosyh1 (I 4 ( 1-p)
a z + ay2zh Sinyzcosyh - - a zh 4 (1-p)
S iny zS inyh - -Y-- S iny ZCO syh] (I (1-ply 4 (1-p)
Simi la r ly ,
1 + [-Sinyz - a' ( ~ i n y z - yzCosyz) 1 ( -ayh Sinyh) F Y 4 ( 1-p) y" 1 -P
which on s i m p l i f i c a t i o n becomes
- t U2 - - a [(l-2p)SinyzCosyh+yzCosyzCosyh+ yhSinyzSinyh1F 2 (1-p)
Theref o r e
u = -.- a [ ( 1-2p) CosyzSinyh-yhCosyz Cosyh - yzSinyz$inyh] @ 2~
The same procedure i s app l i ed t o de r ive t h e express ions f o r t h e o t h e r
q u a n t i t i e s .
In o rde r t o de r ive express ions f o r t h e s t r e s s e s and displacements
a t any p o i n t , we s h a l l f i r s t c a r r y o u t ope ra t ions such a s
~ i n ' y h . Sin am x Sin fin y
b
Expanding t h e ope ra to r w e have
x Sin a xSin fi y rn n
9 4
Carrying o u t t h e o p e r a t i o n s t e r m by t e r m w e ob t a in .
y4h4 zY6hf i ,e s in2yh . s ina x s i n B y =[y'h2 - -- + -- - . . , lS ina xsinBny1
m n 3 4 5 m
1 2 Sin' yh.Sina xsinB y=- [€i2 h' + -6'h4+ - €j6h6+. . .] Sinamxsing y m n 3 4 5 11
= - s i n h 2 6h S i n a xSinB y ... (6) m n
where 6' = a: + f3:
Proceeding a s above, t h e fo l lowing a@ obta ined:
y' ,Sins xSinB y = -B2Sina xSinB y m n m n
y S i n yhCo syh . s i n a xS i n y = -6~inhBhcoshBh . sinamxSinBny m n
Cas2yh.Sina.xSinf3 y = cosh2 6 h ~ i n a xSinB y m n m n
~ o s y h . S i n a x SinB y = cosh6h S i u mxSinBny m n
aos inamxSin8 y - - cos a x Sin 8 y n m IL
-B., S ina x SinBny = 8 sinamxCosB y m n n
... (7 )
, S u b s t i t u t i n g f o r q and t h e func t ion @ and from equat ions (3.42) i n
(3 .30 ) and us ing equat ions ( 7 ) , w e have
C C ['-A2 h-&~inh6hCosh&h-X':inh~ 6h] .$,~ina x ~ i r l e ~ y = - : C C q S ina xSinB y m n m mnmn m n
Simi la r ly , s u b s t i t u t i n g f o r q and t h e funct ion F from equat ions
(3.42) i n t o equat ion (3.40) and using equat ions ( 7 ) , t h e co-
e f f i c i e n t s f a r e o b t a i n e d a s mn
Hence t h e func t ions 0 and F a r e given by
S u b s t i t u t i n g f o r t h e func t ions 4 and P i n equat ion (5 ) and us ing
equqt ions (7) , t h e displacement U a t any po in t (x ,y , z ) is given by
Expressions f o r o t h e r q u a n t i t i e s a r e s i m i l a r l y obtained.
APPENDIX C
COMPUTER PROGRAMMES
1840DEFPROC5TKESX(Ml,Nl) 1850REM..PROCEDURE TO EVRLUATE STRESS 186UHEM.. I N X-DIRECTION l 8 7 O X 1 a0 1BBOFOR t l = l T O M l STEP 2 1 t391:lFOR N= 1 TUN I STEP :.: IYoOPROCHYP'(M, N) 1911:)V5=(D2\2) + ( S 2 + F 2 ) + 2 t L ~ n * ( S 1 " 2 ) + 2 + (D~>,2)& ( L 2 + S l * C l - L 3 + . F l ) 1920V6= (D"%2) + (S2-FZ) +2*L+D+(C1.'2) +2+ (D, -Z) * (L2+Sl *Cl+~..Z*t:l )
1 9 3 0 x 2 ~ ( ( 6 - 2 ) * (CS* ( B i - F i + C i ) + F 3 + 5 3 + S l ) +2*U.~t (HI-?) t C 3 - S l I .iV:i 194i1X3= ( (A'5?) * (SS* ( ( : l -FI+Sl) +F3+C3*Cl ) + 2 9 W (H"':!) xS:?+Cl ) /L'b 1950Xl=Xl+l6*(X3-X2)U((-l)A(((M+N)/2)-l))~(M*N+(W2)) 196ONEXT N 1970NEXT tl 19BClENDF'HOC 1990: ~OCIODEFPROCST'RESXZ (M1. N 1 ) 2OIOREM..PKOCEDURE TO EVALUATE SHEAR S 'TRES 20ZOP=O 2030FOH M=lTOHi STEP 2 2040FOH N=1TON1 STEP 2 ZOSOPRUCHYP (M. N) 2060V7=D+ (S2+F2) +2+L* ( S i " " 2 ) +2*.D+ ( L 2 + S 1 +C 1 -LZ+F l ) 2070VB=D+(S2-F2)+Z+L*(C1"2)+?+U+(L2+Sl+Cl+L3+FI) 2 0 8 0 P l =A* (F3+C3*SI -F l + S Z + C l ) /V7 2090PZ=A+(F3+S9*C1-F1*(33*91)/VB 21OOP=P+(16*(PI-P3)+COS(A+Al)*SIN(B*(Rl/C.) ) ) / (M+N+(B" .Z) 2110NEXT N 212ONEXT M 2 130ENDF'KOC 2 1 4 0 : 2 IX~DEFPRC)CSTRES'/ ( H I , N 1 ) . . 2 16(:!REM. .PROCEDURE TO EVAL..IJATE STKESS Y 217CIY 1=0 ~IU~:IFOR M-ITOMI STEF; :1 2 1 C)l:,F('JF( N.-- 1 'ruf.J 1 C;T[:p 2
2XWF'ROl~HYP (P I , IN) ?311:1E:I . ( I ) . . ; ) * (:.;;> IF;') +;I. I_.. I) + ( I , ; 1.. 2 4.; k (TI.., 2> P , ,: e 5 1 (: 1 .-L.:.+f 1 ) ..,-., iLL~.~t.~,=(D.-.~) - --- . ( ~ z L p 2 ) + 2 + L . r . [ ) . i C 1 ' 3 ) + ~ . * ( D " ' : ' I r : 2*r ! ; l~( : j + I - < t F l )
2 2 3 Y 2 - ( (8.'-~)r.(~;*(Sl--..F1rC1)+fl:rS:5+61)+y'.(I~ ;i.: ')+C':uSl)/[:z ?24c:IY?.-:( (r'l.'':!) * ( ! ~ . " ; # . ( ~ ~ t , . . l : ~ ~ ~ ~ ~ ~ ) + ~ . ~ ~ ~ l . ; . ' ~ , N ~ I ) t i ' ~ l m . ~ t l :') h c ; ~ ~ , h l ' ; l )/I:: , ., ,.., ,~. .: ..... , l l Y l ~ - . ' f l I ! O * ( Y - , . Y z : ) H ( ( I j ' ( ( ( k I l N ) / 2 ) ~ - 1 ) ~ . ! * I J k t t J >!))
::'h(:rlVt:Y T w 227r:)NCXT M 2'2130L IV1)I~'I:iflC
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