Finite Strip Analysis of Bridges [1 ed.] 041919150X, 9780419191506

Table of contents :
Dedication
Contents
Preface
Notation
Part I: Mathematical Approach
1 Introduction
2 Basic concepts of numerical methods
3 Numerical errors
Part II: Finite Strip Method
4 Finite strip method
5 Higher order finite strips
6 Spline finite strip method
7 Compound strip method
8 Finite layer method; finite prism method
9 Vibration and stability analyses
10 Nonlinear analysis
11 Combined analysis
Part III: Finite Strip Analysis of Bridges
12 Slab and slab-on-girder bridges
13 Curved and skewed bridges
14 Box girder bridges
15 Continuous haunched bridges
16 Cable-stayed bridges
17 Finite strip modeling of bridges
References
Index

Citation preview

Finite Strip Analysis of Bridges

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O T H E R B O O K S O N S T R U C T U R E S F R O M E & F N SPO N Aluminium Allo y Structure s F . M . Mazzolan i A l t e r n a t i v e M a t e r i a l s f o r th e R e i n f o r c e m e n t a n d P r e s t r e s s i n g o f C o n c r e t e Edited b y J . L . C l a r k e Bridge Bearing s a n d E x p a n s i o n Joint s D . J. L e e Bridge Dec k Behaviou r E . C. Hambl y Bridge Managemen t 3 Edited b y J . H a r d i n g , G . A . R . Park e an d M . J . R y a l l B u i l d i n g th e F u t u r e Innovation i n design , material s an d constructio n Edited b y F . K . G a r a s , G . S . T . A r m e r and J . L . C l a r k e Computer Method s i n Structura l Analysis J. L . Mee k Concrete Structure s Stresses an d deformation s A . G h a l i an d R . F a v r e Concrete unde r Sever e Condition s Environment an d loadin g Edited b y K . S a k a i , N . Banthi a an d O . E . Gjor v C o n s t r u c t i o n a l Stee l D e s i g n A n internationa l guid e Edited b y P . J . D o w l i n g , J . E . Hardin g an d R . Bjorhovd e Continuous an d Integra l Bridge s Edited b y B . Pritchar d Design o f Prestresse d Concret e R . I . Gilber t and N . C . Mickleboroug h Flexural-Torsional Bucklin g o f Structure s N . S . Trahai r Strip Metho d Desig n Handboo k A . Hillerbor g Structural Desig n o f Polyme r Composite s E U R O C O M P Desig n Cod e an d Backgroun d Documen t Edited b y J . L . C l a r k e S t r u c t u r a l D y n a m i c s fo r th e P r a c t i s i n g E n g i n e e r H . M . Irvin e For details of Boundary Row,

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Finite Strip Analysis of Bridges

M.S. Cheun g Director, Ottawa-Carleton

Bridge Research Institute and Adjunct Professor, Department of Civil Engineering, University of Ottawa, Ottawa, Canada

W. Li Post-Doctoral Fellow, Department of Civil Engineering, University of Ottawa, Ottawa, Canada,

and S.E. Chidia c Research Officer, Structures Laboratory, Institute for Research in Construction, National Research Council Canada, Ottawa, Canada

m E & F N SPON

An Imprin t o f Chapma n & Hal l

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4 1 9 1915 0 X

A p a r t fro m an y fai r dealing fo r th e purpose s o f researc h o r privat e study , o r criticism o r review , a s permitte d unde r th e U K Copyright Design s an d Patent s A c t , 1988 , thi s publicatio n ma y no t b e reproduced , stored , o r transmitted , i n an y form o r b y an y means , withou t th e prio r permission i n writin g o f th e publishers , or i n th e cas e o f reprographi c reproductio n onl y i n accordanc e wit h th e term s o f the licence s issue d b y th e Copyrigh t L i c e n s i n g A g e n cy i n th e U K , or i n accordance wit h th e term s o f licence s issue d b y th e appropriat e Reproductio n Rights Organizatio n outsid e th e U K . Enquirie s concerning reproductio n outsid e the term s state d her e shoul d b e sen t t o th e publisher s a t th e L o n d o n addres s printed o n thi s page . T h e publishe r make s n o representation , expres s o r implied , wit h regar d t o th e accuracy o f th e informatio n containe d i n thi s boo k an d canno t accep t an y lega l responsibility o r liabilit y fo r an y error s o r omission s tha t ma y b e made . A catalogu e recor d fo r thi s boo k i s availabl e fro m th e Britis h Librar y

To our families

Contents Preface x

i

Notation xii

i

Part I Mathematica 1 Introductio 1.1 B r i d g

n3

e analysi s b y refine d method s 3

1.2 Developmen 1.3 Readin 2 Basi

l Approac h 1

t o f t h e finite s t r i p m e t h o d 5

g suggestion s 7

c concept s o f numerica l method s 9

2.1 I n t r o d u c t i o n 9 2.2 P r o b l e m f o r m u l a t i o n 1 2.3 D i r e c t approac h 1 2.4 V a r i a t i o n a l m e t h o d 1

0 3 5

2.5 2.6 2.7 2.8 2.9

7 1 2 3 6

W e i g h t e d residua l m e t h o d 2 F i n i t e elemen t m e t h o d 3 Convergenc e requirement s fo r i n t e r p o l a t i o n function s 3 Generalize d procedur e fo r a finite elemen t s o l u t i o n 3 Finit e stri p metho d 3

3 Numerica l error s 3 3.1 I n t r o d u c t i o n 3 3.2 D e f i n i t i o n o f c o m p u t a t i o n a l error s 3 3.3 Assessin g c o m p u t a t i o n a l error s 3 3.4 Assessin g discretizatio n error s 4 3.5 C o n c l u d i n Part I I Finit 4 Finit

g remark s 4

e Stri p Metho d 5

e stri p metho d 5

4.1 I n t r o d u c t i o 4.2 E n e r g

n5

y approac h fo r a simpl e bea m 5

7 7 8 9 1 9 1 3 3 4

Contents

viii

4.3 P l a t

e stri p 5

4.4 P l a n

e stres s s t r i p 6

4.5 F l a

6 9

t shel l s t r i p 7

4.6 B o u n d a r

5

y c o n d i t i o n s alon g n o d a l line s 7

4.7 C o n t i n u o u

s structures : flexibility

approac

8 h8

0

4.8 C o n t i n u o u

s structures : stiffnes s approac h 8

1

4.9 N u m e r i c a

l example s 8

4

5 Highe

r o r d e r finit e s t r i p s 9

5.1 I n t r o d u c t i o

n9

1 1

5.2 P l a t

e stri p H 0 2 w i t h curvatur e continuit y 9

5.3 P l a t

e s t r i p H 0 3 w i t h a n i n t e r n a l n o d a l lin e 9

3

5.4 Plan

e stres s s t r i p H 0 3 9

7

5.5 F l a

t shel l s t r i p H 0 3 9 e o f approac h 9

5.6 Choic

5.7 N u m e r i c a 6 Splin

l example s 9

e finit e s t r i p m e t h o d 10

6.1 I n t r o d u c t i o 6.2 Splin 6.3 F l a

e f u n c t i o n i n t e r p o l a t i o n 10 t shel l splin e finit e s t r i p 10

6.4 Analysi 7 Compoun 7.1 7.2 7.3 7.4

n 10

s o f a r b i t r a r i l y shape d plate s 11 d s t r i p m e t h o d 12

I n t r o d u c t i o n 12 Rectangula r c o m p o u n d s t r i p 12 Rectangula r B-splin e c o m p o u n d s t r i p 12 N u m e r i c a l example s 12

8 Finit

e laye r m e t h o d ; finit e p r i s m m e t h o d 13

8.1 I n t r o d u c t i o

1

9 9 9 5 5 7 9 2 1 1 2 6 7 3

n 13

3

8.2 F i n i t

e laye r m e t h o d 13

4

8.3 F i n i t

e p r i s m m e t h o d 13

6

8.4 C i r c u l a r l

y c u r v e d finit e p r i s m 14

1

8.5 N u m e r i c a

l example s 14

2

9 Vibratio

n a n d s t a b i l i t y a n a l y s e s 14

9.1 I n t r o d u c t i o 9.2 V i b r a t i o 9.3 Mas

n 14

n finite s t r i p analysi s 14

s m a t r i x o f a f i n i t e s t r i p 15

9 9 9 0

9.4 Mas

s m a t r i x o f a p l a t e s t r i p 15

1

9.5 Mas

s m a t r i x o f a plan e stres s s t r i p 15

2

9.6 Mas

s m a t r i x o f a flat shel l s t r i p 15

2

9.7 B e n d i n

g a n d in-plan e i n t e r a c t i o n 15

3

Contents i

x

9.8 S t a b i l i t

y analysi s o f plate s 15

5

9.9 S t a b i l i t

y analysi s o f t h i n - w a l l e d structure s 15

5

9.10 N u m e r i c a l examples 15

7

10 N o n l i n e a r a n a l y s i s 1 6

7

10.1 I n t r o d u c t i o n 16

7

10.2 N o n l i n e a r s o l u t i o n procedur e 16

7

10.3 Elastoplasti c analysi s 17

1

10.4 A n a l y s i s o f reinforce d concret e slab s 18

0

10.5 G e o m e t r i c a l l y nonlinea r analysi s 19

0

11 C o m b i n e d a n a l y s i s 2 0

3

11.1 I n t r o d u c t i o n 20

3

11.2 C o m b i n e d F S / F E analysi s o f i r r e g u l a r plate s 20

4

11.3 C o m b i n e d F S / B E analysi s o f i r r e g u l a r plate s 20

9

11.4 C o m b i n e d B E / F S analysi s o f sla b girde r bridge s 21

3

P a r t I I I F i n i t e S t r i p A n a l y s i s o f B r i d g e s 22

3

12 S l a b a n d s l a b - o n - g i r d e r b r i d g e s 2 2

5

12.1 I n t r o d u c t i o n 22

5

12.2 Stiffnes s m a t r i x o f a l o n g i t u d i n a l b e a m 22

6

12.3 D e f o r m a t i o n a n d settlemen t o f s u p p o r t 22

8

12.4 N u m e r i c a l examples 22

8

13 C u r v e d a n d s k e w e d b r i d g e s 2 4 13.1 I n t r o d u c t i o n 24 13.2 C i r c u l a r l y c u r v e d p l a t e s t r i p 24 13.3 C u r v e d c o m p o u n d s t r i p 24 13.4 C u r v e d s t r i p fo r b o x bridge s 24 13.5 Skewe d p l a t e s t r i p 25 13.6 A n a l y s i s o f skewe d b o x girde r bridge s 25 13.7 N u m e r i c a l examples 25 14 B o x g i r d e r b r i d g e s 2 6

1 1 1 4 7 0 2 2 3

14.1 I n t r o d u c t i o n 26

3

14.2 Elasti c propertie s o f c o n s t i t u e n t plate s 26

4

14.3 I n t e r m e d i a t e s u p p o r t s an d d i a p h r a g m 26

6

14.4 Prestressin g force s 27

0

14.5 L o c a l b e n d i n g m o m e n t 27

2

14.6 N u m e r i c a l examples 27

3

15 C o n t i n u o u s h a u n c h e d b r i d g e s 15.1 I n t r o d u c t i o n

289 289

Contents 15.2 F i n i t e s t r i p analysi s

289

15.3 Splin e finit e s t r i p analysi s

296

15.4 N u m e r i c a l example s

300

16 C a b l e - s t a y e d b r i d g e s

307

16.1 I n t r o d u c t i o n

307

16.2 G i r d e r s u b s t r u c t u r e

308

16.3 Formula s fo r cable s

309

16.4 Stiffnes s m a t r i x fo r pylon s

313

16.5 Initial-stiffnes s i t e r a t i o n

314

16.6 N u m e r i c a l example s

315

17 F i n i t e s t r i p m o d e l i n g o f bridge s

321

17.1 I n t r o d u c t i o n

321

17.2 Selectin g approac h a n d s t r i p

321

17.3 G e n e r a t i n g a finit e s t r i p m o d e l

322

17.4 N u m b e r i n g n o d a l line s a n d strip s

324

17.5 I n p u t d a t a fil e

324

17.6 O u t p u t file

326

References

329

Index

341

Preface T h e finite s t r i p m e t h o d i s w i d e l y recognize d a s a powerfu l a n d versatil e analysis t o o l , a n d i s v e r y effectiv e i n t h e analysi s o f structure s w h i c h have comple x geometry , m a t e r i a l propertie s an d l o a d i n g condition s b u t w i t h r e l a t i v e l y simpl e s u p p o r t c o n d i t i o n s , suc h a s bridges . T h i s m e t h o d has bee n extensivel y applie d i n t h e stati c a n d d y n a m i c analysi s o f bridg e structures fo r m a n y years . U n d o u b t e d l y on e o f t h e reason s fo r i t s grea t appeal t o bridg e engineer s an d researcher s i s t h e fac t t h a t t h i s m e t h o d provides efficien t a n d accurat e analysi s w i t h m i n i m u m m o d e l l i n g effor t a n d i n p u t requirements . On e objectiv e o f t h i s b o o k i s t o a t t e m p t t o b r i n g togethe r m a n y o f th e specialize d application s fo r differen t type s o f bridges a n d t o presen t t h i s m a t e r i a l , alon g w i t h t h e fundamental s o f t h e m e t h o d , i n a unifie d a n d consisten t manner . T h i s b o o k summarize s t h e current development s a n d advancement s o f t h e finite s t r i p m e t h o d a n d t h e i r application s t o bridg e engineering . A m o n g t h e specia l application s considered ar e t h e following : linea r a n d nonlinea r analysi s o f slab , sla b o n girder, composit e b o x girder , cable-staye d a n d othe r form s o f m e d i u m a n d long spa n bridges . T h e b o o k comprise s o f thre e parts : I ) basi c concept s o f n u m e r i c a l methods; I I ) fundamentals o f t he finite s t r i p m e t h o d ; a n d I I I ) application s of t h e finite s t r i p m e t h o d t o bridg e analysis . I t i s w r i t t e n fo r senio r engineering students , p r a c t i s i n g a n d researc h engineers , a n d other s w h o have acquire d t h e knowledg e o f bridge analysi s an d mechanic s o f materials . T h e reade r doe s n o t nee d a n extensiv e b a c k g r o u n d i n mor e advance d techniques, suc h a s t h e t h e o r y o f elasticity , energ y methods , p l a s t i c i t y a n d n u m e r i c a l analysis . T h e m a t h e m a t i c a l knowledg e require d i s likewis e n o t v e r y great , sinc e t h e presentatio n i s base d u p o n a physica l s t r u c t u r a l p o i n t of v i e w r a t h e r t h a n a m a t h e m a t i c a l one . Differentia l a n d i n t e g r a l equation s are r e q u i r e d t o a l i m i t e d exten t o n l y i n t h e first t w o parts , wherea s applications o f t he t h e o r y i n t he derivation s o f solutions appea r i n t he late r chapters. T h r o u g h o u t , a n a t t e m p t ha s bee n mad e t o presen t t h e m a t e r i a l i n it s entirety , t h a t is , t he development s begi n w i t h f u n d a m e n t a l principles , followed b y applications , n u m e r i c a l example s a n d t h e final result s ar e t r a n s l a t e d i n t o table s o r graph s fo r convenien t referenc e a n d use . M a n y solved example s fo r differen t type s o f bridge s ar e given . P a r t I consist s o f thre e chapter s w h i c h ar e i n t e n d e d a s a n overvie w o f

Preface

xii

t h e fundamental s o f n u m e r i c al m e t h o ds a n d s t r u c t u r a l mechanics, i n c l u d i n g such area s a s basi c concept s o f n u m e r i c a l m e t h o d s , v a r i a t i o n a l p r i n c i p l e s, convergence requirement s a n d erro r e s t i m a t i o n s . P a r t I I concentrates o n t h e t h e o r y o f t h e finite

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

applications i n various areas of analysis. Specifi c details o n c o n v e n t i o n a l a nd spline finite s t r i p m e t h o ds ar e give n i n C h a p t e rs 4 an d 6 , w h i le s o p h i s t i c a t e d finite strip s suc h a s highe r orde r a n d c o m p o u n d finite strip s ar e discusse d i n Chapters 5, 7 a n d 1 1 . T h r e e - d i m e n s i o n al finite strip s i n t he f o r m o f finite layers o r finite prism s ar e als o d e r i v e d a n d discusse d i n d e t a i l i n C h a p t e r 8 . These strip s ca n b e a p p l i e d n o t o n l y t o t h e analysi s o f t h i ck concret e b r i d g e decks b u t als o t o l a m i n a t e d composites , nuclea r structure s a n d concret e dams. C h a p t e r 9 i s devote d e n t i r e l y to t h e d y n a m i c a n d s t a b i l i t y analyse s o f bridge s usin g variou s type s o f finite

s t r i p models . Sinc e d y n a m i c a n d

s t a b i l i t y analyse s ar e essentia l factor s i n b r i d g e design , especiall y fo r stee l bridges, t h i s chapte r provide s i m p o r t a n t guidanc e a n d t h e necessar y tool s for p r a c t i s i n g engineers. M a t e r i a l a n d g e o m e t r y n o n l i n e a r i t y considerations i n t h e finite s t r i p f o r m u l a t i o n ar e presente d i n C h a p t e r 10 . T h e r e m a i n i n g t e x t present s a p p l i c a t i o n s o f t h e finite

s t r i p m e t h o d t o specifi c type s o f

bridge. Thes e i n c l u d e sla b a n d sla b o n girde r bridges , c u r v e d a n d skewe d bridges, b o x girde r bridges , continuou s haunche d bridge s a n d cable-staye d bridges. A n extensiv e lis t o f reference d o c u m e n t s a n d a subjec t i n d e x ar e give n i n t h e appendices . Thes e appendice s hav e bee n c o m p i l e d t o p r o v i d e a quic k reference t o an y subjec t area s discusse d i n t h e book . T h e senio r a u t h o r o f t h i s b o o k ha s devote d ove r 2 5 year s o f researc h t o t h e finite s t r i p m e t h o d . A n u m b e r o f m a j or finite-strip

compute r program s

have bee n develope d b y h i m a n d hi s colleague s d u r i n g t h i s t i m e . I n orde r t o p r o m o t e t h e widesprea d us e o f t h i s m e t h o d b y bridge-desig n engineers , especially i n s m a l l desig n offices , t h e a u t h o r s hav e consolidate d a l l thes e p r o g r a m s i n t o t h e comprehensiv e a n d eas y us e packag e B A F S M w h i c h w i l l be availabl e soon. * T h e a u t h o r s w i s h t o expres s t h e i r g r a t i t u d e t o t h e N a t u r a l Scienc e a n d E n g i n e e r i n g Researc h C o u n c i l o f C a n a d a fo r financial

suppor t o f thei r

research i n t h i s are a d u r i n g t h e las t 1 2 years . T h e author s w o u l d als o lik e t o t h a n k M r G . Franch e o f t h e N a t i o n a l Researc h C o u n c i l o f C a n a d a fo r his hel p i n p r e p a r i n g a l l t he d r a w i n g s a n d table s i n t h i s b o o k .

M.S. Cheun g W. L i S. E . C h i d i a c * F o r informatio n abou t B A F S M software , contac t D r . M . S . C h e u n g , D e p a r t m e n t o f C i v i l E n g i n e e r i n g , U n i v e r s i t y o f O t t o w a , 16 1 L o u i s P a s t e u r , O t t o w a , O n t a r i o , C a n a d a K I N 6NS .

Notation A are {a} vecto b widt

[B] s t r a i

a o f cross-sectio n area o f d o m a i n r o f j o i nt displacement s h o f stri p subscript fo r b e n d i n g analysi s superscript o r subscrip t fo r bea m n matri x

[B] s t r a i D plat

n m a t r i x fo r larg e d e f o r m a t i o n e rigidit y

[D] elasti c matri x [DT] t a n g e n t i a l elastoplasti c m a t r i x [D ] t a n g e n t i a l elasti c m a t r i x o f concret e e prescribe d toleranc e o f i t e r a t i on eccentricity o f b e a m subscript fo r elasti c stat e E modulu s o f e l a s t i c i ty E tangentia l m o d u l u s a t zer o stres s E secan t m o d u l u s a t m a x i m u m compressiv e stres s EA a x i a l stiffnes s o f b e a m EI flexural rigidit y o f bea m f' u n i a x i a l compressiv e s t r e n g t h o f concret e / / uniaxia l tensil e s t r e n g t h o f concret e { / } vecto r o f displacemen t component s {F} vecto r o f unbalance d force s [/], [F] f l e x i b i l i t y matri x G shea r modulu s GJ t o r s i o n a l rigidit y o f bea m [G] slop e matri x h lengt h o f l o n g i t u d i n a l sectio n i n splin e finit e s t r i p thickness o f s t r i p i n v i b r a t i o n analysi s H' s t r a i n h a r d e n i n g paramete r i subscrip t fo r n o d a l lin e N o . i subscript fo r p r i n c i p a l stres s Oi C

Q

c

c

I superscrip

t o r subscrip t fo r finite s t r i p N o . I

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

x i v Notation j subscrip

t fo r node , k n o t o r n o d a l lin e N o . j

k mir/l m

[fc], [K] stiffnes

s matri x

[KG] g e o m e t r i c a

l stiffnes s m a t r i x , i n i t i a l stres s stiffnes s m a t r i x

[ K T ] tangentia

l stiffnes s m a t r i x

/ lengt

h o f stri p

Lj(y) Lagrang

e i n t e r p o l a t i o n expression o f nod e j

m subscrip

t fo r t h e m - t h t e r m o f serie s subscript fo r t h e m - t h k n o t i n splin e finit e s t r i p

M bendin

g momen t o r twistin g momen t

[M] mas

s matri x

[N] m a t r i

x o f shap e function s

p in-plan

e forc e pe r u n i t are a subscript fo r plasti c stat e

P poin

t forc e subscript fo r plan e stres s analysi s

{ p } , {P} l o a

d vecto r

q transvers

e l o a d pe r u n i t are a weight pe r u n i t l e n g t h

r numbe

r o f serie s t e r m s use d i n a n analysi s n u m b e r o f l o n g i t u d i n a l section s i n a splin e s t r i p r e d u n d a n t forc e radius i n r- c o o r d i n a t e syste m c u r v a t u r e r a d i u s o f c u r v i l i n e a r c o o r d i n a t e lin e

{ r } vecto

r o f r e d u n d a n t force s

R curvatur

e radiu s o f b o t t o m flang e

{R} vecto

r o f resistan t force s t o d e f o r m a t i o n

s boundar

y coordinat e

S tota

l numbe r o f strip s i n a structur e

t thicknes

s o f stri p t i m e i n v i b r a t i o n analysi s subscript representin g t h e w h o l e s t r u c t u r e

[t], [T] c o o r d i n a t

e t r a n s f o r m a t i o n m a t r i x fo r displacement s

[T ] coordinat

e t r a n s f o r m a t i o n m a t r i x fo r strain s

e

u,v,w displacement U strai W potentia x , 7/ , z loca x, y, z g l o b a

s i n x , y a n d z directions , respectivel y n energ y l energ y o f e x t e r n a l l o a d i n g

l cartesia n c o o r d i n a t e syste m (fo r i n d i v i d u a l s t r i p ) l cartesia n c o o r d i n a t e syste m (fo r s t r u c t u r e ) skew coordinate s

X x/b y ( y ) th m

e m - t h eigenfunctio n o f beam v i b r a t i o n

spanwise subtende d angl e i n r- plane r a t i o E

6w(x)

dx

3

I

J x= 0

0 or w(x)0

dx

3

where A(w) ar e t h e differentia l equations , an d S\(w) a n d S2(w) p r o v i d e t h e necessary b o u n d a r y conditions . W e hav e therefor e d e m o n s t r a t e d t h a t a se t of Eule r equation s ca n b e establishe d f r o m th e v a r i a t i o n a l p r i n c i p l e . N e x t we shal l sho w w h i c h form s o f differentia l equation s ar e E u l e r equation s o f the f u n c t i o n a l . 2.4-3 Linear

self-adjoint

differential

equations

T h e n a t u r a l v a r i a t i o n a l p r i n c i p l e fo r linea r differentia l equation s ca n b e established p r o v i d e d t h e operato r i s s y m m e t r i c o r sel f adjoint . R e v i s i t i n g E q u a t i o n 2. 3 a n d fo r s i m p l i c i t y i g n o r i n g th e n a t u r a l b o u n d a r y condition s given i n E q u a t i o n 2.4 , i.e . A(u)=

Lu+p=

G ; in

Q

(2.47

)

t h e s y m m e t r y an d therefor e t h e self-adjointnes s o f t h e o p e r a t o r L ca n b e d e m o n s t r a t e d b y showin g t h a t / v LudQ — Jn Jn

\ u LvdQ, +

T

1

boundary (2.48

)

for an y t w o function s u an d v. T h i s p r o o f a u t o m a t i c a l l y allow s t h e establishment o f t h e v a r i a t i o n a l p r i n c i p l e , i.e . J=

J (^u

Lu +

u p^ dQ

T

+ boundary (2.49

T

)

T a k i n g t h e first v a r i a t i o n E q u a t i o n 2.4 9 lead s t o 8J =

J (^6u

Lu +

T

^u S(Lu) +

8u p^ dCt

T

+ boundary (2.50

T

)

a n d recallin g t h a t t h e operato r i s linea r an d self-adjoint , E q u a t i o n 2.5 0 ca n be r e w r i t t e n a s p)dQ 4 - boundary (2.51

8J — I 8u (Lu + Jn T

)

T h e t e r m s insid e t h e bracke t y i e l d th e Eule r equatio n a s required . Consider agai n t h e s i m p l y s u p p o r t e d bea m subjecte d t o a u n i f o r m l y d i s t r i b u t e d loa d p(x) presente d i n E x a m p l e 2.2 . th e differentia l e q u a t i o n i s given b y dw A(w)= EI-j^j p = 0 0