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Title

Author(s)

Analysis of plate-type structures by finite strip, finite prism and finite layer methods

Kong, Jackson.; 江傑新

Citation

Issue Date

URL

Rights

1994

http://hdl.handle.net/10722/34498

The author retains all proprietary rights, (such as patent rights) and the right to use in future works.


Analysis of plate-type structures finite stripďźŒfinite prism and finite layer methods

Jackson Kong B.Sc.(Eng.) London, ILK. ďź› M.A.Sc.

A thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy at the University of Hong Kong

March 1994


DECLARATION

I declare that t h i s thesis represents

my

own work,

except

where due

acknowledgement i s made, and t h a t i t has n o t been p r e v i o u s l y i n c l u d e d i n a t h e s i s , d i s s e r t a t i o n o r r e p o r t s u b m i t t e d t o t h i s U n i v e r s i t y o r t o any o t h e r i n s t i t u t i o n f o r a degree, diploma, o r o t h e r q u a l i f i c a t i o n .

Jacksoi


ACKNOWLEDGEMENTS

I

am g r e a t l y i n d e b t e d t o P r o f e s s o r Y . K . C h e u n g ,

my s u p e r v i s o r , f o r

h i s g u i d a n c e , a d v i c e and t h e i n f l u x o f g r e a t i d e a s t h r o u g h o u t t h e c o u r s e o f t h i s t h e s i s . S p e c i a l t h a n k s a l s o go t o P r o f e s s o r K . P . C h o n g ( D i r e c t o r of

Structures

U.S.A.),

and

Dr.S.H.Lo

Building

Systems,

and Dr.L.G.Tham

National

who

have

Science

served on

Foundation, my d o c t o r a l

e x a m i n a t i o n committee and p r o v i d e d v a l u a b l e comments and s u g g e s t i o n s . I

am a l s o g r a t e f u l t o t h e C i v i l E n g i n e e r i n g Department o f H.K.U.,

w h i c h has p r o v i d e d

me w i t h a s t i m u l a t i n g environment f o r

my s t u d y and

r e s e a r c h s i n c e J a n u a r y o f 1990. T r u l y unbounded t h a n k s a r e due t o

my p a r e n t s . W i t h o u t t h e i r l o v e

and s u p p o r t , t h i s t h e s i s c o u l d n o t have become a r e a l i t y . L a s t , but f a r from l e a s t ,

I thank t h e one t o whom t h i s t h e s i s i s

d e d i c a t e d , f o r h e r c o n t i n u e d encouragement and many s a c r i f i c e s t o h e l p me f i n i s h t h i s work, I s h a l l a l w a y s be g r a t e f u l .


Abstract of thesis e n t i t l e d Analysis of plate-type structures by f i n i t e s t r i p , f i n i t e p r i s m a n d f i n i t e l a y e r methods submitted by J a c k s o n Kong f o r t h e degree o f D o c t o r o f P h i l o s o p h y a t t h e U n i v e r s i t y o f Hong Kong i n March 1994 The f i n i t e s t r i p , f i n i t e p r i s m a n d f i n i t e l a y e r methods, p i o n e e r e d b y Y.K.Cheung, have undergone c o n s i d e r a b l e development d u r i n g t h e p a s t two d e c a d e s . D e s p i t e t h e s e r e s e a r c h e f f o r t s , a p p l i c a t i o n o f t h e methods to

some o f

the v e r y

common p l a t e - t y p e s t r u c t u r e s h a s n o t

met w i t h

c o m p l e t e s u c c e s s . A d d i t i o n a l l y ďźŒ because of the increasing popularity of fabricating the

structural components with laminated composite materials,

analysis

of

laminated

increasingly important.

plates,

in

particular,

is

becoming

It is the main theme of the present studies to

explore the methods of finite strip, finite prism and finite layer for the analysis of isotropic and laminated composite plate-type structures. A new finite strip method

is developed for the analysis of thin

plates , 2-D elastic problems and folded-plate structures. Unlike other finite strips, the displacement functions of the present finite strip method are expressed as the product of the usual 1-D shape functions in the

transverse direction and

longitudinal direction.These

a

set

of

computed

static modes

can

be

static modes easily

in

the

obtained

by-

means of a continuous-beam computer program. The resulting finite strip possesses a few superior features when compared with the semi-analytical finite strip and the spline finite strip. These features include, just to

mention

a

few,

the

capability

of

dealing

with

geometric

discontinuities, multi-point supports, and the ease of implementation. The original spline finite strip is restricted to the analysis of thin plates or thin-walled structures.

In this thesis, a third-order

spline finite strip is formulated using a third-order plate theory. This


s p l i n e f i n i t e s t r i p i s capable o f t r e a t i n g moderately t h i c k i s o t r o p i c o r l a m i n a t e d composite p l a t e s . I t i s a l s o a p p l i e d t o n o n l i n e a r f o r c e d vibration

using

time-stepping

procedures.

In

order

to

reduce

computational c o s t of the time-stepping t r a n s i e n t a n a l y s i s ,

the

a special

r e d u c t i o n method i s proposed. F o r t h e a n a l y s i s o f r e c t a n g u l a r , t h i c k , l a m i n a t e d composite p l a t e s , a l l k i n e m a t i c assumptions a r e r e l a x e d and 3 - D e l a s t i c i t y t h e o r y i s adopted. A new s p l i n e f i n i t e p r i s m and a 3 - D t h i c k - p l a t e f i n i t e element a r e d e v e l o p e d . A l t h o u g h t h e y a r e s i m i l a r t o a c o n v e n t i o n a l 3-D f i n i t e element model i n terms o f i n t e r p o l a t i o n c a p a b i l i t y and p r o b l e m s i z e , i t m a i n t a i n s a 2 - D - t y p e d a t a s t r u c t u r e s i m i l a r t o a c o n v e n t i o n a l 2-D f i n i t e element

model.

The

2-D—type

data

structure

also

allows

faster

f o r m u l a t i o n o f t h e element s t i f f n e s s m a t r i c e s . For f r e e v i b r a t i o n a n a l y s i s o f r e c t a n g u l a r p l a t e s w i t h intermediate line

supports,

the

usual

single-span

beam v i b r a t i o n f u n c t i o n s

are

augmented b y 1-D p o l y n o m i a l s i n s u c h a way t h a t a l l k i n e m a t i c boundary conditions,

in

each o f

the

corresponding

in-plane directions,

are

s a t i s f i e d a p r i o r i . They a r e s u b s e q u e n t l y employed a s t h e i n - p l a n e t r i a l f u n c t i o n s f o r t h e development o f a new f i n i t e l a y e r t e c h n i q u e . Numerous n u m e r i c a l e x p e r i m e n t s a r e c o n d u c t e d i n t h i s s t u d y .

The

purpose o f t h e s e e x p e r i m e n t s i s t o v e r i f y t h e v a l i d i t y o f t h e proposed methods,

by

comparing i t s

solution with

those

obtained

by

other

a n a l y t i c a l o r n u m e r i c a l methods, r a t h e r t h a n p r e s e n t i n g new f a c t s o r r e s u l t s . From t h e s e comparisons, t h e a c h i e v e d a c c u r a c y , convergence and e f f i c i e n c y o f t h e proposed methods a r e f o u n d t o be s a t i s f a c t o r y .


To

Sara


CONTENTS

P DECLARA丁 ION ACKNOWLE DGEMENTS TABLE OF CONTENTS L I S T OF TABLES L I S T OF FIGURES L I S T OF SYMBOLS

CHAPTER 1 : INTRODUCTION 1. 1. 1.

1. 1. 1.

F i n i t e s t r i p , f i n i t e p r i s m and f i n i t e l a y e r methods T h i n p l a t e s , moderately t h i c k p l a t e s and t h i c k p l a t e s General r e v i e w o f f i n i t e s t r i p , f i n i t e p r i s m and f i n i t e layer 1.3.1 F i n i t e s t r i p 1.3.2 F i n i t e prism 1.3.3 F i n i t e layer G e n e r a l r e v i e w o f modern p l a t e t h e o r i e s Scope o f p r e s e n t s t u d i e s Organization o f thesis Remarks

11 12

CHAPTER 2 : FORMULATION OF COMPUTED SHAPE FUNCTIONS 2. 2. Z. 2.

L o n g i t u d i n a l t r i a l f u n c t i o n s : COMSFUN M a t h e m a t i c a l p r o p e r t i e s o f COMSFUN Computation o f i n t e g r a l s Summary A p p e n d i x : Beam shape functions

CHAPTER 3 : A THIN-PLATE FINITE STRIP USING COMSFUN 3.2

Introduction Formulation Numerical Examples Pre』iwinery assessment : Static anaiysis of beams A square plate with span a and subjected to a point load A square plate with span a and subjected to UDL A 3-bay flat-plate wode』 A S-bay flat-slab model A 5-bay flat-slab model A continuous-slab bridge rnnn 1 ncii on

16 18 18 19

20 23 23 23 27 27 29 30 31 31 32 32 33


CHAPTER 4

CHAPTER 5

: VIBRATION AND BUCKLING OF RECTANGULAR PLATES

54

Formulation N u m e r i c a l Examples Free Square plates with homogeneous boundary conditions 2^2 3^3 Square plates with mixed boundary conditions Point-supported plates Square plates with step change in thickness Linear Buckling analysis Square plates under uniaxial loads Square plates subjected to Conclusion

54 55 56 56 56 59 60 60 62 63 63 63 64

:

A 2-D FINITE

STRIP USING COMSFUN

CHAPTER 6

VIBRATION AND BUCKLING OF THIN-WALLED STRUCTURES Formulation N u m e r i c a l Examples Vibration of cantilever folded plates Vibration and buckling of Buckling of a continaous-span stiffened Vibration of a fou.r-stiffener panel Vibration of prismatic square tubes Buckling of a NASA panel Conclusion

CHAPTER 7 : A THIRD-ORDER SPLINE F I N I T E NONLINEAR FORCED VIBRATION

STRIP

Introduction Third-order p l a t e theories Spline f i n i t e s t r i p Equilibrium equation N u m e r i c a l Examples Linear static Buckling of isotropic plates with cut-outs Nonlinear anaJysis* : Transverse ioading Noninear analysis Preliminary Remarks Nonlinear forced-vibration A general review

01

02 03 04 AND 346700344555

6

panel

94 96 97 99 99

111122222222

1 2 3 4 5

03344566799 77777777777

Formulation N u m e r i c a l Examples A deep beams wi th both end clamped A deep cant ilever A deep cantilever with step change in depth A deep cantilever with step change in thickness A bar under tension Free vibration of deep beams Shear waJi supported on portal frame Conclusion Appendix :


78912578912 22233333344

7 . 10 7.11

The present approach F o r m u l a t i o n o f r e d u c e d dynamic e q u i l i b r i u m e q u a t i o n s Transformation Newmark Time i n t e g r a t i o n N u m e r i c a l Examples Conclusion Appendix : Consistent mass matrix Stiffness matrices Modified Bz-spline functions Algorithm for generating Ritz vectors Transformation

6

o

CHAPTER 8 : A GLOBAL-LOCAL SPLINE FINITE PRISM METHOD

015568892 666666667

8. 8. 8.

Introduction Formulation Numerical Examples Static analysis A simply-supported, three-layers composite laminate A clamped, three-layers composite laminate A five-layers composite laminate Vibration and Buckling analysis Conclusion

8.4

6

7

CHAPTER 9 : A 3-D THICK-PLATE FINITE ELEMENT

6 7 1 3 5 8 13 77888899

Introduction Formulation Numerical Examples t of layerwise model Assessment of global-local model Assessment of the combined model Conclusion Appendix : Through-thickness integrals CHAPTER 10: A FINITE LAYER METHOD FOR CONTINUOUS PLATES

4 466

2 2 2 2 2 2 2 2 2 2

67

REFERENCES

C o n c l u d i n g Remarks S u g g e s t i o n s f o r f u t u r e work A higher order thin-plate element Reduction method for nonlinear vibration Analysis of continuous sector plates

2 2

11.1 11.2

2 2 2 2 2 2 2

CHAPTER 11 : SUMMARY

3 3o o 4 5 1 o o 111 1 2

12 3

IX

d d d

i x

10.4

Introduction Formulation Numerical Examples 尸reJiminary Simply-supported laminated plates Continuous rectanguJar plates Conclusion

2

229


TABLES Table

Page Treatment o f boundary c o n d i t i o n s

17

R e s u l t s f o r the p r i s m a t i c beam

28

2

Moment d i s t r i b u t i o n o f

28

3

R e s u l t s o f moments f o r a s q u a r e p l a t e (^=0.3, a/h=100)

29

4

D e f l e c t i o n s and Moments a t t h e c e n t r e o f s q u a r e p l a t e s

30

F r e q u e n c i e s o f square p l a t e s w i t h

57

t h e n o n - p r i s m a t i c beam

various

homogeneous

boundary c o n d i t i o n s . 2

F r e q u e n c i e s o f 2*2 c o n t i n u o u s p a n e l s

58

3

F r e q u e n c i e s o f 3*3 c o n t i n u o u s p a n e l s

59

4

Frequencies o f

square

plates

with various

mixed

60

plates

with

point

61

boundary c o n d i t i o n s . Frequencies o f

square

various

supports. Fundamental

frequencies

of

square

plates

with

62

various

63

step-change i n t h i c k n e s s . Buckling

loads

of

square

plates

with

homogeneous boundary c o n d i t i o n s B u c k l i n g loads o f square p l a t e s w i t h continuous l i n e

64

support. D e f l e c t i o n s o f a deep beam

74

2

D e f l e c t i o n s o f a deep beam w i t h change i n d e p t h

75

3

D e f l e c t i o n s o f a deep beam w i t h change i n t h i c k n e s s

76

4

A x i a l stresses of a non-prismatic bar

76

5

F r e q u e n c i e s o f a deep

beam w i t h

various

boundary

77

conditions 1.

Frequencies o f a c a n t i l e v e r channel

2.

Frequencies

of

cantilever

folded

97 plates

with

98

d i f f e r e n t crank angles. 3

V i b r a t i o n and b u c k l i n g o f c o n t i n u o u s beams

100

4

Buckling o f a stiffened panel

100

5

Frequencies o f a s t i f f e n e d panel.

101

6

Frequencies o f composite tubes

103

7

B u c k l i n g o f NASA example 2

104

Comparisons o f b e n d i n g moments a t t h e square

plate

centre

of

121


Vll R e s u l t s o f b e n d i n g moments and

deflections

at

the

122

centres o f simply supported t h i c k p l a t e s

(。,90》0°)

Results of deflections f o r a three-ply

123

square laminated plate with simply supported edges Buckling loads for simply

supported

square

plates

123

with hole Configuration and loading of the thin

isotropic

133

plate 7.6

Comparison of natural frequencies

7.7

Maximum centre deflections obtained

133 with

different

134

methods and load magnitudes. Comparison of results for a three-ply laminate

167

with span-to-thickness ratio of 4. Comparison of results for a three-ply laminate

with

168

span-to-thickness ratio of 10. 8.3

Comparison of results for a five-ply laminate.

8.4

Comparison buckling

of

results

analysis

of

for

free

cross-ply

vibration laminated

169 and

171

square

plates. 8.5

Comparison of the number of unknowns within a

171

cross-section 9.1

Interpolation across thickness in different regions

179

9.2

Comparison of results for a four-ply laminate with s

184

= 2

and 4. Only Layerwise cubic shape function is

used. 9.3

Comparison of results for a nine-ply laminate with s = 2

185

and 4. Only Layerwise cubic shape function is

used. 9.4

Convergence of centre

deflections

of

the

global-

186

using

187

with

187

local model. 9.5

Comparison of results for a four-ply laminate a mesh of 4x4.

9.6

Comparison of

results

s=100. A mesh of

for

laminated

4x4 with only

plates

through-thickness

global-local interpolation. 9.7

Comparison of results for a four-ply s=2

and

model.

s=4.

A

mesh

of

4x4

using

laminate the

with

combined

189


vui Comparison o f r e s u l t s f o r a n i n e - p l y

laminate

with

190

s=2 and s=4. A mesh o f 4x4 u s i n g the combined model. Comparison o f

results

laminates w i t h

for

s=100.

A

four-ply mesh o f

and

nine-ply

4x4 u s i n g

190

the

combined model. ,10

Comparison o f r e s u l t s o f n i n e - p l y l a m i n a t e s f o r

the

191

e f f e c t s o f z i g - z a g f u n c t i o n s on In-plane s t r e s s e s . A mesh o f 4x4 u s i n g t h e combined model. 10. 1

Beam v i b r a t i o n f u n c t i o n s . ( S . S . = s i m p l y - s u p p o r t e d )

209

10.2

Boundary and i n t e r n a l c o n d i t i o n s o f p l a t e s .

210

10.3

Frequencies o f a

beam w i t h

212

F r e q u e n c i e s o f a f o u r - s p a n c o n t i n u o u s beam w i t h b o t h

213

three-span

continuous

uniform support spacing. EI,p=constant ends 10.5

simply

supported.

Frequencies o f a two-span

EI,p=constant. non-prismatic

continuous

213

beam w i t h b o t h ends s i m p l y s u p p o r t e d . 10.6

Fundamantal f r e q u e n c i e s o f s q u a r e l a m i n a t e s .

214

10.7

Frequencies o f continuous t h i n - p l a t e s .

216

10.8

Frequencies o f continuous t h i c k - p l a t e s .

218

10.9

F r e q u e n c i e s o f c o n t i n u o u s symmetric l a m i n a t e s .

219

10.10

F r e q u e n c i e s o f c o n t i n u o u s skew-symmetric l a m i n a t e s .

220


FIGURES Figure 1.1

Page A

typical

functions.

finite A

strip

and

box g i r d e r i s

its

displacement

divided

into f i n i t e

its

displacement

13

strips. A

typical

functions.

finite A

prism

and

thick-walled

tube i s

divided

14

into

f i n i t e prisms. A

typical

finite

layer

and

its

displacement

15

functions. A f o u r - p l y ( f o u r - l a y e r s ) laminated p l a t e i s divided into f i n i t e layers. A n o n - p r i s m a t i c beam i s d i v i d e d i n t o

standard

beam

22

varying

34

elements. (a)

A

thin-plate

finite

strip

with

longitudinal r i g i d i t y . (b) The f i c t i t u o u s beam and i t s d i s c r e t i z a t i o n . ( c ) COMSFUN Y c o r r e s p o n d i n g t o node m. m

A t y p i c a l f i n i t e s t r i p w i t h f i v e nodes the

two

nodal

lines.

The

nodes

on

are

each o f of

35

unequal

distance apart. Classification of strips for

analyzing f l a t

slabs

35

w i t h drop panels. A s i m p l y - s u p p o r t e d beam w i t h two p o i n t l o a d s . E = 1 y=0

36

Convergence o f c e n t r e d e f l e c t i o n s , NDOF = number

of

37

and

38

d e g r e e s o f freedom Three-bays

flat-plate

model.

(a)

Geometry

c o n f i g u r a t i o n s , (b) f i n i t e s t r i p mesh. (a)

Distribution of

moment a l o n g L I

39

(b) D i s t r i b u t i o n o f moment a l o n g L 2

40

( c ) D i s t r i b u t i o n o f moment a l o n g L 3

41

Three-bays

flat-slab

configurations,

model.

(a)

(b) f i n i t e s t r i p

Geometry

and

42

mesh.

(a) D i s t r i b u t i o n o f moment a l o n g L I

43

(b) D i s t r i b u t i o n o f moment a l o n g L 2

44

( c ) D i s t r i b u t i o n o f moment a l o n g L 3

45

Five-bays f l a t

slab

model.

(a)

Geometry

and

46


mesh

c o n f i g u r a t i o n s , (b) f i n i t e s t r i p 3

Checkerboard l o a d p a t t e r n s . DL=5 kN/m2LL=10 kN/m2

47

3

HG f foorr the the ccaassee o f (a) D i s t r i b u t i o n o f moment a l o n g HG

48

UDL (b) D i s t r i b u t i o n o f moment a l o n g

KL f o r the c a s e

of

48

I J f o r the case

of

49

HG f o r t h e c a s e

of

49

KL f o r t h e c a s e

of

50

UDL ( c ) D i s t r i b u t i o n o f moment a l o n g UDL 3.

(d) D i s t r i b u t i o n o f moment a l o n g checkerboard l o a d p a t t e r n 1 (e) D i s t r i b u t i o n o f moment a l o n g checkerboard l o a d p a t t e r n 2 (a)Two-span

slab

bridge

arrangements (b) F i n i t e (a) P o i n t Load : moment M

strip

and

its

loading

51

meshes

Distributions

of

deflection

along the l o n g i t u d i n a l l i n e a t

and

52

1m from

t h e edge. (b) UDL ďźš Distributions of deflection and moment along the longitudinal line at 1m from the edge.

M y

53

A finite strip with initial inplane stresses.

65

Rectangular plates of various homongeneous

66

boundary

conditions. 2

x

2

continuous

panels

with

various

boundary

66

conditions. 3 x 3

continuous panel

Rectangular plates with mixed

67 boundary

Rectangular plates with various

point

conditions. supports.

67 68

A square plate with step change in thickness.

68

Square plates subjected to uniaxial loads.

69

Square plates with intermediate line supports

69

(a) Cases 1-4 (b) Cases 5-8 (c) Cases 9-12 (a) A 2HD plane finite strip (b) The fictituous beam and its discretization. (c) COMSFUN Y. of node i. i (a) A typical strip with 5 nodes on each nodal line. (b) An assemblage of three finite strips. (c) Connection of COMSFUN at node j.

8 1


XI

A deep beam w i t h b o t h e n d s clamped. H a l f o f t h e beam

81

i s d i v i d e d i n t o 8 s t r i p s and i n c r e a s i n g number o f nodes o n e a c h n o d a l l i n e . Stresses d i s t r i b u t i o n across the

depth

( a ) 〜 a t y=〇,(b) cr^at y=0 ,

82

(c)

S3

xy

(e)

x

< r a t y=l y S t r e s s e s d i s t r i b u t i o n a c r o s s t h e d e p t h (a) cr^at

84 y=0

85

(b) cr a t m i d - s p a n (y=6) A deep

cantilever

with

step-change

y=10. I t i s d i v i d e d i n t o

in

depth a t

86

15 f i n i t e s t r i p s and 7

nodes p e r n o d a l l i n e . Stresses d i s t r i b u t i o n across the depth (a) <r a t y=0 (b) cr a t y = 9 . 5 y y A deep c a n t i l e v e r w i t h s t e p - c h a n g e i n t h i c k n e s s y=10. T h i c k n e s s t

= 0 . 1 f o r y^lO and t

87 at

88

= 0.2 for

y t l O . ( a ) FS1 : A mesh of 10 strips with 9 nodes per nodal line is used, (b) FS2 : A mesh with

of

20

strips

5 nodes per nodal line.

Stresses distribution across the depth (a)

< r at y=10

(c) x

y

xy

(b)

R

at y=10

(d) x

R

c r at y=10

89

at y=10

90

y

L

xy

Finite strip meshes of a

bar

L

with

step-change

in

91

cross-section at mid-span. 11

Vibration modes of a deep beam

12

(a) Geometry

and

91

configuration

of

a

shear

wall

92

supported on portal frames. (b) Finite strip mesh (c) Longitudinal nodes of the present finite strip. 13

(a) Vertical deflection along the height of the wall (b)

Vertical

stresses

distribution

section of the wall at 4in.

across

93

the

from ground surface.

A flat-shell finite strip and its coordinate systems

106

A cantilever channel.

106

3-D view of

the

first

four

mode

shapes

the

107

folded

108

of

cantilever channel in Fig.6.2. Cross-sections plates.

of

one-fold

and

two-fold


XII

Buckling

of

continuous

stiffened

panel;

(a)

108

l o n g i t u d i n a l and t r a n s v e r s e s e c t i o n o f t h e p a n e l , (b)buckling

pattern

(transverse

direction)ďź&#x152;

(c)

three strips idealization Cross-section of a

simply-supported

four-stiffener

109

panel 3-D view of

the

first

seven mode

shapes

of

the

110

stiffened panel in Fig.6.6.

111

Cross-section of the twin-cell tube

109

Loading and dimensions for stiffened panel.

112

Simply-supported on all four edges. Normalized vertical deflections at the centre

of

a

144

(a) A square plate with a square hole of width 0.5a.

145

square plate with increasing number of strips, Exterior

boundaries

simply-supported, and

the

of

the

plate

are

(b) The spline finite strip mesh

corresponding

boundary

conditions

for

a

quadrant Finite strip mesh of a edges

or

square

simply-supported

plate

edges

with

and

clamped

subjected

145

to

uniformly distributed load. Load-deflection of a square y=0.333.

plate

with

a/h=5

and

Inplane displacements are clamped at

146

the

edges for case (a) and free for case (b). (a) clamped (b) simply-supported Load-deflection of a square

plate

with a/h=100.

147

All inplane displacements are clamped at the edges, (a) clamped (b) simply-supported A square plate subjected to edge Out-of-plane compression,

deflection

with

compression increasing

148 edge

149

(aă&#x20AC;&#x2022; Case 1(b) , (b) Case II(b)

(a) Loading function on a plate

150

(b) Nonlinear stiffening behaviour of a plate. Response

of

the

square

plate

subjected

to

a

sinusoidal load of maximum amplitude lOq . t=0.002 0 sec per time step. (a) vertical displacement response

(b)

inplane

displacement

response.

1 5 1


xiii 1 0

Response o f t h e s q u a r e p l a t e s u b j e c t e d

to

a half-

152

s i n e l o a d o f maximum a m p l i t u d e lOq^. t=0.002 s e c p e r time step,

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

(b)

i n p l a n e displacement response. 11

Response

of

the

square

plate

subjected

to

a

153

s i n u s o i d a l l o a d o f maximum a m p l i t u d e 50q . t=0.001 sec

per

time

step.

(a)

O

vertical

displacement

r e s p o n s e (b) i n p l a n e d i s p l a c e m e n t r e s p o n s e . 12

Response o f t h e s q u a r e p l a t e s u b j e c t e d

to

a

half-

154

s i n e l o a d o f maximum a m p l i t u d e 50q . t=0.001 s e c p e r time step,

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

(b)

i n p l a n e displacement response. 13

Response o f t h e s q u a r e p l a t e s u b j e c t e d

to

a

step

155

l o a d o f maximum a m p l i t u d e 50q . t=0.001 s e c p e r t i m e O

s t e p , (a) v e r t i c a l d i s p l a c e m e n t response (b) i n p l a n e displacement response. 14

Response o f t h e s q u a r e p l a t e s u b j e c t e d

to

a

step-

156

change l o a d o f maximum a m p l i t u d e 50q . t=0.001 s e c O

per time step, (b) 15

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

inplane displacement

Response

of

the

square

response. plate

subjected

to

a

157

t r i a n g u l a r l o a d o f maximum a m p l i t u d e 50q . t=0.001 O

sec 16

per

time

response

(b)

Response

of

step.

(a)

vertical

inplane displacement the

square

plate

displacement response.

subjected

to

a

158

e x p o n e n t i a l l o a d o f maximum a m p l i t u d e 50q . t=0.001 O

sec

per

response 7 . 17

time (b)

(a) A t y p i c a l

step. inplane

(a)

vertical

displacement

B3-spline

displacement response,

functions

(b)

a

typical

159

i n t e g r a t i o n r e g i o n and t h e s p l i n e s t h a t i n t e r s e c t it. 1 8

Discretization of

a

multi-connected

domain

s p l i n e f i n i t e s t r i p s w i t h modified s p l i n e A

rectangular

plate

and

its

into

159

function.

cross-sectional

173

d i s c r e t i z a t i o n i n t o 8 - n o d e s elements. G l o b a l - L o c a l F i n i t e P r i s m o f a hollow tube.

174

Results o f transverse

175

shear

stresses

distribution


xiv

f o r a t h r e e - p l y l a m i n a t e w i t h a l l edges clamped. (a) r

yz

a t (a/2, a/12)

(b) x

xz

a t (a/12, a/2)

A l a m i n a t e d c o m p o s i t e p l a t e and i t s d i s c r e t i z a t i o n . 2ig-Zag

function

for

a

nine-ply

laminate

195 195

w i t h t h i c k n e s s =10. Division

of

a

plate

into

three

regions

with

196

regions

196

d i f f e r e n t through-thickness interpolation. D i v i s i o n o f a square p l a t e

into

different

f o r c a l c u l a t i n g the transverse shear s t r e s s p o i n t 1 and t

a t point 2.

yz

D i s t r i b u t i o n o f transverse

shear

stresses

xz

through

197

t h e t h i c k n e s s o f t h e f o u r - p l y l a m i n a t e (a/h=2ďź&#x152; h=10) .(a)

T

xz

(b)T

,

yz

Distribution of transverse

shear

stresses

through

198

the thickness of the four-ply laminate (a/h=4, h^S) .(a) x

x2

, (b)T

yz

Distribution of transverse the

thickness

h=0.2) .(a) x

of

the

shear

four-ply

stresses laminate

through

199

(a/h=100,

, (b) x

xz

yz

Distribution of transverse

shear

stresses

through

200

the thickness of the nine-ply laminate (a/h=2, h=10) .(a) T

xz

, (b)T

yz

Distribution of transverse

shear

stresses

through

201

the thickness of the nine-ply laminate (a/h=4, h=5) .(a) x

xz

, (b)T

yz

Distribution of transverse the

thickness

h=0.2) .(a) T

xz

of

the

, (b) T

A continuus thick

shear

nine-ply

stresses laminate

through

202

(a/h=100,

yz

plate

with all edges

222

simply-supported. (a) 2x2 continuous panel (b) 3x3 continuous panel (c) 1-way 3-span continuous panel

223

(d) 1-way 4ä¸&#x20AC;span

continuous panel A r e c t a n g u l a r t h i n - p l a t e element

using

COMSFUN i n

228

both directions. A

curved f i n i t e

supports.

strip

with

intermediate

line

228


SYMBOLS

General

Magnitudes of uniformly distributed loads Bending rigidity (Et3/12(l-y2)) E G v

Young's modulus,

shear modulus

and Poisson、

ratio of isotropic material M

M M

Bending and Twisting moments of a plate

N

N

In-plane stress resultants of a plate

X

E

X

x:

y

ij

y NX:

(= E iJ

Young's Modulus of orthotropic material Shear Modulus and Poisson's ratio of orthotropic material

X,(x) Y i (y) Z,(z)

Trial functions in the X- » Y- and Z-directions Frequencies Normalized frequencies

[D]

Material constant matrix

[N(s)]

Beam element shape functions

[M(s)]

One-dimensional

linear

or

cubic

Lagrangian

shape functions 8-node Seredipity shape functions 1-D, 2-D

One-dimensional, two dimensional

Finite Strip : Computed Shape Functions

b d

Length and width of a finite strip Thickness and density of a finite strip Number

of

beam

elements

and

nodes

for

computed shape functions Mid-plane displacement variables of a plate Nodal displacement unknowns Nodal rotation of finite strip, (3w/Sx)^ ij

Material constant Inclined angle of a flat-shell finite strip In-plane stresses of plates

the


In-plane s t r a i n s o f p l a t e s Out—〇f一plane s t r e s s e s o f p l a t e s Out-of-plane stresses of plates {a } m

Joint

displacement vector o f

beam element

k

( l e n g t h = l ) o f computed shape f u n c t i o n Y k m External load vector

{F} B}

Strain-displacement vector

B p > {B D }

Membrane and b e n d i n g s t r a i n - d i s p l a c e m e n t v e c t o r of a flat-shell finite strip

k]

Point-spring s t i f f n e s s matrix

K]

Stiffness matrix o f a f l a t - s h e l l f i n i t e s t r i p b

p

K ] [K ]

Membrane and b e n d i n g components o f a f l a t - s h e l l f i n i t e s t r i p s t i f f n e s s matrix Mass m a t r i x o f a f l a t - s h e l l f i n i t e s t r i p

Mp]

[M1

Membrane and b e n d i n g components o f a f l a t - s h e l l f i n i t e s t r i p Mass m a t r i x

G]

Gepmetric m a t r i x o f a f l a t - s h e l l f i n i t e s t r i p

p

b

p

b

G ] [G ]

Membrane and b e n d i n g components o f a f l a t - s h e l l f i n i t e s t r i p Geometric m a t r i x

D ] [D ]

Membrane and b e n d i n g components o f

a finite

s t r i p material matrix T]

Co-ordinate transformation matrix

Spline F i n i t e S t r i p

U V W 0) 0

o)

x

8

y

x

¢/

x

Displacement v a r i a b l e s o f a p l a t e

y

S

C r o s s - s e c t i o n a l warping v a r i a b l e s

y

Generalized rotations

入入

x

y

i - t h term o f t h e B ^ - s p l i n e f u n c t i o n s {a} ^ {/3} j {y} ^ 代} i { O i U n k n o v m d i s p l a c e m e n t p a r a m e t e r s {ip}

Reduced unknown p a r a m e t e r s o f t h e d i s p l a c e m e n t vector

{ {B

} {G

\

l>

{R} s

[D ]

{B

NL} N}

{€: }

0

{<5}

L i n e a r , n o n l i n e a r and i n i t i a l s t r a i n v e c t o r s L i n e a r and n o n l i n e a r s t r a i n - d i s p l a c e m e n t v e c t o r Internal resistance vector T r a n s v e r s e s h e a r component o f

a finite strip


material matrix Reduction matrix

[ĺ&#x17D;&#x201A;] ]

Linear e l a s t i c s t i f f n e s s matrix

]

Large displacement m a t r i x

[K cr ]

Geometric m a t r i x

[K^ ]

Tangent s t i f f n e s s m a t r i x

[K 1

Secant s t i f f n e s s m a t r i x

E

5f(x, f(x,t))

Global-Local Spline F i n i t e Prism

h

Thickness o f p l a t e s

r s

o r d e r o f t h e g l o b a l p o l y n o m i a l i n t h e X and Z directions respectively

s P(x,z)

Cross-sectional global-local interpolation

Q(x,2)

Cross-sectional g l o b a l polynomial

[S]

Global s t i f f n e s s matrix o f plates

3-D T h i c k P l a t e F i n i t e Element

g l o b a l - l o c a l through-thickness interpolation Q(2)

Through-thickness g l o b a l polynomial

F i n i t e Layer

Parameters

of

the

augmented

polynomials

m o d i f i e d beam v i b r a t i o n f u n c t i o n s [B]..

1J

Strain-displacement vector o f a f i n i t e layer

of


CHAPTER 1 INTRODUCTION 1.1 F i n i t e s t r i p , f i n i t e prism and f i n i t e layer methods In this thesis,

t h e f i n i t e s t r i p , f i n i t e p r i s m and f i n i t e l a y e r

methods a r e f u r t h e r e x p l o i t e d w i t h p a r t i c u l a r emphasis o n the a n a l y s i s of

isotropic or

this

thesis,

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

these

methods a r e t r e a t e d a s

displacement-based f i n i t e

element

method.

a special class of To a v o i d a m b i g u i t y ,

the these

methods a r e f i r s t d e f i n e d a s below ďź&#x161; Finite strip

(Fig.1.1) ďź&#x161; It

is a two-dimensional subdomain of a

structure in which one opposite pair of the sides of such a subdomain are in coincidence with the boundaries of the structure. The width of a strip remains constant along the longitudinal direction.One dimensional finite element shape functions are used in one direction while a series of trial functions is adopted in the orthogonal direction. Finite prism (Fig.1.2) : It is a three-dimensional subdomain of a structure in which one opposite pair of the faces of such a subdomain are

in coincidence with the boundaries of

the structure.

The cross-

section of a prism remains constant along the longitudinal direction. Two-dimensional , isoparametric, finite-element shape functions are used within the cross-section of a prism while a series of trial functions is adopted in the orthogonal direction. Finite layer

(Fig.1.3):It

is a three-dimensional subdomain of a

structure in which two opposite pairs of the faces of such a subdomain are in coincidence with the boundaries of the structure. The span and width of a layer remain constant across the thickness. One-dimensional finite-element shape functions are used across the thickness of a layer while

a

series

of

trial

functions

Is

adopted

in

each

of

the

two

orthogonal direction. Throughout this thesis, readers are assumed to be familiar with the basic terminologies, concepts and procedures of the displacement-based finite

element

method.

They

will

be

used

without

citing

specific

references. For details, interested readers may consult standard texts such as [1]. 1.2 Thin plates, moderately thick plates and thick plates In

this

study

special

attention

is

given

to

the

analysis

of

isotropic or laminated composite plates with different span-to-thickness ratio. Although a rigourous definition of the terms such as thin plates,


m o d e r a t e l y t h i c k p l a t e s and t h i c k p l a t e s depends on t h e n a t u r e o f the underlying s o l u t i o n s [2],

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

k i n e m a t i c assumptions i n v o l v e d .

Plates b y s t r a i g h t o r curved edges. F o r a t h i n p l a t e , t h e t h i c k n e s s i s so s m a l l compared t o t h e o t h e r two g e o m e t r i c d i m e n s i o n s t h a t t h e t r a n s v e r s e shear a n d n o r m a l d e f o r m a t i o n s c a n be i g n o r e d . However, f o r a m o d e r a t e l y t h i c k p l a t e o r a shear-deformable p l a t e , t r a n s v e r s e s h e a r d e f o r m a t i o n i s taken i n t o account w h i l e t r a n s v e r s e normal d e f o r m a t i o n i s n eg l ect ed . A t h i c k p l a t e i s t r e a t e d a s a 3 - D e l a s t i c body w i t h b o t h t r a n s v e r s e s h e a r and normal deformations considered.

Laminated that

composite

plates

ďź&#x161;

is formed by combining

scale.

Plates

made

of

Composite materials

two

such

are

or more materials on a macroscopic

materials

commonly

are

formed

termed

in

three

composite

plates.

different

types

ďź&#x161;

fibrous compositesy which consist of fibers of one material in a matrix material

of another;

particles

of

one

particulate

material

in

a

composites matrix

of

another;

and

laminated

composites, which are made of layers of different materials. A laminated composite plate is made of layers of the same orthotropic material that are bonded together with the material coordinates of each layer oriented differently with respect to the plate coordinates. 1.3 General review of finite strip,finite prism and finite layer 1.3.1 Finite Strip The finite strip method pioneered by Y.K.Cheung[3], is an efficient analysis tool for plate-type structures with regular geometry and simple boundary conditions. The method is one of the best choices available to an

engineer

structures.

for It

analyzing

is

and

well-known

designing

that

the

prismatic

method

has

thin-walled

several

major

advantages relative to the standard finite element method, namely, (1) ease

of

data

convergence,

input,

(2)

reduced

(4) elimination of

number

of

unknowns,

in-plane rotation

(3)

rapid

(5) fulfillment of

-continuity for plate-bending problem and (6) ease of implementation. In addition, the basic concept of the method is simple and involves no complicated mathematics. Since the first paper of finite strip[4] appeared in 1968, intense interest

has

been

analysis. Several

focused

different

on

the

approaches

development have

been

of

finite

developed,

strip and

an

enormous literature now exists. The state of the art has been recently


summarized i n Cheung[5-7], Chong b r i e f summary o f t h e method, w i t h r e l e v a n c e t o the p r e s e n t s t u d y , i s g i v e n below. (1)

Various longitudinal t r i a l functions f o r f i n i t e s t r i p s I n e a r l y v e r s i o n s o f t h e f i n i t e s t r i p method, commonly known a s the

s e m i - a n a l y t i c a l f i n i t e s t r i p s , the a n a l y t i c a l f u n c t i o n s o f v i b r a t i o n / s t a b i l i t y o f an E u l e r beam s a t i s f y i n g t h e boundary c o n d i t i o n s a are

used i n t h e

longitudinal direction.

For

either

single-span or

continuous-span structures,these f u n c t i o n s a r e d e f i n e d mathematically i n terms

of

trigonometric

series.

The

continuously

differentiable

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

changing l o n g i t u d i n a l r i g i d i t y ,

since

the

second

d e r i v a t i v e s s h o u l d be d i s c o n t i n u o u s . I n a d d i t i o n , t h e method r e q u i r e s a tedious

flexibility

approach

to

analyse

structures

with

internal

d i s c r e t e supports such a s b r i d g e s supported on m u l t i p l e p i e r s remedy,

known a s

the

compound f i n i t e

strip

method[12],

was

[11].A later

developed f o r the a n a l y s i s o f p l a t e s w i t h i n t e r n a l d i s c r e t e supports, a t t a c h e d beams o r columns. The s t i f f n e s s c o n t r i b u t i o n o f t h e s e a t t a c h e d e l e m e n t s a r e d i r e c t l y added t o t h e p l a t e s t r i p s t i f f n e s s m a t r i x a t the element l e v e l , o r i n e s s e n c e , a s t i f f n e s s method i s a p p l i e d . E x c e p t f o r s t r u c t u r e s w i t h two s i m p l y - s u p p o r t e d o p p o s i t e ends ( i n t h i s c a s e t h e t r i g o n o m e t r i c s e r i e s u s e d i n t h e f o r m u l a t i o n decouples completely the r e s u l t i n g s t i f f n e s s matrix equation), the e a r l y versions o f t h e f i n i t e s t r i p method were l a t e r s u p e r s e d e d b y t h e s p l i n e f i n i t e s t r i p d e v e l o p e d b y Cheung and h i s c o - w o r k e r s [ 1 3 ] . I n t h e s p l i n e f i n i t e s t r i p method, t h e a n a l y t i c a l v i b r a t i o n f u n c t i o n s o f beam a r e r e p l a c e d by t h e C 2 - c o n t i n u o u s B ^ - s p l i n e f u n c t i o n s * T h e s e f u n c t i o n s a r e a c t u a l l y the a n a l y t i c a l s t a t i c f u n c t i o n s o f a n E u l e r beam s u b j e c t e d t o m u l t i p l e p o i n t l o a d s . M a j o r advantages o f t h i s method i n c l u d e ( i ) d i f f e r e n t boundary c o n d i t i o n s c a n be t r e a t e d w i t h t h e same f u n c t i o n and ( i i ) r a p i d and non-oscillatory

convergence f o r p l a t e s

d i s c r e t e supports splines or

by

under

point

loads.

Internal

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

lumping

the

stiffness

contribution of

the

discrete

s u p p o r t s t o the a t t a c h e d s t r i p s . Among t h e d i f f e r e n t v e r s i o n s o f f i n i t e s t r i p s , t h e s p l i n e f i n i t e strip

remains a s

the

most

powerful f i n i t e s t r i p i n

terms o f i t s

v e r s a t i l i t y , a c c u r a c y and e f f i c i e n c y . A l t h o u g h t h e method i s o r i g i n a l l y d e s i g n e d f o r l i n e a r a n a l y s i s o f p r i s m a t i c t h i n - w a l l e d s t r u c t u r e s , i t was l a t e r extended t o l i n e a r / n o n l i n e a r a n a l y s i s o f u s i n g a method o f g l o b a l g e o m e t r i c mapping[14].

irregular structures I n t h i s method,

an


i r r e g u l a r s t r u c t u r e i s f i r s t t r a n s f o r m e d i n t o a r e c t a n g u l a r domain and s u b s e q u e n t l y d i s c r e t i z e d i n t o r e c t a n g u l a r s p l i n e f i n i t e s t r i p s . The i d e a o f g e o m e t r i c mapping has been f u r t h e r e x t e n d e d b y Cheung and Au[15] w i t h i n t h e c o n t e x t o f i s o p a r a m e t r i c mapping; t h a t i s , e a c h s p l i n e f i n i t e s t r i p i s t r e a t e d a s a n i n d i v i d u a l f o r g e o m e t r i c mapping and s p l i n e function i s

used f o r

both displacement

i n t e r p o l a t i o n and geometric

mapping.( I n a s i m i l a r manner, t h e c r o s s - s e c t i o n o f a p r i s m a t i c t h i n w a l l e d tube c a n be a p p r o x i m a t e l y t r a n s f o r m e d i n t o a c i r c u l a r c r o s s s e c t i o n and s u b s e q u e n t l y d i s c r e t i z e d i n t o c u r v e d f i n i t e s t r i p s [ 1 6 ] . ) F o r m u l t i p l y - c o n n e c t e d domains o r domains w i t h r e - e n t r a n t c o r n e r s (e.g. square

plate

with

cut-out),

an

p r o p o s e d b y Cheung and L i [ 1 7 ] .

alternative In t h i s

approach

approach,

was

a

recently

the u s u a l f i n i t e

e l e m e n t s o r boundary elements a r e u s e d t o d i s c r e t i z e t h e r e g i o n s i n the v i c i n i t y o f i r r e g u l a r boundaries o r c u t - o u t s w h i l e t h e remaining area a r e d i v i d e d i n t o f i n i t e s t r i p s . These two r e g i o n s a r e t h e n connected together v i a s p e c i a l l y developed t r a n s i t i o n elements. s u c e s s f u l l y a p p l i e d t o p l a t e problems

The method was

and 2 - D e l a s t i c problems.

It

s h o u l d be n o t e d t h a t a X - s p l i n e f i n i t e s t r i p was a l s o d e v e l o p e d b y Chong et.aă&#x20AC;?.[18,19] for irregular shaped plates. In addition to the beam vibration functions and spline functions, two other types of longitudinal trial functions have been developed :(i) Trigonometric series are superimposed on beam shape functions in such a way that plates with general boundary conditions can be dealt with [20]. Owing to orthogonality relationships of the trigonometric functions used in the formulation, partial uncoupling of the equations can be achieved, and

this

can

be

taken advantage

Gaussian elimination process.

of by using

condensation prior

to

(ii) Orthogonal polynomials generated by

the Gram-Schmidt process [21]. The method was applied to free vibration of plates with general shapes and boundary conditions. The performance in terms of versatility, efficiency and accuracy is, at most, comparable to the semi-analytical finite strip method. (2)

Various plate theories for finite strips The semi-analytical finite strips were originally formulated within

the context of Kirchhoff's thin-plate theory. They were later extended to the analysis of moderately thick plates using the first-order shear deformation

theory,

i.e.

Mindlinâ&#x20AC;&#x2122;s

plate

theory.

In

Hindiin-type finite strip has been successfully applied

recent to

years,

the shape

optimization of variable thickness prismatic folded plates and curved shells[22] and to the study of boundary layer effects in the vicinity of edges and vertices of rectangular and sectorial plates[129].


Although

1-D L a g r a n g i a n

shape

functions

are

commonly

used

to

i n t e r p o l a t e displacements i n the t r a n s v e r s e d i r e c t i o n o f H i n d i i n - t y p e f i n i t e s t r i p s , h i e r a c h i c a l shape f u n c t i o n s o f a n i n c r e a s i n g o r d e r have a l s o been adopted i n c o n s t r u c t i n g a p - v e r s i o n M i n d l i n - t y p e f i n i t e s t r i p f o r postbuckling a n a l y s i s o f layered composites[23]. I n a d d i t i o n t o displacement-based formulation,

an assumed-strain

a p p r o a c h was p r o p o s e d b y C h u l y a and M u l l e n [ 2 4 ] so t h a t s h e a r l o c k i n g o f M i n d l i n - t y p e f i n i t e s t r i p s c a n be a v o i d e d . U s i n g a m o d i f i e d H e l l i n g e r R e i s s n e r v a r i a t i o n a l p r i n c i p l e , a h y b r i d f i n i t e s t r i p based on

s

p l a t e t h e o r y was a l s o p r o p o s e d [ 2 5 ] . T h i s h y b r i d f i n i t e s t r i p g i v e s a b e t t e r p r e d i c t i o n o f s t r e s s e s and r e d u c e s s t r e s s e s jump between a d j a c e n t finite strips. Due t o r e c e n t advances i n modern p l a t e t h e o r i e s ,

some new f i n i t e

s t r i p s a r e f o r m u l a t e d u s i n g modern p l a t e t h e o r i e s s u c h a s t h e n o n l i n e a r s t r e s s - c o n t i n u i t y p l a t e theory by A t l u r i s t r e s s p l a t e t h e o r y b y Tseng and Wang[27] and a t h i r d - o r d e r p l a t e t h e o r y b y Tseng and Wang[28]. I t i s n o t e w o r t h y t h a t a l m o s t a l l o f t h e t h i c k plate f i n i t e strips are restricted to

the a n a l y s i s o f

single-span

p l a t e - t y p e s t r u c t u r e s w i t h two s i m p l y - s u p p o r t e d o p p o s i t e e n d s . I n case o f g e n e r a l boundary c o n d i t i o n s , o n l y t h e M i n d l i n - t y p e f i n i t e s t r i p , w i t h a n a l y t i c a l v i b r a t i o n f u n c t i o n s o f Timoshenko beams a s l o n g i t u d i n a l t r i a l f u n c t i o n s , h a s been a t t e m p t e d [ 2 9 ] . M i n d l i n - t y p e spline finite strip has been devised by Tham[30] and Cheung e<:.aă&#x20AC;?[15] and applied

to the analysis of non-prismatic folded

plates and bridge decks. 1.3.2 Finite Prism The closed-form solution of

the

three-dimensional analysis of a

elastic structure is often extremely tedious and complicated to find, if not

impossible.

In

addition,

the

solution

is

often

restricted

to

structures with very simple geometry and boundary conditions. Although the

standard

finite

three-dimensional computationally three-dimensional

element

method

solution

to

the

expensive.

To

reduce

finite

element

can

generate

problem, the

analysis,

this

an method

computational the

approximate

finite

is cost

prism

usually of

a

method

[31-34] was devised for cost-effective analysis of prismatic structures with two simply supported ends. For prismatic structures with free ends, a special finite prism method was proposed by Fu and Yang[35]. In this method, the end conditions are satisfied in an iterative manner using an equilibrium approach. Additionally, the usual trigonometric series are


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

concentrated forces i s

conditions,

a

complementary

hybrid energy

satisfaction

of

eliminated.

finite

In

case

prism[36]

was

principle.

equilibrium

The

equations

of

developed

method and,

general

boundary using

requires

as

a

the

pointwise

result,

involves

complicated formulation and implementation and therefore has not gained popularity. In addition

to conventional 2-D finite element

shape functions,

spline functions of various order can be used within the cross-section of a finite prism[37]. Finite prism system

or

can be

a Cartesian

formulated

coordinate

within

system.

a

An

cylindrical example

of

coordinate cylindrical

finite prism is due to Yoseph et.al.[38] who proposed a hybrid finite prism for the analysis of singular stress fields in the vicinity of a circular cut-out in a laminated plate.

In this method, equilibrium is

only satisfied in an integral sense via a modified He11inger-Reissner variational principle. 1.3.3 Finite Layer The

finite

layer

method

is

an

efficient

three-dimensional rectangular structures of

analysis

layered

method uses a two-dimensional analytical solution

tool

for

construction. The in

the plane of a

three-dimensional structure and a one-dimensional finite element shape function through the thickness. The method has been used to analyze the structural

and/or

thermal

behaviour

of

rectangular

sandwich

and

isotropic plates with simply-supported edges [39,40], and vibration of rectangular plates with various homogeneous boundary conditions [3]. In addition, the method was employed to study semi-infinite elastic bodies such

as

the

static

analysis

and

consolidation

of

layered

soil

[41,42], interaction between an elastic slab and its foundation [43,44]. In addition

to rectangular

structures,

the method

can be formulated

within a cylindrical co-ordinate system for studying the vibration of thick-walled circular cylindrical tubes with various end supports [3], and recently, the analysis of pile foundation using an infinite layer method

[45]•

When formulation

is done within a spherical

system, the method can be applied most

of

these finite

layer

to 3-D spherical

studies,

structures[46]. In

either harmonic series or beam

vibration functions are adopted as the structures with supports only around

coordinate

trial functions for analyzing

the edges.

internal supports, no attempts have been made.

For

structures with


1 . 4 G e n e r a l r e v i e w o f modern p l a t e t h e o r i e s The i n c r e a s i n g u s e o f l a m i n a t e d c o m p o s i t e m a t e r i a l s i n d i f f e r e n t branches o f engineering s t i m u l a t e s

the

advances i n the a n a l y s i s o f

l a m i n a t e d beams a n d p l a t e s . E x t e n s i v e r e s e a r c h h a s b e e n c o n d u c t e d i n t h e t h e o r e t i c a l d e v e l o p m e n t s o f new p l a t e t h e o r i e s and , c o n s e q u e n t l y , many i n t e r e s t i n g a p p r o a c h e s have b e e n d e v e l o p e d d u r i n g t h e l a s t decade.

No

a t t e m p t i s made t o r e v i e w modern p l a t e t h e o r i e s i n t h i s b r i e f s e c t i o n . 〔 A literature review in modern plate theories would entail in itself a major piece of work ! Comprehensive r e v i e w s o f t h e d e v e l o p m e n t s i n t h e a n a l y s i s o f l a m i n a t e d p l a t e s have b e e n r e c e n t l y summarized b y K a p a n i a

et.al description

of plate theories, which are of relevance to this thesis,

is given below. Many of categories,

the existing modern plate namely,

(1) single-layer

theories fall

theory

(SLT)

and

into

two major

(2)

layerwise

expanded

in power

theory (LT). In

the

SLT,

series of the

the

displacement

thickness

components

coordinate

are

z , giving

u(x,y,z) = iio(x,y) +

(x,y) -z + u 2 (x,y”z2 + u 3 (x,y)*z3 + ....

v(x,y,z) = v 0 (x,y) +

(x,y) »2 + v 2 (x,y”z2 + v 3 (x,y)*z3 +• …

w ( x , y , z ) = w o ( x , y ) + w) ( x , y )

(1.1)

2 + w 2 (x’y) .z +. …

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

z i n the s e r i e s determines the

order o f the corresponding theory. F o r high-order theories, p l a t e s are p h y s i c a l l y t r e a t e d a s 3 - D e l a s t i c b o d i e s . A l l s i x components o f s t r a i n s are

taken

into

consideration.

For

low-order

theories,

however,

t r a n s v e r s e n o r m a l s t r a i n i s u s u a l l y n e g l e c t e d and o n l y t h e o t h e r f i v e s t r a i n components a r e c o n s i d e r e d . A n e x a m p l e o f t h i s i s t h e w e l l - k n o w n f i r s t - o r d e r shear d e f o r m a t i o n t h e o r y due t o H i n d i i n and R e i s s n e r . F o r this

theory,

the

displacement

functions

involve

five

independent

v a r i a b l e s , n a m e l y , t h r e e m i d - p l a n e d i s p l a c e m e n t s a n d two r o t a t i o n s , t h a t is u ( x , y , z ) = u o ( x , y ) + z • (p^ix.y)

v(x,y,2) = vo(x,y) + z • ^ ( x ^ )

(1.2)


w(x’y,z) = w o 〔x,y)

It

is

obvious

that

the

displacement

functions

yield

constant

transverse shear strain across the thickness coordinate z which contradiction with

the usual

parabolic distribution

assumption

of

is in

parabolic distribution

(

is a commonly adopted assumption for thin and

moderately thick , laminated or isotropic plates). To account for this, a shear correction factor is required.This factor, which is crucial to an accurate analysis, is sensitive to the plate gemoetry, configuration, boundary conditions and

loading.

However,

rigorous

and

theoretically

sound procedures for the calculation of this factor is not currently available. Also cross-sectional warping is neglected as "plane sections remain

plane"

is

assumed.

This

theory

can

be

further

reduced

to

Kirchhoff's thin-plate theory by neglecting transverse shear strains, mathematically, u C x ^ . z ) = u o (x,y) — z • dwQ(xiy)/dx v(x,y’z) = v 0 (x,y) - z • dwo (xfy)/dy

(1.3)

w(x,y, z) = w o (x,y) The Mindlin and Kirchhoff plate theories have been coimnonly used for

the

finite

element

analysis

of

moderately

isotropic plates. Kirchhoff's plate theory

thick

and/or

thin

is not capable of dealing

with laminated composite plates, except for very large span-to-thickness

ratio, because the transverse shear moduli of modern composite materials are usually very low compared to the inplane tensile moduli, with the result

that

transverse

shear

deformation

can

be

of

importance compared to homogeneous isotropic materials. these

plate

accurately

theories

predict

neglect

the

cross-sectional

through-thickness

warping

stresses

considerable In addition, and

cannot

distribution

of

laminated plates. In 1984, Reddy [53] proposed a variational consistent third-order

theory which accounts

for

the

parabolic distribution of

transverse shear strains explicitly and hence avoids the use of shear correction factor.

It

also

accounts

for

cross-sectional warping but

neglects transverse normal deformation. Many other forms of third-order plate theories have been proposed in the past decade and they will be discussed in Chapter 7.


Better

accuracy

in

predicting

global

responses

(such

as

d e f l e c t i o n s , n a t u r a l f r e q u e n c i e s and b u c k l i n g l o a d s ) and s t r e s s e s can be a c h i e v e d by i n c r e a s i n g the o r d e r o f i n t e r p o l a t i o n a c r o s s t h e t h i c k n e s s of

the p l a t e .

The

number o f

unknowns

i n v o l v e d i n such an a n a l y s i s

depends on the d e s i r e d l e v e l o f a c c u r a c y and i s

independent o f the

number o f m a t e r i a l l a y e r s . T h i s h i e r a c h i c a l concept was f i r s t proposed by Szabo e t . a l [ 5 4 ] f o r t h e a n a l y s i s o f i s o t r o p i c p l a t e s and s h e l l s , and l a t e r extended by Surana e t . a l [ 5 5 ] , B a r b o n i e t . a i [ 5 6 ] , e£.al[57] for

the analysis

approach, however,

of

laminated

and Hatsunaga

composite plates

.

In

this

all six components of strains are continuous across

the interfaces between layers of different materials, and the condition of continuous transverse shear and normal stresses is totally out of the question.

Consequently,

if

the

transverse

interest,

they are obtained indirectly by integrating the equilibrium

equations of elasticity ( 乙 o e t . a J

[58]

ghear

stresses

are

of

; Chaudhuri [ 5 9 , 6 0 ] ; L a j c z o k

[ 6 1 ] , [ 6 2 ] ) . R e g a r d l e s s o f i t s complex i m p l e m e n t a t i o n , t h i s c o n t r o v e r s i a l procedure

involves

higher-order

in-plane

differentiation

of

the

d i s p l a c e m e n t f u n c t i o n , and therefore, its accuracy is questionable for analyzing plates with highly irregular geometry. In the LT, the displacement variation through the thickness of each layer

of

material

is

interpolated

by

means

of

the

standard

one-

dimensional Lagrangian shape functions. For analyzing thick laminated plates, each layer of material can be modelled by several linear onedimensional elements (Reddy et.al. [63]; S r i n i v a s [ 6 4 ] ; Owen o r a few q u a d r a t i c elements (Robbins and Reddy[66] hierachical

e le me n t ( O r t h

displacement

continuity i s

and

Surana[67]). In

maintained

between

this

) o r a high-order approach,

layers

of

only

different

m a t e r i a l and the c o n t i n u i t y o f t r a n s v e r s e shear and normal s t r e s s e s can s t i l l be s a t i s f i e d i n an i n t e g r a l s e n s e through t h e p o t e n t i a l energy formulation.

A c c u r a t e t r a n s v e r s e s h e a r s t r e s s e s can be o b t a i n e d wit ho ut

i n t e g r a t i n g t h e e q u i l i b r i u m e q u a t i o n s o f e l a s t i c i t y e x c e p t f o r the case i n which

a s i n g l e o n e - d i m e n s i o n a l l i n e a r element i s u s e d f o r each

m a t e r i a l layer(Reddy

[63]).

In addition,

extension to

the

n o n l i n e a r i t y [ 6 8 ] and d e l a m i n a t i o n [ 6 9 ] i s s t r a i g h t f o r w a r d .

study o f

However, i t

s u f f e r s from a s e r i o u s drawback t h a t t h e number o f unknowns i n v o l v e d i n s u c h a f i n i t e element a n a l y s i s depends on t h e number o f l a y e r s o f material. I t i s

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

l a m i n a t e d p l a t e w i t h many l a y e r s o f m a t e r i a l s . 1 . 5 Scope o f P r e s e n t s t u d i e s


10

The

finite

strip,finite

prism

and f i n i t e

layer

undergone c o n s i d e r a b l e development d u r i n g the p a s t

methods

two decades,

have but

s t i l l , d e s p i t e these research e f f o r t s , a p p l i c a t i o n o f the methods t o some o f the v e r y common p l a t e - t y p e s t r u c t u r e s has not met w i t h complete success.

In

addition,

laminated

composite

structures

are

becoming

i n c r e a s i n g l y popular, and a n a l y s i s o f laminated p l a t e s , i n p a r t i c u l a r , i s f r e q u e n t l y necessary. I t i s the main theme o f t h e p r e s e n t s t u d i e s t o e x p l o r e the methods o f f i n i t e s t r i p , f i n i t e p r i s m and f i n i t e l a y e r f o r t h e a n a l y s i s o f i s o t r o p i c o r laminated composite p l a t e - t y p e s t r u c t u r e s . Although most o f the recent r e s e a r c h endeavor has been d i r e c t e d toward the developement o f M i n d l i n - t y p e elements w i t h the c a p a b i l i t y o f thin-plate analysis, i t i s believed that,

f o r t h i n - p l a t e problems,

K i r c h h o f f - t y p e elements a r e more e f f i c i e n t and r e l i a b l e . As t h i n - p l a t e structures

are

very

commonly

development o f h i g h l y e f f i c i e n t

used

in

engineering

practice,

the

- c o n t i n u o u s elements e x c l u s i v e l y f o r

t h e i r a n a l y s i s I s worthwhile. An i d e a l and v e r s a t i l e t h i n - p l a t e f i n i t e s t r i p should be

- c ontinuous such t h a t a p l a t e w i t h a b r u p t l y changing

t h i c k n e s s can be t r e a t e d . When a n a l y z i n g a p l a t e w i t h p o i n t loads, the s t r i p should be a b l e t o g i v e r a p i d l y converged s o l u t i o n s . I n a d d i t i o n , the s t r i p should be a b l e t o take c a r e o f c o m p l i c a t e d support c o n d i t i o n s i n an easy and u n i f i e d manner. Although i t i s n o t a b s o l u t e l y necessary, easy implementation o f the method i s h i g h l y d e s i r a b l e . Based on these criteria,

a new f i n i t e s t r i p i s developed. U n l i k e c l a s s i c a l f i n i t e

s t r i p s , the displacement f u n c t i o n o f t h i s newly developed f i n i t e s t r i p i s expressed as the product o f a s e t o f computed s t a t i c modes i n the l o n g i t u d i n a l d i r e c t i o n and, as u s u a l , the beam shape f u n c t i o n s i n the t r a n s v e r s e d i r e c t i o n . These s t a t i c modes can be e a s i l y o b t a i n e d by means of

a continuous-beam computer

program.

Parallel

development i n the

a n a l y s i s o f 2-D plane problems and f o l d e d - p l a t e s t r u c t u r e s has a l s o been attempted. The o r i g i n a l s p l i n e f i n i t e s t r i p i s r e s t r i c t e d t o the a n a l y s i s o f t h i n p l a t e s o r t h i n - w a l l e d s t r u c t u r e s . Although the method has been a p p l i e d t o l i n e a r a n a l y s i s ( s t a t i c , f r e e v i b r a t i o n and b u c k l i n g ) and nonlinear

analysis

(bending,

postbuckling,

nonlinear

steady-state

v i b r a t i o n ) o f p l a t e - o r s h e l l - t y p e s t r u c t u r e s , i t s a p p l i c a t i o n t o one common and important aspect o f a n a l y s i s namely, n o n l i n e a r t r a n s i e n t s t a t e v i b r a t i o n , remains unexplored. I n t h i s work, a t h i r d - o r d e r s p l i n e f i n i t e s t r i p i s developed as the c o u n t e r p a r t o f the o r i g i n a l s p l i n e f i n i t e s t r i p which i s capable o f t r e a t i n g moderately t h i c k i s o t r o p i c o r laminated composite p l a t e s . The f o r m u l a t i o n o f the present s p l i n e f i n i t e


11

s t r i p i s based on a t h i r d - o r d e r p l a t e t h eo r y which takes i n t o account t r a n s v e r s e shear deformation and c r o s s - s e c t i o n a l warping o f laminated composite p l a t e s . F o r n o n l i n e a r v i b r a t i o n , s t r i p method i s

the present s p l i n e f i n i t e

combined w i t h a r e d u c t i o n method so t h a t

the h i g h

computational c o s t o f t i m e - s t e p p i n g t r a n s i e n t a n a l y s i s can be reduced. F o r the a n a l y s i s o f r e c t a n g u l a r , t h i c k , laminated composite p l a t e s , a l l k i n e m a t i c assumptions a r e r e l a x e d and 3-D e l a s t i c i t y theory i s adopted. A new s p l i n e f i n i t e p r i s m method i s developed u s i n g B3-spline f u n c t i o n s i n the

l o n g i t u d i n a l d i r e c t i o n and

p o l y n o m i a l s i n the c r o s s - s e c t i o n o f

a set

the p l a t e .

of global-local

In addition,

a 3-D

t h i c k - p l a t e f i n i t e element i s developed which i n c o p o r a t e s the advantages o f the LT and SLT i n a n a l y z i n g laminated composite p l a t e s . T h e advantages o f these methods over the c o n v e n t i o n a l 3-D f i n i t e element models are two- f o l d , name 1y , the volume o f i n pu t d a t a i s reduced > and the plane 2-D mesh and the 1-D i n t e r p o l a t i o n can be r e f i n e d independent o f each other w i t h o u t h a v i n g t o r e c o n s t r u c t a 3-D f i n i t e element mesh. The 2-D-type d a t a s t r u c t u r e a l s o a l l o w s f a s t e r f o r m u l a t i o n o f t h e element s t i f f n e s s matrices. F o r f r e e v i b r a t i o n a n a l y s i s o f r e c t a n g u l a r p l a t e s w i t h intermediate line

supports,

polynomials

are

superimposed

on

single-span

beam

v i b r a t i o n f u n c t i o n s and subsequently employed a s the t r i a l f u n c t i o n s f o r the development o f

a new f i n i t e l a y e r technique.

Based on t h i s new

f i n i t e l a y e r method, some new r e s u l t s o f laminated composite p l a t e s are presented. 1.6 Organization o f t h e s i s T h i s t h e s i s i s o r g a n i z e d i n t o e l e v e n chapters.The o r d e r o f chapters i s arranged i n such a way t h a t Chapter two t o Chapter s i x a r e devoted t o a newly developed f i n i t e s t r i p method, f o l l o w e d by Chapter seven which d e a l s w i t h the refinement o f t h e o r i g i n a l s p l i n e f i n i t e s t r i p method. The l a t e r chapters, Chapter e i g h t t o Chapter ten, c o n c e n t r a t e on 3-D f i n i t e pr is m and f i n i t e l a y e r a n a l y s i s o f laminated p l a t e s . The present Chapter g i v e s a b r i e f r e v i e w i n f i n i t e s t r i p , f i n i t e p r i s m , f i n i t e l a y e r methods and modern p l a t e t h e o r i e s . The scope o f the p r e s e n t s t u d i e s i s a l s o presented. The

second Chapter

describes

f u n c t i o n s f o r d e v e l o p i n g the properties

of

the t r i a l

the b a s i c

concept o f

new f i n i t e s t r i p s .

functions w i l l

be

the t r i a l

The d e f i n i t i o n s and

addressed.

These t r i a l

f u n c t i o n s a r e then used t o d e r i v e a t h i n - p l a t e f i n i t e s t r i p w i t h d e t a i l s o f f o r m u l a t i o n and a s e r i e s o f examples g i v e n i n Chapters t h r e e and


12

f o u r . I n the f i f t h Chapter, t h e t r i a l f u n c t i o n s a r e used t o d e v e l o p a 2-D f i n i t e s t r i p . The l a t t e r , w h e n combined w i t h t h e t h i n - p l a t e f i n i t e strip,

forms

a flat-shell f i n i t e s t r i p for vibration

and b u c k l i n g

a n a l y s i s o f t h i n - w a l l e d s t r u c t u r e s , a s g i v e n i n Chapter s i x . I n the seventh Chapter, the o r i g i n a l s p l i n e f i n i t e s t r i p i s r e f i n e d u s i n g a t h i r d - o r d e r p l a t e theory. This t h i r d - o r d e r s p l i n e f i n i t e s t r i p i s a p p l i e d t o the a n a l y s i s o f n o n l i n e a r t r a n s i e n t - s t a t e v i b r a t i o n o f plates.

A

reduction

method

is

also

proposed

for

reducing

the

computational cost o f time-stepping t r a n s i e n t a n a l y s i s . I n Chapter e i g h t , applied

to

the

a

new s p l i n e f i n i t e p r i s m i s d e v e l o p e d and

bending,

vibration

and

buckling

analysis

of

t h i c k - p l a t e f i n i t e element.

The

t h i c k , l a m i n a t e d composite p l a t e s . Chapter n i n e d e a l s w i t h a 3-D,

method i s a p p l i e d t o t h e s t r e s s a n a l y s i s o f l a m i n a t e d p l a t e s . numerical

examples

are

used

A few

t o v e r i f y and demonstrate t h e method.

Chapten t e n d e a l s w i t h a

new f i n i t e l a y e r

t e c h n i q u e which i s

s p e c i a l l y designed f o r rectangular t h i c k p l a t e s w i t h intermediate l i n e s u p p o r t s . T h i s f i n i t e l a y e r t e c h n i q u e employs a s e t o f m o d i f i e d beam v i b r a t i o n f u n c t i o n s w h i c h s a t i s f y the boundary and i n t e r n a l c o n d i t i o n s a p r i o r i . Some new r e s u l t s o f f r e e v i b r a t i o n o f l a m i n a t e d composite p l a t e s a r e presented. C o n c l u s i o n s o f the p r e s e n t s t u d i e s a r e g i v e n i n t h e l a s t Chapter o f t h e t h e s i s . A few remarks on f u t u r e r e s e a r c h a r e a l s o d i s c u s s e d . 1 . 7 Remarks Unless otherwise s p e c i f i e d ,

the f o l l o w i n g n o t a t i o n s a r e adopted

i n t h i s thesis. For d i f f e r e n t i a t i o n o f a s i n g l e - v a r i a b l e f u n c t i o n , say f (x), w i t h respect to the corresponding s p a t i a l v a r i a b l e x : ( f ( x ) ) ’ s df(x)/dx

and

(f(x))"

= d2f(x)/dx2

For differentiation of a multi-variable function with respect to the corresponding spatial variable(s) :e.g. for a 2-D function f(x,y) (2-D = 2-dimensional),

(f(x,y))

x

= f

x

= af(x)/ax

(f(x,y))'

= f'

= a 2 f(x)/ax 2

(f(x,y))

= f

= d f ix)/dxdy etc.

,XX

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16

CHAPTER 2 FORMULATION OF COMPUTED SHAPE FUNCTIONS D e t a i l s o f a new type o f t r i a l f u n c t i o n s a r e d e s c r i b e d i n t h i s chapter. The f u n c t i o n s w i l l be u t i l i z e d f o r the development o f a s e r i e s o f t h i n - p l a t e , 2-D p l a n e and f l a t - s h e l l f i n i t e s t r i p s i n the chapters t o follow. 2 . 1 L o n g i t u d i n a l t r i a l f u n c t i o n s : COMSFUN F i r s t o f a l l , c o n s i d e r an E u l e r beam w i t h v a r y i n g r i g i d i t y along the Y - a x i s as shown i n F i g . 2. l a . The beam i s d i v i d e d i n t o p beam elements which are not n e c e s s a r i l y o f equal l e n g t h .

Among the

p+1 j o i n t s ,

a

number o f them, say r , a r e a s s i gn ed a s nodes.Notice t h a t the j o i n t a t each end o f the beam i s always a s s i g n e d as a node. S i m i l a r t o the concept o f Lagrangian shape f u n c t i o n s , a u n i t d e f l e c t i o n i s imposed t o one o f these nodes w h i l e the remaining nodes a r e c o n s t r a i n e d w i t h zero d e f l e c t i o n ( F i g . 2 . l b ) . T h i s procedure i s then a p p l i e d t o each o f the nodes and a t o t a l o f

r

computed s t a t i c modes can be obtained.These

s t a t i c modes correspond t o the e x a c t d e f l e c t i o n p a t t e r n s o f s u b j e c t e d t o a s e t o f p o i n t l o a d s a t the nodes.

a beam

They can be e a s i l y

obtained by means o f a continuous-beam program.The amount o f computing e f f o r t r e q u i r e d i s i n s i g n i f i c a n t . B e c a u s e o f the s i m i l a r i t y between the Lagrangian shape f u n c t i o n s and t h e s t a t i c modes, they are g i v e n the name o f COMputed Shape FUNctions, o r a b b r e v i a t e d as COMSFUN hereon. A displacement v a r i a b l e , w , can be w r i t t e n i n terms o f the COMSFUN, r

w(y) = ^ w. Y. (y) i=1

(2.1)

I t i s noteworthy t h a t , u n l i k e t r i a l f u n c t i o n s used i n a l l other f i n i t e strips,

each o f the unknown parameters

a c t u a l displacement a t

node i .

In

terms o f

, the

w.

, represent the

usual

beam shape

f u n c t i o n s , the COMSFUN can be expressed i n the form Y (y) = ) [N(y)]{ a } m k=l with

Y (y ) = 1 m

Y

m

m

(y ) = 0 n

m = 1,2,

(2.2) , r ďź&#x203A; r ^ p+1

(2.3)


17

where

[N(y) ] are the u s u a l

displacement v e c t o r o f

beam shape f u n c t i o n s and { a

the j o i n t s a t

the

} i s the mk beam element k

ends o f

corresponding t o the l o n g i t u d i n a l mode Ym â&#x20AC;˘ The summation is carried out with due regard

to the joint arrangement.

In the following chapters,

unless otherwise stated, we assume r=p+l. In the preceding discussions, no rotational restraint is applied to either end of the beam. Put

in other words, the boundary condition at

each end of the beam is either simply supported or free when computing the corresponding shape functions for respectively.

As

a

result,

no

the internal nodes or end node

restraint

is

imposed

on

the

first

derivative of the resulting COMSFUN. For 2-D plane stress problems, the first derivative of the in-plane displacement variables is not involved in the kinematic boundary conditions, and therefore the afore-mentioned COMSFUN can be adopted directly as the corresponding longitudinal trial functions. For thin-plate problems with a transverse clamped or sliding clamped

edge

however,

the

longitudinal

coordinate)

vanishes

the

at

of

first

derivative

the

out-of-plane

corresponding

end

of

the

(with

respect

displacement

strip.

to

the

variable

Accordingly,

the

rotation at the corresponding end of the beam must be constrained

to

zero. For thin-plate problems with a simply-supported or free edge, no rotational restraint Table 2.1 summarizes

is needed at the corresponding end of the beam. the

treatment

of boundary conditions for plate

bending problems.

Kinematic boundary conditions (2)

(1)

(3)

clamped

sliding-clamped

clamped

sliding-clamped

sliding-clamped

clamped

Simply-supported

free

Simply-supported

free

free

Simply-supported

Table 2.1 Treatment of boundary conditions. (1) : actual boundary condition at one end

(y = 0 ,say) of a strip

(2) : boundary condition at y = 0 of the beam

when computing the

shape function for the end node at y=0. (3) : boundary condition at y = 0 of the beam

when computing the

shape function for each of the internal nodes and the end node at y


18

2 - 2 Mathematical p r o p e r t i e s o f COMSFUN I t i s i n t e r e s t i n g and important t o r e c o g n i z e t h a t each o f these computed shape f u n c t i o n s represent, i n s t r u c t u r a l e n g i n e e r i n g terms,the i n f l u e n c e l i n e f o r the r e a c t i o n f o r c e a t the corresponding node. As a consequence, one can e a s i l y d e r i v e the f o l l o w i n g p r o p e r t i e s ďź&#x161; Vertical equilibrium of the beam yields, r

V Ym(y) mil

s l

(2.4)

Moment equilibrium of the beam yields, r

V Y (y) x y = y

(2.5)

2.3 Computation of integrals One of the major difference between COMSFUN and the trigonometric series of semi-analytical finite strips or the Bs-spline functions of spline

finite

numerical

strip

is

integration.

that

The

the

integrals

of

integrals are computed

COMSFUN by

involve

summing up

no the

transformed integrals of the product of beam shape functions, that is Ym Yn

(2.6a)

dy

[N(y)]

[N(y)] dy ]

(2.6b)

Ym" Yn d y

[NMy)]' [N(y)] dy ] 彳 a >L

Ym Yn"

(2.6c)

dy

[N(y)]

[N"(y)] dy


19

Ym"

(2.6d)

Yn"

[ N " ( y ) ] ' [N"(y)] dy ] { a

Ym'

Yn,

>,

dy

(2.6e)

[ N , ( y ) ] T [ M , ( y ) ] dy ] { a

Ym'

Yn"

dy

C2.6f)

[N, ( y ) ] 1

[ă&#x20AC;&#x17D;ďź&#x2C6;y)] dy ] { a >L

The explicit form of integration of the beam shape functions is given in the Appendix.

2.4 Summary The newly developed

trial

applying a concept similar

functions,

COMSFUN,

of

a

generated

by

to that of the Lagrangian shape functions.

However, unlike the Lagrangian shape functions, functions are only

are

these computed shape

-continuous. They can be easily obtained by means

continuous-beam

computer

program.

Only

classical

beam

shape

functions are involved in the formulation. This allows us to obtain all integrals

in an economical

and explicit form without using Gaussian

numerical integration. From a computational point of view, these trial functions are easy to implement as they only involve beam elements and beam shape functions.


20

Appendix

Beam shape functions The beam shape f u n c t i o n s a r e g i v e n by N ( s ) =[(1 - 3S2+ 2s 3 ) ,31( 1 - 2 s + s 2 ) , ( 3 s 2 - 2 s 3 ) , s ( s 2 ! 2 - s )] =[N 1 , N, 1 c

N . N ] 5 4

(2.7)

where 1 = length of element and s is

the

These

element

shape

coordinate running from 0 to 1

functions

also

satisfy

the

following

interesting

properties : N + N,s 1 1

(2.8)

3

N^+ N,』+ N s s 2

3

(2.9)

4

N,』2 + 2 N 1 = s 2 3

The

(2.10)

4

integrals of

the products of

these

shape functions can be

obtained in an analytical form using symbolic computation.

156 [ N ( S ) ] T [ N(S)3

ds = i/420

221

54

4 1

1312-312

SYM.

-13』n

156 -221 4 1

12

61

-12

41

-61

[N" ( s ) ] [ N " ( s ) ] d s =

SYM.

21'

12

4/;

-36

r

r

T [N,(s)] [N,(s)] ds =

SYM.

4I

2

3i

-3』-J 36

2

-3i 4i:

3


T

[ N ( S ) ] [ N " ( S ) ] d s = 1/301

-1/2 T

[N(s)] [N,(s)] ds

.36

-33i

36

- 3J

•31

-4i

2

3i

I2

36

3i

-36

331

•3i

I2

3i

-4i

-1/10

-1/10

0

1/2

-i/10

1/10

-』2/60

-1/2 I 2/60

-1/10

1 [N*(s)] [ N " ( s ) ] d s = 1/1

-1

0

-1

-1/2

-1 1 / 2

一1

1/2


cof

CD

T3 O

O

C

S 一 cd — < D Xi m <D u O O S c T J c w d c d c W T J D < 5 S) O C -o扣

"D CO

O 」

c o

CL

sz CO c: CD M—

o

CD E o "o

•a 2 • •HI ‘ fj^ •o ① cn w Tf芏 ^o -i O £o C log5 巧 ,、w4 o Jag T j ^ o 七『I o 5 g ’o “ w o • t„ fc wn » ’ S D < Tl 2 C 3 sD <c d C r-H < D < CS] C O

ci •H ix.


23

CHAPTER 3 A THIN-PLATE FINITE STRIP USING COMSFUN I n t h i s c h a p t e r a t h i n - p l a t e f i n i t e s t r i p i s proposed w h i c h a l l o w s t h e treatment o f r e c t a n g u l a r p l a t e s t r u c t u r e s w i t h a b r u p t l y changing l o n g i t u d i n a l r i g i d i t y and c o m p l i c a t e d boundary c o n d i t i o n s i n a n e a s y and u n i f i e d manner. Based on K i r c h h o f f ' s t h i n - p l a t e t h e o r y , t h e s t r i p i s conforming w i t h

new f i n i t e

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

s t r i p i s expressed as the product o f t h e beam shape f u n c t i o n s i n t h e transverse

direction

and

the

computed

shape

functions

in

the

l o n g i t u d i n a l d i r e c t i o n . Numerical examples a r e g i v e n t o demonstrate i t s a p p l i c a t i o n t o the s t a t i c a n a l y s i s o f

n o n - p r i s m a t i c beams, square p l a t e s

with point

drop panels

loads, f l a t slabs w i t h

and c o n t i n u o u s s l a b

b r i d g e s w i t h changing t h i c k n e s s . 3.1 Introduction An i d e a l t h i n - p l a t e f i n i t e s t r i p s h o u l d be

-continuous such that

a p l a t e w i t h a b r u p t l y changing t h i c k n e s s c a n be p r o p e r l y t r e a t e d . The i n f l u e n c e o f v a r y i n g r i g i d i t y on t h e d e f o r m a t i o n s h o u l d be t a k e n i n t o a c c o u n t . When a n a l y z i n g a p l a t e s u b j e c t e d t o p o i n t l o a d s ,

t h e element

s h o u l d be a b l e t o g i v e r a p i d l y converged s o l u t i o n s . I n a d d i t i o n , the s t r i p s h o u l d be a b l e t o t a k e c a r e o f c o m p l i c a t e d s u p p o r t c o n d i t i o n s i n a n easy and u n i f i e d manner. A l t h o u g h i t i s n o t a b s o l u t e l y necessary, easy implementation o f t h e method i s h i g h l y d e s i r a b l e . Based on these c r i t e r i a , the present f i n i t e s t r i p i s developed.Unlike c l a s s i c a l f i n i t e s t r i p s , the p r e s e n t f i n i t e s t r i p employs a s e r i e s o f computed s t a t i c modes

instead

of

a

series

of

analytical

vibration

modes

as

the

longitudinal t r i a l functions.

3.2 Formulation A f i n i t e s t r i p w i t h v a r y i n g r i g i d i t y a l o n g t h e Y - a x i s i s shown i n F i g . 3.l a .

fictitious

I n F i g . 3.l b ,

a u n i t w i d t h o f t h e s t r i p i s t a k e n o u t as a

beam

t h e o r i g i n a l s t r i p . I n accordance w i t h t h e boundary c o n d i t i o n s a t each end o f t h e s t r i p , a p p r o p r i a t e boundary c o n d i t i o n s a r e imposed on the f i c t i t i o u s beam, see T a b l e 2 . 1 . A t o t a l o f r computed shape f u n c t i o n s a r e computed from t h e f i c t i t i o u s beam. The i n f l u e n c e o f v a r y i n g r i g i d i t y


and t h e requirement o f

- c o n t i n u i t y a r e n a t u r a l l y taken i n t o account. I t

s h o u l d be n o t e d t h a t t h e s e shape f u n c t i o n s a r e i d e n t i c a l f o r a l l s t r i p s w i t h the same v a r i a t i o n o f t h e l o n g i t u d i n a l r i g i d i t y . The displacement f u n c t i o n o f a f i n i t e s t r i p can be w r i t t e n as r

wCx^)

Lj

ni=l

r

= 7 Lj m= 1 where and

=

[Q ] m

[Q ]{ r > ,

(3.1) with

= [N(x)] Y (y) m

[N(x)] and Y (y) denote r e s p e c t i v e l y t h e u s u a l beam shape f u n c t i o n s m

a l o n g X - a x i s and t h e l o n g i t u d i n a l COMSFUN. Based on t h e mathematical p r o p e r t i e s o f COMSFUN and beam shape f u n c t i o n s , i t can be e a s i l y shown t h a t the r i g i d - b o d y c r i t e r i o n i s s a t i s f i e d f o r t h e p r e s e n t s t r i p when no s u p p o r t s a r e p r o v i d e d . The p r e s e n t s t r i p i s co nfo rm ing w i t h o n l y two degrees o f freedom p e r node. U n l i k e a l l o t h e r f i n i t e s t r i p s , t h e unknowns { r > o f t h e p r e s e n t m s t r i p r e p r e s e n t t h e a c t u a l n o d a l d i s p l a c e m e n t s (w) and r o t a t i o n s ( F i g . 3 . 2 ) . As

a

consequence, s u p p o r t

conditions

involving

vertical

d e f l e c t i o n s and/or r o t a t i o n s about Y - a x i s can be implemented i n the same f a s h i o n a s i s done f o r t h e u s u a l f i n i t e element method, o r i n o t h e r words, t h e y can be t r e a t e d degrees

of

freedom i n

the

global

stiffness

matrix.

However,

for

r o t a t i o n s about X - a x i s , dvi/dy, a t e i t h e r end o f t h e f i n i t e s t r i p , they a r e taken i n t o account when f o r m i n g t h e computed shape f u n c t i o n s , t h a t i s , they are t r e a t e d a Based on K i r c h h o f f ' s t h i n - p l a t e

theory,

the strain-displacement

m a t r i x can be w r i t t e n a s

[B b ] {

(3.2)


25

[B b ]

where

m

z { -a2[Q

=

m

m

F o l l o w i n g s t a n d a r d d i s p l a c m e n t - b a s e d f i n i t e element procedures, i t can be shown t h a t the s t r i p s t i f f n e s s m a t r i x i s g i v e n by [K b ]

[K b ]

11

[K b ],

12

[K b ]

[K b ]

CKb]

[K b ]

r 1

ii

12

Y

Y

dy

Y" Y dy

Integration

n

through

i

dx dy [N]" [N]" dx +

D

[N]

D

[N],

D Y' Y, dy 33 m

[K b ]

r2

[Bb]T[Db][Bb]

w i t h [K ]

(3.3)

[N]" dx +

Y

Y"

dy

[N]" [N] dx

Y" Y"

dy

[N]T[N] dx

12 m

22 m

n

n

[N]' dx

thickness

is

(3.4) taken

into

account

in

the

elasticity matrix, D

[Db]

and

ii

D

12

0

0 D D 21 22 0 〇 D D"

denotes

t/2

where D

dz

(3.5)

t/2 33

the

entries

of

the

plane-stress

elasticity

matrix.For isotropic material, particularly, we have 〕 = D 11

22

12

= D = E t3/12(l-i/2) D

33

= G t / 12

where t = plate thickness and varies along y-axis All integrations are carried out over the entire surface area of the strip. The explicit form of integrations in both directions has been discussed in Chapter 2. Note that [Kb3

is a 4x4 matrix.

Load vector The load vector of a strip with lateral load f(x,y) is given by


26

{F} = { {F} where

{F} = m In

1

A

particular,

transverse

2

f(xďź&#x152;y) J

,

ďź&#x152;{F}

X

r

}T

(3.6)

[N (x) ] Y (y) dx dy m

assuming

direction

{F}

m

that

and

the

stepwise

(3.7)

loading

constant

is

constant

longitudinally

in

the

(

the

loading, denoted by f , is a constant within element k of the fictitious k beam), then the following load vector can be derived, {F} = {d/2 d h z m where

d/2

-dhz)

1 2 /12 k

{ f f (1 /2 L k k k=1

-1 2 /12) { a } } k m k (3.8)

1 /2 k

d = strip width

and {a } = joint displacements of element k of the fictitious beam for m k

the mth COMSFUN. Transit ion Strip As shown

in Fig. 3.3,

transverse direction.

The

there

is a change

longitudinal

shape

in

rigidity across

functions

of

the

the panel

strip are different from those of the column strip. A transition strip is therefore introduced

to satisfy the compatibility requirement along

the nodal lines between adjacent strips. The displacement function of a transition strip is given by r

w(x,y) = V L m=l

[( N

1

N o )(Y ) 2

1 m

( No N 3

4

) (Y ) 3 { r >

2 m

m

(3.9)

where

(Y ) and (Y ) denote respectively the COMSFUN of nodal lines 1 1 m 2 m and 2. Derivation of all element matrices follows the same track as for the original strip. Nodal point springs For nodal point springs corresponding to w and dw/dx, its stiffness can be added directly to their corresponding location along the diagonal in

the

global

stiffness

matrix.

However,

for

a

point

spring

corresponding to a rotation about X-axis (du/dy), the spring stiffness is

superimposed

onto

the

stiffness

of

the

strip using

the compound

finite strip method. Suppose such a rotational point spring is attached to nodal line 1


27

o f a s t r i p , t h e s t i f f n e s s o f the s p r i n g i s t h e n g i v e n by [k]^ [k]”. 11

[k] ‘

12

[k]

(3.10)

[k] [k]

[k] ^ tk]

k

[k]

with

where k

s

location

1 0 0 0

Y* Y’

0 0 0 0

0 0 0 0

0 0 0 0

(3.11〕

is the stiffness of the spring and Y* , Y' are evaluated at the m

of

the

spring.

This

matrix

is

n

superimposed

onto

the

corresponding strip stiffness matrix and then assembled into the global stiffness matrix. Global stiffness matrix equation The procedure of assembling of the strip stiffness matrices and load vectors follows that of Cheung[3]. The final matrix equation is stored

in the form of a ha If-band matrix which Is then solved using

standard Gaussian elimination. 3.3 Numerical Examples Examples are given herein

to demonstrate the application of the

present strip to plate bending problems. The performance of the COMSFUN is first assessed by means of a simple beam. The second and the third examples,which

involve

a

square

plate

with

different

boundary

conditions,illustrate the accuracy and the convergence characteristics of the present strip.A few rectangular plate systems are then analysed and the finite strip results are compared with the usual finite element method[70]. In all cases, unless otherwise specified, the nodes on each nodal line are of equal distance apart. 尸reiiininary To demonstrate t h e a c c u r a c y o f t h e computed shape f u n c t i o n s i n d e a l i n g w i t h p o i n t l o a d s and a b r u p t l y changing t h i c k n e s s ,

two s i m p l y


supported beams are analyzed

(Fig.3.4).

The p o r t i o n s of

the

beams

between the two p o i n t loads are subjected t o constant bending moments ( i . e . i n the constant s t r a i n s t a t e o f bending )â&#x20AC;˘ Although they can be analyzed by

the usual beam elements, a single strip, with different

number of computed shape functions,is used to generate the results given in Tables

3.1 & 3.2.

As

the

beams are

statically determinate,

the

variation in thickness does not affect the moment distribution. Using shape functions 2 & 3,which take into account the presence of the two point loads, accurate displacements, rotations and bending moments can be obtained for the prismatic and

the non-prismatic beams. Using the

first shape function alone, however, the mid-span moments in both cases differ from the exact values by more than 30 percent. As expected, very accurate mid-span deflection can still be obtained by using the first shape function alone, even though the latter does not take into account the presence of the point loads. modes y=l

Deflection y=2

Rotation y=0

Moment y=l. 5 y=2

y=2.5

1

20000

27500

20625

2.063

2.750 2.063

2+3

20000

27500

22500

2.000

2.000 2.000

1+2+3

20000

27500

22500

2.000

2.000 2.000

20000

27500

22500

2.000

2.000 2.000

Beam theory

Table 3.1 : Results for the prismatic beam

modes

Moment y=2.0

y=2.0

y=2.5

2.063

750

2.715

2.063

2+3

2.000

000

2.000

2.000

1+2+3

2.000

000

2.000

2.000

Exact

2.000

000

2.000

2.000

Table 3.2 : Moment distribution

of

the non-prismatic beam

y=2.0 and y=2.0 indicate, respectively, the positions just to L

R

the left and to the right of the centre of the beam.


29

A square

plate

with

span a and subjected

A square p l a t e o f u n i f o r m t h i c k n e s s at

the c e n t r e .

The p l a t e i s

boundary c o n d i t i o n s

namely,

to a point

load

i s subjected to a p o i n t load

treated w i t h three d i f f e r e n t (1)

s i m p l y - s u p p o r t e d edges,

cases o f ⑵

clamped

edges (3) v e r t i c a l l y supported a t t h e f o u r c o r n e r s . O n l y a q u a r t e r o f t h e p l a t e i s d i s c r e t i z e d . For the computed shape f u n c t i o n s , a p r i s m a t i c beam w i t h span a / 2 is. assumed. The beam i s d i v i d e d i n t o 12 i d e n t i c a l beam

elements.

corresponding

Different shape

number

functions

are

of

nodes

are

assigned

computed.Deflections

and

and

the

moments

obtained u s i n g the present f i n i t e s t r i p are normalized w i t h respect t o t h e e x a c t s o l u t i o n s [ 7 1 ] . R a p i d convergence o f d e f l e c t i o n s a t the c e n t r e o f the p l a t e can be seen i n F i g . 3 . 5 . R e s u l t s o f t w i s t i n g and bending moments a r e g i v e n i n T a b l e 3 . 3 .

U s i n g a mesh w i t h 4 s t r i p s and 4 nodes

(i.e.

freedom

number

of

degrees

c o n d i t i o n s , N D O F = 40

of

prior

to

imposing

boundary

), the centre deflections are 99 percent accurate

for all cases, while the moments given in Table 3.3 indicate a maximum error

of

less than 5 percent. All

results compare favorably with

the

exact solutions. Number of strips *

NDOF

Moments

Number of nodes

12*

(1)

(2)

(3)

(4)

24

1.0095

0.9534

0.8172

1.0073

40

1.0006

0.9619

0.9546

1.0318

60

1.0002

0.9743

0.9481

1•0063

102

1.0000

0.9854

0.9773

1.0031

162

0.9999

0.9885

0.9933

0.9998

242

1.0000

0.9945

0.9932

1.0000

Table 3.3 : Results of moments for a square plate (i/»=0.3,a/t=100) (1) = M (2) = M

xy

X

at x=0, y=0 of the simply-supported plate at x=0, y=a/2 of the clamped plate

(3) = H at x=a/2, y=0 of the clamped plate y (4) = M at x=0, y=a/2 of the corner-supported plate y Note : location of centre x=a/2, y=a/2


30

A square

plate

with

span a and subjected

to UDL

A square p l a t e o f u n i f o r m t h i c k n e s s i s s u b j e c t e d t o a u n i f o r m l y distributed

load

over

the e n t i r e

conditions are considered.

domain.

Quart er o f

Three d i f f e r e n t

boundary

the p l a t e s i s d i v i d e d i n t o 4

s t r i p s and v a r i o u s number o f nodes. R e s u l t s o f bending moments and d e f l e c t i o n s a t t h e c e n t r e o f the p l a t e a r e summarized i n T a b l e 3 . 4 . F o r the

simply-supported

or

clamped p l a t e s ,

the

present

finite

strip

s o l u t i o n s converge r a p i d l y t o the e x a c t s o l u t i o n s [ 7 1 ] . However, f o r the corner-supported p l a t e , present f i n i t e s t r i p r e s u l t s a r e c l o s e r to

the

s p l i n e f i n i t e s t r i p r e s u l t s [13] (12 s t r i p s and 12 s e c t i o n s ) t h a n t o the exact solutions.

Number o f nodes

(3)

(2)

(1)

simply-supported p l a t e 0.47848 0.48130 0.48037 0 . 47886qa 2 * 10

0.48102 0.48152 0.48123 0.47886 qa 2 * 10"

Exact[71] Multiplier

0.406394 0.406253 0.406245 0.406236 qa 4 * 10" 2 /D

clamped p l a t e 0.23626 0.23303 0.23142 0.22905 q a * ICf 1

0.23425 0.23331 0.23284 0.22905 qa2* 10"

Exact[71] Multiplier

0.126318 0.126518 0.126525 0.126532 qa 4 * 10**V D

Corner-supported p l a t e (m

9 5

J/

Jf K 3 9 9fv 4 2 9 1

o o o oq

2 1 1 o 1 1 1 1 1 12

o

Exact[71] Spline F.S.[13] Multiplier

0.11546 0.11297 0.11287 0.1090 0.^1190 qa

0.25249 0.25376 0.25419 0.249 0.25506 qa 4 * I C T V d

T a b l e 3.4ďź&#x161; Deflections and Moments at the centre of square plates (1) moment M

x

, (2) moment M

y

and (3) deflection 0=0.3)


31

Z 3-bay Having e s t a b l i s e d t h e a c c u r a c y and v a l i d i t y o f the p r e s e n t f i n i t e strip,

a plate

system w i t h m u l t i p l e

internal discrete

s uppo rt s i s

s t u d i e d i n t h i s example. A 3~bay f l a t p l a t e model i s g i v e n i n F i g . 3 . 6 a . I t i s s u p p o r t e d by

16

columns o f

the

same dimensions.

A uniformly

d i s t r i b u t e d l o a d i s a p p l i e d over t h e e n t i r e p l a t e . Because o f symmetry, a q u a r t e r o f the model i s d i v i d e d i n t o 15 s t r i p s a l o n g X - a x i s and w i t h either

7

nodes

or

13

nodes

along

Y-axis,see

Fig.3.6b.

The

same

l o n g i t u d i n a l shape f u n c t i o n s a r e used f o r a l l s t r i p s . A node i s p l a c e d a t the c e n t r e o f each column which i s m o d e l l e d w i t h one v e r t i c a l p o i n t s p r i n g and two r o t a t i o n a l p o i n t s p r i n g s . F o r t h e edge columns, t h e i r c e n t r e s a r e assumed t o be l y i n g on t h e boundary l i n e s . T h e model was a l s o a n a l y s e d b y the f i n i t e element method. The f i n i t e element mesh c o n s i s t s o f 900 q u a d r i l a t e r a l elements (20x20 elements f o r e a c h p a n e l ) w i t h a t o t a l o f 961 nodes amounting t o 2883 degrees o f freedom and a h a l f bandwidth o f 96. ( The number o f d e g r e e s o f freedom f o r t h e

FSM w i t h 13

nodes i s j u s t 416 w i t h a h a l f - b a n d w i d t h o f 5 2 . ) Bending

moments

along

three

selected lines

on

the

model

are

r e c o r d e d on F i g . 3 . 7 . One can observe t h a t t h e i n t e r i o r b e n d i n g moments along l i n e s agreement.

1 and

3 o b t a i n e d f r o m b o t h methods a r e i n reasonable

Increase i n

siginificantly

affect

the the

number results.

of On

longitudinal column

nodes

line

2,

does

not

however,

a

s i g n i f i c a n t d i s c r e p a n c y i s found a t t h e l o c a l i s e d a r e a i n t h e v i c i n i t y o f t h e c o l u m n - p l a t e c o n n e c t i o n . The s h a r p v a r i a t i o n i n s t r e s s e s i n these l o c a l i s e d area r e v e a l s the s i n g u l a r behaviour near the p o i n t supports. We a l s o n o t e t h a t t h e p o i n t support c o n d i t i o n does n o t r e f l e c t the t r u e s i t u a t i o n o f a column w i t h a f i n i t e c r o s s - s e c t i o n a l a r e a . In t h i s

analysis,

the s t i f f e n i n g e f f e c t

of

a

column on

the

c o n n e c t i n g p o r t i o n o f t h e p l a t e i s n e g l e c t e d . T h i s l o c a l i s e d e f f e c t can be t a k e n i n t o account f o r l a r g e r column b y a s s i g n i n g a l a r g e s t i f f n e s s t o those beam elements w h i c h r e p r e s e n t t h e s t i f f e n e d p o r t i o n o f t h e p l a t e when the l o n g i t u d i n a l shape f u n c t i o n s a r e computed.

A 3-bay flat-slab

model

W i t h t h e a d d i t i o n o f drop p a n e l s t o t h e f l a t p l a t e model, application of

the present s t r i p t o

the

a p l a t e w i t h a b r u p t l y changing

l o n g i t u d i n a l r i g i d i t y i s demonstrated i n t h i s example ( F i g . 3 . 8 ) .

The


32

model i s

a n a l y z e d by

the f i n i t e s t r i p

and f i n i t e

element

method.

T r a n s i t i o n s t r i p s a r e employed because t h e l o n g i t u d i n a l shape f u n c t i o n s o f t h e column s t r i p s a r e d i f f e r e n t from t ho s e o f the panel s t r i p s . U s i n g a f i n i t e element mesh w i t h 576 elements (16x16 elements f o r each p a n e l ) , t h e r e s u l t s o b t a i n e d from b o t h methods a r e shown i n F i g . 3 . 9 . Reasonable agreement between the two s e t s o f r e s u l t s can be observed a l o n g L I and L 3 . S i g n i f i c a n t d i s c r e p a n c y can be seen a g a i n a t the l o c a l i s e d area i n t h e v i c i n i t y o f the c o l u m n - p l a t e c o n n e c t i o n a l o n g L 2 .

A 5-bay flat-slab

model

F i g . 3 . 1 0 d e p i c t s a 5x5 f l a t - s l a b model w i t h 36 s u p p o r t i n g columns. It

is

subjected

to

either

a

uniformly

distributed

load

or

the

checkerboard l o a d s i n F i g . 3 . 1 1 . The model i s d i v i d e d i n t o 50 s t r i p s and 16 nodes.

A node i s p l a c e d a t

the c e n t r e o f

each column which i s

modelled w i t h one v e r t i c a l s p r i n g and two r o t a t i o n a l s p r i n g s . The model was a l s o a n a l y s e d by t h e f i n i t e element method u s i n g a mesh o f 2500 q u a d r i l a t e r a l elements ( i . e . 10x10 elements f o r each p a n e l ) . Comparisons o f bending moments a r e p r e s e n t e d i n F i g . 3 . 1 2 . One c a n observe t h a t the bending moments from b o t h methods a r e i n r e a s o n a b l e agreement except f o r t h e peak moments a t t h e c o l u m n - s l a b c o n n e c t i o n s .

A A two-span b r i d g e w i t h v a r y i n g t h i c k n e s s o v e r t h e i n t e r i o r support i s shown i n F i g . 3 . 1 3 . I t i s s u b j e c t e d t o e i t h e r an e c c e n t r i c p o i n t l o a d o r a u n i f o r m l y d i s t r i b u t e d l o a d . The b r i d g e i s d i v i d e d i n t o 16 s t r i p s and

11

nodes.

To

determine

COMSFUN,

the v a r i a t i o n o f

longitudinal

r i g i d i t y i s approximated b y 40 p r i s m a t i c beam elements w i t h d i f f e r e n t thickness.

In Fig.3.14

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

moments a l o n g t h e span a r e compared w i t h t ho s e o f f i n i t e element u s i n g 16*40 elements. As expected, one c a n e a s i l y see the c o n t r a f l e x u r e p o i n t i n the shorter

span o f

the b r i d g e when s u b j e c t e d t o

UDL.

In t h i s

example, the p r e s e n t f i n i t e s t r i p method can produce r e a s o n a b l e r e s u l t s w i t h a p p r o x i m a t e l y 80 pe r ce n tage r e d u c t i o n i n the number o f unknowns when compared w i t h t h e f i n i t e element method.


33

3 . 4 Conclusion A

new

finite

strip

for

thin-plates

is

presented i n

this

c h a p t e r . U s i n g a s e r i e s o f computed shape f u n c t i o n s i n the l o n g i t u d i n a l direction,

the f l e x u r a l problem o f

a r e c t a n g u l a r p l a t e system w i t h

changing l o n g i t u d i n a l r i d i g i t y and/or c o m p l i c a t e d support c o n d i t i o n s can be s o l v e d i n an easy and u n i f i e d manner . F o r p l a t e s w i t h p o i n t loading, r a p i d convergence can be achieved. Because o f unknowns a t each node,

there i s

the reduced number o f

a c o n s i d e r a b l e s a v i n g i n computing

e f f o r t w i t h r e s p e c t t o the f i n i t e element method. The b a s i c concept o f the method i s simple and r e q u i r e s no c o m p l i c a t e d mathematical f u n c t i o n s . Because the f o r m u l a t i o n i n v o l v e s o n l y t h e u s u a l beam shape f u n c t i o n s , a l l element m a t r i c e s can be o b t a i n e d i n an economical and e x p l i c i t form. The computer implementation o f the method i n v o l v e s the i n c o r p o r a t i o n o f a continuous-beam program i n t o a s t a n d a r d f i n i t e s t r i p program. accuracy

and

the e f f i c i e n c y o f

n u m e r i c a l examples g i v e n .

the s t r i p a r e

The

demonstrated i n the


34

(/) O D c • H > » u a >

』A=A

J C «4-^ O) c 0

. c

H—

O

^ ^

c

0

E

-p-—^ _Q

CD

•-«

-<^V

a

(J)

.E •a

-o g

a -H u

Z

s

D < "O r; o

W

W

n c o

① « p -H C — ^

-o c¢0 C i; m TJ sC §

二 TJ Cfl t 艺—口 0

a ^ §

U ①

><

i i CO

rH -l->

5 t— O ID2: 一 CO -<H < + ( Lu T J ⑴ < P D C 2: P jC O H O G O r^\ rt O XI o w rH ^~• W-

OD t-H

C O C O Uh

"D ①

a

E

^

8


35

panel strip

k column strip

transition strip Fig.3.3

C l a s s i f i c a t i o n o f s t r i p s f o r analyzing f l a t slabs w i t h drop p a n e l s .

(w , 0 )

(w

2 ,v

m

nodal line Fig.3.2

A t y p i c a l f i n i t e s t r i p w i t h f i v e nodes on each o f the two n o d a l l i n e s . The nodes a r e o f unequal distance apart.


0=3

eCDPOIAI

SCDT3OIAI

^CDi


jo

? TJsro

S

ddns j

cn

O

> a CN

< T ) cn

0 0

O C

0

寸 CO

丄 CM

00

to

o

uoi;Da|j.sp peziiotujON

00 00 CT>

UIOP99JJ

pii

jDsdtuop

」0aN

丄 00

d d n s


38 s e p o u ei-

sepou

u-k in TJ O * • * < g C <\i J t C w II w II U ( Tl >.C rt X: U M o bQ -P O rH o < D g o £ H .x; o .c O * Q S jC w P n c „ ^ ^ D < " O j E ^ o a * .t-Jo a

O

< D

X

上 亡 • 6 ^ CN3 ^ II ? 2 丨 | ① uut;©

^ ‘ 广 W e‘ C

e

•^画

/ c 3 r-

^ Q X) P II :DO . D < *r-i < X IM u ( ^ 4 > - e u c (d \ X h a a 乂 v D C O

E

• CO

d t CO 」

T-j

ci •H Du


0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Distance along L1 一 FEM

0

FS(7) — FS(13)

15

10

5—

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Distance along L1

Fig. 3.7 (a) Moment along LI


40

X2}ueEo2

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

D i s t a n c e along L 2 0

FEM

FS(7) — FS(13)

o

>2

-

o

4 -

iueluoi/NJ

6

o •80

00 1,5

2

2.5

3

3.5

4

4.5

5

5.5

6

Distance along L2

Fig. 3.7 (b) Moment along L2


41

xsceluos 1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

D i s t a n c e along L 3 FEM

i

i

1.5

2

Q

FS(7) — FS(1 3)

i

2,5

i

3

i

3.5

i

4

4.5

5

5,5

6

Distance ^lo门g L 3

Fig. 3.7 (c) Moment along L3


sepous l

lefDOUU

sepou i

C/) •9"

"vZ •+—'

¢/) in

cd Q.

"vZ

4-» ¢/)

o 1"•“

•Q

xs o c c d tn w D < 4 > - O D < B O H < D C / 3

0) c O C

c

•r-4

E

Q. CL O u.

O o

0 0 1 & y.rH

C

o

•rH

d ( r"i W I -»-> •-» *+-*

£: to o

E 寸

O C E ^ a

I I S

4- > UiL

•"、 • XI S v_x CO • •O n c C II o •f-i rH Q C 4J 0) (XS C d c u aj Xt P a i b0 O -H <d、a U G O X: o u H O TJ


0

0.5

1

1.5

2

2.5

3

a 5

4

4.5

5

5.5

6

D i s t a n c e along L i 一 FEM

0

FS(7) — FS(13)

15

10

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Distance along L1

Fig. 3.9 (a) Moment along LI


44

- 2 0

x 2

-40

匚 < D

6 0

l 00

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Distance along 0

FEM

FS(7) — FS(13)

40

20

>>

C CD

E o

2

60 80 100 -12 0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Distance along L 2 Fig.3.9

(b) D i s t r i b u t i o n o f moment a l o n g L2


4f

x -4—‘

匚 D <

E o

2

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Distance along L 3 FEM

0

FS(7) — FS(1 3)

-4—'

匚 ①

E o

2

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Distance along L 3 Fig.3.9

(c) D i s t r i b u t i o n o f moment a l o n g L 3


46

Thickness of slab ~0.2rri

5000

Thickness of drop :0.4771

Columns

5000

:0.57n*0.5m

Height of columns =2m 5000

1m*1m

• Centre

drop of

pcme

column

5000

o o 5

o

ri/

F L

• 5000

5QQ0

5000

5QOO

5000

(a)

5?

5?

5?

5?

6<

5?

5?

V,

5?

5?

16 nodes

(b) Fig.3.10

Five-bays flat slab model. (a) Geometry configurations, (b) finite strip m e s h

and


Checkerboard Load Pattern 1

BL only

Checkerboard Load P a t t e r n 2 Fig.3.11

Checkerboard l o a d p a t t e r n s . DL=5 kN/m2 LL=10 kN/m2


48

15

Mx/KNm

10

0

5

2 5

12.5

10

7.5

Distance from point Him + FSM Fig.3.12

(a) Distribution of moment along H G for the case of UDL

Mx/KNm 50

150 2.5

5

7.5

12.5

17.5 20

2Z5

Distance from point Kim Fig.3.12

(b) Distribution of moment along K L for the case of UDL

25


49

50

My/KNm

0

50

100

150 0

5

10

15

20

25

Distance from point Hm

F i g . 3.12

(c) D i s t r i b u t i o n o f moment a l o n g I J f o r t h e c a s e o f UDL

12.5

2 5

Distance from point Him Fig.3.12

(d) D i s t r i b u t i o n o f moment a l o n g HG f o r the c a s e o f checkerboard load p a t t e r n 1


Mx/KNm

50

0

50

100

150 0

2.5

5

7.5

Distance from point K/m â&#x20AC;&#x201D; FEM +FSM Fig.3.12

(e) Distribution of moment along K L for the case of checkerboard load pattern 2

12.5


51

8 m

12m

UDL=25kN/m

(a)

20m • Point load = 100kN

8 m

(b)

8 m

Fig.3.13

(a)Two-span slab bridge and arrangements (b) Finite strip mesh

its

loading


52

Deflectionjm 0.0025 0.002

0.0015 0.001

0.0005

-0.0005 â&#x20AC;˘0.001

0.002 -0.0025 0

2

4

6

8

10

12

14

16

18

20

Distance From Edge/m

MomentJKNm 100

100

0

2

4

6

8

10

12

14

16

18

Distance From Edge/m Fig.3.14

(a) Point L o a d : Distributions o f deflection a n d moment M along the longitudinal line at 1m f r o m the edge.

20


53

0.004

Deflectionjm

0.003 0.002 0.001

,0.001

•0.003 - 0.004 0

4

2

6

10

8

12

14

16

18

20

14

16

18

20

Distance From Edge/m — FSM

a

FEM

Moment!KNm 250 200

150 100

50 0 •50 •100

150

0

2

4

6

8

10

12

Distance From Edge/m Fig-3.14

(b) U D L : Distributions o f d e f l e c t i o n and moment M a l o n g the longitudinal l i n e at 1m f r o m the edge.

y


54 CHAPTER 4 VIBRATION AND BUCKLING OF RECTANGULAR PLATES

In Chapter 3 the new t h i n - p l a t e f i n i t e s t r i p has been s u c c e s s f u l l y a p p l i e d t o the s t a t i c a n a l y s i s o f r e c t a n g u l a r p l a t e - t y p e s t r u c t u r e s w i t h varying

bending

ridigity

a p p l i c a t i o n to v i b r a t i o n

and/or

multiple

discrete

and b u c k l i n g a n a l y s i s o f

supports.

Its

rectangular plates

w i t h c o m p l i c a t e d boundary and i n t e r n a l s u pport c o n d i t i o n s i s presented i n t h i s c h a p t e r . Numerical examples a r e g i v e n t o v e r i f y the v a l i d i t y o f the p r e s e n t f i n i t e s t r i p and to compare i t s v e r s a t i l i t y , e f f i c i e n c y to

the s p l i n e f i n i t e s t r i p o r

the

a c c u r a c y and

semi-analytical f i n i t e

strip.

4.1 Formulation Readers may r e c a l l t h a t the d i s p l a c e m e n t f u n c t i o n o f a t h i n - p l a t e f i n i t e s t r i p can be w r i t t e n a s r w(x,y) = V

tN(x)] Y (y) { ^ }

U

= y U m=l

(4.1)

to ]{

F o r f r e e v i b r a t i o n a n a l y s i s , the c o n s i s t e n t mass m a t r i x o f a s t r i p c a n be w r i t t e n a s

with

[M ]

[Q]

t Y

p t

Y

[Q]

dy

p = d e n s i t y o f the m a t e r i a l

(4.3)

dx d y

[N]

[N] dx

t = varying thickness along Y-axis


55

For l i n e a r buckling a n a l y s i s , with i n i t i a l

in-plane stresses

the geometric matrix

cr , X

w r i t t e n as

cr

and

y

[ [G b ]

[G b ] [G b ]

[G b ]

rl

x

xy

of

a strip

(see F i g ' 4 . 1 ) ,

can be

• (4.4)

mn [G b ]

r2

where

[G b ]

(wt

w,)

Y

Y

dy

Y, Y, dy

+ T

t Y _Y, dy

Y' Y

All

dy

integrations are

strip. The explicit

T

(w,

dx dy

[N]’ T [N], dx

[N] [N]

[N]>

(4.5)

dx

[N]

dx

[N】 T [N]’

dx

carried out

form of

w, )

over

integrations

discussed in Chapter 2. Note that

[M]

global

solved

the entire

surface of

the

in both directions has been

and [G] are only 4x4 matrices. mn mn After assembling the stiffness, mass and geometric matrices, the eigen-equations

are

then

using

the

subspace-iteration

method for the frequenices and load factors.

4.2 Numerical examples In the following examples, free vibration frequencies and buckling loads of rectangular plates with various complexity are computed using


56 the

present f i n i t e s t r i p .

assess

the

accuracy

of

The the

main purpose o f present

finite

these

strip

by

s o l u t i o n s w i t h those o b t a i n e d by s p l i n e f i n i t e s t r i p ,

examples i s

to

comparing

its

semi-analytical

f i n i t e s t r i p ( a l l as quoted i n the paper by Leung and Au [72]) and o t h e r a n a l y t i c a l / n u m e r i c a l methods a v a i l a b l e i n t h e l i t e r a t u r e . In cases where s p l i n e f i n i t e s t r i p solutions are a v a i l a b l e ,

the p r e s e n t f i n i t e s t r i p

meshes a r e chosen i n such a way t h a t ,

for

the

efficiency

the

total

involved

with

in

otherwise

spline

each

finite

method i s

specified,

the

strip,

about

present

the

same.

finite

sake o f number

In a l l

strips

comparing i t s

are

of

unknowns

cases,

unless

running i n

the

Y - d i r e c t i o n o f the p l a t e s and the nodes on e a c h n o d a l l i n e a r e o f equal distance apart.

Free

vibration

Comparison strip

with

spline

finite

strip

and

: Square plates

A number o f square p l a t e s o f v a r i o u s homogenous boundary c o n d i t i o n s a s shown i n F i g . 4 . 2 were a n a l y s e d u s i n g a mesh o f 8 s t r i p s and 7 nodes. Table

4.1

shows

the

results f o r

the

lowest

eight

frequencies.

The

p r e s e n t f i n i t e s t r i p r e s u l t s a r e compared w i t h s e m i - a n a l y t i c a l f i n i t e s t r i p s u s i n g 10 s t r i p s and f o u r terms and t h e s p l i n e f i n i t e s t r i p s u s i n g eight s t r i p s with subdivisions of 6 sections. Rayleigh-Ritz solutions of L e i s s a [ 1 2 6 ] a r e a l s o g i v e n . Good agreement i s o b s e r v e d i n a l l c a s e s .

Comparison

with

panels

various

with

Fig.4.3

shows

semi-analytical

finite

strip

:

2*2

conditions

a series of

2*2 c o n t i n u o u s p a n e l s f o r w h i c h o n l y

s e m i - a n a l y t i c a l f i n i t e s t r i p s o l u t i o n s a r e a v a i l a b l e . B y a s s i g n i n g a node a t t h e l o c a t i o n o f the t r a n s v e r s e i n t e r n a l l i n e s u p p o r t ( p a r a l l e l t o the x - a x i s ) , continuous

plates

with

various

boundary

conditions

can

be

solved. imposed o n the c o r r e s p o n d i n g d i a g o n a l e n t r i e s i n the g l o b a l s t i f f n e s s m a t r i x . E i g h t s t r i p s and 9 nodes a r e u s e d i n a l l c a s e s . T h e d i m e n s i o n l e s s frequencies obtained are g i v e n i n Table 4 . 2 . Leissa[126],

Analytical solutions of

Kim

S p l i n e f i n i t e s t r i p s o l u t i o n s a r e n o t a v a i l a b l e f o r t h e s e continuous p l a t e s . Good agreement i s o b s e r v e d i n a l l c a s e s .


57

D imen s ion les s f r e q u e n c i e s 入 Mode sequence

1

2

3

4

5

6

i

S-S-S-S (Fig.4.2a) Leissa[126]

19.74 49.35

49.35

78.96

98.70

93,.70

128.30

128.30

Finite strip

19.74 49..32

49.34

78.91

98.64

98,• 68 128.17

128.22

Spline s t r i p

19.74 49..36

49.38

78.98

98.80

99 .21

128.40

128.72

Present s t r i p

19.74 49,.36

49.38

78.,98

98.80

99 .28

128.41

128.79

Leissa[126]

23.65 51..67

58.65

86..13

100.27 113 .23 133.79

140.85

Finite strip

23.62 51..62

58.65

86,.16

100.35 113 .24 134.00

140.94

Spline s t r i p

23.65 51..71

58.67

86,.17

100.78 113 .40 134.22

141.01

Present s t r i p

23.65 51,.71

58.67

86. 18

100.86 113 • 42 134.27

141.02

Leissa[126]

22.27 26..53

43.66

61,.47

67.55

79.90

Finite strip

22.29

.08

44.76

61,.53

68.29

81.08

89.55

120.73

Spline s t r i p

22.33 27..14

44.80

61,.72

68.46

81.07

89.67

122.40

Present s t r i p

22.25 26,.53

43.65

61 • 45

67.59

80.18

87.82

120.83

Leissa[126]

9.63

16..13

36.73

38.95

46.74

70.74

75.28

87.99

Finite strip

9.87

16,.91

38.11

39.49

47.72

74.03

76.49

88.94

Spline s t r i p

9.80

17..02

37.90

39.38

48.03

72.82

76.37

89.16

Present s t r i p

9.63

16.13

36.72

38.99

46.78

70.78

75.33

88.64

Leissa[126]

24.02 40,.04

63.49

76.76

80.71

116.80

Finite strip

24. 11 40,.66

63.61

77.32

81.77

118.60 122.87

135.11

Spline s t r i p

24.35 40,.66

63.90

77.41

81.66

118.21 124.62

135.29

Present s t r i p

23.95 40,.04

63.44

76.82

80.76

116.93 123.95

134.80

Leissa[126]

11.68 27.76

41,.20

59.07

61.86

90.29

94.48

108.92

Finite strip

11.92 28 .38

41 .62

59.94

62.94

91.07

96.18

109.99

Spline s t r i p

12.00 28 .37

41 .73

60.01

62.45

91.55

95.77

110.42

Present s t r i p

11.68 27.76

41 .23

59.09

61-88

90.93

94.54

109.50

C-S-S-S (Fig.4.2b)

C-F-C-F (Fig.4.2c)

21,

F-S-F-S (Fig.4.2d)

C - O F - C (Fig.4.2e)

S-S-F-S (Fig.4.2f)

Table

4.1

: Frequencies of

boundary conditions.A ,

c^a

square plates with various homogeneous 2

(p t / D) 1 / 2 , v = 0.3


58

Dimensionless f r e q u e n c i e s A. Mode sequence 1

2

3

4

S-S-S-S (Fig.4.3a) Leissa[126]

19.74

23.63

23.63

27.06

49.35

49.35

Kim[128]

19.74

23.98

23.98

27.70

49.35

49.35

Liew[130]

19.74

23.71

23.71

27. 10

49.37

49.37

Finite strip

19.74

23.67

23.68

27.11

49.35

49.74

Present s t r i p

19.74

23.65

23.66

27.07

49.48

51.77

Finite strip

20.82

24.61

27.16

30.24

50.38

52.22

Present s t r i p

20.82

24.58

27.15

30.19

50.22

52.37

Finite strip

24.61

29.84

30.25

34.69

54.52

57.60

Present s t r i p

24.58

29.74

30.21

34.56

54.63

57.49

Finite strip

21.86

28.00

28.01

33.13

52.84

53.34

Present s t r i p

21.85

27.96

27.97

33.04

52.98

53.11

Finite strip

13.58

21.38

22.36

28.41

31.65

36.18

Present s t r i p

13.65

21.46

22.34

28.38

31.58

36.65

Finite strip

13.60

21.40

26.04

31.53

32.58

37.56

Present s t r i p

13.66

21.47

26.00

31.47

32.49

37.37

Finite strip

13.17

14.09

21.11

21.73

28.75

34.03

Present s t r i p

13.23

14.15

21.18

21.80

28.71

33.93

O S - S - S ( F i g . 4. 3b)

C-C-S-C (Fig.4.3c)

C-C-S-S (Fig.4.3d)

C-S-S-F (Fig.4.3e)

C-C-S-F (Fig.4.3f)

C-F-S-F (Fig.4.3g)

T a b l e 4 . 2 F r e q u e n c i e s o f 2*2 c o n t i n u o u s p a n e l s X = w I 2 (p t / D) 1 / 2 = 0.3 i i For f i n i t e s t r i p s o l u t i o n , f o u r

beam

modes

f o u r s t r i p s between two s u p p o r t s a r e used.

and


59

Comparison

wi th

semi-analytical

finite

strip

panels with various conditions Continuous plates with analyzed

three panels

in

each direction

are

also

(Fig.4.4). The boundary conditions considered are the same as

those of the previous 2*2 continuous panels. Twelve strips and 13 nodes are

used

compared

in with

analytical between

all

those

finite

two

cases.

The

given

present

by

Wu

and

finite Cheung

strip method with four

supports.

The

strip

beam

dimensionless

results

[73]

using

modes

and

frequencies

are the

four

only semi-

strips

obtained

are

summarized in Table 4.3 and good agreement is again observed.

Dimensionless frequencies Types o f s t r1i p

2

3

入 令

4

S-S-S-s Finite strip

19.74

21.64

21.62

23.39

26.06

26.09

Present

19.74

21.61

21.61

23.34

26.01

26.02

Finite strip

20.24

22.08

23.72

25.35

26.46

28.12

Present

20.23

22.06

23.66

25.27

26.39

28.10

Finite strip

22,08

25.38

26.46

29.31

29.42

29.52

Present

22.06

25.27

26.40

29.18

29.32

29.49

Finite strip

20.72

24.12

24.14

27.17

28.49

28.52

Present

20.71

24.07

24.08

27.07

28.44

28.46

Finite strip

12.82

17.17

20.87

22.57

24.28

25.00

Present

12.87

17.23

20.85

22.64

24.21

24.95

Finite strip

12.82

17. 17

22.57

22.57

25.81

27.69

Present

12.87

17.23

22.55

22.64

25.72

27.62

Finite strip

12.74

12.89

17.12

17.23

22,53

22.61

Present

12.80

12.95

17.

17.29

22.60

22.68

C-S-S-S

OOS-C

C-C-S-S

OS-S-F

OC-S-F

C-F-S-F

T a b l e 4 . 3 : Frequencies of 3*3 continuous panels 入 = I 1 T

2

(p t / D) 1 / 2

, y = 0.3


Comparison

wi th

spline

finite

strip

and

semi-analytical

finite

strip ; Using a

mesh o f e i g h t s t r i p s and seven nodes,

frequencies of

the p l a t e s

(Fig.4.5)

are

the

lowest

th ree

summarized i n T a b l e 4 . 4

compared w i t h s e m i - a n a l y t i c a l f i n i t e s t r i p

solutions

and

the

and

spline

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

D i m e n s i o n l e s s f r e q u e n c i e s A. T y p e s 1o f s t r i p

2

3

Fig.4.5a Semi-analytical

22.5

Spline

22.73

50. 15

56.23

Present

22.82

50. 18

56.42

Fig.4.5b Semi-analytical

25.9

Spline

26.37

52.23

61.78

Present

26.62

52. 19

62.29

Fig.4.5c Semi-analytical

28. 1

Spline

28.65

61.06

62.48

Present

28.90

61.02

62.99

Semi-analytical

27.63

52.39

66.24

Spline

27.83

52.41

66.25

Present

28. 13

52.77

66.95

Fig.4.5d

Table 4.4

: F r e q u e n c i e s o f square p l a t e s w i t h v a r i o u s mixed 2 1/2 boundary c o n d i t i o n s - X . = w . a (p t / D) . v = 0.3 F o r s p l i n e f i n i t e s t r i p s o l u t i o n , 9 s t r i p s and 6 s e c t i o n s a r e used i n t h e f i r s t t h r e e c a s e s , w h i l e 8 s t r i p s and 6 s e c t i o n s a r e used i n t h e l a s t c a s e .

Comparison

with

spline

finite

strip

: Point-sapported

plates

The f r e e v i b r a t i o n o f a v a r i e t y o f p o i n t supported p l a t e s a s shown i n F i g . 4 . 6 were s t u d i e d . The c o n d i t i o n o f z e r o v e r t i c a l d e f l e c t i o n s a t


the p o i n t s u p p o r t s i s e a s i l y imposed i n the same f a s h i o n a s i s done f o r the u s u a l f i n i t e element method.The r e s u l t s o b t a i n e d u s i n g 8 s t r i p s and 7 nodes o f the p r e s e n t method a r e g i v e n i n T a b l e 4 . 5 and compared w i t h the s p l i n e f i n i t e s t r i p s o l u t i o n s . the

point

supports

are

taken

In the s p l i n e f i n i t e s t r i p method,

into

account

s p l i n e s a t the c o r r e s p o n d i n g l o c a t i o n .

by

modifying

the

local

No s e m i - a n a l y t i c a l f i n i t e s t r i p

s o l u t i o n s a r e a v a i l a b l e f o r these p o i n t - s u p p o r t e d p l a t e s . R e s u l t s o f the two f i n i t e s t r i p method shows e x c e l l e n t agreement.

D i m e n s i o n l e s s f r e q u e n c i e s X. Types o f s t r i p

Fig.4.6a Spline

7.11

15.77

15.77

19.58

38.43

44.37

Present

7.14

15.78

15.83

19.64

38.62

44.41

Spline

17.852

34.89

34.89

38.43

60.12

68.51

Present

17.98

35.19

35.22

38.62

60.82

68.97

Spline

13.47

17.85

18.79

18.79

26.92

51,73

Present

13.48

17.98

18.88

19.02

27. 16

51.42

Spline

49.35

49.35

52.78

78.96

98.71

128.32

Present

49.36

49.38

53.27

78.98

99.04

128.41

Spline

9.61

17.32

30.60

43.66

51.09

64.40

Present

9,63

17.35

30.63

43.80

51.19

64.45

Spline

15. 17

23.93

39.40

54.16

62.83

77.46

Present

15.21

23.97

39.43

54.33

62.94

77.52

Fig.4.6b

Fig.4.6c

Fig.4.6d

Fig.4.6e

Fig.4.6f

Table

4.5

:

Frequencies

of

square p l a t e s w i t h v a r i o u s 2 1/2 s u p p o r t s . A . = w.a (p t / D ) . v = 0.3

point

For spline f i n i t e s t r i p s solutions,

and 6

s e c t i o n s a r e used.

8 strips


62

Compar ison

with

Kantorovich

method

and

Square plates with step change in thickness The

fundamental

simply-supported

frequencies

edges

or

of

clamped

two

edges

square are

plates

obtained

by

with the

either present

method using a mesh of 8 strips and 7 nodes (Fig. 4.7). No attempts have been made Results

to

are

analyze given

in

these plates using Table

4.6

solutions of Cortinez and Laura

and

other

compared

finite with

strip methods. the

Kantorovich

[74] and finite element results

[75].

The present finite strip results are in good agreement with the finite element

results

one-term

while

the

approximation

Kantorovich solution,

method,

which

slightly

makes

use

of

over-estimated

a the

frequencies.

Boundary

Kantorvich [74]

FEM [75]

Present Strip

Simply supported

16.11

15.38

15.52

Clamped

28.90

28.30

28.32

Conditions

Table

4.6

:

Fundamental

frequencies 2

of

step-change in thickness. X = w a ( p t / D)

1/2

square . v = 0.3

plates

with


63

Linear

Buckling

Comparison

analysis

with

exact

solut ions

loads A

number

of

conditions are

square

subjected

plates

with

to uniaxial

various

homogenous

loads along

the Y-axis(Fig. 4.8).

Critical buckling load factors obtained by the present method strips and 7 nodes) are given in Table 4.7 and solutions

boundary

( using 8

compared with the exact

[76]. No significant difference can be found between the two

sets of results.

Critical buckling load factor, LF Present strip

Exact[76]

Fig.4.8a

39.4758

39.4784

Fig.4.8b

99.7000

99.1895

Fig.4.8c

56.6530

56.6533

Fig.4.8d

66.6563

66.5211

Fig.4.8e

76.0163

75.8973

Table

4.7

ďź&#x161;

Buckling

loads

homogeneous boundary conditions N

cr

=

of

square

plates

with

various

LF ( D / l ) . v = 0 . 3

Comparison with Ritz method ďź&#x161; Square plates subjected to unaxial or biaxial loads The

buckling

of

plates

(Fig.4.9)

with/without

internal

line

-supports and with two simply-supported opposite edges are analyzed. To the author's knowledge, the problem has not been attempted before using other

finite

strip

method.

vertical deflection at usual

finite

compared

with

element the

plates

with

internal

line-supports,

the corresponding nodes are constrained manner.

The

Rayleigh-Ritz

Mindlin plate theory. all cases.

For

present

solutions

finite of

Wang

strip et

In Table 4.8, very good agreement

in the

results ai.[77]

are

using

is observed

in


64

C r i t i c a l b u c k l i n g load f a c t o r LF Cases

B.C.

Loading

R a y l e i g h - R i t z [77]

Present s t r i p

SSFF

0.9523

0.9524

SSSS

4.0000

3.9997

SSFF

0.9322

0.9323

SSSS

2.0000

2.0004

SSFF

2.6725

2.6731

SSSS

16.000

16.010

SSFF

1.1898

1.1894

SSSS

4.9999

5.0022

SSFF

5.6063

5.6096

10

SSSS

15.999

16.010

11

SSFF

4.2204

4.2221

12

SSSS

7.9998

8.0028

T a b l e 4 . 8 : Buckling

loads of

square plates with continuous

line

c r t b 2 / (7i2D) y=0.3 and t/b=0.001 cr For cases 1-4, a mesh of 8 strips and 7 nodes was used. support. LF =

For cases 5-12, a mesh of 8 strips and 9 nodes w a s used. B. C. = Boundary conditions

at y = 0 , 7 = 5 »

x=0

and x=b

(S=simply-supported,F=free) L o a d i n g : U=uniaxial load along Y-axis ;B = b l a x i a l l o a d

4.3 Conclusion In t h i s c h a p t e r , t h e v a l i d i t y o f t h e new t h i n - p l a t e f i n i t e s t r i p i n analyzing f l e x u r a l vibration support examples,

conditions

is

and b u c k l i n g o f

confirmed.

As

p l a t e s with complicated

demonstrated

in

the

numerical

the p r e s e n t f i n i t e s t r i p i s a s v e r s a t i l e as t h e s p l i n e f i n i t e

s t r i p and t h e s e m i - a n a l y t c i a l f i n i t e s t r i p . I t a l s o p o s s e s s e s comparable e f f i c i e n c y and a c c u r a c y w i t h r e s p e c t

to

additionally,

the

the

implementation

of

the s p l i n e f i n i t e s t r i p and, support

conditions

is

much

s i m p l e r f o r t h e p r e s e n t s t r i p i n t h e s e n s e t h a t t h e y c a n be imposed i n t h e same f a s h i o n a s i s done f o r the u s u a l f i n i t e element method.


Fig.4.1

A f i n i t e s t r i p with i n i t i a l inplane stresses.


66

(d) F-S-F-S

(a) s - s - s - s

(b) C—S—S—S

(e) C-C-F~C

(c) C-F-C-F

Fig.4.2

( f ) S-S-F-S

R e c t a n g u l a r p l a t e s o f v a r i o u s homongeneous boundary (F); dashed conditions . L e g e n d : so l i d r u l e , f r e e e,clamped (C) ( S ) ; h a t c h e d r u l l y s u p p o r t e d r u l e , simp

fW

(a) S-S-S-S

(e) C - S - S - F

(b) C—S-S-S

(f) C-C-S-F

(c) O O S - C

(g) C - F - S - F

Z

(d) C—C—S—S

Fig.4.3

2 x 2 continuous p a n e l s w i t h v a r i o u s c o n d i t i o n s . Legend a s f o r F i g - 4 . 2

boundary


67

卜 / 十 / 十 H

Fig.4.4

3 x 3

continuous panel

• Z y y z x / / z / /

-,

Fig.4.5

Rectangular plates with mixed boundary conditions. Legend as for Fig.4.2.


68

(a)

â&#x2018;ˇ

(b)

(e)

(c)

Fig.4.6

::

(f)

R e c t a n g u l a r p l a t e s w i t h v a r i o u s p o i n t s u p p o r t s , (a) Corner p l a t e s u p p o r t e r d a t c o r n e r s and m i d - p o i n t s o f s i d e s , (c) Rectangular p l a t e supported a t m i d - p o i n t s o f s i d e s . (d) R e c t a n g u l a r p l a t e s i m p l y - s u p p o r t e d a t p e r i p h e r y and a t c e n t r e o f p l a t e . ( e ) R e c t a n g u l a r p l a t e s i m p l y - s u p p o r t e d a t two a d j a c e n t s i d e s and point p l a t e clamped a t two a d j a c e n t s i d e s and p o i n t s u p p o r t e d a t o p p o s i t e c o r n e r . Legend: x , p o i n t support; o t h e r legend a s f o r F i g . 4 . 2 .

Fig.4.7

A s q u a r e p l a t e w i t h s t e p change i n t h i c k n e s s . c / a = 0.5 h / h = 0.6 2 i


No

rr三 丨 厂 令i •i

]—

I_ (• No

if -*P- c-=

,

= * n

N.

I

L

丨 丨 丨

! 三

y-岭

f -~-z

N<

i'命 二!

I r ' - i r Sqi fo i F

<_

i•

(IT 々 •

a t e s s u b j e c t e d t o u n i a x i a l l o a d s . Legend as .2.

••-

internal line support at mid-span

(b) Fig.4.9

Square p l a t e s w i t h i n t e r m e d i a t e line (a) C a s e s 1 - 4 (b) C a s e s 5 - 8 ( c ) Cases 9 - 1 2

(c) supports


70

CHAPTER 5 A 2 - D FINITE STRIP USING COMSFUN I n t h i s c h a p t e r , t h e COMSFUN i s adopted a s t h e l o n g i t u d i n a l t r i a l functions of

a

new f i n i t e s t r i p f o r

2 - D e l a s t i c problems.

Unknown

parameters a r e r e p r e s e n t e d by t h e a c t u a l d i s p l a c e m e n t v a r i a b l e s a t t h e nodes. The s t r i p I s

-continuous i n the l o n g i t u d i n a l d i r e c t i o n w i t h i n

i t s e l f and C ^ - c o n t i n u o u s a c r o s s b o u n d a r i e s between a d j a c e n t s t r i p s . Deep beams w i t h i n m u l t i p l y - c o n n e c t e d domains c a n be t r e a t e d i n a s t a n d a r d finite

element

manner.

Several

numerical

examples

are

given

to

demonstrate i t s a c c u r a c y i n s t a t i c and v i b r a t i o n p r o b l e m s . 5-1 Formulation A 2 - D p l a n e f i n i t e s t r i p i s shown i n F i g . 5 . l a . I n F i g . 5 . l b , a u n i t w i d t h o f the s t r i p i s taken out as

a

b e n d i n g r i g i d i t y . The beam i s d i v i d e d i n t o p beam e l e m e n t s w h i c h a r e n o t n e c e s s a r i l y o f e q u a l l e n g t h . A number o f j o i n t s s a y , r , a r e a s s i g n e d as nodes. U n l i k e t h i n - p l a t e p r o b l e m s , t h e f i r s t d e r i v a t i v e o f t h e i n - p l a n e d i s p l a c m e n t s a r e n o t i n v o l v e d i n t h e k i n e m a t i c boundary c o n d i t i o n s o f 2-D

problems.

As

a

consequence,

no

restraint

is

applied

to

the

r o t a t i o n a l d e g r e e o f freedom a t e a c h end o f t h e f i c t i t i o u s beam; t h a t is,

t h e boundary c o n d i t i o n a t

e a c h end o f

the f i c t i t i o u s

beam a r e

e i t h e r , r e s p e c t i v e l y , s i m p l y s u p p o r t e d o r f r e e when computing t h e shape f u n c t i o n s f o r t h e i n t e r n a l nodes and t h e c o r r e s p o n d i n g end node. A t o t a l o f r computed shape f u n c t i o n s a r e u s e d i n f o r m u l a t i n g e a c h o f t h e two t h e i n - p l a n e d i s p l a c e m e n t components. The d i s p l a c e m e n t f u n c t i o n s o f

a 2 - D p l a n e f i n i t e s t r i p can be

written as r

u(x,y) =

Y L( m= 1

Y Lf r

v(x,y) =

m=1

ä¸&#x20AC;

[M(x)] Ym(y) { a

}

(5.1a)

â&#x20AC;&#x201D;

[H(x) ] Ym(y) {

/3

}

m

( 5. 1 b )

where [M(x)] and Y (y) denotes the linear finite-element shape functions m and the longitudinal COMSFUN for 2-D plane problems respectively. One can easily show

that

the rigid-body and constant-strain criteria are

exactly satisfied. It is important to recognize that the unknowns {a} and {/3 } in the m m above expressions represent the actual in-plane displacements at the


nodes ( s e e F i g . 5 . 2 a ) , t h a t i s ioc}

= { u , m 1 a consequence,

As

> and 2 m m 1 2 m the implementation o f support c o n d i t i o n s

and t h e

a s s e m b l y o f f i n i t e s t r i p s c a n be done i n t h e same f a s h i o n a s i s done f o r the

usual

finite

-continuous

element

across

method.

the

Moreover,

boundaries

between

each f i n i t e adjacent

strip i s

elements.

To

c l a r i f y the i d e a o f c o n t i n u i t y , an assemblage o f f i n i t e s t r i p s w i t h 5 l o n g i t u d i n a l nodes p e r s t r i p i s

shown i n F i g . 5 . 2 b .

m a i n t a i n e d a c r o s s t h e i n t e r f a c e between s t r i p

C0-continuity i s

1 and s t r i p 3 because

displacements f o r both s t r i p s along the i n t e r f a c e are uniquely defined b y t h e d i s p l a c e m e n t s a l o n g t h e same n o d a l l i n e p k . The same p r i n c i p l e c a n b e a p p l i e d t o v e r i f y t h e C ^ - c o n t i n u i t y a c r o s s t h e i n t e r f a c e between s t r i p 1 and s t r i p 2; a s COMSFUN i s d e f i n e d i n a manner s i m i l a r t o t h a t o f t h e L a g r a n g i a n s h a p e f u n c t i o n s , t h e d i s p l a c e m e n t s o f t h e two s t r i p s a l o n g the i n t e r f a c e ,

even though t h e y a r e connected end-to-end,

are

d e f i n e d u n i q u e l y b y t h e d i s p l a c e m e n t s a t t h e i r common n o d e s j and k . Put i n o t h e r w o r d s , t h e f i r s t d e r i v a t i v e o f t h e COMSFUN i s d i s c o n t i n u o u s a t node j ,

see F i g . 5 . 2 c .

Each f i n i t e s t r i p i s ,

therefore,

C。- continuous

around all edges. The strain-displacement relations are given by

{ cp

}

=f u

[B p ] { 6 p } m m

(5.2)

[M],?

where

[Bp]

[M]

0 [M] Y'

and

(5.3)

?,

n

[M]'Y

{ 5 P > = { { a } , { / 3 } >T = { ( u , u ) , ( v , v ) }T m m m 1 2 1 2 m

I t c a n be shown t h a t t h e s t r i p s t i f f n e s s m a t r i x i s o f t h e f o r m

11

[Kp]io

[Kp]

[Kp] [Kp] ^ r1

w i t h [Kp]

[KPL

12

[Kp]

[K p ]

o

r2

= f [BP]T[DP] [Bp]

(5.4)

dx dv

(5.5)


72

[A] K 2 ] [ 《 ] [ K ^ ] and

( 5 . 6 )

[K

D

Y Y dy 11 m n

[Mr^Mrdx +

[K^ ] = D 1 Y Y’ dy 12 mn i 12 m n

21

[K1 22

2

H

i

d

y

[M],[M] dx +

[M]T

J

D _ Y , Y , dJy

22 m

[M]

n

[M]^dx

+

U 3 3 m Y,n dy

[M]

[M] dx

D”Y’ Y dy 33 m n

[M]1 [MPdx

| D 3 3 Y m Y; dy

[M]’ 1 [M] dx

D, Y Y dy 33 m n

[M],[M]’dx

J

[M] dx +

For plane-stress analysis of isotropic material, the entries in the elasticity matrix [Dp] are : = D 2 2 = D = E t / (1D

33

12

= G

For vibration, the consistent mass matrix can be written as [M p ]

11

[Mp]

[M p ]

12

[Mp]

[Mp] [M p ] ,

[ I

[Mp]

C

11

with [M p ]“ = [M p ] 11

22

[Mp]

(5.7) [Mp]

r2

(5.8)

0 [M「 22 t Y

Y dy

[M]

[M] dx

(5.9)

where p and t = d e n s i t y and t h i c k n e s s o f t h e m a t e r i a l r e s p e c t i v e l y I n t e g r a t i o n s o f t h e s t i f f n e s s and mass m a t r i c e s a r e c a r r i e d o u t o v e r t h e e n t i r e s u r f a c e o f t h e s t r i p . The e x p l i c i t f o r m o f i n t e g r a t i o n s


73

o f COMSFUN has been d i s c u s s e d i n C h a p t e r 2. L i n e a r shape f u n c t i o n s and i t s i n t e g r a l a r e i n c l u d e d i n t h e A p p e n d i x . Note t h a t [K p ]

and [M p ]

mn

mn

a r e 4x4 m a t r i c e s .

Load vector The l o a d v e c t o r o f a s t r i p w i t h l a t e r a l l o a d f ( y ) per u n i t l e n g t h ( a l o n g X - d i r e c t i o n ) a c t i n g on n o d a l l i n e 1 i s g i v e n by {F} = { {F} {F}

whc 厂 e

, {F}

’ {F}

(5.10)

)1

, { F }

f ( y ) [ 1 0 0 0] Ym(y) d x

(5.11)

I f we assume t h a t t h e l o a d i n g i s s t e p w i s e c o n s t a n t l o n g i t u d i n a l l y ( the loading,

d e n o t e d by

f

k

, is

a c o n s t a n t w i t h i n element

k of

the

f i c t i t i o u s beam ), then the following load vector can be derived,

{F}J

f

where

{ a } m

k

(1 /2

denotes

the

1/12

- 1 / 1 2 ) { oc } }

1 /2

joint displacements

of

the

(5.12)

fictitious beam

element k for the corresponding m-th COMSFUN • 5.2 Numerical Examples A deep beam

with both ends clamped

(1/=0. 15, E=1. 0, thickness-1. 0, span = 2.0, depth = 1. 0) A deep beam with both ends clamped analyzed

( Fig.5.3). The beam

(i.e. u,v = 0 at both ends) is

is subjected

to a uniformly distributed

load ( q=l ) on the top surface. Because of symmetry, half of the beam is

discretized

into

eight

finite

strips

through

the

depth

with

increasing number of nodes along the span. The boundary conditions at the clamped ends ( u, v=0 at y=0) and mid-span ( v=0 at y=l) can be easily imposed in the usual finite element manner. Results of deflections are summarized

in

Table

5,1

and

solutions using 8 strips and both methods are

compared

with

the

spline

finite

strip

11 sections[13]. Results obtained using

in very good

agreement. Convergence of

stresses are

given in Fig.5.4 and compared with the spline finite strip solutions and the finite element

solutions.

At

mid-span of

the

beam, 5 nodes are

sufficient to give very accurate stresses. However,at the top and bottom surfaces

of

the

clamped

end

where

the

stresses

are

theoretically

infinite, 9 nodes are required to pick up the sharp changes in stresses.


I n t h e r e g i o n s away f r o m t h e c o r n e r s o f t h e clamped end, t h e agreement between t h e s o l u t i o n s o b t a i n e d b y t h e t h r e e d i f f e r e n t methods a r e i n v e r y good agreement. I f t h e s t r e s s e s a t t h e c o r n e r s a r e n o t o f i n t e r e s t , t h e n f i v e nodes p e r n o d a l l i n e a r e s u f f i c i e n t t o g i v e a c c u r a t e r e s u l t s . Number o f

D e f l e c t i o n a t x=0,y=l

nodes a l o n g s p a n 5

2.108

7

2.119

9

2.124

spline strip[13]

2.125

T a b l e 5 . 1 . D e f l e c t i o n s o f a deep beam

A deep A deep c a n t i l e v e r (u=0=v a t one e n d a n d f r e e a t t h e o t h e r end) i s analyzed using

12 s t r i p s w i t h

5 or

9 nodes p e r n o d a l l i n e . I t i s

s u b j e c t e d t o a u n i f o r m l y d i s t r i b u t e d l o a d (q=l p e r u n i t l e n g t h ) o n t h e t o p s u r f a c e . L o n g i t u d i n a l s t r e s s e s a r e compared w i t h s p l i n e f i n i t e s t r i p s o l u t i o n s , f i n i t e element s o l u t i o n s and t h e o r e t i c a l s o l u t i o n s ( a l l as q u o t e d b y F a n [ 1 3 ] ) . I t i s a p p a r e n t f r o m F i g . 5 . 5 a t h a t t h e f i n i t e element a n d s p l i n e f i n i t e s t r i p methods p r e d i c t h i g h e r s t r e s s e s a t t h e c o r n e r s of

the

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

Bares

and t h e

p r e s e n t f i n i t e s t r i p s o l u t i o n s ( a c t u a l l y , t h e a n a l y t i c a l s o l u t i o n s and the present f i n i t e s t r i p s o l u t i o n s a r e almost i d e n t i c a l a t the corners). Away

from

these

corners

(Fig.5.5b),

one

can

see

no

significant

d i f f e r e n c e s between t h e s o l u t i o n s o f t h e d i f f e r e n t methods.

A deep cantilever

with

step

change

in depth

(

ÂŁ=2,v=0.3,

To demonstrate the application of the present finite strip to 2-D multiply-connected domainsďź&#x152; a deep cantilever with step-change in depth is analyzed

( Fig. 5.6). The beam

is subjected to a point load at the

free end. It is discretized into 15 finite strips with increasing number of nodes longitudinally. The boundary conditions at the clamped end

(

u,v=0 at y=0) can be easily imposed in the usual finite element manner. Convergence of deflections are summarized in Table 5.2 together with the finite

element

results

using

150

QMS

incompatible

elements[70].

Convergence of stresses are given in Fig.5.7 together with the finite

thickness


75

element r e s u l t s .

The 5 - n o d e s s o l u t i o n s a r e , i n g e n e r a l , s u f f i c i e n t l y

a c c u r a t e when compared w i t h t h e f i n i t e element s o l u t i o n s e x c e p t i n t h e v i c i n i t y o f t h e g e o m e t r i c d i s c o n t i n u i t y where 7 nodes a r e r e q u i r e d t o capture the sharp v a r i a t i o n o f s t r e s s e s . Number o f

D e f l e c t i o n a t x=0,y=20

nodes/element 89.86 92.92 93.06 93.85 T a b l e 5 . 2 . D e f l e c t i o n s o f a deep beam w i t h change i n d e p t h

A deep To assess the importance of C o -continuity in analyzing 2-D plane problems, a beam, with step-change in thickness» is analyzed. Two meshes are used

: in Fig. 5.8a,

within each

10 finite strips are used with

strip; In Fig. 5.8b,

the

beam

is divided

-continuity

into 20 finite

strips with the same number of unknowns as the mesh of Fig.5.8a. Notice that, for

the mesh

in Fig. 5.8b,

it

is only

-continuous across the

edges of each strip and hence (^-continuity can be maintained across the geometric discontinuity. The boundary conditions at the clamped end u,v=0

at

y=0)

can

be

easily

imposed

manner.Deflections are summarized finite

element

results

using

in

the

usual

finite

(

element

in Table 5.3 and compared with the 200

QMS

incompatible

elements[70].

Deflections obtained using both meshes are in very good agreement with that of finite element method. Convergence of

stresses are given

in

Fig.5.9 together with the finite element results. One can see that the stresses obtained by the second mesh are much more accurate than those obtained

by

results; the

the

first

situation

mesh

when

compared

is much more

with

obvious

In

the

finite

terms

of

stresses. This demonstrates that siginificant improvement

element

the

shear

in stresses

can be achieved if C0-continuity is maintained at the location of sudden change in thickness.


76

Mesh

D e f l e c t i o n a t x=0,y=20 360. 1 363. 1 368.5

T a b l e 5 . 3 . D e f l e c t i o n s o f a d e e p beam w i t h change i n t h i c k n e s s

A bar under

tension

(1/=0. 0, E=1. 0)

A b a r , w i t h s t e p change i n t h i c k n e s s , i s c l a mpe d a t one end and s u b j e c t e d t o a u n i f o r m l y d i s t r i b u t e d t e n s i l e f o r c e a t t h e o t h e r end. Two d i f f e r e n t d i s c r e t i z a t i o n a r e used

:

two 3 - n o d e s f i n i t e s t r i p s ; o r

a

5-nodes f i n i t e s t r i p , see F i g . 5 . 1 0 . S t r e s s e s a t d i f f e r e n t l o c a t i o n s are computed a n d compared w i t h t h e t h e o r e t i c a l v a l u e s . From T a b l e 5 . 4 , t h e importance o f ^ - c o n t i n u i t y a t the l o c a t i o n o f d i s c o n t i n u i t y i s c l e a r l y demonstrated. Mesh

S t r e s s e s cr y=0

y=0•5L

two 3-node elements

1.00

one 5-node element Theoretical values

y

y=0•5L

y=12

1.00

0.50

0.50

0.99

0.73

0.73

0.53

1.00

1.00

0.50

0.50

Table 5.4. Axial stresses of a non-prismatic bar (*〕Just left of discontinuity (**) Just right of discontinuity. Thickness of bar = 1.0 for ys 0.5L and 2.0 for y^0.5L where L is the bar length.

Free vibration of deep beams (£=1.0,

1/=0. 3,

For examining

thickness=l. 0, span=10, depth=2.5, density=l. 0)

the accuracy of

the present finite strip for free

vibration problems, the first four vibration frequencies of a deep beam with different boundary conditions are computed and compared with finite element solutions.

The mode

sliding base are also shown

shapes of

the

in Fig.5.11.

clamped-clamped beam with In each case,the beam

is

divided into 5 strips through the depth and 9 nodes along the span. For finite element analysis,

the beam

is divided

into

100 four-node QM6

elements with 5 element layers through the depth. Comparsion of results are given in Table 5.5

; the r e s u l t s obtained by the present f i n i t e


77

s t r i p method, w i t h o n l y 43 p e r c e n t o f t h e number o f unknowns o f the f i n i t e element a n a l y s i s ,

are l e s s than

1 p e r c e n t d i f f e r e n t from the

f i n i t e element r e s u l t s i n most c a s e s ( e x c e p t f o r mode 4 o f t h e second and t h i r d c a s e s where a shapes

of

Fig.5.11

1.5 percent d i f f e r e n c e i s found.)

reveal

the

significance

of

The

transverse

mode

normal

d e f o r m a t i o n i n deep beam a n a l y s i s . T h i s example a l s o demonstrates t h e v e r s a t i l i t y o f t h e new f i n i t e s t r i p i n d e a l i n g w i t h d i f f e r e n t boundary conditions. V i b r a t i o n frequencies Mode sequence Clamped - F r e e F.E.M.

0.02452

0.12486

0.15770

0.28773

0.02457

0.12555

0.15762

0.29059

0.12246

0.27130

0.31625

0.44774

0.12235

0.26963

0.31623

0•44227

0.09284

0.15770

0.24500

0.42320

0.09329

0.15762

0.24727

0.42977

0.15789

0.45989

0.61960

0.64312

0.15786

0.46066

0.62217

0.64345

0.31654

0.58592

0.64661

0.68702

0.31666

0.58817

0.64871

0.68776

Clamped-Clamped F.E.M. Clamped-S.S.

C l a m p e d - S . S . - S l i d i n g base • M. Clamped-Clamped-Sliding base .M.

Table 5.5. Frequencies of a deep beam with various boundary conditions * boundary conditions at y=0 ; Clamped : u=v=0

** boundary c o n d i t i o n s a t y=10.0

S . S . : u=0, v^0

Free : u^O, v^0

Sliding base : u=0 along x=2.5 Shear wall supported on portal frame [Fig.5. 12) The present finite strip is also applied to the analysis of a shear wall with a portal frame connected

to

its base. The portal frame is

modelled with the usual beam elements. To ensure full compatibility of rotations between beam elements of the frame and finite strips of the shear wall, the latter is divided into finite strips with the following modified displacement functions,


78

where

u(x,y)

[M(x)] Ym(y) { a }

(5.13a〕

v(x,y)

[N(x)] Ym(y) {

〔5.13b)

/ 3 }

[N(x)] denotes the beam shape functions. Accordingly, the nodal

displacements are modified as {oc}

= { u., , u n }

and {p}

= { v

dv/dx

m 1 2 m m 1 1 2 2 m where dv/dx. represents the nodal rotation of vertical fibres of the shear wall. Because of the addition of in-plane rotation at each node, the stiffness matrix of each of the beam elements of the portal frame can be

directly

manner,

onto

attaches.

superimposed,

the

(In

in

the

stiffness matrix

the

usual

of

semi-analytical

the

finite

standard

finite

finite

strip

strip

element

to which

method,

the

it

beam

stiffness matrices are first transformed and subsequently added onto the corresponding

finite

strip

stiffness

matrices).

Formulation

of

the

stiffness matrices follows the same track as before. The

accuracy

of

assessed by comparing analytical

finite

Experimental

the

finite

strip

in

this

example

is

its solutions with those obtained by the semi-

strip

results

present method

are

also

and

the

adopted

for

finite

element

comparison.

In

method. the

semi-

analytical finite strip method, the COMSFUN in (5.1) is replaced by the analytical

series

of

vibration

of

a

vertical

cantilever

with

an

elastically-supported base. Furthermore, the finite element results are computed using a 4-node beam-type membrane element with nodal in-plane rotations. Details of the semi-analytcial finite strip and the beam-type finite element can be found in [132]. Vertical deflections

and

stresses

are

depicted

in

Fig.5.13.

As

sharp variation of stresses are expected in the proximity of the wallframe interface, comparison is made at a small distance away from the centre-line using 5

of

the

transfer beam.

The present finite

longitudinal nodes with non-uniform spacing

compared with

the five-terms solutions of

strip results,

(Fig.5.12c), are

the semi-analytical finite

strips. Together with the finite element and experimental results, four sets

of

results

are

shown

deflections can be seen

in

Fig.5.13b.

are

in

agreement

of

in Fig. 5.13a. For the stresses, however, the

present finite strip results, especially columns,

Reasonable

better

agreement

in the area right above the

with

the

finite

element

and

experimental results than those of the semi-analytical finite strips.


79

T h i s may be p a r t l y a t t r i b u t e d t o t h e f a c t t h a t t h e l o n g i t u d i n a l s p a c i n g o f the nodes i n t h e p r e s e n t f i n i t e s t r i p c a n be a d j u s t e d a p r i o r i i n t h e a r e a where s h a r p v a r i a t i o n o f s t r e s s e s i s a n t i c i p a t e d . 5 . 3 Cone1usions The f o r m u l a t i o n and a p p l i c a t i o n s o f COMSFUN a r e

presented.

Its

accuracy

i n t h e g i v e n n u m e r i c a l examples. comparable

accuracy

and

a

2-D f i n i t e s t r i p u s i n g

and v a l i d i t y

are

demonstrated

The p r e s e n t f i n i t e s t r i p possesses

efficiency

with

respect

to

spline/semi-

a n a l y t i c a l f i n i t e s t r i p s , and s u p e r i o r v e r s a t i l i t y i n t h e sense t h a t , unlike a l l other

2~D f i n i t e s t r i p s ,

the present

one i s c a p a b l e o f

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

manner.

We

note

that

the

present

finite

strip

can

be

i n t e r p r e t a t e d a s a r e c t a n g u l a r h i g h - o r d e r f i n i t e element. Appendix

Linear

shape

functions

The l i n e a r shape f u n c t i o n s a r e g i v e n b y M(s) =[(1 - s / i ) , s / I ]

(5.14)

=[M , M ] 1 2 where 』 = l e n g t h of element and s is

the

element

coordinate running from 0 to I

They also satisfy the identity given by M + M = 1 1

The

(5.15)

2

integrals

of

the products of

easily worked out.

[M]

[M] ds

= 1/6

[M]’ [ M p d s

= 1/1

[M] ,T [M] ds

= 1/2

2 1

f-l

-1 1 一

1

these

shape functions

can

be


P0u6jsscd0)

TJ

① u

o 匚

c o 4> -

"o. O L 03

rt N

•沿

」 s

Hh—« CO C CD •田二 CD a> x z <D — V— M — — o c “ 0 O g -I—• a3 J Z "H 0 2 as ^ <D o …

z

B

4> 0 u o M •H T J W •H a •h -d u c -p rt . w e— < D RT < D -P U < Tl -J XI o 仁 •h W (DO ^o C -P ¾( «H >1 rH 4-> a o s: a ^ £ i cn <N 0 X X: o < H U J T < VP-/ XI o

W


edols

SOJOCCJIJCUC!

OJaツァ

SSOJOOS

^

v V

O

ine

d!縲考s

(l

(U

CO

a

CD

(a)

p i c a l s t r i p w i t h 5 nodes on ssemblage o f t h r e e f i n i t e s e c t i o n o f COMSFUN a t node j

Fig.5.2

pedEBo6up|s

TJedlucclo

髣ィode

5 nodes 7 nodes Fig.5.3

A deep beam with both ends clamped. Half of the beam Is divided into 8 strips and increasing number of nodes on each nodal line.


82

,6

0.5 .4 0.3

2

(/) LJ if) {/)

U o t cr

1 0 1 •0.2

•0.4 •0.5

0

01

0.4

0.2

0.5

0.6

0.7

0.8

0.9

THROUGH-THICKNESS COORDINATE 口

5 NODES

+

7 NODES

9 NODES

FEM

X

SFS

(b)

4

2

1 (/) d I U) if) u if)

0 1 2

4

0

0.1

0.6

0.4

0.7

0.8

0.9

THROUGH-THICKNESS COORDINATE 口

5 NODES

Fig.5.4

+

Stresses distribution across the depth (a) < r atJ y=0 , (b) c r at y=0 , 〔 c ) y x y xy (d) o r at y=l , (e) < r at y=l X

V

at y=0 ,


(c)

2 .9 .8

.7 .6

,5 .4 .3 .2

(/) U (/) 00 Ld U1

1 1 0.9 0.8 0.7

0.6 0.5 0.4

0.2

1 0

0

01

0.3

0.2

0.4

0.5

0.6

0.7

1

0.8

THROUGH-THICKNESS COORDINATE 口

5 NODES

7 NODES

9 NODES

FEM

X

SFS

(d) i 0.9 0.8 .7

{/) u (D (/) U O C

.5 .4

2 . 1 0

01

0.4

0.2

0.5

0.6

0.7

0.8

THRDl ir:H-THir.kNF^^ nOORDlNATF 5 NODES

NODES

9 NODES

FEM

X

SFS


sbjssbjcels

0.1

0.2

0.3

0.4-

0.5

0.6

0.7

THROUGH-THICKNESS COORDINATE 口

0.8

0.9


- 1 0

9

10

11

12

X— c o o r d i n a t e a c r o s s d e p t h 5 nodes

9 nodes

^

FEM

A

Bares

X

SFS(12*8)

SFS(12*5)

10

11

12

X—coordinate a c r o s s depth nodes

nodes

Fig.5.5

'

FEM

Bares

SFS(12^5)

S t r e s s e s d i s t r i b u t i o n across the depth (a)

SFS(12 + 8)


Fig.5.6

A deep c a n t i l e v e r w i t h s t e p - c h a n g e i n d e p t h a t y=10. I t i s d i v i d e d i n t o 15 f i n i t e s t r i p s a n d 7 nodes p e r nodal l i n e .


w 0 ) w w 0 )

X—coordinote a c r o s s d e p t h •

5 nodes

+•

7 nodes

O

9 nodes

fern

(b〕

0.5

•0.5

4

S nnH#ac; Fig.5.7

7

6

8

9

10

X — coordinate across depth

+ 7 node?;

9 nodes

A

fem

Stresses distribution across the depth (a) < r at y=0 (b) c r at y=9.5 y y


C o- c o n t i n u o u s (a)

20

10

Co-continuous â&#x20AC;˘ = node (b) 5.8

A deep cantilever with step-change in thickness at y=10. Thickness t = 0.1 for y^lO and t = 0.2 for y>:10. (a) FS1 : A mesh of 10 strips with 9 nodes per nodal line is used, (b) FS2 : A mesh of 20 strips with 5 nodes per nodal line.


8S

(a)

4

3

2

1

U) )V /( tn 0 u 7 ()

0 1

• 2

3

.4

0

1

2

4

5

6

7

8

9

10

X—coordinate •

FEM

FS1

O

FS2

(b) 6 5 4

2

1 0 0 ))/ C U) U< L. 75 (

1

0

1

2

7

4

9

10

X-coordinate

Fig.5.9

FEM

+

FS1

O

FS2

Stresses d i s t r i b u t i o n across the depth (a) cr a t y=10 (b) cr a t y=10 ( c ) x a t y=10 y H y L xy R (d) x a t y=10 xy J L 10^ = j u s t t o t h e l e f t o f y=10 10

= j u s t t o t h e r i g h t o f y=10


90

X—coordinate FEM

FS1

O

FS2

(d)

S0SS0 J

X—coordinate FEM

FS1

FS2

L


5-é&#x2014;¨odes strip

t w o 3 - n o d e s strips Fig.5.10

F i n i t e s t r i p meshes o f a b a r w i t h step-change i n c r o s s - s e c t i o n a t mid-span.

Fig.5.11

V i b r a t i o n modes o f a deep beam


92

s

M

s-p JO

•GoBJJns

Jcdtulls03

PJe-pori

95 jo do XIctiM

i u e l u e l a )

(

q

.aTJ4->sCDwfudJ:+^d0s9Ja

)

I

sg.e 9 L ' Q l

C

1

i cS

! >

u!s.0 u!o.e

u!sm

S21S0*Z SZ89•廿 9/89*17

spou IB


93

(a) 0.9 0.8

0.7 0.6

0.5

o 6 * C D <

E 0 ) u 0 a 0 ) Q

0.4 0.3 0.2

0.1

10

12

14

16

18

20

22

Height above ground,in. FS

FEM

SAFS

Experiment

(b) 90 80 70 60 50 40 30 20 in 0 ) 0 0( ( 0 ) to

0 V 亡 . 0 ) >

10

0 - 1 0 -20

-30 -40 -50 -60

-70 -80

-90

D i s t a n c e , in. SAFS

Fig.5.13

FEM

Exper.

FS

(a) Vertical deflection along the height of the wall fb)

Vertical

stresses

s e c t i o n o f the w a l l a t 4in.

distribution

across

from ground s u r f a c e .

the


94

CHAPTER 6 VIBRATION AND BUCKLING OF THIN-WALLED STRUCTURES F o r a n a l y z i n g and d e s i g n i n g p r i s m a t i c t h i n - w a l l e d s t r u c t u r e s , t h e f l a t - s h e l l f i n i t e s t r i p method i s one o f t h e b e s t c h o i c e s a v a i l a b l e t o a n e n g i n e e r . The advantages o f t h e method r e l a t i v e t o t h e f i n i t e element method have been d i s c u s s e d i n C h a p t e r 1. The d i s p l a c e m e n t f u n c t i o n s o f a f l a t - s h e l l f i n i t e s t r i p a r e made u p o f two p a r t s , namely, t h e i n - p l a n e d i s p l a c e m e n t f u n c t i o n s and the o u t - o f - p l a n e d i s p l a c e m e n t f u n c t i o n . The d i s p l a c e m e n t f u n c t i o n s , based on COMSFUN, have been i n d i v i d u a l l y s t u d i e d and

reported i n previous

chapters.

In this

combined and e x t e n d e d t o t h e development o f

chapter,

they w i l l

be

a flat-shell finite strip

w h i c h w i l l be s u b s e q u e n t l y used t o s t u d y t h e v i b r a t i o n and b u c k l i n g o f folded p l a t e s w i t h varying complexity. 6.1 Formulation By r e f e r r i n g t o t h e l o c a l c o - o r d i n a t e s y s t e m shown i n F i g . 6 . 1 , t h e displacement f u n c t i o n o f

a flat-shell finite strip

element

can be

w r i t t e n as

u(x,y) =

Y

[M(x) ] Y ( y ) { a }

(6. l a )

v(x,y) =

Y

[M(x) ] Y ( y ) { p }

(6.1b)

f o r in-plane displacements and

w(x,y) =

r

Y U

[ N ( x ) ] Y (y) {

>

(6.1c)

f o r out-of-plane displacement where Y ( y ) and Y ( y ) d e n o t e r e s p e c t i v e l y t h e COMSFUN f o r t h e i n - p l a n e m m and o u t - o f - p l a n e d i s p l a c e m e n t components.[M(x)] and [ N ( x ) ] a r e 1-D linear

shape

functions

and

d i r e c t i o n . Notice that the

beam

shape

functions i n

the

local x -

same number o f nodes and i d e n t i c a l n o d a l

spacings a r e involved i n the i n - p l a n e

and o u t - Q f - p l a n e d i s p l a c e m e n t

f u n c t i o n s . The d i s p l a c e m e n t f u n c t i o n s have been d i s c u s s e d i n p r e v i o u s c h a p t e r s . T h e p r e s e n t s t r i p c o n t a i n s f o u r d i s p l a c e m e n t unknowns a t each node.


95

Having d e f i n e d the displacement f u n c t i o n s o f a f l a t - s h e l l f i n i t e s t r i p , one c a n w r i t e down i t s s t i f f n e s s m a t r i x i n l o c a l c o o r d i n a t e s .

[ K ]

t K ]

1 1

[K]

l

2

[K]

[K] [K] ,

[K]

(6.2) [K]

r2

[K P ]

w i t h [K]

[Ku]

0

(6.3) mn

S i m i l a r l y , t h e mass m a t r i x c a n be g i v e n a s [MP]

w i t h [M]

0

0

mn

[M b ]

(6.4) mn

where t h e e x p l i c i t f o r m s o f t h e s u b m a t r i c e s [K b ] P

, [KP] mn

, [Mb mn

and [M ] have been d e r i v e d i n p r e v i o u s c h a p t e r s . mn For b u c k l i n g a n a l y s i s , the strain—displacement r e l a t i o n s h i p taken as 1/2 { (u

) 2 + (v

) 2 + (w (w

1/2 { (u

):

(6.5a)

)'

(6.5b) (6.5c)

For a s t r i p subjected t o in-plane stresses

and T

, the

g e o m e t r i c m a t r i x c a n be e a s i l y d e r i v e d , g i v i n g [G]

[G P ]

0 (6.6)

IG1

0 where

K

[GP]= and L[G

p

] 11

[G^ ] 22

] 0

(6.7a) [G 22 t Y

Y dy Y, Y, dy

[ M r T C M ] ’ dx [M]T[M]

(6.7b)

dx

The out-of-plane component [Gb] has been presented in Chapter 4. mn The unknowns {a} ,{B } and in equations (6.1) represent the


96

p h y s i c a l nodal displacements i n l o c a l coordinates, that i s {a} {/3}

= ( v , v

{r}-

= ( w ,

)

0 , w ,

ä¸&#x20AC; 0

with 0

These l o c a l d i s p l a c e m e n t unknowns can be c o l l e c t i v e l y arranged i n the form o f {d}

m

= { u ,v ,v/ ,0 , u ,v ,w ,0 } 1 1 1 1 2 2 2 2 n

(6.8)

and then t r a n s f o r m e d t o the g l o b a l c o - o r d i n a t e system by the standard transformation, g i v i n g

{d}

where [ t ]

[t]

0

0

[t]

{D}

[THD}

cosa 0 s i n a 0 01 0 0 - s i n a ă&#x20AC;&#x2021; cosa 0 0 0 0 1

(6.9a)

(6.9b)

and {D} = nodal displacement unknowns in global coordinates. m The corresponding matrices are transformed in the usual manner, that is [K]*

= [T]T[K]

[T]

(6.10a)

[G]*

= [T]T[G]

[T]

(6.10b)

[T]

(6.10c)

[M]

[T]ä¸ [M]

A f t e r t r a n s f o r m a t i o n , the f i n i t e s t r i p m a t r i c e s a r e assembled and boundary c o n d i t i o n s a r e imposed. Subspace i t e r a t i o n was used t o e x t r a c t e i g e n - p a i r s from t h e r e s u l t i n g e i g e n - v a l u e e q u a t i o n . 6 . 2 N u m e r i c a l Examples The

main purpose o f

t h e f o l l o w i n g examples i s

t o v e r i f y the

v a l i d i t y o f t h e new f l a t - s h e l l f i n i t e s t r i p and t o a s s e s s i t s a c c u r a c y and e f f i c i e n c y by comparing i t s s o l u t i o n w i t h those o b t a i n e d by f i n i t e


97

element/finite vibration

strip

frequencies

method and

and/or

other

buckling

loads

structures w i t h varying complexity

numerical of

methods.

numerous

Free

folded-plate

and w i t h i s o t r o p i c o r

laminated

composite m a t e r i a l s a r e computed u s i n g t h e proposed f i n i t e s t r i p . The f i n i t e s t r i p meshes a r e chosen i n t h e l i g h t o f the e x p e r i e n c e gained i n p r e v i o u s c h a p t e r s . I n a l l c a s e s , u n l e s s o t h e r w i s e s p e c i f i e d , the nodes on each n o d a l l i n e a r e o f e q u a l d i s t a n c e a p a r t .

Vibration

of

Three t y p e s o f c a n t i l e v e r f o l d e d - p l a t e s a r e demonstrated i n t h i s example t o show t h e a c c u r a c y and e f f i c i e n c y o f t h e p r e s e n t f l a t - s h e l l f i n i t e s t r i p i n analyzing folded-plate

structures with significant

b e n d i n g a c t i o n i n t h e i r component f l a t p l a t e s . The l o w e s t f i v e f r e q u e n c i e s o f F i g . 6 . 2 are

t h e c a n t i l e v e r c h a n n e l shown i n

computed u s i n g 8 s t r i p s and 6 nodes.

The f i n i t e s t r i p

r e s u l t s a r e compared w i t h f i n i t e element s o l u t i o n s [70] u s i n g a mesh o f 160 f o u r - n o d e s h e l l elements, see T a b l e 6 . 1 . The f i n i t e s t r i p r e s u l t s a r e w i t h i n 1 p e r c e n t d i f f e r e n c e from t h e f i n i t e element r e s u l t s (except f o r mode 4 w i t h a d i f f e r e n c e o f 1 . 6 p e r c e n t ) . I t i s noteworthy t h a t the number o f unknowns f o r the f i n i t e s t r i p method i s o n l y 1 p e r c e n t o f the f i n i t e element a n a l y s i s . The f i r s t f o u r mode shapes o f t h e channel a r e a l s o g i v e n i n F i g . 6 . 3 from w h i c h one c a n observe t h e s i g n i f i c a n t bending a c t i o n i n each o f t h e f l a t p l a t e s . Mode i

Natural frequencies finite strip

f i n i t e element

0.5260

0.5230

0.9043

0.9041

1.1963

1.1988

1.4993

1.4758

1.8402

1.8480

Table 6 . 1 . F r e q u e n c i e s o f a c a n t i l e v e r channel I n a d d i t i o n t o the c a n t i l e v e r c h a n n e l , v i b r a t i o n o f o n e - f o l d f o l d e d plates

and t w o - f o l d f o l d e d p l a t e s w i t h d i f f e r e n t

crank

angles are

c o n s i d e r e d ( see F i g . 6 . 4 ) . U s i n g t h e p r e s e n t f i n i t e s t r i p , the lowest f i v e f r e q u e n c i e s a r e compared w i t h t h e f i n i t e e l e m e n t - t r a n s f e r m a t r i x s o l u t i o n s o f L i u [ 7 8 ] and summarized i n T a b l e 6 . 2 . F o r o n e - f o l d f o l d e d


98

p l a t e s , s i x s t r i p s and 6 nodes a r e u s e d w h i l e n i n e s t r i p s and 6 nodes a r e used f o r t h e t w o - f o l d f o l d e d p l a t e s . Good agreement i s observed; the maximum d i f f e r e n c e between t h e two s e t s o f r e s u l t s i s 2 p e r c e n t , and i n most c a s e s t h e d i f f e r e n c e i s w i t h i n 1 p e r c e n t . The number o f degrees o f freedom i n t h e s e examples a r e o n l y 168 and 240 f o r t h e o n e - f o l d and two-fold folded plates respectively.

Crank a n g l e , a

Mode i

F r e q u e n c i e s A.

(degrees) Present s t r i p one-fold folded plates 90 1

0.0491 0.0976 0.1788 0.2096 0.3508

Liu[78]

49ďź&#x161; 97ďź&#x161;

78< 08ă&#x20AC;? 55J

120

0.0491 0.0947 0.1789 0.2076 0.2933

0.0491 0.0943 0.1787 0.2065 0.2971

150

0.0492 0.0809 0.1790 0.1908 0.2207

0491 0812 1787 1912 2210

0.1245 0.1258 0.2598 0.2710 0.3222

1249 1260 2579 2692 3286

0.0980 0.1249 0.2589 0.2643 0.2927

1000 1241 2571 2630 2986

0.0675 0.1145 0.2078 0.2407 0.2585

0687 1145 2109 2415 2571

two-fold folded plates 90 1

2 3

4 5

Table 6.2. Frequencies of cantilever folded plates with different crank angles. A.= w.L [ p{l-v2) / E ]1/2. v=0.3


99

Vibration and hackling ( ÂŁ = 1 9 2 fv=0.3 )

of

The free vibration and buckling under uniform compressive stresses of continuous beams are considered. Although the beams can be analyzed by the usual beam elements, they are treated as folded plates in this example. The cross-section of

the beam, which was modelled with five

finite strips, is an I-section made up of two equal flanges of width 0.5 and web of 1.0 and a constant thickness of 0.1 throughout. In contrast to

the

cantilever

folded-plates

in

previous

examples,

significant

membrane action is expected in the component flat plates of the beams. The boundary conditions at

the end

supports and at

the

intermediate

support are modelled as rigid in-plane diaphragm. These conditions can be easily imposed, in the usual finite element manner, by constraining the in-plane degrees of freedom at the corresponding locations in the global

stiffness

conditions splines priori

require

while for

matrix

(

tediuos

In

modification

continuous-beam

the

spline

vibration

semi-analytical

finite

finite of

strip

the

method,

these

corresponding

local

functions strip

are

determined

method.)

Results

a are

compared with beam theory[76] in Table 6.3. Agreement between the two sets of results is found

to be satisfactory, as only a maximum of 2

percent difference is found. It is worth mentioning that in cases where the internal diaphragms at the supports are flexible, the in-plane stiffness can be taken into account by dividing the diaphragms into the usual 2-D finite elements or boundary elements. After applying condensation, the stiffness matrix of the

diaphragms,

corresponding

to

the

degrees

of

freedom

of

the

connecting nodes between the diaphragms and the folded-plate structure, can be directly superimposed onto the global stiffness matrix. Unlike semi-analytical

finite

strips,

no

transformation

of

the

diaphragm

stiffness matrix is required as the nodal degrees of freedom between the diaphragms and the folded-plate structure are fully compatible. ä¸&#x2018; To demonstrate t h e e f f i c i e n c y and accuracy o f the p r e s e n t f i n i t e s t r i p with respect t o

the s e m i - a n a l y t i c a l f i n i t e s t r i p ,

a two-span

c o n t i n u o u s s t i f f e n e d p a n e l i s examined. D e t a i l s o f t h e s t r u c t u r e i s g i v e n i n F i g . 6 . 5 . I n t h e s t u d y conducted by D e l c o u r t and Cheung [79] using

the

semi-analytical

finite

strip

method,

a

three-strip

i d e a l i z a t i o n was adopted because o f t h e r e p e t i t i v e b u c k l i n g p a t t e r n s .


100

Because o f symmetry about t h e i n t e r m e d i a t e support, o n l y one span, w i t h d i f f e r e n t span l e n g t h , i s s t u d i e d .

The present f i n i t e s t r i p r e s u l t s

u s i n g t h r e e s t r i p s and 9 nodes a r e compared w i t h the s e m i - a n a l y t i c a l f i n i t e s t r i p u s i n g t h r e e s t r i p s and n i n e terms o f the v i b r a t i n g beam s e r i e s . Only

1 p e r c e n t d i f f e r e n c e i s found between the

r e s u l t s i n Table 6 . 4 .

two s e t s o f

The p r e s e n t f i n i t e s t r i p , t h e r e f o r e , possesses

comparable e f f i c i e n c y and a c c u r a c y w i t h r e s p e c t t o the s e m i - a n a l y t i c a l f i n i t e s t r i p i n t h i s example. Present f i n i t e s t r i p V i b r a t i o n frequencies

Beam theory[76]

two-span c o n t i n u o u s beam 0.5718 0.8903 2.2488 t h r e e - s p a n c o n t i n u o u s beam 0.2265 0.5803

mode 1 mode 2 mode 3 mode 1 mode 2 Buckling stresses

two-span c o n t i n u o u s beam 0.8281 1.6760 t h r e e - s p a n c o n t i n u o u s beam 0.4323

mode 1 mode 2 mode 1

0.5692 0.8896 2.2770 0.2224 0.5692 0.8210 1.6773 0.4350

T a b l e 6 . 3 . V i b r a t i o n and b u c k l i n g o f c o n t i n u o u s beams F o r t h e two-span beam L i = F o r t h e t h r e e - s p a n beam L1 = L3 = 5 and L2 = 10 spanwise n o d a l arrangement ďź&#x161; 3 nodes within each span and one node at each support, that is, 9 nodes and 13 nodes per nodal line for the two-span and three-span beams, respectively. All nodes within each span are of equal distant apart.

Critical stresses (N/mm ) Span(mm) 2770 2900

Present finite strip 258.8 258.6

Table 6.4. Buckling of a stiffened panel

Delcourt [79] 256.14 255.75


101

Vibrat ion of a four-stiffener A

simply-supported

Although

it

is

more

pane

four-stiffener

panel

efficient

analyze

to

is

given the

in

panel

Fig.6.6. by

the

semi-analytical finite strip (with complete decoupling of the stiffness matrix corresponding to each term of the trigonometric series),

it is

used to examine the accuracy of the present finite strip for computing higher

frequencies

of

prismatic

thin-walled

structures.

The

present

results of the lowest twenty frequencies are given in Table 6.5 together with

the

' exact'

finite

strip

solutions

due

to

Wittrick

and

Williams[80]. 3-D view of the first seven mode shapes are also given in Fig.6.7 from which one can see the coupling of the transverse bending and longitudinal bending. Despite the fact that a relatively coarse mesh with twelve strips and 6 nodes is intentionally used

to predict the

deformation patterns of the complicated mode shapes, reasonable results can still be obtained. Except for modes 17,19 and 2 0 , t h e differences between the two sets of results are less than 6 percent. A maximum of 10 percent difference is found at mode 20.

Mode

Frequencies W i t t r i c k [80]

入 彳 Present s t r i p

61921504956125794184 89569910153444445360

22333345556666666778 o ooooooooooooooooooo oooooooooooooooooooo

01234567890 2345678911111111112

0.0302 0.0307 0.0384 0.0385 0.0413 0.0414 0.0431 0.0530 0.0547 0.0584 0.0669 0.0677 0.0679 0.0684 0.0684 0.0692 0.0706 0.0772 0.0826 0.0888

Table 6 . 5 . Frequencies o f a s t i f f e n e d panel. A.

= w . b / (E/p)1/2


102

Vibrat ion of prismat ic square To

demonstrate

the

tubes

validity

of

the

present

finite

strip i n

a n a l y z i n g s y m m e t r i c a l l y - l a m i n a t e d t h i n - w a l l e d s t r u c t u r e s , t h e lowest t e n f r e q u e n c i e s o f a s i n g l e - c e l l tube and a t w i n - c e l l tube ( F i g . 6 . 8 ) a r e computed u s i n g t h e p r e s e n t f i n i t e s t r i p s . I t i s noteworthy t h a t f o r a n a l y z i n g p r i s m a t i c t h i n - w a l l e d s t r u c t u r e s w i t h s y m m e t r i c a l l y laminated composite m a t e r i a l ,

a minor m o d i f i c a t i o n o f

r e q u i r e d i n the c a l c u l a t i o n o f

the m a t e r i a l matrix i s

the s t i f f n e s s matrix.The orthotropic

m a t e r i a l m a t r i x f o r e a c h l a y e r o f m a t e r i a l s i s f i r s t t r a n s f o r m e d t o the global

co-ordinate

system

and

subsequently

integrated

through

the

t h i c k n e s s t o f o r m a smeared m a t e r i a l m a t r i x . T h i s smeared approach i s commonly used i n the a n a l y s i s o f l a m i n a t e d composite p l a t e s and s h e l l s . The tubes a r e formed o f s y m m e t r i c a l l y - l a m i n a t e d , o r t h o t r o p i c p l a t e f l a t s . Each f l a t has t h e p r o p e r t i e s o f a f i v e - l a y e r , c r o s s - p l y p l a t e w i t h a 0^90^0^90^0°lay up in which the thickness of each of the 0oplies is h/6 and that of each of the 90oplies is h/4 ( where h is the total thickness

of

the

flat).

The

material

properties

of

all

plies

are

identical and correspond to a typical high-modulus fibre composite with

E

1

/

= 30

2

G

/ E,= 0.6 12 2

u

12

0.25

They are simply-supported at both ends with rigid in-plane diaphragms. Each of these tubes has a wall breadth B ( measured between the centre lines of a pair

of opposite walls), wall

thickness 0.01B,

and

span

length 10B for the single-cell tube and 5B for the twin-cell tube. As the wall thickness is relatively small with respect to the span of the tubes, significant out-of-plane bending of the wall is anticipated. In the finite strip analysisďź&#x152;each tube wall

is modelled by

two finite

strips. In Table 6.6 the finite strip results, using 8 strips x 9 nodes for the single-cell tube and 14 strips x 9 nodes for the twin-cell-tube, are compared with the finite element results using 80 four-node shell elements for

the single-cell tube and

70 elements for

the

twin-cell

tube. For the single-cell tube, the lowest eight frequencies obtained by the finite strip method, with 55 percent of the number of unknowns of the finite element analysis, are within 1 percent difference from the finite

element

solutions.

For

the

results are in very good agreement.

twin-cell

tube,

the

two

sets

of

In this case, the same number of

unknowns is involved in each of the two analyses.


103

Mode i

F r e q u e n c i e s (x 0.01) single-cell

twin-cell

1

8.793

8.831

8.870

8,878

2

8.848

8.882

9.262

9.233

3

8.978

8.999

10.43

10.30

4

9.240

9.229

11.37

11.41

5

9.647

9.636

11.68

11.68

6

9.647

9.636

12.64

12.51

7

9.677

9.651

12.71

12.61

8

9.718

9.651

12.86

12.77

9

10.55

10.29

13.25

13.30

10

13.16

11.26

14.15

14.04

T a b l e 6 . 6 : F r e q u e n c i e s o f composite tubes ( 8 = 1 ,

Buckling

of a NASA panel

1)

(Fig.6.9)

S t r o u d e t . a i . [ 8 1 ] o f NASA have p r e s e n t e d a d e t a i l e d s t u d y o f the b u c k l i n g o f seven s t i f f e n e d p a n e l s w h i c h have diaphragm ends and a r e s u b j e c t e d t o combined l o n g i t u d i n a l compression ă&#x20AC;&#x201D; Ny ) and shear ( Nxy )â&#x20AC;˘ Of particular interest is the NASA example 2 which has been studied in detail by Stroud et al. using hybrid finite element method, and by Lau and Hancock

[82] using the spline finite strip method. This example

provides a good test case for evaluating the efficiency and accuracy of the present finite strip. In

the

finite element analysis, a very fine mesh of four-node

assumed-stress hybrid element was used to provide a benchmark solution. The panel is divided into two elements along the depth of the stiffener, four elements between stiffeners, 1296 elements,

36 elements along

length,

total

unknowns.

In the spline finite strip analysis, six spline sections is

longitudinally with a

method,

the stiffened

panel

total

1369 nodes and

the

making a used

of

and

of

approximately 8000

504 unknowns.

is divided

into

For

18 strips

the

present

( two strips

between stiffeners and one strip for each stiffener) and seven nodes longitudinally. There are a total of 436 unknowns after imposing the boundary conditions. A comparison of the three sets of results is given in Table 6.7. The results obtained by the present finite strip model, with only 5.5% of the number of unknowns of

the NASA finite element


104

model, a r e o n l y 3 t o 4 % h i g h e r . I n a d d i t i o n , the agreement between the two f i n i t e s t r i p models i s v e r y good. I t i s worth n o t i n g t h a t s i n c e the present NASA p anel i s s i m p l y supported around t h e edges,

one

may

think o f using semi-analytical

f i n i t e s t r i p method which i s more e f f i c i e n t f o r a n a l y z i n g s t r u c t u r e s w i t h two s i m p l y - s u p p o r t e d o p p o s i t e ends. Al tho ug h the method has proved t o be a c c u r a t e and e f f i c i e n t f o r a n a l y z i n g the b u c k l i n g o f compressed s t r u c t u r e s , i t has

some d i s a d v a n t a g e s f o r a n a l y z i n g t h e b u c k l i n g of

s t r u c t u r e s loaded i n compression and/or shear ďź&#x161; (i ) Difficulties are experienced when dealing with non-periodic buckling mode. (ii) the buckling analysis of plate assemblies loaded in shear is problematic[133]. (iii)

For

thin-walled

structures

loaded

in

compression,

many

buckling modes are possible. Since it is not known in advance which mode is critical, it will be necessary to solve the same eigenvalue problem for different terms of Fourier series buckling

load.

Especially

in

case

in order of

to obtain the minimum

structures

with

complex

cross-sections, this repetitive process may be time-consuming. The present method, however, does not

suffer from

the problems

mentioned above, and therefore seems more suitable to study buckling of prismatic thin-walled structures under arbitrary loading.

Loading Ny (kN/m)

70.05 175.13 350.26 875.65 17S.12

Load factor

Nxy ă&#x20AC;&#x201D;kN/m)

175.13 175.13 175.13 175.13

NASA[81]

Spline strip[82] Present strip

0.7195

0.7371 0.6183 0.4542 0.1977 1.0003

0.6061

0.4444 0.1929 0.9759

0.7490 0.6304 0.4612 0.1990 1.0070

T a b l e 6 . 7 . B u c k l i n g o f NASA p a n e l 6 . 3 Conclusion A f l a t - s h e l l f i n i t e s t r i p i s p r e s e n t e d f o r a n a l y z i n g v i b r a t i o n and b u c k l i n g problem o f p r i s m a t i c f o l d e d p l a t e s . The acc uracy o f the present f i n i t e s t r i p i s a s s e s s e d by comparing i t s s o l u t i o n s w i t h those o f the s e m i - a n a l y t i c a l f i n i t e s t r i p , s p l i n e f i n i t e s t r i p and f i n i t e element


105

methods. I t possesses comparable accuracy, e f f i c i e n c y and v e r s a t i l i t y as those o f

the s p l i n e f i n i t e s t r i p .

unknowns a t each node, effort

with

respect

to

there i s the

Because o f

a c o n s i d e r a b l e s a v i n g i n computing

finite

element

comparison w i t h t h e s p l i n e f i n i t e s t r i p , relatively

simpler

the reduced number o f

i n dealing with

method.

Moreover,

in

the present f i n i t e s t r i p i s

boundary

and

internal

support

c o n d i t i o n s ; the support c o n d i t i o n s can be e a s i l y imposed i n the usual f i n i t e element manner. A l s o , the method can be e a s i l y implemented by i n c o r p o r a t i n g a s t a n d a r d f i n i t e s t r i p program w i t h a continuous-beam program.The computation o f the s t i f f n e s s m a t r i x i n v o l v e s no numerical i n t e g r a t i o n because o n l y

beam shape f u n c t i o n s a r e

i n v o l v e d i n the

c a l c u l a t i o n s . A drawback o f the method, which i s common t o the semia n a l y t i c a l f i n i t e s t r i p (except f o r the case o f two simply-supported o p p o s i t e edges ) o r s p l i n e f i n i t e s t r i p , i s t h a t the s t i f f n e s s matrices corresponding t o d i f f e r e n t terms o f the l o n g i t u d i n a l t r i a l f u n c t i o n are coupled w i t h each o t h e r . T h i s disadvantage however, would not become a major

drawback

of

the

method

because

f u n c t i o n s , u s u a l l y l e s s than 10,

only

a

few

computed

shape

a r e s u f f i c i e n t t o g i v e acceptable

r e s u l t s , as demonstrated i n the n u m e r i c a l examples.


106

Local

• X

Global

Nodal Variables u,v,w,0 Fig.6.1

A flat-shell finite strip and its coordinate systems

Fig.6.2

A cantilever channel. E=1.0, 1^=0.3, a=50, b=152, c=150, density=l.0,thickness=3.2


107

Mode 2

Fig.6.3

3-D view o f t h e f i r s t f o u r mode shapes o f the c a n t i l e v e r channel i n F i g . 6 . 2 .


108

Fig.6.4

C r o s s - s e c t i o n s o f o n e - f o l d and t w o - f o l d f o l d e d plates. span = L , t h i c k n e s s = 0 . 0 2 L For one-fold folded plates, F o r t w o - f o l d f o l d e d p l a t e s , b ĺšżb ,

L / 3 , a^= oc^a

(a)

(b)

(c) Fig.6.5

Buckling of

continuous

s t i f f e n e d panel;

(a)

l o n g i t u d i n a l and t r a n s v e r s e s e c t i o n o f the panel, (b)buckling

pattern

(transverse

three s t r i p s i d e a l i z a t i o n

direction),

(c)


109

Fig.6.6

C r o s s - s e c t i o n o f a simply-supported f o u r - s t i f f e n e r panel

Fig.6.8 Cross-section of a twin-cell tube


lie

Mode 2

Fig.6.7

3-D v i e w o f t h e f i r s t seven mode shapes o f the s t i f f e n e d panel i n F i g . 6 . 6 .


Mode 5


112

Uimeasions and Material Properties of Square Stiffened Panel Skin Thickness 2.1336mm Length (LI 0.7620m Stiffener Thickness 1-C732mm Width (bl 0 7620m Young's Modulus 0.724 * 10^ HP a Stiffener Spacing 0.1270m 0.32 Stiffener Oepfh 0.03434m Poissoo's Ratio

F i g . 6. 9

L o a d i n g and dimensions f o r s t i f f e n e d p a n e l . S i m p l y - s u p p o r t e d on a l l f o u r edges. X=0,L

v=0=w

Y=0,b

u=0=w


113

CHAPTER 7 A THIRD-ORDER SPLINE FINITE STRIP AND NONLINEAR FORCED VIBRATION I n t h i s c h a p t e r the o r i g i n a l s p l i n e f i n i t e s t r i p , which i s based upon the t h i n - p l a t e t h e o r y , i s m o d i f i e d i n such a way t h a t the present spline

finite

strip

is

formulated w i t h i n

third-order plate theory. finite strip

is first

The performance of

the

nonlinear

analysis

procedures. For reducing analysis, a special number

of

equations

context

of

Reddyďź&#x152;s

this third-order spline

investigated with particular attention on thin

plates as well as shear-deformable plates. to

the

of

forced

It is subsequently extended

vibration

using

the computational cost of

reduction method is

solved

is employed

within

each

time-stepping

the so

time

time-stepping

that a smaller

step

and

hence

a

reduction of computing time can be achieved. 7.1 Introduction The

transverse

shear

moduli

of

usually very

low compared with the

result

transverse

that

importance compared the usual assumption

shear

modern

composite

materials

Inplane tensile moduli, with the

deformation

can

be

of

to homogeneous isotropic materials. of

are

considerable Additionally,

"plane section remains plane"

is no

longer

appropriate for thick isotropic plates and moderately thick laminated composite plates because of the significant cross-sectional warping. The classical Kirchhoff plate theory is only capable of dealing with plates with very large span-to-thickness ratio. Although the shear deformation theory developed by Mindlin and Reissner has been commonly used for the finite

element

warping gives

analysis

of

moderately

thick

plates,

cross-sectional

is not taken into account. Furthermore, Mindlin plate theory constant

transverse

shear

strain

distributions

through

the

thickness and hence a shear correction factor is required to account for the actual parabolic distribution.( For moderately thick laminated or isotropic platesďź&#x152; transverse shear strains are commonly assumed as a parabolic distribution through thickness). This factor depends on the geometry

of plates

conditions and

as well

loading.

as

its

laminate configuration,

It plays a paramount role

boundary

in the analysis of

laminated composite plates. However, accurate and reliable analysis of


114

composite laminates can be c a r r i e d out by u s i n g a t h i r d - o r d e r p l a t e theory which t a k e s i n t o account, i n an e x p l i c i t manner, c r o s s - s e c t i o n a l warping

and

parabolic

transverse

shear

strain

distributions.

A

t h i r d - o r d e r p l a t e theory can be g e n e r a l l y w r i t t e n as u(x,y’z) = uo(x’y) + ui (x,y) -z + u2(x,y)*z2 + u3(x,y) •z3

(7.1a)

v(x,y,z) = vo(x,y) + vl (x,y) •z + v2(x,y) •z2 + v3(x,y)*z3

〔7.1b)

(x,y) *2 + vr2(x,y”z2

w(x,y,z) = w0(x,y) +

(7.1c)

By selecting different terms from these expressions, a very large number of third-order plate theories have been developed(see e.g.[84] and references therein). To reduce the number of independent variables, one can

impose the the

traction boundary conditions on the top and

bottom surfaces of a plate and neglect transverse normal deformation. A third-order plate theory, which contains the same number of dependent variables as the Mindlin plate theory, has been developed by Reddy[85], Details

of Reddy's

third-order

plate

theory

and

its

implementation

will be discussed in the following section. 7.2 Third-order plate theories By neglecting transverse normal deformation, the displacement field of a plate starts off with U(x, y, z) = u(x,y)+ 2 0

X

V(x,y, z) = v(x,y) + z ^

y

W(x’y,z) = w(x,y)

+ z2co + z 3 5 X X + z2o) + z 3 6 y

y

(7,2a) (7.2b) (7.2c)

where u,v and w denote the displacements of a point (x,y) on the midplane, and

b l

x

and

y

iIj

are the rotations of normals to mid-plane about

the Y- and X-axes, respectively. Cross-sectional warping is represented by a) x

to)

y

,6 x

,5 y

.

First consider the transverse shear strains which are given by (7.3a)


115

C

xz

= U

z

+ W

(7.3b)

,x

where p a r t i a l d i f f e r e n t i a t i o n i s denoted by a comma such as ()

ďź&#x152;

=~ 2

The t r a c t i o n boundary c o n d i t i o n s on the top and bottom surfaces o f a p l a t e w i t h t h i c k n e s s t can be expressed as cr

yz

= 0 = cr a t 2 : = Âą t/2 xz

(7 4) *J

where

c r and c r denote the transverse shear stresses. Usine the 0 yz xz strain-displacement relationships of (7.3) and assuming linear elastic

material, the four traction-free conditions of (7.4) can be applied to eliminate four out of the nine variables in (7.2). Hence the following displacement field of the Reddy's third-order plate theory (TOP) can be derived :

U(x,y,z)

ip - 4/3 ( z/t ) 2 (

V(x,y,z) = v +

4/3 ( z/t ) 2 ( 0 + w

)]

(7.5)

W(x,y,z) = w(x,y)

It is clear that (7.5) contains the same number of independent variables as the Mindlin plate theory. By substituting (7.5) into (7.3), it can be easily shown that the transverse shear strains at mid-plane are actually represented by

(7.6a) (7.6b) Substituting (7.6) into (7.5), we obtain the following modified thirdorder plate theory ( MODTOP ),


116

U(x,y,z)

1 一 4/3 (z/t)'

V(x,y,z) W(x,y)

From

(7.7) =

w(x,y)

a theoretical

(7.5) and

point

(7.7) are

in

of

view,

fact

the

the displacement same.

However,

fields given by

for

finite

element

implementation, they are not identical ;t h i s w i l l be e l a b o r a t e d i n the following section. 7.3 Spline f i n i t e s t r i p I t can be e a s i l y observed t h a t the f i r s t d e r i v a t i v e o f the o u t - o f plane

displacement

variable i s

involved i n

TOP

or

MODTOP.

As

a

consequence, second d e r i v a t i v e o f w appears i n the s t r a i n - d i s p l a c e m e n t r e l a t i o n s h i p , i n an e x p l i c i t form, f o r TOP, 4/3 2(z/t) v

+ z

< /

(0

+ w

4/3 z(z/t)2 ( t j y» x

)

+ w ,yy

(7.8a)

)

(7.8b)

f o r MODTOP

• yy Using

displacement-based

4/3(z/t)2 ]A

(7.9a)

4/3(2/1)

(7.9b)

]A y,y

formulation,

function for

the displacement variable w

displacement

variables,

sufficient.

To

this

end,

Co-continuous each

of

variables of TOP is interpolated as

the

Ci-continuous displacement is required. For

displacement five

the other

functions

independent

are

displacement


117

u(x,y)

cc}.

]

v(x,y)

/3}.

w(x,y)

¢: y

,(x,y) = p(x,y)=

x E

where [M(x)] and

(7.10)

y

€>;

y

C};

[N(x)] are the one-dimensional linear shape functions

and the Hermite cubic polynomial respectively. of

the

B3-spline

function

with

¢. (y) is the i~th member

{a>, {p>, {y>,

and

{<>

as

the

corresponding unknown parameters. It has been shown by many researchers that the transverse shearing strains

must

be

interpolated

consistently

in

order

to

avoid

shear

locking for finite element analysis of thin plates. In the spline finite strip model based on TOP,it is obvious that, by substituting (7.10) into (7.6), the contributions to the transverse shear strains at mid-plane are

of different

order polynomials.

This causes a problem

for

thin

plates where both G and e must tend towards zero. A consistent xz yz interpolation scheme would be one in which identical polynomial terms must be used for both the rotations the

vertical

interpolation

displacement scheme

is

field

possible,

xjj (or xb )and the derivative of x y w ( or w ). While such an .x

it

,y

would

not

be

convenient

to

implement. However, with the modified displacement field (MODTOP) this problem does not exist because the transverse shear strains e

xz

or

and

and c

yz

(

can be interpolated consistently by replacing 〜 a n d 〜

of equation (7.10) by A

and A

respectively.

7.4 Equilibrium Equation Taking

into

consideration

the

von-Karman nonlinear

strains

and

initial imperfection, the strain-displacement expressions can be written as


118

w

V w ,y 0,)

(7.11)

where w。 denotes the out-of-plane initial imperfection. In a condensed form, the above relationships can be written as the summation of linear strains, nonlinear strains and initial strains ; that

(7.12)

{E> = {G} + {€>.. + {G}.

Substituting the spline finite strip displacement functions into the above expressions, we obtain the strain-displacement equations given by {e> = [Bl+ B 1 (8)]{6} = [B(5)]{6}

(7.13)

where u

B.(6) = 2

+ 9 、

0,> [NCx)]^. (y)

[N(x)]

where B l denotes the linear strain-displacement matrix . The nonlinear matrix B 1 (S) is a function of the the current displacement {5}. Taking


119

v a r i a t i o n w i t h r e s p e c t t o {5}, i t can be shown t h a t the incremental s t r a i n - d i s p l a c e m e n t e q u a t i o n i s g i v e n by

d{c} = [B + B + B (6)]d{6} L

0

(7.14)

N

= [ B + B ⑷ ] d { < 5 } = [B(6)]d{5} L

N

where

[ N ( x ) ] ^ t (y)

.(6)

[N(x)] 0: (y)

[M(x)],¢. (y)

B_(6)

o,:

[N(x)] ¢:(y)

and

[Ntx)]'^.(y)

B (6)

o,:

[N(x) ] (p* (y) o,:

Using iterative

the

Total

approach

Lagrangian [86-88],

the

formulation following

and

the

incremental-

dynamic

incremental

equilibrium equation at time t + At can be derived,

[M] t + A t


120

今[M]

where

t

+

t+At

A

t

+ 1[KT]d{5}k =t+At{F} -

k

{ 6 }

{<5} k =

{R}

k

-1

(7.25)

+ d{5} k 1

and “ ^ 仴 广

t+At

=

t + A t

[K广

1

E

The l e f t s u p e r s c r i p t r e p r e s e n t the time o f the c o n f i g u r a t i o n i n which the

quantity t+

t

c o u n t . ^ {p} i

occurs

and

the

right

superscripts

the e x t e r n a l load v e c t o r

s

1+

14

denote

iteration

1

{R} "" i s the i n t e r n a l

r e s i s t a n c e e q u i v a l e n t t o the element s t r e s s e s t h a t correspond t o the displacements

t+

^t{6}k

1

1

; and

{6} k i s t h e d i s p l a c e m e n t v e c t o r a t

t h e end o f i t e r a t i o n k and time t+At. t+At

0

{6} =

t + A t

When the i t e r a t i o n count k = l ,

0

{ R } = ^ R } and

The m a t r i c e s [K ] and [K ] L

N

r e p r e s e n t the l i n e a r e l a s t i c s t i f f n e s s

m a t r i x and the l a r g e displa cem ent m a t r i x r e s p e c t i v e l y . m a t r i x [K ] depends on the c u r r e n t i n p l a n e s t r e s s e s cr (T

sum o f the se m a t r i c e s

x

The geometric ,cr

y

and cr

xy

. The

[K^] i s t h e s o - c a l l e d tangent s t i f f n e s s m a t r i x .

S i n c e l i n e a r e l a s t i c m a t e r i a l i s considered i n t h i s study, the i n t e r n a l r e s i s t a n c e {R} can be e xpr e ssed i n terms o f a secant s t i f f n e s s m a t r i x [K ] . E

The d e t a i l s o f the s t i f f n e s s m a t r i c e s and t h e mass m a t r i x a r e

g i v e n i n t h e Appendix.

The i n e r t i a term on the l e f t - h a n d - s i d e o f the

e q u i l i b r i u m e q u a t i o n drops

out i n cas e o f

static analysis

and the

s t a n d a r d Newton-Raphson i t e r a t i o n w i t h l o a d i n c r e m e n t a t i o n i s employed t o s o l v e t h e s t a t i c e q u i l i b r i u m e q u a t i o n . A r e s i d u a l f o r c e convergence c r i t e r i o n i s used i n the s t a t i c a n a l y s i s . A l l i n t e g r a l s a r e c a r r i e d out over

the

original

volume

of

each

section of

a s t r i p with

three

i n t e g r a t i o n p o i n t s i n each o f t h e X and Y d i r e c t i o n s . 7 . 5 Nu me r ica l Examples

Linear

static

analysis

of plates

To i n v e s t i g a t e the performance o f the p r e s e n t s p l i n e f i n i t e s t r i p f o r a n a l y z i n g t h i n p l a t e s as w e l l a s shear-deformable p l a t e s , a quadrant o f a square p l a t e , w i t h e i t h e r s i m p l y - s u p p o r t e d edges o r clamped edges, i s d i v i d e d i n t o a number o f s t r i p s w i t h 4 s p l i n e s e c t i o n s i n each s t r i p . The p l a t e i s

subjected to

an u n i f o r m l y d i s t r i b u t e d l o a d .

Vertical

d e f l e c t i o n a t t h e c e n t r e o f t h e p l a t e i s n o r m a l i z e d w i t h r e s p e c t t o the e x a c t s o l u t i o n [ 7 1 ] . The convergence o f the s t r i p based on TOP and MODTOP i s demonstrated i n F i g . 7 . 1 . From the r e s u l t s o b t a i n e d by the TOP model, one can e a s i l y see the r a t e o f convergence d e t e r i o r a t e s r a p i d l y as the


121

span-to-thickness

increases. On

the contrary,

perfect deflections however large

is

the MODTOP model gives

the span-to-thickness ratio. The

convergence of centre bending moments, obtained by the MODTOP model, are given in Table 7.1. Again, very rapid convergence can be observed. Results

of

analyses

of

moderately

thick

isotropic

plates

and

laminated composite plates with different span-to-thickness ratio are presented in Tables 7.2 and 7.3.

In all cases, the plates are subjected

to uniformly distributed loads. A quadrant of each isotropic plate is divided into strips with 4 spline sections. Results given in Table 7.2 indicate the good agreement between the spline finite strip results and the

thick-plate solutions by Kant

[89] for

isotropic plates. Unlike

thin-plate problems, both models show rapid convergence For

composite

laminates,

Reddy[85] are used

analytical

for comparison.

solutions It

of

in this case.

TOP

is apparent

that

developed there

by

is no

significant difference among the three sets of results given in Table 7.3.

It is important to see that there

is no major difference between

the results of the two spline finite strip models in analyzing sheardeformable plates.

Number of strips

M

Centre bending moments M M

Simply supported 4816 4804 4800 4798 4816 4804 4800 4798 .4816 Analytical[71] Multiplier Table 7.1

.4816 .4789

a/t=200 0.4815 0.4811 0.4809 0.4809 a/t=500 0.4815 0.4811 0.4809 0.4809 a/t=10Q 0.4815 a/t=100ă&#x20AC;&#x2021; 0.4815 0.4789 0.1 aa2

Clamped 2331 2312 2306 2304

0.2325 0.2323 0.2322 0.2322

2331 2312 2306 2304

0.2325 0.2323 0.2322 0.2321

0.2332

0.2324

0.2331 0.2291

0.9291

ďź&#x161; Comparisons of bending moments at square

0.2325

the centre of a

plate ( ĺ?Ł = 0 . 3 ), 4 s p l i n e s e c t i o n s per s t r i p ,

a / t = span-to-thickness r a t i o


122

Number o f s t r i p s

Deflection

Moment (* qa 2 )

(* q a V D)

Span-to-thickness r a t i o = 5 0.00490 0.00490 0.00490 0.00490

4 6 8 1

0.4809 0.4798 0.4795 0.4793 MODTOP 0.4812 0.4801 0.4797 0.4796 0.4840

0.4822 0.4806 0.4799 0.4796 o*â&#x20AC;&#x201D;i o o o 5 8 8 8 8 8 4 4 44-4:

10

Kant[89]

0.00490 0.00490 0.00490 0.00490 0.00480

TOP

S p a n - t o - t h i c k n e s s r a t i o = 10

8 8 8 7 8

0.

0.4820 0.4806 0.4801 0.4798 3 o 9 9 o 1X 101o 8 8 8CD 8 8 4 4 4 4 4

0. 4802 4797 4794 0. 4793 MODTOP

6 4 o 8 o 1 o o 9 o

10 Kant[89]

0.00427 0.00427 0.00427 0.00427 0.00424

TOP

4 4 4

10

0.00426 0.00427 0.00427 0.00427

T a b l e 7 . 2 R e s u l t s o f bending moments and d e f l e c t i o n s a t the centres o f s i m p l y supported t h i c k p l a t e s , y = 0 . 3 , q = magnitude of uniformly distributed load, D = bending rigidity


123

Span-to-thickness ratio

Deflection w

10 10(

Table 7 . 3

TOP

MODTOP

Analytical[85]

7.7663 2.9090 1.0898 0.6699

7.7613 2.9078 1.0900 0.6705

7.7661 2.9091 1.0900 0.6705

R e s u l t s o f d e f l e c t i o n s f o r a t h r e e - p l y (0/90/0 ) square l a m i n a t e d p l a t e w i t h s i m p l y supported edges 8 s t r i p s x 4 s p l i n e s e c t i o n s a r e used. w = wt 3 E2 * 100 / qa 4 t = t o t a l thickness of plate 6 E i = 25 * 10 6 p s i E 2 = 10 p s i 6 Gi2= Gi3= 0 . 5 * 10 p s i G23= 0 . 2 * 10 6 p s i

Buckling The

of isotropic

plates

with

cut-outs

MODTOP model i s a l s o a p p l i e d t o

(v=0.3)

the a n a l y s i s o f

isotropic

square p l a t e s w i t h square c u t - o u t s a t the c e n t r e s ( F i g . 7 . 2 ) . F o r p l a t e s with cut-outs,

a m o d i f i e d B 3 - s p l i n e f u n c t i o n i s adopted

( details of

B s - s p l i n e f u n c t i o n and i t s m o d i f i c a t i o n a r e summarized i n t h e Appendix ).The c r i t i c a l b u c k l i n g l o a d f a c t o r s a r e compared w i t h the h i g h e r - o r d e r f i n i t e element s o l u t i o n s [90] o f M i n d l i n p l a t e t h e o r y , see T a b l e 7 . 4 . I t can be seen t h a t t h e p r e s e n t f i n i t e s t r i p b u c k l i n g l o a d s a r e lower than those

obtained

by

the

higher-order

finite

element;

this

may

be

a t t r i b u t e d t o t h e f a c t t h a t a l t h o u g h h i g h e r o r d e r f i n i t e elements a r e used i n [90], t h e c u t - o u t s a r e t r e a t e d w i t h an approximate " n e g a t i v e s t i f f n e s s " approach. U n i a x i a l c r i t i c a l b u c k l i n g l o a d f a c t o r LF span-to-thickness r a t i o

10

100

MODTOP

1.823

1 â&#x20AC;˘ 987

FEM[90]

1.860

2.090

Table 7.4 Buckling loads for simply supported square plates with hole c r

= LF re2 E t 2 / [ 12 ( l-p2) a 2 ]


124

Nonlinear

Analysis

Two isotropic square plates with different thickness ( a/h=5 and 100) are subjected to uniform distributed loads. All edges of the plates are either simply-supported or clamped. Only a quadrant of the plate is analysed with four finite strips and

four spline sections.Information

for the thin plate with a/t=100 is provided in Fig. 7.3. For presentation of results in Fig.7.4 and Fig.7.5 the following

normalised quantities

are used : Q = q a 4 / E t4

and

From the results it

W = w(0,0) / t

is clear that the present finite strip results are

in reasonable agreement with the Levy* s solution for thin plates [91] and

Mindlin-type

between

the

two

higher-order spline

strip

element models

solutions is

not

[92]. The

siginificant

difference in

these

analyses. Nonl inear zlmiysis : In-plane loading The nonlinear behaviour of two isotropic square plates with small initial

curvatures

loaded

in

edge

compression

are

analysed.

For

comparison, Yamaki's cases 1(b) and 11(b) are examined[93]. The boundary conditions

for

the

4-strips mesh

(with 4 spline sections) shown

Fig.7.6 being as follows :

Case 1(b) Simply supported x= 土 a / 2

u,v,0 free, X

y= ± a/2

u,0

y

w = 0 = 0 y

free, v constant, w =

i / = 0 父

Case 11(b) S i m p l y supported/clamped x - ± a/2 u, v free, w = y= ± a/2 u, i / y

p i = i / = 0 X y

free, v constant, w =

X

i / = 0

The initial imperfection in each case is given by

Case 1(b) Case II (b)

w o = 0.11 cos(nx/a) cos(7ry/a) w = 0.05t ( 1 + cos(27rx/a) )( sin(27iy/a)) o

in


125

The f o l l o w i n g n o r m a l i s e d q u a n t i t i e s a r e used f o r p r e s e n t a t i o n o f results i n Fig.7.7 V = v a / t

2

and

W = w(0,0) / t

for case I(b)

W = w(0,a/4) / t for case 11(b)

During the initial stage of edge compression, the loading is mainly carried b y the bending resistance of the plate. As the MODTOP model has been

shown

to

perform

better

in

especially with clamped edges, one that

a

finer

mesh

is

required

for

bending

problems

of

thin

plates,

can easily observe from Fig.7.7(b) the

TOP

model

to

achieve

similar

accuracy as the MODTOP model during

the initial stage of loading. When

the

the

edge

compression

resistance

becomes

transverse

shears

increases,

significant.

affects

only

importance

As the

consistent

bending

of

the

membrane

interpolation

behaviour

of

the

of

strip

models, the difference between results of the two models reduces as the edge compression increases. o n MODTOP

are

in

In general,

reasonable

agreement

the finite strip results based with

those

provided

b y Yamaki

[93].

Preliminary

Remarks

The third-order spline finite strip is applied to a few linear and nonlinear static problems. For the analysis of shear-deformable plates, there is n o major difference between the performance of the finite strip models based o n TOP and MODTOP. However, the MODTOP model in

terms of

analysis.

the

The

rate of

MODTOP

convergence, to

model,

therefore,

is superior,

the TOP model

in

thin-plate

will

in

subsequent

be

used

analysis. It

should

also be noted

that, unlike

the semi-analytical or

classical spline finite strips, the present spline finite strip, o n modified B^-spline function,

is also applicable

to

the

based

the analysis of

rectangular plates with cut-outs.

7 . 6 Nonlinear forced vibration A general

review

The understanding of geometrically nonlinear, forced vibration of plates with time-varying loads is important of

structures.

Because

of

the

membrane

in many engineering design

action

of

an

elastic

plate


126

undergoing l a r g e d e f l e c t i o n ,

t h e n o n l i n e a r s t i f f e n i n g b e h a v i o u r has t o

be t a k e n i n t o a c c o u n t i n many s i t u a t i o n s . F o r l a r g e - a m p l i t u d e v i b r a t i o n of

a

plateďź&#x152;

the

response

can

be

obtained

using

an

implicit

time

integration scheme w i t h equilibrium iterations for nonlinearity. Within each

time

step,

the

dynamic

and

solved

using

established process

consumes

tangent

stiffness

internal

force

a

lot matrix

h i g h l y desirable can be

and

equilibrium

equilibrium

computing

and

vector

preliminary analysis o r

method

of

incremental

the

study,

iterations.

time

in

for

of

is

However,

evaluation the

equilibrium

this

of

the

out-of-balance

iterations.

For

a reduction of computational effort

frequently necessary.

applied.

the

calculation

required

equation

This method

aims

To

at

this

end,

savings

in

a

is

reduction

computational

c o s t s b y u s i n g a smaller n u m b e r of generalised variables to represent the original problem w i t h assumed that sufficient number

the v e c t o r of

accuracy,

of

larger n u m b e r of degrees o f freedom.

global

coefficients

of

is

discrete fundamental unknowns c a n be,with

represented

as

approximation

the

It

linear

a

linear

combination of

functions

combination

(or

are

basis

a

small

vectors).The

determined

by

solving

a

reduced system o f e q u a t i o n s w h i c h is the projection o f the full system of

equations

onto

the

subspace

spanned

by

the

basis

vectors.The

effectiveness of this approach depends, to a great extent, o n the proper selection o f the reduced basis vectors.Using this method, b e t w e e n the p r e d i c t i o n o f complex behaviour and can be

achieved.

The

method

isďź&#x152; therefore,

a compromise

the computational cost

suitable

for

preliminary

a n a l y s i s or study o f structures using a micro- or mini- computer. A n u m b e r o f r e d u c t i o n methods f o r geometrically nonlinear dynamic analysis have

b e e n proposed

since

the

mid

seventies[94-99].

Most

these approaches are based o n either the tangent stiffness method o r the pseudo-force m e t h o d

(PFM).

In the TSM,

of

(ISM)

the change o f basis is

performed at e a c h time step or at a certain pre-defined interval using the m o d e shapes corresponding to the instantaneous system matrices- The updating of mode

shapes

(by

solving

the

corresponding

eigen-problem)

d u r i n g time-stepping analysis w a s a time-consuming process until, very recently,

an

efficient

et.al[100], R e g a r d l e s s o f

updating

procedure

was

proposed

this efficient updating procedure,

by

Chen

it

still

s u f f e r s from the d r a w b a c k that the incompatibility between the old and the

new

basis

introduces

a

truncation

error

in

velocities

and

accelerations e v e r y time a b a s i s change is made. This error produces a


127

c o n t i n u o u s l y growing l a c k o f e q u i l i b r i u m i n the affects

the f i n a l

shapes,

based on the l i n e a r

analysis. complete

The

linear

response

pseudo-force

results

on

[101]

the

PFM,

matrices

calculation, right

and

hand

are

the side

is

set

of

mode

used throughout

employed

the

the

throughout

nonlinearities of

system t h a t

a single

system m a t r i c e s ,

system

the

. In

complete

are

dynamic

the

taken

as

equilibrium

e q u a t i o n s . T h i s a p p r o a c h i s e f f i c i e n t because i t r e q u i r e s n e i t h e r the solution of

the e q u i l i b r i u m e q u a t i o n n o r

the updating o f

the tangent

s t i f f n e s s m a t r i x a t e a c h time s t e p . However, the method i s a p p l i c a b l e t o o n l y weakly n o n l i n e a r systems.

Recently,

Idelsohn

[101,102]

[103] proposed a new approach which i s based on a s e t o f

and Chang

mode shapes o r

R i t z v e c t o r s t o g e t h e r w i t h t h e i r p a t h d e r i v a t i v e s a s the b a s i s v e c t o r s f o r n o n l i n e a r dynamic a n a l y s i s o f derivatives

was

originally

beams

developed

and a r c h e s . T h e

by

perturbation f o r nonlinear s t a t i c analysis

Noor

using

[104]

idea of

the

path

concept

of

.An e v a l u t i o n o f t h i s

approach was r e c e n t l y conducted by K a p a n i a e t a l . [ 1 0 5 ] . By combining the p e r t u r b a t i o n method w i t h the harmonic b a l a n c e t e c h n i q u e , n o n l i n e a r f r e e v i b r a t i o n o f p l a t e s was i n v e s t i g a t e d by Noor[106]. approach, p r o p o s e d by Chan and Yao

Another i n t e r e s t i n g

[107] and Das e t . a l .

[108]ďź&#x152; involves

a number of displacement vectors which are collected at different times during

the

first

half

cycle

of

the

complete

vectors are orthonormalised w i t h respect Gram-Schmidt

process. Subsequent

analysis

system

analysis.

These

to the mass matrix using the is

then

carried

out

in

the

subspace spanned b y these vectors. Other researchers such as Bathe [109] and Sheu

[110] have suggested

computational effort solutions for the

The present The

the use of

while Liu and Lin

substrueturing for

reducing

[111] employed known elastic

dynamic analysis of elastic-plastic beams and plates.

approach

efficiency

and

accuracy

of

reduced

basis

technique

in non-

linear dynamic problems depends o n (i) the number of vectors required to accurately simulate the response, basis vectors,

(ii) the efficiency of generating the

(iii) the frequency of updating the basis vectors,

(iv)

the efficiency of the algorithm for transforming the complete system to the reduced system. In this work, the R i t z vectors are adopted because they have been shown to b e accurate, efficient and relatively easy to generate when compared with vibration mode shapes for vibration

analysis

[112].

Their

performance

in

linear forced-

nonlinear

dynamic


128 a n a l y s i s i s examined i n t h i s work. method,

To maximize the e f f i c i e n c y o f

the

a s i n g l e s e t o f v e c t o r s i s g e n e r a t e d and then used throughout

t h e complete response a n a l y s i s . To t h i s end, a s e t o f R i t z v e c t o r s u s i n g t h e l i n e a r e l a s t i c s t i f f n e s s m a t r i x i s f i r s t generated. N o n l i n e a r i t y i s only partially matrix

and

taken i n t o

account

the e f f e c t i v e

by

updating

load vector

during

the

tangent s t i f f n e s s

time-stepping a n a l y s i s .

Depending on the degree o f n o n l i n e a r i t y and the q u a n t i t i e s o f i n t e r e s t , t h e s e l i n e a r R i t z v e c t o r s a r e augmented by another s e t o f R i t z v e c t o r s computed

from

the

tangent

stiffness

d i s p l a c e m e n t c o r r e s p o n d i n g t o the v a r y i n g l o a d . As a r e s u l t ,

matrix

based

on

the

maximum l o a d i n t e n s i t y o f

static

the time-

the c o u p l i n g o f bending and membrane a c t i o n s

( i . e .

the n o n l i n e a r e f f e c t

before

the

time-stepping

) can be i n t r o d u c e d i n t o the b a s i s v e c t o r s

analysis.In

this

case,

a

nonlinear

static

a n a l y s i s o f the p l a t e i s c a r r i e d out a p r i o r i w i t h the maximum amplitude of

t h e t i m e - v a r y i n g l o a d a p p l i e d on i t

j u s t i f i e d because i n p r a c t i c e often, into

precede the

distribution of

the

response applied

extract useful information

full

system

a nonlinear s t a t i c analysis

may

,very

a n o n l i n e a r dynamic a n a l y s i s i n o r d e r t o g a i n i n s i g h t

nonlinear

efficiency of

( T h i s p r i o r a n a l y s i s can be

of

the

loading.

system

to

The p r e s e n t

a

given

spatial

method attempts

from the n o n l i n e a r s t a t i c a n a l y s i s . )

to The

t h e method can be f u r t h e r enhanced by t r a n s f o r m i n g the equations

to

the

subspace

prior

to

the

time-stepping

a n a l y s i s so t h a t the u p d a t i n g o f the tangent s t i f f n e s s m a t r i x

and the

i n t e r n a l f o r c e v e c t o r c a n be done c o m p l e t e l y w i t h i n t h e subspace. T h i s special

transformation

method w i l l

be

elaborated i n

the

following

sections.

7 . 7 Formulation of reduced dynamic equilibrium equations A s i n g l e s e t o f R i t z v e c t o r s i s f i r s t g e n e r a t e d . I t c o n s i s t s o f two s u b s e t s o f v e c t o r s n a m e l y , ( i ) p l i n e a r v e c t o r s computed from t h e i n i t i a l l i n e a r e l a s t i c s t i f f n e s s m a t r i x , and ( i i ) q n o n - l i n e a r v e c t o r s computed f r o m t h e tangent s t i f f n e s s m a t r i x based on the n o n l i n e a r displacement c o r r e s p o n d i n g t o the

maximum l o a d i n t e n s i t y o f the t i m e - v a r y i n g load.

These v e c t o r s a r e used throughout the response a n a l y s i s . T o gen erate the nonlinear basis vectors, first

performed w i t h t h e

a nonlinear s t a t i c analysis o f maximum amplitude o f

the

the p l a t e i s

time-varying load

a p p l i e d on i t ďź&#x152; see Fig.7.8. The algorithm for generating Ritz vectors is given

in

the Appendix for reference. The displacement vector

(with m


129

unknowns, say) c a n be approximated by the n (n=p+q) b a s i s v e c t o r s v i a the following transformation

{ 3 } = [ 门 { 0 } = Z^ i=l

(7.16)

1

where

[r] = [r

r

厂]

i s an vector of generalised variables with n entries (7.16)

into

the

( i72>>n ) . B y s u b s t i t u t i n g

dynamic e q u i l i b r i u m e q u a t i o n

(7.15)

and s e t t i n g the

r e s i d u a l t o be o r t h o g o n a l t o t h e subspace spanned by the b a s i s v e c t o r s ( i.e.

a G a l e r k i n approach a p p l i e d t o

equations,

the d i s c r e t e dynamic e q u i l i b r i u m

see A p p e n d i x ) t h e f o l l o w i n g reduced system o f e q u a t i o n s can

be o b t a i n e d :

[M]*

t + A t W

k

t4

[Kj*d{0}k

"At{F}*_

t+At{R}*k-i

(7.27)

where

[M]# = [r] T [M] #

[K ] = [T]1[K ] [r]

[n

T

{ F } = [r] {F}

T

T

{r}*= [ n T { R }

and

w i t h the superscripts denoting time and

iteration omitted for clarity.

Details of the transformation are described in the following subsection.

7 . 8 Transformation To

update

the

tangent

time-stepping

analysis,

system

then

and

transformation computing time.

they

and can

transformed

matrix

secant

[F] . This

be

stiffness

first

back

to

procedure

matrices during

evaluated the is

in

the

subspace expensive

complete

using in

the

the

terms

of

In this study the tangent and secant stiffness matrices

are transformed immediately after the basis vectors are established. For linear elastic material, this can b e easily done prior to time-stepping analysis.

The

procedure

of

this

transformation

are

elaborated

follows. The linear stiffness matrix is transformed in the usual manner :

as


N e g l e c t i n g i n i t i a l deformation,

the

l a r g e displacement matrix i s

g i v e n by [K ] = J N putting

[ B「DPB ⑷ L N

(7.16)

into

it,

+

B : (3)DPBt + N L

the

large

(5)D P B (5) N N

displacement

] dV

matrix

(7.19)

can

be

w r i t t e n a s t h e summation o f two p a r t s ;

[K ]*= [r] T [K j a

[r]T z

[r]

(7.20)

S i b^bpb ( r

+ [r] T E I i=lj=l

S

dV

il

E { [r] T j [ b^dpb ( r . ) + B T (r.)d p b

] dv [r] }

{ [r] T i=lj=l

E

( [Q] 1

i

i

[0]

2

i=l

1 1 = 1

[Q]

3i j

w i t h t h e f i r s t and second p a r t s b e i n g a l i n e a r and a q u a d r a t i c f u n c t i o n s of

current

[Q]

3i j matrix.

displacement,

represent

the

respectively.

components

of

The

the

matrices [ Q ] ,

reduced

large

[Q] 2 i

an

d

displacement

These m a t r i c e s a r e computed and s t o r e d a f t e r the b a s i s v e c t o r s

a r e g e n e r a t e d . They w i l l be r e t r i e v e d d u r i n g the time—stepping analysis and

, b y multiplying w i t h

the corresponding generalised variables,the

updating of the large displacement matrix can be done without going back to

the

full

system

of

equations.

The

secant

stiffness matrix

can be

transformed in a similar manner. For the geometric matrix, however, the current

inplane stresses can be written explicity

in terms o f inplane

strain, that is

{(rp} = D p {c p }

(7.21)


131

where the r i g h t s u p e r s c r i p t denotes i n - p l a n e components. By r e f e r r i n g to e q u a t i o n ( 7 . 1 1 ) , i t can be seen t h a t the i n p l a n e s t r a i n s can be f u r t h e r subdivided into function of

two p a r t s

namely,

a linear

the c u r r e n t d i s p l a c e m e n t s .

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

function

and

a quadratic

U s i n g the procedure s i m i l a r t o the t r a n s f o r m a t i o n and updating

o f the g e o m e t r i c m a t r i x can be accomplished. It

is

varying

noteworthy

functions,

distribution,

the

that

but

for

external

identical

loads

maximum

with d i f f e r e n t amplitude

time-

and

spatial

same s e t o f v e c t o r s and transformed m a t r i c e s can be

used f o r the t i m e - s t e p p i n g a n a l y s i s .

7 . 9 Newmark time i n t e g r a t i o n As one o f the aims o f the study i s to d e v e l o p a f o r m u l a t i o n which r e q u i r e s reduced run time, d i s c r e t i s a t i o n o f the time domain i s c a r r i e d o u t by Newmark* s i m p l i c i t time i n t e g r a t i o n scheme because i t a l l o w s f o r the use o f l a r g e time s t e p s and i s u n c o n d i t i o n a l l y s t a b l e f o r l i n e a r and n o n l i n e a r problems[1]. without

( In a d d i t i o n , f o r s t r u c t u r a l dynamic problems

impact o r i m p u l s i v e - t y p e o f

efficient

than

the

explicit

loadings,

scheme.)

implicit

The

scheme i s more

relationships

between

d i s p l a c e m e n t , v e l o c i t y and a c c e l e r a t i o n a r e g i v e n by

t + A t

{0}

t + A t

t+At

=

{,/,}

{^}k=

1

=

{0} A +

1

t+At

t

(

t + At

+

+ At(

{(//> )/2

+

(7.22a)

t + At

{»//} )/2

(7.22b)

{i/,} k " 1 + d{0} k

(7.22c)

Substituting (7.22) into (7.17), w e finally obtain

( 广 . 4 / ( A t )

2

[ M ]

#

) d{^}k = t +A t { F } ^ t +A t

{ R }

^i„

[ M ]

^

[ (

t

+

At

w

^i

T

- 4八缸 - )七{0}

]

(7.23)

F o r e q u i l i b r i u m convergence, a d i s p l a c e m e n t c r i t e r i o n i s adopted i n t h i s study.


132

7 . 1 0 N u m e r i c a l examples The p r e s e n t method was coded w i t h F o r t r a n 77 and implemented on a 4 8 6 - c o m p a t i b l e PC.

In order

to t e s t i t s accuracy

and e f f i c i e n c y

,

a

s q u a r e p l a t e under a u n i f o r m l y d i s t r i b u t e d l o a d i s a n a l y z e d . T h i s p l a t e has

been

examined

by

many

other

researchers

[e.g.113-115]

and

the

d e t a i l s are given i n Table 7.5. H a l f o f t h e p l a t e i s d i v i d e d i n t o f o u r s t r i p s and e i g h t s e c t i o n s . To e s t a b l i s h c o n f i d e n c e I n t h i s model, t h e f i r s t few n a t u r a l f r e q u e n c i e s a r e p r e d i c t e d and compared w i t h t h e a n a l y t i c a l s o l u t i o n . I t i s a p p a r e n t , from

Table

7.6,

that

the

spline

finite

strip

mesh g i v e s

reliable

f r e q u e n c i e s . The t i m e i n c r e m e n t u s e d i n t h e a n a l y s e s a r e , r e s p e c t i v e l y , A t = 0 . 0 0 2 s e c and 0 . 0 0 1 s e c f o r t h e maximum l o a d a m p l i t u d e s o f l O q

and 0 and

50q . T h e s e t i m e increments are a b l e t o g i v e accurate,s t a b l e O e f f i c i e n t s o l u t i o n f o r f u l l system a n a l y s i s o f t h e p l a t e ; s e e T a b l e 7 . 7 for

the case o f

step-loading.

system a n a l y s i s .

Zero i n i t i a l

They a r e a l s o conditions

adopted f o r t h e

(displacement,

reduced

velocity

and

a c c e l e r a t i o n ) a r e assumed. I n a l l casesďź&#x152; the tangent stiffness matrix is updated

on

every

fifth

time

step

and

remains

unchanged

during

equilibrium iterations within each time step. To

compare

the

reduced

basis

solutions, the vertical displacement the

inplane displacement

solutions

with

the

full

system

(w) at the centre of the plate and

(v) at quarter-span along the centre-line of

the p l a t e are monitored.The comparison of computing time is measured in terms o f time r e d u c t i o n ďź&#x161; % reduction in computing time= ( 1 -

) x 100

w h e r e Ti = computing time f o r a reduced basis analysis which e m b r a c e s the time required t o generate the basis vectors, to carry

out

transformation

and

the

subsequent

analysis and T 2 = computing time for a full system analysis

time-stepping


Span = 2 4 3 . 8 cm

t h i c k n e s s = 0 . 6 3 5 cm

Y o u n g ' s Modulus = 7 . 0 3 1 x l 0 5 kg/cm 2 d e n s i t y = 2.547x10

6

y=0.25

kgsec2/cm4

q(x,y,t) = q o f (t)

q 。 = 4.882 x

IC)"*4 k g / c m 2

Fundamental period of vibration = 0.1905 sec Boundary condition : simply supported edges with unyielding supports In the horizontal plane

S i x different types of time-varying functions

are considered :

1. sinusoidal load

f(t)=sin(7rt/t

)

0

<oo

2. half-sine load

f (t)=sin(7rt/t

)

0

< t

=0

t ==0. 125T s i t =0.1257s

t

f(t)=l

3. step load 4. step-change load

5. triangular load

f(t)=l一t/t

t < t

t =〇•15s

t 〈 t

t =0.15s

t/、)

0 ^ t ^ t

t <

t

t 6. exponential load

f(t)=(l-t/ti)exp(-入 =0

Table 7.5

t

oo

1

= 0.15

Configuration and loading of the thin isotropic plate

Frequencies 入 Mode

Spline f i n i t e s t r i p

Analytical[13]

1

19, 74

19.74

2

49, 35

49.35

5

98.,77

98.69

6

98.,80

98.69

7

128.24

128.30

2 o. 5 T a b l e 7 . 6 : Comparison o f n a t u r a l f r e q u e n c i e s A = a) a ( p t / D ) '


134

Load l e v e l

_

time s t e p s (sec)

0

maximum c e n t r e d e f l e c t i o n s F i n i t e s t r i p (MODTOP) Akay[113]

0.0050

0.5961

0.6014

0.0025

1.2570

1.2607

0.0020

1.6470

1.6337

Table 7 . 7

Maximum c e n t r e d e f l e c t i o n s o b t a i n e d w i t h d i f f e r e n t methods and l o a d magnitudes.

S i x d i f f e r e n t types o f time-varying f u n c t i o n s a r e considered, Table 7.5.

Results o f

the f i r s t

two

types o f

see

time v a r y i n g f u n c t i o n s

namely, s i n u s o i d a l and h a l f - s i n e l o a d s , w i t h d i f f e r e n t l o a d i n t e n s i t i e s a r e d e p i c t e d i n F i g s . 7 . 9 and 7 . 1 0 .

One can e a s i l y observe from these

graphs

prediction

that,

although

reasonable

of

the

peak

vertical

d i s p l a c e m e n t c a n be o b t a i n e d when u s i n g the l i n e a r v e c t o r s a l o n e , better prediction of a c h i e v e d by u s i n g latter

case,

slightly

a

further

much

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

4 l i n e a r vectors

55% r e d u c t i o n i n reduction of

and

4 nonlinear vectors.

computing

computing

time i s

time

is

achieved.

In the Only

achieved using

a

the

l i n e a r v e c t o r s a l o n e . By v a r y i n g the number o f l i n e a r v e c t o r s from 2 t o 8,

the

reduced system s o l u t i o n s t i l l d e v i a t e s p r e g r e s s i v e l y from the

f u l l system s o l u t i o n a s the time s t e p s proceed,

and no improvement o f

the

agreement

vertical

displacment

can

be

made.

The

of

inplane

d i s p l a c e m e n t between the f u l l system s o l u t i o n and t h a t o b t a i n e d by u s i n g 4 l i n e a r v e c t o r s and 4 n o n l i n e a r v e c t o r s i s s a t i s f a c t o r y . I t s h o u l d be noted

that

disregarded,

because

the

coupling

effect

in

the

basis

vectors

is

the r e d u c e d system s o l u t i o n s , w i t h l i n e a r v e c t o r s a l o n e ,

g i v e zero in-plane displacements. When the maximum l o a d amplitude i s i n c r e a s e d from lOq^to SOq^, the reduced system s o l u t i o n s w i t h 8 l i n e a r v e c t o r s a r e s t i l l f a r away from, satisfactory,

see F i g s . 7 . 1 1 and 7 . 1 2 .

U n l i k e the p r e v i o u s c a s e s ,

even

t h e peak v a l u e s o f v e r t i c a l d i s p l a c e m e n t cannot be p r e d i c t e d u s i n g the l i n e a r vectors alone. vectors,

the

However,

w i t h 4 l i n e a r v e c t o r s and 4 n o n l i n e a r

reduced system s o l u t i o n s ,

in

terms o f

the v e r t i c a l and

i n - p l a n e d i s p l a c e m e n t s , a r e v e r y much improved w i t h a c o r r e s p o n d i n g 74%


135

r e d u c t i o n i n computing time.

These r e s u l t s i n d i c a t e the

importance o f

i n t r o d u c i n g the c o u p l i n g e f f e c t i n t o the b a s i s v e c t o r s . A closer

examination o f F i g . 7 . 9

and F i g . 7 . 1 1

F i g . 7 . 1 2 ) r e v e a l s t h a t , u n l i k e l i n e a r dynamics,

( or Fig.7.10

and

the response frequency

o f the system v a r i e s w i t h the amplitude o f the e x t e r n a l load. In a d d i t i o n t o the r e s u l t s o f harmonic loads and h a l f - s i n e loads discussed

above,

comparison

are

also

made

for

other

types

of

t i m e - v a r y i n g f u n c t i o n s (t y p e s 3 t o 6 i n T a b l e 7 . 5 ) . I t can be seen t h a t the

vertical

obtained by

displacement-time the p r e s e n t

history

(Fig.7.13(a)

to

Fig.7.16(a))

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

r e s p e c t t o the f u l l system s o l u t i o n .

Furthermore,

the peak v a l u e s and

the o v e r a l l p a t t e r n o f the i n - p l a n e d i s p l a c e m e n t response can s t i l l be predicted

with

satisfactory

Fig.7.13(b) to Fig.7.16(b).

accuracy

However,

by

the

present

method,

one can a l s o n o t i c e t h a t ,

f o r the s t e p - l o a d response i n F i g . 7 . 1 3 ( b ) ,

see

except

the reduced system s o l u t i o n

s t a r t s t o d e v i a t e from the f u l l system s o l u t i o n s d u r i n g the t r a n s i t i o n period of

time

attributed to

when the l o a d i n t e n s i t y approaches z e r o . the f a c t

that

the p l a t e undergoes

This

may be

a t r a n s i t i o n from a

state of nonlinear forced v i b r a t i o n to nonlinear f r e e v i b r a t i o n .

For

h i g h e r a c c u r a c y o f r e s u l t s d u r i n g the t r a n s i t i o n p e r i o d , u p d a t i n g o f the b a s i s v e c t o r s o r m o d i f i c a t i o n o f the i n i t i a l b a s i s v e c t o r s i s r e q u i r e d .

7.11 Conclusion For

nonlinear

dynamic

analysis

of

a

thin

isotropic

d i f f e r e n t l o a d i n t e n s i t i e s and t i m e - v a r y i n g f u n c t i o n s ,

p l a t e with

the a c c u r a c y and

e f f i c i e n c y o f t h e r e d u c t i o n method a r e demonstrated. F o r moderate l o a d amplitudes,

satisfactory

results

can

be

obtained with R i t z

vectors

computed from the l i n e a r e l a s t i c s t i f f n e s s m a t r i x a l o n e , p r o v i d e d t h a t v e r t i c a l d i s p l a c e m e n t i s the o n l y q u a n t i t y o f i n t e r e s t .

The e f f i c i e n c y

and a c c u r a c y o f the method c a n be c o n s i d e r e d as the r e s u l t o f ( i ) the good performance o f R i t z v e c t o r s , ( i i ) the i n c l u s i o n o f n o n l i n e a r e f f e c t s i n the R i t z v e c t o r s b y performing a n o n l i n e a r (iii)

a

single

set

of

static

analysis a p r i o r i ,

vectors

being

used

throughout

the

complete

response and t h e r e f o r e r e q u i r e s no u p d a t i n g o f the b a s i s v e c t o r s , ( i v ) the s p e c i a l t r a n s f o r m a t i o n t e c h n i q u e . Although e n c o u r a g i n g r e s u l t s a r e o b t a i n e d i n the examples

shown,

the f u l l p o t e n t i a l o f method i n a n a l y z i n g g e n e r a l p l a t e - t y p e s t r u c t u r e s


136

with

complicated

general f i n i t e method

to

boundary

elements

non-linear

conditions

instead o f softening

can o n l y

finite

be

strips.

explored

by u s i n g

Extension o f

(e.g.elastic-plastic)

s u b j e c t e d t o loadings o f h i g h frequency i s a l s o f e a s i b l e .

the

structures


137

Appendix

Consistent mass matrix The k i n e t i c energy o f the p l a t e i s g i v e n by

( U ‘ ,tt

1/2 {

,t •

)dV }

,tt

(7.24)

Substituting equation (7.5) into the above equation, and after some algebraic

manipulation

and

simplification,

we

obtain

the

following

expression for TOP,

= 1/2 {

J

{ ip2 + ih2 )

[ u 2 + v 2 + w 2 + 17 t 3 / 315

x

y

3

- 4

t / 315 (

x , x

+ t 3 / 252 ( w 2 +

w2

,xx

y

, y

(7.25)

, y y

) ] dA }p

Similarly, we can derive

the following kinetic energy expression

for MODTOP,

=1/2 { -

J

[ u 2 + v 2 + w 2 + 17 t 3 / 315 ( A 2 + 入2 ) x

t 3 / 15

+ t 3 / 12

( A w x

(w2 +

,x

w2

+ w

A

,x x

+ A w y

,y

y

(7.26)

入 )

• y y

) ] dA }p

The derivation assumes that the density (p) is uniform through the thickness

and

initial

imperfection

is

ignored.

The

consistent

mass

matrix can be easily formed by putting the displacement functions into the corresponding kinetic energy expression.


138

Stiffness

matrices

If

the

D is

elasticity

t r a n s v e r s e shear component D

s

matrix

with

inplane

components

Dp

and

, that i s

0

(7.27)

the components o f t h e t a n g e n t i a l s t i f f n e s s m a t r i x can be w r i t t e n as

[K ]= L

L

[K ]= J N [K ]= (T

(7.28a)

L

[ B T D P B (5) L N

J

( w

,x

(a)DpB (6) ] dV

(5)DPB

, w

dV

,y

(7.28b)

(7.28c)

with

[ N ( x ) ] > (y)

[N (x )]殄:(y )

w h i l e the s e c a n t s t i f f n e s s m a t r i x i s g i v e n by

[K ] = 工 [ B 丁 D B + B

T

(6)DPB

+ B

T

D P B (5)

+B

T

For linear buckling analysis o f plates,

(5) D P B (5)

] dV

(7.29)

the g e o m e t r i c m a t r i x o f a

s t r i p can be w r i t t e n a s

(w

where

cr

,

< r

and cr

w ) dx dy ,y

r e p r e s e n t the i n i t i a l i n p l a n e s t r e s s e s .

(7.30)


139

Modified Suppose a d i s p l a c e m e n t v a r i a b l e u, which i s d e f i n e d i n t h e domain y €

[0,L]

, is

divided

into

m

equal

spline

sections,

see

Fig.7.17.

A

variable u can be written as the summation of m+3 spline functions, that is,

u(y) =

where

{a}

and

{0}

Y,

¢. =

{¢) {a}

(7.31)

> 0

The B 3 - s p l i n e f u n c t i o n , 、 , i s

(7.32)

given by

(y-y,

h

+ 3h (y~y

h

+ 3h (y

3h(y-y

f -

3(y_y

)、

(p (y) =l/6h" - y ) + 3h(y

- y ) - 3(y.

-y)、

(y … - y ) .

h = L / m = l e n g t h o f each s p l i n e s e c t i o n The unknown parameters o f

( 7 . 3 1 ) c a n be transformed t o a s e t o f

p h y s i c a l v a r i a b l e s which i n v o l v e t h e d i s p l a c e m e n t second d e r i v a t i v e s a t e a ch end o f t h e domain.

u and i t s f i r s t and

(The b a s i c concept o f the

m o d i f i e d s p l i n e f u n c t i o n i s f i r s t i n i t i a t e d by Fan and Luah f o r a n a l y s i s o f s h e l l s o f r e v o l u t i o n [116].

the

The concept i s now g e n e r a l i s e d

and a p p l i e d t o a s p l i n e f i n i t e s t r i p . ) The t r a n s f o r m a t i o n t a k e s the form

{oc} = [T] {p}

where {8}

(7.33)

'yy2

W i du/dy a t y=0

and

,yy2

d u/dy

a t y=L

etc.


140

r

1

h2/3

-h

1

0

-h2/6

1

h

h /3 0

zero 1

[T] 1

0 h /3

zero

-h2/6 h2/3

S u b s t i t u t i n g (7.33) i n t o (7.31),

we o b t a i n

u ( y ) = {(j>} [T] {/3} u(y) =

(7.34)

where The

set

displacement

of

unknowns

variables

at

{/3} the

of

equation

ends

and

(7.34)

internal

contains

physical

unknown parameters.

I n t e g r a t i o n o f the m o d i f i e d s p l i n e f u n c t i o n can be transformed i n the f o l l o w i n g manner, f o r i l l u s t r a t i o n ,

S (h T

For

analyzing

(h

the

multiply-connected

domain

in

Fig.7.18Âť

the

displacement variable u of a spline finite strip element is interpolated b y the standard 1-D shape functions in the transverse direction and the modified

Bs-spline

functions

in

the

longitudinal

direction.

The

displacement across the interface between elements 1 and 2, which are connected end-to-end, can be defined uniquely displacement variables at

the

in terms of the physical

two connecting nodes. As a consequence,

the modified spline finite strip can be assembled manner

with

element 2.

C 2 -continuity

across

the

in a finite element

boundary between element

1 and


Algorithm

for

generating

Ritz

vectors

T

i

G i v e n mass, s t i f f n e s s m a t r i c e s M,K and l o a d Solve f o r f i r s t vector M—normalize

K

厂* = 1

厂 二 1

S t a r t i n g w i t h 1=2 Compute a d d i t i o n a l v e c t o r s For

1,2’

-1

M-orthogonalize using the modified Gram-Schmidt process

M—normalize where

c

= rTM

j J 广*(j+i>


142

Redaction The

fundamental

idea

of

reduction

can

be

interpretated

as

the

a p p l i c a t i o n o f the G a l e r k i n method t o a d i s c r e t e system o f e q u a t i o n s . C o n s i d e r the s t i f f n e s s matrix e q u a t i o n ,

[K]{u} :

where

{f}

(7.36)

[K] i s a MxM s t i f f n e s s m a t r i x ,

the f o r c e v e c t o r .

In the f o l l o w i n g d i s c u s s i o n , t h i s e q u a t i o n i s d e f i n e d

i n mathematical sense as

K u =f ,

u € D , f € R

where

: D

K

{u} the displacement v e c t o r and {f}

岭 R is

an o p e r a t o r e q u a t i o n ,

that i s

(7.37)

linear,

positive definite

and s e l f - a d j o i n t .

o p e r a t o r K c a n be c o n s i d e r e d as a mapping from a

The

v e c t o r space D ( the

domain ) t o another v e c t o r space R ( the range ) . F i r s t t h e i n n e r p r o d u c t o f any two v e c t o r s ,

v and w, i s d e f i n e d as

a s c a l a r q u a n t i t y denoted by

(7.38)

The reduced b a s i s t e c h n i q u e s t a r t s o f f w i t h an a p p r o x i m a t i o n t o u o f t h e form (7.39)

where {0}Ni^ ―i1s with N « The

the

set

of

basis

vectors

spanning

the

subspace S

M. Galerkin

approximation

minimizing the error

seeks

for

the

parameters

N {a} i=i

by

(u - U ) under the energy norm

|丨 u - U 丨| k = [ < K ( u - U ) , ( u - U)> ] 1 / 2 U s i n g the fundamental p r o p e r t i e s o f

inner products

(7.40) and s e l f - a d j o i n t

l i n e a r o p e r a t o r s , i t can be shown t h a t

| | u - U | |^ = <K i u 〉 - 2 < Ku,U > + < KU,U >

(7.41)


143

S u b s t i t u t i o n o f ( 7 . 3 9 ) i n t o (7.41) y i e l d s

11

u

2 =

U

K u

< ,

u >

N

一 2[

N

、 < i =1

N

Ku,、〉

i =1k = l

U ||2 has a minimum a t the same p o i n t as j| u -

Since 丨 卜 -

U

| |

K

K

t h e s q u a r e d measure i s u s e d i n the s e a r c h f o r the

minimum p o i n t . T h i s

measure

coefficients

can

be

minimized

with

respect

to

the

s e l e c t i n g t h a t s e t o f v a l u e s f o r which

cc

j

bvy

d Combination o f equations

( 7 . 4 2 ) and

(7.43)»

and making use of equation

(7.37),leads to N

,<!> i i j j Jj i=l Furthermore, this equation can be written in the usual form : [K]

= {f}

where

(7.45)

[ K T = [r] T [K] [ r ] , { f } ^ [ r ] T { f } and [F] = N and

1

2

N


144

O

O C o E 3 Z

f t c

o § 2 ^

o

+

h-

o 〇

T-

S §

UOIiOO|^©p

*

扪 H

o O

§

h2

CM H

+ i O ?

^ H 2 }

O o 广 H 木

O O • •H D < m u a-p 4-^ •H | C uO o -P -P O W I s c D < Im dc x o a 4 > w u +> D < s o W D t„ CC o —SP C L i c 5 D <W 「 r-4 O S DU, tw 0) c -O O u 5H TJ O P Q C : O r-4 ^ S J t C ! 1 tog 一—ii 0) S > < D " O 4-> u O T? 0 3 '-‘ 0 ) •-» N a „ •H II O i—( C) Ja U O g ( 0O • * " * < ^ O -p o tr^H 2 W H J t

f I E z

UOI1O0U0D POZIIBIUJON

D <

C

M

o

H2

t n t n

ci •H u.


145

S . S

Hole

S . S

â&#x2018;ť

s.c. (b)

S . S

s.c. S . S .

Fig.7.

Fig.7.3

(a) A s q u a r e p l a t e w i t h a s q u a r e h o l e o f w i d t h 0 . 5 a . Exterior boundaries o f the p l a t e are s u p p o r t e d , ( b ) The s p l i n e f i n i t e s t r i p m e s h and t h e c o r r e s p o n d i n g boundary c o n d i t i o n s f o r a q u a d r a n t S S . = s i m p l y - s u p p o r t e d ; S . C . = s l i d i n g clamped; F=Free

F i n i t e s t r i p mesh o f a s q u a r e p l a t e w i t h d a m p e d edges o r s i m p l y - s u p p o r t e d e d g e s a n d s u b j e c t e d t o u n i f o r m l y d i s t r i b u t e d l o a d , a = 3 0 0 i n , t =3 i n , 0.3e8 l b / i n 2 , v = 0.316


146

2.2

0.6

100

300

200

TOP (4*4)

400

500

HIGH ORDER ELEMENT

(b)

2.6

0

90

10

Fig.7.4

丁OP (4*4-)

HIGH ORDER ELEMENT

L o a d - d e f l e c t i o n o f a square p l a t e w i t h a/h=5 and p=0.333. Inplane displacements a r e clamped a t t h e edges f o r c a s e (a) and f r e e f o r case ( b ) . (a) clamped (b) simply-supported


147

〔a)

0.9

0.8 0.7 0.6 0.5 0.4 0.3 0.2

0

100

200

+

TOP ( 4 * 4 )

300

400

500

MODTOP (4*4>

(b) 7

6

5

4

3

2

1

0 Q (*1000) 口

Fig.7.5

LEVY

+

TOP (4*4)

MODTOP (4^4)

Load-deflection of a square plate with a/h=100. All inplane displacements are clamped at the edges, (a) clamped (b) simply-supported


148

1(b)

11(b)

S=simply supported C=clamped

square plate with span = 400in thickness =1.5in E=30e6 lb/in2 Fig.7.6

Poisson ratio = 1/3

A square p l a t e s u b j e c t e d t o edge compression along Y-axis.


149

(a) 4.5

4

3.5 3

±N323uvldsla

2.5 2

5 1 0.5 0

0

20

40

30

50

UNIFORM EDGE COMPRESSION V Yomaki

+

TOP (4>4)

O

TOP

(8*8)

A

MODTOP (4*4)

(b) 2.4 2.2

0.8 0.6

0.2

10

0

20

50

40

30

60

UNIFORM EDGE COMPRESSION V 口

Fig.7.7

Yamaki

+

TOP (4M)

Out-of-plane compression,

O

TOP (8^8)

deflection (a) Case 1(b)

A

MODTOP (4*4)

with Increasing , (b) Case 11(b)

edge


Loading P(x,y,t) = P(x,y)T(t) T(t) = time varying function P(xďź&#x152;y) = spatial distribution Load

Deflection Fig.7.8

(a) Loading function o n a plate (b) Nonlinear stiffening behaviour of a plate" K = Initial linear elastic stiffness matrix o K = tangent stiffness matrix corresponding displacement at the maximum load level.


151

)C©UJ9 0

I 10

20

30

40

I 60 50

70

80

I 100 90

I 120 丨 140 110 130

I 150

Time steps •

V

Full

p=4

(b) 0.0005

0 -0.0005 一0 . 0 0 1

-0.0015

ICQUJSuoldsj'Docoldcl

一0 . 0 0 2

-0.0025 -0.003 -0.0035

" 1 " 2 0 I 4 0 “ I 6 0 I 10 30 50 70

80

I 1 00 I 1 20 I 90 110 130

140

I 150

Time steps

Full

p=4

& c q=4

Fig.7.9 Response of the square plate subjected to a sinusoidal load of maximum amplitude lOq • t=0.002 sec p e r time step. O (a) vertical displacement response (b) inplane displacement response, p = n u m b e r of Ritz vectors (linear vectors) computed from the initial linear elastic stiffness matrix, q = number of Ritz vectors (nonlinear vectors) computed from the tangent stiffness matrix at maximum load level.


Time steps 口

full

+

p=2 .

p=3

p=4

X

p=8

V

p=4

q=4

(b) 0.0005

•0.0005

0.001

】 0.0015

0.002

0.0025

0.003

0.0035

20

40 50

30

60

70

80

90

100

120

140 130

150

Time — s t e p s •

Full

p = 4 8c q = 4

Fig.7.10 Response of the square plate subjected to a half-sine load of maximum amplitude lOq^. t=0.002 sec per time step, (a) vertical displacement response (b) inplane displacement response. Legend same as Fig.7.9

CSU


c 0 )

E

u O OaC • 6 o V L .

0

50

100

150

200

250

300

Time steps 口

Full

+

p=8

p = 4 Sc q = 4-

Cb]

U.UU I

一0 . 0 0 1

一0 . 0 0 2

一 0.003 -0.004

C >a E 0 ) KJ 0 a ( / ) • " 5 0 ) c o a c

—0.005 一0 . 0 0 6

—0.007 一0 . 0 0 8

-0.009 -0.01 一0 . 0 1

1

Time s t e p s 口

Full

p = 4 8cq = 4

Fig.7.11 Response of the square plate subjected to a load of maximum amplitude 50q . t=0.001 sec per time step. O (a) vertical displacement response (b) inplane displacement response. Legend same as Fig.7.9 sinusoidal


100

50

150

200

250

300

Time steps full

X

p=8

V

p=4

&c q=4

(b) 0.001

-0.001 -0.002 -0.003

c o E O KJ 0 Q. (/)

• " 6 Q) c 0 a c

- 0.0 0 4 -0.005 -0.006 -0.007 -0.00S -0.009 - n

01

-0.011

0

50

100

150

200

250

30C

Time s t e p s

full

p=4 &. q=4

Fig.7.12 Response of the square plate subjected to a ha If-sine load of maximum amplitude 50q . t=0.001 sec per time step, (a) O vertical displacement response (b) inplane displacement response.Legend same as Fie.7.9


155

h 2

l j

5 u 5 O (/)l 5

<

o u >

(b]

.002

.001

0 .001 .002 .003 •0.004 •0.005 2 u i 2 Ld o

0.007

l c n ( Q d L 2

•0.01

5

C L 2

0.012

0.014

100 TIME STEPS Full

V

d= 4

<S£ 0 = 4

Fig.7.13 Response of the square plate subjected to a step load of maximum amplitude 50q • t=0.001 sec per time step, (a) vertical O displacement response (b) inplane displacement response.Legend same as FIR.7.9


156

(a)

4

3

2

z u 2 u o

1

CL

0) 5

<

o

0

u 1

2

0

50

100

150

200

250

300

TIME STEPS Full

p = 4 &c. q = 4

(b)

0.002 0.001

0 •0.001 •0.002

0.004

H bJ 2 Ld 0 1 0 ( / ) D d L Z

5 z 0-

-0.009 •0.01 •0.01 1

0.013 •0.014

0

50

100

150

TIME STEPS Full

p = 4 & q= 4

Fig.7.14 Response of the square plate subjected to a step™ load of maximum amplitude 50q。. t=0.001 sec per time step, (a) change v e r t i c a 1 displacement response (b) inplane displacement response, Legend c;a mA Fiec.7.9


157

l~N3230vndsla

0

50

TIME STEPS Full

V

p = 4 Sc q = 4

(b) 0.002 0.001 —

一0 . 0 0 1

一0 . 0 0 2

一0 . 0 0 3

l23so3dso LL1NV1CINI

- 0 . 0 0 4

- 0 . 0 0 5

-0.006 - 0 . 0 0 7

-0.008 - 0 . 0 0 9

-0.01 -0.01 1 -0.012 5 0

1 0 0

1 5 0

2 0 0

2 5 0

3 0 0

TIME STEPS Full

Fig.7.15 t r i a n g u l a r load o f

V

p= 4

& < q=4

Response of the square p l a t e s u b j e c t e d t o a maximum amplitude 50q。. t=0.001 sec per time step.

(a) vertical displacement response response.Legend same as Fig.7.9

(b)

inplane

displacement


158

z J L 2 u Q.

n c o < o s u

5 0

1 0 0

1 5 0

2 0 0

2 5 0

3 0 0

TIME STEPS Full

V

p=4 & q = 4

Cb)

o.ooi 0 •0.001 •0.002

bJ

2 id O 5

•0.004

C L

.0.006

5

•0.007

(/)

J U z Q.

z •0.01 •0.01 1 •0.012

100

1 5 0

TIME STEPS Full

p=4

q= 4

Fig.7.16 Response of the square plate subjected to a exponential load of maximum amplitude SOq^. t=0.001 sec per time step. (a) vertical displacement response response•Legend same as Fig.7.9

(b)

inplane

displacement


•Integration reQion (b)

Fig.7.17

(a) A t y p i c a l Ba-spline f unc t i o ns (b) a t y p i c a l Integration region and the splines that intersect i t .

‘‘

CNJ -M

c

0)

E0)

-f-*

c ①

TT"

CO

C ①

c ①

m

CO

£ ① l O

E0) E0)

C(2)-continuous Fig.7.18

Discretization of a multi-connected domain into spline finite strips with modified spline function. Each finite strip can be treated as a finite element for assembly. Displacement unknowns at node k of element i are (u) , rI cI


160

CHAPTER 8 A GLOBAL-LOCAL SPLINE FINITE PRISM METHOD 8.1 Introduction A g l o b a l - l o c a l s p l i n e f i n i t e prism method i s presented to analyze t h i c k laminated composite p l a t e s . In t h i s study, a t h i c k laminated plate is

treated

as

a

3-D

nonhomogeneous

anisotropic

elastic

body.

The

m a t e r i a l v a r i a t i o n w i t h i n the c r o s s - s e c t i o n can be e a s i l y taken into account

by

dividing

elements,see F i g . 8 . 1 . interpolated

by

the

the

cross-section

Displacements B3-spline

into

along the

function

such

conventional

span o f that,

8-node

the p l a t e are

unlike

classical

f i n i t e prism method, boundary c o n d i t i o n s other than two simply-supported o p p o s i t e ends can be treated. spline f i n i t e

prism

can

be

The displacement f u n c t i o n s o f an 8-node

expressed

as

the

product

of

isoparametric shape f u n c t i o n s and the s p l i n e f u n c t i o n .

the

8-node

However, f o r a

laminated t h i c k p l a t e w i t h a large number o f l a y e r s , the method becomes computationally expensive as the number o f s p l i n e f i n i t e prisms depends on the number o f l a y e r s o f m a t e r i a l .

To circumvent t h i s problem,

a set

o f g l o b a l polynomials o f an appropriate order i s used to transform the nodal

variables

of

the

cross-section

to

a

much

smaller

set

of

g e n e r a l i z e d parameters a s s o c i a t e d w i t h the polynomials. The t o t a l number o f unknowns involved , which i s independent o f the number o f l a y e r s o f m a t e r i a l s , i s d r a s t i c a l l y reduced when compared with a complete 8-node s p l i n e f i n i t e prism a n a l y s i s . The philosophy o f t h i s method i s t o t r e a t the l o c a l v a r i a t i o n o f m a t e r i a l s (or geometry such as the hollow tube i n F i g . 8 . 2 ) u s i n g a f i n e mesh o f d i s c r e t i s a t i o n while the g l o b a l behaviour i s captured by means of

some g l o b a l functions(see[131]

and references t h e r e i n ) .

The nodal

v a r i a b l e s are transformed to the g e n e r a l i z e d parameters a s s o c i a t e d with the g l o b a l f u n c t i o n s .

As the number o f

the generalized parameters i s

much smaller than the number o f nodal v a r i a b l e s , the number o f unknowns t o be s o l v e d i s r e l a t i v e l y small. Because the l o c a l and g l o b a l behaviour are

embedded i n a small set o f g e n e r a l i z e d parameters,

a

compromise

between computational cost and accuracy can be accomplished. Numerical examples a r e g i v e n to demonstrate the accuracy and e f f e c t i v e n e s s o f the global-local

spline

fundamental n a t u r a l

finite

prism

frequencies

t h i c k , laminated composite p l a t e s .

in

and

predicting the c r i t i c a l

the

stresses

buckling

, the

loads

of


161

8. 2 Formulation A t h i c k p l a t e w i t h a r b i t r a r y c o n d i t i o n s of

support i s given In

F i g . 8 . 1 . The c r o s s - s e c t i o n i s f i r s t d i v i d e d i n t o the usual 2-D elements. The i n t e r p o l a t i o n w i t h i n each element i s d e f i n e d by the isoparametric 8-node shape f u n c t i o n s ,

.The nodal displacements o f the e n t i r e

c r o s s - s e c t i o n are then assumed to f o l l o w a p a t t e r n of v a r i a t i o n which i s d e f i n e d by the g l o b a l f u n c t i o n s Q ( x , 2 ) .

Along the spanwise d i r e c t i o n ,

(p~3) e q u a l l y - s p a c e d B s - s p l i n e s e c t i o n s a r e used. The displacements o f

the g l o b a l - l o c a l s p l i n e f i n i t e prism can be

w r i t t e n as p u = J] {oc} P ( x , z ) •沴.(y)

(8.1a)

v = E {p}.P(x,z)^ (y) i i i =1

(8.lb)

p

w = J i =1

p(x’z).0 (y) i i

where

P(x,z) = { P

and

P

1

, P

2

, P

(8.1c)

,

3

P

}

(8. Id)

q

ne j

= E [ k=i

m

刀 ) ] {

qk}.

J

{ q } = nodal displacements of the element k computed from k j the j-th term of the global function Q(x, 2). ne = number of elements

(y) represents the i-th term of the spanwise Bs-spline function with ,

and

.

being

the

corresponding

unknown

displacement

parameters. Kinematic boundary conditions at y=〇 and y=a are taken into account by modifying the corresponding local splines. The global polynomial

functions

are expressed as

the Kronecker

product of two series, Q ( x , 2 ) = X(^)-ZCz) where

X(>:) = {

and

Z(2)

=

, (1+^)/2

{ 1 j2 , 2 , . . . ’ 2

(8,2) , (/-¾)

2 € [0,h]

c)


162

with

x = 2 x / b - 1

义€

c = 1

if r is even

c = ^

if r Is odd

[-1,1]

x e [0,b]

q = (r+1)(s+1) = t o t a l number of terms in the global function The series in X-direction is so chosen that the kinematic boundary conditions at the two ends of X-axis can be easily treated by removing appropriate unknown displacement parameters corresponding

to the first

two terms in the series. Note that no kinematic boundary conditions are imposed on the top and bottom surfaces of the plate and a simple power series

in

Z-direction

suffices.

By

varying

the

parameters

r and

s,

different orders of polynomials can be used in the X- and 2-directions. As thick plates are treated as 3-D elastic bodies

in this study,

the strain-displacement functions can be written as

(8.4)

Using displacement-based formulation, banded

nature

of

B^-spline

functions,

and taking advantage o f the one

can

easily

derive

the

s t i f f n e s s matrix o f the e n t i r e p l a t e , g i v i n g , [S].

[引〗

[s]

[S]

[S]

[B] T [D][B]

w i t h [S]

[

[ S

u

]

[V L[ V

and

[S

(8.5) ^

[S]

p,p_3

I

dV

12

]

[So 1 22

[S_] 32

[S ] 13 [S

23j

1

^331 J

(8.6)


163

[S

0

11

D P P dx dz + 11 ,X ,X p (

dy

D

[s 1 2

<p (p* dy m n J

[S

沴0

[S. 21

f

m n

22

p (

mn

D

> ! <

mn

each o f

D

¢), dy

P

22

P

23

31

D

33

P

,2

P

P

J

,z

T

P

. z

, z

P ,z d xd z

, z P

P

D P 44

P dx dz

D 44

m n

dx dz +

0

dy

66

+

p

P

,2

dx dz

,x

P

4

dx dz

x

D P P dx dz 44 ,X

沴沴,dy

+

P dx dz

P ,z

dy

dx dz

d x d z

D P 3 2 , z

(j) dy

0

Unlike

dy

P

D 55 D

P

,x

P

D

.2

,x

P

13

dy

"

32

33

^

(p } (j>

[S 一 23

31

D

dy

¢) d y

[S.

12

<j6,¢, dy

[S_]

P

66

0' 0'

dy

n

D

J

44

P

P

,x

,x

dx dz

dx dz

+

「 沴 沴 , d y「

J

m

dx dz + , x

P dx dz +

dx dz + , z

^

m

D

n

n ti

n

0 ' 沴

T

dy

d y

j

P d xd z

55

dy J

n

沴 , 余

P

J

D 66 D

P 55

, z

P P ,x ,z P

dx dz

dx dz ^z

D P T P dx dz 55

D P P dx dz 66 ,x ,x

dy

isotropic thick plates,

the m a t e r i a l c o n s t a n t s , D " ,

for

the 8-node s p l i n e f i n i t e prisms are not n e c e s s a r i l y I d e n t i c a l

and v a r y from l a y e r to l a y e r i n a laminated composite p l a t e . For f r e e v i b r a t i o n of plates,

the c o n s i s t e n t

mass matrix can be

d e r i v e d from the same displacement f u n c t i o n s , g i v i n g

[M]

[M]

(8.7)

[M]

[M]

[M]

[M [M]

i j

ii

] 0

P’P-3

[M]

p,p

0

[M

0

0

22

]

0 [M ] 33

(8.8)


164

and [M . ]

For

• buckling

P

0

of

a

plate

, cr

initial stresses ( c r [G]

P dx dz

under

constant,

uniaxial

or

biaxial,

) , the geometric matrix can be written as

,[G],

[G]

[G] [G]

[G]

[G 0 J

(8.9) [G]

〕,p-3

0

22

0

(8.10)

[G

33*

and [G.

dy

¢)沴

xO

P

dx

P T P dx dz

¢, 0' dy All

P

integrations are carried out throughout

the entire volume of

the plate. During computation, integration of the spline functions is done

in

sectional

the

usual

manner

interpolation

[13,14]

P(x,z)

is

while

integration

carried

out

by

of an

the

cross-

elementwise

summation of the transformed integral of the 8-node shape functions. For example,

1 1

12

(8.11)

P T P dA

q2

qq

and P dx dz = V {5} ( j ^ ik J k=l where

[Mvm

dA)

>

(8.12)

ne= the total number of elements in the cross-section, b = width of plate h = thickness of plate A = area of element k


165

The

vectors

{

6,

and

{ 6.

contains

the

nodal

displacements

of

element k computed from the i - t h and j - t h terms of the g l o b a l functions Q(x,z) respectively. The integral of the shape functions [N] is carried out

numerically.

At

this

stage,

it

is

worth

pointing

out

that

the

formation of the stiffness matrix of the 8-node spline finite prism is not necessary. For plates subjected

to a vertical

load, f(x,y)=f(x)*giy), acting

on the top surface, the corresponding load vector is given by

{F}={{F} i ,{F} 1 2 where {F}

m

=

{

「 H x ) J

F

m

}

}

T

(8.13a)

p

P(x,z)| dx「茗(y) 1 z=h J m

A f t e r s o l v i n g the matrix equation f o r s t a t i c a n a l y s i s , the s t r e s s e s a r e c a l c u l a t e d at the Gaussian i n t e g r a t i o n p o i n t s w i t h e x t r a p o l a t i o n to nodal

points

vibration

via

the

standard

and b u c k l i n g a n a l y s i s ,

bilinear

shape

functions.

the e i g e n p a i r s a r e

For

free

computed by

the

subspace i t e r a t i o n method.

8 . 3 Numerical Examples

Static To e s t a b l i s h the accuracy o f the present method,

we s h a l l consider

a s e r i e s o f b i d i r e c t i o n a l laminated square p l a t e s w i t h edges o f length a.

Emphasis i s p l a c e d on the a n a l y s i s of p l a t e s w i t h s m a l l span-to-

thickness

ratio.

Results

are

compared

with

the

exact

elasticity

s o l u t i o n s [117,118] and a three-dimensional f i n i t e element s o l u t i o n due to Barker [119]. In the f o l l o w i n g examples, the top s u r f a c e loading i s d e f i n e d as

q = q s i n(

where q i s the

maximum i n t e n s i t y o f the double s i n u s o i d a l load at the

c e n t r e o f the p l a t e . A l l edges o f the p l a t e s are e i t h e r simply supported o r clamped.

By symmetry, o n l y a q u a r t e r of the p l a t e i s analyzed. Each

model i s d i v i d e d i n t o f o u r s p l i n e s e c t i o n s of equal l e n g t h along the span.

Each p l y

composes

of

orthotropic material

with

the

material

p r i n c i p a l a x i s 1 o r i e n t e d a t an angle o f 0° or 90° to the global X-axis.


166

The material p r o p e r t i e s are defined as follows, E l l =25 * 10 o psi

E22 =E33 = 1 * 1(1)6p s i

G12 =Gl3 = 0 . 5 * 10 6 psi

G23 = 0 . 2 * 10 6 psi

v^2 =P23 = V13 = 0 . 2 5 where

1 signifies

the d i r e c t i o n p a r a l l e l

to

the f i b e r s ,

and 2 the

transverse d i r e c t i o n . The

following

normalized

quantities

are

defined

for

the

presentation o f r e s u l t s . (crx

cry

Txy) = (<rx

(Tyz

Txz)

= (Tyz

cry

TxyJ/qs1

Txz) /qs

w =Tr 4 Qw / 12s4hq Q

= 4G

[E

12 +

11 + E 2 2 ( 1 ^ 2 2 ) ] / ( 1 - , 1 2 , 2 1 )

s=a/h z = z/h - 1/2

A laminated

composite

plate

with

three

layers

( s = 4

)

Is used

edges A three-ply symmetric laminate convergence of

the present method. All

to

test

the

layers are of equal thickness

with ply angles 0°/ 90°/ 0°. Particular attention

is focused

on

the

order of the global functions (s) used across the thickness. A fourthorder polynomial

(r = 4) is used across the width of the plate. In order

to approximate the nonlinear layerwise distribution of transverse shear stresses,

the

cross-section

thickness of each

is

layer divided

divided into

into

a

two element

6x6

mesh

with

the

layers. Results are

shown in Table 8.1. An approximate solution is obtained when s=7 with all stresses within 6 percent difference from the exact solutions. this case,

In

the number of unknowns within the cross-section is reduced

from 399 nodal variables, if 8-node spline finite prisms are used alone, to

96

generalised

parameters,

which

implies a 76 percent

reduction.


167

Higher accuracy can be obtained by increasing the order o f the global functions,

and the

number of

unknowns w i l l ,

therefore,

depend on the

d e s i r e d l e v e l o f accuracy. To demonstrate the importance of the order of g l o b a l functions i n p r e d i c t i n g s t r e s s e s f o r p l a t e s with d i f f e r e n t thickness, an analysis i s a l s o performed on the same p l a t e with reduced t h i c k n e s s 〔 s = 1 0 〕 . The same discretisation and global functions are employed except that only a quintic polynomial

(s=5) across

the

thickness

is used

in

this case.

Comparisons of the results with the exact solutions are shown in Table 8.2. A good agreement between the two sets of results can be observed; all stresses are within 5 percent difference from the exact solutions. An 82 percent

reduction

in

the number of unknowns within

the cross-

section is achieved in this analysis.

士丄)"U'土;)Txy(0,0,±;) TxzCo, ? # 0 ) w(?f ? # 0 )

Exact

0.750

0.482

1. 171

-0.0479

-0.705

-0.504

-0.166

0.0473

0.767

0.494

1.010

-0.0490

-0.723

-0.519

-0.005

0.0485

0.787

0.502

1.016

-0.0503

-0.743

-0.527

-0.015

0.0498

0.801

0.534

-0.755

-0.556

1.0

.210

1.896

•2 4 3

1.926

• 270

-0.0511

.961

2.02

.256

0.0505

Table 8.1 Comparison of results for a three-ply laminate with span-to-thickness ratio of 4. *

The

due

to

deflections Liou

and

are Sun

compared with using

a

the

hybrid

results element

approach. In this case, the normalized deflection is defined as w = 100 E22 w / q h s 4


168

cx(H±;)

s = 5 Exact

cry (?,

) crz(?#?,±^) T x y i O . O ^ p Txz(0,?f 0) w(? f ? f o)

0.586

0.277

0.983

-0.0284

-0.585

-0.281

-0.022

0.0285

0.590

0.285

1.0

-0.0289

一 0 . 5 9 0

-0.288

0.0

0.0289

0.340

0.7323

0.357

0.7546^

Table 8.2 Comparison of r e s u l t s f o r a three-ply laminate with span-to-thickness r a t i o of 10. *

The d e f l e c t i o n s

due

to

Liou

and

are

compared with the r e s u l t s

Sun

using

a

hybrid

element

approach. In t h i s case, the normalized d e f l e c t i o n i s defined as w = 100 E22 w / q h s 4

A To demonstrate

the a p p l i c a t i o n of

the present method i n dealing

with boundary c o n d i t i o n s other than simply-supported edges, ply

laminate

clamped.

(s=4)

in

the

last

example i s

Analytical e l a s t i c i t y solution i s

the three-

analyzed with a l l not

edges

a v a i l a b l e , r e s u l t s are

compared with a three-dimensional f i n i t e element s o l u t i o n due to Barker [119] . The

same g l o b a l polynomial with orders r=4 and s=7 i s used i n

t h i s example.

In order to obtain an accurate transverse shear stresses

d i s t r i b u t i o n , a f i n e r mesh of 9x9 elements, with each p l y layer divided i n t o three element l a y e r s , i s used to d i s c r e t i s e the cross-section. By r e f e r r i n g to F i g . 8 . 3 , i t is apparent that the agreement between the two sets

of

results

is reasonably good.

In

this example,

the

number

of

unknowns within the cross-section is reduced from 840 to 96 compared to a full 8-node spline finite prism analysis. A Laminated composite plate with five layers A five-ply laminate( 5

=4 )

with ply angles 0°/ 90°/ 0 / 90 /

0 0 i s analyzed. Unlike the previous examples, the total thickness of the 0° and 90° layers are the same, whereas layers at the same orientation have equal thicknesses. Under these conditions, the effective laminate stiffness in the X- and Y- directions are the same. The cross-section is discretised into a 10x10 mesh with each layer of materials divided into two element layers. Using a global polynomial of orders r=4 and s=7 in


169

t h i s analysis,

a reasonable agreement w i t h the exact s o l u t i o n s can be

observed from Ta ble 8 . 3 ; a l l s t r e s s e s are w i t h i n 4 percent d i f f e r e n c e from the exact s o l u t i o n s f o r 5=4. The same f i v e - p l y laminate w i t h a s p a n - t o - t h i c k n e s s r a t i o o f 2 i s a n a l y z e d . Although a f i n e r mesh w i t h h i g h e r - o r d e r g l o b a l polynomials i s more a p p r o p r i a t e t o t h i s r e l a t i v e l y t h i c k e r p l a t e , i t i s I n t e n t i o n a l to use the same d i s c r e t i s a t i o n and g l o b a l polynomials as the previous case. D e s p i t e the f a c t

that

a r e l a t i v e l y c o a r s e mesh w i t h low-order g l o b a l

p o l y n o m i a l i s used, r e s u l t s i n T ab l e 8 . 3 show that the s t r e s s e s obtained by the p r e s e n t method a r e s t i l l w i t h i n 6 percent d i f f e r e n c e from the e x a c t s o l u t i o n s , except f o r the l o n g i t u d i n a l s t r e s s e s cr d i f f e r e n c e o f 9 . 9 percent i s found. nodal

variables

within

the

where a maximum y a t o t a l o f 1023

In t h i s example,

cross-section i s

contracted

down

to

96

g e n e r a l i z e d parameters a s s o c i a t e d w i t h t h e g l o b a l polynomials.

±1)

T x y (0^

|

:)Txz(0

))v/(:

!。 , )

span-to-thickness ratio

0.612

0.997

-0.0378

-0 638

-0.606

-0.004

0.0369

,685

0.633

-0.0394

,651

-0.626

0.0384

672

Exact

.245

• 070

.238

• 291

.226

11.631

• 227

12.278

span-to-thickness ratio = 2

Exact

1. 286

0.902

1.001

-0.0792

-0. 863

-0.776

0.001

0.0595

332

1.001

903

-0.848

1.0

-0.0836 0.0634

Table 8.3 Comparison of results for a five-ply laminate.

7ibration and Suclc』ing anaiysis We

shall

consider

another

series

of

cross-ply

laminated

square

plates with available elasticity solutions [120,121】 for free vibration


170

and l i n e a r e l a s t i c s t a b i l i t y a n a l y s i s . A l l simply

supported.

uniaxially.

For

buckling

Each p l y i s

global X-axis. The

analysis,

composed of

material p r i n c i p a l a x i s

edges the

of

the plates

plates

are

orthotropic material

1 oriented at

an angle of

total thickness of

loaded

with

0° or

the 0° and 90°

are

90°

the

to

the

layers are

the

same, whereas layers at the same orientation have equal thicknesses. The material

properties

and

the

geometry

of

the

plates

are

defined

as

follows,

e l 1 = 40 e22 = 40 e33 G12 = G13 = 0.6 Ezz

G23 = 0.5 £22

V12 = P23 = P13 = 0.25 span-to-thickness ratio = 5 =10 where

1 signifies

the

direction

for vibration analysis for buckling analysis parallel

to

the

fibers,

and

2

the

transverse direction. In all cases, only a quarter of the plate is analyzed because of the

symmetric

properties.

Each

model

is

divided

into

four

spline

sections of equal length along the span.As we are only interested in the prediction

of

the

global

responses

of

plates with

relatively

large

span-to-thickness ratio, when compared to those in the static analysis, we

anticipate

that

a

relatively

low-order

polynomial

across

the

thickness is sufficient. To see this, a global function, with a quartic polynomial (r=4) across the width and

a cubic polynomial (s=3) through

the thickness of the plates, is used. The

following

normalized

quantities

are

defined

for

the

and

the

Very

good

presentation of results. Natural frequency

a) = 10

c o h V(p

/

E22)

N = Nx a 2 / ( E22 h 2 )

Critical buckling load

where p is the density of the composite material. Results elasticity agreement

of

comparisons

solutions between

the

between

[120,121] two

sets

the

are

listed

of

results

present in

method

Table

can be

8.4.

seen;

all

present

results are within 2 percent difference from the elasticity solutions of Noor.

The efficiency of

referring

to

the

figures

the present contained

method in

are also demonstrated by

Table 8.5 which compares

the

amount of reduction in the number of unknowns involved in the analyses.


171

Lamination

NL

Frequency to

Buckling load N

Present method

Present method

Elasticity solution [40]

Elasticity solution [41]

Antisymmetric

10 Symmetric

3.4918 4.3289 4.5225 4.6398

4250 2719 5091 6498

11.0298 21.7346 23.9024 25.0857

23.6689 25.3436

4.3056 4.5577 4.6602

4.3006 4.5374 4.6679

23.1172 24.9463 25.5184

22.8807 24.5929 25.3436

Table 8 . 4 Comparison o f

10.8167

r e s u l t s f o r f r e e v i b r a t i o n and buckling

a n a l y s i s o f c r o s s - p l y laminated square p l a t e s .

ND0F2

ND0F1

Mesh c o n f i g u r a t i o n

Number o f l a y e r s

Number o f 8â&#x20AC;&#x201D;node elements

x 4

195

48

8 x 8

675

48

675

48

10 x 10

1023

48

6 x 6

399

48

9 x 9

840

48

10 x 10

1023

48

8â&#x20AC;&#x201D;

10

x

8

Table 8.5 Comparison

of

the

number

of

unknowns within a

number

of

nodal

variables

cross-

section Note :

ND0F1= section

for

a

complete

8-node

within

spline

the

cross-

finite

prism

within

each

analysis; ND0F1 = number of nodes x 3 ND0F2

=

the

cross-section

number

using

the

polynomial of r=4 and s=3

of

unknowns

present

method

for all cases

with

a

global


172

8 . 4 Conclusions It solutions

has

been

for

a

demonstrated rectangular

that

approximate

laminate

with

three-dimensional

arbitrary

conditions

of

support can be obtained by the present g l o b a l - l o c a l s p l i n e f i n i t e prism. When

the

number

of

layers

i nvo l ve d i s

large,

the

number

of

nodal

v a r i a b l e s w i t h i n the c r o s s - s e c t i o n f o r a f u l l 3-D a n a l y s i s using 8-node spline f i n i t e

prism

alone

is

extremely

large

and

the

global-local

approach would be more e f f e c t i v e i n the sense that a compromise between computational c o s t and accuracy can be

achieved. From a computational

p o i n t o f view, the assumed displacement f u n c t i o n s are constructed on the i d e a o f s e p a r a t i o n o f v a r i a b l e s such that the i n t e g r a t i o n o f the s p l i n e function

can

programming

be

and

done

separately.

accelerates

the

This

facilitates

establishment

of

the

the

final

computer matrix

equation. It

should

also

be

mentioned

that

if

the

cross-sectional

displacement i n t e r p o l a t i o n s o f a p r i s m a t i c 3-D s t r u c t u r e are defined i n terms o f

the g l o b a l polynomials alone,

the complicated 2-D plane o f

the r e s u l t i n g i n t e g r a t i o n over

the c r o s s - s e c t i o n

(such as an i r r e g u l a r

shaped t h i c k - w a l l e d tube) would be much more d i f f i c u l t than that using the g l o b a l - l o c a l i n t e r p o l a t i o n s .


z

Cross-section of a 4-layers (4-ply) laminated plate

Layer 1 Layer 2 Layer 3 Layer 4

Fig.8.1

A rectangular plate and its cross-sectional d i s c r e t i z a t i o n i n t o 8-nodes elements.


suollouni

Hole

JO

|BiuiouA|od japjoij6!Li

• le_LUOUAIOd

T


Present

.5

Barker

â&#x20AC;˘3

.1 0.3

1

3

0.5

(a) x

yz

at (a/2, a/12, z )

Present Barker 5

â&#x20AC;˘1

(b) Fig.8.3

at (a/12, a/2, z )

Results of t r a n s v e r s e s h e a r s t r e s s e s d i s t r i b u t i o n f o r a t h r e e - p l y laminate (S=4) with a l l edges clamped


176 CHAPTER 9 A 3-D t h i c k - p l a t e f i n i t e element 9.1 I n t r o d u c t i o n In Chapter 8, analysis

of

the s p l i n e f i n i t e prism method i s applied to

rectangular

plates.

The

assumed

displacement

the

functions

combine a 2-D f i n i t e element d i s c r e t i z a t i o n with a 1-D s p l i n e functions. T h i s allows the p o t e n t i a l energy f u n c t i o n a l to be separately integrated w i t h respect to the spanwise d i r e c t i o n . A l t h o u g h the method i s s i m i l a r to a

conventional

capability

3-D f i n i t e

element

and problem s i z e , i t

method i n

maintains

terms o f

interpolation

a 2-D-type data structure

s i m i l a r t o a conventional 2-D f i n i t e element model. The advantages over the conventional volume

of

3-D f i n i t e element models are two-fold,

input

interpolation

and

data

is

the

1-D

reduced, spanwise

and

the

2-D

interpolation

namely,

the

cross-sectional can

be

refined

independently o f each other without having to reconstruct a 3-D f i n i t e element mesh. These advantages f a c i l i t a t e p r a c t i c a l modelling o f p l a t e s . The

2-D-type data

structure

also

allows

faster

formulation o f

the

element s t i f f n e s s matrices. In t h i s chapter, the a f o r e - s a i d f e a t u r e s o f the f i n i t e prism method are r e t a i n e d i n d e v i s i n g a 3-D t h i c k - p l a t e element.

The displacement

f u n c t i o n s o f the element are expressed as the product o f the in-plane 8-node shape f u n c t i o n s and a through-thickness i n t e r p o l a t i o n . P a r t i c u l a r a t t e n t i o n i s focused on the p r e d i c t i o n o f transverse shear s t r e s s e s f o r t h i c k p l a t e s and the choice o f the through-thickness i n t e r p o l a t i o n i s c r u c i a l to the a n a l y s i s .

The philosophy o f the present method i s that

layerwise l o c a l shape f u n c t i o n s are used i n the regions where transverse shear

stresses

are o f

interest

and

importance.

The

layerwise

model

ensures the c o n t i n u i t y o f transverse shear and normal s t r e s s e s to be s a t i s f i e d i n an i n t e g r a l sense through the p o t e n t i a l energy f u n c t i o n a l and hence a r e a l i s t i c p r e d i c t i o n o f transverse s t r e s s e s can be made. Because o f the l a r g e number o f unknowns involved i n a layerwise model, a s p e c i a l through-thickness g l o b a l i n t e r p o l a t i o n i s used i n the areas of the p l a t e where o n l y g l o b a l response are o f i n t e r e s t , and a compromise between computational cost and accuracy can be reached. For s a t i s f y i n g the displacement c o m p a t i b i l i t y between these two regions, r e g i o n i s introduced.

a transition

The three d i f f e r e n t regions a r e t r e a t e d with the

same 2-D mesh. The method, t h e r e f o r e , incorporates the advantages of the


177

l a y e r w i s e p l a t e theory and the s i n g l e - l a y e r p l a t e theory, in

Chapter

p r e s e n t ed

1. in

Details the

of

the

following

method

sections

and

its

together

as discussed

formulation w i l l

with

a

few

be

numerical

examples f o r v e r i f y i n g i t s accuracy i n a n a l y z i n g p l a t e s w i t h l a r g e or s m a l l spanâ&#x20AC;&#x201D;to-thickness ratio.

9.2 Formulation Fig.9.1 depicts a

laminated plate with multi-layers of composite

materials. To describe the behaviour of an element on the X-Y plane, the displacement

components

within

each

conventional 8-node shape function

element

is

defined

by

the

[//(x,y)] ( other 2-D shape functions

are equally valid), mathematically,

E

Z^ ( 2 )

where

neglected

(9.1a)

[A/(xďź&#x152;y) ] 2.(2) {/3},

(9.1b)

[N(x,y)] 2 ( 2 )

(9.1c)

represents

interpolation displacement

[iV(x,y) ] Z. (z) {a};

the

with {a} ^, unknowns.

j-th

{/3}, and

Transverse

in many of the

term

of

the

through-thickness

^ being the corresponding nodal normal

deformationďź&#x152;which

is

often

proposed single-layer or layerwise theories,

is taken into account in this study.

It is worth noting that the general

format

functions

prism

of

the assumed displacement

method

, the

layerwise

theory. The choice of

theory

and

the

is common

to

hierachical

the finite single-layer

the through-thickness interpolation, however,

is

different from each other. In this study, the interpolation function across the thickness of the

plate

material

is layer

first such

defined that

by

1-D

parabolic

cubic or

shape

functions

higher-order

for

each

variation

of

transverse shear stresses and of normal stresses can be accounted for. Mathematically,

the

cubic

shape

functions,

for

the

i-th

layer

of

material, are given by

M(<) = { M

M

M

M

}

(9.2)


178

where

M l (<)= -9 ( 1/3 -

1

- <

)( 1 / 3 +

M 2 (<〕= 27 ( 1/3 - 《 ) ( 1 - <

; < ) / 16

)( 1 + < ) / 16

M 3 ( 0 = 27 ( 1/3 + 《 ) ( 1 - < )( 1 + < ) / 16 M 4 (C)= -9 ( 1/3 - C

and

)( 1 + C

)( 1/3 + < ) / 16

C, is the layerwise coordinate ranging from -1 to 1 for z€[z z 1 i-r i It is computationally inefficient to analyze the entire plate with

this layerwise model because of the large number of unknowns involved. To

reduce

the

computational

cost,

a

special

form

of

constraint

is

introduced. In the area where only the global response are concerned, a set of through-thickness global functions is selected

to transform the

nodal variables of the layerwise cubic shape functions to a small set of generalized parameters associated with the global functions. In essence, the layerwise cubic shape function is enforced to follow a small number of

pre-defined

patterns.

With

a

limited

sacrifice

number of unknowns is thereby reduced with respect

of

accuracy,

the

to a full layerwise

model. The j-th term of the global-local interpolation can be written as

<p ^(z)

= tM(z) ] {

forz € [ z, , z ] i -1 i

(9.3a)

is the nodal displacements of layer I computed from the j-th

where

term of the global functions given by Q(z) = { 1 - z/h, z / h , s i n ( T r z / h ) , sin(27i2/h), . . . s i n (j i t z / h ) ( 9 . 3 b )

and the

number o f

unknowns increases w i t h i n c r e a s i n g number o f

taken from (9.3b). global-local appropriate

terms

One can e a s i l y observe the h i e r a c h i c a l nature o f the

interpolation. global

functions

It

is

can

also be

noteworthy

introduced

that

into

any

(9.3b)

other without

changing other p a r t s of the formulation. To

clarify

the

idea,

Fig.9.3

shows

two

simple

transverse s t r e s s e s are o f s i g n i f i c a n c e and i n t e r e s t .

cases

where

In the proximity

o f geometric d i s c o n t i n u i t y ( e . g . f r e e edges or holes i n F i g . 9 . 3 ) where the

transverse

shear

and/or

normal

stresses

layerwise cubic shape f u n c t i o n i s adopted.

are

required,

only

the

To s a t i s f y the displacement


179

c o m p a t i b i l i t y between these

two areas,

a t r a n s i t ion zone i s introduced

, i n which the i n t e r p o l a t i o n f u n c t i o n across the thickness i s defined by superimposing

the

layerwise

shape

interpolation.

The b a s i c features of

function

to

the

these regions are

global-local summarized i n

Table 9 . 1 . Displacement c o m p a t i b i l i t y can be enforced by r e s t r a i n i n g the appropriate

nodal

degrees

of

freedom

on

the

boundary

between

the

corresponding regions, These r e s t r a i n t s can be e a s i l y imposed ( a f t e r the g l o b a l s t i f f n e s s matrix i s formed) i n the

same way as i s done f o r the

boundary c o n d i t i o n s .

Regions

Interpolation

2

Number of terms p

(*)

across thickness

I

layerwise cubic shape f u n c t i o n , M

II

global-local interpolation,

III

H +w

(*)

3 nl

p (

r

(*)

3 nl+l+r

Table 9.1 ďź&#x161; Interpolation across thickness in different regions p = number of terms used in equation (9.1) nl = number of layers of material r = number of terms taken from the global functions given in expression (9.3) and r ^ 3 nl + 1.

Having

defined

the

displacement

functions

of

the

plate,

the

strain-displacement relation can be written down as

N N

Z N Zf

N

Z

N t

xz^

and

(9.4)

Z N 2'

or

w:

{c} = Y [B].{6}.

the

element

takes the form,

stiffness matrix

for

each

of

the

three

regions


180 [K]

[K]

[K]

[K]

[K]

(9.5)

[K] with [K]

[b]t[d][b]

tK]

i

dV

[K^]

[K

] [K

J

[K

]

[K

] [K

]

]

[K

] [K

]

11

[K

21 31

p,p

12

22

32

13

23

33

(9.6)

and D

11

2 2

D

[Ki J 12 i

D

D

13•

[K.

D

[K J

D

21

22 r

[K

[K

[K

D

23"

D

31

32

]

mn

D

D

33 '

Corresponding above

Z

13

2

m

n

dz

N

dz

N

Z

m

n

n

m

m n

n

dz

dz

2' 2 m

n

m

Z

n

M

dx dy

D

N

dx dy

dx dy +

dx dy +

N

dx dy + J

dz

N

N

through-thickness

expressions

+

M N

m n

Z

dz

Z

Z

44 m

N

D

dz

n

2' 2

6 6 m

N 7 N dx dy + »y

Z'Z^ dz

66

dx dy + D

D

N' M

dz

Z Z

44 m n

N

dx dy

N

N dx dy ,y , x

44

44

Z

Z

m

m

N1 N

dz

n

Z

2

n

n

dx dy

,X

dz

N1 M

dz

NT N

dz

N N

»X

, y

dx dy

dx dy

iVT

n

r

m

D

N dx dy +

,x

Z’ Z* dz

2’Z

33

2

N

- - - -

dz

Z

32

V i

Z

m

dx dy +

, X

NTN dx dy

N

55

23

N

X

Z 21 dz

Z

22

31

n

m n

12

D

N

2’ 2’ dz

66 m

12

D

[K

dz

11 m n

according

to

D D

D

2' 2

55

m

n

D

55

Z

m

r

Z

m

dz

n

Z

n

dx dy

Nl N dx dy

Z Z’ dz

66 m n

55

,y

dz

NT N »y NT N

dx dy

dx dy

interpolation the

region

is

substituted

concerned.

The

into

the

in-plane


181 eight-node shape f u n c t i o n s , denoted by integrated

using

2x2

Gaussian

points.

In

the

through-thickness

d i r e c t i o n , however, i n t e g r a t i o n can be done a n a l y t i c a l l y f o r each of the three

regions.

This

is

f u n c t i o n s are a c t u a l l y

because

involved

only and

integrations

they

u s i n g symbolic compuatation. D e t a i l s of g i v e n i n the Appendix. takes i n t o account

It

cubic

shape

can be evaluated e x p l i c i t l y

the integration precedure are

should be noted that

the various

of

the assembly process

number of nodal degrees o f

freedom i n

d i f f e r e n t regions o f the p l a t e . For

a

loading

f (x, y)

distributed

over

the

top

surface

of

an

element, the corresponding element load vector i s given by {F}T = { { f } i , { f } o , … … { f } 1 2 m

}T =

where

2 1|

m

m 2=h

J " A

{f}

p

>T

〔9.7)

f(x,y) [//(x,y)]T dx dy

(9.8)

9.3 Numerical Examples To establish the accuracy of the present method, we shall consider two

symmetric

laminated

square

plates

with

4

(0^90°)s

(〇?90?0?90?0o)s layers of material. The plates are formed of orthotropic material

and

9

layers of

with the material principal axis 1 oriented at an

angle of 0° or 90° to the global X-axis. The total thickness of the 0° and 90° layers are the same, whereas layers at the same orientation have equal

thicknesses.

stiffness

in

the x

Under and

these

conditions,

y directions

are

the

the

effective

same.

All

laminate

edges of

the

plates are simply supported. In the following examples, the top surf ace loading is defined as

q where q

= q sin(7rx/a)sin(7ry/a) is the maximum intensity of the double sinusoidal load at

centre of the plate. The material properties are defined as follows,

En = 25 * 10 6 psi

E22 = E33 = 1 * 10 6 psi

G12 = Gis = 0.5 * 10 6 psi

G23 = 0.2 * 106psi

V12 = V 2 3 = y i 3 = 0.25

the


182

where

1 signifies

the d i r e c t i o n p a r a l l e l to the f i b e r s ,

and 2 , 3

the

transverse d i r e c t i o n . By symmetry, o n l y a q u a r t e r o f the p l a t e i s analyzed.Emphasis i s p l a c e d on the a n a l y s i s o f p l a t e s w i t h s m a l l span-to-thickness r a t i o as most o f the low-order p l a t e t h e o r i e s a r e not capable o f g i v i n g accurate s t r e s s p r e d i c t i o n s f o r such p l a t e s . F o r n i n e - p l y laminates, the in-plane displacements f o l l o w c l o s e l y a z i g - z a g v a r i a t i o n through the thickness [118] and, therefore, an additional zig-zag function is included in the global

functions

of

(9.3).

Because

through-thickness distribution

of

in-plane stresses depends on the corresponding distribution of in-plane displacements,

more

accurate

in-plane

stresses

can

be

predicted

by

including the zig-zag function in (9.3b) ; t h i s w i l l be demonstrated i n the numerical examples.

Suppose t h a t the mid-surface o f the laminated

p l a t e i s l o c a t e d i n the L - t h l a y e r , the z i g - z a g f u n c t i o n can be w r i t t e n as, f o r layer k, . Z(z) where

l

k-l

.

,

i=l L-1 i

i=i

d

and z i s the d i s t a n c e between the laminate mid—surface and the lower d s u r f a c e o f the L - t h l a y e r , h.the t h i c k n e s s o f l a y e r i , < the layer n a t u r a l c o o r d i n a t e which ranges from - 1 to +1. The z i g - z a g f u n c t i o n f o r a n i n e - p l y laminate i s g i v e n i n F i g . 9 . 2 . S t r e s s e s a r e c a l c u l a t e d and e x t r a p o l a t e d from the Gaussian p o i n t s of

the

eight-node

elements. R e s u l t s

are

compared

with

the

exact

e l a s t i c i t y s o l u t i o n s o f Pagano [118]. The

following

normalized

presentation of results. (crx

(Tyz

cry

Txy) = (crx

Txz)

= (Tyz

crz = crz / q

w =7r Qw / 12s hq

try

Txy)/qs

丁 X2)/qs

quantities

are

defined

for

the


183 Q

= 4 G

12

+

[E

11

+

E

22

(l+2i,22)]/(l^i2i,2i)

s = a/h z = z / h - 1/2 In a few i n s t a n c e s ( s m a l l s ) the maximum v a l u e s o f t r a n s v e r s e shear s t r e s s e s do n o t o c c u r a t m i d - s u r f a c e , hence i n these cases there are two e n t r i e s i n t h e r e s p e c t i v e columns. The upper v a l u e g i v e s the f u n c t i o n a t m i d - s u r f a c e , w h i l e the lower number r e p r e s e n t s the maximum v a l u e .

Assessment

of Layerwise

Each p l a t e i s f i r s t

i720(iei(LM)

analyzed using

the p r e s e n t

method w i t h the

l a y e r w i s e c u b i c shape f u n c t i o n s a l o n e f o r the e n t i r e p l a t e . Convergence o f t h e l a y e r w i s e model a r e summarized i n T a b l e s 9 . 2 , 9 . 3 . One can e a s i l y o b s e r v e t h a t a mesh o f 4x4 elements i n the X-Y p l a n e .together w i t h a c u b i c element p e r m a t e r i a l l a y e r i s s u f f i c i e n t t o g i v e a c c u r a t e In-plane and

transverse

shear

stresses;

all

stresses

are

within

3

percent

d i f f e r e n c e t o t h e e x a c t e l a s t i c i t y s o l u t i o n s f o r b o t h c a s e s o f 4 - and 9ä¸&#x20AC;ply l a m i n a t e s .

Furthermore,

the exact s o l u t i o n s .

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


184

Mesh

crx (:

664

1.404

444

224

663

443

223

exact

664

444

exact

K“猛

)

Txz(0,墓,0)

T y z ( 鑑 , 0 , 0 ) Txy(0 # 0 # :

w(;#;f0)

0.844

0. 154

0.289

-0.0872

11.764

-0.922

-0.803

(0.265)

(0.300)

0.0681

1.424

0.856

0.157

0.293

-0.0885

-0.935

-0.814

(0.271)

(0.304)

0.0690

1.518

0.917

0.168

0.313

-0.0949

-1.001

-0.872

(0.291)

(0.326)

0.0739

1.404

0.841

0.155

0.306

-0.0872

-0.922

-0.805

(0.268)

(0.320)

0.0680

1.423

0.853

0.157

0.310

-0.0884

-0.935

-0.816

(0.271)

(0.329)

0.0690

1.517

0.913

0.168

0.332

-0.0948

-1,000

-0.873

(0.291)

(0.338)

0.0738

0.153

0.295

-0.0863

(0.264)

(0.298)

0.0673

0.292

-0.0472

1.388

0.835

一0.912

一0.795

0.729

0.670

0.222

-0.693

-0.673

(0.224)

0.739

0.679

0.225

-0.702

-0.683

(0.227)

0.720

0.663

0.219

-0.684

-0.666

(0.222)

11.764 11.754

11.763

11.763

11.689

11.767

.491

0.0464 -0.0479

.296

,491

0.0470 -0.0467

.292

4.491

0.0458

T a b l e 9 . 2 . Comparison o f r e s u l t s f o r a f o u r - p l y laminate w i t h s = 2 and 4. Only Layerwise c u b i c shape f u n c t i o n i s used.

(*) Mesh : first two

digit denotes the number of elements in each of the in-plane direction. The third digit denotes the number of 1-D cubic element in the thickness coordinate.

Values

transverse stress.

in

bracket

are

the

maximum

of

the

corresponding


185

Mesh

crx

669

1.274

1.063

0.206

0. 199

-0.0730

-0.876

-0.834

(0.225)

(0.210)

0.0541

1.291

1.078

0.209

0.196

-0.0740

-0.888

-0.846

(0.228)

(0.211)

0.0548

1.368

1.154

0.224

0.213

-0.0795

—0.951

-0.905

(0.245)

(0.225)

0.0587

1.260

1.051

0.204

0.194

-0.0722

-0.866

一 0 . 8 2 4

(0.224)

(0.211)

0.0534

0.692

0.635

0.226

-0.0341

»0.656

-0.619

(0.228)

0.0332

0.701

0.644

0.229

-0,0346

-0.666

-0.627

(0.231)

0.0336

0.684

0.628

0.223

-0,0337

-0.649

-0.612

(0.225)

0.0328

449

229

exact

669

449

exact

#

)

cry (;

xxz (0# , 0 )

( 言 , 0 , 0 ) TxyCoj

.226

.229

.223

w(;

.0)

12.288 12.288 12.280

12.288

.079

.079

.079

Table 9.3. Comparison of results for a nine-ply laminate with s = 2 and 4. Only Layerwise cubic shape function is used.(*) Mesh : first two digit denotes the number of elements in each of the in-plane direction. The third digit denotes the number of 1-D cubic element in the thickness coordinate.

Values

in

bracket

are

the

maximum

of

the

corresponding

transverse stress.

Assessment of global-local

/wodeJ (GLM)

To examine the accuracy of the global-local model, the plates are first

discretized

into

element per material from

the

a

mesh

layer.

through-thickness

of

4x4

B-nodes

elements

Increasing number of global

functions

and

terms used

and

a

cubic

is then taken throughout

the

entire plate. The centre deflections of the resulting global-local model are compared with the solutions of the corresponding layerwise model and the

exact

solutions,

see

Table

9.4.

Considering

the

results of

the

four-ply laminates obtained by taking 5 terms from the global functions, one

can

see

that

more

than

60

percent

reduction

in

the

number

of


186 unknowns i s a c h i e v e d w i t h l e s s than 3 percent l o s s i n accuracy o f the corresponding d e f l e c t i o n s .

F o r the n i n e - p l y laminates,

9 terms o f the

GLM a r e s u f f i c i e n t t o r e t a i n an a c c u r a t e p r e d i c t i o n o f d e f l e c t i o n s w i t h less

than

1 percent l o s s i n accuracy,

and the

number o f

unknowns i s

reduced by more than 60 p e r c e n t i n comparison w i t h a complete layerwise model. To f u r t h e r v e r i f y the implementation o f the g l o b a l - l o c a l model, the s t r e s s e s o f the f o u r - p l y l a m i n a t e a r e o b t a i n e d by u s i n g 13 terms o f the global functions.

I t i s found t h a t t h e y a r e i d e n t i c a l t o those u s i n g 4

l a y e r w i s e c u b i c shape f u n c t i o n s , see T a b l e 9 . 5 . T h i s i s expected because i d e n t i c a l number o f unknowns i s u s e d i n b o t h c a s e s which i m p l i e s that no c o n s t r a i n t i s e f f e c t i v e l y a p p l i e d t o the l a y e r w i s e model. F o r l a m i n a t e d p l a t e s w i t h s=100,

we a n t i c i p a t e t h a t the f i r s t two

l i n e a r terms o f t h e g l o b a l t h r o u g h - t h i c k n e s s f u n c t i o n s a r e s u f f i c i e n t to g i v e a c c u r a t e d e f l e c t i o n s as w e l l a s i n - p l a n e s t r e s s e s . As demonstrated i n Table 9.6,

the present r e s u l t s ,

i n c l u d i n g d e f l e c t i o n s and s t r e s s e s ,

a r e w i t h i n 2 p e r c e n t d i f f e r e n c e from the e x a c t s o l u t i o n s . (*)

Centre d e f l e c t i o n w GLM

%e r r o r

GLM

%e r r o r

%r e d u c t i o n

5

11,.432

f o u r - p l y laminate .395

2.1

62

9

11,.475

.413

1.7

31

13

11.,764

.491

LM

11..764

,491

exact

11..767

,491

5

11.547

n i n e - p l y laminate .924

82

9

12.188

,047

68

12

12.216

,054

57

LM

12.288

,079

exact

12.288

,079

T a b l e 9.4.Convergence o f c e n t r e d e f l e c t i o n s o f the g l o b a l - l o c a l model. (*) p = number o f terms t a k e n from e x p r e s s i o n ( 9 . 3 ) . GLM=globall o c a l model , LM=layerwise model â&#x20AC;˘ % error = percentage difference between the GLM and the corresponding LM(4x4 mesh with one cubic element per material layer.) % reduction = percentage reduction in the number of unknowns of the GLM with respect to the corresponding LM.


187

cyd士!)Txz(0,言,0)

Tyz(言,0,0) Txy (o, 0, ±; w(!

1.424

0.856

0. 157

0.293

-0.935

-0.814

(0.271)

(0.304)

0.0690

1.424

0.856

0. 157

0.293

~0. 0885

-0.935

-0.814

(0.271)

(0.304)

0,.0690

1.388

0.835

0. 153

0.295

-0,.0863

-0.912

一 0 . 7 9 5

(0.264)

(0.298)

0..0673

0.739

.679

s = 4 0.225

0.296

""0 .0479

-0.702

- 0 ..683

(0.227)

0.739

0.( .679

0.225

-0.702

-0.683

(0.227)

0.720

0.663

-0.684

一 0 . 6 6 6

【(墓’言‘土^)

LM

GLM

exact

GLM

exact

Table 9.5.

- 0 . 0885

,0)

11 .764 11 .764 11 .767

4. 491

0.0470 0.296

- 0 .0479

4. 491

0.0470

0.219

.292

(0.222)

-0.0467

.491

0.0458

Comparison o f r e s u l t s f o r a f o u r - p l y laminate u s i n g a

mesh o f 4x4. G L M = g l o b a l - l o c a l model , LM=layerwise model

2 2

2

;y(H土1)

Txy (0,0,土!)

4 黨 '

0 )

f o u r - p l y laminate

exact

±0.550

±0.276

+0.0216

0.993

±0.553

±0.277

+0.0219

1.007

±0.539

±0.271

+0.0213

1.008

nine-ply laminate

exact

±0.550

±0.440

+0.0216

0.991

±0.553

±0.443

+0.0218

1.005

±0.553

±0.443

+0.0218

1.005

±0.539

±0.431

+0.0213

1.008

Table 9.6. Comparison of results for laminated plates with s=100. A mesh of 4x4 with only through-thickness global-local interpolation. p = number of terms taken from expression (9.3).

(*)


188

Assessment

of

the combined

model

A p l a t e i s d i v i d e d i n t o t h r e e d i f f e r e n t r e g i o n s i n which d i f f e r e n t i n t e r p o l a t i o n a c r o s s t h e t h i c k n e s s i s used, a s shown i n F i g . 9 . 4 . i n the a r e a where t r a n s v e r s e s h e a r s t r e s s e s a r e r e q u i r e d , a l a y e r w i s e model i s adopted. the

zone

The r e m a i n i n g p o r t i o n i s d i v i d e d i n t o the t r a n s i t i o n zone and where

the

global-local

model

is

employed.

Comparison

of

s t r e s s e s a r e summarized i n T a b l e s 9 . 7 - 9 . 9 . F o r the f o u r - p l y laminates w i t h s=2 and s=4, obtained by using

v e r y good r e s u l t s o f

t r a n s v e r s e shear s t r e s s e s are

5 terms i n t h e g l o b a l - l o c a l i n t e r p o l a t i o n zoneďź&#x203A; a

maximum o f 5 p e r c e n t d i f f e r e n c e i s f o u n d between the p r e s e n t method and the exact s o l u t i o n s . Reasonable i n - p l a n e s t r e s s e s a r e a l s o obtained.

A

39 p e r c e n t r e d u c t i o n i n t h e e f f e c t i v e number o f d e g r e e s o f freedom

(

i.e.

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

c o n s t r a i n t s on t h e i n t e r f a c e s between d i f f e r e n t zones) i s a c h i e v e d when comparing t h e combined model w i t h r e s p e c t t o the c o r r e s p o n d i n g complete l a y e r w i s e model. For

the

n i n e -p l y

laminates,

the

agreement

of

the

solutions

o b t a i n e d b y the p r e s e n t method and t h e a n a l y t i c a l method i s r e a s o n a b l y good,

see T a b l e 9 . 8 .

In t h e s e c a s e s ,

a

49 p e r c e n t r e d u c t i o n i n the

e f f e c t i v e number o f d e g r e e s o f freedom i s a c h i e v e d . R e s u l t s o f l a m i n a t e s w i t h s=100 a r e a l s o p r e s e n t e d i n T a b l e 9 . 9 . The p r e d i c t i o n o f b o t h i n - p l a n e and t r a n s v e r s e s t r e s s e s by t h e combined model i s r e a s o n a b l y a c c u r a t e . To demonstrate t h e e f f e c t o f z i g - z a g f u n c t i o n i n p r e d i c t i n g i n p l a n e s t r e s s e s f o r the n i n e - p l y laminates, sets of

results

obtained

by u s i n g

the

comparison between t h e

same

two

combined model w i t h o r

w i t h o u t t h e z i g - z a g f u n c t i o n i s g i v e n i n T a b l e 9 . 1 0 . I t i s apparent t h a t the

inclusion

of

zig-zag

function

predicting in-plane stresses, s=2 and 3=4. F o r 3=100,

makes

a

positive

however,

remain p l a n e "

on

e s p e c i a l l y f o r the t h i c k l a m i n a t e s w i t h no s i g n i f i c a n t d i f f e r e n c e between the

two s e t s o f r e s u l t s c a n be f o u n d because, i n t h i s c a s e , "plane sections

influence

the c l a s s i c a l

assumpti on i s s u f f i c i e n t l y a c c u r a t e t o

d e s c r i b e t h e v a r i a t i o n o f i n - p l a n e d i s p l a c e m e n t s through t h e t h i c k n e s s of

plates.

In

other

words,

the

in-plane

displacements

follows

a

s t r a i g h t - l i n e v a r i a t i o n th r o u gh t h e t h i c k n e s s . To examine t h e d i s t r i b u t i o n o f t r a n s v e r s e shear s t r e s s e s o b t a i n e d by t h e combined model and t h e g l o b a l - l o c a l model, r e s u l t s f o r the f o u r


189 and n i n e - p l y laminates w i t h s=2,4 and 100 are g i v e n i n F i g s . 9 . 5 - 9 . 1 0 . I t can

be

seen t h a t

results

of

the g l o b a l - l o c a l

model e x h i b i t s

s t r e s s e s jump a c r o s s the i n t e r f a c e s between m a t e r i a l l a y e r s . and bottom s u r f a c e s o f

the p l a t e s ,

moreover,

severe

On the top

the g l o b a l - l o c a l

model

shows s i g n i f i c a n t r e s i d u a l t r a n s v e r s e shear s t r e s s e s which v i o l a t e s the t r a c t i o n - f r e e c o n d i t i o n s on these s u r f a c e s . The combined model, however, performs

much

better

in

terms

i n t e r f a c i a l s t r e s s jump and, the

continuous

transverse

of

the

traction-free

conditions

and

t h e r e f o r e , i s more r e l i a b l e i n p r e d i c t i n g shear

stresses

distribution

in

laminated

plates.

cry (;

) T x z (0,

1.265

0.713

0. 157

0.281

-0.849

-0.710

(0.267)

(0.296)

0.0673

1.388

0.835

0. 153

0.295

-0.0863

-0.912

-0.795

(0.264)

(0.298)

0.0673

0.721

0.641

0.222

0.287

-0.0476

-0.686

-0.647

(0.222)

0.720

0. 663

0.219

-0.684

-0.666

(0.222)

exact

exact

,

/

)

, 0)

T y z( ; , 0 # 0 ) T : x y ( 0 , 0 , ± 2 ) W(:

-0.0861

11.503

11.767

.422

0.0466 .292

-0.0467

.491

0.0458

Table 9.7. Comparison of results for a four-ply laminate with s=2 and s=4. A mesh of 4x4 using the combined model. ( * 〕 p = number of terms taken from expression (9.3b).


190

cryU,土宴)

exact

exact

1.237

1. 126

0.209

0.198

-0.863

-0.908

(0.234)

(0.210)

0.0524

1.260

1.051

0.204

0.194

-0.0722

-0.866

-0.824

(0.224)

(0.211)

0.690

0.658

-0.654

-0.641

0.684

0.628

一 0 . 6 4 9

-0.612

Table 9.8.

.229

.228

-0.0720

12.215

12.288

0.0534 -0.0340

.059

0.0330 .223

.223

-0.0337

4.079

0.0328

Comparison o f r e s u l t s f o r a n i n e - p l y laminate w i t h s=2

and s=4. A mesh o f 4x4 u s i n g the combined model.

(*) p = number o f terms

taken from e x p r e s s i o n ( 9 . 3 b ) .

(Ty (? ?

) Txz (o # ? 0 )

Tyz (^,0, 0) Txy(0,0,土】)W(:

,0)

four-ply 4

±0.553

±0•277

0.339

.145

+0.0219

1.007

exact ±0.539

±0.271

0.339

• 138

+0.0213

1.008

nine-ply 4

±0.553

±0.443

0.281

• 242

+0.0218

1.004

5

±0.553

±0.443

0.258

• 230

+0.0218

1.004

exact ±0.539

±0.431

0.259

.219

+0.0213

1.005

Comparison

of

for

Table

9.9.

results

four-ply

and

laminates with s=100. A mesh of 4x4 using the combined model. number of terms taken from expression (9.3b).

nine-ply (*) p =


191

2'2'

cry(

2

TxyCO.O^;)

2'2'-

1.237

1 . 126

-0.0720

-0.863

-0.908

0.0524

1. 1 2 3

0.914

-0.0697

-0.713

-0.676

0.0501

1.260

1.051

-0.0722

-0.866

-0.824

0.0534

0.690

0.658

-0.0340

-0.654

-0.641

0.0330

0.633

0.576

-0.0339

-0.597

-0.558

0.0329

0.684

0.628

-0.0337

-0.649

-0.612

0.0328

exact

exact

3 = 100

4 、 '

+0.0213

±0.431

±0.539

Table 9.10.

+0.0219

±0.442

±0.553

exact

+0.0218

±0.443

±0.553

Comparison of results of nine-ply

laminates for

the

effects of zig-zag functions on in-plane stresses. A mesh of 4x4 using the

combined

model.

(*)

p

=

number

of

terms

taken

from

expression

(9.3b). (**) Results obtained without using zig-zag function.

9 . 4 Conclusions It

has

solutions

been

for

a

demonstrated laminate

that

can

be

approximate obtained

by

three-dimensional using

a

combined

latyerv/ise-single layer itiodel. Restsonable prediction of in-plane stresses and

transverse shear stresses

compromise between Additionally, separately

the

computational

integration and

can. b e

in

the

analytically.

nia.de and

, at

cost

accuracy

and

thickness This

the same time,

coordinate

facilitates

is can the

achieved. be

done

computer

implementation of the method. The accuracy and application of the present method are demonstrated

a.


192 b y s o l v i n g s i m p l e problems o f square laminated p l a t e s . F o r g e n e r a l use, one s h o u l d r e p l a c e the 8-node i n - p l a n e shape f u n c t i o n s w i t h L a g r a n g i a n shape f u n c t i o n s a s therefore

the

they a r e

more

c o m p l i c a t e d geometry

not a f f e c t e d by a n g u l a r d i s t o r t i o n s

reliable [122].

elements

for

For rectangular

analyzing

plates

and with

and p a r a l l e l o g r a m p l a t e s ,

however, s e r e d i p i t y elements a r e more c o s t e f f e c t i v e a s t h e y have the same a c c u r a c y w i t h t h e L a g r a n g i a n elements but w i t h o u t t h e a d d i t i o n a l degrees o f small

freedom a s s o c i a t e d w i t h the i n t e r n a l nodes.

problems

are

dealt

with

in

this

chapterÂť

Although very

the

savings

in

computational cost ( reduction in number of unknowns) would b e much more significant

for

complicated

plates

with

only

a

small

area

where

transverse shear stresses are of interest. In this case, the number of unknowns involved in a full layerwise model is extremely large and the present approach would be more cost-effective.


193 Appendix

Through-thickness

integrals

F o r r e g i o n I , i n t e g r a t i o n o f the product of l a y e r w i s e c u b i c shape f u n c t i o n s can be w r i t t e n a s , nl I = I I;

"128 99 [M]

where

[M] d z = h./1680

-36

19

648 - 8 1 - 3 6 m.

648

99 128^

The

summation

is

carried

out

with

arrangement i n t h e t h i c k n e s s c o - o r d i n a t e .

due

regard

to

the

Other i n t e g r a l s o f the

nodal cu b ic

shape f u n c t i o n s a r e a s f o l l o w s .

[M],

[M] d z = 1/80

-40

-57

57

0

-24

81

7

-24

24

-7

-81

24*

0

-57

57

40

148 -189 5 4 - 1 3 [M]â&#x20AC;&#x2122; [M]â&#x20AC;&#x2122;dz = l/40h. T

432-297 sym.

54

432 -189 148

For region the product

of

transformation

II, with reference the global-local of

the

integral

to equation

(9.3a),

integration of

interpolation can be written as of

layerwise

cubic

shape

the

functions,

that is

I k l= 1 i=1 where I

J

[M] T [M] d z {6 i } l h .

denotes the integral of the product of the k-th and l-th terms kI of the global-local interpolation.


194

F o r r e g i o n I I I , i n a d d i t i o n t o t h e two types o f i n t e g r a l s presented above, t h e i n t e g r a l o f the product o f the j - t h term o f the g l o b a l - l o c a l interpolation

and

illustration,

I. = ^

the

layerwise

shape

function

is

needed.

For

we have

{ 5.} 1 J [ M ] T [ M ] dz

i=1

h. i

w i t h the summation b e i n g c a r r i e d out w i t h due r e g a r d t o the nodal arrangement i n t h e t h i c k n e s s c o o r d i n a t e .


195

2-D shape functions

Trial functions i

Z

Fig.9.1

A l a m i n a t e d composite p l a t e and i t s d i s c r e t i z a t i o n .

0 ) 0 c

V

0 0

u cn tn

0) xcu

Z 1 r C P D _o v H h -

-0.8

-0.6

-0.4

—0.2

0.2

0

I

0,4

0.6

0 8

Zig — Z o g f u n c t i o n

Fig.9.2

Zig-Zag

function

ui th thickness =10.

for

a

nine-ply

laminate


196

Free

Line 1 !

\ Line 2

Line 1 '

« Line 2

Fig.9.3 Division of a plate into three regions with different through-thickness interpolation. Region I : Layerwise cubic shape function (M) Region II : through-thi ckness global-local interpolation {(p) R e g i o n III : Transition zone C M + <p ). Line 1 •• Nodal degrees of freedom which correspond to the through-thi ckness global-local interpolation are restrained in the global stiffness matrix equation. Line Z : Nodal degrees of freedom which correspond to the layerwise cubic shape functions are restrained in the global stiffness matrix equation.

Fig. 9 . 4 Division of a square plait© for calculating the transverse shear stress point 2.

into different regions at point 1 and 了口 a t


197

(a)

〇7

0, 6

0.5

0.4

0.3

0.2

01

Z—coordinate •

GLM

Combined

(b) 0.7

0.6

0.5

0.4

Z—coordinate 口

Fig 9 5

GLM

+

Distribution of

Combined

transverse

shear

stresses

through

t h e t h i c k n e s s o f t h e f o u r - p l y l a m i n a t e (a/h=2, h=10) . ( a ) 、 , ( b 〕 i : combined = combined model ( s u b d i v i s i o n i n t o t h r e e r e g i o n s ) ; g l o b a l - l o c a l i n t e r p o l a t i o n i s used f o r the e n t i r e p l a t e .

y z

GLM

一 =


198

Z—coordinate 口

GLM

+

Combined

(b)

Z—coordinate •

GLM

Combined

Fig.9.6 D i s t r i b u t i o n o f t r a n s v e r s e shear s t r e s s e s t h e t h i c k n e s s o f the f o u r - p l y laminate (a/h=4, h=5) . (a) , Legend same a s F i g . 9 . 5

through •


199

(a)

60

50

30

4) (/) (/)

20

Q> (n u

0.02

0.04

0.06

0.08

01

0.12

0.16 0.14

0.18

0.2

2—coordinate •

GLM

Combined

Cb)

6

4

2

1 0 9

8 7

</)

)0 / )的 (Q > u ij)

4

2

1 0 —1 0. 1

0.16

0.12 0.14

Z — coordinate •

GLM

+

Combined

Fig.9.7 Distribution of transverse shear stresses through thickness o f the four-ply laiuinate ( a / h = 1 0 0 , h - 0 . 2 ) - ( a 〕 (b) x

vz

• Legend same a s Fig.9.5


200 (a)

0.2

Z—coordinate 口

GLM

+

Combined

(b) 0.6

0.4

0.2

Z—coordinate •

GLM

+

Combined

Fig.9.8 Distribution of transverse shear stresses through the thickness of the nine-ply laminate (a/h=2, h=10) .(a) ! : 口 ’(b)*r y z Legend same as Fig.9.5


201

Ca) 0.9 0.8

0.5

0.3

0

1

2

4

Z—coordinate 口

GLM

Combined

(b)

0.7

0.4

.0.2

-0.3 •0.4

Z— c o o r d i n a t e 口

GLM

+•

Combined

pig,9^9 Distribution of transverse shear stresses through the thickness of the nine-ply laminate (a/h=4> h=5) • ( a 〕 t , (b)T • Legend same a s F i g . 9 . 5


202

(a)

35

30

25

20

0.04

0.06

0.02

0.08

01

0.16

0.12

0.2

0.14

Z—coordinote •

GLM

Combined

(b) 28 26 24 22

20

0.18 Z—coordinate 口

the (b)

GLM

Combined

F i g . 9 . 1 0 D i s t r i b u t i o n o f t r a n s v e r s e shear s t r e s s e s through t h i c k n e s s o f the n i n e — p l y laminate (a/h=100, h=0.2) .(a) t

yz

. Legend same a s F i g . 9 . 5


203 CHAPTER 10 A FINITE LAYER METHOD FOR CONTINUOUS PLATES 10.1 I n t r o d u c t i o n In

this

study,

intermediate l i n e First,

free

vibration

of

s u p p o r t s i s s t u d i e d by

rectangular a

plates

new f i n i t e l a y e r

with

method.

a p l a t e i s t r e a t e d as a t h r e e - d i m e n s i o n a l e l a s t i c body and the

displacement

field

f u n c t i o n s i n the

is

approximated

plane o f

by

the p l a t e .

a

To

set

of

admissible

satisfy

the

trial

boundary

and

i n t e r n a l c o n d i t i o n s , s i n g l e - s p a n beam v i b r a t i o n f u n c t i o n s a r e augmented by

a polynomial,

vibration

and t o g e t h e r

functions[123].

they form m u l t i - s p a n o r m o d i f i e d beam

These

modified

beam

functions

can

be

c o n s i d e r e d a s t h e approximate v i b r a t i o n modes o f a continuous beam. The m o d i f i e d beam v i b r a t i o n f u n c t i o n s a r e employed as the t r i a l f u n c t i o n s for

subsequent f i n i t e

layer

formulation.One-dimensional

linear

shape

f u n c t i o n s a r e adopted t o s i m u l a t e the v a r i a t i o n o f d i s p l a c e m e n t s through the

thickness.

Stiffness

and

mass

matrices

are

formulated v i a

the

c o n v e n t i o n a l d i s p l a c e m e n t method and the e i g e n - e q u a t i o n i s then s o l v e d . A number o f n u m e r i c a l examples a r e p r o v i d e d f o r which s o l u t i o n s a r e not a v a i l a b l e i n open l i t e r a t u r e .

10.2

Formulation

Displacement

functions

F i g . 1 0 . 1 shows a r e c t a n g u l a r t h i c k p l a t e w i t h q and q

intermediate y Y~ and X - a x i s r e s p e c t i v e l y . I t i s d i v i d e d x

l i n e s u p p o r t s p a r a l l e l t o the into

a

number o f f i n i t e l a y e r s through the t h i c k n e s s c o o r d i n a t e .

The

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

i(x,y,z)

X , ( x ) . Y ( y ) â&#x20AC;˘ [M(z)] {a}.

(10.1a)

r

X(x). Y,(y). [M(z)] {p}.

(10.1b)

X(x). Y(y).

(10.1c)

(x,y,z)

r(x,y, z)

[M(z) ]

and [M(z)] = 1-D linear shape functions


204 The d i s p l a c e m e n t unknowns a r e d e n o t e d by { o c } . 1 . . J i j i ] trial f un c t i o n s and their first derivatives are represented b y X ( x ) , 《 ' ( x 、 , Y (y) . , Y, (y)j,It is important to recognize that the first term o f the trial f u n c t i o n s d o e s not necessarily start with 1; the selection of

the

starting

term will b e demonstrated

in

the numerical examples.

Furthermore, it is noteworthy that X' (x) . and Y, (y) . are incorporated in the

expressions

for

u

interpolation c a n b e

and

v

achieved

respectively and

d e a l i n g w i t h thin-plate problems

shear

such

locking

that

can be

a

consistent

avoided when

[124]. This can b e easily observed by

w r i t i n g d o w n the transverse shear strains explicitly,

Gx2 =

Z E

(

(x). YCy).

[M' (2)]{a}.. +

,

^

G y2 = E E 1

X' (x). Y(y). [M(z)] {y}..)

( X(x).

Y,

+

X(x).

Y ' ( y ) . [Mfz)]

J

(10.2b)

The t r i a l f u n c t i o n s f o r e a c h o f t h e t r a n s v e r s e s h e a r s t r a i n component a r e c o n s i s t e n t i n t h e s e n s e t h a t t h e y a r e o f e q u a l o r d e r i n t h e X - and Y - d i r e c t i o n . The i n c o n s i s t e n c y between t h e f i n i t e element shape f u n c t i o n in

the

2-direction

thickness-span r a t i o

becomes

relatively

becomes s m a l l .

insignificant

Application of

when

the

these displacement

f u n c t i o n s t o t h i n - p l a t e v i b r a t i o n w i l l be demonstrated i n t h e n u m e r i c a l examples. A l t h o u g h t h e f i n i t e l a y e r a p p r o a c h was d e v e l o p e d I n t h e s e v e n t i e s , i t i s i n t e r e s t i n g to recognize that that

of

the

generalised

the b a s i c concept c o i n c i d e s w i t h

layerwise

laminate

plate

theory

which

was r e c e n t l y p r o p o s e d by Reddy and h i s c o - w o r k e r s [ 6 8 ] .

Modified

beam vibration

F i r s t of a l l , the

out-of-plane

functions

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

component,

w,

in

the X - d i r e c t i o n .

f u n c t i o n s a r e r e q u i r e d t o s a t i s f y t h e boundary c o n d i t i o n s a t

The

x=0 and

x=L and t h e z e r o d e f l e c t i o n c o n d i t i o n s a t t h e i n t e r m e d i a t e l i n e s u p p o r t s x p a r a l l e l t o t h e Y - a x i s . I n g e n e r a l , t h e i - t h term o f t h e f u n c t i o n s c a n be w r i t t e n a s

X(x) . = X(x).+ i i

X(x).

i

(10.3a)


205

where X ( x ) . = ^

X

p=o

A.

i P

and

p

xp

(10.3b〕

= constants to be determined

X(x). is the i-th term of the conventional beam vibration functions

w h i c h satisfies the corresponding boundary conditions at the ends, see Table 10.1. To determine the constants A. ^ of the augmented polynomial, the

four

(10.3).

boundary The

conditions

remaining

deflection, for the

at

both

q^constants

ends

are

are

found

internal supports, on

imposed

by

on

imposing

equation the

zero

(10.2). The same procedure

is repeated for each term of the function. To illustrate this procedure, consider a continuous plate with simply-supported ends at x=0,

and

w i t h a n intermediate

line support at x=x . The boundary conditions at q both ends can be expressed in the form

= 0 = X(L )

and X" (0). = 0 = X11 (L

今 X(0). = 0 = X(L )

and X" (0). = 0 = X11 (L

X(0)

1

X

i

1

x

X

x

)

)

(10.4)

For the intermediate line-support, we have

X(x

q

). = X(x ).+ i q i

X(x )

= » X(x

q i

Substituting

X(x

). = 0 q 1

). q i

(10.3)

into

(10.5)

(10.4)

and

(10.5),

the

five

unknown

coefficients of the augmented polynomial can be determined b y solving the following matrix equation :

0

0

0

L2

L3

0

2

0

L; > 0

0

2

6L

1

0

1

L

0 0

x

x

X

2

q

q

X

3 X

x

q

12L 4 X

q

(10.6) x

-X(x )•


206 The f i r s t f o u r rows r e f e r to the boundary c o n d i t i o n s a t the ends w h i l e the

last

one

refers

to

the

condition

of

zero

deflection

at

the

intermediate l i n e support. The same p r o c e d u r e a p p l i e s t o the Y - d i r e c t i o n . A f t e r the constants o f the augmented p o l y n o m i a l s i n b o t h d i r e c t i o n s a r e found , the modified beam f u n c t i o n s a r e s u b s t i t u t e d i n t o the displacement f u n c t i o n s g i v e n i n (10.1).

On a c l o s e r e x a m i n a t i o n o f e q u a t i o n ( 1 0 . 1 ) , one can then e a s i l y

o b s e r v e t h a t the c r o s s - s e c t i o n o f the p l a t e a t t h e corresponding l i n e s u p p o r t a r e a c t u a l l y assumed as a r i g i d i n - p l a n e diaphragm,

see Table

10.2. Having

defined

the

d i s plac emen t

field,

the

strain-displacement

r e l a t i o n i s g i v e n by

e e

X': Y.[M] 0

x

y

e

(10.7] .Y': [M] X.Y.tM], i

z

J

r

{a> )

m X'.Y,. [M]

i J

ir}

0

X.Y,. [M], X.Y’. [M]

y2 e

X U M ] ,

X2

Because

X'.Y,. [M]

of

the

double

i J X,-Y.[M]

i J

0

summation,

the

layer

stiffness matrix

is

written in the form of

[K]

[K]

[K]

[K

[K]

[K

[K]

[K]

[K]

[K]

where [K].

jkl

[K

(10.8)

[D][B] kl dV

[B]

[、][k

1 2

][

V

[

v

[

v

[K,

V [ ‘

[

V

[K

(6x6) m a t r i x

3 3 ] ijkl


207 and

(10.9) 11 i j k l

Y'.

44

X,. X; dx

dy

Y . Y i dy

1 2 ijkl

Y:_

12

^13 ijkl

D

66

13

'21 ijkl

44

21

[K

2

]

2 ijkl

44

dy

^3 J ijkl

55

23

V

31

66

ijkl 一

31

] = 32 ijkl

D

55 +D

33

i ikl

X1: X, dx

Y . Y, dy j t

J

Xi

X>

X’ dx

X. X, dx

Y,. Y; dy

X. X, dx

Y; dy t

Y': Y, dy j I Y. Y l dy 、 Y . Y, dy j l

Y,.

Y;

Y' d y

[M] dz

[M] dz

[M]’ [M] dz

dx X ;

dy

J

[M]

[M],dz

X: dx f [M] 丁[M] d z

X:

Y,. Y: dy

Y,-

[M]'dz

X': X, d x 「 [ M ] i k J

Y,. Y' dy

Y'; Y l dy

[M],

[M]

X: X; dx i k J

Y. Y i dy

32

55

X: dx

X’. X: dx

Y . Y': dy

22

55

X,.

[M] dz

[M]丨[M] dz

X: dx

Y . Y l dy

k

[M]

,

X

i

[M]T

[M] d z

[M] dz

[M] dz

[M],

[M],dz

_ [M]

X, X: dx i k J X

[M] dz

[M]

X. dx

X. X. dx

[M]T

T

[M]

[M] , dz

T X'; d x 「 [ M ] , [ M ] dz k J

X. X- dx

f [M]

[M],dz

Y" d y 「 X . X, dx

T [M], [ M ] dz

Y; dy

[M]

X

X k dx

[M] dz


208

+D

66 J

Y

jY

l

+D

3 3 J YjY

l

d

x

办J 、

X

y J

k

d x

|

[ m ] t [ M ]d z

d x

J [M],T[M”dz

In a similar manner, one can derive the consistent mass matrix,

where [M”] 0

[M].

[M22]

0

i ikl

o

0

0

is a 6x6 matrix

[M 33 ]

(10.10)

ijkl

and [M

11 ijkl

[M

22

ijkl

33^ ijkl All plate

Y

X,. X: dx

[M] T [M] dz

Y’.Y:dy

X. X, dx

[M] [M] d z

Y.Ydy

X. X. dx

[M] [H] dz

integrations are carried out

and

through

the

thickness

assembling the submatrices

of

[K]. .^^and

over each [M]

the entire surface of layer.

The

procedure

the of

follows that of Cheung[3].

For each span in either direction of the X-Y plane, the modified beam vibration functions are numerically integrated using 10 Gaussian points. Integration of the 1-D linear shape functions has been discussed in the Appendix of Chapter 5.


209

Boundary c o n d i t i o n s

Beam v i b r a t i o n f u n c t i o n s

sin ( X(0) _

=0

i

X"(0)

, X ( L ) =0

L )

Iti

x i

=0 , X " ( L ) =0

sin X(0) 一

=0

i

X' (0)

, X(L ) =

t.x / L ) - sinh ( a 1

¢, (cos (

x i

i

= 0 , X , (L ).

L

X

(i + 0.5

i x x / L ) - cosh (

i )7T

L ))

x

=sin / i - sinh ji / cos ji 一 c o s h i

i

=sin ( X(0)

1

=0

= ( i + 0.25

, X, (3L ) =C

= s i 〜n

X

1

=sin X" (0) = 0

, X"(L ) = 0

X' , ’(0)

= 0 , X’,’(L ) = 0

i

x

, X ( L ) =0 _

X" (0〕 = 0

i

x

i

/

i

sinh

l

i

x

ju^

( (j. x / i

L ) + sinh

x

( u x / i

L )

x

- (p• (cos i fi x / L ) + cosh ( fi x / L )) 1

I

X

= ( i - 1. 5

i

x

)7r

=sin 严 u i - sinh u产i / cos a严i - cosh u 严i X = s i n ( /i x / L ) - s i n h i

,X"(L

X(0) (0)

) =0

=0 , X … ( L

x

-沴(cos (

i = ( i - 1.5

i

)71

= s i n 产i a + s i n h a产i / c o s u

+ cosh u

= s i n ( fi x / L ) + 0 s i n h ( i

X(0) =0

, X"(L )

X" (0) = 0

,X , "

i

x

T a b l e 10.1

(L

i

= ( i + 0.25 = s i n fj.

x

i

/ u x / L ) i

)7r

/ s i n h p.

: Beam v i b r a t i o n f u n c t i o n s .

F r e e , C = Clamped)

x

fx x / L ) - c o s h ( u x / L ) ) i x i x

( S = simply-supported,

x


210

L o c a t i o n and t y p e o f supports

displacement c o n s t r a i n t s o f the corresponding cross—section

For edges parallel

to the Y-axis

simply-supported

v=0=w and

clamped

u=v=w=0

< r =0 x

For simply-supported

u=0=w and cr =0 y u=v=w=0

clamped

For

intermediate

parallel For

to the Y-axis

intermediate

parallel

v=0=w

line

line

supports

u=〇=w

to the X-axis

Table 10.2 : Boundary and internal conditions of plates

10.3 Numerical examples Pre』i历Jinary To

assess

the

accuracy

and

efficiency

of

the

modified

beam

functions, w e will first consider the following continuous beams with various geometric configuration. (I) A three-span continuous beam with uniform support spacing and different end supports. Bending rigidity and mass density are constant along the overall span. (II) A four-span continuous beam with non-uniform support spacing and

simply-supported

ends.

Bending

rigidity

and

mass

density

are

constant along the overall span. (III) A two-span continuous beam with non-uniform support spacing and simply-supported ends. The beam is non-prismatic in the sense that it is made up of two prismatic spans with different bending rigidity and mass density. In

these examples,

the stiffness matrix and

the mass matrix are

defined as

[ K 1

U = I

X

i

EI(x) X': dx j

[M]..= u

J X. p(x) X. dx

(10.11)


211 where E I ( x )

and p ( x ) denote r e s p e c t i v e l y the v a r y i n g bending r i g i d i t y

and mass p e r u n i t l e n g t h a l o n g the span. In a d d i t i o n to the i n c r e a s i n g number o f

terms used i n each case,

the s t a r t i n g term o f the t r i a l f u n c t i o n s a l s o varies; for a beam with q intermediate supports, and if the lowest nf frequecies are sought, then starting term r = 1 or 1+q nf ^ number of terms taken, nt ^ 2 x nf For

comparison

with

analytical

solutions

[125],

the

following

normalized frequency (X) is adopted,

o) = (A/L)

2

(EI/p)

1 /2

where L = beam overall length

The present results are summarized in Tables 10.3 to 10.5.

In all

cases, except for the second non—prismatic two-span beam in Table 10.5, 2 x nf terms are sufficient to give an accuracy of less than 1 percent difference between the present results and the analytical solutions. For the

second

non-prismatic

two-span

beam,

a

maximum

of

difference is found at mode 4 when 2xnf terms are used

2.7

percent

to compute the

frequencies. This relatively larger difference can be attributed to the fact

that

the

mass

density

influence of

the

of

the sharp changes of bending rigidity and

beam

are

not

taken

into

account

in

the

trial

functions• In general rapidly trial

to

one

can

observe

the analytical

function

is.

When

that

the

present

solution whatever the

frequencies

the

are

solution

starting

computed

converges

term of the

using

only nf

terms, however, the higher frequencies in most cases are more accurately predicted b y starting expected

as

the

appropriately

conditions;

higher

the

Unfortunately,

mode

faster

for

the

the

simply-supported-free

trial

modes

function from

of

shapes

a

of

convergence

cases of

the

term

single-span beam the

corresponding

is not

guaranteed

1+q.

This

is

approximate more multi-span for

all

beam.

boundary

three-span beams with free-free ends or

ends, unreasonable

results,

which

are

excluded

from Table 10.3, are obtained by starting the trial functions from the term

1+q.

In

these

cases,

the

first

two

terms

of

the

corresponding

single-span beam vibration modes cannot be neglected in determining the trial

functions.

Therefore,

one

must

starting term of the trial functions.

be

cautious when

selecting

the


lll:x 」

3 2

议23.130.22.22.22.2222:22.1

40542642 82112111 C\JC\I

92 62 52 42 62 52 42 43 2 2 7352328152429161 9 9 9 9 9 9 9 9 9 444d-4444

750781841 733233222

Ki22.22u22.l22z2:22.26.瓜瓜23.23.1

_

*

OJ

18651654 32223222 66666666 44444444

4 B7 27 17 17 17 17 17 17 17 80 89 79 71 80 89 7 9了

944354332 888888888

777777777 •

999999999

u U1514S

%

%312721

15543 32 2 2 2

CsJCNJ

84 332 42 8 8 8 8 8 9 81 87 74 7 2 2222 3333.3.

81152 3 2 2 11

况23.瓜23.23.23,20.肌肌肌20.23.2121^21.

7 4 3 442

3^^219

oooooo

11 51 41 61 41 2 2

*

*

*

«

«

«

o 〇 〇 o o

S7

\

4

/

44 4 44

CvJ

})

3 7553 o 3z 2o 54)42*4?2au^!51

oo

49 1 9o 9 91 9o 98 «

〇 〇

1 11 1 1 1

887887 6 6655 5 1 oo 9 666 6 66 33333 1111 o 22222 6666 6 44444 44444

111111

f

C\J OO

Table 10.3 Frequencies of a three-span continuous beam.

弘 沾 4 5 4 5 8 1 4 6 4 6 3 8 S 2 0 5 5幻 ½6 9 ¾ M 2 7 1 8 4 8 4 2 3 9 3 3 4 2 5 : ^ ¾ 7 8 W 7 6 9 8 花 沾

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lD025M17%sjKll

C\1 5^ 58 58 58 58 58 58 58 5 ^ _j

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9.8 9.8 9.8 9.8 9.8

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,,,,,,,ac

))02))0t 6811681 c

«

666266319 387387320 20762765 3 8 6 87 9 5 4 4 4 4 4o o 9 8 099999998 988888888 99889888 322221 77777 88877 « » • • • • • • • • • • • • • • • • 3 -• 2•2 2 2 2«2•2 2 3 3 3 3 3 3 3 3 3 •2•2•2•2•2•2•2«2 4 4 4 4 4 4 • • 11111 1 1 1 11 1 1 1 1 11 111111 1 11 11111 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1

\j,

1

* • _ • • • • • • • • * • • • • o o o1 o o o 1 1 1 11 1 1 1 1 o o o o 11 1111 1 11 1 1 1 1 1 1 11 1 1

.

V

f v

4.9 4.9 4.9 4.9

29 7 68 67 69 68 68 67 67 6 •

.

/IV

\1/

555555555 222222222 444444444 999999999

〇 〇 〇1 1 1 1 1 1

N

y l

))02))02t ))02))02t 68116811 c 68116811 c cj 1,1,1,1>3>3,3,3 , a X

s-s

Frequencies 入 Mode Boundary condition (r,nt) 1

2222

-


213

Frequencies A Mode

(rďź&#x152;nt)

Exact[125]

9.665

13.68

17. 07

Zl. 91

9.658

13.48

16. 56

19. 19

9.654

13. 47

16. 43

19.01

9.668

13. 49

16.,54

19.,51

9.643

13. 47

16,,50

19,,17

9.638

13.47

16,.49

19 .16

9.630

13.42

16,.46

19 .00

Table 10.4 Frequencies of a four-span continuous beam with both ends simply supported. EI,p=constant. Li=0.1, L2=0.3, L3=0.4, L4=0.2 where Li = length of span i / overall beam length.

Frequencies A Mode

(r.nt)

7 2 2 16 2 4 5 4 3

IX

i x 1X

1

1X

1

o

A x

il

12 7 8 3 19 6 8 5

IX

i x

6 o 8 8 3 2 1 2 o 9

IX

1

v

%

7.1 1 1 Lt 1X iri 1X 1

1X

Ii

5 4 4 4 4

IX

1 o 5 o 7 7 6 6 6 5

IX

= 0.1296 = 3 1 1 o 3 9 3 o 2 3 8 8 9 8 7

/

1 3 1 5 o 3 4 4 2 1 7 6 7 6 5

1

â&#x20AC;&#x201D;I

8 7 7 7 6 3 3 3 3 3

8 8 4 8 1) 5 4 5 4 4 J 2 2 2 2 2 E

ly X

4 8 ~ 4v 8 c 4848 c , , , , f y I > 1 1 0 4 2I X I C N J C S J X r v /1 r v /wV.

(EI)

11

,6561 2 1 6 17

(EI) / (EI)

Table 10.5 Frequencies of a two_span non-prismatic continuous beam

with both ends simply supported. Li=0.3, L2=0.7 where Li = length of span i / overall beam length. For calculating normalized frequencies, (EI)i and p

are used.


214

Simply-supported

To

verify

laminated

the

plates

formulation

fundamental frequencies of

of

the

single-span,

finite

layer

method,

the

symmetric and skew-symmetric

square laminates are computed using one term of the t r i a l functions i n each of the two i n - p l a n e d i r e c t i o n s ( r=s=m=n=l ). They are compared with the three-dimensional f i n i t e d i f f e r e n c e s o l u t i o n of Noor[120]. The f i b r e o r i e n t a t i o n s o f the d i f f e r e n t laminas a l t e r n a t e between 0Oand 90 O with respect t o the X - a x i s , and i n the symmetric laminates the 0°layers are the outer surfaces. The total thickness of the aligned layers are the same,whereas the individual layers are taken to be of equal thickness. Geometric and material constants are given as follows : span—thickness ratio E

/ E = 40

G

/ E =〇•

12 12

G

23

2

.25

In 丁able 10.6,

13

/ E

2

density

13

23

1.0

one can e a s i l y observe the good agreement between

the two s e t s o f r e s u l t s . In most cases, the present f i n i t e layer r e s u l t s are about 1 percent l e s s than those given by Noor.

No. o f m a t e r i a l Layers

Frequency 入 Noor [120]

Present

0.34250

0.33900

No. of f i n i t e layers

skew- symmet r i e 0.33758 0.42719

0.42379

0.43006

0.42666

0.45374

0.44681

symmetric

Table 10.6 : Fundamantal frequencies of square laminates. 入= ( w / h ) ( p / E

2

)1/2


215

Continuoas rectangular plates A series

of

rectangular

plates

with d i f f e r e n t

arrangement

of

intermediate l i n e supports are shown i n F i g . 10.2. The p l a t e s are made of isotropic

material

or

laminated

composite

material.

In

case

of

laminates, the m a t e r i a l constants used are the same as those given i n the preceding s e c t i o n except that

=

i s replaced by

=G

in

the f o l l o w i n g examples. In a l l cases, the r a t i o between the shorter span and the t h i c k n e s s i s taken e i t h e r as 5 or 100. Support conditions along the s i d e s of the p l a t e are i n d i c a t e d by four l e t t e r s that define the type

of

support

used

along

the

x=0ďź&#x152;

x=L , y=0 and y=L sides x y respectively. The letters S and C denote simply supported and clamped boundary

conditions.

For

the

2-way

3-span

plates

with

S-S-S-S

in

Fig. 10.2(b), the locations of the intermediate line supports on X- and Y- axis are replaced by (0.3b,0.65b) and (0.45b.0.85b) respectively. For isotropic plates, the natural frequencies are normalized as A2 = u)2 phb4 / D

where D = Eh3/ 12(1-1^2) and v-0.3

while for composite plates, X

2

= a)2 phb4 / ( D

11

D

22

) 1/2 where D. = 4E h/45 ii

xi

The starting terms for the present example are taken as r^l+q^ and s=l+q , y In order

to assess

the accuracy

of

the modified

â&#x20AC;˘ beam functions

relative to other methods available in the literatures, the lowest six frequencies of plates with the shorter span-to-thickness ratio of 100 (although finite layer method is more appropriate for analyzing plates with small span-to-thickness ratio) are computed and summarized in Table 10.7. Using five finite layers and five terms of trial functions in each of the two in-plane directions, the present results for the 2 way-2 span plate differ by a maximum of 4 percent from those due to Leissa[126]. Results

for

other

cases

also

compare

Zhou[123], Azimi et.al. [127] ďź&#x152;and Kim

well

with

those

provided

by


216

Frequencies 入 Mode sequence 1

2

3

2 9 7 7

0 0 8 1^

2

o 8 1 9 2 16 3

2 5 5 3 2 8 5 4 2 3 9 6

49.52 49.52

2 2 16 3

o 2

2 9 7 7 o 9 9 9

o o 9 9 9

2 o

2 5

214.62 212.14

4 4

24.83 23.65

9 9

21. 16

5

20.81

5

29.70 28.07

o o

6

24.37

9 9 4 4

7

6 o o

8

30.77 28.95

2 21 1 1

3〇9

108.22 9 4

2 2 o

118.55 115.04 113.8 113.18

4

9 4

2 o 4 4

4

—*

26.00

2 2 2

4

Is

27. 10

8 7 4 9 9 9

7

.

22. 16 22. 16 21.60

9 9

o

3 o

8

7

9

7

7

o 4 • 2

8

A

9

0

8

2

9

8 7

1 9999

o o 8 z 9)

9

8

11 • 9 9 s2

2 (Fig.10.2a) S-S-S-S 2*3*3Ca) 2*4*4 3*5*5 5*5*5 Leissaf126] 1 way-3 span S-S-S-S 5*4*4 5*5*5 Azimi[127] S-S-C-C 5来5来5 21.92 Azimi[127] 21.63 S-S -O S 5*5*5 20.41 Azimi[127] 20.22 1 way-4 span (Fig.10.2c) S-S-S-S 5*5*5 19.82 Azimi[127] 19.74 2 way-3 span (Fig.10.2b) C-C-S-S 5*5 来5 186.38 Kim[128] 184.30 O S - C-S 5来5米5 191.98 190.69 Kim[128] OC-OC 5来5来5 201.77 198,55 Kim[128] S-S-S-S 140.08 5*5*5 139.09 Zhou[123] C-C-C-S 196.92 5*5*5 193.78 Kim[128]

51. 15 51.15

54.13

. . . . d

51.24 50.69

52.45

30.21 27.12

49.52 49.35

49.52 49.98

235.67 231.35

255.37 252.17

259.65 255,60

291.16 286.95

230.85 226.87

231.18 227.18

260.47 259.99

260.60

265.88

266.09 265.93

248.35 243.27

248.35 243.32

288.35 282.31

302.05 297.20

302.05 297.20

172.03 169.04

179.33 175.60

207.19 202.06

217.65 210.78

247.97 239.84

235.06 229.59

244.51 237.58

264.14 254.80

277.35 268.75

299.00 288.79

Table 10.7 : Frequencies of continuous thin-plates. (a) = no. of finite layers * no. of terms

in X-direction * no.of

terms in Y-direction. Having

established

the

confidence

in

the

method

in

analyzing

isotropic thin plates, the lowest three frequencies of thick isotropic plates and laminated composite plates (span-to-thickness ratio =5) with


217

i d e n t i c a l geometric arrangements are summarized i n Tables 10.8,

10.9

and 10.10. These sets of r e s u l t s are presented f o r the f i r s t time i n open

literature,

and

no

comparison with

other

methods

are

made.

Increasing number of terms was used to obtain a converged solution i n a l l cases. For square p l a t e s of 2 way-2 span and 2 way-3 span, equal number of terms of t r i a l f u n c t i o n s are used i n each of the two in-plane d i r e c t i o n s . For 1-way continuous p l a t e s , however, more terms are taken i n the continuous-span d i r e c t i o n . Based on the r e s u l t s shown i n Tables 10.8-10.10, a few remarks can be made. ( i ) By comparing the frequencies of Table 10.8 w i t h the those i n Table 10.7, one can observe that, because of the increased f l e x i b i l i t y due to transverse

shear

isotropic plates

and

normal

deformation,

the

frequencies of

thick

( Table 10.8) are lower than those o f t h i n plates

(Table 10.7). I t i s apparent that the amount of d i f f e r e n c e depends on the types of boundary conditions and the arrangement of intermediate l i n e support. ( i i ) Results i n Table 10.9 i n d i c a t e that the frequencies of the 3-layers p l a t e s are higher than those of the 5 - l a y e r s p l a t e s despite what types of boundary conditions are concerned. Table

10.10

The

same trend can be seen i n

by comparing the r e s u l t s of

the 2 - l a y e r s and 4-layers

p l a t e s . I t i s a l s o noteworthy that the same amount o f material i s used i n the 2 - l a y e r s and 4ä¸&#x20AC;layers p l a t e s or the 3 - l a y e r s and 5 - l a y e r s plates. This implies that the frequencies of a p l a t e w i l l increase by d i v i d i n g the material i n t o more layers. ( i i i ) I t i s well-known that the bending-stretching coupling of skewsymmetric laminates w i l l , i n general, buckling

loads

and

natural

cause larger d e f l e c t i o n ,

frequencies

because

of

lower

increased

f l e x i b i l i t y [ 4 7 ] : comparison between the two sets o f r e s u l t s i n Table 10.9 and Table 10.10 reveals that laminates

(e.g.2-layers plates)

are

the frequencies of skew-symmetric indeed lower than those of

corresponding symmetric laminates (e.g. 3 - l a y e r s p l a t e s ) .

the


218

Frequencies Mode sequence

1

)

*

3 5 6

a

a 一 来

2s

y Is

WS356WS356S356S356 一 *** _*** 一 来 来 米 一 米 来 米 2S5551S555S555S555

*

3 S c s ys334c334c334

a

64.80 63. 26 62.57

64.80 63.26 62.57

17.63 17.63 17.63

19.37 19.37 19.21

23.64 22.63 22.63

22.78 22. 49 22.41

24.92 24.62 24.62

18.08

21. 11

18.08

20.81

25.00 24.03 24.00

i

56.25 56.25 56.25

* 来 米 一 一

1X

6 3 3

1X

2 11

IX

*

9 9 9

来 米 来 来

18.04

来 来 来 一 来 来

ys356c356c356s356c356

WC356S356C356S356C356 l 5 *5 C *5 5 5 C 5 5 5 S 5 5 5 C 5 5 5 2C5

5*6*4

3 3 3 6 6 6

1 X I X r H

5*5*3

7 7 7

1 way-4 span (Fig.10.2c) S-S-S-S 5*3*3

20.78

21.28 21.28

102.44 101.48 101.36

111.27 109.66 109.47

112.10

104.93 103.63 103.59

114.34 112.73 112.45

114.50 112.91 112.65

107.25 105.55 105.55

116.00

114.65 114.62

116.05 114.68 114.64

97.57 97.51 97.25

107.59 105.73 105.44

107.64 105,76 105.47

106.05 104.56 104.54

114.70 113.57 113.50

115.64 113.84 113.60

20.84

3

*

i

18.65 18.64 18.64

110.57 110.47

Table 10.8 : Frequencies of continuous t h i c k - p l a t e s . ( a ) = no. of f i n i t e layers * no.of terms i n X - d i r e c t i o n * no.of terms i n Y-direction


219

Frequencies A Mode sequence

1

2 way-2 span (Fig.10.2a) S-S-S-S 3 layers 4来3米3⑷

35.08 4*5*5 35.08 4*6*6 35.08 5 layers 5*3*3 36.36 5*5*5 36.36 5*6*6 36.36 1 way-3 span (Fig.10.2d) S-S-S-S 3 layers 4*3*3 14.54 4*5*3 14.54 4*6*4 14.54 5 layers 15.25 5*3*3 15.25 5*5*3 15.25 5*6*4 1 way-4 span (Fig.10.2c) S-S-S-S 3 layers 14.54 4*3*3 14.54 4 〒 3 14.54 4*6*4 5 layers 15.25 5*3*3 15.25 5*5*3 15.25 5*6*4 2 way-3 span (Fig.10.2b〕 s-s-s-s 3 layers 54. 18 4*3*3 54. 15 4*5*5 54.01 4*6*6 5 layers 55.50 5*3*3 55.47 5*5*5 55.33 S*6*6

38.09 37.55 37.22

38.72 37.68 37.33

39.27 38.61 38.33

39.87 38.87 38.45

15.71 15.71 15.53

17. 15 17. 15

18. 18

16.41 16.41 16.25

18.81

15. 25 15.24 15.24

16.84 16.84 16.84

15.95 15.95 15.95

17.51 17.51 17.51

57.88 57.27 57.12

58.78 57.37 57.28

59. 12 58.31 SR. 17

59.94 58.61 S8. 51

17.85 17.85

Table 10.9 : Frequencies of continuous symmetric laminates. ( a ) = no, of finite layers * no.of in Y-direction

terms in X-direction * no.of terms


220

Frequencies A Mode sequence

1

f

n

o 印

4 -

o o o

1.66

1.66 1.66

12.72 12.72 12.63

15.04 14.42 14.42

14.47 14.47 14.47

15.46 15. 46 15.34

16.81 16.81

11.66

12.29 12.29 12.29

18.40 13.79 13.79

14.47 14.47 14.47

15.07 15.07 15.07

16.42 16.42 16.42

51.79 51.77 51.66

56.52 55.71

56.53 55.71 55.59

54.05 54.02 .S^.89

58.19 57.22 57.10

58.19 57.22 57.10

c

? d

a

s

r

Sp

)

d

IX

ti

« 1

F

a

c

n

s

z o

8 88

38. 37 58 37 25

e

3 3 3

oo o

38.37 37.58 37.25

555 3 33

36 01 30 35 00

¾

36.01 35.30 35.00

s

2 22 3 33

3

s s

p -

)

F

r

s

m

4

p

-

(Fig.10.2c)

17.56

a

S

s

s

-

11.66 11. 66

e r D

^

s

^

p

s

(Fig.10.2b)

S a

p

7 " - s e r e r

外 Tayr 气: f3^y6*§s-h3*y^>hyyJ>§-sla^3*5%lM*5*6w-sl*3*5%l*3*5*6 gs-h 沪 y^>h

2^2^^^4^^^1^2^^^4^^51S255545552S2444455S

(Fig.10.2a)

Table 10.10: Frequencies of continuous skew-syininetrie laminates. (a) = no. of finite layers * no.of terms in X-direction * no.of terms i n Y—direction

町 T


221

10.4 Conclusion A f i n i t e layer method i s used to analyze v i b r a t i o n o f p l a t e s w i t h intermediate l i n e supports. The p l a t e s are treated as three-dimensional e l a s t i c bodies.

A modified v i b r a t i o n f u n c t i o n i s adopted as the t r i a l

f u n c t i o n f o r computing the n a t u r a l fequencies of i s o t r o p i c and laminated composite p l a t e s and the r e s u l t s are compared w i t h a v a i l a b l e s o l u t i o n s f o r the cases of t h i n p l a t e s . For the cases o f t h i c k i s o t r o p i c and laminated composite p l a t e s , r e s u l t s are presented f o r the f i r s t time i n open l i t e r a t u r e . I t should be noted that the layerwise laminate p l a t e theory r e c e n t l y proposed by Reddy e t . a l . [ 6 8 ] can be considered as a g e n e r a l i z a t i o n of the f i n i t e layer method.


222

c

o SDu Xw.

Q=3 a> ai丨>丨丨"» •‘ 丨 -55 a>

E o

_丨•丨丨»

o --① «

[

T3

<o

5 O

^

2

o

5

CO

X

n:

¾

cc o ¾

X ca

0) u 0 a a d n c 1 >>

r-H

g;oi> 5 - «4——

3 ^ <D (/) £ T 3

W

«3

D ) O "5

a s • r H cn

cn

0 W) -O (D rH cd

X:

4 J •rH

U 0 4-» U 0

a a

8 w 名 a

-E

< w

0)

as r H a 乂

o x:

D 1 d o c

co

o < T—i

o rH * n Hti


223

2

- D

Houcdd

snonufwuoo

n

snonulfwuoa (

o

rH9uled

cud

qsd

j Q

i i i i

qscod

JQ 本

n 令 丨 士

/-—S

cS

n

ucdds-对 Ho Ac fc d d M a IX

73


224

CHAPTER 11

11.1 Concluding remarks In t h i s study f u r t h e r development o f the f i n i t e s t r i p , f i n i t e prism and

finite

layer

methods

are

presented.

Throughout

this

thesis,

displacement-based formulation i s adopted. S p e c i a l a t t e n t i o n i s g i v e n to the a n a l y s i s o f i s o t r o p i c or laminated composite p l a t e - t y p e s t r u c t u r e s . The

methods

are

formulated w i t h i n

the

frameworks

of

Kirchhoff's

t h i n - p l a t e theory, Reddyďź&#x152;s third-order plate theory, and 3-D continuum theory. A

set

of

computed

shape

functions,

COMSFUN,

is

developed

and

adopted as the spanwise trial, functions for a new flat-shell finite strip. This finite strip has been successfully applied to the flexural vibration and buckling of plates with complicated support conditions, bending of rectangular plates with changing longitudinal ridigity and vibration and buckling problem of prismatic folded plates. In addition, 2-D plane problems with non-rectangular domains can be treated

in a

standard finite element manner. The present finite strip retains the advantages of the classical finite strip and the spline finite strip method such as rapid convergence for point-loaded plates, reduced number of nodal variables when compared with the finite element method and ease of modelling. Additionally, the present finite strip method possesses a few

superior features when compared with

the semi-analytical finite

strip and the spline finite strip such as the capability of dealing with geometric

discontinuities

problems,

its

simplicity

conditions, and

in in

plate-bending handling

problems

boundary

and

and

2-D

Internal

plane support

its ease of implementation. Furthermore, only simple

shape functions are involved in the calculations and computation of the stiffness matrix involves no numerical Integration. The original spline finite strip is restricted to the analysis of thin

plates.

In

this

work,

a

third-order

spline

finite

strip

is

developed as the counterpart of the original spline finite strip which is capable of treating moderately thick isotropic or laminated composite plates. The formulation is done within the framework of a third-order plate

theory.

Its

accuracy

in

performing

linear/nonlinear

plates with different span-to-thickness ratios standard examples. For nonlinear vibration,

analysis

is assessed by a few

a. special reduction incthod


225

i s proposed f o r reducing the computational cost of the time-stepping a n a l y s i s . The success of the reduction method r e l i e s on the b e n e f i c i a l features of the R i t z vectors, a pre-transformation technique. A new s p l i n e f i n i t e prism method i s developed and a p p l i e d to the a n a l y s i s of rectangular, t h i c k , laminated composite p l a t e s . No kinematic assumptions are imposed and 3-D e l a s t i c i t y theory i s adopted. The method i s formulated using B3-spline functions i n the l o n g i t u d i n a l d i r e c t i o n and a set of g l o b a l - l o c a l polynomials i n the c r o s s - s e c t i o n o f plates.

The

rectangular

method i s

validated

by

means

of

a

few

examples

the of

laminated plates w i t h a v a i l a b l e a n a l y t i c a l s o l u t i o n s or

f i n i t e element s o l u t i o n s . Rectangular p l a t e s w i t h d i f f e r e n t boundary c onditions can be analyzed at a resonable cost w i t h s u f f i c i e n t accuracy f o r preliminary design. The 2-D-type data structure of the f i n i t e prism method i s retained i n d e v i s i n g a 3-D t h i c k - p l a t e element. In terms of the s i m p l i c i t y of modelling, the advantages of the present method over the conventional 3-D f i n i t e element models are two-fold,namely, the volume o f Input data is

reduced,

and

the

in-plane

2-D

mesh

and

the

transverse

1-D

i n t e r p o l a t i o n can be refined independently of each other without having to reconstruct a 3-D f i n i t e element mesh. The 2-D-type data structure a l s o allows f a s t e r

formulation of

the

element s t i f f n e s s matrices.

A d d i t i o n a l l y , the method retains the accuracy of p r e d i c t i n g transverse shear stresses of layerwise model as w e l l as the computational economy o f the h i e r a c h i c a l s i n g l e - l a y e r model. The method i s v e r i f i e d through a few examples w i t h a n a l y t i c a l s o l u t i o n s a v a i l a b l e . For f r e e v i b r a t i o n analys i s of rectangular p l a t e s w i t h intermediate l i n e supports, beam v i b r a t i o n functions are augmented by polynomials and subsequently employed as the t r i a l f u n c t i o n s f o r the development of a new f i n i t e layer technique.

The n a t u r a l fequencies are computed f o r

i s o t r o p i c and laminated composite p l a t e s and the r e s u l t s are compared w i t h a v a i l a b l e s o l u t i o n s f o r the cases of t h i n p l a t e s . For the cases of t h i c k i s o t r o p i c and laminated composite p l a t e s , r e s u l t s are presented for

the f i r s t

time i n

open l i t e r a t u r e ,

to

the

best

of

author's

knowledge. I t should be noted that the computed shape functions developed f o r t h i n - w a l l e d structures i n e a r l i e r chapters are equally a p p l i c a b l e to the a n a l y s i s o f moderately thick p l a t e s and laminated composite p l a t e s . In


226

p a r t i c u l a r , the s p l i n e functions of the s p l i n e f i n i t e s t r i p and spline f i n i t e prism i n Chapter 7 and 8 r e s p e c t i v e l y , can be replaced by the computed shape f u n c t i o n s . For the f i n i t e layer method i n Chapter 10, COMSFUN

can

satisfying

be the

used i n

each o f

boundary

and

the

two

intermediate

in-plane d i r e c t i o n s f o r line

support

conditions.

However, the computed shape f u n c t i o n s are not i n t e n t i o n a l l y designed as a s u b s t i t u t e f o r s p l i n e f u n c t i o n s , but rather as a counterpart, even though they are superior In d e a l i n g w i t h some o f the common plate-type s t r u c t u r e s as demonstrated i n the e a r l i e r chapters. Furthermore, the e x p l o r a t i o n of other types of t r i a l f u n c t i o n s , such as the modified beam func tio ns of

the

new f i n i t e

layer

method i n Chapter

10,

and the

i n v e s t i g a t i o n of s p l i n e functions i n new areas o f a p p l i c a t i o n are p a r t l y our u l t i m a t e goals i n e x p l o i t i n g f u r t h e r the f i n i t e s t r i p , f i n i t e prism and f i n i t e layer methods. By no means can t h i s work be considered exhaustive. Each of the l a t e r chapters i s amenable to expansion. A few suggestions f o r future research are discussed i n the f o l l o w i n g s e c t i o n . 11.2 Suggestions f o r f u t u r e work

A Higher-order

thin-plate

element

The computed shape functions can be s l i g h t l y modified and be used i n each of the two orthogonal d i r e c t i o n s of a rectangular t h i n - p l a t e element(Fig.11.1). The r e s u l t i n g element can be assembled i n standard finite

element

manner

and

C^-continuity i s

maintained

across

the

boundaries between adjacent elements. S p e c i f i e d displacement constraints can be imposed on the corresponding nodal displacement v a r i a b l e s . The same idea can be applied to

2~D e l a s t i c problem and a higher-order

f l a t - s h e l l element can be e a s i l y f i g u r e d out.

Redaction

method for nonlinear

vibration

The high computational cost o f

a nonlinear v i b r a t i o n a n a l y s i s

stimulates the development of an e f f i c i e n t reduction method. Only a preliminary investigation

was

conducted

on the

performance of

the

reduction method f o r geometrically nonlinear v i b r a t i o n of p l a t e s . I t s a p p l i c a t i o n to general plate-type s t r u ct u r e s w i t h a r b i t r a r y geometry, loading,

and boundary conditions i s f e a s i b l e

investigation.

and requires f u r t h e r


227

i4naiysis1 For

cont inuous

sector

plates

with

circular

planform,

corresponding single-span, curvedâ&#x20AC;&#x201D;beam vibration functions modes)

can

be

augmented

coordinate system,

by

polynomials, expressed

to form a set

of

spanwise

trial

the

(or buckling in

a

circular

functions for a

circular thin-plate finite strip. This finite strip can be applied the

analysis

of

circular

bridge

slabs

or

other

sector

to

plates with

continuous line supports in the radial directions (see Fig.11.2). Adaptive finite strip methods Error estimation for plate bending problems can be developed for finite strips

in such a way that mesh refinement can be automatically

carried out in the area of rapid stress gradient. Combination of finite strip and finite prism methods In a previous study conducted by Chong et.ai-[40], finite strips are combined with finite prisms for

the analysis of sandwich panels.

This approach can be extended to the calculation of transverse stresses in a laminated material

layers

plate; where

finite prisms transverse

are used

stresses

to model each

are

required

of

while

the the

remaining layers are smeared Into finite strips. When compared with a complete finite

prism

analysis

of

computational cost can be achieved.

laminated

plates, a reduction

of


F ig .11.1

A rectangular t h i n - p l a t e element using COMSFUN both d i r e c t i o n s .

a continuous and curved finite strip with four intermediate radial supports


229

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238

AUTHOR'S PUBLICATIONS

Papers Published 1.

or accepted

for

publication

Y.K.Cheung and J.Kong " F i n i t e s t r i p a n a l y s i s o f f l a t s l a b u s i n g

computed shape f u n c t i o n s

"

PUBLICATION ) 2.

Y.K.Cheung and J.Kong" Application of a new finite strip to the

free vibration of rectangular plates of varying complexity" J.Sound and Vibration (ACCEPTED FOR PUBLICATION) 3.

Y.K. Cheung

vibration

and

and

丨 • Application of

J.Kong

buckling

of

folded-plate

a

new f i n i t e s t r i p t o

structures 1 '

Structures 4.

Y.K.Cheung, F.T .K.A u and J.Kong

丨 • S t r u c t u r a l a n a l y s i s by f i n i t e

s t r i p u s i n g computed shape f u n c t i o n s “

Computational 5.

methods in Engng.

Y.K.Cheung and J.Kong" A p p l i c a t i o n o f a new f i n i t e s t r i p t o the

analysis of plates with varying r i g i d i t y “

East

Asia

Pacific

Conf.on

Structural

Engg. and

Construction,

Korea, 1993 6.

Y.K. Cheung and J . Kong " A n a l y s i s of f l a t s l a b s by a

new f i n i t e

strip" (ACCEPTED FOR PUBLICATION) 7.

J.Kong and Y.K.Cheung " A p p l i c a t i o n o f the s p l i n e f i n i t e s t r i p t o

the a n a l y s i s o f

shear-deformable p l a t e s

4 6 ( 6 ) , 985-988, 1993

8.

Y.K.Cheung

and

J . Kong

Linear

elastic

stability

deformable p l a t e s u s i n g a modified s p l i n e f i n i t e s t r i p

of

shear

丨 •

Structures 9.

J.Kong and Y.K.Cheung "A g e n e r a l i s e d s p l i n e f i n i t e s t r i p f o r the

a n a l y s i s of p l a t e s " 10.

J.Kong and Y.K.Cheung " Nonlinear dynamic a n a l y s i s o f p l a t e s u s i n g

a reduction method " J. 11.

J.Kong

and

Y.K.Cheung"

The

Reduced-Basis

technique

and

its

application ”

Computational 12.

Mechanicst

Y.K.Cheung and J.Kong

丨 丨 Approximate three-dimensional a n a l y s i s o f

rectangular t h i c k laminated composite p l a t e s : Bending, Vibration and Buckling " Computers and Structures 47(2), 193-199,1993


239

13.

J.Kong and Y.K.Cheung " V i b r a t i o n a n a l y s i s o f t h i c k p l a t e s w i t h

continuous

line

support

using

a finite

layer

method."

Vibration (ACCEPTED FOR PUBLICATION) 14.

Y.K.Cheung

and

J. Kong

"Three-dimensional

analysis

of

thick

laminated plates using finite prism and finite layer methods" Invited Lecture, Third World Congress of Computational Mechanics, Japan, 1994 Papers submitted and under review 1.

J. Kong

and

Y.K. Cheung

‘丨 V i b r a t i o n o f

continuous

beams

using

modified beam v i b r a t i o n functions" (

in Engineering) 2.

Y.K. Cheung and J.Kong

丨 •A new f i n i t e s t r i p f o r a n a l y z i n g deep beams

and shear w a l l s " C 3.

Y.K.Cheung and J.Kong" Approximate 3-D f i n i t e element a n a l y s i s of

t h i c k laminated p l a t e s " (

Analysis of plate-type structures by finite strip, finite prism and finite layer methods  

PhD Thesis, The University of Hong Kong

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