Linear and nonlinear finite strip analysis of bridges

1+1

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Canad!l

LINEAR AND NONLINEAR FINITE STRIP ANALYSIS OF BRIDGES

A thesis submitted to the School of Gradunte Studies and Research in partial fulfillment of the thesis reqllirements fol' the degree of Doctor of Philosophy in the Department of Civil Engineering

PhoDo Candidate: Wenchang Li Thesis SlIpervisor: Mo So Cheung

Department of Civil Engineering Fnculty of Engineering University of Ottawa Ottawa, Canada o

~wenChang Li,

Ottawa, Canada, 1991

•••

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Direction des acquisitions et des services bibliographiques

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ISBN

Canada

~.

-315-80004-&

~::J UNIVERSITÉ D'OlTAWA UNIVERSITY OF OITAWA

ABSTRACT The annl~'~iR of highway bridges such ns slnb-on-gil'der bl'idges, box-girder bridgcs, cable-stayed bridges etc. is a very complicuted undert,nking. Analyticnl methods are applicable only for the simplest st,ructures. Finite clcmcnt \\lcthod is thc most powerful and versatile tool, which can be applicd to IUlnlyzc /Uly types of bridgc and any load cases. However, thc cfficicncy of that mcthod nccds to be improvcd bccause the finite element solutions usunlJy rcquirc too \\luch computer time, too large core storage and too many input datu. If a structure has a unifonn cross-scction and lillc end support,s (in fnct, a high

proportion of bridges can be simplified to such It st.l'lIctmc), the finite stl'ip method has proven to be the most cfficient numcricnl st.l'lIct.lIml ILnnlysis mcthod, which employs a series of functions to sil11ulate the variation of displacel11flnts in thc longitudinal direction of the structure. Thus, the nllmbcr of dil11ensions of /UlIllysis is reduced by at least one. Consequcntly, the comput,cl' time, storagc and input data are reduced significantly. Since this method

WIIS

first publishcd in 1968, it

has been extensively used for linear and nonlinear, static and dynamic anluysis of rectangu1ar, skew and curved slab bridgcs, slab-on gil'der bridges box-girder bridges etc. In the present study, the following efforts arc made:

1. Extending the finite strip rnethod to the aIlruysis of continuous haullchcd alah-

on-girder bridges and hox-girder bridges.

ii 2. Exl.!mding the splinc finitc strip mcthod to thc Iwalysis of continuous haunched slab-on-girdcr bridges Iwd box-girder bridgc8. :\. Extcnding thc finitc strip mcthod to nonlincar analysis of cable-stayed bridges. 4. hllPIYlving thc cfficicncy of gcomctrically nonlinear finite strip analysis of plates. 5. IJIlproving thc nccuracy of materiaIly nonlinear finite strip analysis of reinforccd concrctc slabs. 6. Combining the finite strip method with finite element method and boundary clement mcthod for analysis of rectangular plates with some irregularities.

A nl1l11bcl' of IlUmerical eXlUnples will show the accuracy and efficiency of the lIIethods devcloped in the present study.

iii

ACKNOWLEDGEMENTS The author wishes to express his sincere appl'cciation to his I'CNCIU'ch N1\pcl'visor, Dl'. M.S. Cheung, for his constructive suggcstions, vllluablc disCIIHsionN lUHI continucd assistance throughout thc coursc of thc study. Sinccre thanks are also exprcsscd to Dl'. L.G .•1ncgcr, Dl'. A.G. Rmmpm, Dr. M. Saatcioglu, Dr. S.F. Ng and Dr. TlUmka for t.heil' important information, valuable advice IUld gcnerous nssistlUlcC in c1lOosing topics, solving c1iflicult qucstions, revicwing the Thesis Proposal ctc. The fimUlcial support from the Natural Scicnccs lUul Engincming Rescarch COl1ndl of Callada is gratefully nckllowledgcd.

I

Contents

ABSTRACT .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . ..

iii

Tahlc of ContcIlts. . . . . . . . . . . . . . . . . . . . . . . • . . . . . .,

iv

List. of Figllrcs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

viii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

xii

. NOMEMCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xiv

1 INTRODUCTION 1.1

1

ANALYSIS OF HIGHWAY BRIDGES . . . . . . . . . . . . . . ..

1.2 DEVELOPMENT OF FINITE STRIP METHOD 1.3 SCOPE OF STUDY

.... . . ......... ... . . . .....

i\'

1 2

6

CONTENTS

2

v

FINITE STRIP METHOD 2.1

CONVENTlONAL FINITE STRlP METHOD

8

· .........

.

8

2.1.1

SERlES PART OF DISPLACEMENT FUNCTION . . . .

2.1.2

DISPLACEMENT FUNCTIONS . . . . . . . . . . . . . "

12

2.1.3

STRAINS............................

13

2.1.4

STRESSES...........................

14

2.1.5

MINIMIZATION OF TOTAL POTENTIAL ENERGY

14

2.1.6

COORDINATE TRANSFORMATION

...........

17

2.1.7

FLEXIBILlTY METHOD . . . . . . . . . . . . . . . . . ..

18

2.2

COMPOUND FINITE STRlP METHOD . . . . . . . . . . . . . ,

ID

2.3

EIGENFUNCTIONS OF CONTINUOUS DEAMS . . . . . . . .,

21

2.4 ANALYSIS OF CONTINUOUS HAUNCHED DRIDGES . . . ..

28

D

2.4.1

STRAIN-DISPLACEMENT RELATIONSHIP .. . . . ..

28

2.4.2

DISPLACEMENT FUNCTIONS . . . . . . . . . . . . . .,

30

2.4.3

SOLUTION PROCEDURES . . . . . . . . . . . . . . . ..

34

vi

CONTENTS

2.4.4

NUMERICAL EXAMPLES . . . . . . . . . . . . . . . . ..

35

2.4.5

CONCLUSION

38

........................

3 SPLlNE FINITE STRIP METHOD 3.1

47

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . ..

47

3.2 SPLINE FUNCTION INTERPOLATION . . . . . . . . . . . . ..

49

3.3

ANALYSIS OF CONTINUOUS HAUNCHED BRIDGES . . . "

51

3.3.1

STRAIN-DISPLACEMENT RELATIONSHIP . . . . . . .

51

3.3.2

DISPLACEMENT FUNCTIONS . . . . . . . . . . . . . "

53

3.3.3

PENALTY FUNCTION APPROACH . . . . . . . . . . ..

56

3.3.4

NUMERICAL EXAMPLES . . . . . . . . . . . . . . . . ..

57

4 NONLINEAR ANALYSIS 4.1

68

NONLINEAR ANALYSIS OF CABLE-STAYED BRIDGES . . ..

69

4.1.1

FINITE STRIP ANALYSIS OF GIRDER . . . . . . . . ..

70

4.1.2

FORMULAS FOR CABLE . . . . . . . . . . . . . . . . ..

72

4.1.3

STIFFNESS MATRIX OF THE PYLON . . . . . . . . ..

77

CONTENTS

4.2

4.1.4

INITIAL-STIFFNESS ITERATION

78

4.1.5

NUMERICAL EXAMPLES ..

80 84

DISPLACEMENT FUNCTIONS AND INITlAL STIFFNESS MATRIX. . . . . . . . . . . . . . . . . . . . . . . . . . ..

87·

4.2.2

GEOMETRICAL NONLINEAR SOLUTION . . . . . . ..

89

4.2.3

NUMERICAL EXAMPLES . . . . . . . . . . . . . . . . ..

93

NONLINEAR ANALYSIS OF REINFORCED CONCRETE SLABS 95 4.3.1

MATERIAL MODEL OF CONCRETE . . . . . . . . . ..

96

4.3.2

MATERIAL MODEL OF REINFORCEMENT . . . . . ..

99

4.3.3

FINITE PLATE STRIP . . . . . . . . . . . .. , . . . . .. 101

4.3.4

NONLINEAR SOLUTION . . . . . . . . . . . . . . . . .. 102

4.3.5

NUMERICAL EXAMPLE . . . . . . . . . . . . . . . . . . 105

5 COMBINED ANALYSIS 5.1

...............

GEOMETRICAL NONLINEAR ANALYSIS OF PLATES ... " 4.2.1

4.3

vii

118

FINITE STRIP METHOD FOR REGULA R PART .. . . . . .. ll!l

viii

CONTENTS !i.2

5.3

COMJ3JNED WITH FINITE ELEMENT METHOD . . . . . . . , 120 [i.2.1

FINITE ELEMENT METHOD FOR IRREGULAR PART

120

5.2.2

TRANSITION ELEMENT . . . . . . . . . . . . . "

5.2.3

NUMERICAL EXAMPLES . . . . . . . . . . . . . . . . . . 122

. . . 121

COMBINED ANALYSIS WITH BOUNDARY ELEMENTMETHOD123 5.3.1

BOUNDARY ELEMENT ANALYSIS FOR IRREGULAR REGION ... ' . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.3.2

TRANSITION STRIP AND COMBINED SOLUTION .. 126

5.3.3

NUMERICAL EXAMPLES . . . . . . . . . . . . . . . . .. 128

6 CONCLUSIONS AND RECOMMENDATIONS

138

6.1

CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.2

RECOMMF.NDATIONS........................ 141

REFERENCES

142

List of Figures

2.1

39

Structure Analyzcd by F.S.M..

o'

••••••••••••

39

2.3 Individual and Common Coordinate System .

............

40

2.4 Continuolls Berun . . . . . . . . . . . . . . . . . . . . . . ; . . . . ,

40

. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .

41

.......................

41

.........................

42

...............................

42

....................

43

....................

43

2.11 The Mesh of Shell Elements . . . . . . . . . . . . . . . . . . . . . ,

44

2.2 Folcled Plate Strip· . . . .

2.5 Span i ..

2.6 Support i . . . . . . . . 2.7 Reguli-Fhlsi Iteration . 2.8 Web Strip . 2.9

•••••••••

Shell Strip . . . . . . . . . . . .

2.10 Continuous Box-Girdcr Bridgc

IX

•

x

LIST OF FIGUR.ES

..............

44

.....................

45

2.12 Five Span Composite Box-Girder Bridge . 2.13 Whccl Wcight of Two Trucks

2.14 Division of Strips . . . . . . . . . . . . . . . . . . . . . . ., . . 2.15 Longitudinal Stresscs in Steel Girder at Section X-X (in MPa)

3.1

45 ..

Splinc Function and Its Derivatives .

46

63

........ . .

64

3.3 Will) Strip in Individual System ..

..................

64

3.4 Shell Strip . . . ...

•

•

65

3.5 Continl\ous Beanl . . . . . . . . . . . . . . . . . . . . . . . . . . ..

65

3.2 Plate Strip . . . . . . . . . . . . ..

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

o.'

•

•

•

•

3.6 Hmlllched Continuous Bridge . . . . . ..

........ ......

3.7 Haullched Continuous Box-Girder Bridge

.'

. .... ...... ..

66

3.8 Division of Strips . . . . . . . . . . . . . . . . . . . . .

67

3.9 Longitudinal Stresses at Cross-Sectioll X-X (in MPa) .

67

4.1

Cablc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.2 Pylon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

LIST OF FIGURES

4.3

xi

Initial Stiffncss Method . . . . . . . . . . . . . . . . . . . . . . . . 108

4.4 Single PhUle Cable-Stnyed Bl'idge . . . . . . . . . . . . . . . . . "

108

4.5

Double Planc Cable-Stayed Bl'idgc . . . . . . . . . . . . . . . . .. 100

4.6

Deflection of Girder and Pylon .. . . . . . . . . . . . . . . . . .. 100

4.7 Longitudinal Strcsses at Cl'oss-Section B (in MPII) . . . . . . . .. 110 4.8 Longitudinal Stresses at Cl'oss-Sect.ion F (in MPa) . . . . . . . "

111

4.9 Possible Divergence . . . . . . . . . . . . . . . . . . . . . . . . . "

112

4.10 Equivalent Uniaxial Stress-Strain Model . . . . . . . . . . . . . .. 113 4.11 Biaxial Strength Envelope . . . . . . . . . . . . . . . . . . . . . .. 114 4.12 Material Model of Steel . . . . .

. . . . . . . . . . . . . . . . . 115

4.13 Layers of Strip

. . . . . . . . . . . . . . . . . 115

4.14 Taylor Slab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 116 4.15 Deflection of Taylor Slab . . . . . . . . . . . . . . . . . . . . . . "

5.1

117

Rectangular Finite Element . . . . . . . . . . . . . . . . . . . . .. 131

5.2 Transition Element . . . . . . . . . . . . . . . . . . . . . . . . . .. 131

LIST OF FIGURES

xii

5.3 Squarc Platc . . . . . . . . . . . . . . . , 5.4

............

132

Platc Supported by Walls and Columns . . . . . . . . . . . . . .. 132

5.5

Dcflcction and Bcnding Moments of Plate in Fig.5.4

5.ü

Double Nodes ..

.. . . .. .. .. . ... . . .. . . .. .

. .. 133 . .. 134

5.7 Transition Strip. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 134 5.8 Simply Supported Square Plate under Uniform Load . . . . . . . , 135 5.0 Plntc wlth Opcning and Skew Corner 5.10 Bcnding Moments along A-B-C

. . . . . . . . . '... 136 . . . . . . . . . . . . . . 137

List of Tables

2.1

Thc Proper Number of Scgmcnts and Gauss Points ..

35

2.2

Longitlldinal !)tresscs in Two Span Box-Gil'dcl' Bridgc . . . . . "

37

3.1

Vallles of Spline F\mction at l(nots . . . . . .

50

3.2 Deflection and Longitudinal Stl'esscs in Continuous Deam

58

3.3 Longitudinal Strcsscs in Two Span Slab-oll-Girdcl' Bridgc

59

3.4

Longitudinal Stresscs in Two Span Dox-Gil'dcr Bridgc . . . . . ..

61

4.1

Deflections (in lllcters) of Girder at Cabl,! Attachment Points and Vertical Forces (in MN) of Cablcs . . . . . . . . . . . . . . . . . .

81

...............

81

4.2

Bending Moment (in MN.m) of Girdcl'

4.3

Vertical Forces of Cablcs on Girdcr und Horizontal Forccs of Cublcs on Pylon (in MN) . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

84

DcHcction and Strcsses in Cylindrical Platc Bending

94

4.5 DeHection and Strcsscs in Clamped Squarc Plate . . . .

94

4.6 Cocfficicnt K for Stress of Stecl after Concrete Cracking

. . .. 100

4.4

5.1

DcHcction and Moments in Clampéd Square Plate

. . . . . . 122

5.2 DcHcctioll and Stt-csscs in Simply Supported Squarc Plate . . . .. 129

xiv

NOMENCLATURE {a} b

[B] D

[D] E f~

f:

{F} [F]

h

[K] l

L(y) m

[N]

{P} q T

Til T2

R {R} [T], [t] [T,] u,v,w

U W

displncclllcnt vcctor width of strip strnÍn matrix = Eh 3 j12(1- v 2 ) c1astic matrix Young's modulus uninxial compressivc stl'cngth of concrctc uninxial tensilc stl'cngth of concretc force vcctor flexibility matrix thickness of stl'ip, length of longitudinal scction stiffness matrix length of strip Lagrange interpoIntion cxprcssion nU1l1ber of longiturunal scctions in a strip matrix of shapc function load vcdor loading per unit lU'cn, weight pel' unit length nUlllber of serics tcnns uscd in IUlnlysis curvatl1l'e radii of curvilincnl' coordinatc Iincs curvnturc radius of bot tom fllUlgc vector of redundlUlt forces or resistant forccs coordinnte transfonnntion matriccs for displncclllcnts coordinate transformation mntrices for strnÍns displacements in x,y and z directions strain energy potential energy of exterual londing

xv

Y,,,(y) a

{6}",

l'", lj i

{u}

frec vihration cigcnfullction of heam IUnplification factor of flcxura! stiffness, ratio ut! U2 vcctor of displaccmcnt parruneters cquivalcnt IIniaxial strain strain vcctor free viImItion cigenvalllc of berun Poisson '8 mti o total potcntia! encrgy B3 splinc function ccntcred at Yi vec tor of strcsscs and moments principal strcsscs maximum compressivc stress and corresponding strain of concrete transversc slope 8w / 8x

""

xvi

Chapter 1

INTRODUCTION 1.1

ANALYSIS OF HIGHWAY BRIDGES

In service, highway bridgcs such as slab on gil'clcl' bl'iclgcs, box gil'der bridgcs, cable-stayed bridges etc. undergo not only longitudinni bcnding but al80 transverse bending, torsion, distortion and shcar dcformation. Thc lond distribution runong the girders, the support reactions and cable-tcnsions arc highly static!llly indeterminate and material and gcometricnl nonlincaritics duc to coneretc cracking, cable sagging and p-~ effcet etc. arc oftcn significant. Thercfore, thc Imalysis of a bridge is acomplex undertaking. There exist SOllle analytical methods for bridge analysis, such

liS

the load distri-

bution technique for right simply supported slab-type bridges [1,2,3], thc stiffness method of analysis for stecl orthotropic deck sJ'stellls [4] and thc cxtcnded foldcdplate theory for box-girder bridges [5]. However, bccause thcsc analytical mcthods can only be used to analyze highly simplificd structurcs, their applicability is liml

2

CHAPTER 1. INTRODUCTION

it.ed. The fini te element method is the most powerful and versatile tool for analysis of hridges [6,7J. That mcthod clm be applied to dcal with any specific configuration of bridgc structure and supports. It is suitablc for analysis involving alI types of "t.lltical and dynamical loads and alI kinds of elastic and inelastic deformation. Ncverthdess, the finite clement solution usually requires a

signi~cant

arnount of

computer time, large corc storage and tedious and lengthy input data files. Therefore, t.he efficiency of this method nccds to be improved. In fnct, the simply supported right deck of uniform section ( or a structure which iliny be rClllistically nnalyzed as such ) constitutes a high proportion of the large nU111ber of bridges being built. For analyzing this type of bridge, the finite strip lIIethod has proven to be the most efficient numerical methodj it uses a series of orthogonal functions in the longitudinal direction, y, combined with the conventional finite elcment polynomial shape function in the transverse direction, x, t.o simulatc ali the displaccment components of the structure. In this way, the 1111Inber of dimensions of the analysis is reduced by at least one. Consequently, cOlllputer time, storage and input data requirements are reduced significantly.

1.2

DEVELOPMENT OF FINITE .STRIP METHOD

The fillite strip method was first published by Y.ICCheung [SJ for analysis of simply supported bridge deck structures in 1965. The finite strip method for rectangular

CHAPTER 1. INTRODVCTION

3

slnb-typc bridgc dccks was also suggested independent.\y by Powell IInd Ogden 19J in 1969. Since t.hen, considcrable reseru'ch and development. on thllt method have bcen carricd out in many cOlIntrics. In the late 1960' IUld em'ly 1970', t,he field of rescllr("h cxt,enclcd to nuuly types of bridge mIClloading concli tions~ slIch ll.~:

• rcctanguhu' slabs with end bOllndnry conditions other thllll simple SlIpports,

110J in 1968, • simply supported box girder bridges, I11J in 1969, • curved slab and box girder bridgcs, [12J in 1969 ami [13J in 1971, • slab-type bridgcs with intermediate column slIpport,s Ilsing the Ikxibilit.y lIppronch, 114J in 1970, • rectallgular slabs wi th variable cross section in the splll1\vise directiolI, 114J in 1970, • the frequency analysis of some simple and continuous rcctangulllr slabs, 115J in 1971, • skew slab bridges, [16J in 1972, • skew box girder bridges, [17J in 1975, • the initial buckling analysis of box-type structures, [18J in 1973 and I19J in

1974, • continuous box girder bridgcs with trnnsverse diaphragms, using the HcxibilitY approach, [20J in 1976,

CHAPTER 1. INTRODUCTION

• slab and hox girder

bridg~s

4

continuous over rigid supports, using continuous

bmuu eigeufullctiollS alld a direct stiffness method, [21,22] in 1974 and 1978, • allalysis of general plates, [23] in 1978.

Y.I
whcrc KA is the axinl stiffness of columni Ker and Key ru'e the flexural stiffness of colullll1 in the dircctions x and y rcspectivclYi

Ne

is the number of the columns

SlIppol'ting the strip. Tll1IS, thc total strain encrgy of the compound strip, Us, is the sum of following it.clIIs: (2.26) whcrc UI' is the flexural strain energy in the plate strip. SlIbstituting thc expression of displacements (2.3) into (2.26), the strain energy

us can bc cxprcssed in terms of the nodal displacement parameters of the strip in the form:

(2.27) The stilfness submatrix of thc compound strip, [Klmn, is readily obtained. This IIlcthod has been successfully applied to the analysis of a variety of continnaus platc structures.

CHAPTER 2. FINITE STRIP METHOD

2.3 In the

21

EIGENFUNCTIONS OF CONTINUOUS BEAMS anal~'sis

of continllolls foldcd pintcs, til
(2.35), the following equntion CIUl bc obtnined: (2.36) 2. The bending moment and, conscquently, t.hc curvnturcs must also bc eontillllous over support i. From equations (2.34) and (2.32) thc following cquntioll enn similarly be obtnined: (2.37) The values of p. and Ai; (i=1...n, j=I,2) have to satisfy thc continuity eonditions (equation (2.36) and (2.37)) at ali the intermediate supports. Furthcr, /' must also satisfy the end conditions of the beam : Au

=O ,

Ani

=O

Equation[2.38] states that M~ = O at both ends of thc beam.

(2.38)

CHAPTER 2. FINITE STRIP METHOD

24

The Ai; represent only the mode shape, not its size; hence the following values mny be adopted, without loss of generality,

Au = 1 , At2 = O

(2.39)

For n given p., Ait and A i2 can be evaluated from the known values Ai_I,1 and

Ai-t,2 by using equations (2.36) and (2.37). Doing this span by span, finally, ali the Ai; up to Ant and An2 , which are the functions of p., can be calculated.

If 11.11 the Ai; arc not simultaneously zero, and the value of p. makes (2.40) then p. and Ai; have satisfied both the continuity conditions and the end conditions, thercfol'e these values give a valid solution. In solving equations (2.36) and (2.37), different procedures must be followed, depending on whether or not sin(p.1i)

= O and sin(p.1i_t) = O.

1. If sin(/tli) '" O, from equations (2.36) and (2.37)

(2.41) (2.42) 2. If sin(/11i) = O , /tli =

ln'/!'

(m=1,2 ... ): From equation (2.32), it can then be

seen that ai = O and

li = Ait sin(p.Yi) + Ai2 sin(p.y') = Ai! sin(/IYi) + Ai2 sin (In'/!' - P.Yi)

CHAPTER 2. FINITE STRIP METHOD

25

= (Ail - ( -l)'" Ai2) sin{JIYi)

This means that the cocfficicnts Ait 11m! Ai2 arc dcpclldcllt variablcs. Hcncc, lU1t! again without loss of gcnemlity, .4i2 = O can bc I1SSl1111Cd, so that (2.43)

It is noted, howevcl', that the calcl1lation of .4 i1 still dcpcnds on thc VIUUC of sin(JL1i_l) associatecl wi th thc pl'evious span (i-l), and it lIIUst. be cOllsidcred scp-

arately for the two cascs of sin(/Lli_l) = O I1Jld sin(JL1i_l) a. If sin(JLli_l ) = O, then

Yi-l

i= O.

= Ai_I,1 sin(!tYi_I)' Ai-I,2 = O I1Jld cquation (2.37)

becomes an "identity". Then, from the slopc continl1ity conditionluonc (2.36) wc can obt.ain (2.44) b. If sin(JLli_l)

i=

O, Ai_I,1 must bc

ZC1'O,

othcl'wisc it docs not satisfy equu-

tion(2.37). Thcn from the slope continuity condition alonc (2.36), wc

CM

obtaill (2.45)

However, in the case ofthe first interior support (betwccn SpMS l IUld 2), if sin(JL11) is not equal to zero but sin(JLI2) = O, it cl1Jll'csult in Ai'; = O (i = 2oo.lI,j = 1,2) from equations (2.39) (2.43) (2.44) etc. In this case, although Ani = O, JL is still not a solution of equation (2.28), because it violates the curvature continuity condition (equation (2.37)) at the first interior supports. In the case of the last span, if sin(JLl,,) = O, thc curvature at the last cnd of bcam is automatically equal to zero. Therefore, in this casc, even though equation (2.40) is not satisfied, JL may still be a solution of the equation (2.28).

26

CIiAPTER 2. FINITE STRIP METHOD

TIli! first-order Reguli-Faisi iteration can be employed to solve the p. and Ajj ,(Fig.2.7).

StlLrting with the point /jo, /j is increased by a small increment 8,1 in evel'y step. At IJvm'Y tria! point p., Anl(p.) is determined from A 11

= 1 and AI2 = O by using

elluationH (2.41),(2.42), etc. However in some cases, if Ani changes its sign between two trittl points

p.(0)

= p. -

8p. (the previous trial point) and /-1(1)

= /-I (the present

trinl point), the straight line is drawn between the points (/-1(0), Ant (/-1(0))) and (/L(I), Anl(/-I(I))) as shown in Fig. 2.7. The intersection of the straight line with the /L ruds is nt /-1(2) which is doser to the solution

From /1(2) the point (/I(U), Ani (/L(O)))

(,L(2),

il than /-1(1).

Ani (/-12)) is located and the straight-line is connected to.

or (/1(1), Ant (p.(t))) depending on at which point Ant has different

sign from Ani (/-1(2))). This locates /-1(3) on the /L 8.'Cis,which is doser to the desired resu!\. thllJl /L(2).The process is repeated unti!

(2.46) The itcmtion formula is

(2.47) Continuing to increase /-I by 8/-1 for every step, eventually, the necessary eigenvalues, /1"" and modcs,

Alj)

are identified, these being as many as required by the finite

strip nnalysis. When thc eqllntion (2.41) was applied to determine Ai2 fron. t. ' , both sides of cqllntion (2.37) were divided by sin(/-Ili ) , which might result in missing the solution of sin(lLl j ) = O. Therefare, if during

IL

particular increment of /-I, 5in(/-Ili ) change5

CHAPTER 2. FINITE STRIP METHOD

27

its sign, thcn II = inTr/li nltlst bc used as the t,rial point, t.o l:nknlnt,c .4ij IUlCl scc if this is one of the eigenvalucs. Howcver, in any ot,her ensc, it. must uot I,akc lUly trial value of II equal to inTr/li so as to avoid sihmtions sneh ns 5in(Jl./2) = Owhite sin(pll)

1= O, which will result in IUl C1'1'OnCOU5 vnluc of .4"1(/1).

In the computer program, the following valucs \Vere used: Óp

= Tr /30li",ar

EI

=

lQ-IO

and it is assumed that sin(pli) = O if 1sin(pli)

1< lQ-lO.

If the lengths of alI the spans arc not exactly thc samc, but slightly difl'crcnt from each other, then,p = inTr/li is not a solution, and thc sign of A"I(/L) changcs drastically near p = inTr/li, In this case, snmllcr increment,s of Óli lUust hc uscd l :'

in order to prevent the possibility of missing modcs. For cxlt1nplc, if ali of thc

!

spans havc magnitudc cithcr lmar or Imi,,, and if O < (lm"x - l"'in)/l",ar.

!t

it requircs great care not to miss thc solution ncar rmr / l",ar' Thc rccollllllcnded

~

,,

< 0.05,

remedialllleasure is that if there has bccn mTr/lm"r - /L < Ó/L, thc incremcnt is rcduced to óp' = m11'(l/lmin - 1/lma:)/4 and the ncxt trial value of /L is changcd into p = mTr Ilma: +óp' /2. Aftcr the eigcnvaluc ncar mTr / l..,,: has hecn found, thc original increment ÓIl = mTr /301ima: is resumcd.

28

CIIAPTER 2. FINITE STRIP METIlOD

2.4

ANALYSIS OF CONTINUOUS HAUNCHED BRIDGES

III the

prc~cllt

stllcly, the finitc stri]> method was extended to the analysis of

hlLllllChcd, continuous sIab-on-girdcl' bridges ([49], 1988) and haunched continuaus bux-girder bridges ([50], 1989). In such nn analysis, three types of strip arc used, these being the top flange plate stri]> (Fig.2.2), the vcrtical web strip (Fig.2.8) and the bottom flange shell strip (Fig.2.9).

2.4.1

STRAIN-DISPLACEMENT RELATIONSHIP

Fol' thc top flI1nge plnte strip, the Cartesian coordinate system is used and the strnin-displacement relationships (2.5) are applicable. For the web strip of variable depth, the curvilinear orthogonal coordinate system (~-I/)

is morc convcllient, in which.,., has units of length. The web can be divided

into n numbcr of equal or unequal width strips. The width of cach strip is taken ILS

b = c x b,., where

/1..

is the depth of web, with O < c

~

1. In the web, the

most important deformation is in-plane bending, and thc most significant strain componcllts are thc longitudinal normal strain E~ and the longitudinal shear strain

re./.

According to thc actual proportion of ordinary haunched box-girder bridges,

CHAPTER 2, FINITE STRIP METHOD

29

the relationship bctwcen displaccmcnts nlld stnuns can bc writtcn

Dv

u

f

-

,,-

+1'2 8'1

--

Du

')'(" =

81}

1 Dv

'/I

+ bD~

- ;:;-

18211)

(2.48)

X( = - b2 D~2 X~

!IS:

82 11) = -

1 81 2

2 D211) X(" = bD~ 8,}

where

l',

nlld

1'2

arc the radii of curvaturc of coordinatc lincs (Fig,2,8), Imd thc

following approxilllate geollletrical rclationship can bc takcll with rcusonnble accuraey: 1 1'2

x' cl2 b," = b," cl112 b,"

1't

=

(2.49) (2,50)

~.

1Ir,

For the bot tom flange shell strip, cylindrical surfucc coordinntcs nrc used, and thc relationships between displacelllents and stnuns [59] arc 8u

E:z:=-

8x

f

8v

w

8u

8v

---U - 8y R

')'ru

= -8y +-8 x

82 w

(2,51)

Xr = - 8x2

82 w

W

Xu = - 8y2 - R2

30

Cl1APTER 2. FINITE STRIP METHOD

EJlw Xry = 2 8x8y

28v

+ R8x

whcre R is the mclius of Clll"vature of the middle surface in the longitudinal direct.ion, this is usually a function of y. In ordinary haunched box-girder bridges it can he

!LqSUmcc\

timI. (2.52)

2.4.2

DISPLACEMENT FUNCTIONS

The choice of suitable displacement functions is the most crucial and difficult part of the analysis. The right choice should lead to satisfactory results for maximum deflection

IUlCl

muximum longi tudinal stress using only a very few terms of series

if thc mmlyzed bridge is subjected to uniform load (according to our experience, the rcquh'cd number of terms in series should be less than 5 times the number of spans). Aftcr comprehcnsive research and experiments, the following formulations are. worked out. The strip c1isplucclllcnt paranleters for the m-th term are taken as:

wherc

VI

IUld

V2

mainly represent the longitudinal movements due to in-plane

bcndillg IUld VI IUld V2 mainly represcnt displacements in the y direction caused

CHAPTER 2. FINITE STRIP METHOD

31

by the longitudinal elongation or compression of the stdp. Thc c1isplllcCllIcnts wi thi n a plate strip are as follows: r

II

L «1 -

=

X)lIf'

+ Xu;")Y,,,(y)

m=1 r

V

=

L«l- X)v;" + Xv~')Y,:(y)+ m=) r

L «(1- X)v;" + Xv;T':.•(y)

(2.53)

r

W

=

L «1 -

3X 2 + 2X 3 )w;" + :t:(1 - 2X + X 2 )9;"

m=)

where Y,,, is the eigenfunction of a contilluollS bell.lll which hlls t.he span lengt.hs of the actual briclge, Imcl Y:' (y) = (b,u dY.[m (y

Ym(y) =

_

1

dib," Y,,,) (y

(Ym(y)/R)dy

(2.5

L=40m

1=40m

_____o

----t

12cL-j

1--- 20Il1--t

.K

l

_-,---::---;,------71 Th. L. ~ m I c=2.4ul ~

~

1--.- Yz A

II

hl .. O.2DI

I

'::=-.=i1!

.

D

i

h

:- b, _0.l.5111

T c

h -O.16m

b,.311

FiglU'C 2.10: COlltilluOUS Dox·Gil·dcr Dridgc

44

Cll.·IJ>TER 2, FISITE STRIP MET.fIOD

6.0

9.0

40.0 Y 2

Figlll'c 2.11: Thc Mc~h of Shell Elcmcllts

X-I l ... í

1. S4

3.H

F ~-r- ~

-;l

=td:

x-1 ~I

I .~4.H&

.44

,I

3&.~6

I

. \I , J.K "

,u. J

~

I

Trm'k
111

is t.he number of sections, i is the Ba spline expression wi th

y = Yi IL~ the ccnter, j is thc local nwnbcr of knot i in the corresponding pair of

sections, Ilnd Lj l\.I'e quadrntic interpolations in the following form:

LI(Y) = (1 - Y)(l - O.5Y)

La(Y) = O.5Y(Y -1) ill which, Y =

y'/ It (sec Fig.3.2).

Eq. (3.5)

IU'C

x,y,z

local Cl\.I'tcsilUl coordinatcs, IUld u,v,w arc corresponding displacement

IU'C

also applicnblc fol' thc vert.ical web strip. However, in this case, the

componcnts. In addition, X is not only a function of coordinate x, but also of the coordinlltc y in the form:

x=

x-

XI

=

b

X

CI

cbw(Y)

C

where ilu.(Y) is thc Vl\.l'iable dcpth nf thc wcb, b = cbw is the width of the strip and ;rl

=

is thc coordinntc of nodallinc 1, wi th c Rnd CI being constant (Fig. 3.3).

CI" IU

Substituting t.hc displllcclllcnt fundions of the web strip into the expression for in (2.5) yiclds the following relationship: f

U

-

av ay

-

EV

CH1~PTER

3. SPLINE FINITE STRIP METHOD

55

(3.6) It

C!Ul

hc sccn thnt

dcpcnds not only ou thI' first
61

GHAPTER :J. SI'UNE FU-UTE STnIP METHOD !,,,illl. is

24.47 MPa [GOJ.

•

62

03

CH.-lPTER 3. SPLINE FlNITE STRlP METlIOD c>

..t ~(y)

c>

..; c>



..: c>

ci -3.0

-2.0

-1.0

0.0

1.0

2.0

3.oh

2.0

3.oh

Y od ......... c>

d~(y)/dy

..t c>



ci

3.0

-2.0

-1.0

c>

~

I

N

od

~

d·~(y)/dya

.;

c>

ci+-----~~----~~--_.--~~~----~----__,

3.0

-2.0

2.0

c>

.; I c>

,I."1'I11L1 appli",1 load. being nssumed to be the same as the actual girdel'. 2. Double- P hUl" Cable-Stayed Concrete Box-Girder Bridge. A douhl"-plallc cnble-stayed concwte box-girder bridge is shown in Fig.4.5. The load. arc the self-weight of thc girdcr and a live load, dis tribu ted uniformlyon the whole deck hctwcen towers, of intensity 4.4881(N/m2. The details of the bridge arc ns follows: For t.hc girdcr, Eg = 25 x 1061(N/m2, Poisson's ratio v = 0.2, specific weight

"fg = 241\ N / nl3 j for the pylons, Elv = 2.5 x 108 m 2J( Nj for the cables, Ee _ Li

X

108 1\N/1I/2, "fc = i81\N/m 3, the arens AI = A4 = 0.04m 2, A 2 = A3 -

0.024m 2 j the vert.ical components of cable prestressing forces under the dead load arc VOI = i849.01(N, V02 = 9066.51(N, V03 = 8121.2J(N,

Vo4

= 8322.7J(N, which

make the deflections of the gil'der at the cable attachment points under the dead load c()ual t.o zero. Because of symmetry, only half of the bridge, which is divided into 6 strips by 5 nodnl lilll!S, needs to be annlyzed. In tbc finite strip analysis of the girder, the lIlIIuber of terms used in the eigenfunction series is nl + n2 = 21 + 9. Convergence is obtlunccl wi thi n 8 iterations. The deflect.ioll of the girdcl' nlollg the noda! line 5 is depicted in Fig. 4.6. The vcrt.icnl forccs V, applicd to the girder by the cables, and the horizonta! components

H of cablc tcnsion arc givcn in Tablc 4.3. Tbc maximum negative bending moment

CH.4.PTER 4. NONLINE.4.R .4.N.4.LYSIS Cnbl" l

2 3 4

84

V H 10.730 25.985 9.831 11.841 11.315 13.419 10.521 25.103

Table 4.3: Vert.icnl Forces of Cnbl!'s on Ginl"r IInd Horizont.al Flln'es Ilf Cablc.: on Pylon (in IvIN) of thc girdcr is nt cross section B , Illul !.ll(! lllnxillllllll posil.iv" hl'llding nl