load distribution in edge-stiffening beam of a simply supported

LOAD DISTRIBUTION IN EDGE-STIFFENING

BEAM OF A SIMPLY SUPPORTED BRIDGE

DECK

IWAN PERMADI KUSUMA

UNIVERSITI TEKNOLOGI MALAYSIA

PSZ 19:16 (Pind. 1/97)

UNIVERSITI TEKNOLOGI MALAYSIA

BORANG PENGESAHAN STATUS TESIS LOAD DISTRIBUTION IN EDGE-STIFFENING BEAM OF ASIMPLY SUPPORTED BRIDGE DECK

SESI PENGAJIAN: 2004/2005

IWAN PERMADI KUSUMA (No. Passport : AG 674328)

Prof. Dr. AZLAN ABDUL RAHMAN

Prof. Dr. AZLAN ABDUL RAHMAN

LOAD DISTRIBUTION IN EDGE-STIFFENING BEAM OF A

SIMPLY SUPPORTED BRIDGE DECK

IWAN PERMADI KUSUMA

A project report submitted in partial fulfilment of the

requirements for the award of the degree of

Master of Engineering (Civil – Structure)

Fakulty of Civil Engineering

Universiti Teknologi Malaysia

March 2005

ii

I declare that this thesis entitled “Load Distribution In Edge-Stiffening Beam of a

Simply Supported Bridge Deck” is the result of my own Project

except as cited in the references. The thesis has not been accepted for any degree and

is not concurrently submitted in candidature of any other degree.

Tandatangan : ………………………

Nama Penulis : IWAN PERMADI KUSUMA

Tarikh : 18 March 2005

iii

ACKNOWLEDGEMENT

I am very grateful to Allah SWT for giving me the strength and the capability

to finish my master project dissertation. I wish to my sincere gratitude to my

supervisor, Prof. Dr. Azlan Abdul Rahman, of Faculty of Civil Engineering for his

guide, invaluable advice and useful suggestions during the conduct of this project.

Thanks are also due Ade and Wandi to helping me for any thing, to mak siti’s

family (along, anem, dura, vivi and ame), and to my friend in PPI for their generous

support. Many thanks to Nasir and Dony for help in the printing of the document.

Specially thanks to my beloved wife Sita Nurmari Putri Utami in yogyakarta,

and heartiest gratitude to my parent (Murtadi,- Sri Permata and Soegijanto – Sudi

Nurtini) as well as to my siplings (Mbak Angie- Mas Zain – Tole Ilham , Dik Epi

and Dik Inda) for their moral and the spiritual support throughtout my studies.

IWAN PERMADI KUSUMA

MARCH, 2005

iv

ABSTRACT

This study involves two distinct methods for analysis of a simply supported

bridge deck with or without edge-stiffening beam and the effect of edge-stiffening on

the longitudinal and transverse moments along the deck. An analysis of a slab bridge

with or without edge-stiffening beams is made and the result obtained from LUSAS

programs (Finite Element Method) are compared with those derived theoretically

(Load Distribution Method). The analysis considered HB loading acting on deck

bridge. The analysis is confined to the case where the effective depth of the bridge is

constant between the edge-stiffening. The degree of accuracy to be expected from the

theoretical analysis and the difference in the longitudinal and transverse bending

moment due to the effect of edge-stiffening beams are estimated. Part of the analysis

for the above problems involves the determination of longitudinal moment in the

corner or edge-stiffening beam.

v

ABSTRAK

Studi ini melibatkan dua metoda yang beda untuk analisa suatu jembatan

yang disokong secara mudah, dengan atau tanpa pengukuhan tepi (ketebalan di

samping jembtan) dan efek dari pengukuhan tepi pada gaya momen yang membujur

dan melintang sepanjang geladak itu. Suatu analisa dari papan jembatan dengan atau

tanpa pengukuhan tepi dibuat dan hasilnya diperoleh dari program LUSAS (Metode

Unsur Tak Terhingga) dibandingkan dengan hasil yang diperoleh secara teoritis

(Metoda Distribusi Beban). Analisa mempertimbangkan beban HB yang berada di

atas jembatan. Analisa terbatas pada kasus di mana ketebalan efektif dari jembatan

adalah tetap diantar pengukuhan tepi. Derajat ketepatan yang diharapkan dari analisa

secara teoritis dan perbedaan pada gaya momen yang membujur dan melintang

akibat dari adanya pengukuhan tepi dapat diperkirakan. Bagian dari analisa untuk

permasalahan di atas melibatkan penentuan dari gaya momen membujur di sudut

atau balok untuk pengukuhan tepi.

vi

CONTENTS

CHAPTER ITEM PAGE

ACKNOWLEDEMENT iii

ABSTRACT iv

ABSTRAK v

CONTENTS vi

LIST OF TABLES x

LIST OF FIGURE xi

LIST OF SYMBOLS xiv

LIST OF APPENDICES xv

PART ONE

INTRODUCTION

CHAPTER I INTRODUCTION 1

1.1. General 1

1.2. Importance of Study 2

1.3. Objectives of Study 2

1.4. Scope of Study 3

1.5. Methodology 3

vii

PART TWO

LITERATURE REVIEW

CHAPTER II METHODS FOR BRIDGE DECK ANALYSIS 6

2.1. Introduction 6

2.2. Types of Bridge Deck Construction 6

2.2.1 Solid Slab Deck 7

2.2.2 Voided Slab Deck 8

2.2.3 Beam-and-Slab Deck 9

2.3. Bridge Loading 10

2.4. Load Distribution Method 12

2.4.1 Distribution Coefficients 15

2.4.2 Maximum Longitudinal Moments 17

2.4.3 Maximum Transverse Moments 19

2.5. Finite Element Method 21

2.5.1 General Description of Method 22

2.5.2 Finite Element Program (LUSAS) 23

2.5.3 Input and Output 24

PART THREE

METHODOLOGY

CHAPTER III EDGE-STIFFENING IN CONCRETE BRIDGE DECKS 26

3.1. Introduction 26

3.2. Forms of Edge-Stiffening 26

3.3. Analytical Solution for Effect of Edge Stiffening 28

3.4. General Effects of Edge-Stiffening 31

viii

PART FOUR

RESULT AND DISCUSSION

CHAPTER IV CASE STUDY ON EFFECT OF EDGE-STIFFENING USING

LOAD DISTRIBUTION METHOD 35

4.1. Introduction 35

4.1.1 Deck Geometry (Case Study 1, 2, 3) 35

4.1.2 Loading 37

4.1.3 Moments 38

4.2. Results for Case 1 39



CHAPTER V ANALYSIS OF EFFECT OF EDGE-STIFFENING USING

FIFNITE ELEMENT METHOD 54

5.1 Introduction 54

5.1.1 Description of LUSAS Program 54

5.1.2 Deck Idealization (Case Study 1a, 2a, 3a) 55

5.1.3 Finite Element Model and Mesh Layout 55

5.1.4 Element Properties 55

5.1.5 Loading 56

5.1.6 Moment 56

5.2 Results for Case 1a 57



ix

CHAPTER VI DISCUSSION OF RESULTS 60

6.1 Result of both method 60

6.2 General Effect of Edge-Stiffening 63

6.3 Form of Edge-Stiffening 63

6.4 Method of Analysis 63

PART FIVE

CONCLUSION

CHAPTER VII CONCLUSIONS 65

7.1 General Conclusions 65

7.2 Recommendations for Future Work 66

PART SIX

REFRENCES

REFRENCES 67

PART SEVEN

APPENDICES

APPENDICES 68

x

LIST OF TABLE

TABLE NO. TITLE PAGE

2.1

4.1

4.2

4.3

4.4

4.5

5.1

5.2

5.3

6.1

6.2

Unit of HB loading

Value of Kmx0.25

Statical equivalent load position

Value of 25.0

Distribution coefficient

Statical equivalent load position

Maximum the bending moment for slab without edge

stiffening

Maximum bending moment for slab with edge stiffening

for Case 2a

Maximum bending moment for slab with edge stiffening

for Case 3a

Result of longitudinal and transverse moment with Load

Distribution Method

Result of longitudinal and transverse moment with Finite

Element Method

11

40

41

42

43

45

57

58

59

60

60

xi

LIST OF FIGURE

FIG. NO. TITLE PAGE

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

2.10

2.11

2.12

2.13

3.1

3.2

Cross section of bridge and solid slab deck

Cross section of bridge and voided slab deck

Cross section of bridge and slab deck: (a) T - beam bridge,

(b) Slab on steel beams, (c) Slab on prestress concrete

beams

HB Loading

HB in 1 National Lane

HB in 2 National Lane

Actual and effective width of deck

Standard positions or Reference Station

The positions of load: (a) The positions of the effective

width 2b, (b). Equivalent load positions

Typical distribution coefficient profiles for abnormal

loading

Position of wheels for maximum transverse moment at

centre of bridge

Elements type

Finite Element System

The form of edge stiffening of slab. (a) Edge beam

centroids on mid plane of slab. (b) edge beam centroids

above mid span of slab.

The modifity form of edge stiffening of slab. (a) Concrete

7

8

10

10

11

11

12

13

16

18

18

20

22

24

27

xii

3.3

3.4

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

5.1

5.2

5.3

5.4

5.5

6.1

box girder. (b) Steel buckle plate (c) Prestressed inverted

T-beam (composite) (d) Steel through deck

(a) Bridge considered in analysis. (b) Forces and moments

acting on bridge and edge beams

Superposition of symmetrical and asymmetrical edge

moments to obtain desired edge moments

Shows the dimension of the plan of concrete bridge deck

(a) Case study 1and (b) Case study 2 and 3

Cross section of bridge slab deck (a) Case study 1, (b) Case

study 2, (c) Case study 3

The arrangement of live load on plan on the deck for

longitudinal

The arrangement of live load on plan on the deck for

transverse

The condition maximum for longitudinal moment

The condition maximum for transverse moment

Distribution coefficient for actual load positions under the

wheels

Force acting on slab bridge at mid-span

Load positions and equivalent loads for maximum

transverse moment

The finite element model for the plan deck (a) without and

(b) with edge-stiffening

The model of HB loading in finite element (a) position for

longitudinal moment and (b) position for transverse

moment

Moment distribution in both directions for Case 1a



Shows the comparison between Case 1, 2 and 3 in

maximum bending moment of the longitudinal beams and

transverse bending moment for load distribution method

27

28

29

36

36

37

38

38

39

43

45

48

55

56

57

58

59

61

xiii

6.2

6.3

Shows the comparison between Case 1, 2 and 3 in

maximum bending moment of the longitudinal beams and

transverse bending moment for finite element method

Shows the comparison between load distribution method

and finite element method for Case 1, 2 and 3 in maximum

bending moment of the longitudinal beams and transverse

bending moment

61

62

xiv

LIST OF SYMBOLS

2a - Span Bridge

2b - Width Bridge

d - the slab depth

E - Young's modulus of the material of deck

G - modulus of rigidity or torsional modulus of the material of the deck

i - longitudinal second moment of area of the equivalent deck per

unit width

I - second moment of area of each longitudinal girder

io - longitudinal torsional stiffness per-unit length

j - transverse second moment of area of the equivalent deck

per unit length

J - the second moment of area of each transverse diaphragm or

cross beam

jo - transverse torsional stiffness per unit length

Ka - the distribution coefficient for the actual value of a.

Ko - distribution coefficient for a equal to 0.

K1 - distribution coefficient for a equal to 1.

lo - torsional stiffness constant of a longitudinal girder

p - spacing of longitudinal girders

q - spacing of stiffners i.e. diaphragms or cross beams

xv

LIST OF APPENDICES

APPENDIX TITLE PAGE

A

B

C

Coefficient of Lateral Distribution

Distribution Coefficient

Transfer Moment Coefficient

68

71

81

CHAPTER I

INTRODUCTION

1.1 General

A bridge is a permanent raised structure which allows people or vehicles to

cross an obstacle such as a river without blocking the way of traffic passing

underneath (Heinz Kurth 1976). And the construction of bridge comprises one

section of the work of the civil engineering which was has an immediate impact upon

the public. The reasons of the impact are not hard to find since most major bridge

combine a strong visual impression together with obvious benefit in the way of

improved communication.

Many bridges are designed to incorporate some forms of edge stiffening for

the need to accommodate services of various types, or deal with narrow or negligible

footpaths, (which permit excessive eccentricities of abnormal loads) or, in the case of

a railway over bridge, to provide a parapet specifically to prevent accidents.

Whatever the reason for its presence, the edge stiffening will considerably affect the

behaviour of the bridge structure under load and, of more importance; it can improve

the distribution characteristics of the bridge with regard to longitudinal moments.

However, the beneficial effects of edge stiffening can only be obtained by

ensuring a positive structural connection between any edge stiffening and the main

bridge structure and by analyzing in detail the entire structure. On the other hand

cases exist where the effective transverse stiffness of the main bridge structure is

maintained to the parapet beams. Then the effect of the stiffness beams at the edges

2

should be ignored. No theoretical analysis at present available to cover this particular

aspect of bridge design, though Massonnet [4] has extended his distribution analysis

to allow for the effect of edge beams in which no torsional stiffness is present.

Load distribution analysis includes the effect of torsion and covers the range

from a no torsion grillage to a full torsion slab. From this analysis it is possible to

assess the effect of torsion in the edge beams at the design stage and hence it will be

for the designer to use his judgement in deciding whether or not to include the

torsional effect. If the torsion is neglected, the analysis will yield results which would

be identical to those obtained by Massonet.

1.2 The Importance of the Study

The study will address some of the important aspect as follows:

The development of load distribution in edge-stiffening beam of a simply

supported bridge deck.

The analysis of bending moment in a bridge deck using load distribution

coefficients method and finite element method for comparison.

The example of design of a selected structure is illustrated.

To comparison of bending moments in bridge deck with edge-stiffening and

without edge stiffening.

1.3 Objectives

The objectives of this study are as follows:

To study various forms of edge-stiffening in concrete bridge deck

3

To analyse the bending moment in a bridge deck with and without edge

stiffening using load distribution coefficient (manual method) and finite

element method

To compare maximum moments in a bridge deck with and without edge

stiffening

1.4 Scope of the Study

In the beginning, complete sets of information on concrete deck bridge being

used and applied in Malaysia is collected. The public Work department ministry of

work and Malaysia provide all data about standard design of concrete deck bridge.

And the standard design is the studied. Several parameters which need to be studies

are finalized. Finally, the structures are modeled, in the putted with the different sets

of parameter and analysis using finite element software. LUSAS finite element

software has been used in the study. The scope of this study is:

A single span simply supported bridge

Effect of 45 unit HB live load only (BS 5400).

1.5 Methodology of the Study

In this project, the steps taken in studying load distribution in Edge-stiffening

beam of a simply supported bridge deck can be summarized into several steps as

below:

4

Problem Identification and Definition

Identify the problem through reading, discussion and observation of the

area studied.

Understand the background of the problem through literature studies.

Study the feasibility and the needs to carry out the research topic and the

scope.

Identify the title, scope, aim and objectives of the project.

Plan the methodology for the project.

Literature Survey

Search information from book, journals, articles, thesis, seminar notes or

conference paper, and internet.

Review of the various type of the bridge deck.

Understand the principles of load distribution.

Understand the basic principles of maximal bending moment in

longitudinal and transverse.

Understand the basic principles of the application of LUSAS programs.

Edge-Stiffening in Concrete Bridge Deck Analysis

Review the form of edge-stiffening.

Understand the basic steps analytical solution of edge-stiffening

Understand the principles effected of edge stiffening.

Design of deck

Deck without edge-stiffening beam.

Deck with edge-stiffening beam use form edge-stiffening 1.

Deck with edge-stiffening beam use form edge-stiffening 2.

5

Case Studies

Carry out the edge-stiffening on three selected case studies.

Calculate using load distribution method.

Calculate using Finite Element Method

Interpret the result of the analysis

Discussion

Discuss of comparison of the case studies.

Conclusions and Recommendations

Write an overall conclusion for this project. Give some recommendations for future research or project.

CHAPTER II

METHOD FOR BRIDGE DECK ANALYSIS

2.1 Introduction

2.2 Type of bridge Deck Construction

2.2.1 Solid Slab deck

Fig 2.1

2.2.2 Voided Slab Deck

Fig. 2.2

2.2.3 Beam and Slab Deck

Fig 2.3

2.3 Bridge Loading

1 m

3.5 moverallwidth

1 m

whichever dimensionproduces the most severeeffect on the memberunder consideration

6, 11, 16, 21 or 25 m1.8 m 1.8 m

axleUnits

1 m

CL

WheelaxleLC

axleLC

axleLC

Fig 2.4

Table 2.1

Central reserve No loading for global analysis

Overall vehiclelength for axlespacing having

most severe effect

Loaded length for intensity of HA UDL

Lane

Load

ing

are

inte

rcha

nge

a ble

for

mos

tse

vere

effe

ct

1/3 HA

1/3 HA

1/3 HA

No LoadingFull HA UDL HB vehicle

Full HA

1/3 HA

25 m

Nationallanes

No Loading Full HA UDL

25 m Nationallanes

Fig 2.5

are

inte

rcha

nge

abl e

for

mos

tse

vere

eff e

ct

1/3 HA

1/3 HA

1/3 HA


Full HA UDL

Full HA UDL


most severe effect

1/3 HA

HB vehicleNo Loading 3.5 No Loading


25 m 25 m

Nationallanes

Full HA UDL

Full HA UDL

Nationallanes

Or vice-versaNational lanes

Nationallanes

are

inte

rcha

nge

able

for

mos

tse

vere

e ffe

ct


1/3 HA

1/3 HA

1/3 HA


Full HA UDL

1/3 HA UDLNo Loading

Or vice-versa 25 m


most severe effect

No Loading3.5HB vehicle

Full HA

Full HA UDL

1/3 HA UDL

25 m

or

Fig 2.6

2.4 Load Distribution Method

Fig. 2.7

j

ix

a

b

a

b

p

Ii

q

Jj

dji

ixjxEx

jiGx oo

p

looi

2.4.1 Distribution Coefficients

Fig. 2.8

KKKK o

2.4.2 Maximum Longitudinal Moments

P1 32P P 4P

P P P P

b

andnn

(a)

(b)

Fig. 2.9

2.

Fig. 2.10 5

2.0

1.5

Ko-0.5

VA

LUE

OF

K

1.0

0.5

0

K1

K

REFERENCE STATION

b 3/4b b/4b/2-1.0

-b/2-b/40 -3/4b -b

b

p

b

p

2.4.3 Maximum Transverse Moments

Fig. 2.11

n a

xnbHnny

n n

a

xn

n

a

un

a

un

a

un

a

un

aHn

u u u u

a

u

a

u

a

u

a

u

a

bMy

a

u

a

u

a

u

a

u

etca

u

a

u

a

u

a

u

nnnn

2.5 Finite Element Method

2.5.1 General Description Of Method

Fig 2.12

2.5.2 Finite Element Program (LUSAS)

Volumes Surfaces Lines

Combined Lines Points

Pre-Processing Results-Processing

Finite Element Solver

Fig. 2.13

2.5.3 Input and Output

Ü Ü Ü

Ü

Ü

CHAPTER III

EDGE-STIFFENING IN CONCRETE BRIDGE DECKS

3.1 Introduction

A slab deck is able to carry a load near an edge if the edge is stiffened with a

beam. Additional of edge stiffening beam can influence the behaviour of bridge

structure.

3.2 Forms of Edge-Stiffening

Many contemporary bridges are constructed with some form edge-stiffening

in the form of deeper edge beams, fascia beams and increased structural depth for the

sidewalk slab. Figure 3.1 (a) shows a slab deck with edge stiffening beam which

have their centroids on the mid plane of the slab. The bending inertias of such beams

are calculated about the mid plane of the slab and the beam sections are fully

effective. Improved edge stiffening is achieved if the beams do not have their

centroids on the mid plane of slab as in figure 3.1(b) because the beams then act as L

beams with the slab deck acting to some extent as a flange. Under bending action, the

neutral axis remains near the mid plane of the slab in the central region and rises

towards the edges. The width of the slab that acts as flange to the edge beam is

restricted by the action of shear lag. And the modification of form edge stiffening as

show in figure 3.2

27

Figure 3.1 The form of edge stiffening of slab. (a) Edge beam centroids on midplane of slab. (b) edge beam centroids above mid span of slab.

(a)

(b)

(c)

(d)

Figure 3.2 The modifity form of edge stiffening of slab. (a) Concrete box girder. (b) Steel buckle plate (c) Prestressed inverted T-beam (composite)

(d) Steel through deck

28

3.3 Analytical Solution for Effect of Edge Stiffening

Consider the bridge with edge stiffening beams in Figure 3.3 (a). Let the span

of the bridge be 2a, the width of the uniform section of the bridge (i.e. excluding

width of edge beams) be 2b, the stiffness per unit length of the longitudinal section

be the second moment of area of the edge beams be and the torsional stiffness

of the edge beams be . The applied loading is represented by the four equal

loads P.

E E

EGJ

The edge beams can be isolated from the remainder of the bridge and edge

shear forces and moments introduced as shown in Figure 3.3 (b). These shear forces

and edge moments are assumed to be distributed sinusoid ally along the span.

The bridge can now be analysed using the results obtained in the previous

sections and the unknown shear forces F1 and F2 edge moments M1 and M2

determined from the compatibility equations for deflection and slope at the edges y=

+ b.

(a)

(b)

Figure 3.3 (a) Bridge considered in analysis. (b) Forces and moments acting

on bridge and edge beams

The shear forces F1 and F2 can be treated as applied loads on the bridge and

hence the deflection at the edge y=b due to all the applied loads can be written:

29

bbP

bby b

F

b

F

a

xaduetoP

a

x

b

aW

222sin

16

2sin

2

16 214

41

4

4

1 ……… (3.1)

Where H1 is the amplituded of the first term in the Fourier series for the applied

loads P.

The edge M1 and M2 can be considered in the form shown in Figure 3.4. This

method of superposing symmetrical edge moments enables the coefficients to be

used. Thus the deflection at the edge y=b due to the edge moments can be written:

a

xbbW b

Eb

Eby 2

sin'22

221

221

1 ……..…… (3.2)

Figure 3.4 Superposition of symmetrical and asymmetrical edge moments to obtain

desired edge moments.

The total deflection, give by the sum of equations (3.1) and (3.2), must equal

that of the edge beam at y=b under the action of the loading F1a

x

2sin . Therefore for

compatibility of deflection the following equations must be satisfied:

bbbEE b

F

b

FduetoP

ba

xa

a

xF

EI

a

2222sin

16

2sin

16 2114

4

14

4

ba

xbb

E

'2

sin2 2121

2

……..…… (3.3)

30

Similarly at y = -b:

bbbE b

F

b

FduetoP

ba

xa

a

xF

EI

a

2222sin

16

2sin

16 2114

4

24

4

ba

xbb

E

'2

sin2 2121

2

……..…… (3.4)

For slope compatibility at the edge y=b, it is found that for the bridge:

bbbaEby

FFduetoPx

s

a

y

W'''

2sin

`211

21

bbaE

xb'

2sin

2 2121 ………….…..…… (3.5)

For the torsional of the edge beam subjected to a twisting moment varying

sinusoid ally over the span, the angle of twist at any point may be determined by

considering the equilibrium condition of the edge beam with restraining couples

applied at its junction with the support diaphragms. Thus the edge beam rotation at a

distance x from the abutment may be shown to be.

a

a

E

x

GJ 2sin

22

1 ……………………….…….…..…… (3.6)

The compatibility equation for slope at y=b is therefore:

bbbaEa

a

E

FFduetoPxax

GJ'''

2sin

2sin

2211

22

1

bbaE

xb21212

sin2

…….…..… (3.7)

And similarly at y=-b:

bbbaEa

a

E

FFduetoPxax

GJ'''

2sin

2sin

2211

22

2

bbaE

xb21212

sin2

…….….…… (3.8)

31

Equations (3.3), (3.4), (3.7) and (3.8) enable the unknown edge effects, F1, F2,

M1and M2 to be determined. For this purpose it is sufficient to consider the mid-span

section of the bridge, i.e. x = a. Once these values have been determined the

deflections, the longitudinal and transverse moments at any point in the bridge, and

the bending and torsional moments at any point in the edge beams may be

determined by superposing the various effects.

To obtain a fully rigorous solution it would be necessary to consider the

various terms in the Fourier series for the load ; this would imply using values of the

coefficients K,K’, ,,,', and ' appropriate to values of the flexural parameter

of .,3,2, etc This would involve solving the compatibility for each term. The

values of the coefficient are given in Graph 14-19.

At the junction of the edge stiffening and the main bridge structure an

implicit assumption has been made concerning the neutral axis. This that the natural

axis coincides in both the bridge and the edge stiffening member ; if any design it

does not then additional stresses will be set up locally which will modify, to a limited

extent, those given by the above analysis. Calculation of these stresses is not

practicable and nominal reinforcing steel should be included to cover this effect. In

general, however, the primary stresses in both the bridge and the edge beam will be

in accord with the above analysis.

In certain cases where the edge stiffening members is of considerable width

the edge shear forces, F, will induce both deflection and rotation of the edge

stiffening member. This effect can obviously be included in the above analysis if it

appears to be of significance.

3.4 General Effects of Edge-Stiffening

The detailed method of analysis given in the preceding sections can deal

satisfactorily with any of the edge conditions met in practice. However, in some

32

cases the full analysis may not be necessary and it then is sufficiently accurate to

consider only the effect of the edge shear forces F. In this case only deflection

compatibility need be considered at the two edges y = + b for this analysis only the

distribution coefficients K are required and the analysis may be out by omitting the

terms involving M1 and M2 in equations (3.3) and (3.4)

From experimental work that has been carried out it appears that the edge

moments may be neglected in the analysis when the ratio of the flexural to the

torsional stiffness of the edge beam is greater than about 5 provided that the flexural

stiffness/unit width of the edge beam is not considerably greater than that of the

equivalent orthotropic plate.

In general the effect of moderate edge stiffening is to reduce the maximum

longitudinal moments by between 20 and 30 per cent and to increase the maximum

transverse moments by about 5 per cent. The reduction in the longitudinal moments

is attributing able to :

(a) the decrease in the “ mean “ moment caused by the additional stiffness at the

edges, and

(b) the reduction in the maximum distribution coefficient due primarily to the edge

shear forces.

As in most engineering problems, the compatibility equations lead to a set of

simultaneous equations which tend to be ill-conditioned. Care should therefore be

exercised in solving these equations ; it is best to determine F1 and F2 first and

subsequently to back-substitute these values in the original equations to determine

M1 coefficients are approximately unity or less. It is also preferable to carry out the

solution on a desk calculating machine rather than a slide rule since a greater number

of decimal places can be used.

The method of analysis presented is in series form and therefore a rigorous

solution may only be obtained by considering the various terms in the series for the

applied load, edge shear forces and edge moments. By considering the centre of the

bridge, i.e. = a only the odd terms need be summed. While the first, third and fifth

terms of the series will led to a reasonably accurate solution, this implies solving

there sets of compability equations. This procedure may, in special circumstances, be

33

warranted but for most practical problems it is sufficiently accurate to consider only

the first term of the various series and apply certain correcting factors to the solution

so found.

If only the first term is used for the applied loads, edge shears and edge

moments and the compatibility equations for deflection and slope are solved, then the

deflection of both the edge beams and the bridge structure can be predicted with an

accuracy of within about 3 per cent. This follows from the consideration of Fourier

representations of various types of loading.

For the longitudinal moments, the Fourier representation does not given the

same accuracy as for deflection owing to the double differentiation involved.

However, the relation between the first term of the Fourier series for the “mean “

moment and the exact “ mean “ moment derived in the manner. Hence a factor can

be introduced to correct for the consideration of the first term only in the “mean “

moment. The same factor will be applicable to the edge shear forces F since Fouries

series for these are of the same form as that for the applied load. The second

correcting factor arises because only the first term is considered for the distribution

coefficient K; this factor of 1.1, but it must be noted that it only applies to the region

of maximum moment under or near the applied loads.

The design office analytical procedure for longitudinal moments may be

summarized as follow:

a) For the specific bridge and edge beams, derive the deflections and slope

compatibility equations and solve for the values of F and M applicable to the

first term of all the series involved;

b) Knowing F and H1derive the longitudinal moments at the standard positions due

to both the applied loads edge shear forces using the “ mean “ moment

appropriate to the first term of the series ; in this procedure the edge moments

are ignored since they give rise to no significant longitudinal moments;

c) Obtain the relation between the “ mean “ moment for the applied loading

deduced from the first term of the series, i.e. H1, and the exact procedure ;

d) Increase or decrease the total moments in (b) by the correcting factor found in

(c)

34

e) Increase the maximum moment found in (d) by 10 per cent.

For the transverse moment, therefore the relation between the total transverse of

the various terms in the series and therefore the relation between the total transverse

moment and that given by the first term of the series is known. This same relation

must apply to the transverse moment caused by the first term of the edge shear force

series; the total transverse moment can be derived.

CHAPTER IV

CASE STUDY ON EFFECT OF EDGE-STIFFENING USING LOAD

DISTRIBUTION METHOD

4.1 Introduction

4.1.1 Deck Geometry (Case study 1,2,3)

Fig. 4.1

Fig. 4.2

4.1.2 Loading

Fig. 4.3

Fig. 4.4

4.1.3 Moment

Fig. 4.5

Fig. 4.6

4.2 Result for Case 1

a

b

Longitudinal bending moment, Mx

Table 4.1

Table 4.2

xPxRA

P

Transverse bending moment, My

Table 4.3

Fig. 4.7

Table 4.4

xxxx

My


bhi

mx

i

xhxbj

mxxx

Ex

xE

di

mi

bi

mi

Fig. 4.8

a

u

a

PH

xa

PH

bKbFKbbFbKpHI

ibF

bKbFKbbFbKpHI

ibF

xFxFxxPF

xFxFxxPF

Longitudinal bending moment, My

Kb

HaMx

a

x

xa

PH

a

Px

bx

aMx

PxxMx

b

Fx

a

xx

b

Fx

a

Pxx


Fig. 4.9

Table 4.5

xFxFxxPF

xFxFxxPF

a

xxxFFbxM y


bhi

mx

i

xhxbj

mmxxx

Ex

xE

di

mi

bi

mi

a

u

a

PH

xa

PH

xFxFxxPF

xFxFxxPF

Longitudinal bending moment, My

Kb

HaMx

a

x

xa

PH

a

Px

bx

aMx

Pxx

xMx

b

Fx

a

Pxx

b

Fx

a

Pxx


xFxFxxPF

xFxFxxPF

a

xxxFFbxM y

CHAPTER V

ANALYSIS OF EFFECT OF EDGE-STIFFENING USING FINITE

ELEMENT METHOD

5.1 Introduction

Today’s computer technology allows designs to be assessed much more

easily and quickly. Evaluating a complex engineering design by exact mathematical

models, however, is not a simple process.

5.1.1 Description of LUSAS Program

Compare to other FEM soft wares, LUSAS finite element software is very

user friendly. This software works in the window version, which is much simpler and

easier to use compared to other software. In the windows, the structures can be

modeled graphically and this is very helpful, especially to enable the user to view

and understand the models thoroughly.

By using the LUSAS software, both linear and non-linear analyses can be

performed. The attributes of the models are easy to create and they can be easily

assigned to the models by simply dragging the attributes’ icons to the models. This

finite elements software can model the concrete and steel structures, as long as their

material properties are known.

55

5.1.2 Deck Idealization (Case Study 1a, 2a, 3a)

A simply supported bridge structure consisting of an in-situ slab deck is to be

analysed. The geometry of the deck is as shown in Figure 4.1 and 4.2 in chapter IV.

5.1.3 Finite Element Model and Mesh Layout

LUSAS models are defined in terms of geometric features which must be

sub-divided into finite elements for solution. In this case regular meshing is used

with uniform spacing for beam and slab. Quadrilateral elements are used in this case

analysis, while for the generic element type thick shell is used and linear interpolar

linear order. Transitions pattern is used for the regular mesh. Figure 5.1 shows the

finite element model for the deck with and without edge-stiffening beam.

XY

Z (a) (b)XY

Z

Fig 5.1 The finite element model for the plan deck (a) without and (b) with edge-stiffening

5.1.4 Element Properties

Every part of a finite element model must be assigned a material property

dataset. LUSAS material datasets are defined from the Attributes menu. This case

use isotropic material with young modulus: 30E6, poison ratio: 0.2, mass density: 2.4

and coefficient of thermals expansions: -0.01E3. For the geometry of the deck model

surface element are used to represent the deck slab and the edge stiffening beam at

56

different level of eccentricities. Due to the ‘upstand ‘of the edge beam, Case 2 has an

eccentricity of -0.1 m, while Case 3 has no eccentricity due to symmetry of the

‘upstand’ and ‘downstand’ beam.

5.1.5 Loading

A HB load consists of coordinates defining the vertices (different to

geometric points defining the model), load intensity and local x, y and z position.

Any geometric points selected when the discrete loading dialog is initiated are

entered as coordinates. Point load defines a general set of discrete point loads in 3D

space. Each individual point load can have a separate load value. This case uses 16

distinct load values with the axle spacing as 6 and number of HB unit as 45. Figure

5.2 shows the HB position on the bridge deck.

XY

ZXY

Z (a) (b)

Fig 5.2. The model of HB loading in finite element (a) position for longitudinal moment and (b) position for transverse moment.

5.1.6 Moments

Several forms of result are be provided by the LUSAS solver. However, in

this study the consideration is given to maximum longitudinal bending moment and

maximum transverse bending moment in the bridge model Case 1, 2 and 3.

57

5.2 Results for Case 1a

Table 5.1 Maximum the bending moment for slab without edge stiffening

Moment Maximum kNm/m

Longitudinal bending Moment (Mx) 669.255

Transverse Bending Moment (My) 142.152

Mx:

My:

Fig 5.3 Moment distribution in both directions for Case 1a

58


Table 5.2 Maximum bending moment for slab with edge stiffening for Case 2a




Mx:

My:


59


Table 5.3 Maximum bending moment for slab with edge stiffening for Case 3a




Mx:

My:


CHAPTER VI

DISCUSSION OF RESULTS

6.1 Result of both method

The calculated values of longitudinal and transverse moments are

summarized in Table 6.1 for load distribution method and table 6.2 for finite element

method. In general the presence of edge-stiffening beam resulted in significant

reduction of longitudinal moment but a slight increase of transverse moment. Figure

6.1 and 6.2 shows comparisons of maximum longitudinal bending moment and

transverse bending moment obtains for deck bridge between both case for load

distribution method and finite element method.

Table 6.1 Result of longitudinal and transverse moment with Load Distribution Method

Moment CASE 1 CASE 2 % Difference CASE 3 % Difference % Difference

(kN/m / m-width) (kN/m / m-width) Cas e1-Cas e2 (kN/m / m-width) Cas e1-Cas e3 Cas e2-Cas e3

Longitudinal 754.8255 515.296 31.73 313.0875 58.52 39.24Transverse 178.312 186.7696 4.74 206.6878 15.91 10.66

Table 6.2 Result of longitudinal and transverse moment with Finite Element Method Moment CASE 1 CASE 2 % Difference CASE 3 % Difference % Difference

(kN/m / m-width) (kN/m / m-width) Cas e1-Cas e2 (kN/m / m-width) Cas e1-Cas e3 Cas e2-Cas e3

Longitudinal 669.345 592.619 11.46 465.803 30.41 21.40Transvere 142.512 153.779 7.91 170.0173 19.30 10.56

61

735.6814

515.296

313.0875

0100200300400500600700800

Mom

ent

Comparasion Longitudinal Moment

case 1 Case 2 Case3

178.312186.7696 206.6878

0

50

100

150

200

250

Mom

ent

Comparasion Transverse Moment

case 1 Case 2 Case3

Fig 6.1 shows the comparison between Case 1, 2 and 3 in maximum bending

moment of the longitudinal beams and transverse bending moment for load

distribution method

669.345

592.619465.803

0100200300

400500600700

Mom

ent


case 1 Case 2 Case3

142.512153.779 170.0173

0

50

100

150

200M

omen

t


case 1 Case 2 Case3

Fig 6.2 shows the comparison between Case 1, 2 and 3 in maximum bending

moment of the longitudinal beams and transverse bending moment for finite element

method

An analysis of a slab bridge with or without edge-stiffening beams is made

and the result obtained from LUSAS programs (Finite Element Method) are

compared with those derived theoretically (Load Distribution Method). Results for

the analysis of longitudinal and transverse moment are obtained from load

distribution method and finite element method. This will allow the assessment be

made on effect of edge-stiffening to both longitudinal and transverse moments.

Comparison between load distribution method and finite element method is show in

figure 6.3.

62

754.83669.345

515.296 592.619

313.0875

465.803

0100

200

300

400

500600

700

800

Mom

ent


case 1 case 2 case 3

178.312142.512

186.7696153.779

206.6878

170.0173

0

50

100

150

200

250

Mom

ent


case 1 case 2 case 3

Fig 6.3 shows the comparison between load distribution method and finite element

method for Case 1, 2 and 3 in maximum bending moment of the longitudinal beams

and transverse bending moment.

63

6.2 General Effect of Edge-Stiffening

In general the effect of edge-stiffening beam is to reduce the maximum

longitudinal moment and to increase the maximum transverse moment. From Table

6.1 and figure 6.1 it can be seen that the effect of edge-stiffening with a simple

‘upstand’ reduces the longitudinal moment by 31.73% and increases the transverse

moment by 4.74%. For Case 3, with the combination of ‘upstand’ and ‘downstand’,

there is further reduction of longitudinal moment by 58.52% and increase in

transverse moment by 10.66%. Table 6.2 and figure 6.2 are shows that effect edge-

stiffening with a simple ‘upstand’ reduces the longitudinal moment by 11.46% and

increase the transverse moment by 7.91%. By providing an additional ‘downstand’,

further reduction of longitudinal moment by 30.41% and increase the transverse

moment by 19.30%.

6.3 Form of Edge-Stiffening

The different form and size of edge-stiffening will influence of the

longitudinal and transverse moment. In Case 2 with ‘upstand’ (200x600mm)

compared With Case 3 with ‘upstand’ and ’downstand’ (200x1000mm) there is a

reduction in longitudinal moment by 39.2% for load distribution method and 21.4%

for finite element method, while the transverse moment increased by 10.66% for load

distribution method and 10.56% for Finite Element method.

6.4 Method of Analysis

For analysis of deck without edge-stiffening (Case 1), the values of

longitudinal and transverse moments obtained by load distribution method are bigger

that those obtained by finite element method. The difference is about 11.32 % for

longitudinal moment and 20 % for transverse moment. However, for the deck with

64

edge-stiffening (Case 2 and Case 3) the load distribution method given lower values

of longitudinal moment by about 15-48.7 % compared to finite element method. The

inverse is true for the transverse moment, whereby the values given by load

distribution method are higher by 17 %.

CHAPTER VII

CONCLUSIONS

7.1 General Conclusions

1. The general effect of edge-stiffening in bridge deck is to reduce the longitudinal

moment significantly. Although the transverse moment shows an increase, this

however, is less significant. The increase in transverse moment is offset by

advantageous reduction in longitudinal moment. Results from both method show

similar trend.

2. An edge-stiffening beam with a simple ‘upstand’ (200x600mm) reduced the

longitudinal moment by 11.46%-31.73% while the transverse moment increased

by 4.74%-7.91%. By providing an additional ‘downstand’ (200x1000mm),

further reduction of longitudinal moment is achieved by 30.41%-58.52%, which

is very significant. However this also results in significant increase in transverse

moment by 15.91%-19.3%. Although different form and size of edge-stiffening

beam will influence the moment, the choice will be dependent on various factors

such as technical requirement, nature of service to be provided and cost.

3. The effect of edge-stiffening is due to the modifying effects of the edge shear and

edge moment due to the presence of the edge beam.

4. In general, the values of moments obtained by load distribution method are

higher that those obtained by finite element method except for longitudinal

moments in edge-stiffening Case. The load distribution method uses simplified

66

equations of a complex edge shear forces and moment acting at edge beam.

Whereas finite element method is influenced by the quality of modeling, meshing

and choice of element type, material and support conditions.

7.2 Recommendations for future study

To improve the study on the effect edge-stiffening in bridge decks, some

recommendations are proposed to produce more comprehensive finding in future

study.

i. The comparison between different procedures used for other program finite

element such as COSMOS-M, NASTRAN, ABAQUS and ANSYS [5]

ii. Laboratory tests are can be used to carry out and for comparison and

verification with theoretical calculation (load distribution method) or finite

element method.

67

REFERENCE

LITTLE, G. and ROWE, R.E. The effect of edge-stiffening and eccentric

transverse prestress in bridge. London, Cement and Concrete Association, November

1957. Technical Report TRA/279.

ROWE, R.E. Concrete Bridge Design. London, C.R.Brooks Ltd.,1962.

BECKETT, D. An Introduction to Structural Design (1) Concrete Bridge.

Surrey University Press.,1973.

L.A. CLARK. Concrete Bridge Design to BS 5400. Construction Press., 1983

CUSENS, A.R. and PAMA, R.P. Bridge Deck Analysis. A Wiley-

Interscience Publication. 1975.

EDMUND C. HAMBLY. Bridge Deck Behaviour. Chapman and Hall Ltd 11

New Fetter Lane, London EC4P 4EE. 1976.

AZLAN, A.R (2004), Method of Load Distribution Coefficients for Bridge Deck Analysis,

Bridge Engineering Notes, Faculty of Civil Engineering, University Technology

Malaysia.

68

APPENDIX A

71

APPENDIX B

82

APPENDIX C

load distribution in edge-stiffening beam of a simply supported

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