comportamiento al warping de vigas en voladizo con aberturas

7/27/2019 Comportamiento Al WARPING de Vigas en Voladizo Con Aberturas

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WARPING BEHAVIOUR OF

CANTILEVER STEEL BEAM WITH OPENINGS

TAN YU CHAI

A thesis submitted in fulfillment

of the requirements for the award of the degree

of Master of Engineering (Civil-Structure)

Faculty of Civil Engineering

Universiti Teknologi Malaysia

OCTOBER, 2005


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I declare that this thesis entitled WARPING BEHAVIOUR OF CANTILEVER

STEEL BEAM WITH OPENINGS is the results of my own research except as

cited in references. This thesis has not been accepted for any degree and is not

concurrently submitted in candidature of any degree.

Signature :

Name of Candidate : Tan Yu Chai

Date : 30 October 2005


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The journey of a thousand miles begins with a single step.

A Man without dream is nothing at all.

Dare to dream!!


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ACKNOWLEDGEMENTS

First of all, the author wishes to express the deepest gratitude to his

supervisors Professor Ir. Dr. Abdul Karim Mirasa and Associate Professor Ir. Dr.

Mohd. Hanim Osman, for their insight and greatest guidance during this project.

Without their noble approach, this study will never finish so smoothly.

Acknowledgement is extended to Mr. Koh Heng Boon for his great advice

which helped author to complete his study especially in understanding LUSAS

software. The author is thankful to Puan Fatimah Denan for her encouragement and

help. Acknowledgements are also due to Mr. Moumouni Moussa Idrissou, Mr.

Felix Ling Ngee Leh, Mr. Tan Che Siang and Mr. Sia Chee keong for their advice

and helpful cooperation during this research. Besides, appreciation is

acknowledged for those who ever direct or indirectly involved in the completion of

this project.

The author will never forget the internal supports from his family members

especially the countless blessing from his parents which have always been the source

of motivation in achieving success to a higher level. Last but no least, the author

wishes to acknowledge the most important people in his life, Ms. Loke Chai Yee for

her endless support and motivation.


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ABSTRACT

This project presents a short study on warping behaviour of cantilever steel

beam with openings subjected to coupled torsional force at the free end. Thus far

there has not been any research regarding the relationship between warping and

webs openings. Finite element software, LUSAS 13.6, was used to perform

analysis on seven groups of modelling. The analysis of the results showed that

opening has a close relationship with warping since opening can reduce web stiffness.

When warping resistance decrease, warping displacements and warping normal

stress will increase. Opening with bigger size, installed at the free end and central

of the web will induce greater warping and vice versa. Simple approximation of

installing stiffeners is proposed in this study to provide sections warping resistance

effectively.


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ABSTRAK

Projek sarjana ini mengkaji kelakuan ledingan (warping) pada rasuk julur

keluli berlubang yang mana hujung bebasnya dipiuhkan terhadap paksi membujurnya.

Hingga kini, tiada sebarang kajian mengenai hubungan antara kelakuan ledingan

dengan lubang pada web rasuk. Perisian LUSAS 13.6 digunakan untuk mengkaji 7

kumpulan model di dalamprojek ini. Keputusan yang diperolehi menunjukkan

lubang boleh mempengaruhi kelakuan ledingan rasuk dengan mengurangkan

kekukuhan web rasuk. Anjakan dan tegasan paksi akan meningkat berikutan

dengan pengurangan kekukuhan ledingan. Lubang berbentuk lebih besar yang

dipasang pada hujung bebas dan tengah web akan membentuk piuhan yang lebih

ketara dan sebaliknya. Fahaman ringkas terhadap pemasangan pengukuh turut

dikaji bagi meningkatkan keupayaan ledingan rasuk dengan berkesan.


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TABLE OF CONTENTS

CHAPTER TITLE PAGE

DECLARATION ii

DEDICATION iii

ACKNOWLEDGEMENTS iv

ABSTRACT v

ABSTRAK vi

TABLE OF CONTENT vii

LIST OF TABLE x

LIST OF FIGURES xi

NOTATION xiii

LIST OF APPENDIX xv

I INTRODUCTION 1

1.1 Introduction 1

1.2 Background of Study 3

1.3 Problem Statement 5

1.4 Objectives of Study 5


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1.5 Scope of Study 6

1.6 Research Significance 6

II REVIEW OF LITERATURE STUDIES 8

2.1 Introduction 8

2.2 Review of Published Work 10

2.3 Basic Theory 11

2.3.1 Torsion 11

2.3.2 Significance of Warping Constant 16

2.4 Conclusion 19

III LINEAR FINITE ELEMENT ANALYSIS 22

3.1 Introduction 22

3.2 Modeling 23

3.2.1 Model Geometry 25

3.2.2 Types of Elements 26

3.2.3 Meshing 26

3.2.4 Material Properties 27

3.2.5 Support Conditions 28

3.2.6 Loads Arrangement 28

3.3 Convergence 28

3.4 Model Validation 29

3.5 Conclusion 31


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IV RESULTS AND DISCUSSIONS 37

4.1 Introduction 37

4.2 Stress Concentration Zone 38

4.3 Stress Distribution 38

4.4 Stress Across Flange Width 39

4.5 Discussion for every group of models 39

4.5.1 Various Horizontal Location of Opening 40

4.5.2 Various Vertical Location of Opening 40

4.5.3 Various Sizes of Opening 41

4.5.4 Various Numbers of Openings 41

4.5.5 Comparison between Circular and Square

Openings 41

4.5.6 Various Spacing between Two Openings 42

4.5.7 Comparison between Two Types of Stiffener 42

4.6 Summary 43

V CONCLUSION AND SUGGESTION 54

5.1 Conclusion 54

5.2 Suggestion 55

REFERENCES 56

BIBLIOGRAPHY 58

APPENDIX A 59

APPENDIX B 64


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LIST OF TABLES

NO. OF TABLE TITLE PAGE

3.1 Results obtained from various numbers of elements 29


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LIST OF FIGURES

FIGURE NO. TITLE PAGE

1.1 Torsional shear flow in a solid bar by Englekirk 7

1.2 Torque induced shear flow by Gorenc, Tinyou & Syam. 7

2.1 Uniform and non-uniform torsion of an I-section member 20

2.2 Bimoment and stresses in an I-section member 20

2.3 Deformation u and associated with lateral-torsional buckling 21

3.1 Different positions of opening along the beam 32

3.2 Different position of opening along the web 32

3.3 Twelve openings along the beam 32

3.4 Various spacing between two 300mm square openings 33

3.5 Two types of stiffeners 33

3.6 Geometry specifications and load arrangement 33

3.7 Linear and quadratic shell element types 34

3.8 Fixed-end support 34

3.9 Graph of deflection on y-direction against number of elements 35

3.10 Graph of angle of twist against number of elements 35

3.11 Graph of normal stress against number of elements 36

3.12 Converged model with appropriate element size 36

4.1 Stress concentration zone of control specimen 44

4.2 Stress concentration zone of model with 12 openings 44

4.3 Stress distribution of model with 100mm square opening 45

4.4 Stress distribution of model with 500 mm square opening 45

4.5 Stress across flange width of model with 500mm square opening 45


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4.6 Deflection on Y-direction for models with different horizontal

location of opening along the beam 46

4.7 Angle of twist for models with different horizontal location of

opening along the beam 46

4.8 Maximum normal stress for models with different horizontal

location of opening along the beam 47

4.9 Deflection on Y-direction for models with different vertical

location of opening along the web 47

4.10 Angle of twist for models with different vertical location of

opening along the web 48

4.11 Maximum normal stress for models with different vertical

location of opening along the web 48

4.12 Deflection on Y-direction for models with different sizes of

opening at fix location 49

4.13 Angle of twist for models with different sizes of opening at fix

location 49

4.14 Maximum normal stress for models with different sizes of

opening at fix location 50

4.15 Deflection on Y-direction for models with different number of

openings along the beam 50

4.16 Angle of twist for models with different number of openings

along the beam 51

4.17 Maximum normal stress for models with different number of

openings along the beam 51

4.18 Deflection on Y-direction for models with different spacing

between two same size of openings along the beam 52

4.19 Angle of twist for models with different spacing between two

same size of openings along the beam 52

4.20 Maximum normal stress for models with different spacing

between two same size of openings along the beam 53


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NOTATIONS

B - Width of flange

D - Overall depth of girder

t - Thickness of web

T - Thickness of flange

E - Youngs modulus

H - Warping Constant

Iw - Waripng Constant

J - Torsional Constant

G - Shear modulus

- Angle of twist

L - Length of the section subject to T

T - Applied torque

w - Warping normal Stress

w - Warping shear stress

Wns - Normalized warping function at the particular point S in the cross

Section

Wws- Warping statical moment at the particular point S in the cross section.

a - Distance of effective flange restraint

ME - Elastic critical moment

Mp - Plastic moment capacity of section

Mb - Buckling resistance moment

Pb - Bending strength

Sx - Plastic section modulus


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n - Slenderness correction factor

u - Buckling parameter

v - Slenderness factor

x - Torsional index

py - Design strength

A - Cross-sectional area of a member

- Slenderness of a beam

1 - Constant for a particular grade of steel

LT - Equivalent slenderness

D LT- Non-dimensional effective slenderness, ratio ofLT / 1

h - Distance between shear centre and the flanges

K - Global stiffness matrix


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LIST OF APPENDICES

APPENDIX TITLE PAGE

A Finite Element Models 62

B Raw data obtained from finite element analysis 67


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CHAPTER I

INTRODUCTION

1.1 Introduction

Steel was first produced in the Middle Ages, but it was just used for

structural engineering over a century ago. Steel is one of the most important

construction materials available in Malaysias market due to its strength-to-volume

ratio, wide range of possible applications, availability of many standardized parts,

reliability of the material and its ability to give shape to nearly all the architectural

wishes. Numerous researches had been carried out to study various strength

properties of steel sections. BS 5950 for example has been introduced to provide a

guideline in designing steel structures. The main reason of using standard in design

work is structural safety.


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In design of beam, various strength properties of steel beam need to be taken

into consideration. One of them is lateral-torsional buckling strength of beam. In

order to understand the lateral-torsional buckling (LTB), it is essential to develop the

knowledge about torsional behaviour of the section including the torsional properties

i.e. torsional constant (J) and warping constant (H). A general idea of lateral-

torsional buckling including the torsional properties i.e. torsional constant (J) and

warping constant (H) can be obtained through Appendix B BS5950: Part 1: 2000.

Frequently torsion is a secondary, though not necessarily a minor effect that

must be considered in combination with the action of other forces. The shapes that

make good columns and beams, i.e. those that have their material distributed as far

from their centroids as practicable, are not equally efficient in resisting torsion.

Thin-wall circular and box sections are stronger torsionally than sections with the

same area arranged as channel I, tee, angle, or zee shapes. When a simple circular

solid shaft is twisted, the shearing stress at any point on a transverse cross-section

which is initially planar remains a plane and rotates only about the axis of the shaft.

The development of cellular beams was initially for architectural application,

where exposed steelwork with circular openings in the webs was considered

aesthetically pleasing. It was recognized that their application could be extended to

floor beams and that, due to the high price of curtain walling, savings in the total

building cost were attainable through the use of long span cellular beams. They

would allow floor zones to be kept to a minimum, without increasing the cost of the

steel frame, and enable services to pass through the circular openings, obviating the

need for underslung services.


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However, the effect of warping due to openings is not stated in BS 5950.

The purpose of this study is to assess the warping behaviour of the cantilever steel

beam with web openings using finite element modeling. Warping normal stress,

displacements on longitudinal axis and angle of twist obtained through finite element

analysis were used as comparison parameters between section with and without

openings.

1.2 Background of Study

The aim of structural design should be to provide a structural capable of

fulfilling its intended function and sustaining the specified loads during its service

life. Any features of the structure that have a critical influence on its overall

stability should be identified and taken account of in the design. In structural

design, torsional moment may, on occasion, be a significant force which provision

must be made because the stability of a flexural member is very often a function of

its torsional stiffness. The theory of torsion would be considerably simpler if the

planar surfaces assumed to be remained plane after twisting. In fact, only

cross-sectional surfaces of round shapes remain planar after twisting. In 1853, the

French engineer Adhemar Jean Barre de Saint-Venant showed that when a

noncircular bar is twisted, it will not remain plane. The original cross-section plane

surface becomes a warped surface.

Warping is a difficult phenomenon to visualize. A variable shear flow will

occur around the perimeter of a square bar if the shear stress distribution postulated


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by using membrane analogy as illustrated in Figure 1.1. This variation in shear

stress in terms of magnitude and direction induces flexural stresses provided the

member subjected to torsion was constrained from warping. If the plate is not

constrained, the induced flexural stresses cause warping.

For closed sections such as tubes and box sections, the sections remain plane

after twisting within practical limits of accuracy, and the torsional resistance

contributed by the parts of the cross-section is proportional to the distance from the

centre of twist. While I-section member under uniform torsion such that flange

warping is unrestrained, the pattern of shear stress is shown in Figure 1.2. Open

sections are substantially less rigid torsionally than sections of the same overall

dimensions and thickness with flanges restrained against warping [1].

The development of cellular beams was initially for architectural application,

where exposed steelwork with circular openings in the web is considered

aesthetically pleasing. Furthermore, this application will allow floor zones to be

kept to a minimum, without increasing cost of the steel frame, and enable services to

pass through the circular openings, obviating the need for underslung services. But

there is no reference available for the warping effect due to the openings. Therefore,

this project is carried out.


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1.3 Problem Statement

Nowadays, the use of steel beams with openings is commonly used since it

makes ducting and services work much more easily. Despite the advantages of

flexibility in construction or better outlook, the introduction of opening may reduce

the strength of the section if it was not properly designed. Ward (1990) [2] shows

that the overall flexural capacity is assessed by considering the plastic moment

capacity of the cross section through the centre line of the opening. This reflects

that opening can influence the webs strength properties. Hence, it is essential to

carry out a study to determine the warping behavior for the steel beam with web

opening. Cantilever steel beam was chosen in this research since the nature of

cantilever steel beam which restrained at one side makes it vulnerable for torsion.

1.4 Objectives of Study

The objectives of study are as below:

1. To determine the warping behaviour of cantilever steel beam with

openings

2. To observe the effect of installing intermediate stiffeners.

3. The use of finite element method in the study of warping behaviour.


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1.5 Scope of Study

The scope of study can be divided into several areas which stated as below:

1. Verification of the FEM modal by analytical method.

2. To identify the warping displacement.

3. To identify the angle of twist

4. To identify the warping normal stress.

5. Two types of stiffeners was studied

.

1.6

Research Significance

The significance of the study is that the establishment of warping behaviour

of cantilever steel beam with openings and guideline for installing intermediate

stiffeners on cantilever steel beam with openings with respect to warping behaviour.

This new understanding will then pave way to the development of accurate use of

transverse stiffeners on cantilever steel beam with openings as a fundamental

engineering problem-solving methodology.


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Figure 1.1: Torsional shear flow in a solid bar by Englekirk

Figure 1.2: Torque induced shear flow by Gorenc, Tinyou & Syam


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CHAPTER II

REVIEW OF LITERATURE STUDIES

2.1 Introduction

The concept of stability as it applies to structures is best understood by

considering conditions of equilibrium. Englekirk (1994) [3] showed that a

structural system, which is in equilibrium if disturbed by a force, has two basic

alternatives to remove the disturbing force:

1.

It could return to its original position in which case we refer to the system

as being stable.

2. It could continue to displace, and as a consequence be incapable of

supporting the load it is supported before the disturbance occurred, in

which case we refer to the system as being unstable.


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Instability is then characterized as a change in geometry, which results in a loss

of ability to support load. The stability of a member subjected to a flexural load is

impacted significantly by its torsional stiffness. Torsion is seldom relied upon as a

significant load path in the design of steel buildings. This is primarily because open

shapes are torsionally very flexible and twist readily when torque is applied [3]. It

is very important to the steel structures designers to understand torsion since the

stability of a flexural member is very often a function of its torsional stiffness.

The torsional warping function for a thin-walled open-section beam may contain

two parts: the contour warping function (the primary warping) and the thickness

warping function (the secondary warping). Vlasov (1961) [4] and Timoshenko

(1934) [5] for example only consider the contour warping function as the real

warping function. Due to some thin sections where the contour warping is much

larger than the thickness warping and the contribution of the thickness warping to the

warping constant may be small, the vast majority of researchers only consider the

contour warping function as the warping function [6]. Vlasovs [4] theory for open

cross sections is presented in terms of generalized stresses and strains by assuming

the cross section completely rigid in its own plane and neglect the effect of shearing

deformations. However, the cross section is affected by out-of-plane strains in

general case, defined by a warping function.

A detailed historical review of the early development of thin-walled beam

theories is presented by Nowinski (1959). An improvement of the elementary

thin-walled beam theory was proposed by Goodier in 1942 by the addition of

Kirchhoffs hypothesis for shells to the previous assumptions, as discussed and

enhanced by Gjelsvik (1981). The resulting warping function consists of two

components: the warping of the middle surface and the warping of the section

relative to the middle surface. [7]


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2.2 Review of Published Work

Numbers of research works had been done on warping. However, there are no

work has been done relevant warping of beam with opening. Ward (1990) [2]

showed the load carrying capacity of a cellular beam is the smaller of its overall

strength in flexure and lateral torsional buckling and the local strength of the web

posts and the upper and lower tees. In many practical applications, the beams will

be laterally restrained, causing local effects around the openings to control the design,

as indicated by experimental tests at the University of Bradford.

It indicated that warping resistance of section might reduce with the presence of

openings. Several researchers have dealt with beams of variable cross section

ignoring the warping effects resulting from the corresponding restraints at the ends of

the member. If the aforementioned structures are analyzed or designed for torsion

considering only the effect of Saint-Venant torsion resistance, the analysis may

underestimate the torsion in the members and the design may be not conservative. [8]


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2.3 Basic Theory

2.3.1 Torsion

Torsion is seldom relied upon as a significant load path in the design of steel

buildings. This is primarily because open shapes are torsionally very flexible and

twist readily when torque is applied. Accordingly, the compatibility-induced

torsion introduced into these girders will be small and the experienced rotation easily

determined by an analysis of the beam. Further, torsional stresses, however induced,

are seldom calculated, for they have little or no impact on the strength limit state of a

member. Anyway, torsion is nevertheless very important to the designer of steel

structures because the stability of a flexural member is very often a function of its

torsional stiffness. [3]

Torsion can be categorized into two types: pure torsion or Saint-Venants

torsion, and warping torsion. Pure torsion assumes that a cross-sectional plane prior

to application of torsion remains a plane and only element rotation occurs during

torsion. Warping torsion is the out-of-plane effect that arises when the flanges are

laterally displaced during twisting, analogous to bending from laterally applied loads.

[10] Hence, torsion may be thought of as being composed of two parts: (1) rotation

of elements, the pure torsion part, and (2) translation producing lateral bending, the

warping part. [10]

The total resistance of a member to torsional loading is composed of the

sum of two components known as uniform torsion and warping torsion. When


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the rate of twist (twist per unit length), and the longitudinal warping deflections are

constant along the member, it is in a state of uniform torsion as shown in Figure

2.1(a). If the rate of twist, and warping deflections are varies along the member, it

is in a state of non-uniform torsion as Figure 2.1(b) & (c). In this case, an

additional set of shear stresses may act in conjunction with those due to uniform

torsion to resist the torque acting. The stiffness of the member associated with these

additional shear stresses is proportional to the warping rigidity. [9] Whether a

member is in a state of uniform or non-uniform torsion also depends on the loading

arrangement and the warping restraints.

In uniform torsion, the applied torque is resisted entirely by shear stresses

distributed throughout the cross section. The ratio of the applied torque to the twist

per unit length is equal to the torsional rigidity, GJ of the member, where G is the

shear modulus and J is the torsional constant. J is sometimes called the St Venant

torsion constant.

In non-uniform torsion, both direct and shear stresses are generated which

are additive to those due to bending and pure torsion respectively. The stiffness of

the member associated with these additional stresses is proportional to the warping

rigidity, EH, where E is the modulus of elasticity and H is the warping constant.

When the torsional rigidity, GJ of the section is very large compared with the

warping rigidity, EH, the member will effectively be in a state of uniform torsion.

Closed sections, angles and tee sections behave in this manner as do most flat plates

and all circular sections. Conversely, if the torsional rigidity of the section is very

small compared with the warping rigidity, the member will effectively be in a state of

warping torsion. This condition is closely approximated for very thin walled open

section such as cold formed sections. Between these two extremes, the members

will be in a state of non-uniform torsion and the loading will therefore be resisted by


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a combination of uniform and warping torsion. This is the condition which occurs

in hot rolled I, H and channel sections.

The shear centre of a cross section lies on the longitudinal axis about which

the section would twist if torsion acts on the section. If the resultant force acts

through the shear centre, no twist will occur and the torsional stresses will be zero.

The shear centre and the centroid are not necessarily coincident. However, in a

rolled I or H section, which is symmetrical about both principal axes, the shear centre,

s, coincides with the centroid, c.

In most engineering type structures, displacement will occur due to an

applied torque. The out-of-plane distortions do not induce any normal stresses

providing these displacements are not restrained or altered along the axis of the

section. If the warping is restrained, warping normal stress will be induced. [11]

The induced warping phenomena can be explored by consideration of behavior of a

cantilever I-steel beam. Assuming web of section remains undeformed, the applied

torque is resisted by flanges, and the shear deformations in the flanges are caused by

cross-bending.

In elastic non-uniform torsion, both the rate of change of the angle of twist

d/dz and the longitudinal warping deflections w vary along the length of the

member. The varying warping deflections induce longitudinal strains and stresses. [9]

Warping moment or bimoment and stresses induced are shown in Figure 2.2.

According to Salmon and Johnson, three kinds of stresses arise in any

I-shaped or channel section due to torsional loading:


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(a) Shear stresses in web and flanges due to rotation of the elements of the

cross-section (Saint-Venant torsional moment)

(b) Shear stresses in the flanges due to lateral bending (warping torsional

moment)

(c) Normal stresses (tension and compression) due to lateral bending of the

flanges (lateral bending moment on flange)

In general, warping normal stresses are direct stresses (tension or compression)

resulting from the bending of the element due to torsion. In the case of an I beam,

the stresses occur in the flanges. They act perpendicular to the surface of the cross

section and are constant across the thickness but vary along the length of an element.

While shear stresses are in-plane shear stresses that are constant across the thickness

of the element but vary in magnitude along the length of the element and act in a

direction parallel to the edge of the element. Each stress is associated with the

angle of twist () or its derivatives. Hence, when is determined for different

position along the girder length, the corresponding stresses can be evaluated at each

position.

The total angle of twist is given by:

GJ

TL= (2.1)

where T is the applied torque

L is the length of member subject to T

G is the shear modulus of the material

J is the torsion constant for the section.


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With angle of twist obtained, we can find magnitude of warping normal

stress and warping shear stress at any point s in the cross section with Equation

(2.2) and (2.3)

z = ''nEW (2.2)

where = the second derivative of equation (2.1)

Wn = b

oo wtdswA 0

1

= the normalized unit warping of the cross section.

w =t

wES ''' (2.3)

where = the third derivative of equation (2.1)Sw =

s

tdsWn0

= the warping statical moment

Distance of effective flange restraint, a is given by formula as below:

GJ

wEIa = (2.4)


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Besides, Equation (2.1) also provides a convenient method for determining the

modulus of rigidity of a given material. The corresponding values of the angle of

twist, at length, L of the specimen can be indicated by applying increasing

magnitude of torque, T. By plotting against T, a straight line will be obtained.

And the slope of this straight line represents JG/L. And with that, torsion constant J

can be calculated. (For further details, see[13])

2.3.2 Significance of Warping Constant

Beams of open section bent in their stiffer principal plane are susceptible to

an analogous type of buckling involving a combination of lateral deflection and twist

as Figure 2.3. This is known as lateral-torsional buckling. The buckling

deformation that appears depends upon the initial shape of the beam and the way in

which the loading is applied. As its deformations are coupled, it increases the

complexity of its analysis. [14] An analysis is similar to that for the Euler buckling

of struts may be used. For the idealized case of loading and support, taken to be

uniform single curvature bending and ends that cannot deflect laterally or twist (but

are provided with no other restraining effects), the expression for the elastic critical

moment is obtained as:

GJL

EIGJEI

LM

WYE

2

2

1()(

+= (2.5)

Where EIy = minor (or y-y) axis flexural rigidity of the section

GJ = torsional rigidity of the section


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EIw = warping rigidity of the section (Iw is defined as H in BS5950:Part 1)

The value of the equivalent slenderness of the beam LT is defined as:

E

p

y

LT

M

M

p

E2 = (2.6)

In which 1 is a constant for a particular grade of steel. The ratio LT/1 is

often termed the non-dimensional effective slenderness LTD .

Substituting4

2hI

Iy

w = for an I-section into Equation (2.5) and re-arranging

Equation (2.6) gives:

2

1

2

2

21 2

12

+

==

h

L

EI

GJh

L

EI

pS

y

y

yxLTLT

D (2.7)

where Iy is the second moment area about the minor axis = B3T / 6

Sx is the plastic modulus

+BT

thBTh ws411

J is the torsion constant 0.33(hwt3

+ 2BT3)

G is the shear modulus E / 2.6

hs = D - T

hw = D - 2T


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Equation (2.7) may be presented in terms of the following parameters:

vuLT .1=

D (2.8)

where2

12

=Ah

Su

x

and4

12

2011

+=x

v

where2

1

566.0

=J

Ahx

A is the cross sectional area.

Hence, uvLT = (2.9)

Where is the slenderness of the beam

u is the buckling parameter

v is the slenderness factor


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2.4 Conclusion

In this chapter, warping and torsion of girder in general were review and

summarized. From the literature, it can be seen that effect of warping cannot be

neglected in obtaining the steels strength properties since warping constant, Iw or H

plays an important role in finding the value of the equivalent slenderness of the beam,

LT which is a main parameter in lateral torsional buckling. Subsequently, the

bending strength, pb and buckling resistance moment, Mb can be obtained through

relevant formula stated in BS 5950. Since the warping constant for beam with and

without opening are different, it is important to study the warping behaviour of steel

beam with web opening to understand further about the defect of strength properties

due to openings.


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Figure 2.1: Uniform and non-uniform torsion of an I-section member.

Figure 2.2: Bimoment and stresses in an I-section member.


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Figure 2.3: Deformation u andassociated with lateral-torsional buckling.

T

T


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CHAPTER III

LINEAR FINITE ELEMENT ANALYSIS

3.1 Introduction

The finite element method is a numerical analysis technique for obtaining

approximate solutions to a wide variety of engineering problems. The principle of

discretization is used in the basic concept of finite element analysis. It is governed

by a master equation, written in matrix notation stated as below where Kis the

master stiffness matrix, also called global stiffness matrix, assembled overall

stiffness matrix,

{ } [ ]{ }uKF = 3.1


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Where F = Load vector

K = Global stiffness matrix

U = Displacement

In the field of design, finite element method is a very powerful tool since it

can evaluate a complex engineering design in much more easier and economic way

with the promise of more powerful computers. There is plenty of software

available in the market, including LUSAS, MSC. Patran/Nastran, and SAMCEF.

LUSAS 13.6 software was used in this study to obtain the value of warping normal

stress, warping displacement and angle of twist. Through these parameters, we can

understand more about the warping behaviour of cantilever steel beam with web

openings.

3.2 Modeling

The steps of modeling are almost similar for all of the finite element

software which stated as below:-

a. Creating model geometry

b. Meshing

c. Assign the geometry and material properties

d. Assign the boundary condition (support condition)

e. Assign the loading applied.


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In this project, a control specimen is modeled, converged and validated.

By using the same model, seven groups of modeling will be analyzed:-

i) Various horizontal location of opening

Single 300mm square opening was installed at different horizontal location

along the beam from position 1 at free end to position 12 at the fix end as

illustrated in Figure 3.1.

ii) Various vertical location of opening

Single 100mm square opening was installed at different horizontal location

along the web from position 1 at bottom to position 5 as shown in Figure

3.2.

iii) Various sizes of opening

Single opening with difference sizes was installed at fix location at web

which is 700mm from free end.

iv) Various numbers of openings

Different numbers of 300 mm square opening was installed at the section

from 1 to 12. Figure 3.3 showed the example of twelve openings along the

beam.

iv) Comparison between circular and square opening

Single circular opening with 300mm diameter was installed and compared

with 300mm square opening at the same location which is 700mm from free


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end.

v) Various spacing between two openings

Two 300mm square openings were installed with 100mm spacing. The

results were then compared with increased spacing as illustrated in Figure

3.4.

vii) Comparison between two types of stiffener

Two types of stiffener were installed to compare their effective in reducing

warping. Type A stiffened around the opening while type B stiffener

connected the upper and bottom flange as shown in Figure 3.5

3.2.1 Model Geometry

Usually, surface-like elements used to represent thin-walled structures since

detail stress distribution is required while the thickness is much smaller compare to

the sections depth. The units used in the modeling are Newton and millimeter.

Modification has been made by considering models complexity which ignores the

small curves that connecting the web and flange in real section. Details of the

model dimensions are shown in Figure 3.6.


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3.2.2 Type of Elements

Shell elements can be quadrilaterals or triangles. Generally, a quadrilateral

mesh is more accurate than a mesh of similar density based on triangles. Linear or

first order shell elements are normally planar and degrade in accuracy as their initial

definition deviates from planar. Higher order shell elements can provide accurate

results with curved initial geometries. In fact, one of the benefits of using higher

order elements is that positioning the mid-side nodes on the actual curved geometry

increases the models accuracy. Parabolic elements can be defined to actually

represent bidirectional curvature and can smoothly represent initial geometry as

illustrated in Figure 3.7. Linear quadrilateral thick shell element has been chosen to

be used in this study.

3.2.3 Meshing

Building a shell model requires mid-plane surfaces in one form or another.

A good technique for starting shell models is to sketch the part first to identify the

key features required in the model. A well-prepared, underlying surface model will

improve the efficiency of making changes. It is good practice to try to break the

geometry into four-sided patches with corner angles as close to 90 as possible.Surfaces are entered into the model when the curves in an area are completely

defined.


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When the surface model is complete and all edge curves of adjacent surfaces

are merged, mesh the entire model with a consistent element size. This technique

will ensure that the modes at the edges of the surfaces will align and can be merged

cleanly. Preliminary analysis was carried out by applying minimum mesh density

for the model. Convergence step was carried out to provide a good representation

of the geometry and response to the loads. Besides from the accuracy of the model,

economic consideration in term of processing time also governed the choice of finite

mesh density.

.

3.2.4 Material Properties

Most materials behave differently under different conditions. Even steel,

one of the more predictable engineering materials has different failure properties

depending on alloying, heat treatment, cold working, or manufacturing method. If

accurate stress data are required for a failure or a fatigue calculation, independent

testing the material should be performed.

Since only elastic state of the material was being studied and accuracy not

playing the governing role, the material property can assigned directly to the model

from LUSAS material library. This whole I-beam was assigned ungraded mild steel

for its material property with Youngs modulus, E and Poissons ratio, v were taken

as 209x103N/mm

2and 0.3 respectively.


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3.2.5 Support Conditions

Since cantilever I-beam was studied, it is important to make sure the model

is fully restrained at one side since support condition acts critical role in obtaining the

results. To achieve it, we need to fix the translation and rotation in all of the

direction X, Y and Z for all of the nodes at one side as illustrated at Figure 3.8.

3.2.6 Loads Arrangement

There are few alternative patterns of loading in order to create torsion and

hence produce warping effect. Simplified loading method was used where one

coupled point load were applied at the edge of upper and bottom flange in opposite

direction as illustrated in Figure 3.6

3.3 Convergence

Convergence was done by increasing the element mesh density of the model

beam section. Table 3.1 shows results created with different numbers of elements.

2 comparison parameter, which are angle of twist, and normal stress, x were

plotted against number of elements as shown in Figure 3.9, 3.10 and 3.11. It was


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clearly indicated that a convergence solution has been obtained for the mesh with

1492 elements. Since the model involved is very simple, mesh with 4564 elements

was chosed to enhance the accuracy with little increase of computer processing time.

Anyway, for further increment in number of elements, the results obtained might be

more accurate. But the difference can be neglected since it is not fulfill the

economic consideration. Figure 3.12 shows the converged model with appropriate

elements.

Number of

elements

y (mm) (rad) x (N/mm2)

175 -22.259 0.0944 79.13296 -25.006 0.1060 80.54

459 -25.897 0.1102 90.03

630 -26.163 0.1109 89.01

1492 -26.728 0.1132 93.66

1800 -26.799 0.1135 93.38

3562 -27.027 0.1145 96.23

4564 -27.068 0.1147 97.40

Table 3.1: Results obtained from various numbers of elements.

3.4 Model Validation

In order to verify the modeling technique of torsional problem, one control

specimen is modeled to determine the angle of twist, by using finite element

analysis and compared with theory. Below are the calculation examples to calculate

the angle of twist and compare it with the result obtained from finite element

analysis.


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theory =

+

)1(

)1(1 /2

/2

aL

aL

eL

ea

GJ

TL

Since a =GJ

EIw= 124

812

10/)1006.60)(80385(

10/10176.1209000(

1

= 2.256

where,

J = 0.33 (hwt3

+ 2BT3) = 0.33 ((500 - 2 x 15)(10)

3+ (2 x 200 x 15

3))

= 60.06 x 104

Iw = (Iyh2s) / 4 =

4

)15500(6

15200 23

= 1.176 x 1012

G =6.2

E=

6.2

209000= 80385

And e-2L/a

= e-2x5/2.256

= 0.01188

theory =

+

)01188.01(

)01188.01(

5

29.21

10/1006.601080385

51021246

3

= 0.1145 radians


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46

Considering the result of node 2 which locate at the centre of upper flange,

FEA = tan-1

z

y

= tan-1

152

500

07.27

= 0.1147 radians

The difference between theory and FEA is (0.1147 0.1145 = 0.0002 radians) or

0.17 %

3.5 Conclusion

With the difference of 0.17%, it shows that the results predicted by finite

element modeling are acceptable as long as all groups of model were compared

based by the same meshing. All groups of model are assumed to have a difference

of nearly zero percents with the theoretical value as the roof of comparison.

Warping normal stress, deflection on y-direction (across the flange) and angle of

twist for node 2 were the parameters used in this study. By comparing these

parameters within the models, we can understand the warping effect due to openings.


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T

T Fig 3.1: Different positions of opening along the beam

5

4

3

2

Fig 3.2: Different position of opening along the web.

Fig. 3.3: Twelve openings along the beam.

2 3 4 5 6 7 8 9 10 11 121

1


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L

x

10KNB

Y

10KN Z

T

t

T

D

Where, t = 10mm

T=15mm

B=200mm

D=500mm

L=5000mm

Type BType A

Fig 3.4: Various spacing between two 300mm square openings.

Fig 3.5: Two types of stiffeners.

Fig 3.6: Geometry specifications and loads arrangement.

500mm 4100 mm1 3 12


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Fig 3.7: Linear and quadratic shell element types.

Fig 3.8: Fixed-end support.

Linear Triangle

(3 nodes)

Parabolic Quadrilateral

8 nodes

Parabolic Triangle

6 nodes

Linear Quadrilateral

4 nodes


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22

24

26

28

0 1000 2000 3000 4000 5000

Number of elements

De

flec

tion

(mm

)

Fig. 3.9: Graph of deflection on y-direction against number of elements.

0.09

0.11

0.13

0 1000 2000 3000 4000 5000

Number of elements

Ang

le

of

tw

is

t

(ra

d)

Fig. 3.10: Graph of angle of twist against number of elements.


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77

82

87

92

97

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Number of elements

Norma

l

St

ress

(N/mm

2)

Fig. 3.11: Graph of normal stress against number of elements.

Fig. 3.12: Converged model with appropriate element size

XY

Z

XY

Z


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CHAPTER IV

RESULTS AND DISCUSSIONS

4.1 Introduction

In this chapter, results obtained from LUSAS linear finite element analysis

were stated clearly and presented by graphically. Comparisons were made with

control specimen without opening and models with different size, location and

numbers of openings. Models studied in this project are shown in Appendix A.

Besides, effects of stiffeners were observed by comparing model with and without

stiffeners. By assuming web of section remain undeformed and the applied torque

is resisted by flanges, result will only be observed on flange. (Please refer to the

appendix B)

In order to study the warping behaviour of cantilever steel beam with

openings, few parameters were observed, including stress concentration zone, stress


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distribution, stress across flange width, deflection on y-direction, angle of twist and

warping normal stress. The results were described briefly in the following parts.

4.2 Stress Concentration Zone

Since the section was restrained at one end, stress will induced at the flange.

From the observation, the pattern of normal stress concentration zone is same for the

control specimen and model with 12 openings as shown in Figure 4.1 and 4.2. This

had proven that openings will affect the magnitude of stress which has close

relationship with warping but not affect the pattern of normal stress concentration

zone.

4.3 Stress Distribution

If only the left side of upper flange to be considered, the observation showed

that the maximum normal stress occurred at the fix end while the minimum normal

stress at the free end. Figure 4.3 and 4.4 showed the stress distribution of model

with 500mm square opening and 100mm square opening. It showed that opening

did affect the stresss curve on the flange above the webs opening. There is no

difference for the curve shape after that.


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4.4 Stress Across Flange Width

As warping react under symmetric loading in this study, there is no normal

stress at the centre of flange. Maximum normal stress occurred at both edge of

flange with opposite direction symmetrically. Figure 4.5 illustrated an example of

normal stress diagram across the upper flange width of model with 500 mm square

opening. This showed that the left flange was experienced tensile stress and

compressive stress at the right flange.

4.5 Discussion for every group of models.

As mentioned in the previous chapter, deflection on y-direction, angle of

twist and normal stress are the parameters used to compare within the models.

Every models deflection was obtained through the node at the centre of the upper

flange. Angle of twist can be calculated by the formula shown previously.

Anyway, the normal stress mentioned here was represent the maximum value of

every model.


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4.5.1 Various Horizontal Location of Opening

Base on the observation, it is clear that the effect of opening decrease when it

was installed far from the free end. When the opening installed at position 12, the

deflection on y-direction and the angle of twist are almost the same with the control

model. Figure 4.6 and 4.7 showed that the deflection and angle of twist were

reduced when the opening was installed nearer to the fixed end. From Figure 4.8,

maximum normal stress increased when the opening was installed from 1st

to 8th

position. After that, it was decreased dramatically.

4.5.2 Various Vertical Location of Opening

Figure 4.9 and 4.10 showed that the results obtained are symmetrical where

the deflection on y-direction and maximum angle of twist occurred when opening

installed at the centre of the web. Results for normal stress also symmetrical in a

way of the maximum normal stress occurred when opening installed at the centre of

the web as shown in Figure 4.11.


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4.5.3 Various Sizes of Opening

All of the three comparison parameters are governed by the size of opening.

It is clear that bigger size opening will produce greater warping as illustrated in

Figure 4.12, 4.13 and 4.14. Deflection due to 500mm square opening is greater

2.8% than the deflection of 100mm square opening.

4.5.4 Various Numbers of Openings

Same as the size factor, the number of openings will caused greater deflection,

angle of twist and also warping normal stress. (Refer to Figure 4.15, 4.16 and 4.17)

Anyway, it can be observed that when the opening located nearer to the fix end, the

effect of warping was decreased. For example, the difference within eleven and

twelve openings in this study is very small where only 0.7% of deflection, 0.8% of

angle of twist and 0.1% of normal stress.

4.5.5 Comparison Between Circular and Square Openings

As mentioned in previous chapter, warping is influenced by the cross-section

area of the section. This theory has been proven where the beam with square

opening which has lesser cross section area than the beam with circular opening has


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been observed to have greater warping. Beam with 300mm square opening has

0.11% more deflection and 0.05% more normal stress than the beam with 300mm

diameter circular opening.

4.5.6 Various Spacing Between Two Openings

Similar to the effect of horizontal location where the second opening installed

nearer to the fixed end, the warping effect was decreased. Figure 4.18 and 4.19

showed that the deflection on y-direction and the angle of twist was decreased when

the spacing between two openings increased. It has been observed that when the

openings installed at position 1 and 12, the deflection on y-direction is -27.22mm

which is almost the same with only one opening at position 1 with deflection on

y-direction is -27.21mm. Figure 4.20 showed that the maximum normal stress will

increase when the spacing between two openings increase. When the spacing more

than 2500mm, maximum normal stress will decrease dramatically.

4.5.7 Comparison Between Two Types of Stiffener

From the observation, stiffeners can reduce warping effectively. Stiffener

type B will provide better warping resistance by reduce 9.3% of deflection, 11.12%

of angle of twist and 4.18% of normal stress. This is because of stiffener type B

provides greater flange resistance by connecting upper and bottom flange.


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4.6 Summary

From the results discussed above, the effect of opening cannot be neglected in

studying the warping behaviour of steel beam. Anyway, the results above only

represent the cantilever steel beam where one end fixed. Or else, the results of

spacing and numbers of opening may differ. From the simple study on stiffeners, it

shows that stiffeners with proper designed at suitable location can reduce warping

effectively and economic.


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Figure 4.1: Stress Concentration zone of control specimen

Figure 4.2: Stress concentration zone of model with 12 openings.


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Figure 4.3: Stress distribution of model with 100mm square opening.

Figure 4.4: Stress distribution of model with 500mm square opening.

Figure 4.5: Stress across flange width of model with 500mm square

opening


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27.07

27.17

27.27

27.37

0 1 2 3 4 5 6 7 8 9 10 11 12

Horizontal Position of Opening

De

flec

tion

on

Y-d

irec

tion

(mm

)

Figure 4.6: Deflection on Y-direction for models with different horizontal

location of opening along the beam.

0.1145

0.1155

0 1 2 3 4 5 6 7 8 9 10 11 12


Ang

le

of

tw

is

t(

ra

d)

Figure 4.7: Angle of twist for models with different horizontal



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97.4

97.7

98

0 1 2 3 4 5 6 7 8 9 10 11 12


Norma

l

St

ress

(N/mm

2)

Figure 4.8: Maximum normal stress for models with different horizontal


27.03

0 1 2 3 4 5

Vertical Position of Opening

De

flec

tion

on

Y-

direc

tion

(mm

)

Figure 4.9: Deflection on Y-direction for models with different vertical

location of opening along the web.


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0.1143

0 1 2 3 4 5

Vertical Position of Opening

Ang

le

of

tw

is

t

(ra

d)

Figure 4.10: Angle of twist for models with different vertical


97.3

97.4

97.5

0 1 2 3 4 5

Norma

l

Stres

s

(N/mm

2)

Figure 4.11: Maximum normal stress for models with different vertical



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27

27.5

28

0 100 200 300 400 500

Size of Opening(mm2)

De

flec

tion

on

y-

direc

tion

(mm

)

Figure 4.12: Deflection on y-direction for models with different sizes of

opening at fix location.

0.1147

0.1167

0 100 200 300 400 500


Ang

le

of

tw

i

st

(ra

d)

Figure 4.13: Angle of twist for models with different sizes of



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97

98

99

0 100 200 300 400 500


Norma

l

Stress

(N/mm

2)

Figure 4.14: Maximum normal stress for models with different sizes of


27.1

28.1

29.1

0 1 2 3 4 5 6 7 8 9 10 11 12

Numbers of Opening

De

flec

tion

on

Y-d

irec

tion

(mm

)

Figure 4.15: Deflection on y-direction for models with different number of

openings along the beam.


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0.1147

0.1197

0.1247

0 1 2 3 4 5 6 7 8 9 10 11 12

Numbers of Opening

Ang

le

of

tw

is

t

(ra

d)

Figure 4.16: Angle of twist for models with different number of


97

100

103

0 1 2 3 4 5 6 7 8 9 10 11 12

Numbers of Opening

Norma

l

Stress

(N/mm

2)

Figure 4.17: Maximum normal stress for models with different number of



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27

27.3

27.6

0 1000 2000 3000 4000

Spacing between Openings(mm)

De

flec

tion

on

Y-

di

rec

tion

(mm

)

Figure 4.18: Deflection on Y-direction for models with different spacing

between two same size of openings along the beam

0.1145

0.1165

0 1000 2000 3000 4000


Ang

le

of

twi

st

(ra

d)

Figure 4.19: Angle of twist for models with different spacing



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97.4

97.9

98.4

0 1000 2000 3000 4000


Norma

l

Stress

(N/mm

2)

Figure 4.20: Maximum normal stress for models with different spacing



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CHAPTER V

CONCLUSION AND SUGGESTION

5.1 Conclusion

Study on warping behaviour of cantilever steel beam with openings had

been conducted using finite element analysis. From the study, it can be concluded

that opening has a close relationship with warping behaviour. In fact, the effect of

opening depends on its location, numbers, size and shape since opening will reduce

webs stiffness and hence reduce warping rigidity.

From the observations, opening did not change the pattern of stress

concentration zone but affect the magnitude of normal stress. Anyway, effect of

opening in term of warping can be reduced if it is installed near to the fixed end.

By simple approach, stiffener can reduce warping effectively with appropriate type

and location. Lastly, it is clear that finite element method can be used satisfactorily


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in the study of warping behaviour of cantilever steel beam with openings.

5.2 Suggestion

For future researches, experiment test to study the warping behaviour of

cantilever steel beam with various location, sizes and numbers of openings should be

carried out. Anyway, only cantilever steel beam studied in this project, further

researches should be carried out to study the effect of opening to the uniform

warping behaviour of simply supported steel beam. Besides, study on stiffeners in

more detail should be conducted since this project only studied on two types of

stiffeners due to time constraint.


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REFERENCES

1. Gorenc, B., Tinyou,R. & Syam,A. Steel Designers handbook (Sixth Edition).

Australia: University of New South Wales Press. 1996.

2. Ward, J.K.Design of Composite and Non-composite Cellular Beams. Silwood

Park: The Steel Construction Institute. 1990.

3. Englekirk Robert. Steel Structures. Controlling Behaviour through Design. USA:

John Wiley & sons, Ltd. 1994.

4. Vlasov VZ. Thin Walled Elastic Beams. English translation published for US

Science Foundation by Israel for Scientific Translations. 1961.

5. Timoshenko, S. Theory of Elasticity. New York: McGraw-Hill Book Co. Inc.

1934.

6. W.Y. Lin and K.M. Hsiao.More General Expression or the Torsional Warping of

a Thin-walled Open-Section Beam. International Journal of Mechanical Sciences,

Vol. 45, Elsevier Ltd. 2003.

7. Mauro Schulz and Filip C. Filippou. Generalized Warping Torsion Formulation.

Journal of Engineering Mechanics, Vol. 124, No.3, ASCE. 1998.


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8. E.J. Sapountzakis and V.G. Mokos.Nonuniform Torsion of Bars of Variable

Cross Section. Journal of Computers and Structures, Vol.82, Elsevier Ltd. 2004.

9. Trahair, N.S & Bradford, M.A. The Behaviour and Design of Steel Structures.

2nd

edition. New York: Harper Collins College Publishers. 1988.

10. Salmon, C.G. & Johnson, J.E. Steel structures.Design and Behaviour,

Emphasizing Load and Resistance Factor Design. 4th

edition. USA: Harper

Collins College Publishers. 1996.

11. Heins,C.P.Bending and Torsional Design in Structural Members. Canada: D.C.

Health and Company. 1975.

12. Vince Adams & Abraham Askenazi.Building Better Products with Finite

Element Analysis. Santa Fe, USA: OnWord Press. 1999.

13. Salman Ullah Sheikh. Warping Behaviour of Trapezoidal Web Profile. Master

Thesis. University Teknology Malaysia. 2002

14. Yong Jitt Ching. Warping Constant. Master Thesis. University Technology

Malaysia. 2003.

15.British Standard Institution.Lateral Stability of Steel Beams and Columns

Common cases of restraint. 1992.

16.British Standard Institution.BS5950: Part 1: 2000, Structural Use of Steelwork in

Building. 2000.


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BIBLIOGRAPHY

Tirupathi R. Chandrupatla and Ashok D. Belegundu (1997). Introduction To Finite

Elements in Engineering. New Jersey: Prentice Hall.

Wilson E. L. (1997). An Introduction to The Analysis of Linear Elastic Structures

by Finite Element Method. University of Colorado Online Technical Support,

http://www.colorado.edu (20-8-2005).

L.P. Yong and N.S.Trahair (2000). Distortion and Warping at Beam Supports.

Journal of Structural Engineering, Vol. 126, ASCE.

P.A. Kirby & D.A.Nethercot (1979). Design for Structural Stability. New York:

John Wiley & Sons.

W.F. Chen and E.M.Lui (1987). Structural Stability (Theory and Implementation).

New York: Elsevier Science Publishing.


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Appendix A

Figure A-1: Different location of opening along the beam


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Figure A-2: Different location of opening along the web

Figure A-3: Various sizes of openings at fix location.


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Figure A-4: Various numbers of openings along the beam


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Figure A-5: Circular and square shape of openings at fix location.

Figure A-6: Various spacing of two openings


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-Type A

-Type B

Figure A-7: Two types of stiffeners.


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Appendix B

Table B-1: Different location of opening along the beam

Table B-2: Different location of opening along the web

Position y(mm) (rad) x(N/mm2

)

1 -27.21 0.1153 97.59

2 -27.38 0.1160 97.84

3 -27.38 0.1160 97.87

4 -27.36 0.1160 97.89

5 -27.34 0.1158 97.92

6 -27.31 0.1157 97.94

7 -27.27 0.1155 97.96

8 -27.23 0.1154 97.97

9 -27.19 0.1152 97.96

10 -27.15 0.1150 97.90

11 -27.12 0.1149 97.78

12 -27.08 0.1147 97.58

Position y(mm) (rad) x (N/mm2)

1 -27.05 0.1146 97.37

2 -27.09 0.1148 97.42

3 -27.11 0.1149 97.45

4 -27.09 0.1148 97.42

5 -27.05 0.1146 97.37


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Table B-3: Various sizes of openings at fix location.

Table B-4: Various numbers of openings along the beam

Size y(mm) (rad) x(N/mm2)

Control -27.07 0.1147 97.43

100mm -27.12 0.1149 97.47

200mm -27.24 0.1154 97.62

300mm -27.41 0.1161 97.85

400mm -27.62 0.1170 98.12

500mm -27.89 0.1181 98.48

Position y(mm) (rad) x(N/mm2)

1 -27.21 0.1153 97.59

2 -27.51 0.1165 97.97

3 -27.83 0.1179 98.42

4 -28.14 0.1192 98.90

5 -28.43 0.1204 99.42

6 -28.70 0.1215 99.96

7 -28.93 0.1225 100.5

8 -29.12 0.1233 101.1

9 -29.26 0.1239 101.6

10 -29.37 0.1243 102.1

11 -29.44 0.1246 102.4

12 -29.46 0.1247 102.5


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Shape y(mm) (rad) x(N/mm2)

Control -27.07 0.1147 97.43

Circular -27.35 0.1159 97.79

Square -27.38 0.1160 97.84

Table B-5: Circular and square shape of openings at fix location.

Spacing y(mm) (rad) x(N/mm2

)

100 -27.51 0.1165 97.97

500 -27.53 0.1166 98.04

900 -27.51 0.1165 98.06

1300 -27.48 0.1164 98.09

1700 -27.45 0.1163 98.11

2100 -27.42 0.1162 98.13

2500 -27.38 0.1160 98.14

2900 -27.34 0.1158 98.12

3300 -27.29 0.1156 98.07

3700 -27.26 0.1155 97.94

4100 -27.22 0.1153 97.74

Table B-6: Various spacing of two openings


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Type y(mm) (rad) x(N/mm2)

Without stiffeners -27.38 0.1160 97.84

Type A -24.32 0.1053 94.47

Type B -24.84 0.1031 93.75

Table B-7: Two types of stiffeners.

comportamiento al warping de vigas en voladizo con aberturas

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