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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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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
Horizontal Position of Opening
Ang
le
of
tw
is
t(
ra
d)
Figure 4.7: Angle of twist for models with different horizontal
location of opening along the beam.
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97.4
97.7
98
0 1 2 3 4 5 6 7 8 9 10 11 12
Horizontal Position of Opening
Norma
l
St
ress
(N/mm
2)
Figure 4.8: Maximum normal stress for models with different horizontal
location of opening along the beam.
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
location of opening along the web.
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
location of opening along the web.
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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
Size of Opening(mm2)
Ang
le
of
tw
i
st
(ra
d)
Figure 4.13: Angle of twist for models with different sizes of
opening at fix location.
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97
98
99
0 100 200 300 400 500
Size of Opening(mm2)
Norma
l
Stress
(N/mm
2)
Figure 4.14: Maximum normal stress for models with different sizes of
opening at fix location.
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
openings along the beam.
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
openings along the beam.
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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
Spacing between Openings(mm)
Ang
le
of
twi
st
(ra
d)
Figure 4.19: Angle of twist for models with different spacing
between two same size of openings along the beam
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97.4
97.9
98.4
0 1000 2000 3000 4000
Spacing between Openings(mm)
Norma
l
Stress
(N/mm
2)
Figure 4.20: Maximum normal stress for models with different spacing
between two same size of openings along the beam
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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.