UNIVERSITY OF BELGRADE
FACULTY OF MECHANICAL ENGINEERING
Hasan Mehdi Nagiar
STRESS ANALYSIS OF CORNER WELDED
JOINTS STRUCTURE BY MODERN
NUMERICAL-EXPERIMENTAL APPROACH
Doctoral Dissertation
Belgrade, 2013
UNIVERZITET U BEOGRADU
MAŠINSKI FAKULTET
Hasan Mehdi Nagiar
NAPONSKA ANALIZA UGAONIH
ZAVARENIH SPOJEVA NOSAČA 
KONSTRUKCIJA SAVREMENIM
NUMERIČKO-EKSPERIMENTALNIM 
PRISTUPOM
doktorska disertacija
Beograd, 2013
Dedicated to all my family
Supervisor of doctoral dissertation
Dr. Tasko Maneski, full professor
University of Belgrade, Faculty of Mechanical Engineering
__________________________
Committee members for the doctoral dissertation
Dr. Aleksandar Sedmak, full professor
University of Belgrade, Faculty of Mechanical Engineering
__________________________
Dr. Vesna Milosevic-Mitic, full professor
University of Belgrade, Faculty of Mechanical Engineering
__________________________
Dr. Dragan Ignjatovic, full professor
University of Belgrade, Faculty of Mining and Geology
__________________________
Dr. Nenad Gubeljak, full professor
University of Maribor, Faculty of Mechanical Engineering
__________________________
Date of defense:
ii
Acknowledgment
I wish to record my deep gratitude. Thanks and sincere appreciation to my supervisor Prof.
Tasko Maneski for his supervision, guidance, advice, continuous help, interest and
encouragement thought the different stages of my Ph.D. study.
I would like to thank all the staff members of the Mechanical Engineering Faculty for their
help and guidance during my Ph.D. study.
A special thank is given to my family for their continuous support and help during my Ph.D.
study.
In the last, I would like to record my thanks to all who contributed in any way to this work.
Author
Belgrade, November 2013
iii
STRESS ANALYSIS OF CORNER WELDED JOINTS STRUCTURE BY MODERN
NUMERICAL-EXPERIMENTAL APPROACH
Abstract:
Hollow and non hollow section members are widely used in industrial applications for the
design of many machine and structural components. These components are often fabricated
by welding rather than by casting or forging. For example, in agricultural machinery, the
hollow tubes are typically connected together through welding to form a corner welded joints.
Such joint connections are also employed in other engineering applications such as
construction machinery, offshore structures, bridges, and vehicle frames. In this dissertation,
the behavior of tubular (box and circle profile) and non tubular (L, Z, C and X profiles) joint
connections profiles, subjected to static loads were studied both experimentally and
numerically.
From a structural analysis point of view, despite of the wide use of corner welded joints as
efficient load carrying members, there is no available practical, simple and accurate approach
for their design and analysis. For this purpose, engineers must often prepare relatively
complicated and time consuming Finite Element models made up of shell or solid elements.
This is because unlike solid-section members, when hollow section members are subjected to
general loadings, they may experience severe deformations of their cross-sections that results
in stress concentrations in the connection’s vicinity. One of the objectives/contributions of
this research work is the better understanding of the behavior of the corner welded joint
connections under out-of- plane bending and torsion loading conditions. Through a detailed
Finite Element Analysis (FEA) using shell and solid elements, the stress distribution at the
connection of the tubular and non tubular corner welded joints were obtained for different
loading conditions. It is observed that at a short distance away from the connection of the
corner welded joints, the structure behaves similar to beams when subjected to loadings.
Finite element models with different modeling techniques and meshing with various size and
types of elements were created and analyzed. The full displacement field results were
obtained experimentally by using the digital image correlation (DIC) technique. Experimental
tests were performed to validate numerical simulations in order to investigate the mechanical
performance of a series of fillet-welded connections under combined loading. The full
displacement field results show good agreement comparing with the experimental results.
iv
In the numerical study, the connections were modeled using beam, shell and solid elements
using software for Finite Element Analysis - Abaqus. The performance of the connections
under combined loading was further studied by a simple analytical method aiming for rapid
information on the influence of the weld modeling on prediction of the structural
performance. Experiments of many beam joints subjected to bending were used for further
assessment of the FE models developed in the present study. The study was performed with
focus on the prediction of the structural strength of fillet welded joints.
The finite element results of the solid element model show that a generally linear stress
distribution across the thickness of the beams is observed. Therefore, the use of shell
elements for the analysis of the beam-joint connection is appropriate. Also, at a relatively
short distance from the connection, the effects of the local stresses and deformations
disappear.
The results show that the normal stresses in the 1 or 2-direction are the highest at the weld
path (top of the path) on the top surface, and at horizontal weld path of the beam. It can be
observed that the 1 or 2-direction component of normal stress causes the most damage. The
values of the Von-Mises equivalent stresses are very close to the values of the stresses in the
1 or 2-direction (absolute value). This is expected as the stresses in the 1 or 2-direction are
much larger than the other components of stress. This means that the strains and stresses
normal to the weld (i.e., 1 or 2 -direction in this study) are mainly responsible for
plasticity/crack initiation and propagation.
Keywords:
Corner welded joints, Finite element method, 3D optical analysis, stress, strain
Scientific field:
Doctor science, Mechanical engineering
Narrow scientific field: Strength of structures
UDC number: 621.791.052:624.014(043.3)
vNAPONSKA ANALIZA UGAONIH ZAVARENIH SPOJEVA NOSAČA 
KONSTRUKCIJA SAVREMENIM NUMERIČKO-EKSPERIMENTALNIM 
PRISTUPOM
Rezime:
Otvoreni i zatvoreni tankozidi preseci imaju široku primenu u industrijskim aplikacijama za
dizajn mnogih mašina i strukturnih komponenti. Ove komponente su često fabrikovane 
zavarivanjem, a ne livenjem ili kovanjem. Tankozidi profili su obično povezani pomoću 
ugaonih zavarenih spojeva. Takvi spojevi su takođe korišćeni u drugim inženjerskim 
aplikacijama kao što su građevinske mašine, mostovi, ramovi, šasije vozila i dr.. U ovoj 
disertaciji ponašanje zatvorenih profila (kutija, okrugla cev) i otvoreniih profila (L, Z, C i K
profili) šavnih profila, proučavani su tako što su izlagani statičkim opterećenjima i 
numeričko-eksperimentalnim pristupom. 
Sa tačke gledišta strukturne analize, uprkos široko rasprostranjenim ugaono zavarenim 
spojevima koji efikasno nose opterećenja na elementima, ne postoji praktičan, jednostavan i 
precizan pristup za njihov dizajn i analizu. U tu svrhu , inženjeri često moraju pripremiti 
relativno komplikovan model konačnih elemenata ploče ili zapremine. To je zato što 
navedeni eleementi izloženi opštim opterećenjima proizvode koncentracije napona u blizini 
spojeva. Jedan od doprinosa ovog istraživanja je bolje razumevanje ponašanja ugaono
zavarenih spojeva izloženi naprezanju savijanja i naprezanja savjanja sa uvijanjem. Primena
metode konačnih elemenata predstavlja osnovni numerički pristup. Posmatrani profili su 
medelilrani različitim konačnim elementima (greda, ploča i zapremina) i sa različitim 
slučajevima opterećenja. Primećuje se da na kratkom rastojanju od spoja putem ugaono 
zavarenih spojeva  struktura nosećeg elementa pod izloženim opterećenjem ponaša se kao 
greda.
Zavareni spojevi su generisani različitim tehnikama modelrianja sa primenom različitih 
tipova i veličina konačnih elemenata elemenata. Kompletni rezultati deformacija su, takođe,  
eksperimentalno dobijeni pomoću tehnike primene korelacije digitalne slike (DIC). 
Eksperimentalni testovi se izvode za proveru numeričkih simulacija u cilju ispitivanja 
mehaničkih performansi zavarenh spojeva izloženi kombinovanim opterećenjima. Rezultati 
numeričkog prroačuna pokazuju dobro slaganje sa eksperimentalnim rezultatom . 
vi
U numeričkom modeliranju primenom metode konačnih elelemenata zavareni spojevi se 
modeluju pomoću greda, ploča i zapremina korišćenjem softvera - Abakus. Performanse 
zavarenih spojeva dalje su proučavani korišćenjem jednostavnih analitičkih metoda sa ciljem 
dobijanja brzih informacija o uticaju zavarenog spoja na predviđanju strukturalnih 
performansi. Eksperimenti na mnogim spojevima na gredi koji su podvrgnuti savijanju se
koriste za dalju procenu FE modela razvijenih u ovoj studiji. Studija se obavlja sa fokusom na
predviđanje nosivosti strukture zavarenih spojeva. 
Rezultati proračun na grednom modelu pokazuju prostiranje linearnog napona posmatranog  
preko debljine grede. Zato je, upotrebna ploča za analizu spojeva odgovarajuća. Takođe, na 
relativno maloj udaljenosti od spoja, efekti lokalnih napona i deformacija nestaju.
Rezultati pokazuju da normalan napon u jednom ili dva pravca su najviši na ivici zavarenog
spoja na gornjoj površini i na horizontalnoj liniji zavarenog spoja poprečnog preseka profila. 
Primetno je da komponenta normalnog napona u jednom ili dva pravca ima najveću vrednost. 
Normalni naponi u jednom ili dva pravca mnogo veći od ostalih komponenti napona. To 
znači da su naponi istezanja u zavarenom spoju uglavnom  odgovorni za  pojavu plastičnosti i  
naprslina, kao i njihovo širenje.
Ključne reči : 
Ugaoni zavareni spoj, Metoda konačnih elemenata, 3D optička analiza,  napon, deformacija 
Naučna oblast : 
Doktor nauka, Mašinsko inženjerstvo
Uža naučna oblast : 
Otpornost konstrukcija
UDK broj : 621.791.052:624.014 ( 043.3 )
vii
Contents
Acknowledgment .......................................................................................................................ii
Abstract:................................................................................................................................... iii
Nomenclature...........................................................................................................................xv
Chapter 1....................................................................................................................................1
General Introduction ..................................................................................................................1
1.2. Background.....................................................................................................................1
1.2 Review of the Literature ..................................................................................................1
1.3 Objective and scope .........................................................................................................5
1.4 literature review of experimental research.......................................................................6
1.5 Fundamental Theory of Digital Image Correlation Techniques ......................................8
1.5.1 Two-dimensional digital image correlation method .................................................8
1.5.2 Two-Dimensional Image Correlation: Mathematical Formulation ........................12
1.5.3 Three-Dimensional Video Image Correlation ........................................................13
Chapter 2..................................................................................................................................16
Analytical Analysis of Welding...............................................................................................16
2.1 Introduction....................................................................................................................16
2.2 Assumption ....................................................................................................................17
2.3 Fillet Weld case(a) .........................................................................................................18
2.3.1 Rectangular section................................................................................................22
2.3.1.1 The effect of F1 ................................................................................................22
2.3.1.2 The effect of F2 .................................................................................................23
2.3.2 I section.................................................................................................................24
2.3.2.1 The effect of the force F1 ..................................................................................24
2.3.2.2 The effect of the force F2 ..................................................................................25
2.3.3 X section ................................................................................................................27
2.3.3.1 The effect of the force F1 ................................................................................27
2.3.3.2 The effect of the force F2 ................................................................................27
2.3.4 Z Section ...............................................................................................................28
2.3.4.1 The effect of the force F1 ...............................................................................28
2.3.4.2 The effect of the force F2 ...............................................................................29
viii
2.3.5 C section.................................................................................................................30
2.3.5.1 The effect of the force F1 ..................................................................................30
2.3.5.2 The effect of the force F2 .................................................................................31
2.3.6 circular section .......................................................................................................32
2.3.6.1 The effect of the force F1 ................................................................................32
2.3.6.2 The effect of the force F2 ................................................................................33
2.4. V Weld Case (b)...........................................................................................................34
2.4.1. Rectangular section...............................................................................................35
2.4.1.1. The effect of F1 ...............................................................................................35
2.4.1.2. The effect of F2 ...............................................................................................36
2.4.2 I section...................................................................................................................37
2.4.2.1 The effect of the force F1 ..................................................................................37
2.4.2.2. The effect of the force F2 .................................................................................37
2.4.3. X section ...............................................................................................................38
2.4.3.1. The effect of the force F1 .................................................................................38
2.4.3.2. The effect of the force F2 ................................................................................39
2.4.4. Z Section ..............................................................................................................40
2.4.4.1. The effect of the force F1 ...............................................................................40
2.4.4.2. The effect of the force F2 ...............................................................................41
2.4.5. C section................................................................................................................42
2.4.5.1. The effect of the force F1 .................................................................................42
2.4.5.2 The effect of the force F2 ................................................................................42
2.4.6 circular section ......................................................................................................43
2.4.6.1 The effect of the force F1 ...............................................................................43
2.4.6.2 The effect of the force F2 ...............................................................................44
Chapter 3..................................................................................................................................46
Three dimensional optical strain measurements ......................................................................46
3.1 Brief introduction to the Aaramis system ......................................................................47
3.2 Main hardware and software components .....................................................................48
3.3 Principle of deviation measurements .............................................................................48
3.4 Facet computation..........................................................................................................50
3.5 Steps to carry out a typical measuring procedure ..........................................................51
3.6 Calibration......................................................................................................................52
ix
3.6.1 Calibration objects ..................................................................................................52
3.6.2 When is calibration required? .................................................................................53
3.6.3 Background information about calibration theory..................................................54
3.7 Measuring ......................................................................................................................54
3.7.1 Selecting the correct measuring volume .................................................................54
3.7.2 Preparing a specimen ..............................................................................................54
3.7.3 Spraying a stochastic pattern ..................................................................................55
3.7.4 Computations ..........................................................................................................57
3.8 Facets .............................................................................................................................57
3.8.1 Facet shapes ............................................................................................................57
3.8.2 Rectangular facets...................................................................................................57
3.8.3 Quadrangular facets ................................................................................................58
3.9 Computation masks........................................................................................................58
3.10 Start point.....................................................................................................................60
3.11 Strain computation .......................................................................................................61
3.11.1 Linear strain ..........................................................................................................61
3.11.2 Spline strain ..........................................................................................................61
Chapter 4..................................................................................................................................62
Finite element method..............................................................................................................62
4.1 Physical problems, mathematical models and the finite element solution ....................63
4.2 General steps of the finite element method....................................................................64
4.2.1 Step 1 Discretize and select the element types .......................................................64
4.2.2 Step 2 Selection of a proper interpolation or displacement model .........................66
4.2.3. Step 3 Define the strain displacement and stress strain relationships....................66
4.2.4. Step 4 Derive the element stiffness matrix and equations .....................................66
4.2.4. 1 Direct equilibrium method...............................................................................67
4.2.4.2 Work or energy methods...................................................................................67
4.2.4.3 Methods of weighted residuals .........................................................................67
4.2.5. Step 5 Assemble the element equations to obtain the global or total equations ....68
4.2.6 Step 6 Solve for the unknown degrees of freedom (or generalized displacements)
..........................................................................................................................................68
4.2.7 Step7 Computation of element strains and stresses ................................................69
4.2.8 Step 8 Interpret the results ......................................................................................69
x4.3 Advantages of the finite element method ......................................................................69
4.4 Guidelines on element layout.........................................................................................70
4.4.1 Mesh refinement .....................................................................................................70
4.4.2 Element aspect ratios ..............................................................................................71
4.4.3. Physical interfaces .................................................................................................71
4.4.4. Preferred shapes .....................................................................................................71
4.5 Plane stress and plane strain stiffness equations............................................................72
4.5.1 Basic concepts of plane stress and plane strain ......................................................73
4.5.1.1 Plane stress........................................................................................................73
4.5.1.2 Plane strain........................................................................................................73
4.5.2 Two-dimensional state of stress and strain .............................................................74
4.5.3 Stress/strain relationships........................................................................................75
4.6. Finite element theory ....................................................................................................76
4.6.1. Constant strain triangle ..........................................................................................76
4.6.2 Plate bending element .............................................................................................82
4.6.2.1 Finite elements for plates ..................................................................................82
4.7 Shells..............................................................................................................................86
4.7.1 Shell element...........................................................................................................86
4.7.2 Finite elements for shells ........................................................................................87
Chapter 5..................................................................................................................................90
Finite element modeling of the welded joints using beam elements .......................................90
5.1 Fillet weld ......................................................................................................................91
5.1.1 Finite element results of beam shell coupling of fillet weld in case of bending.....91
5.1.1.1 Rectangular Section ..........................................................................................91
5.1.1.2 I section.............................................................................................................92
5.1.1.3. C- section ........................................................................................................93
5.1.1.4 Circular- section..............................................................................................95
5.1.1.5 Z- section .......................................................................................................96
5.1.1.6 X- section .......................................................................................................97
5.1.2 Finite element results of beam shell coupling of fillet weld in case of twisting.....98
5.2. V welded joint.............................................................................................................100
5. 2.1 Finite element results of beam shell coupling of V-welded joint in case of
bending...........................................................................................................................100
xi
5. 2.2 Finite element results of beam shell coupling of V-welded joint in case of
twisting...........................................................................................................................101
5.4 Comparison ..................................................................................................................102
5.4.1 Comparison between V and fillet welded joints in case of bending.....................102
5.4.2 Comparison between V and fillet welded joints in case of twisting.....................104
5.5 Discussion ....................................................................................................................105
Chapter 6................................................................................................................................106
V-weld modeling using solid and shell elements ..................................................................106
6.1 V-weld modelling using solid elements.......................................................................106
6.1.1 Finite element results of solid element model of V- weld in case of bending......108
6.1.1.1 Rectangular cross-section ...............................................................................108
6.1.1.2 I section...........................................................................................................109
6.1.1.3 C section..........................................................................................................110
6.1.1.4 Circular section ...............................................................................................110
6.1.1.5 Z – section......................................................................................................111
6.1.1.6 X – section .....................................................................................................111
6.1.2 Finite element results of solid element model of V- weld in case of twisting......112
6.1.2.1 Rectangular cross-section ...............................................................................112
6.1.2.2 I-section...........................................................................................................112
6.1.2.3 C-section .........................................................................................................113
6.1.2.4 Circular-section...............................................................................................113
6.1.2.5 Z-section .........................................................................................................114
6.1.2.6 X-section.........................................................................................................115
6.2 V-Weld modelling using shell elements ......................................................................115
6.2.1 Finite Element Results of Shell element model of V- weld in case of bending. ..116
6.2.1.1 Rectangular cross-section ...............................................................................116
6.2.1.2 I cross-section ................................................................................................116
6.2.1.3 C cross-section................................................................................................117
6.2.1.4 Circular cross-section......................................................................................118
6.2.1.5 Z cross-section ...............................................................................................119
6.2.1.6 X cross-section...............................................................................................120
6.2.2 Finite element results of shell element model of V- weld in case of twisting. .....120
6.2.2.1 Rectangular section.........................................................................................120
xii
6.2.2.2 I section..........................................................................................................121
6.2.2.3 C section.........................................................................................................122
6.2.2.4 Circular section ...............................................................................................123
6.2.2.5 Z section........................................................................................................123
6.2.2.6 X section .........................................................................................................124
6.3 Comparison between shell and solid elements ............................................................125
6.3.1 Case (a) Bending...................................................................................................125
6.3.1.1 Rectangular cross-section ...............................................................................125
6.3.1.2 I- section..........................................................................................................125
6.3.1.3 C- section ........................................................................................................126
6.3.1.4 Circular- section..............................................................................................126
6.3.1.5 Z- section ........................................................................................................127
6.3.1.6 X - section.......................................................................................................128
6.3.2 Case (b) Twisting..................................................................................................128
6.3.2.1 Box- section ....................................................................................................128
6.3.2.2 I - section.........................................................................................................129
6.3.2.3 C - section .......................................................................................................129
6.3.2.4 Circular - section.............................................................................................130
6.3.2 .5 Z - section ......................................................................................................130
6.3.2.6 X - section.......................................................................................................131
6.4 Discussion ....................................................................................................................131
Chapter 7................................................................................................................................133
Fillet weld modeling using solid and shell elements .............................................................133
7.1 Fillet weld modelling using solid elements .................................................................133
7.1.1 Finite element results of solid element model of fillet- weld in case of bending. 134
7.1.1.1 Rectangular cross-section ...............................................................................134
7.1.1.2 I-section..........................................................................................................134
7.1.1.3 C-section ........................................................................................................135
7.1.1.4 Circular-section..............................................................................................135
7.1.1.5 Z-section ........................................................................................................136
7.1.1.6 X-section........................................................................................................136
7.1.2 Finite element results of solid element model of fillet- weld in case of twisting. 137
7.1.2.1 Rectangular cross-section ...............................................................................137
xiii
7.1.2.2 I-section...........................................................................................................138
7.1.2.3 C-section .........................................................................................................138
7.1.2.4 Circular -section..............................................................................................139
7.1.2.5 Z –section.......................................................................................................139
7.1.2.6 X –section ......................................................................................................140
7.1.3 Discusion...............................................................................................................140
7.2 Fillet weld modeling using oblique shell elements......................................................140
7.2.1 Finite element results of shell element model of fillet- weld in case of bending. 142
7.2.1.1 Rectangular -section........................................................................................142
7.2.1.2 I-section..........................................................................................................142
7.2.1.3 C-section ........................................................................................................143
7.2.1.4 Circular –section ............................................................................................143
7.2.1.5 Z -section .......................................................................................................144
7.2.1.6 X –section ......................................................................................................144
7.2.2 Finite element results of shell element model of fillet- weld in case of twisting. 145
7.2.2.1 Rectangular cross-section ...............................................................................145
7.2.2.2 I-section...........................................................................................................146
7.2.2.3 C-section .........................................................................................................146
7.2.2.4 circular section ................................................................................................147
7.2.2.5 Z- section ........................................................................................................148
7.2.2.6 X- section........................................................................................................148
7.3 Comparison between solid and shell elements of fillet welds .....................................149
7.3.1 Case(a) Bending....................................................................................................149
7.2.1.1 Rectangular cross-section ...............................................................................149
7.3.1.2 I-section...........................................................................................................149
7.3.1.3 C-section ........................................................................................................150
7.3.1.4 Circular-section..............................................................................................150
7.3.1.5 Z-section .........................................................................................................151
7.3.1.6 X-section.........................................................................................................152
7.3.2 Case (b) Twisting..................................................................................................152
7.3.2.1 Rectangular cross-section ...............................................................................152
7.3.2.2 I-section...........................................................................................................153
7.3.2.2 C-section .........................................................................................................153
xiv
7.3.2.2 Circular-section...............................................................................................154
7.3.2.5 Z-section .........................................................................................................154
7.3.2.6 X-section.........................................................................................................155
7.4 Comparison between V and fillet welds ......................................................................155
7.4.1 Box section............................................................................................................155
7.4.2 I section.................................................................................................................156
7.4.3 C section................................................................................................................156
7.4.4 Circular section .....................................................................................................157
7.4.5 Z section................................................................................................................157
7.4.6 X section ...............................................................................................................158
7.5 Discusion......................................................................................................................158
Chapter 8................................................................................................................................160
Experimental results using digital image correlation (DIC) ..................................................160
8.1 Rectangular section......................................................................................................161
8.2 I-section........................................................................................................................164
8.3 C-section ......................................................................................................................167
8.4 Z-Section......................................................................................................................171
8.5 X-Section .....................................................................................................................175
Chapter 9................................................................................................................................180
Conclusion .............................................................................................................................180
References..............................................................................................................................183
xv
Nomenclature
ୄ the normal stress perpendicular to the critical plane of the throat
∥ the normal stress parallel to the axis of the weld
ୄ the shear stress perpendicular to the weld axis,
∥ the shear stress parallel to the weld axis.
ଵ The force that makes bending
ଶ The force that makes bending and twisting
∥,୊భ direct shear stress due to load ଵ
୶,ూభ bending moment due to load ଵ
∥,୊మ direct shear stress due to load ଶ
୑ ౮,ూమ bending stress due to bending moment due to load ଶ
୲ twisting moment
୑ ౪
shear stress due to the turning moment ୲
ଡ଼ the moment of inertia about neutral bending x-axis
୴ the moment of inertia about neutral bending x-axis for vertical weld
୦ the moment of inertia about neutral bending x-axis for horizontal weld
ଡ଼ the section modulus of rectangular weld section
ୋ the polar moment of inertia about the centroid of the weld group G
l weld length
a throat thickness
୫ mean radius
୊ଵ the resultant stress due to the force ଵ
୊ଶ the resultant stress due to the force ଶ
CST the constant-strain triangular element
DIC digital image correlation
୸ the normal stress
୶୸ , ୷୸ the shear stress
ୣ the element area
୶, ୷ the normal strain
xvi
୶୷ the shear strain
ୣ element stiffness matrix
thickness
୬
ୣ element stiffness matrix due to normal stress
ୱ
ୣ element stiffness matrix due to shear stress
E Young modulus
ⱴ Poisson ratio 
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
1
Chapter 1
General Introduction
1.2. Background
Welded structures made from steel are used in several applications. Some applications
are buildings, bridges, and other structures. Among these structures, Structural hollow
sections, circular (CHS) and rectangular (RHS) are widely used in all kinds of structures
under different types of loading. Many examples in nature show the excellent properties
of the tubular shape with regard to loading in compression, torsion and bending in all
directions. These excellent properties are combined with an attractive shape for
architectural applications. Furthermore, the closed shape without sharp corners reduces
the area to be protected and extends the corrosion protection life. Another aspect which
is especially favorable for circular hollow sections is the lower drag coefficients if
exposed to wind or water forces [1].
1.2 Review of the Literature
The aim of this literature review is to outline the progress of research and development
work in welded joints over the years.
In the recent years, various aspects and interests in the numerical modeling of welding
stresses and distortions have been done by using finite element method.
Butler and Kulak [2] showed that the strength and ductility of fillet welds in shear are
markedly dependent upon the orientation of the weld with respect to the line of action of
the load. Welds placed parallel to the direction of the load have the lowest .Also his
results showed that the increase in strength of the fillet welds tested in this program was
approximately 44% as the angle of load changed from zero degrees (longitudinal weld)
to 90 deg (transverse weld) strength and highest ductility. Also, they presented that
fillet welds loaded in shear do not exhibit any well defined yield point. The load-
deformation response of fillet welds cannot be generally represented as elastic or
elastic-perfectly plastic. Dawe, J. L., and Kulak, G. L.[3] investigated the behavior of
weld groups subjected to shear and out-of-plane bending. The test results were used to
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
2
validate an analysis procedure presented by Butler, L. J., and Kulak [2]. Lesik, D. F.,
and D. J. L. Kennedy [4] established a simplified expression for predicting the ultimate
strength of fillet welds loaded in shear at any angle of loading. Vanam B. C. L.,
Rajyalakshmi M., Inala R.[5] in their research work , they analyzed the static analysis
of an isotropic rectangular plate with various boundary conditions and various types of
load applications. Finite element analysis has been carried out for an isotropic
rectangular plate by considering the master element as a four node quadrilateral
element. Finite element analysis (FEA) has been carried out by developing
programming in mathematical software MATLAB and the results obtained from
MATLAB are giving good agreement with the results obtained by classical method -
exact solutions. Later, for the same structure, analysis has been carried out using finite
element analysis software ANSYS. The results were obtained not only at node points
but also the entire surface of the rectangular plate. Comparison has been done between
the results obtained from FEA numerical analysis, and ANSYS results with classical
method - exact solutions. Their numerical results showed that, the results obtained by
finite element analysis and ANSYS simulation results are in close agreement with the
results obtained from exact solutions from classical method. During their analysis, the
optimal thickness of the plate has been obtained when the plate is subjected to different
loading and boundary conditions. M. F. Ghanameh, D. Thevenet [6] analyzed tubular
elements that used to design offshore platforms which are subjected to combined
loading conditions. These loading conditions generate, in the vicinity of joints, zones of
strong stress concentration. Determination of stresses at hot-spot points by finite
element analysis gives results in good agreement with literature of simple loading,
which validates the modeling approach employed. Pawel Bilous, Tadeusz Lagoda [7]
discussed stress concentrators occurring in welded joints. Methods of fatigue life
evaluation for welded joints have been presented. Moreover, their work contains a
description and methods of determination of the fatigue notch coefficient. Histories of
the fatigue notch coefficient (Kf) were plotted for a double-Vee butt weld. N. S.
Raghavendrantg , M. E. Fourney [8] illustrated the use of the two dimensional finite
element method (FEM) in finding the stress distribution across a butt-welded joint for
two types of loading conditions, tension and bending. In their work, they used two
dimensional photo-elastic analysis (PEA) of a butt weld under plane stress conditions.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
3
S. Krscanski, G. Turkalj [9] presented the concentrations factors for fillet welded CHS-
Plate T-Joint. They modeled the problem with finite element method. The modeling of
the mesh was according to IIW (International Institute of Welding) recommendations
and their results were compared to experimental data from other authors and to a simple
analytical solution. They found stress concentration factors calculated by finite element
model analysis to be higher than those interpolated from experimental data.
P.M.Gedkar, D.V. Bhope [10] presented stress analysis of two pressurized cylindrical
intersection using finite element method. Their results showed that the lower value of
branch pipe thickness are preferable for lower stress values at intersecting junction
provided the stresses in branch pipe are within the safe limits. This reduction of stresses
at the intersection with lesser branch pipe thickness may be due to the less constraining
effect of branch pipe on the run pipe. X.W. Ye, Y.Q. Ni, J.M. Ko [11] described an
experimental study on determining the stress concentration factor (SCF) and its
stochastic characteristics for a typical welded steel bridge T-joint. A full-scale segment
model was fabricated and tested. With the measurement data obtained from strain
gauges deployed on the web and flanges, the hot spot strain at the weld toe is
determined by a linear regression method. The SCF can be calculated as the ratio
between the hot spot strain and the nominal strain which is obtained directly by
measurement. To fully account for the effect of predominant factors on the scatter of
SCF, the experiments are carried out under different moving load conditions.
Shao YB.[12] presented general remarks of the effect of the geometrical parameters on
the stress distribution in the hot spot stress region for tubular T- and K-joints subjected
to brace axial loading. His parametric study has also been carried out to investigate the
effect of three popularly used joint geometrical parameters on the stress distribution.
From parametric study, he found that chord thickness has remarkable effect on the stress
distribution for both T- and K-joints, while brace thickness has no effect on such stress
distribution. Hellier AK, Connolly MP, Dover WD [13] presented parametric study
using finite elements of the stress concentration factors in tubular K joints commonly
found in offshore platforms. Their study covers a comprehensive range of geometric
joint parameters for unbalanced out-of-plane moment loading. They proposed new
equations for the maximum SCFs on the chord and the brace, and they have been found
to provide a good fit to the large finite element database and to meet the proposed
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
4
assessment criteria. Chang E, Dover WD [15] conducted systematic thin shell finite
element (FE) analyses for 330 different tubular X and DT-joints (Double T joint),
typical of those used in offshore structures, subjected to six different modes of loading.
They derived a set of parametric equations to predict the stress distributions along both
chord and brace toes in tubular X and DT-joints under each mode of loading.
Karamanos SA, Romeijn A, Wardenier J [16] showed the prediction of stress
concentration factors (SCFs) of multi-planar welded tubular (CHS) XX-joints in steel
structures. They proposed a set of parametric equations to determine the SCFs for multi-
planar welded CHS XX-connections. In their study, weld profile was modeled using 20-
node solid elements while eight-node shell elements were used to model the chord and
brace. Their research covered the various loading modes.
Karamanos SA, Romeijn A, Wardenier J. [17] repeated the previous study for tubular
gap K joints. Chiew SP, Soh CK, Wu NW. [18] presented parametric stress analysis of
steel multi-planar tubular XX-joints using the finite element method (FEM). Five
different load cases of basic brace and chord axial load, in-plane-bending (IPB) and out-
of-plane-bending (OPB) moments were considered. Through the parametric study, they
established a set of SCF design equations for general end loading. Karamanos SA,
Romeijn A, Wardenier J.[19] developed a methodology for the calculation of the
maximum local stress, referred to as ‘‘hot-spot stress’’, in a multi-planar DT-
joint(Double T joint), with particular emphasis on the effects of bending moments on
the braces and the chord. Gho WM, Gao F. [20] conducted finite element models to
predict the stress concentration factors (SCF) of completely overlapped tubular K-joints
under lap brace axial compression. In their analysis, they found suitable both 8-node
thick shell and 20-node solid elements for modelling the joint. Gao F. [21] investigated
experimentally the stress and strain concentrations of a completely overlapped tubular
joint specimen under lap brace out-of-plane bending (OPB) load. The experimental
results showed that the strain distribution near the weld toe is fairly linear. The
comparison of SCF showed that the existing T/Y-joint parametric equations for
predicting the SCF of completely overlapped tubular joints are inappropriate under OPB
load. Gao F, Shao YB, Gho WM.[22] repeated the previous study for completely
overlapped tubular K(N)-joints under lap in-plane bending loading. Woghiren CO,
Brennan FP [23] proposed a set of parametric equations to predict the values of stress
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
5
concentration factor in multi-planar stiffened tubular KK-joints. The equations describe
the SCF at the different locations as a function of the non-dimensional ratios. The
equations not only allow the rapid optimization of multi-planar joints but also can be
used to quickly identify the location of maximum stress concentration. Lie ST, Lee CK,
Wong SM [24] described a systematic method of modelling the weld thickness of a
tubular Y joint. N Diaye A, Hariri S, Pluvinage G, Azari Z [25] conducted the finite
element method (FEM) on a welded tubular T-joint, in order to analyze stress
distribution in the vicinity of the weld fillet. The finite element method analysis shows
that stresses are very high on the brace member, in the vicinity of the weld, and decrease
gradually, in the direction of the brace extremity. N Diaye A, Hariri S, Pluvinage G,
Azari Z [26] conducted the finite element analysis to predict the location of hot-spot
stresses in a welded tubular T-joint. T. Wanga, O.S. Hopperstada , O.-G. Lademoa, P.K.
Larsena [27] performed finite element analyses to predict the structural behaviour of
welded and un-welded I-section aluminum members subjected to four-point bending.
Their modelling procedure was using shell elements. Their numerical results were
compared with existing experimental data, and, in general, good agreement with the
experimental results was obtained. Seng-Keat Yeoh, Ai-Kah Soh , Chee-Kiong Soh
[28] describes the behaviour of tubular T-Joints subjected to combined loadings. The
stress concentration factor (SCF) obtained for these load cases were compared with
those obtained by other authors. His results showed that the peak SCF under axial load
was slightly higher than those other authors.
1.3 Objective and scope
This thesis focuses on the development of a predictive methodology for fillet welded
structures using beam, shell and solid elements. However, three-dimensional finite
element analysis of complete structural hollow sections can be complex and time-
consuming. Due to the complex nature of the finite element analysis codes, this method
has limited application. It can be used in research area but cannot be widely used by
structural engineers in their real-world projects. Therefore, there is a need to develop a
simplified modeling method that can be implemented by using commonly available
commercial software and easily employed.
The objective is to eventually provide a validated finite element method -based
procedure for large-scale analyses of such structures. Also to establish an appropriate
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
6
level of finite element modeling and in order to determine the optimal quality of
meshing for stress analysis of welded steel structures. Finite element models with
different modeling techniques and meshing with various size and types of elements
were created and analyzed. Experimental tests are performed to validate numerical
simulations in order to investigate the mechanical performance of a series of fillet-
welded connections under combined loading. Figure 1.1 shows pictures of the
investigated specimens. In the numerical study, the connections are modelled using
beam, shell and solid elements in Abaqus [59]. The performance of the connections
under combined loading is further studied by a simple analytical method aiming for
rapid information on the influence of the weld modeling on prediction of the structural
performance. Experiments of many beam joints subjected to combined loading are used
for further assessment of the FE models developed in the present study. The study is
performed with focus on the prediction of the structural strength of the fillet welded
joints.
Figure 1.1 Pictures of the Investigated Specimens
1.4 literature review of experimental research
Strain measurements are very important in mechanical sciences. A strain in any material
can be defined as the coefficient of the change in length and the initial length. Strains
are involved in many important material properties and parameters. Recently, new and
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
7
more complex investigations are requiring strain measurements at any point inside an
area of interest to improve the study of the behavior of materials and structural
components. For this reason, researchers are interested on a strain map over an entire
specimen surface. Some conventional instruments which measure strains (i.e. strain
gage) are not accessible to create strain maps, because it would be very expensive and
not practical [29].
In the past, there are two methods for surveying strains developed in an object which in
subject to external forces. One method is to measure the relative displacement between
two specific points on the surface of an object, and then estimate the strain between
these two points. The disadvantage of this method is that the global strain distribution of
the object cannot be determined directly. The other method is to map meshes on the
surface of an object before deformation occurs, and then survey the displacement of
nodes surrounding these meshes after deformation. Further, the strain distribution of an
object is derived from the displacement field. Contrarily, this technique is a complex
and time-consuming process [30].
Due to the fact that strain maps are needed to perform new investigations, a new
technology was developed to obtain these desired results. This technology is the digital
image correlation, which provides a contour map of strains of an entire specimen
surface subject to mechanical tests.
The digital image correlation (DIC) method is probably one of the most commonly used
methods, and many applications can be found in the literature1 [31–40].
When it is used with a single camera (classical DIC) , the DIC method can only give in-
plane displacement/strain fields on planar objects. By using two cameras (stereovision),
the 3-D displacement fields and the surface strain field of any 3-D object can be
measured. Using stereovision in conjunction with DIC leads to so called digital image
stereo-correlation technique (DISC), also called 3-D DIC [41].
The Developed Digital Image Correlation (DIC) technique is an image identification
method for measuring object deformation. The digital images of an object before and
after deformation that are captured using an optic instrument are subject to correlation
analysis. The corresponding positions recorded on the image are obtained by calculating
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
8
the correlation coefficient of images so that the displacement function and strain
distribution of an object can be inferred. This technique is non-destructive for inspecting
the whole displacement and strain field [29].
Thus, developing a monitor system to inspect the mechanics behaviour of a welded
joints member is an essential and important study. The feasibility of applying DIC
technique to monitor welded joints is tested in this research.
1.5 Fundamental Theory of Digital Image Correlation Techniques
The fundamental theory of the digital-image-correlation technique is to compare the
digital images before and after deformation. The analysis procedure of the DIC method
is shown in figure 1.2. The analysis region is divided into several sub-images. The
grayscale distribution of image is used to identify the relative positions of the same
point on the specimen surface in two images. Correlation analyses of images are carried
out to search the point that has the highest grayscale correlating with the initial position
displacement vector so that the displacement field of specimen can be derived. After
that the strain field can be calculated [29].
1.5.1 Two-dimensional digital image correlation method
The basis of two-dimensional video image correlation for the measurement of surface
displacements is the matching of one point from an image of an object’s surface before
loading (the undeformed image) to a point in an image of the object’s surface taken at a
later time/loading (the deformed image). Assuming a one-to-one correspondence
between the deformations in the image recorded by the camera and the deformations of
the surface of the object, an accurate, point-to-point mapping from the undeformed
image to the deformed image will allow the displacement of the object’s surface to be
measured. Two main requirements must be met for the successful use of DIC-2D. First,
in order to provide features for the matching process, the surface of the object must have
a pattern that produces varying intensities of diffusely reflected light from its surface.
This pattern may be applied to the object or it may occur naturally. Secondly, the
imaging camera must be positioned so that its sensor plane is parallel to the surface of
the planar object, as shown in Figure 1.2 [42].
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
9
Figure 1.2 two-dimensional video image correlations[42].
The imaging process of the camera converts the continuous intensity field reflected
from the surface O( X,Y ) into a discrete field I( X,Y ) of integer intensity levels. In a
CCD camera (a charge-coupled- device which converts light into a digital signal), this
transformation occurs when the light incident on a sensor (commonly known as a pixel)
is integrated over a fixed time period. The rectangular array of sensors in a charge
coupled device (CCD) array converts the continuous intensity pattern into a discrete
array of integer intensity values, I(i,j), where i denotes the row number and j denotes the
column number in the sensor plane. The displacement field for an object is obtained at a
discrete number of locations by choosing subsets from the initial image and searching
throughout the second image to obtain the optimal match. Details of this process are
outlined in the following paragraphs.
The process of deformation in two dimensions is shown schematically in Figure 1.3.
The functions are defined as follows; (a) O(X,Y) denotes the continuous intensity
pattern for the undeformed object, (b) is the continuous intensity pattern for the
deformed object, (c) I(X,Y) is the discretely sampled intensity pattern for the
undeformed object and (d) is the discretely sampled intensity pattern for the
deformed object. It is important to note that a basic tenet of the DIC-2D method is that
points in I(X,Y) and are assumed to be in one-to-one correspondence with
points in O(X,Y) and , respectively. Thus, one can use I(X,Y) and to
determine the displacement field for the object O(X,Y). We should note that obtaining
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
10
accurate estimates for surface deformations using I(X,Y) and requires an
interpolation scheme to reconstruct a continuous intensity function.
Figure 1.3 Analysis Procedure of Digital Image Correlation Method [29].
୧୨ న఩ഥ
୧୨
ଶ
న఩ഥ
ଶ
Proceed to the correlation analysis
Search corresponding sub-
image in the deformed image
Determine the parameters of
corresponding displacement
function
Define the displacement function as
Mesh the undeformed image
into several sub-images
Calculate the displacement
and strain field of the
specimen
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
11
Figure 1.4 The relative position of the sub images before and after deformation.
Figure 1.5. Gray-scale correlation between deformed and undeformed sub-image.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
12
1.5.2 Two-Dimensional Image Correlation: Mathematical Formulation
For the small subset centered at (X,Y) on the undeformed object in Figure 1.4, the
discretely sampled and continuously interpolated intensity pattern at points P and Q,
located at positions (X,Y) and ( X + dX ,Y + dY ) respectively, can be written as
I(P ) = I(X, Y ) ,
I(Q) = I(X + dX, Y + dY ) , (1.1)
Where (dX, dY) represent small distances in the (X,Y) coordinate system. Note that, if
the values for dX and dY are integer pixel values, then no interpolation is required for
the undeformed image. As shown in Figure 1.4, after deformation of an object, points P
and Q are deformed into positions p and q, respectively. Assuming that the intensity
pattern recorded after deformation is related to the undeformed pattern by the object
deformations, and defining {u(X, Y),v(X, Y)} the displacement vector field, we can
write
so that
,
డ௨
డ௑
డ௨
డ௒
డ௩
డ௒
(1.2)
Assuming that the subset is sufficiently small so that the displacement gradients are
nearly constant throughout the region of interest, each subset undergoes uniform strain
resulting in the parallelogram shape for the deformed subset shown in Figure 1.5.
Conceptually, determining for each subset is simply a matter of determining all six
parameters so that the intensity values at each point in the undeformed and deformed
regions match. To obtain these values, there are several measures of intensity pattern
correspondence that could be used, including the following summations for a series of
selected points Qi [42].
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
13
a- magnitude of intensity value difference
௜ ௜௜ (1.3)
b- sum of squares of intensity value differences
௜ ௜
ଶ
௜ (1.4)
c- normalized cross-correlation
∑ 
൫ூ̅(௤೔)ିூ(ொ೔)൯
೔
ඥ(∑ 
ூ̅(௤೔)మ
೔ ) (∑ 
ூ(ொ೔)మ
೔ ) (1.5)
d- cross-correlation
௜ ௜௜ (1.6)
In order to get optimal estimates for all six parameters (u, v, ∂u/∂X, ∂u/∂Y, ∂v/∂X, 
∂v/∂Y), minimization of (a,b) and maximization of (c,d) for each subset should be done.
1.5.3 Three-Dimensional Video Image Correlation
Basic Concepts
Single camera DIC systems are limited to planar specimens that experience little or no
out-of-plane motion. This limitation can be overcome by the use of a second camera
observing the surface from a different direction. Three-dimensional Digital Image
Correlation (DIC-3D) is based on a simple binocular vision model. In principle, the
binocular vision model is similar to human depth perception. By comparing the
locations of corresponding subsets in images of an object’s surface taken by the two
cameras, information about the shape of the object can be obtained. In addition, by
comparing the changes between an initial set of images and a set taken after load is
applied, full-field, three-dimensional displacement can be measured. Both the initial
shape measurement and the displacement measurement require accurate information
about the placement and operating characteristics of the cameras being used. To obtain
this information, a camera calibration process must be developed and used to accurately
determine the model parameters. In the following paragraphs, key aspects of the DIC-
3D method are outlined [42].
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
14
Figure 1-6 A complete three-dimensional measurement system [42].
Figure 1.6 shows a typical DIC-3D system .As already indicated in previous paragraphs,
based on the discretization operation of finite element method (FEM), the undeformed
image is meshed into several sub-images for image analyses. As shown in figure1.4, the
central point prior to deformation is point P; it is shifted to point q after deformation.
The functional relationship is expressed as:
௤ ொ , ௤ ொ (1.7)
Where, ொ ொ and ௤ ௤ are the coordinates of point P before and after
deformation. u(x, y) and v(x, y) are the displacement functions in x- and y-direction
separately. Calculating the grayscale correlation coefficient will lead to find the
corresponding relationship between deformed and undeformed sub-images for
establishing the deformation of sub-image as shown in figure 1.5. The correlation
coefficient is defined as:
∑୥౟ౠ୥෤ഠഡഥ
ට∑୥౟ౠ
మ.∑୥෤ഠഡഥమ (1.8)
Where, ௜௝and పఫഥ are the grayscale of the undeformed sub-image on coordinate (i, j)
and deformed sub-image on coordinate i,j respectively. When the deformed sub-
image corresponds exactly to the undeformed sub-image, the correlation coefficient
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
15
between both sub-images equals 1. Accordingly, the optimal parameters of equation
(1.7) are determined based on the results of correlation analyses. Thus, the displacement
and strain field can be computed individually.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
16
Chapter 2
Analytical Analysis of Welding
2.1 Introduction
Fillet welded joints are widely used in civil engineering construction due to their
relatively high strength and the ease of surface preparation required for such welds. In
many joint configurations used in practice in-plane or out-of-plane eccentricity are
unavoidable, creating more complex stress conditions in the joint than concentrically
loaded joints where the welds are generally subjected to shear in only one direction [43].
A welded joint is a permanent joint which is obtained by the fusion of the edges of the
two parts to be joined together, with or without the application of pressure and a filler
material. The heat required for the fusion of the material may be obtained by burning of
gas (in case of gas welding) or by an electric arc (in case of electric arc welding). The
latter method is extensively used because of greater speed of welding [44]. Figure 2.1
shows the effective length of a fillet weld with a throat thickness a, V and fillet joints.
(a)
Figure 2.1 a) Definition of throat thickness a, b) V joint , c)fillet joint
(b) (c)
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
17
In the analysis procedure, the internal force on the fillet weld is resolved into
components parallel and transverse to the critical plane of the weld throat, see Figure
2.2. A uniform stress distribution is assumed on the critical throat section of the weld,
leading to the following normal stresses and shear stresses:
ୄ the normal stress perpendicular to the critical plane of the throat,
∥ the normal stress parallel to the axis of the weld, it should be neglected when
calculating the design resistance of a fillet weld,
ୄ the shear stress (in the critical plane of the throat) perpendicular to the weld axis,
∥ the shear stress (in the critical plane of the throat) parallel to the weld axis.
Figure 2.2 Stress in critical plane of fillet weld
In this chapter, the analytical analysis of fillet welded beam of different corss-sections
is discussed.
2.2 Assumption
1. For calculating moments of inertia of the weld linear segments, the effective weld
width in the weld plane is the same as the throat length .
2. The transverse shear stress is given by V/A where V is the shear force and A is the
weld throat area.
3. The shear force is carried by the vertical weld [45].
4. The resultant shear stress acting in the plane of the weld throat is the resultant of the
bending and transverse shear stresses [46].
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
18
2.3 Fillet Weld case(a)
Figure 2.3 a-f shows specimens; Rectangular, I, X, Z, C and circular cross sections are
welded to a support plate by means of fillet weld on their one end. The other ends are
loaded by ଵ ଶ .The throat size of weld is 5mm and the dimensions ଵ and
ଶ are 1000mm and 310mm, respectively.
a- Box beam profile
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
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b- I beam profile
c- X beam profile
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
20
d- Z beam profile
e- C beam profile
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
21
f- Circular beam profile
Figure 2.3 Welded cantilever beams subjected to transverse loading.
A The effect of the force F1
The forces acting on the all welded section joints due to the effect of the force ଵ are:
1-Direct shear stress ( ∥,୊భ) due to load ଵ
2-Bending stress due to bending moment ( ୶,ూభ ଵ ଵ ଻ )
B The effect of the force F2
Also for all specimens the forces acting on the welded joint due to the effect of the force
ଶ are:
1- Direct shear stress ( ∥,୊మ) due to load ଶ
2-Bending stress ( ெ ೣ,ಷమ) due to bending moment ( ୶,ూమ ଶ ଵ ଻ )
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
22
3-Shear stress ( ୑ ౪) due to the turning moment ( ×6 )
2.3.1 Rectangular section
The joint, as shown in figure 2.4, is subjected to direct shear stress and the bending
stress.
Figure 2.4 Rectangular section subjected to direct shear force and bending moment for
fillet weld.
To find maximum stress in the weld at support
Solution
2.3.1.1 The effect of F1
The loading involves direct shear plus bending, with the bending moment being
୶ ଵ ଵ
τ∥,୊భ
ଵ
ଷ
The rectangular moment of inertia about neutral bending x-axis consists of contributions
made by the two vertical and two horizontal welds; that is,
ଡ଼ ୴ ୦
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
23
The section modulus of rectangular weld section is ௑
ூ೉
௟ ଶ⁄
ଶ൬
ೌ೗య
భమ
ା௔௟(௟ ଶ⁄ )మ൰
௟ ଶ⁄
ଡ଼
ଶ
ଶ
ହ ଷ
The (tensile) bending stress on the horizontal weld is,
୑ ౮,ూభ ୶,୊ଵ
ଡ଼
଻
ହ ଷ
ଶ
the stress magnitude ୊ଵ is the Pythagorean combination
ிଵ ெ ೣ,ಷభଶ ∥,ிభଶ = ଶ ଶ
2.3.1.2 The effect of F2
The welded joint, as shown in figure 2.5, is subjected to twisting moment ( ୲),shear
force ଶ as well as bending moment ( ୶,୊ଶ).
Figure 2.5 Rectangular section subjected to direct shear force, bending and twisting
moment for fillet weld.
 direct shear stress,
∥,ிమ
ଶ
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
24
The (tensile) bending stress on the horizontal weld is,
ெ ೣ,ಷమ ௫,ிଶ
௪
Note that the magnitudes of the stresses at the upper and lower horizontal weld are the
same signs of the maximum normal stress indicated tension or compression, while the
sign of the maximum shear stress is of no consequence since design is based on the
magnitude [47].
The moment at the support produces secondary shear or torsion stress of the welds, and
this stress is given by the following equation:
୑ ౪ୀ
t max
ୋ
Where, ୫ ୟ୶
And the polar moment of inertia about the centroid of the weld group G is
ୋ ୋଵ ଵ ଵ
ଶ
ୋଶ ଶ ଶ
ଶ ଻ ସ
Where, the subscripts 1 and 2 are for vertical and horizontal welds respectively.
୑ ౪ୀ
6 7
It is clear that °
ிଶ ெ ೣ,ಷమଶ ∥,ிమ ெ ೟ ଶ ெ ೟ ଶ
2.3.2 I section
2.3.2.1 The effect of the force F1
The joint, as shown in figure 2.6, is subjected to direct shear stress and the bending
stress due to the effect of ଵ.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
25
Figure 2.6 A beam of I cross section subjected to direct shear force and bending
moment for fillet weld.
The shear stress due to the effect of the force ଵ is,
∥,ிభ
ଵ
ଷ
The Bending stress due to bending moment ( ௫,ிభୀ ଵ× ଵ)
௫
௟భ௔
య
ଵଶ ଵ
௛భ
మ
ସ
௟మ௔
య
ଵଶ ଶ
௛మ
మ
ସ
௔௟య
ଵଶ
= ଻ ସ
௠ ௔௫ , ௪
ூೣ
௬೘ ೌೣ
ହ ଷ
ெ ௫,ிభ ௫,ிభ
௪
In order to find the resultant stress, the primary and secondary shear stresses are
combined vectorially. The resultant stress due to the force ଵ is:
୊ଵ ୑ ౮,ూభଶ ∥,୊భଶ
2.3.2.2 The effect of the force F2
The joint, as shown in figure 2.7, is subjected to direct shear stress, bending stress and
Shear stress due to the turning moment.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
26
Figure 2.7 A beam of I cross- section subjected to direct shear force , bending and
twisting moment for fillet weld.
The shear stress due to the effect of the force ଶ
∥,ிమ
ଶ
ଷ
The Bending stress due to bending moment ( ௫,ிమ),
ெ ௫,ிమ ௫,ிమ
௪
The Shear stress due to the turning moment is:
ெ ௧
௧ ௠ ௔௫
ீ
Where, ଷ ௠ ௔௫ ீ ଻ ସ
The resultant stress due to the effect of the force ଶ is:
୊మ ୑ ౮,ూమଶ ெ ೟ ଶ ∥,୊మ ெ ೟ ଶ
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
27
2.3.3 X section
2.3.3.1 The effect of the force F1
The joint, as shown in figure 2.8, is subjected to direct shear stress and the bending
stress.
Figure 2.8 A beam of X cross section subjected to direct shear force and bending
moment for fillet weld.
The shear stress due to the effect of the force ଵ is , ∥,ிభ
ிభ
ଶ௔௟
The Bending stress due to bending moment ( ௫,ிభ) is:
୑ ୶,୊భ ୶,୊భ
୵
Where, l=160mm, ୫ ୟ୶ , ୶ ଺ ସ, ୵ ସ ଷ
In order to find the resultant stress, the primary and secondary shear stresses are
combined vectorially. The resultant stress due to the force ଵ is
୊ଵ ୑ ౮,ూభଶ ∥,୊భଶ
2.3.3.2 The effect of the force F2
The joint, as shown in figure 2.9, is subjected to direct shear stress , bending stress and
Shear stress due to the turning moment.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
28
Figure 2.9 A beam of X cross- section subjected to direct shear force, bending and
twisting moment for fillet weld.
The shear stress due to the effect of the force ଶ ( ∥,ிమ) and the bending stress ( ெ ௫,ிమ)
due to bending moment ( ௫,ிమ) are and respectively.
The Shear stress due to the turning moment ( ୲)
ெ ௧
௧ ௠ ௔௫
ீ
Where, ௠ ௔௫ , ீ ଺ ସ
We note that the maximum stress is at point A. The resultant stress due to the effect of
the force ଶ
୊మ ୑ ౮,ూమଶ ெ ೟ଶ ∥,୊మଶ
2.3.4 Z Section
2.3.4.1 The effect of the force F1
The joint, as shown in figure 2.10, is subjected to direct shear stress and the bending
stress.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
29
Figure 2.10 A beam of Z cross-section subjected to direct shear force and bending
moment for fillet weld.
The shear stress due to the effect of the force ଵ ,
∥,ிభ
ଵ
ଵ
The section is antisymmetrical with its centroid at the mid-point of the vertical web.
Therefore, the direct stress distribution is given by either of the equation below [48],
where the point of interest has coordinates (0,80mm)
ெ ௫,ிభ ௫,ிభ ௬௬ ௫௬
௫௫ ௬௬ ௫௬
ଶ
Where, ଵ , ୫ ୟ୶ , ୶ ଻ ସ ୷ ଺ ସ
In order to find the resultant stress, the primary and secondary shear stresses are
combined vectorially. The resultant stress due to the force ଵ is
୊ଵ ୑ ౮,ూభଶ ∥,୊భଶ
2.3.4.2 The effect of the force F2
The joint, as shown in figure 2.11, is subjected to direct shear stress, bending stress and
Shear stress due to the turning moment.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
30
Figure 2.11 A beam of Z cross- section subjected to direct shear force , bending and
twisting moment for fillet weld.
The shear stress due to the effect of the force ଶ, ∥,ிమ
ிమ
ଶ௔௟భ
The Bending stress due to bending moment ( ௫,ிమ), ெ ௫,ிమ ெ ೣ,ಷమ௓ೢ
The Shear stress due to the turning moment ( ୲), ெ ೟
ெ ೟×௥೘ ೌೣ
௝ಸ
Where, ଵ , ௠ ௔௫
, ௫ ଻ ସ ௬ ଺ ସ ீ ଻ ସ °
The resultant stress due to the effect of the force ଶ ,
୊మ ୑ ౮,ూమଶ ெ ೟ ଶ ∥,୊మ ெ ೟ ଶ
2.3.5 C section
2.3.5.1 The effect of the force F1
The joint, as shown in figure 2.12, is subjected to direct shear stress and the bending
stress.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
31
Figure 2.12 A beam of C cross-section subjected to direct shear force and bending
moment for fillet weld.
The shear stress due to the effect of the force ଵ, ∥,ிమ
୊మ
ଶୟ୪భ
The Bending stress due to bending moment ( ௫,ிభ), ெ ௫,ிభ ெ ೣ,ಷభ௓ೢ
Where, ଵ , ௠ ௔௫ , ௫ ଻ ସ
In order to find the resultant stress, the primary and secondary shear stresses are
combined vectorially. The resultant stress due to the force ଵ is
୊ଵ ୑ ౮,ూభଶ ∥,୊భଶ
2.3.5.2 The effect of the force F2
The joint, as shown in figure 2.13, is subjected to direct shear stress ,bending stress and
Shear stress due to the turning moment.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
32
Figure 2.13 A beam of C cross- section subjected to direct shear force , bending and
twisting moment for fillet weld.
The shear stress due to the effect of the force ଶ, , ∥,ிమ
୊మ
ଶୟ୪భ
The Bending stress due to bending moment , ெ ௫,ிమ ெ ೣ,ಷమ௓ೢ
The Shear stress due to the turning moment , ெ ೟
ெ ೟×௥೘ ೌೣ
௝ಸ
Where, ଵ , ௠ ௔௫ , ௫ ଻ ସ ீ ଻ ସ°
The resultant stress due to the effect of the force ଶ is
୊మ ୑ ౮,ూమଶ ெ ೟ ଶ ∥,୊మ ெ ೟ ଶ
2.3.6 circular section
2.3.6.1 The effect of the force F1
The joint, as shown in figure 2.14, are subjected to direct shear stress and the bending
stress.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
33
Figure 2.14 A beam of circular cross-section subjected to direct shear force and
bending moment for fillet weld.
The shear stress due to the effect of the force ଵ, ∥,ிభ
ிభ
ଶగ௔ோ೘
The Bending stress due to bending moment , ெ ௫,ிభ ெ ೣ,ಷభ௓ೢ
Where, ௠ , ௠ ௔௫ , ௫ ଺ ସ ௪
ହ ସ
In order to find the resultant stress, the primary and secondary shear stresses are
combined vectorially. The resultant stress due to the force ଵ is
୊ଵ ୑ ౮,ూభଶ ∥,୊భଶ=98.7MPa
2.3.6.2 The effect of the force F2
The joint, as shown in figure 2.15., are subjected to direct shear stress ,bending stress
and Shear stress due to the turning moment.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
34
Figure 2.15. A beam of circular cross- section subjected to direct shear force , bending
and twisting moment for fillet weld.
The shear stress due to the effect of the force ଶ, ∥,ிమ
ிమ
ଶగ௔ோ೘
The Bending stress due to bending moment, ெ ௫,ிమ ெ ೣ,ಷమ௓ೢ
The Shear stress due to the turning moment , ெ ೟
ெ ೟×௥೘ ೌೣ
௝ಸ
Where, ௠ , ௠ ௔௫ , ௫ ଺ ସ ீ
଻ ସ
௪
ହ ଷ °
The resultant stress due to the effect of the force ଶ
୊మ ୑ ౮,ూమଶ ெ ೟ ଶ ∥,୊మ ெ ೟ ଶ=100.3MPa
2.4. V Weld Case (b)
In this case, the weld thickness is equal to the beam thickness (it’s similar to no weld).
These specimens (Rectangular, I, X, Z, C and circular cross section) are welded to a
support plate by means of bevel weld on their one end. The other ends are loaded by
ଵ ଶ , and the dimensions ଵ and ଶ are 1000mm and 310mm respectively.
A-The effect of the force F1
The forces acting on the all welded section joints due to the effect of the force ଵ are:
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
35
1-Direct shear stress ( ∥,୊భ) due to load ଵ
2-Bending stress due to bending moment ( ୶,ూభ ଵ ଵ ଻ )
B- The effect of the force F2
Also for all specimens the forces acting on the welded joint due to the effect of the force
ଶ are:
1- Direct shear stress ( ∥,୊మ) due to load ଶ
2-Bending stress ( ெ ೣ,ಷమ) due to bending moment ( ୶,ూమ ଶ ଵ ଻ )
3-Shear stress ( ୑ ౪) due to the turning moment ( ×6 )
2.4.1. Rectangular section
The data for rectangular section are : , ௠ ௔௫ , ௠ ௔௫
௫
଻ ସ
ீ
଻ ସ
௪
ହ ଷ
The joint, as shown in figure 2.16, is subjected to direct shear stress and the bending
stress.
Figure 2.16 . A beam of rectangular cross-section subjected to direct shear force and
bending moment for V weld.
2.4.1.1. The effect of F1
The direct shear stress is, ∥,୊భ
ிభ
ଶ.௔.௟
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
36
The (tensile) bending stress on the horizontal weld is, ୑ ౮,ూభ ୑ ౮,ూభ୞౔ ଶ
The stress magnitude ୊ଵ is the Pythagorean combination
ிଵ ெ ೣ,ಷభଶ ∥,ிభଶ
2.4.1.2. The effect of F2
The welded joint, as shown in figure 2.17, is subjected to twisting moment ( ୲),shear
force ଶ as well as bending moment ( ୶,୊ଶ).
Figure 2.17. A beam of rectangular cross- section subjected to direct shear force ,
bending and twisting moment for V weld.
 direct shear stress is, ∥,୊మ
ிమ
ଶ௔.௟
The (tensile) bending stress on the horizontal weld is, ெ ೣ,ಷమ ெ ೣ,ಷమ௓ೢ
The moment at the support produces secondary shear or torsion stress of the welds, and
this stress is given by the equation
୑ ౪ୀ
t max
ୋ
It is clear that °
ிଶ ெ ೣ,ಷమଶ ∥,ிమ ெ ೟ ଶ ெ ೟ ଶ
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
37
2.4.2 I section
The data for I section are : , ௠ ௔௫ , ௠ ௔௫ ௫
଺ ସ
ீ
଺ ସ
௪
ହ ଷ
2.4.2.1 The effect of the force F1
The joint, as shown in figure 2.18., is subjected to direct shear stress and the bending
stress. Due to the effect of ଵ
Figure 2.18. A beam of I cross-section subjected to direct shear force and bending
moment for V weld.
The shear stress due to the effect of the force ଵ is , ∥,ிభ
ிభ
ଶ௔௟య
The Bending stress due to bending moment is, ெ ௫,ிభ ெ ೣ,ಷభ௓ೢ
In order to find the resultant stress, the primary and secondary shear stresses are
combined vectorially. The resultant stress due to the force ଵ is
୊ଵ ୑ ౮,ూభଶ ∥,୊భଶ
2.4.2.2. The effect of the force F2
The joint, as shown in figure 2.19, is subjected to direct shear stress ,bending stress and
Shear stress due to the turning moment.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
38
Figure 2.19. A beam of I cross- section subjected to direct shear force , bending and
twisting moment for V weld.
The shear stress due to the effect of the force ଶ is, ∥,ிమ
ிమ
ଶ௔௟య
The Bending stress due to bending moment ( ௫,ிమ), ெ ௫,ிమ ெ ೣ,ಷమ௓ೢ
The Shear stress due to the turning moment is, ெ ௧
ெ ೟×௥೘ ೌೣ
௝ಸ
Where,
The resultant stress due to the effect of the force ଶ
୊మ ୑ ౮,ూమଶ ெ ೟ ଶ ∥,୊మ ெ ೟ ଶ
2.4.3. X section
The data for X section are : , ௠ ௔௫ , ௠ ௔௫ ௫
଺ ସ
ீ
଺ ସ
௪
ସ ଷ
2.4.3.1. The effect of the force F1
The joint, as shown in figure 2.20, is subjected to direct shear stress and the bending
stress.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
39
Figure 2.20 . A beam of X cross-section subjected to direct shear force and bending
moment for V weld.
The shear stress due to the effect of the force ଵ is , ∥,ிభ
ிభ
ଶ௔௟
The Bending stress due to bending moment , ெ ௫,ிభ ெ ೣ,ಷభ௓ೢ
In order to find the resultant stress, the primary and secondary shear stresses are
combined vectorially. The resultant stress due to the force ଵ is
୊ଵ ୑ ౮,ూభଶ ∥,୊భଶ
2.4.3.2. The effect of the force F2
The joint, as shown in figure 2.21, is subjected to direct shear stress ,bending stress and
Shear stress due to the turning moment.
Figure 2.21. A beam of X cross- section subjected to direct shear force , bending and
twisting moment for V weld.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
40
The direct shear stress due to the effect of the force ଶ ( ∥,ிమ) and the bending stress
( ெ ௫,ிమ) due to bending moment ( ௫,ிమ) are and ,respectively.
The Shear stress due to the turning moment ( ୲) is, ெ ௧
ெ ೟×௥೘ ೌೣ
௝ಸ
We note that the maximum stress is at point A.The resultant stress due to the effect of
the force ଶ is, ୊మ ୑ ౮,ూమଶ ெ ೟ଶ ∥,୊మଶ
2.4.4. Z Section
The following data area for Z section are : , ௠ ௔௫ , ௠ ௔௫
௫
଺ ସ
௬
଺ ସ
௫௬
଺ ସ
ீ
଺ ସ
2.4.4.1. The effect of the force F1
The joint, as shown in figure 2.22, is subjected to direct shear stress and the bending
stress.
Figure 2.22 . A beam of Z cross-section subjected to direct shear force and bending
moment for V weld.
The shear stress due to the effect of the force ଵ is , ∥,ிభ
ிభ
ଶ௔௟భ
The section is antisymmetrical with its centroid at the mid-point of the vertical web.
Therefore, the direct stress distribution is given by the equation below [48], where the
point of interest has coordinates (0,80mm)
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
41
ெ ௫,ிభ ௫,ிభ ௬௬ ௫௬
௫௫ ௬௬ ௫௬
ଶ
In order to find the resultant stress, the primary and secondary shear stresses are
combined vectorially. The resultant stress due to the force ଵ is
୊ଵ ୑ ౮,ూభଶ ∥,୊భଶ
2.4.4.2. The effect of the force F2
The joint, as shown in figure 2.23, is subjected to direct shear stress ,bending stress and
Shear stress due to the turning moment.
Figure 2.23. A beam of Z cross- section subjected to direct shear force , bending and
twisting moment for V weld.
The shear stress due to the effect of the force ଶ, ∥,ிమ
ிమ
ଶ௔௟భ
The Bending stress due to bending moment ( ௫,ிమ), ெ ௫,ிమ
The Shear stress due to the turning moment ( ୲), ெ ೟
ெ ೟×௥೘ ೌೣ
௝ಸ
Where °
The resultant stress due to the effect of the force ଶ ,
୊మ ୑ ౮,ూమଶ ெ ೟ ଶ ∥,୊మ ெ ೟ ଶ
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
42
2.4.5. C section
The following data area for C section are : , ௠ ௔௫ , ௠ ௔௫
௫
଺ ସ
ீ
଺ ସ
௪
ହ ସ
2.4.5.1. The effect of the force F1
The joint, as shown in figure 2.24, is subjected to direct shear stress and the bending
stress.
Figure 2.24 . A beam of C cross-section subjected to direct shear force and bending
moment for V weld.
The shear stress due to the effect of the force ଵ, ∥,ிమ
୊మ
ଶୟ୪భ
The Bending stress due to bending moment ( ௫,ிభ), ெ ௫,ிభ ெ ೣ,ಷభ௓ೢ
In order to find the resultant stress, the primary and secondary shear stresses are
combined vectorially. The resultant stress due to the force ଵ is
୊ଵ ୑ ౮,ూభଶ ∥,୊భଶ
2.4.5.2 The effect of the force F2
The joint, as shown in figure 2.25, is subjected to direct shear stress ,bending stress and
Shear stress due to the turning moment.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
43
Figure 2.25. A beam of C cross- section subjected to direct shear force , bending and
twisting moment for V weld.
The shear stress due to the effect of the force ଶ, , ∥,ிమ
୊మ
ଶୟ୪భ
The Bending stress due to bending moment , ெ ௫,ிమ
The Shear stress due to the turning moment , ெ ೟
ெ ೟×௥೘ ೌೣ
௝ಸ
Where, °
The resultant stress due to the effect of the force ଶ is
୊మ ୑ ౮,ూమଶ ெ ೟ ଶ ∥,୊మ ெ ೟ ଶ
2.4.6 circular section
The following data area for circular section section are : ௠ , ௠ ௔௫
௫
଺ ସ
ீ
଻ ସ
௪
ସ ସ
2.4.6.1 The effect of the force F1
The joint, as shown in figure 2.26, are subjected to direct shear stress and the bending
stress.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
44
Figure 2.26 . A beam of circular cross-section subjected to direct shear force and
bending moment for V weld.
The shear stress due to the effect of the force ଵ is , ∥,ிభ
ிభ
ଶగ௔ோ೘
The Bending stress due to bending moment is , ெ ௫,ிభ ெ ೣ,ಷభ௓ೢ
In order to find the resultant stress, the primary and secondary shear stresses are
combined vectorially. The resultant stress due to the force ଵ is
୊ଵ ୑ ౮,ూభଶ ∥,୊భଶ=123.3MPa
2.4.6.2 The effect of the force F2
The joint, as shown in figure 2.27, are subjected to direct shear stress, bending stress
and Shear stress due to the turning moment.
Figure 2.27. A beam of circular cross- section subjected to direct shear force , bending
and twisting moment for V weld.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
45
The shear stress due to the effect of the force ଶ, ∥,ிమ
ிమ
ଶగ௔ோ೘
The Bending stress due to bending moment, ெ ௫,ிమ ெ ೣ,ಷమ௓ೢ
The Shear stress due to the turning moment , ெ ೟
ெ ೟×௥೘ ೌೣ
௝ಸ
Where, °
The resultant stress due to the effect of the force ଶ
୊మ ୑ ౮,ూమଶ ெ ೟ ଶ ∥,୊మ ெ ೟ ଶ=124.8MPa
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
46
Chapter 3
Three dimensional optical strain measurements
The measurement and analysis of the strain distribution is very important in civil and
mechanical engineering, especially in the stress concentration problem in plastic and
fracture mechanics. The traditional strain measurement technique can be divided into
two types. Type 1: The relative displacement of two points will be measured. Then we
can calculate the average strain between these two points by means of this relative
displacement. The disadvantage of this method is that the surface strain distribution
cannot be obtained. Type 2: A grid will be established on a sample surface before the
deformation. We can measure the displacements of the grid after the deformation and
analyze the strain distribution. The disadvantage of this method lies in its time
consuming at analyzing the data. There are also precise measurement instrument, e.g.
Raman-Spectroscopy and electrical speckle pattern interferometer, which can be used to
measure the surface strain filed without contact. But due to the expensive cost and the
stability problem those methods cannot be widely used in the scientific research and
engineering. A precise measurement of the surface strain distribution can be achieved
with the help of the low-cost digital-image-correlation technique. It is an efficient tool
to verify the result of the numerical calculation.
The fundamental theory of the digital-image-correlation technique is to compare the
digital images before and after the deformation. Therefore, the “structural fleck” will be
established on the sample surface before the deformation. The grayscale distribution
offers an important character at recognizing the corresponding facets between before
and after deformation. The relative position before and after the deformation can be
obtained with the help of the grayscale distribution comparison. Then we can calculate
the displacement vector and the strain in x- and y-direction, shear strain and von Mises
strain[49].
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
47
3.1 Brief introduction to the Aaramis system
Aaramis is a non-contact optical 3D deformation measuring system. Aaramis analyzes,
calculates and documents deformations. The graphical representation of the measuring
results provides an optimum understanding of the behavior of the measuring object.
Aaramis recognizes the surface structure of the measuring object in digital camera
images and allocates coordinates to the image pixels. The first image in the measuring
project represents the undeformed state of the object. After or during the deformation of
the measuring object, further images are recorded. Then, Aaramis compares the digital
images and calculates the displacement and deformation of the object characteristics. If
the measuring object has only a few object characteristics, like it is the case with
homogeneous surfaces, you need to prepare such surfaces by means of suitable
methods, for example apply a stochastic color spray pattern, Stochastic pattern, see
figure 3.1 [50].
Figure 3.1 Stochastic pattern.
Aaramis is particularly suitable for three-dimensional deformation measurements under
static and dynamic load in order to analyze deformations and strain of real components.
One of the fields of application is verification of FE models
New products are designed and optimized with numerical simulation methods. Material
parameters and components deformation behavior have a significant influence on the
accuracy of simulation calculations and their reliability. Therefore Aramis is used for
the validation of numerical simulations by calculating the differences between
experimental measurements ad FE data.
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3.2 Main hardware and software components
• Sensor with two cameras (only for 3D setup).
• Stand for secure and steady hold of the sensor.
• Sensor controller for power supply of the cameras and to control image recording.
• High-performance PC system.
• Aaramis application software v6.1 and GOM Linux 10 system software or higher.
3.3 Principle of deviation measurements
In general, the Aaramis sensor unit is operated on a stand in order to optimally position
the sensor with respect to the specimen. For a 3D measurement setup, two cameras are
used (stereo setup) that are calibrated prior to measuring. The specimen needs to be
within the resulting measuring volume (calibrated 3D space). After creating the
measuring project in the software, images are recorded (monochrome, right camera, left
camera) in various load stages of the specimen. After the area to be evaluated is defined
(computation mask) and a start point is determined, the measuring project is computed.
During computation, Aaramis observes the deformation of the specimen through the
images by means of various square or rectangular image details (facets). The following
figure 3.2 shows 15 x 15 pixel facets with a 2 pixel overlapping area in stage 0.
Figure 3.2. 5x15 facets with 2 pixels overlapping
You may adjust the facet size in pixels in the software. In the different load stages, the
facets are identified and followed by means of the individual gray level structures. In the
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
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following, we show a pair of facets (15 x 15 pixels) of the right and left camera, the
gray values of which were observed through six deformation stages (Stage 0 to Stage 5).
Stage 0 is the undeformed reference state and Stage 5 is the final deformation state. In
these images, the white dashed line visualizes the undeformed state in order to make
clear the factual relation between facets and deformation.
Figure 3.3. Relation between facets and deformation.
The system determines the 2D coordinates of the facets from the corner points of the
green facets and the resulting centers. Using photogrammetric methods, the 2D
coordinates of a facet, observed from the left camera and the 2D coordinates of the same
facet, observed from the right camera, lead to a common 3D coordinate.
After successful computation, the data may undergo a postprocessing procedure in order
to reduce measuring noise or suppress other local perturbations.
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The measuring result is now available as 3D view. All further result representations like
statistical data, sections, reports, etc. are derived thereof.
3.4 Facet computation
Using a single facet (enlarged representation) as example, we explain the computation
principle for a 3D point through several deformation stages.
Figure 3.4.Computation principle
• The facet computation requires start points in all stages.
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Figure 3.5. Start point definition (automatically or manually)
• The size of the green facet results from the facet field definition when creating the
project.
• Computation is started in Stage 0, Left Image.
• Due to the start point definition, the software in principle knows the position of the
facets and their adjacent facets in the 2D image. By identifying the individual spray
pattern of a facet in the right and left image, the facet quadrangle is optimized. From the
resulting 2D image coordinates of the facet (central point of the facet) in the right and
left camera image, the software now calculates the 3D position of the facet.
• After the computation of the 3D positions of one stage, the software automatically
continues with the next stage. Here as well, in principle the position of the facet is
known because of the start point definition. Now, computation of the 3D position of the
facet starts again.
• The strain computation results from the displacements of the 3D points.
3.5 Steps to carry out a typical measuring procedure
• Determination of measuring volume and preparation of the specimen. Prior to start
measuring, make sure the measuring object fits into the selected measuring volume in
all its deformation stages.
• Specimen preparation if the specimen has only little surface structure.
• Calibration of the measuring volume in case of a 3D measuring project.
• Creating a new project (2D or 3D) in the software and defining the project parameters
(Facets, Strain, Keywords, Stage Parameter...).
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• Adjusting the image recording mode, e.g. simple or fast measurement.
• Recording images during measurement (e. g. tensile test).
• Defining the computation mask in the measuring images, so that only deviation
relevant areas of the specimen will be computed.
• Defining a start point for the computation process.
• Compute project.
• Selecting the result representation (Major Strain, Minor Strain, ...).
• Transformation of the project into a defined coordinate system.
• Data post processing to suppress unwanted measuring noise, interpolate missing 3D
points, emphasize local effects .
• Defining analysis elements, sections or stage points for evaluation.
• Documentation of the results (creating reports, export).
3.6 Calibration
Calibration is a measuring process during which the measuring system with the help of
calibration objects is adjusted, such that the dimensional consistency of the measuring
system is ensured.
3.6.1 Calibration objects
For the Aaramis measuring system, two different calibration objects are used,
calibration panels for small measuring volumes and calibration crosses for large
measuring volumes. Calibration panels are also available cube-shaped in order to
calibrate particularly small measuring volumes (10x8 to 66x44 mm). They are available
in different sizes. Depending on the type of the system and the size they may differ
slightly in their appearance. Calibration objects are equipped with so called reference
points.
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Figure 3.6. Calibration panel and calibration cross
The calibration object contains the scale bar information. Depending on the type, a
calibration panel has one or two scale bars. A scale bar is the specified distance between
two specific points. A calibration cross always has two scale bars (one scale bar on each
cross section) [50].
3.6.2 When is calibration required?
• Before starting measurements for the first time, the respective measuring volume
needs to be calibrated.
• Also, if the adjustment of the camera lenses or the position of the cameras with respect
to each other is changed (e.g. when changing the camera support to a different length),
the system requires calibration again.
• If the Aaramis system shows many yellow facets in the camera images after
computation, the system might be decalibrated.
Figure 3.7. Camera image of a stage in a decalibrated state.
The system automatically creates yellow image areas (yellow facets) if the intersection
error of a 3D point (facet) is larger than 0.3 pixels (factory-adjusted setting). The
average intersection error of all 3D points should not be larger than 0.1 pixels.
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3.6.3 Background information about calibration theory
During calibration, the sensor configuration is determined. This means that the distance
of the cameras and the orientation of the cameras to each other are determined. In
addition, the image characteristics of the lenses are determined (e.g. focus, lens
distortions). Based on these settings, the software calculates from the reference points of
the calibration object in the 2D camera image their 3D coordinates. The calculated 3D
coordinates are then “calculated back” again into the 2D camera images. For the
position of the reference points, this results in the so-called reference point deviation
(intersection error)[50].
3.7 Measuring
3.7.1 Selecting the correct measuring volume
The measuring volume depends on the size of the measuring object or on the size of the
area you would like to analyze. Choose a measuring volume in which the measuring
object or the measuring area fills the entire image as best as possible. Ensure that the
measuring object or the measuring area remains within the measuring volume in all
deformation stages! If it is necessary to adapt the measuring volume, you need to read
just and calibrate the sensor again.
3.7.2 Preparing a specimen
The surface structure is important for carrying out a measurement. The specimen’s
surface must meet the following requirements:
• The surface of the measuring object must have a pattern in order to clearly allocate the
pixels in the camera images (facets). Thus, a pixel area in the reference image can be
allocated to the corresponding pixel area in the target image.
• The surface pattern must be able to follow the deformation of the specimen. The
surface pattern must not break early.
• The optimum specimen surface is smooth. Highly structured surfaces may cause
problems in facet identification and 3D point computation.
• The pattern on the object should show a good contrast because otherwise such an
allocation (matching) does not work.
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• The surface pattern must be dull. Reflections cause a bad contrast and brightness
differences between the right and left camera which prevent the facet computation in the
areas of reflection.
• On one hand, the size of the surface characteristics should be small enough to allow a
fine raster of calculation facets during evaluation. On the other hand, the pattern should
be large enough to be completely resolved by the camera.
• Best suitable are stochastic patterns which are adapted to the measuring volume,
camera resolution and facet size. In addition, for calculation, it is advantageous if the
patterns do not have larger areas of constant brightness, e.g. large spots. Structures with
changing gray values as they occur with random patterns are more appropriate. The left
figure shows a pattern that is not really suitable. The right figure shows a good and
clearly better pattern [50].
Figure 3.8 a)Unsuitable low
contrast stochastic pattern
b) High contrast stochastic pattern
with disturbing large sports
c) Good high contrast stochastic
pattern
3.7.3 Spraying a stochastic pattern
Before you start spraying, make sure the surface of the specimen is free of grease and
oil. In a first step, you need to apply a white and dull base layer. In a second step, spray
a black stochastic pattern. In order to check if your spray pattern is suitable for your
measuring volume, specific reference patterns are available for the measuring volumes.
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Figure 3.9. Spray pattern reference
To compare the spray patterns, use your computer screen and the spray pattern reference
for the volume 100x80 mm, regardless of the real measuring area. If you do not have an
original spray pattern reference, you may also use this page of the user manual [50].
• Position your specimen with the spray pattern in the measuring distance in front of the
camera.
• On your computer screen, adjust the window of the camera image to the same
dimensions as the spray pattern reference 100x80 mm (see figure 3.10).
Figure 3.10 Window set to 100mm
• Compare the pattern of the camera image with the spray pattern reference for volume
100x80mm.
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3.7.4 Computations
After having recorded an image series, explains the strain computation, and shows
corresponding 3D representations.
3.8 Facets
Aramis observes the deformation of the specimen through the images by means of
various square or rectangular facets. Figure 3.11 shows 15x15 pixel facets with a facet
step of 13 pixels (corresponds to a 2 pixel overlapping area, default setting).
Figure 3.11.A 15x15 pixel facets with a facet step of 13 pixels
3.8.1 Facet shapes
The facet shape (square, rectangular, quadrangular) influences the computability of the
measuring project. Square and rectangular facets in the reference stage are always
aligned according to the x-y orientation of the 2D image.
3.8.2 Rectangular facets
For strain measurements where the specimen is subject to high levels of strain, use
rectangular facets according to the figure below in order to get evaluable facet fields.
Figure 3.12 a- Undeformed state b-Deformed state with considerable material
necking
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3.8.3 Quadrangular facets
If, in case of quadrangular specimens you would like to create valid facets up to the
edge, you may create a facet field manually which follows the specimen’s geometry.
The individual facets result within this field. The x-y orientation is based on this field as
well. As computation here is only done within the facet field, it is not necessary to mask
the specimen.
Figure 3.13. a- Undeformed state with
quadrangular facet field
b-individual facets in a quadrangular facet
field
3.9 Computation masks
Computation masks allow the software to carry out facet computations in defined areas
of the 2D camera images only. To define computation masks, the software provides
extensive tools.
Figure 3.14. a-Computation mask
during definition
b-Finished computation mask (only the green
area will be computed)
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Only areas on a specimen that are relevant for deformation shall be calculated. Thus, for
example, fixtures, backgrounds, specimen edges and contour jumps are not be included
in the computation.
As the 3D computation of the measuring points is based on facets that need to be seen
from the right and left camera with the individual facet pattern, a correct 3D
computation and strain computation is not possible for specimen edges and contour
jumps in specimens [50].
Figure 3.15 shows measuring images of tensile test specimen with circular hole.
Figure 3.15. Tensile test specimen with circular hole
The images show a tensile test specimen with a hole in stage 1 of a strain measuring
project seen from the left and right camera. While the surface pattern almost looks the
same, considerable differences result at the hole edges due to the different camera
locations.
Figure 3.16 strain of red framed area cannot be computed.
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The red framed area shows strain which cannot be computed correctly because of the
mentioned problems.
Figure 3.17.The image shows a tensile test specimen with hole and a computation
mask.
a) In the blue area, facets will not be
computed
b) Excluding the facet problems in
the edge region.
3.10 Start point
A start point is used to calculate facet. The start point creation may fail in the following
stages because the facet content shows only little stochastic pattern structure figure.
Also it can fall back on a well perceptible stochastic pattern structure figure.
Figure 3.18 Example of a Poor Start Point
a b
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3.11 Strain computation
For strain computation, Aaramis distinguishes between two methods, linear strain and
spline strain computation. In Aaramis measuring projects normally only, linear strain
computation is used. For specimen of small curvature radii, the spline strain
computation method can be used.
3.11.1 Linear strain
Principle of a specimen in the undeformed (left) and deformed (right) state is shown in
figure 3.19. The strain is computed in connection with the surrounding measuring points
which are directly derived from the facets.
Figure 3.19. A specimen in the undeformed (left) and deformed (right) state
3.11.2 Spline strain
Figure 3.20 shows a specimen in the undeformed (left) and deformed (right) state. The
black points are measuring points which are directly derived from the facets. The white
points were interpolated from the black points using the spline function. Here, strain
computation also considers the interpolated (white) points [50].
Figure 3.20 A specimen in the undeformed (left) and deformed (right) state.
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Chapter 4
Finite element method
The finite element method is a numerical method for solving problems of engineering
and mathematical physics. Typical problem areas of interest in engineering and
mathematical physics that are solvable by use of the finite element method include
structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic
potential [51].
The finite element method became popular with the advancements in digital computers
since they allow engineers to solve large systems of equations quickly and efficiently.
The finite element method is a useful tool for the solution of many types of engineering
problems such as the analysis of the structures, heat transfer and fluid flow. The method
is also used in the design of air frames, ships, electric motors, heat engines and
spacecraft.
In most structural analysis applications, it is necessary to compute displacements and
stresses at various points of interest. The finite element method is a valuable tool for
studying the behavior of structures. The finite element model can be created by dividing
the structure in to a number of finite elements. Each element is interconnected by nodes.
The selection of elements for modeling the structure depends upon the behavior and
geometry of the structure being analyzed. The modeling pattern, which is generally
called mesh for the finite element method, is very important part of the modeling
process. The results obtained from the analysis depend upon the selection of the finite
elements and the mesh size. Although the finite element model does not behave exactly
like the actual structure, it is possible to obtain sufficiently accurate results for most
practical applications. Once the finite element model has been created, the equilibrium
equations can easily be solved using digital computers. The deflections at each node of
the finite element model are obtained by solving the equilibrium equations. The stresses
and strains then can be obtained from the stress-strain and strain-displacement relations.
The finite element method is ideally suited for implementation on a computer with the
advancements in digital computers, the finite element method is becoming the method
of choice for solving many engineering problems, and is extensively used for structural
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
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analysis. The structure can be discretized using frame elements, plane elements, plate
elements, or shell elements according to the behavior of the structure. The structure can
also be modeled by combining different types of elements to approximate different
aspects of structural behavior.
Figure4.1. The process of finite element analysis [52].
4.1 Physical problems, mathematical models and the finite element solution
The finite element method is used to solve physical problems in engineering analysis
and design. Figure 4.1 summarizes the process of finite element analysis. The physical
problem typically involves an actual structure or structural component subjected to
certain loads. The idealization of the physical problem to a mathematical model requires
certain assumptions that together lead to differential equations governing the
mathematical model. The finite element analysis solves this mathematical model.
It is clear that the finite element solution will solve only the selected mathematical
model and that all assumption in a model will be reflected in the predicted response. We
cannot expect any more information in the prediction of physical phenomena than the
Refine analysisInterpretation of results
Assessment of accuracy of finite
element solution of mathematical model
Finite element solution choice of
 Finite elements
 Mesh density
 Solution parameters
Representation of
 Loading
 Boundary conditions
 Etc
Improve
mathematical
model
Mathematical model
Governed by differential
equations Assumptions on
 Geometry
 Kinematics
 Material law
 Loading
 Boundary conditions
 Etc.
Physical problem Change of physicalproblem
Design improvements
Structural optimization
Refine mesh, solution
parameters, ect.
Finite
element
solution of
mathemati
cal model
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
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information contained in the mathematical model. Hence the choice of appropriate
mathematical model is crucial and completely determines the insight into the actual
physical problem that we can obtain by the analysis [52].
4.2 General steps of the finite element method
In the finite element method, the actual continuum or body of matter, such as a solid,
liquid, or gas, is represented as an assemblage of subdivisions called finite elements.
These elements are considered to be interconnected at specified joints called nodes or
nodal points. The nodes usually lie on the element boundaries where adjacent elements
are considered to be connected. Since the actual variation of the field variable (e.g.,
displacement, stress, temperature, pressure, or velocity) inside the continuum is not
known, we assume that the variation of the field variable inside a finite element can be
approximated by a simple function. These approximating functions (also called
interpolation models) are defined in terms of the values of the field variables at the
nodes. When field equations (like equilibrium equations) for the whole continuum are
written, the new unknowns will be the nodal values of the field variable. By solving the
field equations, which are generally in the form of matrix equations, the nodal values of
the field variable will be known. Once these are known, the approximating functions
define the field variable throughout the assemblage of elements.
4.2.1 Step 1 Discretize and select the element types
This step involves dividing the body into an equivalent system of finite elements with
associated nodes and choosing the most appropriate element type to model most closely
the actual physical behavior. The choice of elements used in a finite element analysis
depends on the physical makeup of the body under actual loading conditions and on
how close to the actual behavior the analyst wants the results to be.
Many practical problems in engineering are either extremely difficult or impossible to
solve by conventional analytical methods. Such methods involve finding mathematical
equations which define the required variables. For example, the distribution of stresses
and displacements in a solid component might be required [53].
The finite element method is used to solve physical problems in engineering analysis
and design. Figure 4.1 summarizes the process of finite element analysis. The physical
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
65
problem typically involves an actual structure or structural component subjected to
certain loads. The idealization of the physical problem to a mathematical model requires
certain assumption that together leads to differential equations governing the
mathematical model. Since the finite element solution technique is a numerical
procedure. Judgment concerning the appropriateness of one-, two-, or three-dimensional
idealizations is necessary. Moreover, the choice of the most appropriate element for a
particular problem is one of the major tasks that must be carried out by the
designer/analyst. Elements that are commonly employed in practice are shown in
figures 4.2-5.
Figure 4.2 . A bar or beam element a)Simple two-noded line element
b) The higher-order line element
Figure 4.3. Simple two-dimensional elements with corner nodes
a) Triangular Elements b) Quadrilateral elements
The following elements typically used to represent three-dimensional stress state
Figure 4.4. Simple three-dimensional elements and higher-order three-dimensional
elements with intermediate nodes along edges a) Tetrahedral Elements
b)Regular hexahedral c)Irregular hexahedral
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Figure 4.5 Simple axisymmetric used for axisymmetric problems
a) Triangular ring b) Quadrilateral ring
4.2.2 Step 2 Selection of a proper interpolation or displacement model
This step involves choosing a displacement function within each element. The function
is defined within the element using the nodal values of the element. Linear, quadratic,
and cubic polynomials are frequently used functions because they are simple to work
with in finite element formulation. However, trigonometric series can also be used. For
a two-dimensional element, the displacement function is a function of the coordinates in
its plane (say, the x-y plane). The functions are expressed in terms of the nodal
unknowns (in the two-dimensional problem, in terms of an x and a y component). The
same general displacement function can be used repeatedly for each element. Hence the
finite element method is one in which a continuous quantity, such as the displacement
throughout the body, is approximated by a discrete model composed of a set of
piecewise-continuous functions defined within each finite domain or finite element [51].
4.2.3. Step 3 Define the strain displacement and stress strain relationships
Strain/displacement and stress/strain relationships are necessary for deriving the
equations for each finite element. In addition, the stresses must be related to the strains
through the stress/strain law generally called the constitutive law. The ability to define
the material behavior accurately is most important in obtaining acceptable results.
4.2.4. Step 4 Derive the element stiffness matrix and equations
From the assumed displacement model, the element stiffness matrix ୣ of element e is
to be derived by using either equilibrium conditions or a suitable variational principle.
We now present alternative methods used in finite element analysis.
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4.2.4. 1 Direct equilibrium method
According to this method, the stiffness matrix and element equations relating nodal
forces to nodal displacements are obtained using force equilibrium conditions for a
basic element, along with force/deformation relationships. Because this method is most
easily adaptable to line or one-dimensional elements (spring, bar, and beam elements).
4.2.4.2 Work or energy methods
To develop the stiffness matrix and equations for two- and three-dimensional elements,
it is much easier to apply a work or energy method. The principle of virtual work (using
virtual displacements), the principle of minimum potential energy, and Castigliano’s
theorem are methods frequently used for the purpose of derivation of element equations.
The principle of virtual work is applicable for any material behavior, whereas the
principle of minimum potential energy and Castigliano’s theorem are applicable only to
elastic materials. Furthermore, the principle of virtual work can be used even when a
potential function does not exist. The principle of minimum potential energy probably is
the best known of the three energy methods.
4.2.4.3 Methods of weighted residuals
The methods of weighted residuals are useful for developing the element equations;
particularly popular is Galerkin’s method. These methods yield the same results as the
energy methods wherever the energy methods are applicable. They are especially useful
when a functional such as potential energy is not readily available. The weighted
residual methods allow the finite element method to be applied directly to any
differential equation. Using any of the methods will produce the equations to describe
the behavior of an element. These equations are written conveniently in matrix form as
ଵ
ଶ
ଷ
ସ
௡
ଵଵ ଵଶ ଵଷ
ଶଵ ଶଶ ଶଷ
ଷଵ ଷଶ ଷଷ
ଵସ ଵ௡
ଶସ ଶ௡
ଷସ ଷ௡
ସଵ ସଶ ସଷ
௡ଵ ௡ଶ ௡ଷ
ସସ ସ௡
௡ସ ௡௡
ଵ
ଶ
ଷ
ସ
௡
or in compact matrix form as
(4.1)
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68
Where is the vector of element nodal forces, is the element stiffness matrix
(normally square and symmetric), and is the vector of unknown element nodal
degrees of freedom or generalized displacements, n. Here generalized displacements
may include such quantities as actual displacements, slopes, or even curvatures.
4.2.5. Step 5 Assemble the element equations to obtain the global or total equations
and introduce boundary conditions
In this step the individual element nodal equilibrium equations generated in step 4 are
assembled into the global nodal equilibrium equations.
The final assembled or global equation written in matrix form is,
(4.2)
Here is the vector of global nodal forces, is the structure global or total stiffness
matrix, (for most problems, the global stiffness matrix is square and symmetric) and
is now the vector of known and unknown structure nodal degrees of freedom or
generalized displacements. It can be shown that at this stage, the global stiffness matrix
is a singular matrix because its determinant is equal to zero. To remove this
singularity problem, we must invoke certain boundary conditions (or constraints or
supports) so that the structure remains in place instead of moving as a rigid body [51].
4.2.6 Step 6 Solve for the unknown degrees of freedom (or generalized
displacements)
Equation (4.2) is modified to account for the boundary conditions, is a set of
simultaneous algebraic equations that can be written in expanded matrix form as
ଵ
ଶ
ଷ
௡
ଵଵ
ଶଵ
ଷଵ
௡ଵ
ଵଶ
ଶଶ
ଷଶ
௡ଶ
ଵଷ
ଶଷ
ଷଷ
௡ଷ
ଵ௡
ଶ௡
ଷ௡
௡௡
ଵ
ଶ
ଷ
௡
(4.3)
Where now n is the structure total number of unknown nodal degrees of freedom. These
equations can be solved for the displacements u by using an elimination method (such
as Gauss’s method) or an iterative method (such as the Gauss–Seidel method). The
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
69
are called the primary unknowns, because they are the first quantities determined
using the stiffness (or displacement) finite element method.
4.2.7 Step7 Computation of element strains and stresses
From the known nodal displacements , the element strains and stresses (or moment
and shear force) can be computed by using the necessary equations of solid or structural
mechanics.
4.2.8 Step 8 Interpret the results
The final goal is to interpret and analyze the results for use in the design/analysis
process. Determination of locations in the structure where large deformations and large
stresses occur is generally important in making design/analysis decisions. Postprocessor
computer programs help the user to interpret the results by displaying them in graphical
form.
4.3 Advantages of the finite element method
The finite element method has a number of advantages that have made it very popular.
They include the ability to
1-Model irregularly shaped bodies quite easily.
2-Handle general load conditions without difficulty.
3. Model bodies composed of several different materials because the element
equations are evaluated individually.
4. Handle unlimited numbers and kinds of boundary conditions.
5. Vary the size of the elements to make it possible to use small elements where
necessary.
6. Alter the finite element model relatively easily and cheaply.
7. Include dynamic effects.
8. Handle nonlinear behavior existing with large deformations and nonlinear
materials.
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The finite element method of structural analysis enables the designer to detect stress,
vibration, and thermal problems during the design process and to evaluate design
changes before the construction of a possible prototype. Thus confidence in the
acceptability of the prototype is enhanced. Moreover, if used properly, the method
can reduce the number of prototypes that need to be built [51].
4.4 Guidelines on element layout
4.4.1 Mesh refinement
Use a relatively fine (coarse) discretization in regions where you expect a high (low)
gradient of strains and/or stresses. Regions to watch out for high gradients are:
•Near entrant corners, or sharply curved edges.
•In the vicinity of concentrated (point) loads, concentrated reactions, cracks and cutouts.
•In the interior of structures with abrupt changes in thickness, material properties or
cross sectional areas [54].
Figure 4.6 shows some situations where a locally refined finite element discretization.
Figure 4.6. Some situations where a locally refined finite element discretization (in the
red-colored areas) is recommended.
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4.4.2 Element aspect ratios
When discretizing two and three dimensional problems, try to avoid finite elements of
high aspect ratios: elongated or “skinny” elements, as the ones illustrated in figure 4.7.
As a rough guideline, elements with aspect ratios exceeding 3 should be viewed with
caution and those exceeding 10 with alarm.
Figure 4.7. Elements of good and bad aspect ratios.
4.4.3. Physical interfaces
A physical interface, resulting from example from a change in material, should also be
an interelement boundary. That is, elements must not cross interfaces. See figure 4.8.
Figure 4.8 Illustration of the rule that elements should not cross material interfaces.
4.4.4. Preferred shapes
In two-dimensional FE modeling, if you have a choice between triangles and
quadrilaterals with similar nodal arrangement, prefer quadrilaterals. Triangles are quite
convenient for mesh generation, mesh transitions, rounding up corners, and the like. But
sometimes triangles can be avoided altogether with some thought. In three dimensional
FE modeling, prefer strongly bricks over wedges, and wedges over tetrahedra. The latter
should be used only if there is no viable alternative. The main problem with tetrahedral
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
72
and wedges is that they can produce wrong stress results even if the displacement
solution looks reasonable [54].
The aim of this section is not to provide a detailed description of FE theory which can
be found in many textbooks but to give a brief outline of the basic principles involved to
enable the reader with little or no knowledge of the FE method to have some
appreciation of the method in relation to simple problems and the procedure by which
more complex FE models are constructed and analyzed. Brief characteristics of the
elements used within the work in this thesis will also be discussed.
To simplify the explanation of the theory a two-dimensional plate problem will be
considered, taking one element from that. It can be appreciated that for larger problems,
the formulation can be simply built up by repetition.
4.5 Plane stress and plane strain stiffness equations
Two-dimensional (planar) elements are defined by three or more nodes in a two-
dimensional plane (that is, x-y). The elements are connected at common nodes and/or
along common edges to form continuous structures such as those shown in figure 4.3
[51].
Membrane elements are among the simplest elements to develop. These elements are
used for analyzing structures subjected to inplane forces. Assuming that the structure is
in the xy plane, the displacements at any point of the structure are u, the translation in
the x direction and v, the translation in the y direction. The stresses of interest are the
normal stresses σx and σy and the shearing stress xy. The normal stress in the direction
perpendicular to the plane of structure is considered to be zero. Membrane elements are
used to model the behavior.
We begin this section to give a reader some idea about with the development of the
stiffness matrix for a basic two-dimensional or plane finite element, called the constant-
strain triangular element. We consider the constant-strain triangle (CST) stiffness matrix
because its derivation is the simplest among the available two-dimensional elements.
The element is called a CST because it has a constant strain throughout it. We will
derive the CST stiffness matrix by using the principle of minimum potential energy
because the energy formulation is the most feasible for the development of the equations
for both two- and three-dimensional finite elements.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
73
4.5.1 Basic concepts of plane stress and plane strain
4.5.1.1 Plane stress
Plane stress is defined to be a state of stress in which the normal stress ( ௭ ) and the
shear stresses ( ௫௭ , ௬௭) directed perpendicular to the plane are assumed to be zero.
Generally, members that are thin (those with a small z dimension compared to the in-
plane x and y dimensions) and whose loads act only in the x-y plane can be considered
to be under plane stress. Plane stress analysis, which includes problems such as plates
with holes, fillets, such as illustrated in figure 4.9.
Figure 4.9. Plane stress problems: (a) Plate with hole; (b) Plate with fillet
4.5.1.2 Plane strain
Plane strain is defined to be a state of strain in which the strain normal to the x-y plane
௭ and the shear strains ௫௭ and ௬௭are assumed to be zero. The assumptions of plane
strain are realistic for long bodies (say, in the z direction) with constant cross-sectional
area subjected to loads that act only in the x and/or y directions and do not vary in the z
direction. Some plane strain examples are shown in figure 4.10. In these examples, only
a unit thickness (1 mm. or 1 m) of the structure is considered because each unit
thickness behaves identically (except near the ends).
Figure 4.10. Plane strain problems: (a) dam subjected to horizontal loading; (b) pipe
subjected to a vertical load
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
74
4.5.2 Two-dimensional state of stress and strain
Consider an infinitesimal element of the two-dimensional state of stress in the xy plane
with thickness t along the z direction subjected to inplane stresses as shown in figure
4.11. The infinitesimal element has normal and shear stresses ௫, ௬, ௫௬ and ௬௫.
Figure 4.11. Plane differential element subjected to stresses
The equilibrium of forces in x direction ( ௫ ) after simplifying and canceling
terms, we obtain
డఙೣ
డ௫
డఛ೤ೣ
డ௬ ௕
(4.4)
Similarly, the equilibrium of forces in y direction ( ௬ )
డఙ೤
డ௬
డఛೣ ೤
డ௫ ௕
(4.5)
Equilibrium of moments of the element results in ௫௬ being equal in magnitude to ௬௫;
௫௬ ௬௫ (4.6)
The three independent stresses can be represented by the vector column matrix as
follows:
௫
௬
௫௬
(4.7)
Figure 4.12 shows an infinitesimal element used to represent the general two-
dimensional state of strain at some point in a structure. The element is shown to be
displaced by amounts u and v in the x and y directions at point A.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
75
Figure 4.12. Displacements and rotations of lines of an element in the x-y plane.
From the general definitions of normal and shear strains and the use of figure 4.12, we
get;
௫௫
డ௨ೣ
డ௫ ௬௬
డ௨೤
డ௬ ௫௬
ଵ
ଶ
డ௨ೣ
డ௬
డ௨೤
డ௫
(4.8)
The strains given by Eqs. (6.1.4) can be generally represented as the vector column
matrix,
௫
௬
௫௬
(4.9)
4.5.3 Stress/strain relationships
The three-dimensional stress/strain relationship for isotropic materials are given by [55],
௫ ௫ ௬ ௭
௬
ଵ
ா ௬ ௫ ௭
௭
ଵ
ா ௭ ௫ ௬
(4.10)
௫௬
ఛೣ ೤
ீ ௬௭
ఛ೤೥
ீ ௭௫
ఛ೥ೣ
ீ
The stress/strain relationships for isotropic materials for plane stress can be derived as
follows:
As we mentioned in previous sections that for plane stress, the following stresses are
zero,
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
76
௭ ௫௭ ௬௭ (4.11)
The shear strains are:
௫௭ ௬௭ but ௭
For plane stress conditions, we then have
௫
௬
௫௬
௫
௬
௫௬
(4.12)
ா
ଵିణమ ଵିణ
ଶ
Where, is called the stress/strain matrix (or constitutive matrix), E is the modulus of
elasticity, and is Poisson’s ratio.
For plane strain, we assume the following strains to be zero,
୸ ୶୸ ୷୸ (4.13)
Applying Eq. (4.13) to the three-dimensional stress/strain relationship Eq. (4.10), the
shear stresses ୶୸ ୷୸ , but ୸ . The stress/strain matrix then
becomes,
ா(ଵାణ)(ଵିଶణ) ଵିଶణ
ଶ
(4.14)
4.6. Finite element theory
The aim of this section is to provide some description of FE theory and to give a brief
outline of the basic principles involved to enable the reader with little or no knowledge
of the FE method to have some appreciation of the method in relation to some problems
and the procedure by which more complex FE models are constructed and analyzed.
Brief characteristics of the elements used within the work in this thesis will also be
discussed. To simplify the explanation of the theory a two-dimensional plate problem
will be considered, taking one element from that.
4.6.1. Constant strain triangle
The simplest triangular plane stress element is the constant strain triangle (CST). We
use triangular elements because boundaries of irregularly shaped bodies can be closely
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
77
approximated in this way .This element has two inplane degrees of freedom at each
node for a total of six degrees of freedom per element. The constant strain triangle is
widely used for various analysis purposes. The nodes of the CST element are numbered
in a counterclockwise direction as shown in figure. 4.13. Remember that a consistent
numbering procedure for the whole body is necessary to avoid problems in the
calculations such as negative element areas.
Figure 4.13. Constant Strain Triangle (CST).
Here ଵ ଵ , ଶ ଶ and ଷ ଷ are the known nodal coordinates of nodes 1; 2, and 3,
respectively. The procedure for deriving the stiffness matrix of the constant strain
triangle element is as follows [58],
The nodal displacement matrix is given by
ଵ
ଶ
ଷ
ଵ
ଶ
ଵ
ଶ
ଵ
ଶ
(4.15)
The assumed displacement field can be written as,
ଵ ଶ ଷ
ସ ହ ଺
(4.16)
Where and describe displacements at any interior point of the
element. The above equations can be written in matrix form as,
ଵ
ଶ
ଷ
ସ
ହ
଺
(4.17)
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78
Or
From strain-displacement relationships the following results can be obtained,
௫
డ௨
డ௫ ଶ
,
௬
డ௩
డ௬ ଺
,
And,
௫௬
డ௨
డ௬
డ௩
డ௫ ଷ ହ
(4.18)
The element is called the constant strain triangle, because it is clear from the above
equations that the term ଶ represents a constant strain in the x direction, and the term ହ
represents a constant strain in the y direction. Also the term ଷ + ହ represents a
uniform shear strains. The terms ଵ and ସ represents rigid body translation.
To obtain the a’s in Eqs. (4.17), we begin by substituting the coordinates of the nodal
points into Eqs. (4.17) to yield
ଵ ଵ ଵ ଵ ଶ ଵ ଷ ଵ
ଵ ଵ ଵ ସ ହ ଵ ଺ ଵ
ଶ ଶ ଶ ଵ ଶ ଶ ଷ ଶ
ଶ ଶ ଶ ସ ହ ଶ ଺ ଶ (4.19)
ଷ ଷ ଷ ଵ ଶ ଷ ଷ ଷ
ଷ ଷ ଷ ସ ହ ଷ ଺ ଷ
We can put Eqs. (4.19) in matrix form as
.
ଵ
ଵ
ଶ
ଶ
ଷ
ଷ
ଵ ଵ
ଵ ଵ
ଶ
ଷ
ଶ
ଷ
ଶ
ଷ
ଶ
ଷ
ଵ
ଶ
ଷ
ସ
ହ
଺
(4.20)
(4.21)
Where,
ଵ ଵ
ଵ ଵ
ଶ
ଷ
ଶ
ଷ
ଶ
ଷ
ଶ
ଷ
(4.22)
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79
It should be noted that all the terms in matrix are known since they simply consist
of the coordinates of the element nodes.
The coefficients are obtained by inverting Equation (4.21),
ିଵ (4.23)
From equation (4.17),
(4.24)
Thus,
ିଵ (4.25)
Where, ିଵ represents the shape functions .
Hence,
ିଵ (4.26)
Then, Equation (4.25) becomes
(4.27)
Inverting the matrix in Equation (4.21) and solving for gives,
ଵ
ଶ
ଷ
ସ
ହ
଺
௘
ଶ ଷ ଷ ଶ ଵ ଷ ଷ ଵ
ଶ ଷ ଷ ଵ
ଷ ଶ ଵ ଷ
ଵ ଶ ଶ ଵ
ଵ ଶ
ଶ ଵ
ଶ ଷ ଷ ଶ
ଶ ଷ
ଷ ଶ
ଵ ଷ ଷ ଵ ଵ ଶ ଶ ଵ
ଷ ଵ ଵ ଶ
ଵ ଷ ଶ ଵ
ଵ
ଵ
ଶ
ଶ
ଷ
ଷ
(4.28)
Where, ௘ is the area of the triangle element and can be expressed as,
௘ ଵ
ଶ
ଵ ଵ
ଶ ଶ
ଷ ଷ
(4.29)
௘ ଵ
ଶ ଶ ଷ ଷ ଶ ଵ ଷ ଷ ଵ ଵ ଶ ଶ ଵ
(4.30)
by combining Equations (6.2.8) and (6.2.9), we can obtain the shape functions as
follows;
×
ଵ
ଶ஺೐
ଶ ଷ ଷ ଶ ଵ ଷ ଷ ଵ
ଶ ଷ ଷ ଵ
ଷ ଶ ଵ ଷ
ଵ ଶ ଶ ଵ
ଵ ଶ
ଶ ଵ
ଶ ଷ ଷ ଶ
ଶ ଷ
ଷ ଶ
ଵ ଷ ଷ ଵ ଵ ଶ ଶ ଵ
ଷ ଵ ଵ ଶ
ଵ ଷ ଶ ଵ
(4.30)
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80
Equation (4.30) can be written in a simple form as follows;
ଵ ଶ ଷ
ଵ ଶ ଷ
(4.31)
Where,
ଵ
ଶ
ଷ
ଵ
ଶ஺೐
ଶ ଷ ଷ ଶ ଶ ଷ ଷ ଶ
ଵ ଷ ଷ ଵ ଷ ଵ ଵ ଷ
ଵ ଶ ଶ ଵ ଵ ଶ ଶ ଵ
(4.32)
The displacements can now be rewritten in terms of the shape functions as,
ଵ ଶ ଷ
ଵ ଶ ଷ
ଵ
ଶ
ଵ
ଶ
ଵ
ଶ
(4.33)
The normal and shearing strains can be obtained from the strain-displacement
relationships,
୶
ப୳
ப୶
ப
ப୶ ୧ ୧
ଷ
୧ୀଵ
௫௬
డ௩
డ௬
డ
డ௬ ௜ ௜
ଷ
௜ୀଵ (4.34)
௫௬
డ௨
డ௬
డ௩
డ௫
డ
డ௬ ௜ ௜
ଷ
௜ୀଵ
డ
డ௫ ௜ ௜
ଷ
௜ୀଵ
The above equations can be written in matrix form as,
௫
௬
௫௬
ଵ
ଶ஺೐
డேభ
డ௫
డேమ
డ௫
డேయ
డ௫
డேభ
డ௬
డேమ
డ௬
డேయ
డ௬
డேభ
డ௬
డேభ
డ௫
డேమ
డ௬
డேమ
డ௫
డேయ
డ௬
డேయ
డ௫
ଵ
ଶ
ଵ
ଶ
ଵ
ଶ
(4.35)
Eq.4.35 can be rewritten as follows,
୶ ଷ×ଵ ଷ×଺ ୧ ଺×ଵ (4.36)
By taking the derivatives of the shape functions (Equation 4.32) with respect to x and y,
gives the strain-displacement matrix [B],
ଵ
ଶ஺೐
ଶ ଷ ଷ ଵ
ଷ ଶ
ଷ ଶ ଶ ଷ ଵ ଷ
ଵ ଶ
ଵ ଷ ଶ ଵ
ଷ ଵ ଶ ଵ ଵ ଶ
(4.37)
The stiffness matrix of the element ௘ can be found by using the strain-displacement
matrix [B] and the material matrix [D],
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
81
௘
଺×଺ ்଺×ଷ ଷ×ଷ ଷ×଺௏೐ (4.38)
Where ௘ denotes the volume of the element. If the plate thickness is taken as a
constant (t), the evaluation of the integral in Eq. (4.38) presents no difficulty since the
elements of the matrices [B] and [D] are all constants (not functions of x and y). Hence,
Eq.( 4.38) can be rewritten as,
௘
଺×଺ ்
஺೐
௘ ் ௘ ்
஺೐
(4.39)
Where, t is the element thickness.
Although the matrix products involved in Eq. (4.39) can be performed conveniently on
a computer, the explicit form of the stiffness matrix is given below for convenience:
௘
௡
௘
௦
௘ (4.40)
Where the matrix ௘ is separated into two parts: one due to normal stresses. ௡௘ , and
the other due to shear stresses, ௦௘ . The components of the matrices ௡௘ and ௦௘ are
given by
௡
௘
௘ ଶ
ଷଶ
ଶ
ଷଶ ଷଶ ଷଶ
ଶ
ଷଶ ଷଵ ଷଶ ଷଵ ଷଵ
ଶ
ଷଶ ଷଵ ଷଶ ଷଵ ଷଵ ଷଵ
ଷଶ ଶଵ ଷଶ ଶଵ ଷଵ ଶଵ
ଷଶ ଶଵ ଷଶ ଶଵ ଷଵ ଶଵ
ଷଵ
ଶ
ଷଵ ଶଵ ଶଵ
ଶ
ଷଵ ଶଵ ଶଵ ଷଵ ଶଵ
ଶ
And ,
௦
௘
௘
ଷଶ
ଶ
ଷଶ ଷଶ ଷଶ
ଶ
ଷଶ ଷଵ ଷଶ ଷଵ ଷଵ
ଶ
ଷଶ ଷଵ ଷଶ ଷଵ ଷଵ ଷଵ
ଷଶ ଶଵ ଷଶ ଶଵ ଷଵ ଶଵ
ଷଶ ଶଵ ଷଶ ଶଵ ଷଵ ଶଵ
ଷଵ
ଶ
ଷଵ ଶଵ ଶଵ
ଶ
ଷଵ ଶଵ ଶଵ ଶଵ ଶଵ
ଶ
(4.41)
Where,
௜௝ ௜ ௝ and ௜௝ ௜ ௝
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82
4.6.2 Plate bending element
The plate element is one of the more important structural elements and is used to model
and analyze such structures as pressure vessels, chimney stacks, and automobile parts. A
plate can be considered the two-dimensional extension of a beam in simple bending.
Both beams and plates support loads transverse or perpendicular to their plane and
through bending action (see figure 4.14). A plate is flat (if it was curved, it would
become a shell). A beam has a single bending moment resistance, while a plate resists
bending about two axes and has a twisting moment [51].
The plate surfaces are at z =±t/2, and its midsurface is at z= t/2. The assumed basic
geometry of the plate is as follows:
(1) The plate thickness is much smaller than its inplane dimensions b and c (that is, t
b or c). (If t is more than about one-tenth the span of the plate, then transverse
shear deformation must be accounted for and the plate is then said to be thick.)
(2) The deflection w is much less than the thickness t (that is, w 1).
Figure 4.14. Thin plate subjected to transverse loading.
4.6.2.1 Finite elements for plates
A plate is a thin solid and might be modeled by 3D solid elements (figure 4.15 -a). But
a solid element is wasteful of d.o.f, as it computes transverse normal stress and
transverse shear stress, all of which are considered negligible in a thin plate [56].
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
83
Figure 4.15 (a) A 3D solid element (b) Plate Element.
In figure 4.15- b, thickness t may appear to be zero, but it is used in formulating element
stiffness matrices.
Figure 4.16. Rectangular plate element with nodal degrees of freedom
The flat-plate bending element has three degrees of freedom at each node, namely two
rotations and the transverse deflection. The lateral deflection is denoted by , the
rotation about x-axis is denoted by ௫ and the rotation about y-axis is denoted by ௬.
The element has twelve degrees of freedom as shown in figure 4.16. The nodal
displacement matrix at node is given by
௜
௜
௫௜
௬௜
(4.42)
The two slopes ௫ and ௬the rotations are related to the lateral displacement by the
expressions,
௫
డ௪
డ௬
and ௬
డ௪
డ௫
(4.43)
The total element displacement matrix is now given by,
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
84
௜
௝
௠
௡
(4.44)
Since the element has 12 degrees of freedom, 12 undetermined constants must be
employed in the polynomial expression chosen to represent . A suitable function is
given in Eq 4.45,
ଵ ଶ ଷ ସ
ଶ
ହ ଺
ଶ
଻
ଷ
଼
ଶ
ଽ
ଶ
ଵ଴
ଷ
ଵଵ
ଷ
ଵଶ
ଷ (4.45)
We can get the constants ଴ through ଵଵ by expressing the 12 simultaneous equations
linking the values of w and its slopes at the nodes when the coordinates take up their
appropriate values. First, we write,
௜
௫௜
௬௜
௝
௫௝
ଶ ଶ ଷ ଶ ଶ ଷ ଷ ଷ
ଶ ଶ ଷ ଶ
ଶ ଶ ଶ ଷ
ଵ
ଶ
ଷ
ସ
ହ
଺
଻
଼
ଽ
ଵ଴
ଵଵ
ଵଶ
(4.46)
or in simple matrix form the degrees of freedom matrix is,
ଷ×ଵଶ ଵଶ×ଵ (4.47)
Next, we evaluate Eq. (4.46) at each node point as follows;
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85
ଶ ଶ ଷ ଶ ଶ ଷ ଷ ଷ
ଶ ଶ ଷ ଶ
ଶ ଶ ଶ ଷ
ଵ
ଶ
ଷ
ସ
ହ
଺
଻
଼
ଽ
ଵ଴
ଵଵ
ଵଶ
(4.48)
Equation (4.48) can be written in a simple form as follows,
(4.49)
Where, is the 12 ×12 matrix.Therefore, the constants (a’s) can be solved for by
ିଵ (4.50)
Equation (4.47) can now be written as ିଵ (4.51)
Or r (4.52)
Where, ିଵis the shape function matrix. The strain (curvature)/
displacement and stress (moment)/curvature relationships are given by [51],
୶
୷
୶୷
ସ ଻ ଼
଺ ଽ ଵ଴ ଵଶ
ହ ଼ ଽ ଵଵ
ଶ
ଵଶ
ଶ
(4.52)
Eq.( 4.52) can be expressed in matrix form as follows,
(4.53)
Where, is the coefficient matrix multiplied by the ᇱ in Eq. (4.52). Using
Eq. (4.50) for , we express the curvature matrix as,
(4.54)
Where,
ିଵ (4.55)
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
86
The moment/curvature matrix for a plate is given by
௫
௬
௫௬
୶
୷
୶୷
(4.56)
Where the matrix is the constitutive matrix given for isotropic materials by
ா௧య
ଵଶ(ଵିణమ) ଵିణ
ଶ
(4.57)
The Element Stiffness Matrix can be derived by the usual form of the stiffness matrix as
௘
ଵଶ×ଵଶ ் (4.58)
Where ௘ is a rectangular element stiffness matrix.
The surface force matrix due to distributed loading q acting per unit area in the z
direction is obtained using the standard equation, see [51].
ୱ ୱ
୘ (4.59)
For an element of dimensions subjected to a uniform acting over the
surface, Eq. (4.59) yields the forces and moments at node i as
௪ ௜
ఏೣ೔
ఏ೤೔
(4.60)
Similarly for the nodes j,m and n. The element equations can be written as follows,
௪ ௜
ఏೣ೔
ఏ೤೔
ఏ೤೙
ଵଵ ଵଶ ଵଷ
ଶଵ ଶଶ ଶଷ
ଷଵ ଷଶ ଷଷ
ଵସ ଵ,ଵଶ
ଶସ ଶ,ଵଶ
ଷସ ଷ,ଵଶ
ସଵ ସଶ ସଷ
ଵଶ,ଵ ଵଶ,ଶ ଵଶ,ଷ
ସସ ସ,ଵଶ
ଵଶ,ସ ଵଶ,ଵଶ
௜
௫௜
௬௜
௬௡
(4.61)
The rest of the steps, including assembling the global equations, applying boundary
conditions, and solving the equations for the nodal displacements and slopes (note three
degrees of freedom per node).
4.7 Shells
4.7.1 Shell element
Flat shell elements are obtained by combining plate elements with plate stress elements.
The geometry of a shell is defined by its thickness and its midsurface which is curved
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
87
surface in space. Load is carried by a combination of membrane action and bending
action. A thin shell can be very strong if membrane action dominates. A shell of a given
shape can carry a variety of distributed loadings by membrane action alone.
4.7.2 Finite elements for shells
The most direct way to obtain a shell element is to combine a membrane element and a
bending element. Thus a simple triangular shell element can be obtained by combining
the plane stress triangle with the plate bending triangle. The resulting element is flat and
has five or six degrees of freedom per node, depending on whether or not the shell-
normal rotation ௭௜ at node i is present in plane stress element. A quadrilateral shell
element can be produced in similar way, by combining quadrilateral plane and plate
elements .The shell element can also be formulated using the usual method of defining
shape functions, substituting into the constitutive equations, and thus obtaining the
element matrices.
Advantages of a flat element include simplicity of element formulation, simplicity in the
description of element geometry, and the elements ability to represent a rigid body
motion without strain. Disadvantage includes the representation of a smoothly curved
shell surface by flat of slightly warped facets, so that there are fold lines where elements
meet. There is discritization error associated with the lack of coupling between
membrane and bending actions within individual elements. Membrane – bending
coupling arises globally because adjacent elements are not coplanar: membrane force in
one element is transferred to neighboring element with an element normal component,
which produces bending. Discretization error can be reduced by using smaller elements.
Common advice is that a flat shell element should span no more than roughly ° of the
arc of the actual shell [56].
Since the plate structure can be treated as a special case of the shell structure, it is
common practice to use a shell element offered in a commercial FE package to analyze
plate structures.Shell structures are usually curved. We assume that the shell structure is
divided into shell elements that are flat. The curvature of the shell is then followed by
changing the orientation of the shell elements in space. Therefore, if the curvature of the
shell is very large, a fine mesh of elements has to be used [57].
There are six DOFs at a node for a shell element: three translational displacements in
the x, y and z directions, and three rotational deformations with respect to the x, y and z
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
88
axes. Figure 4.17 shows the middle plane of a rectangular shell element and the DOFs at
the nodes. The generalized displacement vector for the element can be written as
௘
௘ଵ
௘ଶ
௘ଷ
௘ସ
(4.62)
௘௜
௜
௜
௜
௫௜
௬௜
௭௜
(4.63)
Where, ୧ୣ ) are the displacement vector at node .
Figure 4.17. The middle plane of a rectangular shell element.
The stiffness matrix for the shell element can be formulated by combining the element
stiffness matrix for membrane and element stiffness matrix for bending.
The membrane stiffness matrix can thus be expressed in the following form using sub-
matrices according to the nodes:
௘
௠
ଵଵ
௠
ଵଶ
௠
ଶଵ
௠
ଶଶ
௠
ଵଷ
௠
ଵସ
௠
ଶଷ
௠
ଶସ
௠
ଷଵ
௠
ଷଶ
௠
ସଵ
௠
ସଶ
௠
ଷଷ
௠
ଷସ
௠
ସଷ
௠
ସସ
௠
(4.64)
Where, the superscript stands for the membrane matrix. Each sub-matrix will have a
dimension of , since it corresponds to the two DOFs and at each node.
The stiffness matrix for a rectangular plate element is used for the bending effects,
corresponding to DOFs of , and ௫, ௬. The bending stiffness matrix can thus be expressed in
the following form using sub-matrices according to the nodes [57]:
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
89
௘
௕
ଵଵ
௕
ଵଶ
௕
ଶଵ
௕
ଶଶ
௕
ଵଷ
௕
ଵସ
௕
ଶଷ
௕
ଶସ
௕
ଷଵ
௕
ଷଶ
௕
ସଵ
௕
ସଶ
௕
ଷଷ
௕
ଷସ
௕
ସଷ
௕
ସସ
௕
(4.65)
Where, the superscript b stands for the bending matrix. Each bending sub-matrix has a
dimension of
The stiffness matrix for the shell element in the local coordinate system is then formulated by
combining equations. (4.64) and (4.65):
௘
ଵଵ
௠
ଵଶ
௠
ଵଷ
௠
ଵସ
௠
ଵଵ
௕
ଵଶ
௕
ଵଷ
௕
ଵସ
௕
ଶଵ
௠
ଶଶ
௠
ଶଷ
௠
ଶସ
௠
ଶଵ
௕
ଶଶ
௕
ଶଷ
௕
ଶସ
௕
ଷଵ
௠
ଷଶ
௠
ଷଷ
௠
ଷସ
௠
ଷଵ
௕
ଷଶ
௕
ଷଷ
௕
ଷସ
௕
ସଵ
௠
ସଶ
௠
ସଷ
௠
ସସ
௠
ସଵ
௕
ସଶ
௕
ସଷ
௕
ସସ
௕
(4.66)
The stiffness matrix for a rectangular shell matrix has a dimension of 24 × 24. Note that
in Eq. (4.66), the components related to the DOF ௭, are zeros. This is because there is
no ௭ in the local coordinate system. If these zero terms are removed, the stiffness
matrix would have a reduced dimension of 20 × 20. However, using the extended 24 ×
24 stiffness matrix will make it more convenient for transforming the matrix from the
local coordinate system into the global coordinate system [57].
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
90
Chapter 5
Finite element modeling of the welded joints using beam elements
The purpose of this chapter is to develop a finite element model and obtain the analysis
result that can be compared with analytical and experimental data. Hence, in most
structural analysis applications, it is necessary to compute displacements and stresses at
various points of interest. The finite element method is a very valuable tool for studying
the behavior of structures. The finite element model is created by dividing the structure
in to a number of finite elements. Each element is interconnected by nodes. The
selection of elements for modeling the structure depends upon the behavior and
geometry of the structure being analyzed.
There are always situations where the models contain regions with complexity in
geometry such as fillet welds which are extensively used in various fields such as in
construction industry. Therefore, it is important to be able to represent the mechanical
capacity of fillet-welded structures in numerical analyses for design purposes. Here a
numerical investigation was undertaken on the performance of a series of fillet-welded
connections in steel.
Accuracy and efficiency are two major concerns in any finite element analysis that are
forcing engineers and design analysts to seek reliable and accurate yet economical
methods to determining the behavior of structural components. These element types
(beams, plates and shells) produce more computationally efficient models, thus reducing
analyses time and cost. They also require less data and are easier to construct than solid
models.
Beam elements allow for bending and stretching and are most useful in modelling of
whole structure. They are available within Abaqus[59] in two different formats, in-
plane (two displacement degrees of freedom and one in-plane rotational degree of
freedom) and three dimensional where all six degrees of freedom are active. Numerical
analyses were performed using the Abaqus[59]. The connections were modelled with
beam and shell elements, see Figure 5.1. The beam and weld were modelled as beam
element, while base plate was modelled using shell element.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
91
In the modeling certain assumption were made to simplify the model. Parent metal and
welded metal had the same material properties which are steel (
ଷ
5.1 Fillet weld
The stiffness of the welds in the fillet welded joints can be represented by increasing the
thickness of the beam in the region of welded joints. Coupling of beams and shells has
been considered and used in this thesis. Based on this technique for analyzing the stress
variation over the cross-section of an interface, the Coupling constraint technique can be
generated by Abaqus[59].
5.1.1 Finite element results of beam shell coupling of fillet weld in case of bending.
Beam elements require less data and are easier to construct than solid models and are
thus especially suitable for conceptual design evaluation and optimization.
5.1.1.1 Rectangular Section
Figure 5.1 shows a cantilever rectangular hollow beam under the given loading of 10KN
acting upward at the free end. Coupling of beam elements and shell elements were used
with a full integration scheme.
Figure 5.2 shows that the finite element results of stress in the fillet weld is 55MPa, and
the displacement at the free end is 1mm. From chapter 2, the analytical results of this
beam is 59MPa. We can see that the finite element results are in good agreement with
the analytical results. We can see that the analytical solution can be replaced by beam
element solution to represent the fillet weld.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
92
Figure 5.1. General view of the Beam-shell coupling of rectangular section
(a) (b)
Figure 5.2. FE Results for a rectangular section a) Displacement b)Von Mises Stress
5.1.1.2 I section
Figure 5.3 shows a cantilever beam of I section profile under the given loading of 10KN
acting upward at the free end. Coupling of beam elements and shell elements were used
with a full integration scheme.
Figure 5.4 shows that the finite element results of stress in the fillet weld is 72.7MPa,
and the displacement magnitude at the free end is 2.1mm.From chapter 2, the analytical
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
93
results of this beam is 71.2MPa. We can see that the finite element results verify the
analytical solution. It is clear that the analytical solution can be replaced by beam
element solution to represent the fillet weld.
Figure 5.3. General view of the Beam-shell coupling of I- section
a) b)
Figure 5.4. FE Results for a I- section a) Displacement b)Von Mises Stress
5.1.1.3. C- section
Figure 5.5 shows a cantilever beam of C section profile under the given loading of
10KN acting upward at the free end. Coupling of beam elements and shell elements
were used with a full integration scheme.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
94
Figure 5.6 shows that the finite element results of stress in the fillet weld is 78.8MPa,
and the displacement magnitude at the free end is 3.3mm.From chapter 2, the analytical
results of this beam is 73 MPa. We can see that the finite element results verify the
analytical solution. It is clear that the analytical solution can be replaced by beam
element solution to represent the fillet weld.
Figure 5.5. General view of the Beam-shell coupling of C- section
a) b)
Figure 5.6. FE Results for a C- section a) Displacement b) Von Mises Stress
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
95
5.1.1.4 Circular- section
Figure 5.7 shows a cantilever beam of circular section profile under the given loading of
10KN acting upward at the free end. Coupling of beam elements and shell elements
were used with a full integration scheme.
Figure 5.8 shows that the finite element results of stress in the fillet weld is 94.4MPa,
and the displacement magnitude at the free end is 2.8mm.From chapter 2, the analytical
results of this beam is 98.7 MPa. It is clear that the finite element results verify the
analytical solution. We can see that the analytical solution can be replaced by beam
element solution to represent the fillet weld.
Figure 5.7. General view of the Beam-shell coupling of Circular- section
a) b)
Figure 5.8. FE Results for a Circular- section a) Displacement b) Von Mises Stress
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
96
5.1.1.5 Z- section
Figure 5.9 shows a cantilever beam of Z section profile under the given loading of
10KN acting upward at the free end. Coupling of beam elements and shell elements
were used with a full integration scheme.
Figure 5.10 shows that the finite element results of stress in the fillet weld is 156.5MPa,
and the displacement magnitude at the free end is 9.9 mm. From chapter 2, the
analytical result of this beam is 149.2MPa. It is clear that the finite element results
verify the analytical solution. We can see that the analytical solution can be replaced by
beam element solution to represent the fillet weld.
Figure 5.9. General view of the Beam-shell coupling of Z- section
a)
b)
Figure 5.10. FE Results for a Z- section a) Displacement b)Von Mises Stress
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
97
5.1.1.6 X- section
Figure 5.11 shows a cantilever beam of X section profile under the given loading of
10KN acting upward at the free end. Coupling of beam elements and shell elements
were used.
Figure 5.12 shows that the finite element results of stress in the fillet weld is 207.1MPa,
and the displacement magnitude at the free end is 7.6 mm. From chapter 2, the
analytical result of this beam is 181 MPa. It is clear that the finite element results verify
the analytical solution. It can be seen that the analytical solution can be replaced by
beam element solution to represent the fillet weld.
Figure 5.11. General view of the Beam-shell coupling of X- section
a) b)
Figure 5.12. FE Results for a X- section a) Displacement b)Von Mises Stress
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
98
We can summarize the numerical results in Figures 5.1-12, in the following table:
Table. 5.1 Analytical and FE Results of beams of fillet weld (Bending Moment)
Specimen
Fillet joint
Analytic
Stress (MPa)
FE (Beam Element)
Stress (MPa)
box 59.1 55.0
I 71.2 72.7
C 73.0 78.8
circle 98.7 94.4
Z 149.2 156.5
X 181.0 207.1
Numerical analyses were carried out to represent the mechanical performance of the
welded beam-sections under bending. The study is summarized and concluded as
follows. The predicted stress curves are in good agreement with the analytical results for
all six cases of the beam members. As can be seen from the test Figures 5.13 the finite
element results verify the analytical results.
Figures 5.13 stress of Analytic and finite Element Results of Various Beam Sections of
fillet weld (Bending)
5.1.2 Finite element results of beam shell coupling of fillet weld in case of twisting
Figure 5.14 shows a cantilever beam of rectangular section profile under the given
loading of 10KN acting upward at the free end. Coupling of beam elements and shell
elements were used.
0.0
50.0
100.0
150.0
200.0
250.0
box I C circle Z X
St
re
ss
(M
Pa
)
Beam Section
Analytic
FE (Beam Element)
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
99
Figure 5.14. General view of the beam-shell coupling of rectangular cross- section
(twisting)
The finite element and analytical results of box, I, C, circular, Z and X of fillet weld in
case of twisting were summarized in figure 5.15.
Figure 5.15. Stress of analytic and finite element results of various beam sections of
fillet weld (twisting)
It is seen from Figure 5.15 that the predicted stress curves are in good agreement with
the analytical results for all six cases of the beam members in case of twisting. It is
observed from the same figure that the largest stress is found at the X beam cross
section while the box beam profile has the smallest stress.
0.0
50.0
100.0
150.0
200.0
250.0
box I C circle Z X
St
re
ss
(M
Pa
)
Beam section
Analytic
FE
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
100
5.2. V welded joint
In this case, the weld thickness is equal to the beam thickness. Coupling of beams and
shells has been considered and implemented in this section. The Coupling constraint
technique of beam to shell can be generated by Abaqus [59].
5. 2.1 Finite element results of beam shell coupling of V-welded joint in case of
bending.
Figure 5.16 shows a cantilever beam of rectangular section profile under the given
loading of 10KN acting upward at the free end. Coupling of beam elements and shell
elements were used.
Figure 5.16. General view of the Beam-shell coupling of Rectangular- section of V
welded joint
Figure 5.17 shows the summary of the finite element and analytical results of box, I, C,
circular, Z and X of fillet weld in case of twisting.
Table 5.2 shows that the finite element results of stress in the fillet weld are varied from
40.5MPa for box beam profile and 323MPa for X beam profile. From chapter 2, the
analytical results of stress in the fillet weld are varied from 45.8MPa for box beam
profile and 290MPa for X beam profile. It is clear that the finite element results verify
the analytical solution. It can be seen that the analytical solution can be replaced by
beam element solution to represent the fillet weld.
Stress Analysis of Corner welded Joints Structure by Modern Numerical
Figures 5.
Table. 5.2 Analytical and FE Results of beams of V weld (Bending Moment)
Specimen
V joint
box
I
C
circle
Z
X
5. 2.2 Finite
twisting.
Figure 5.18 shows the summary of the finite element and analytical results of box, I, C,
circular, Z and
Table 5.3 shows that the finite element results of stress in the V weld are varied from
43MPa for box beam profile and
analytical results of stress in the fillet weld are varied from 47MPa for box beam profile
and 293MPa for X beam profile. It is clear that the finite element results verify the
analytical solution. It can b
element solution to represent the fillet weld.
St
re
ss
(M
Pa
)
17 str
Analytic
Stress MPa
45.8
88.0
100.1
123.3
207.8
289.9
element
X of
0.0
50.0
100.0
150.0
200.0
250.0
300.0
350.0
St
re
ss
(M
Pa
)
ess of analytic and finite
results of
V weld in case of twisting.
box I
weld (Bending)
FE (Beam Element)
Stress MPa
40.5
82.3
95.0
118.3
202.1
323.0
beam
323.8MPa for X beam profile. From chapter 2, the
e seen that the analytical solution can be replaced by beam
C
Beam
element
shell coupling of V
circle
Section
results of v
-welded joint in case of
Z X
-Experimental Approach
arious beam
Analytic
FE (Beam Element)
sections of V
.
101
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
102
Figure 5.18 stress of Analytic and finite Element Results of Various Beam Sections of
V weld for twisting moment
Table. 5.3 Analytical and FE Results of beams of V weld (twisting moment)
Specimen
V joint
Analytic
Stress (MPa)
FE (Beam Element)
Stress (MPa)
box 47.4 43.0
I 92.8 82.3
C 104.0 95.0
circle 124.8 122.5
Z 228.6 202.1
X 293.4 323.8
5.4 Comparison
5.4.1 Comparison between V and fillet welded joints in case of bending.
Figure 5.19 shows the comparison between V and fillet welded joints of box, I, C,
circular, Z and X of V weld in case of bending.
It is seen from Figure 5.19-20 and Table. 5.3 that for rectangular beam profile ,stress
was minimized about 36% in case of using V weld while for the other sections , the
stress was increased from 11.7% for I beam profile to 55% for X beam profile in case
of using V weld .
Figure 5.19 shows the comparison between V and fillet welded joints of box, I, C,
circular, Z and X of V weld in case of bending.
0.0
50.0
100.0
150.0
200.0
250.0
300.0
350.0
box I C circle Z X
St
re
ss
(M
Pa
)
Beam Sections
Analytic
FE (Beam
Element)
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
103
Figure 5.19 finite Element Results of Various Beam Sections of V weld and fillet weld
of bending moment.
Figure 5.20. Stress change of various beam sections (bending)
Table. 5.3. FE Results of V and fillet joints of bending.
Specimen
(Bending)
FE Results, Stress of
V Joint (MPa)
FE Results, stress of
Fillet Joint (MPa)
box 40.5 55.0
I 82.3 72.7
C 95.0 78.8
circle 118.3 94.4
Z 202.1 156.5
X 323.0 207.1
0.0
100.0
200.0
300.0
400.0
box I C circle Z X
St
re
ss
(M
Pa
)
Beam Section
V joint
Fillet joint
-40.0
-30.0
-20.0
-10.0
0.0
10.0
20.0
30.0
40.0
box I C circle Z X
st
re
ss
di
ffe
re
nc
e
(V
-F
ill
et
)/
V
x1
00
Beam Section
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
104
5.4.2 Comparison between V and fillet welded joints in case of twisting.
It is observed from figures. 5.21-22 that the largest stress is found at the X beam cross
section. Also, it is shown from these figures that Fillet welds is better for beam sections
I, C, Circular, Z and X. we can see also the stress was increased from 11.7% for I
section to 55% for X section in case of using V weld instead of fillet weld. This means:
these cross sections are recommended to be welded using fillet weld while a rectangular
cross section is recommended to be welded by using V weld .
Figure 5.21. Finite element results of various beam sections of fillet and V weld
(twisting)
Figure 5.22. Stress change of various beam sections (twisting)
0.0
50.0
100.0
150.0
200.0
250.0
300.0
350.0
box I C circle Z X
St
re
ss
(M
Pa
)
Beam Section
V Joint
Fillet Joint
-40
-30
-20
-10
0
10
20
30
40
box I C circle Z X
st
re
ss
di
ffe
re
nc
e
(V
-F
ill
et
)/
V
x1
00
Beam Section
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
105
5.5 Discussion
The finite element simulations were performed for all the six specimens. The numerical
analyses were carried out to represent the mechanical performance of the fillet weld and
V weld beam-sections under bending. The study is summarized and concluded as
follows. The predicted stress curves are in good agreement with the analytical results for
all six cases of the beam members. The finite element results verify the analytical
results.
Also, the results show that V weld gives high stress for sections I,C , Circular, Z, and
X cross sections while a rectangular cross section gives lower stress when it welded by
using V joint.
Based on numerical simulation beam element technique for V and fillet welded joints,
we can conclude that the analytical approach can be replaced by using a beam element
numerical approach.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
106
Chapter 6
V-weld modeling using solid and shell elements
This chapter focuses on the development of a predictive methodology for V welded
structures using shell and solid element. However, three-dimensional finite element
analysis of complete structural hollow sections can be complex and time-consuming.
Due to the complex nature of the finite element analysis codes, this method has limited
application. It can be used in research area but cannot be widely used by structural
engineers in their real-world projects. Therefore, there is a need to develop a simplified
modeling method that can be implemented by using commonly available commercial
software and easily employed.
Finite element models with different modeling techniques and meshing with various
size and types of elements were created and analyzed. In the modeling certain
assumption were made to simplify the model. Parent metal and welded metal had the
same material properties which are steel ( ଷ
6.1 V-weld modelling using solid elements
Modelling welds with deformable solid elements are widely used because of its
simplicity in modelling work and its accuracy in results since the stiffness of welds can
be modelled accurately. In solid element models, the geometry and stiffness of the
welds in a welded joint can easily be represented by using solid elements, see Figure
6.1. But, three-dimensional finite element analysis of a complete structural hollow
sections can be complex and time-consuming.
Three-dimensional finite element analysis of complete structural hollow sections can be
complex and time-consuming. Therefore, there is a need to develop a simplified
modeling method that can be implemented by using commonly available commercial
software and easily employed.
When modeling the welds with solid elements in shell element models, a special
technique is necessary to connect these two different types of elements. The reason for
that is the fact that solid elements have three degrees of freedom in each node while
shell elements have five degrees of freedom in each node. The bending moments from
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
107
shell elements need to be transferred to solid elements. There are a couple of techniques
available to perform it. One method when connecting the solid elements to the shell
elements is using shell-to-solid-coupling see figure 6.1. However this procedure can be
done automatically by Abaqus[59],a finite element software.
Figure 6.1 Modelling of the V-weld with coupling of shell and solid elements.
Figure 6.2 Stress distribution along weld path for different mesh size and shape
function.
Figure 6.2 shows that the data of interest are substantially independent of the mesh
and/or the polynomial degree of elements. Generally, good agreement of the stress
distribution was obtained between the mesh size 5mm with quadratic polynomial and
mesh size 2.5mm with linear polynomial.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
108
6.1.1 Finite element results of solid element model of V- weld in case of bending.
6.1.1.1 Rectangular cross-section
Figures 6.3-4 show the stress distribution of top and vertical welds of a cantilever
rectangular hollow section joint under the given loading of 10KN acting upward at the
free end, respectively. Solid elements and shell elements were used in this analysis.
The figures show that the stress components S33, S13 and S23 are small compared with
S11, S22 and S12. We conclude that the model cab be modeled by using shell elements.
Also, the figures show stress concentrations in the edges of the box profile. The
distribution cannot be captured by using theoretical formula or by using beam elements.
Figure 6.3. Stress distribution along horizontal weld path of box profile –bending.
Figure 6.4 . Stress distribution along vertical weld path of box section-bending.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
109
6.1.1.2 I section
Figures 6.5-6 illustrate the stress distribution of top and vertical welds of a cantilever
beam of I profile under the given loading of 10KN acting upward at the free end,
respectively. Solid elements were used to model welded joints and the area near welded
joints while shell elements were used away from welded joints.
Also, we can see that the stress components S33, S13 and S23 are small compared with
S11, S22 and S12. We conclude that the model can be modeled by using shell elements.
Also, the figures show stress concentrations in the edges of the box profile. The
distribution cannot be captured by using theoretical formula or by using beam elements.
Figure 6.5 . Stress distribution along horizontal weld path of I section-bending.
Figure 6.6 . Stress distribution along vertical weld path of I section-bending.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
110
6.1.1.3 C section
Figure 6.7 shows the stress distribution in case of bending on the C profile for solid
elements of top weld. Also, we can see that the stress components S33, S13 and S23 are
small compared with S11, S22 and S12. We conclude that the model can be modeled by
using shell elements. The figures show stress concentrations in the side of the stiffened
edge of the C profile.
Figure 6.7 . Stress distribution along horizontal weld path of C section-bending.
6.1.1.4 Circular section
The stress distribution in case of bending on the circular profile for solid elements of top
weld is shown in figure 6.8. Also, we can see that the stress components S33, S13 and
S23 are small compared with S11, S22 and S12. We conclude that the model can be
modeled by using shell elements.
Figure 6.8 . Stress distribution along top weld path of circular section-bending.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
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6.1.1.5 Z – section
The stress distribution in case of bending on the Z profile for solid elements of top weld
is shown in figure 6.9. Also, we can see that the stress components S33, S13 and S23
are small compared with S11, S22 and S12. We conclude that the model can be modeled
by using shell elements. The figures show stress concentrations in the side of the
stiffened edge of the Z profile.
Figure 6.9 . Stress distribution along horizontal weld path ofZ section-bending.
6.1.1.6 X – section
The stress distribution in case of bending on the X profile for solid elements on the weld
path is shown in figure 6.10. Also, we can see that the stress components S33, S13 and
S23 are small compared with S11, S22 and S12. We conclude that the model can be
modeled by using shell elements. The figures show stress concentrations in the top of
the X profile.
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Figure 6.10 . Stress distribution along horizontal weld path of X section-bending.
6.1.2 Finite element results of solid element model of V- weld in case of twisting.
6.1.2.1 Rectangular cross-section
Figures 6.11 show the stress distribution of top weld of a cantilever rectangular hollow
section joint under the given loading of 10KN acting upward at the free end. Solid
elements and shell elements were used in this analysis. The figure shows stress
concentrations in the side of the stiffened edges of the box profile.
Figure 6.11 . Stress distribution along horizontal weld path of box section-twisting.
6.1.2.2 I-section
Figures 6.12 illustrate the stress distribution of top weld of a cantilever beam of I profile
under the given loading of 1KN acting upward at the free end. Solid elements were used
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to model welded joints and the area near welded joints while shell elements were used
away from welded joints. The figure shows stress concentrations in the edges of the I
profile.
Figure 6.12. Stress distribution along horizontal weld path of I section-twisting.
6.1.2.3 C-section
Figures 6.13 illustrate the stress distribution of top weld of a cantilever beam of C
profile under the given loading of 1KN acting upward at the free end. The figure shows
stress concentrations in the stiffened edge of the C profile
Figure 6.13 . Stress distribution along horizontal weld path of C section-twisting.
6.1.2.4 Circular-section
Figure 6.14 shows the stress distribution of top weld of a cantilever beam of circular
profile under the given loading of 10KN acting upward at the free end. We can see that
the stress distribution is maximum on top of the pipe.
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Figure 6.14 . Stress distribution along top weld path of circular section-twisting.
6.1.2.5 Z-section
Figure 6.15 illustrates the stress distribution of top weld of a cantilever beam of Z
profile under the given loading of 1KN acting upward at the free end. Solid elements
were used to model welded joints and the area near welded joints while shell elements
were used away from welded joints. The figure shows stress concentrations in the edges
of the Z profile.
Figure 6.15. Stress distribution along horizontal weld path of Z section-twisting.
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6.1.2.6 X-section
Figure 6.16 illustrates the stress distribution of top weld of a cantilever beam of X
profile under the given loading of 1KN acting upward at the free end. Solid elements
were used to model welded joints and the area near welded joints while shell elements
were used away from welded joints. The figure shows stress concentrations in the top
edge of the X profile.
Figure 6.16. Stress distribution along weld path of the X section - twisting.
6.2 V-Weld modelling using shell elements
Shell elements with specified thickness in the intersection region of welded joints can
be used to represent the stiffness of the welds in V-welded joints as shown in figure
6.17. In this analysis, the weld thickness is equal to the plate thickness as shown in
figure below.
Figure. 6.17 Modelling welds using shell elements.
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6.2.1 Finite Element Results of Shell element model of V- weld in case of bending.
6.2.1.1 Rectangular cross-section
Figures 6.18 and 6.19 show the stress distribution of top and vertical weld of a
cantilever rectangular hollow section joint under the given loading of 10KN acting
upward at the free end, respectively. Shell elements were used in this analysis. The
figures show stress concentrations in the side of the stiffened edges of the box profile
Figure 6.18.Stress distribution along horizontal weld of rectangular box-bending.
Figure 6.19.Stress distribution along vertical weld path of rectangular box-bending.
6.2.1.2 I cross-section
Figures 6.20and 6.21 show the stress distribution of top and vertical weld of a cantilever
I section joint under the given loading of 10KN acting upward at the free end,
respectively. Shell elements were used in this analysis. The figure shows stress
concentrations in the side of the edges of I profile.
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Figure 6.20. Stress distribution along horizontal weld path of I section-bending.
Figure 6.21 Stress distribution along vertical weld path of I section- bending.
6.2.1.3 C cross-section
Figures 6.22 and 6.23 show the stress distribution of top and vertical weld of a
cantilever C section joint under the given loading of 10KN acting upward at the free
end, respectively. Shell elements were used in this analysis. The figure shows stress
concentrations in the side of the stiffened edge of the C profile.
Figure 6.22. Stress distribution along horizontal weld path of C section-bending.
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Figure 6.23 Stress distribution along vertical weld path of C section-bending.
6.2.1.4 Circular cross-section
Figures 6.24and 6.25 show the stress distribution of top and vertical weld of a cantilever
C section joint under the given loading of 10K N acting upward at the free end,
respectively. Shell elements were used in this analysis. The figure shows stress
concentrations on the top of the pipe.
Figure 6.24. Stress distribution along top weld path of Circular section-bending.
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Figure 6.25 Stress distribution along vertical weld path of Circular section-bending.
6.2.1.5 Z cross-section
Figures 6.26 and 6.27 show the stress distribution of top and vertical weld of a
cantilever Z section joint under the given loading of 10KN acting upward at the free
end, respectively. Shell elements were used in this analysis. The figure shows stress
concentrations in the side of the stiffened edge of the Z profile.
Figure 6.26. Stress distribution along horizontal weld path of Z section-bending.
Figure 6.27 Stress distribution along vertical weld path of Z section-bending.
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6.2.1.6 X cross-section
Figures 6.28 shows the stress distribution on the weld path of a cantilever X section
joint under the given loading of 10KN acting upward at the free end. Shell elements
were used in this analysis. The figure shows stress concentrations on the top of the X
profile.
Figure 6.28. Stress distribution along weld path of X section -bending
6.2.2 Finite element results of shell element model of V- weld in case of twisting.
6.2.2.1 Rectangular section
Figures 6.29 and 6.30 show the stress distribution of top and vertical of weld path of a
cantilever box section joint under the given loading of 10KN acting upward at the free
end, respectively. Shell elements were used in this analysis. The figures show stress
concentrations in the side of the stiffened edges of the box profile.
Figure 6.29. Stress distribution along horizontal weld path of box profile-twisting.
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Figure 6.30. Stress distribution along vertical weld path of box profile -twisting.
6.2.2.2 I section
Figure 6.31 and 6.32 show the stress distribution of top and vertical of weld path of a
cantilever I section joint under the given loading of 1KN acting upward at the free end.
Shell elements were used in this analysis. The figure shows stress concentrations in the
side of the stiffened edges of the I profile.
Figure 6.31. Stress distribution along horizontal weld path of I section-twisting.
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Figure 6.32. Stress distribution along vertical weld path of I section-twisting.
6.2.2.3 C section
Figure 6.33 and 6.34 show the stress distribution of top and vertical of weld path of a
cantilever C section joint under the given loading of 1KN acting upward at the free end,
respectively. Shell elements were used in this analysis. The figure shows stress
concentrations in the side of the stiffened edge of the C profile.
Figure 6.33. Stress distribution along horizontal weld path of C section-twisting.
Figure 6.34. Stress distribution along vertical weld path of C section-twisting.
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6.2.2.4 Circular section
Figure 6.35 and 6.36 show the stress distribution of top and vertical of weld path of a
cantilever C section joint under the given loading of 10KN acting upward at the free
end, respectively. Shell elements were used in this analysis. The figure shows stress
concentrations on the top of the pipe.
Figure 6.35. Stress distribution along top weld path of Circular section-twisting.
Figure 6.36. Stress distribution along vertical weld path of circular section-twisting.
6.2.2.5 Z section
Figure 6.37 and 6.38 show the stress distribution of top and vertical of weld path of a
cantilever Z section joint under the given loading of 1KN acting upward at the free end,
respectively. Shell elements were used in this analysis. The figure shows stress
concentrations in the edge of the Z profile.
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Figure 6.37. Stress distribution along horizontal weld path of Z section-twisting.
Figure 6.38. Stress distribution along vertical weld path of Z section-twisting.
6.2.2.6 X section
Figure 6.39 shows the stress distribution along weld path of a cantilever X section joint
under the given loading of 1KN acting upward at the free end. Shell elements were used
in this analysis. The figure shows stress concentrations in the edges of the X profile.
Figure 6.39. Stress distribution along weld path for X section-twisting.
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6.3 Comparison between shell and solid elements
6.3.1 Case (a) Bending
6.3.1.1 Rectangular cross-section
Figure 6.40 shows the stress distribution for shell and solid elements along weld path of
a cantilever box section joint under the given loading of 10KN acting at the free end. It
is clear that stress curves of solid element are in good agreement with stress curves of
shell element.
Figure 6.40 . Comparison of stress distribution between shell and solid elements along
horizontal weld for box section-bending.
6.3.1.2 I- section
Figure 6.41 shows the stress distribution for shell and solid elements along weld path of
a cantilever I section joint under the given loading of 1KN acting at the free end. We
can see that stress curves of shell element are in good agreement with stress curves of
solid element.
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Figure 6.41 . Comparison of stress distribution between shell and solid elements along
horizontal weld path of I section-bending.
6.3.1.3 C- section
Figure 6.42 shows the stress distribution for shell and solid elements along weld path of
a cantilever C section joint under the given loading of 1KN acting at the free end. We
can see that stress curves of shell element are in good agreement with stress curves of
solid element.
Figure 6.42 . Comparison of stress distribution between shell and solid elements along
horizontal weld path of C section-bending.
6.3.1.4 Circular- section
Figure 6.43 shows the stress distribution for shell and solid elements along weld path of
a cantilever circular section joint under the given loading of 10KN acting at the free
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end. We can see that stress curves of shell element are in good agreement with stress
curves of solid element.
Figure 6.43 . Comparison of stress distribution between shell and solid elements along
top weld of circular section-bending.
6.3.1.5 Z- section
Figure 6.44 shows the stress distribution for shell and solid elements along weld path of
a cantilever Z section joint under the given loading of 1KN acting at the free end. We
can see that stress curves of shell element are in good agreement with stress curves of
solid element.
Figure 6.44 . Comparison of stress distribution between shell and solid elements along
horizontal weld path of Z section-bending.
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6.3.1.6 X - section
Figure 6.45 shows the stress distribution for shell and solid elements along weld path of
a cantilever X section joint under the given loading of 1KN acting at the free end. We
can see that stress curves of shell element are in good agreement with stress curves of
solid element.
Figure 6.45. Comparison of stress distribution between shell and solid elements along
top weld path of X section -bending.
6.3.2 Case (b) Twisting
6.3.2.1 Box- section
Figure 6.46 shows the stress distribution for shell and solid elements along weld path of
a cantilever box section joint under the given loading of 10KN acting at the free end.
We can see that stress curves of shell element are in good agreement with stress curves
of solid element.
Figure 6.46 . Comparison of stress distribution between shell and solid elements along
horizontal weld of box section -twisting.
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6.3.2.2 I - section
Figure 6.47 shows the stress distribution for shell and solid elements along weld path of
a cantilever I section joint under the given loading of 1KN acting at the free end. We
can see that stress curves of shell element are in good agreement with stress curves of
solid element.
Figure 6.47 . Comparison of stress distribution between shell and solid elements along
horizontal weld path of I section-twisting.
6.3.2.3 C - section
Figure 6.48 shows the stress distribution for shell and solid elements along weld path of
a cantilever C section joint under the given loading of 1KN acting at the free end. It is
clear that stress curves of shell element are in good agreement with stress curves of solid
element.
Figure 6.48 . Comparison of stress distribution between shell and solid elements along
horizontal weld path of C section-twisting.
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6.3.2.4 Circular - section
Figure 6.49 shows the stress distribution for shell and solid elements along weld path of
a cantilever beam of circular section joint under the given loading of 10KN acting at the
free end. We can see that stress curves of shell element are in good agreement with
stress curves of solid element.
Figure 6.49 . Comparison of stress distribution between shell and solid elements along
top weld path of circular section-twisting.
6.3.2 .5 Z - section
Figure 6.50 shows the stress distribution for shell and solid elements along weld path of
a cantilever beam of Z section joint under the given loading of 1KN acting at the free
end. We can see that stress curves of shell element are in good agreement with stress
curves of solid element.
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Figure 6.50. Comparison of stress distribution between shell and solid elements along
horizontal weld path of Z section-twisting.
6.3.2.6 X - section
Figure 6.51 shows the stress distribution for shell and solid elements along weld path of
a cantilever beam of X section joint under the given loading of 1KN acting at the free
end. We can see that stress curves of shell element are in good agreement with stress
curves of solid element.
Figure 6.51 . Comparison of stress distribution between shell and solid elements along
weld path of X section-twisting
6.4 Discussion
The numerical modeling of the v-welded joint connection was performed using Abaqus
[59 ], the commercial finite element package. The developed FE model is based on the
assumption of linear elasticity and small strains/displacements. Shell and solid elements
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have been used for modeling both the beams and the welds in this study. Quadratic solid
elements (20-noded) with three translational degrees-of-freedom at each node and
quadratic shell elements (8-noded) with 6 degrees-of-freedom at each node were used to
model the weld and the beam structure in order to accurately capture any non-linear
stress gradients on the weld path. The study is summarized and concluded that stress
curves of solid element are in good agreement with stress curves of shell element.
From the FEA results of the solid element model, a generally linear stress distribution
across the thickness of the beams is observed. Therefore, the use of shell elements for
the analysis of the beam-joint connection is appropriate.
The results show that the normal stresses in the1 or 2-direction are the highest at the
weld path (top of the path) on the top surface, and at horizontal weld path of the beam.
It can be observed that the 1 or 2-direction component of normal stress causes the most
damage. The values of the Von-Mises equivalent stresses are very close to the values of
the stresses in the 1 or 2-direction (absolute value). This is expected as the stresses in
the1 or 2-direction are much larger than the other components of stress.
This means that the strains and stresses normal to the weld (i.e., 1 or 2 -direction in this
study) are mainly responsible for plasticity/crack initiation and propagation.
Also the results show that the stress concentrations increase in the stiffened areas.
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Chapter 7
Fillet weld modeling using solid and shell elements
This chapter focuses on the development of a predictive methodology for fillet welded
structures using shell and solid element. However, three-dimensional finite element
analysis of complete structural hollow sections can be complex and time-consuming.
Due to the complex nature of the finite element analysis codes, this method has limited
application. It can be used in research area but cannot be widely used by structural
engineers in their real-world projects. Therefore, there is a need to develop a simplified
modeling method that can be implemented by using commonly available commercial
software and easily employed.
Finite element models with different modeling techniques and meshing with various
size and types of elements were created and analyzed. In the modeling certain
assumption were made to simplify the model. Parent metal and welded metal had the
same material properties which are steel ( ଷ
7.1 Fillet weld modelling using solid elements
Modelling welds with deformable solid elements are widely used because of its
simplicity in modelling work and its accuracy in results since the stiffness of welds can
be modelled accurately. In solid element models, the geometry and stiffness of the
welds in a welded joint can easily be represented by using solid elements, see Figure
7.1. But, three-dimensional finite element analysis of a complete structural hollow
sections can be complex and time-consuming.
Figure 7.1 Modelling of the welds with solid elements.
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Three-dimensional finite element analysis of complete structural hollow sections can be
complex and time-consuming.
7.1.1 Finite element results of solid element model of fillet- weld in case of bending.
7.1.1.1 Rectangular cross-section
Figures 7.2 shows the stress distribution in top weld path of a cantilever rectangular
hollow section joint under the given loading of 10K N acting upward at the free end.
Solid elements and shell elements were used in this analysis.
The figure shows that the stress components S33, S13 and S23 are small compared with
S11, S22 and S12. We conclude that the model can be modeled by using shell elements.
Also, the figures show stress concentrations in the edges of the box profile. The
distribution cannot be captured by using theoretical formula or by using beam elements.
Figure 7.2 .Stress distribution along horizontal weld path of box profile-bending.
7.1.1.2 I-section
Figure7.3 shows the stress distribution in top weld path of a cantilever beam of I profile
under the given loading of 10KN acting upward at the free end. Solid elements and shell
elements were used in this analysis. The figure shows that the stress components S33,
S13 and S23 are small compared with S11, S22 and S12. We conclude that the model
can be modeled by using shell elements.
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Figure 7.3 .Stress distribution along horizontal weld path of I section-bending.
7.1.1.3 C-section
Figure7.4 shows the stress distribution in top weld path of a cantilever beam of C profile
under the given loading of 10KN acting upward at the free end. Solid elements and shell
elements were used in this analysis. The figure shows that the stress components S33,
S13 and S23 are small compared with S11, S22 and S12. We conclude that the model
can be modeled by using shell elements.
Figure 7.4 .Stress distribution along horizontal weld of C section-bending.
7.1.1.4 Circular-section
Figure7.5 shows the stress distribution in top weld path of a cantilever beam of circular
profile under the given loading of 10KN acting upward at the free end. Solid and shell
elements were used in this analysis.
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Figure 7.5 .Stress distribution along top weld path of circular section-bending.
The results show stress concentration on the top of the v-weld.
7.1.1.5 Z-section
Figure7.6 shows the stress distribution in top weld path of a cantilever beam of Z profile
under the given loading of 10KN acting upward at the free end. Solid elements and shell
elements were used in this analysis. The figure shows that the stress components S33,
S13 and S23 are small compared with S11, S22 and S12. We conclude that the model
can be modeled by using shell elements.
Figure 7. 6.Stress distribution along horizontal weld path of Z section-bending.
7.1.1.6 X-section
Figure7.7 shows the stress distribution in top weld path of a cantilever beam of Z profile
under the given loading of 10KN acting upward at the free end. Solid elements and shell
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elements were used in this analysis. The figure shows that the stress components S33,
S13 and S23 are small compared with S11,S22 and S12. We conclude that the model
can be modeled by using shell elements.
Figure 7. 7.Stress distribution along weld of X section-bending.
7.1.2 Finite element results of solid element model of fillet- weld in case of twisting.
7.1.2.1 Rectangular cross-section
Figure7.8 shows the stress distribution in top weld path of a cantilever beam of box
profile under the given loading of 10KN acting at the free end. Solid elements and shell
elements were used in this analysis. It is clear that the normal stress S22 is maximum.
Also, there is stress concentration in the edges of the weld.
Figure 7.8 .Stress distribution along horizontal weld path of box section-twisting.
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7.1.2.2 I-section
Figure7.9 shows the stress distribution in top weld path of a cantilever beam of I profile
under the given loading of 1KN acting at the free end. Solid elements and shell elements
were used in this analysis. It is clear that the normal stress, S22 is maximum. Also, there
is stress concentration in the edges of the weld.
Figure 7.9 .Stress distribution along horizontal weld path of I section-twisting.
7.1.2.3 C-section
Figure7.10 shows the stress distribution in top weld path of a cantilever beam of C
profile under the given loading of 1KN acting at the free end. Solid elements and shell
elements were used in this analysis. It is clear that the normal stress, S22 is maximum.
Also, there is stress concentration in the edges of the weld.
Figure 7.10 .Stress distribution along horizontal weld path of C section-twisting.
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7.1.2.4 Circular -section
Figure7.11 shows the stress distribution in top weld path of a cantilever beam of circular
profile under the given loading of 10KN acting at the free end. Solid elements and shell
elements were used in this analysis. It is clear that the normal stress, S22 is maximum.
Also, there is stress concentration in the edges of the weld.
Figure 7. 11.Stress distribution along top weld of circular section-twisting.
7.1.2.5 Z –section
Figure7.12 shows the stress distribution in top weld path of a cantilever beam of I
profile under the given loading of 1KN acting at the free end. Solid elements and shell
elements were used in this analysis. It is clear that the normal stress, S22 is maximum.
Also, there is stress concentration in the edges of the weld.
Figure 7. 12.Stress distribution along horizontal weld path of Z section-twisting.
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7.1.2.6 X –section
Figure7.13 shows the stress distribution along weld path of a cantilever beam of X
profile under the given loading of 1KN acting at the free end. Solid elements and shell
elements were used in this analysis. It is clear that the normal stress, S22 is maximum.
Also, there is stress concentration on top of the weld.
Figure 7.13 .Stress distribution along weld path of X section-twisting.
7.1.3 Discusion
All of the 6 components of the stress distributions (i.e., normal and shear) and the Von-
Mises equivalent stresses at the connection are considered. It is important to examine
these stresses in detail in order to obtain insight into the components of stress that cause
the most damage. The dominating role of the normal stresses is in the 1 and 2 -direction
in the thesis.
7.2 Fillet weld modeling using oblique shell elements
Shell elements generally have five degrees of freedom per node. Shell elements used in
the analyses contained within this thesis contain six degrees of freedom per node. Shell
elements are the main elements used in this section analysis.
In shell element models, the fillet welds in a welded joint can be represented using
oblique shell elements. Both the stiffness and geometry of the welds can be correctly
represented by utilizing this weld modelling technique. The attached plate should be
joined to the main plate in the intersection as shown in Figure 7.14. The length of
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inclined shell elements can be chosen as shown in this figure7.14. The thickness of
oblique shell elements can be defined same as the throat thickness of welds.
Rectangular cross section
I cross-section
Figure 7.14 Weld modelling using oblique shell elements Shell Elements
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7.2.1 Finite element results of shell element model of fillet- weld in case of bending.
7.2.1.1 Rectangular -section
Figure7.15 shows the stress distribution in top weld path of a cantilever beam of box
profile under the given loading of 10KN acting at the free end. Shell elements were used
in this analysis. It is clear that the normal stress S22 is maximum. Also, there is stress
concentration in the edges of the weld.
Figure 7.15 .Stress distribution along horizontal weld path of rectangular box-bending.
7.2.1.2 I-section
Figure7.16 shows the stress distribution in top weld path of a cantilever beam of I
profile under the given loading of 10KN acting at the free end. Shell elements were used
in this analysis. It is clear that the normal stress S22 is maximum. Also, there is stress
concentration in the stiffened area of the weld.
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Figure 7.16 .Stress distribution along horizontal weld path of I section-bending.
7.2.1.3 C-section
Figure7.17 shows the stress distribution in top weld path of a cantilever beam of C
profile under the given loading of 10KN acting at the free end. Shell elements were used
in this analysis. It is clear that the normal stress S22 is maximum. Also, there is stress
concentration in the stiffened side of the weld.
Figure 7.17 .Stress distribution along horizontal weld path of C section-bending.
7.2.1.4 Circular –section
Figure7.18 shows the stress distribution in top weld path of a cantilever beam of circular
profile under the given loading of 10KN acting at the free end. Shell elements were used
in this analysis. It is clear that the normal stress S22 is maximum. Also, there is stress
concentration in the top of the weld.
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Figure 7.18 .Stress distribution along top weld path of cricular section-bending.
7.2.1.5 Z -section
Figure7.19 shows the stress distribution in top weld path of a cantilever beam of Z
profile under the given loading of 10KN acting at the free end. Shell elements were used
in this analysis. It is clear that the normal stress S22 is maximum. Also, there is stress
concentration in the stiffened edge of the weld.
Figure 7.19 .Stress distribution along horizontal weld path of Z section -bending.
7.2.1.6 X –section
Figure7.20 shows the stress distribution in weld path of a cantilever beam of X profile
under the given loading of 10KN acting at the free end. Shell elements were used in this
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analysis. It is clear that the normal stress S22 is maximum. Also, there is stress
concentration in the top area of the weld.
Figure 7. 20.Stress distribution along weld path of X section-bending.
The figure shows that the normal stresses in the 2-direction are the highest with a
maximum value of approximately 800MPa at the weld path (top of the path) on the top
surface, and 100 MPa at horizontal weld path of the beam. It can be observed that the 2-
direction component of normal stress causes the most damage. The values of the Von-
Mises equivalent stresses are very close to the values of the stresses in the 2-direction
(absolute value). This is expected as the stresses in the 2-direction are much larger than
the other components of stress.
This means that the strains and stresses normal to the weld (i.e.,1 or 2 -direction in this
study) are mainly responsible for plasticity/crack initiation and propagation.
7.2.2 Finite element results of shell element model of fillet- weld in case of twisting.
7.2.2.1 Rectangular cross-section
Figure7.21 shows the stress distribution in top weld path of a cantilever beam of box
profile under the given loading of 10KN acting at the free end. Shell elements were used
in this analysis. It is clear that the normal stress S22 is maximum. Also, there is stress
concentration in the edges of the weld.
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Figure 7.21 .Stress distribution along horizontal weld path of box section -twisting.
7.2.2.2 I-section
Figure7.22 shows the stress distribution in top weld path of a cantilever beam of box
profile under the given loading of 1KN acting at the free end. Shell elements were used
in this analysis. It is clear that the normal stress S22 is maximum. Also, there is stress
concentration in the edges of the weld.
Figure 7.22 .Stress distributions along horizontal weld path of I section-twisting.
7.2.2.3 C-section
Figure7.23 shows the stress distribution in top weld path of a cantilever beam of C
profile under the given loading of 1KN acting at the free end. Shell elements were used
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in this analysis. It is clear that the normal stress S22 is maximum. Also, there is stress
concentration in the edges of the weld.
Figure 7.23 .Stress distribution along horizontal weld path of C section-twisting.
7.2.2.4 circular section
Figure7.24 shows the stress distribution in top weld path of a cantilever beam of pipe
profile under the given loading of 10KN acting at the free end. Shell elements were used
in this analysis. It is clear that the normal stress S22 is maximum. Also, there is stress
concentration in the top of the weld.
Figure 7.24 .Stress distribution along horizontal weld path of pipe section-twisting.
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7.2.2.5 Z- section
Figure7.25 shows the stress distribution in top weld path of a cantilever beam of Z
profile under the given loading of 1KN acting at the free end. Shell elements were used
in this analysis. It is clear that the normal stress S22 is maximum. Also, there is stress
concentration in the edges of the weld.
Figure 7.25 .Stress distribution along horizontal weld path of Z section-twisting.
7.2.2.6 X- section
Figure7.26 shows the stress distribution in weld path of a cantilever beam of X profile
under the given loading of 1KN acting at the free end. Shell elements were used in this
analysis. It is clear that the normal stress S22 is maximum. Also, there is stress
concentration in the edge of the weld.
Figure 7.26 .Stress distribution along weld path of X section -twisting.
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7.3 Comparison between solid and shell elements of fillet welds
7.3.1 Case(a) Bending
7.2.1.1 Rectangular cross-section
Figure 7.27 shows the stress distribution for shell and solid elements along weld path of
a cantilever beam of box section joint under the given loading of 10KN acting at the
free end. We can see that stress curves of shell element are in good agreement with
stress curves of solid element.
Figure 7.27 . Comparison of stress distribution between shell and solid elements along
horizontal weld path of box section-bending.
7.3.1.2 I-section
Figure 7.28 shows the stress distribution for shell and solid elements along weld path of
a cantilever beam of I section joint under the given loading of 1KN acting at the free
end. We can see that stress curves of shell element are in good agreement with stress
curves of solid element.
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Figure 7.28 . Comparison of stress distribution between shell and solid elements along
horizontal weld path of I section-bending.
7.3.1.3 C-section
Figure 7.29 shows the stress distribution for shell and solid elements along weld path of
a cantilever beam of C section joint under the given loading of 1KN acting at the free
end. We can see that stress curves of shell element are in good agreement with stress
curves of solid element.
Figure 7.29 . Comparison of stress distribution between shell and solid elements along
horizontal weld path of C section -bending.
7.3.1.4 Circular-section
Figure 7.30 shows the stress distribution for shell and solid elements along weld path of
a cantilever beam of circular section joint under the given loading of 10KN acting at the
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free end. We can see that stress curves of shell element are in good agreement with
stress curves of solid element.
Figure 7.30 . Comparison of stress distribution between shell and solid elements along
horizontal weld path of Circular section -bending.
7.3.1.5 Z-section
Figure 7.31 shows the stress distribution for shell and solid elements along weld path of
a cantilever beam of Z section joint under the given loading of 1KN acting at the free
end. We can see that stress curves of shell element are in good agreement with stress
curves of solid element.
Figure 7.31 . Comparison of stress distribution between shell and solid elements along
horizontal weld path of Z section-bending.
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7.3.1.6 X-section
Figure 7.32 shows the stress distribution for shell and solid elements along weld path of
a cantilever beam of X section joint under the given loading of 1KN acting at the free
end. We can see that stress curves of shell element are in good agreement with stress
curves of solid element.
Figure 7.32 . Comparison of stress distribution between shell and solid elements along
weld path of X section -bending.
7.3.2 Case (b) Twisting
7.3.2.1 Rectangular cross-section
Figure 7.33 shows the stress distribution for shell and solid elements along weld path of
a cantilever beam of box section joint under the given loading acting at the free end. We
can see that stress curves of shell element are in good agreement with stress curves of
solid element.
Figure 7.33 . Comparison of stress distribution between shell and solid elements along
horizontal weld path of box section -twisting.
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7.3.2.2 I-section
Figure 7.34 shows the stress distribution for shell and solid elements along weld path of
a cantilever beam of I section joint under the given loading of 1KN acting at the free
end. We can see that stress curves of shell element are in good agreement with stress
curves of solid element.
Figure 7.34 . Comparison of stress distribution between shell and solid elements along
horizontal weld path of I section-twisting.
7.3.2.2 C-section
Figure 7.35 shows the stress distribution for shell and solid elements along weld path of
a cantilever beam of C section joint under the given loading of 1KN acting at the free
end. We can see that stress curves of shell element are in good agreement with stress
curves of solid element.
Figure 7.35 . Comparison of stress distribution between shell and solid elements along
horizontal weld path of C section-twisting.
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7.3.2.2 Circular-section
Figure 7.36 shows the stress distribution for shell and solid elements along weld path of
a cantilever beam of circular section joint under the given loading of 10KN acting at the
free end. We can see that stress curves of shell element are in good agreement with
stress curves of solid element.
Figure 7.36 . Comparison of stress distribution between shell and solid elements along
top weld path of circular section-twisting.
7.3.2.5 Z-section
Figure 7.37 shows the stress distribution for shell and solid elements along weld path of
a cantilever beam of Z profile under the given loading of 1KN acting at the free end. We
can see that stress curves of shell element are in good agreement with stress curves of
solid element.
Figure 7. 37. Comparison of stress distribution between shell and solid elements along
horizontal weld path of Z section-twisting.
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7.3.2.6 X-section
Figure 7.38 shows the stress distribution for shell and solid elements along weld path of
a cantilever beam of X profile under the given loading of 1KN acting at the free end.
We can see that stress curves of shell element are in good agreement with stress curves
of solid element.
Figure 7.38 . Comparison of stress distribution between shell and solid elements along
weld path of X section-twisting.
7.4 Comparison between V and fillet welds
This section introduces the comparison between fillet and V welds in case of bending
for different beam sections.
7.4.1 Box section
The Comparison of Mises stress distribution of v-weld, fillet weld and beam along the
top of weld path box profile in case of bending is shown in figure 7.39. The results
show that the fillet weld has high stress gradient while v-weld has low stress gradient.
The stress concentration at the edges of the v-weld is higher than the stress
concentration of the fillet weld.
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Figure 7.39 . Comparison of Mises stress distribution of v-weld ,fillet weld and beam
along horizontal weld path of box profile-bending.
7.4.2 I section
It is seen from Figure 7.40 which represents Comparison of Mises stress distribution of
v-weld, fillet weld and beam along weld path of I profile in case of bending. The v
weld has higher stress concentration than the fillet weld. From this figure, we can
conclude that the fillet weld is better than v-weld for I profile.
Figure 7.40 . Comparison of Mises stress distribution of v-weld ,fillet weld and beam
along weld path of I profile-bending.
7.4.3 C section
Figure 7.41 shows the comparison of Mises stress distribution of v-weld ,fillet weld and
beam along weld path of C profile in case of bending. We can see form this figure that
v weld has higher stress than the fillet weld. This comparison leads us to conclude that
the fillet weld is better that v-weld for C profile.
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Figure 7.41 . Comparison of Mises stress distribution of v-weld ,fillet weld and beam
along weld path of C profile-bending.
7.4.4 Circular section
Figure 7.42 shows the comparison of Mises stress distribution of v-weld ,fillet weld and
beam along weld path of circular section profile in case of bending. It is clear from the
figure that v-weld has high stress concentration while the fillet weld has lower stress
concentration. This means fillet weld is better than v weld for this profile.
Figure 7.42 . Comparison of Mises stress distribution of v-weld ,fillet weld and beam
along weld path of circular profile-bending.
7.4.5 Z section
Figure 7.43 shows the comparison of Mises stress distribution of v-weld ,fillet weld and
beam along weld path of Z profile in case of bending. It is clear from the figure that v-
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weld has high stress concentration while the fillet weld has lower stress concentration.
This means fillet weld is better than v weld for Z profile.
Figure 7.43 . Comparison of Mises stress distribution of v-weld ,fillet weld and beam
along weld path of Z profile-bending.
7.4.6 X section
Figure 7.44 shows the comparison of Mises stress distribution of v-weld ,fillet weld and
beam along weld path of X profile in case of bending. It is clear from the figure that v-
weld has high stress concentration while the fillet weld has lower stress concentration.
This means fillet weld is better than v weld for X profile.
Figure 7.44 . Comparison of Mises stress distribution of v-weld ,fillet weld and beam
along weld path of X profile-bending.
7.5 Discusion
The numerical modeling of the fillet welded joint connection was performed using
Abaqus [59 ], the commercial finite element package. The developed FE model is based
on the assumption of linear elasticity and small strains/displacements. Shell and solid
elements have been used for modeling both the beams and the welds in this study.
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Quadratic solid elements (20-noded) with three translational degrees-of-freedom at each
node and quadratic shell elements (8-noded) with 6 degrees-of-freedom at each node
were used to model the weld and the beam structure, in order to accurately capture any
non-linear stress gradients on the beam. The study was summarised and concluded that
stress curves of solid element are in good agreement with stress curves of shell element.
The results show that the normal stresses in the1 or 2-direction are the highest at the
weld path (top of the path) on the top surface, and at horizontal weld path of the beam.
It can be observed that the 1 or 2-direction component of normal stress causes the most
damage. The values of the Von-Mises equivalent stresses are very close to the values of
the stresses in the 1 or 2-direction (absolute value). This is expected as the stresses in
the1 or 2-direction are much larger than the other components of stress.
This means that the strains and stresses normal to the weld (i.e., 1 or 2 -direction in this
study) are mainly responsible for plasticity/crack initiation and propagation.
The results of the box profile show that the fillet weld has high stress gradient while v-
weld has low stress gradient. Stress concentrations at the weld ends in case of v-weld is
higher than the fillet weld. The other sections profiles, I,C,Z,X and circular profiles ,
show that v weld has higher stress at edges than the fillet welds. We can conclude that
the fillet welds are better than v-welds for these profiles. Also the results show that there
are stress concentrations in the stiffened areas.
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Chapter 8
Experimental results using digital image correlation (DIC)
To assess the effectiveness and accuracy of the modeling, the numerical and
experimental results were compared to verify the validity of the developed FE models.
Static loads were applied in the experiments. The same loads were also applied in the
FE models. The experimental and FE displacements for both loading conditions and for
all types of beam profiles were compared. The displacements obtained from the
simulations and those from the experiments are depicted in graphs by Abaqus [59 ] and
Armis [50] respectively. The results show that in general a good agreement exists
between the experimental and numerical displacements. The finite element method was
used to obtain results there not measured in the experiments like stresses and strains,
giving insight into the welding that cannot be seen in the laboratory.
As the validity of the FE models were verified through the comparison of the FE and
experimental displacements, the developed models were used to obtain a more thorough
understanding of the stress distribution around the connection of the welded joints for
the beams. It should be noted that with the existence of sharp corners at the weld toes,
the elastic stress is theoretically infinite at these locations. This means that one cannot
simply obtain the stresses at the weld toe from the FE results file, as they do not
converge and are mesh sensitive.
Figure 8.1. General view of experimental measuring using DIC.
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Figure.8.2 Shows the measuring areas (white regions) used in DIC.
8.1 Rectangular section
In the experimental study, a test fixture with a load cell is capable of applying static
loads (see Figures8.1-2), in order to verify the FEA results of the numerical study. It is
clear that the measuring object has homogeneous surface, we need to prepare such
surface by means of suitable methods. On this specimen, the corner surface was
prepared by applying a stochastic color spray pattern (see chapter 3). The load was
applied into 6 stages (see table 8.1). The maximum Uz displacement at the end of the
beam is 1.75mm which corresponds to force 6.5KN.
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Table 8.1 The input data for the box profile
Figure.8.3 shows the partial view of a box beam profile test: description of 29mm
length ) which is considered to calculate the distribution of the displacement y by using
the 3-D DIC .It is clear that the displacement y on the top surface of the box beam
profile is 0.12mm
Figure.8.3 Partial view of the experimental results of the displacement y of 29mm
length of the box profile.
Figure 8.4-a. shows the section length (see figure.8.3 ) versus displacement y. it is clear
that there is linear distribution of y displacement along the section path. While figure
8.4-b shows four stage points along the vertical weld of the beam box profile. Each
stage has linear distribution of the displacement y. Where the stage point 0 which is
located at the top surface on the beam has maximum displacement.
Stage number Displacement Uz mm
0 0
1 0.7
2 1.35
3 1.75
4 1.35
5 0.7
6 0
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Figure.8.4 The experimental results of the displacement y of the box profile using
aramis a) Section length versus displacement y b) Strain stage vs displacement y
c) Distribution of displacement y.
Figure 8.5. General view of the displacement distribution Uz of the box profile beam
Figure 8.5. shows the general view of the displacement distribution Uz of the box
profile beam. The applied force that makes deflection 1.75mm at the end of the beam is
6.5KN. Figure.8.6 shows partial view of the FE results of the displacement Uy of 29mm
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of the box profile. The figure shows that the displacement distribution at the top of box
beam is 0.11mm which is similar to the displacement that obtained from DIC (Aramis).
Figure.8.6 Partial view of the FE results of the displacement Uy of 29mm of the box
profile.
Experimental and numerical results of the displacement Uy are shown in Figure 8.3-6.
Generally, good agreement of the displacement distribution, Uy was obtained between
the FE analyses and experiments.
8.2 I-section
The top surface of the I beam profile was prepared by applying a stochastic color spray
pattern ( see figure 8.2). The load was applied into 6 stages (see table 8.2). The
maximum Uz displacement at the end of the beam is 3mm which corresponds to force
8.5KN.
Table 8.2 The input data for the I profile.
Stage number Displacement Uz mm
0 0
1 1
2 2
3 3
4 2
5 1
6 0
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Figure.8.7 shows the partial view of a I beam profile test: description of 30mm length
which is considered to calculate the distribution of the displacement z by using the 3-D
DIC .It is clear that the displacement z on the top surface at the end of 30mm of the I
beam profile is 0.16mm.
Figure.8.7 Partial view of the experimental results of the displacement z of 30mm
length of the I profile.
Figure 8.8-a. shows the section length (see figure.8.7) versus displacement z. It is clear
that there is linear distribution of z displacement along the section path. While figure
8.4-b shows four stage points , where stage point 0 is located on the weld and stage
point 3 on 30mm apart on the top surface of the I beam profile(see figure 8.7). We can
see that each stage has linear distribution of the displacement z. Stage point 0 has
minimum displacement (.09mm) because it is near the support while stage point 3 has
the maximum displacement (0.15mm) because it is locate about 30mm form stage point
0 (see figure8.7).
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Figure.8.8 The experimental results of the displacement z of the I profile using aramis.
a) Section length versus displacement y b) Strain stage vs displacement y c) Distribution
of displacement y.
Figure 8.9 shows the general view of the displacement distribution Uz of the I beam
profile. The applied force that makes deflection 3mm at the end of the beam is 8.5KN.
Figure.8.10 shows partial view of the FE results of the displacement Uz of 30mm of I
beam profile. The figure shows that the displacement distribution at the top of box beam
is 0.155mm which is similar to the displacement that obtained from DIC (aramis).
Figure 8.9. General view of the displacement distribution Uz of the I profile beam.
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Figure.8.10 Partial view of the FE results of the displacement Uz of 30mm of the box
profile.
Experimental and numerical results of the displacement Uz are shown in Figure 8.7,10.
Generally, good agreement of the displacement distribution,Uz was obtained between
the FE analyses and experiments.
8.3 C-section
The corner top surface of the C beam profile was prepared by applying a stochastic
color spray pattern. The load was applied into 8 stages (see table 8.3). The maximum Uz
displacement at the end of the beam is 4mm which corresponds to force 8.6KN.
Table 8.3. The input data for the C profile.
Figure.8.11 shows the partial view of a C beam profile test: description of 53mm length
which is considered to calculate the distribution of the displacement z by using the 3-D
Stage number Displacement Uz mm
0 0
1 1
2 2
3 3
4 4
5 3
6 2
7 1
8 0
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DIC .It is clear that the displacement z on the top surface at the end of 53mm of the C
beam profile is 0.35mm.
Figure.8.11 Partial view of the experimental results of the displacement z of 53mm
length of the C profile.
Figure 8.12-a. shows the section length (see figure.8.11) versus displacement z. It is
clear that there is linear distribution of z displacement along the section path. While
figure 8.4-b shows four stage points, where the point 0 is located on the flange and point
3 is locate 53mm from the flange (see figure 8.11). We can see that each stage has
linear distribution of the displacement z. Stage point 0 has displacement (.24mm)
because it is near the support while point 3 has displacement (0.34mm) because it is
locate about 53mm form point 0 (see figure8.11).
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Figure.8.12 The experimental results of the displacement z of the C profile using
aramis. a) Section length versus displacement z b) Strain stage vs displacement z
c) Distribution of displacement z.
Figure 8.13 shows the general view of the displacement distribution Uz of the C beam
profile. The applied force that makes deflection 4mm at the end of the beam is 8.6KN.
Figure.8.14 shows partial view of the FE results of the displacement Uz of 53mm of the
C beam profile. The figure shows that the displacement distribution at distance 53mm
from the flange is 0.303mm (see figure 8.11) which is similar to the displacement that
obtained from DIC (Aramis ).
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Figure 8.13. General view of the displacement distribution Uz of the C profile beam
Figure.8.14 Partial view of the FE results of the displacement Uz of 53mm of the C
profile.
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Experimental and numerical results of the displacement Uz are shown in Figure 8.11and
figure 8.14, respectively. Generally, good agreement of the displacement distribution,
Uz was obtained between the FE analyses and experiments.
8.4 Z-Section
The corner top surface of the Z beam profile was prepared by applying a stochastic
color spray pattern. The load was applied into 8 stages (see table 8.4). The maximum Uz
displacement at the end of the beam is 4mm which corresponds to force 5.3KN.
Table 8.4 The input data for the Z profile.
Figure.8.15 shows the partial view of a Z beam profile test: description of 38mm length
which is considered to calculate the distribution of the displacement z by using the 3-D
DIC .It is clear that the displacement z on the top surface at 38mm from the flange of
the Z beam profile is 0.288mm.
Stage number Displacement Uz mm
0 0
1 1
2 2
3 3
4 4
5 3
6 2
7 1
8 0
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Figure.8.15 Partial view of the experimental results of the displacement z of 38mm
length of the Z beam profile.
Figure 8.16-a. shows the section length (see figure.8.15) versus displacement z. It is
clear that there is linear distribution of z displacement along the section path. While
figure 8.4-b shows four stage points , where the point 0 is located on the flange and
point 3 is locate 38mm from the flange (see figure 8.15). We can see that each stage
has linear distribution of the displacement z. Stage point 0 has displacement (.2mm)
because it is near the support while point 3 has displacement (0.288mm) because it is
locate about 38mm form point 0 (see figure 8.15).
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Figure.8.16 The experimental results of the displacement z of the Z profile using
aramis. a) Section length versus displacement z b) Strain stage versus displacement z
c) Distribution of displacement z.
Figure 8.17 shows the general view of the displacement distribution Uz of the Z beam
profile. The applied force that makes deflection 4mm at the end of the beam is 5.3KN.
Figure.8.18 shows partial view of the FE results of the displacement Uz of 38mm of the
Z beam profile. The figure shows that the displacement distribution at distance 38mm
from the flange is 0.24mm which is close to the displacement that obtained from DIC
(Aramis ).
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Figure 8.17. General view of the displacement distribution Uz of the Z profile beam
using Abaqus.
Figure.8.18 Partial view of the FE results of the displacement Uz of 38mm of the Z
profile by using Abaqus [59].
Experimental and numerical results of the displacement Uz are shown in Figure 8.15-
18. Generally, good agreement of the displacement distribution Uz was obtained
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between the FE analyses and experiments. We note that this experiment was done
without constraints on the vertical plate.
8.5 X-Section
The corner surface of the X beam profile was prepared by applying a stochastic color
spray pattern. The load was applied into 8 stages (see table 8.5). The maximum Uz
displacement at the end of the beam is 4mm which corresponds to force 1.9KN.
Table 8.5 The input data for the X profile
Stage number Displacement Uz (mm)
0 0
1 1
2 2
3 3
4 4
5 3
6 2
7 1
8 0
Figure.8.19 shows the partial view of a X beam profile test: description of 50mm length
which is considered to calculate the distribution of the y displacement by using the 3-D
DIC .It is clear that the displacement y on the top surface at 50mm from the flange of
the X beam profile is 0.4mm .
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Figure.8.19 Partial view of the experimental results of the displacement z of 50mm
length of the X profile.
Figure 8.20-a. shows the section length (see figure.8.19) versus displacement z. It is
clear that there is linear distribution of y displacement along the section path. While
figure 8.4-b shows five stage points , where the point 4 is located on the top of the
beam and 38mm along y direction (see figure 8.19). We can see that each stage has
linear distribution of the displacement z. Stage point 0 has displacement (.3mm)
because it is near the middle of the beam while point 4 has displacement (0.4mm)
because it is located on top of the beam (see figure8.19).
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Figure.8.20 The experimental results of the displacement z of the Z profile using
aramis. a) Section length versus displacement z b) Strain stage versus displacement z
c) Distribution of displacement z.
Figure 8.21 shows the general view of the displacement distribution Uz of the X beam
profile. The applied force that makes deflection 4mm at the end of the beam is 1.9KN.
Figure.8.22 shows partial view of the FE results of the displacement Uy of 50mm of the
X beam profile. The figure shows that the displacement distribution at distance 50mm
from the flange is 0.4mm which has displacement distribution close to the displacement
that obtained from DIC (Aramis ).
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Figure 8.21. General view of the displacement distribution Uz of the Z profile beam
using Abaqus [59].
Figure.8.22 Partial view of the FE results of the displacement Uz of 50mm of the X
profile by using Abaqus [59].
Experimental and numerical results of the displacement Uy are shown in Figure
8.19,22. Generally, good agreement of the displacement distribution of Uy was obtained
between the FE analyses and experiments while no good agreement of the values of the
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displacement, Uy was obtained between the FE analyses and experiments. From figure
8.19, it is clear that the boundary conditions were not fixed as we applied in the finite
element model. This has effect on the loading and constraints. These constraints involve
mechanical contact which depends on the compliances of the structures that impose the
load and constraints and the physical properties of the contacting surfaces.
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Chapter 9
Conclusion
In this dissertation, the behavior of tubular (box and circle profile) and non tubular (L,
Z, C and X profiles) joint connections profiles, subjected to static loads were studied
both experimentally and numerically. From a structural analysis point of view, despite
of the wide use of corner welded joints as efficient load carrying members, there is no
available practical, simple and accurate approach for their design and analysis. For this
purpose, engineers must often prepare relatively complicated and time consuming finite
element (FE) models made up solid elements. Three-dimensional finite element analysis
of complete structural hollow sections can be complex and time-consuming. Due to the
complex nature of the finite element analysis codes, this method has limited application.
It can be used in research area but cannot be widely used by structural engineers in their
real-world projects. Therefore, there is a need to develop a simplified modeling method
that can be implemented by using commonly available commercial software and easily
employed. This is because unlike solid-section members, when hollow section members
are subjected to general loadings, they may experience severe deformations of their
cross-sections that results in stress concentrations in the connection’s vicinity. One of
the objectives/contributions of this research work is the better understanding of the
behavior of the corner welded joint connections under out-of- plane bending and torsion
loading conditions. Through a detailed Finite Element Analysis (FEA) using shell and
solid elements, the stress distribution at the connection of the tubular and non tubular
corner welded joints are obtained for different loading conditions. It is observed that at a
short distance away from the connection of the corner welded joints, the structure
behaves similar to beams when subjected to loadings.
The finite element results of the solid element model show that a generally linear stress
distribution across the thickness of the beams is observed. Therefore, the use of shell
elements for the analysis of the beam-joint connection is appropriate. Also, at a
relatively short distance from the connection, the effects of the local stresses and
deformations disappear.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
181
In chapter 5, the finite element simulations were performed using beam element for all
the six specimens. The Numerical analyses were carried out to represent the mechanical
performance of the fillet weld and V weld beam-sections under bending. The study is
summarized and concluded as follows. The predicted stress curves are in good
agreement with the analytical results for all six cases of the beam members. The finite
element results are in good agreement with the analytical results. Also, the obtained
results show that the V weld gives high stress for sections I,C , Circular, Z, and X cross
sections while a rectangular cross section gives lower stress when it welded by using V
joint. Based on numerical simulation beam element technique for V and fillet welded
joints, we can conclude that the analytical approach can be replaced by using a beam
element numerical approach. So that for large structures, the beam element can be used
to represent the stiffness of the fillet welded joints by increasing the thickness of the
beam in the region of welded joints.
In chapter 6 and 7, the numerical modeling of the v-welded and fillet welded joint
connections are performed using Abaqus [59 ], the commercial finite element package.
The developed FE model is based on the assumption of linear elasticity and small
strains/displacements. Shell and solid elements have been used for modeling both the
beams and the welds in this study. Quadratic solid elements (20-noded) with three
translational degrees-of-freedom at each node and quadratic shell elements (8-noded)
with 6 degrees-of-freedom at each node were used to model the weld and the beam
structure in order to accurately capture any non-linear stress gradients on the weld path.
The study is summarized and concluded that stress curves of solid elements are in good
agreement with stress curves of shell elements. The simulations using shell elements
were found generally efficient and accurate compared with solid elements. So that for
large structures, the oblique shell element can be used to represent the stiffness of the
fillet welded joints.
In chapter 8, the numerical and experimental results were compared to verify the
validity of the developed FE models. Static loads were applied in the experiments. The
same loads were also applied in the FE models. The experimental and FE displacements
for both loading conditions and for all types of beam profiles were compared. The
displacements obtained from the simulations and those from the experiments are
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
182
depicted in graphs. The results show that in general a good agreement exists between
the experimental and numerical displacements.
Finite element models of the welding experiments were carried out to verify the
validity of the numerical results. The finite element method was used to obtain results
that were not measured in the experiments like stresses and strains, giving insight into
the welding that cannot be seen in the laboratory.
From the FEA results of the solid element model, a generally linear stress distribution
across the thickness of the beams is observed. Therefore, the use of shell elements for
the analysis of the beam-joint connection is appropriate.
The results show that the normal stresses in the1 or 2-direction are the highest at the
weld path (top of the path) on the top surface, and at horizontal weld path of the beam.
It can be observed that the 1 or 2-direction component of normal stress causes the most
damage. The values of the Von - Mises equivalent stresses are very close to the values
of the stresses in the 1 or 2-direction (absolute value). This is expected as the stresses in
the1 or 2-direction are much larger than the other components of stress.
This means that the strains and stresses normal to the weld (i.e., 1 or 2 -direction in this
study) are mainly responsible for plasticity/crack initiation and propagation.
The results of the box profile show that the fillet weld has high stress gradient while v-
weld has low stress gradient while stress concentrations at the weld ends are high for v-
weld. The other section profiles I, C, Z, X and circular profiles, show that v weld has
higher stress at edges than the fillet welds. We can conclude that the fillet welds are
better than v-welds for these profiles. Also, there is stress concentration near the
stiffened region so the designer should take more care near stiffened regions.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
183
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Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
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Biography of the Author
Hasan Mehdi Nagiar was born on 16th of February 1971 in Tripoli Libya, Libyan
nationality. He finished his primary school in Asaad Ben Alforat school. He finished his
secondary education from Shohada Aldamor school, Tripoli Libya in 1989. In spring
1995, he received his B.Sc. Degree from Tripoli University, department of Aeronautical
engineering. In 2001, Nagiar received his M.Sc. Degree from Tripoli University,
department of Aeronautical Engineering. His master thesis was entitled “Buckling of
Axially Loaded Cylindrical Sandwich Shells”. Since March 2009, he has been Ph.D
candidate at the University of Belgrade, Faculty of Mechanical Engineering.
In the period between 1998 and 2003, he worked in research and development center
(RDC) in Tripoli Libya. In 2002, he worked as a part time lecturer at Tripoli University,
department of Aeronautical engineering and Souk Al-Gumma High Institute. In the
period between 2003 and 2007, he worked as a full time lecturer at Tripoli University,
department of Mechanical Engineering. He taught some subjects such as: Strength of
Material, Mechanics of Machines, Stress Analysis and Finite Elements. He engaged in
his Ph.D. research and worked under supervision of Professor Tasko Maneski in the
field of Strength of Structures.
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
189
Прилог 1. 
Изјава о ауторству 
Потписани-a   Хасан Мехди Нагиар 
број индекса   D40/09 
Изјављујем 
да је докторска дисертација под насловом  
НАПОНСКА АНАЛИЗА УГАОНИХ ЗАВАРЕНИХ СПОЈЕВА НОСАЧА 
КОНСТРУКЦИЈА САВРЕМЕНИМ НУМЕРИЧКО-ЕКСПЕРИМЕНТАЛНИМ 
ПРИСТУПОМ
 резултат сопственог истраживачког рада, 
 да предложена дисертација у целини ни у деловима није била предложена 
за добијање било које дипломе према студијским програмима других 
високошколских установа, 
 да су резултати коректно наведени и  
 да нисам кршио/ла ауторска права и користио интелектуалну својину 
других лица.  
                                                                        Потпис докторанда
У Београду, 8. 11. 2013.  
_________________________
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
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Прилог 2. 
Изјава o истоветности штампане и електронске верзије докторског рада 
Име и презиме аутора  Хасан Мехди Нагиар  
Број индекса    D40/09 
Студијски програм  ____________________________________________________ 
Наслов  НАПОНСКА АНАЛИЗА УГАОНИХ ЗАВАРЕНИХ СПОЈЕВА 
НОСАЧА КОНСТРУКЦИЈА САВРЕМЕНИМ НУМЕРИЧКО-
ЕКСПЕРИМЕНТАЛНИМ ПРИСТУПОМ
Ментор   проф. Др Ташко Манески
Потписани /а    Хасан Мехди Нагиар
Изјављујем да је штампана верзија мог докторског рада истоветна електронској 
верзији коју сам предао/ла за објављивање на порталу Дигиталног 
репозиторијума Универзитета у Београду.  
Дозвољавам да се објаве моји лични подаци везани за добијање академског звања 
доктора наука, као што су име и презиме, година и место рођења и датум одбране 
рада.  
Ови лични подаци могу се објавити на мрежним страницама дигиталне 
библиотеке, у електронском каталогу и у публикацијама Универзитета у Београду. 
Потпис докторанда 
У Београду, 8. 11. 2013.  
_________________________
Stress Analysis of Corner welded Joints Structure by Modern Numerical-Experimental Approach
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Прилог 3. 
Изјава о коришћењу 
Овлашћујем Универзитетску библиотеку „Светозар Марковић“ да у Дигитални 
репозиторијум Универзитета у Београду унесе моју докторску дисертацију под 
насловом:
НАПОНСКА АНАЛИЗА УГАОНИХ ЗАВАРЕНИХ СПОЈЕВА НОСАЧА 
КОНСТРУКЦИЈА САВРЕМЕНИМ НУМЕРИЧКО-ЕКСПЕРИМЕНТАЛНИМ 
ПРИСТУПОМ
која је моје ауторско дело.  
Дисертацију са свим прилозима предао/ла сам у електронском формату погодном 
за трајно архивирање.  
Моју докторску дисертацију похрањену у Дигитални репозиторијум Универзитета 
у Београду могу да користе  сви који поштују одредбе садржане у одабраном типу 
лиценце Креативне заједнице (Creative Commons) за коју сам се одлучио/ла. 
1. Ауторство 
2. Ауторство - некомерцијално 
3. Ауторство – некомерцијално – без прераде 
4. Ауторство – некомерцијално – делити под истим условима 
5. Ауторство –  без прераде 
6. Ауторство –  делити под истим условима 
(Молимо да заокружите само једну од шест понуђених лиценци, кратак опис 
лиценци дат је на полеђини листа). 
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____________________
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