ALEXANDRE ALMEIDA DEL SAVIO A COMPONENT METHOD … forças axiais de compressão ou tração serão...
Transcript of ALEXANDRE ALMEIDA DEL SAVIO A COMPONENT METHOD … forças axiais de compressão ou tração serão...
ALEXANDRE ALMEIDA DEL SAVIO
A COMPONENT METHOD MODEL FOR SEMI-RIGID STEEL JOINTS
INCLUDING BENDING MOMENT-AXIAL FORCE INTERACTION
Ph.D. Thesis
Thesis presented to the Post-graduate Program in Structural Engineering of Department of Civil Engineering, PUC-Rio, as partial fulfillment of the requirements for the Ph.D. Degree in Structural Engineering.
Supervisors: Prof. Sebastião Arthur Lopes de AndradeProf. Pedro Colmar Gonçalves da Silva Vellasco
Prof. David Arthur Nethercot
Rio de Janeiro
June, 2008
ALEXANDRE ALMEIDA DEL SAVIO
A COMPONENT METHOD MODEL FOR SEMI-RIGID STEEL JOINTS
INCLUDING BENDING MOMENT-AXIAL FORCE INTERACTION
Thesis presented to the Post-graduate Program in StructuralEngineering, of Department of Civil Engineering, PUC-Rio, aspartial fulfillment of the requirements for the Ph.D. Degree in Structural Engineering.
Dr. Sebastião Arthur Lopes de AndradeSupervisor
Civil Engineering Department - PUC-Rio
Dr. Pedro Colmar Gonçalves da Silva VellascoCo-Supervisor
Structural Engineering Department - UERJ
Dr. Luiz Fernando Campos Ramos MarthaCivil Engineering Department - PUC-Rio
Dr. Luciano Rodrigues Ornelas de LimaStructural Engineering Department - UERJ
Dr. Deane de Mesquita RoehlCivil Engineering Department - PUC-Rio
Dr. Eduardo de Miranda BatistaCOPPE - UFRJ
José Eugênio LealCoordinator of the Scientific Technical Centre - PUC-Rio
Rio de Janeiro, 13th June 2008
All rights reserved. It is prohibited to reproduce either all or part of this work without authorisation from the university, author and supervisor.
Alexandre Almeida Del Savio
B.Sc. by University of Passo Fundo (2002) and M.Sc. by Pontifical Catholic University of Rio de Janeiro (2004). Ph.D. academic visitor at Imperial College of Science, Technology and Medicine, London (2006-2007). The author is a structural engineer with main interests in: steel structures; semi-rigid joints; non-linear formulations and analysis; mechanical model; component method and finite element method. The author has a number of papers published in international journals and conferences related to the structural engineering field.
Card Catalog
Del Savio, Alexandre Almeida
A component method model for semi-rigid steel joints including bending moment-axial force interaction / Alexandre Almeida Del Savio; supervisors: Prof. Sebastião A. L. Andrade, Prof. Pedro C. G. da S. Vellasco and Prof. David A. Nethercot.
v. 177 p. : il. ; 30 cm
Thesis (Ph.D.) – Pontifical Catholic University of Rio de Janeiro (PUC-Rio), Civil Engineering Department
This thesis includes references.
Steel structures; Semi-rigid joints; Joint behaviour; Axial versus bending moment interaction; Mechanical model; Component method; Rotational stiffness.
Dedicated to God for having illuminated me throughout my way, my wife, Janaíne, my parents, Libório and Berenice, and my sisters, Letícia and Patrícia, for
their love and support.
Acknowledgements
I would like to express my sincere gratitude to my supervisor, Prof. Sebastião A.
L. de Andrade, Prof. Pedro C. G. da S. Vellasco and Prof. Luiz Fernando Martha,
for their brilliant joint supervision, continuous encouragement, support and
overall contribution throughout the entire duration of this study.
Prof. David A. Nethercot is also gratefully thanked for his close support and
expert advice during my valuable experience as an Academic Visitor at the
Department of Civil and Environmental Engineering, Imperial College of Science,
Technology and Medicine, London.
Furthermore, I would like to acknowledge the financial support provided for this
work by the Brazilian Scientific and Technological Developing Agencies: CNPq
and CAPES.
I specially thank to the Civil Engineering Department, PUC-Rio – Pontifical
Catholic University of Rio de Janeiro, including its Professors and staff.
Last but not least, I am grateful to my Brazilian friends and colleagues, Fernando
Ramires, Ricardo Araújo, Diego Orlando, Juliana Viana and Luciano Lima, as
well as my Imperial College London’s friends, José Miguel Castro, Daisuke Saito,
Stylianos Yiatros, Michal Jandera, Ken Chan, Ka Ho Nip (Alan) and Jason
Treadway, for their precious help and companionship during the development of
my thesis. I am also grateful to all other friends not mentioned here, but that many
times contributed to my thesis.
Abstract
Del Savio, Alexandre Almeida; Andrade, Sebastião Arthur Lopes de (Supervisor). A component method model for semi-rigid steel joints including bending moment-axial force interaction. Rio de Janeiro, 2008. 177p. Ph.D. Thesis - Civil Engineering Department, PUC-Rio – Pontifical Catholic University of Rio de Janeiro.
The correct knowledge of the joint moment-rotation characteristic is an
essential prerequisite for the use of the so called semi-continuous approach to
steel and composite frame design.
Although the axial force transferred from the beam is frequently low, so that
its effect on the moment-rotation characteristic may often be neglect, certain
circumstances do exist in which axial compression or tension forces will be
sufficiently large that it is no longer reasonable to ignore their influence.
The current thesis is centred on the development of a generalised
component-based model for semi-rigid beam-to-column joints including the full
axial force versus bending moment interaction. The detailed formulation of the
proposed analytical model is fully described in this work, as well as all the
analytical expressions used to evaluate the model properties. Detailed examples
demonstrate how to use this model to predict moment-rotation curves for any
axial force level. Numerical results, validated against experimental data, were also
performed in order to verify the accuracy and validity of the proposed model. A
tri-linear approach to characterise the force-displacement relationship of the joint
components is also proposed to model the joint model structural response.
Comparisons of the present development to key prior studies of this topic was also
made and commented in detail.
A series of parametric and sensitivity studies were executed varying several
key parameters that influence on the joint structural behaviour. The axial force-
bending moment interaction was also carefully analysed and the axial force effect
on the joint response was discussed. The proposed model and associated
analytical studies form the basis of important design considerations, involving the
presence of the axial force, which are suggested in this work to be included in
future improvements of structural design codes.
Finally, in addition to the proposed model and due to the fact of relatively
few experimental results have been reported to investigate the axial force effect,
an alternative method is presented herein. This alternative approach extends the
range of application of available experimental data to generate moment-rotation
characteristics that implicitly make proper allowance for the presence of
significant levels of either tension or compression at the adjacent beams. The
applicably and validity of the proposed methodology is demonstrated through
comparisons against several tests on endplate joints and baseplate arrangements.
Keywords
Steel structures; Semi-rigid joints; Joint behaviour; Axial versus bending
moment interaction; Mechanical model; Component method; Rotational stiffness.
Resumo
Del Savio, Alexandre Almeida; Andrade, Sebastião Arthur Lopes de (Supervisor). Um modelo mecânico baseado no método das componentes para ligações semi-rígidas de aço incluindo a interação momento fletor-força axial. Rio de Janeiro, 2008. 177p. Tese de Doutorado – Departamento de Engenharia Civil, PUC-Rio – Pontifícia Universidade Católica do Rio de Janeiro.
A compreensão correta da curva característica momento-rotação de uma
ligação é uma condição essencial para a utilização das chamadas abordagens
semi-contínuas para o aço e o projeto de estruturas mistas.
Embora a força axial proveniente da viga seja freqüentemente baixa de
modo que o seu efeito sobre a curva característica momento-rotação da ligação
possa muitas vezes ser negligenciado, existem certas circunstâncias nas quais as
forças axiais de compressão ou tração serão suficientemente grandes, não sendo
mais possível ignorar sua influência.
Esta tese é centrada no desenvolvimento de um modelo mecânico
generalizado, baseado no método das componentes para conexões semi-rígidas do
tipo viga-coluna incluindo a interação completa entre a força axial e o momento
fletor. A formulação detalhada do modelo analítico proposto é descrita totalmente
neste trabalho bem como todas as expressões analíticas utilizadas para avaliar as
propriedades do modelo mecânico. Exemplos detalhados demonstram como
utilizar este modelo para prever curvas momento-rotação para qualquer nível de
força axial. Resultados numéricos validados contra dados experimentais também
foram realizados a fim de verificar a exatidão e a validade do modelo proposto.
Uma abordagem tri-linear para caracterizar a relação força-deslocamento das
componentes de uma ligação também é proposta para modelar a resposta
estrutural do modelo de conexões. Comparações do atual desenvolvimento com
estudos fundamentais realizados anteriormente sobre este tema também foram
feitas e comentadas em detalhes.
Uma série de estudos paramétricos e sensitivos foram executados variando
os parâmetros principais que influenciam no comportamento estrutural da
conexão. A interação força axial-momento fletor também foi cuidadosamente
analisada e seu efeito sobre a resposta da ligação foi discutido. O modelo proposto
associado aos estudos analíticos formaram a base para as considerações, que
envolvem a presença da força axial, sugeridas neste trabalho para ser incluídas em
futuras melhorias de normas de projetos estruturais.
Por fim, além do modelo proposto e devido ao fato de que relativamente
poucos resultados experimentais foram relatados investigando o efeito da força
axial, um método alternativo é apresentado. Este método estende o leque de
aplicações dos dados experimentais disponíveis para gerar curvas características
momento-rotação que consideram implicitamente a presença de níveis
significativos de tração ou compressão nas vigas adjacentes. A aplicabilidade e
validade da metodologia proposta é demonstrada através de comparações com
vários ensaios de ligações com placas de extremidade e com placas de base.
Palavras-Chave
Estruturas metálicas; Ligações semi-rígidas; Comportamento estrutural de
ligações; Interação momento fletor versus força axial; Modelo mecânico; Método
das componentes; rigidez rotacional.
Table of Contents
Acknowledgements 5
Abstract 6
Resumo 8
List of Figures 14
List of Tables 20
Notation 22
1 Introduction 32
1.1. Background 32
1.2. Scope of the Present Work 35
1.3. Thesis Layout 36
2 Literature Review 38
2.1. Introduction 38
2.2. Conventional Design Practice 38
2.2.1. Global Analysis 38
2.2.2. Classification of the Joints 39
2.2.2.1. Classification by Stiffness 39
2.2.2.2. Classification by Strength 40
2.2.3. Design Moment-Rotation Characteristic of Joints 40
2.2.4. Component method 41
2.2.4.1. Welded Connections 44
2.2.4.2. Bolted Connections 44
2.2.4.3. Equivalent T-stub 46
2.2.4.3.1. Equivalent T-stub in Tension 46
2.2.4.3.2. Equivalent T-stub in Compression 50
2.2.4.4. Design of the Joint Basic Components 50
2.2.4.4.1. Column Web Panel in Shear 50
2.2.4.4.2. Column Web in Transverse Compression 51
2.2.4.4.3. Column Web in Transverse Tension 55
2.2.4.4.4. Column Flange in Transverse Bending 56
2.2.4.4.5. Endplate in Bending 59
2.2.4.4.6. Beam Flange and Web in Compression 62
2.2.4.4.7. Beam Web in Tension 63
2.2.4.4.8. Bolts in Tension 63
2.2.4.5. Axial Force 64
2.3. Theoretical Models 64
2.3.1. Mathematical Formulations (Empirical Models) 65
2.3.2. Simplified Analytical Models 67
2.3.3. Finite Element Analysis 69
2.3.4. Mechanical Models 72
2.4. Experimental 80
3 Generalised Mechanical Model for Beam-to-Column Joints Including the
Axial-Moment Interaction 81
3.1. Introduction 81
3.2. Characterisation of the Joint Components 82
3.3. Generalised Mechanical Model Formulation 84
3.3.1. Analytical Expressions: Displacements and Rotations 87
3.3.2. Limit Bending Moments 89
3.3.3. Moments that Cause the Joint Rows and the Joint to Yield and
Failure 90
3.4. Prediction of Bending Moment versus Rotation Curve for any Axial
Force Level 92
3.5. Lever Arm d 93
3.5.1. Lever Arm Evaluation for the Complementary Cases Disregarding
Axial Forces and/or Considering Tensile Forces Applied to the Joint 94
3.5.2. Lever Arm Evaluation for Compressive Forces Applied to the Joint95
4 Application of the Proposed Mechanical Model and Its Validation against
Experimental Tests 96
4.1. Introduction 96
4.2. Application of the Proposed Generalised Mechanical Model 96
4.2.1. Extended Endplate Joints 97
4.3. Results and Discussion 107
5 Parametric Investigations 109
5.1. Introduction 109
5.2. Joint Layout 109
5.3. Preliminary Studies 110
5.3.1. Discussion of the Results 114
5.4. Joint Key Parameters 119
5.5. Beam Profile Investigations 120
5.5.1. Discussion of the Results 124
5.6. Column Profile Investigations 125
5.6.1. Discussion of the Results 129
5.7. Endplate Thickness Investigations 130
5.7.1. Discussion of the Results 134
5.8. Bolt Investigations 136
5.8.1. Discussion of the Results 139
5.9. Axial Force Effect 140
5.10. Notes about the Incremental Solution of the Analytical Bending
Moment versus Axial Force Interaction Diagram 142
6 An Alternative Methodology to Extend the Range of Application of
Available Experimental Data so as to Produce Moment-Rotation
Characteristics 143
6.1. Introduction 143
6.2. General Concepts of the Correction Factor 143
6.3. Extension of the Correction Factors for Both Bending Moment and
Rotation Axes 144
6.4. An alternative methodology 146
7 Applicably and Validity of the Proposed Alternative Methodology 149
7.1. Introduction 149
7.2. Application of the Alternative Methodology 149
7.2.1. Flush endplate joints 150
7.2.2. Column bases 155
7.3. Results and Discussion 161
7.3.1. Flush Endplate Joints 161
7.3.2. Column Bases 162
8 Summary and Conclusion 164
8.1. Generalised Mechanical Model 164
8.2. Alternative Methodology 167
8.3. Design Considerations 168
8.4. Main Contributions and Developments of the Present Investigation169
8.5. Future Research Recommendations 170
References 172
List of Figures
Figure 1 - Schematic illustration of a typical staggered-truss system and the
structural system, Ritchie et al. (1979). 33
Figure 2 - Pitched-roof portal frame joint, Lima (2003). 33
Figure 3 - Sub-structural levels for progressive collapse assessment. (a) Bays
adjacent to the lost column; (b) Floors above de lost column; (c) Single floor
system; (d) Individual beams. Vlassis et al. (2006). 34
Figure 4 - Structural progressive collapse real example. Nethercot et al. (2007). 34
Figure 5 - Design moment-rotation characteristic for a joint. 41
Figure 6 – Joints and their associated mechanical models. 43
Figure 7 - T-stub identification and orientation for bolted extended endplate
connections. 46
Figure 8 – Failure modes of a T-stub. 48
Figure 9 – Dimensions of an equivalent T-stub flange (EC3-1-8, 2005). 48
Figure 10 – Collapse mechanisms of the bolt-row outside the beam flange (Faella
et al., 2000). 49
Figure 11 – Yield line models of bolt row group (Faella et al., 2000). 49
Figure 12 – Forces and moments acting on the joint. Direction of forces and
moments are considered as positive in relation to equations presented in this
section. 52
Figure 13 - Transverse compression on an unstiffened column. 54
Figure 14 - Definitions of e, emin, rc and m. 57
Figure 15 - Modelling an extended endplate as separate T-stubs. 60
Figure 16 – Values of for stiffened column flanges and endplates. 61
Figure 17 - Bolt elongation length. 64
Figure 18 - Connection and mechanical model for web cleat connections, Wales &
Rossow (1983). 72
Figure 19 - Mechanical model for flange and web cleated connections,
Chmielowiec & Richard (1987). 73
Figure 20 - Mechanical model for full welded joints, Tschemmernegg (1988). 74
Figure 21 - Mechanical model for bolted joints, Tschemmernegg & Humer
(1988). 74
Figure 22 - Idealization of beam-to-column connection, Madas (1993). 75
Figure 23 - Mechanical model, Jaspart el al. (1999). 75
Figure 24 - Spring model for extended endplate joints, Lima (2003). 77
Figure 25 - Spring model for flush endplate joints, Lima (2003). 77
Figure 26 - Nonlinear spring connection model, Ramli-Sulong (2005). 77
Figure 27 - Moment-rotation curves for the extended endplate joints tested by
Lima (2003) and obtained from numerical simulations, Lima (2003). 78
Figure 28 - Moment-rotation curves for the flush endplate joints tested by Lima
(2003) and obtained from numerical simulations, Simões da Silva et al.
(2004). 78
Figure 29 - Proposed generalised mechanical model for semi-rigid joints. 81
Figure 30 - Constitutive laws of the endplate joint components, Simões da Silva et
al. (2002). 82
Figure 31 - Force-displacement curve for components in tension and compression.
83
Figure 32 - Proposed prediction of the bending moment versus rotation curve for
any axial force level. 93
Figure 33 - Proposed generalised mechanical model for semi-rigid joints – lever
arm d. 94
Figure 34 - Extended endplate joint, Lima et al (2004). 97
Figure 35 - Proposed mechanical model. 98
Figure 36 - Proposed mechanical model for each analysis stage. 101
Figure 37 - Comparison between experimental EE1 moment-rotation curve (Lima
et al., 2004) and predicted curve by using the proposed mechanical model.
104
Figure 38 - Comparison between experimental EE2 moment-rotation curve (Lima
et al., 2004) and predicted curve by using the proposed mechanical model.
104
Figure 39 - Comparison between experimental EE3 moment-rotation curve (Lima
et al., 2004) and predicted curve by using the proposed mechanical model.
105
Figure 40 - Comparison between experimental EE4 moment-rotation curve (Lima
et al., 2004) and predicted curve by using the proposed mechanical model.
105
Figure 41 - Comparison between experimental EE6 moment-rotation curve (Lima
et al., 2004) and predicted curve by using the proposed mechanical model.
106
Figure 42 - Comparison between experimental EE7 moment-rotation curve (Lima
et al., 2004) and predicted curve by using the proposed mechanical model.
106
Figure 43 - Prediction of six moment-rotation curves for different axial force
levels. 107
Figure 44 - Extended endplate joint, Lima et al (2004). 109
Figure 45 - Proposed mechanical model. 110
Figure 46 - Comparison between experimental EE1 moment-rotation curve (Lima
et al., 2004) and predicted curve by using the proposed mechanical model.
110
Figure 47 - Comparison between experimental EE2 moment-rotation curve (Lima
et al., 2004) and predicted curve by using the proposed mechanical model.
111
Figure 48 - Comparison between experimental EE3 moment-rotation curve (Lima
et al., 2004) and predicted curve by using the proposed mechanical model.
111
Figure 49 - Comparison between experimental EE4 moment-rotation curve (Lima
et al., 2004) and predicted curve by using the proposed mechanical model.
112
Figure 50 - Comparison between experimental EE6 moment-rotation curve (Lima
et al., 2004) and predicted curve by using the proposed mechanical model.
112
Figure 51 - Comparison between experimental EE7 moment-rotation curves
(Lima et al., 2004) and predicted curve by using the proposed mechanical
model. 113
Figure 52 - Prediction of six moment-rotation curves for different axial force
levels. 113
Figure 53 - Prediction of the bending moment versus axial load interaction
diagram using the proposed mechanical model for the joint yield and ultimate
resistances. 114
Figure 54 - Investigated EE1 (N = 0.0 kN) moment-rotation curves involving the
beam profile variations. 121
Figure 55 - Investigated EE2 (N = +10% Npl = 135.95 kN) moment-rotation
curves involving the beam profile variations. 121
Figure 56 - Investigated EE3 (N = +20% Npl = 193.30 kN) moment-rotation
curves involving the beam profile variations. 122
Figure 57 - Investigated EE4 (N = +27% Npl = 259.20 kN) moment-rotation
curves involving the beam profile variations. 122
Figure 58 - Investigated EE6 (N = -10% Npl = -127.20 kN) moment-rotation
curves involving the beam profile variations. 123
Figure 59 - Investigated EE7 (N = -20% Npl = -257.90 kN) moment-rotation
curves involving the beam profile variations. 123
Figure 60 - Analytical moment-axial load interaction diagram at different beam
profiles. 123
Figure 61 - Investigated EE1 (N = 0.0 kN) moment-rotation curves involving the
column profile variations. 126
Figure 62 - Investigated EE2 (N = +10% Npl = 135.95 kN) moment-rotation
curves involving the column profile variations. 127
Figure 63 - Investigated EE3 (N = +20% Npl = 193.30 kN) moment-rotation
curves involving the column profile variations. 127
Figure 64 - Investigated EE4 (N = +27% Npl = 259.20 kN) moment-rotation
curves involving the column profile variations. 128
Figure 65 - Investigated EE6 (N = -10% Npl = -127.20 kN) moment-rotation
curves involving the column profile variations. 128
Figure 66 - Investigated EE7 (N = -20% Npl = -257.90 kN) moment-rotation
curves involving the column profile variations. 129
Figure 67 - Analytical moment-axial load interaction diagram at different column
profiles. 129
Figure 68 - Investigated EE1 (N = 0.0 kN) moment-rotation curves involving the
endplate thickness variations. 131
Figure 69 - Investigated EE2 (N = +10% Npl = 135.95 kN) moment-rotation
curves involving the endplate thickness variations. 132
Figure 70 - Investigated EE3 (N = +20% Npl = 193.30 kN) moment-rotation
curves involving the endplate thickness variations. 132
Figure 71 - Investigated EE4 (N = +27% Npl = 259.20 kN) moment-rotation
curves involving the endplate thickness variations. 133
Figure 72 - Investigated EE6 (N = -10% Npl = -127.20 kN) moment-rotation
curves involving the endplate thickness variations. 133
Figure 73 - Investigated EE7 (N = -20% Npl = -257.90 kN) moment-rotation
curves involving the endplate thickness variations. 134
Figure 74 - Analytical moment-axial load interaction diagram at different endplate
thicknesses. 134
Figure 75 - Investigated EE1 (N = 0.0 kN) moment-rotation curves involving the
bolt variations. 136
Figure 76 - Investigated EE2 (N = +10% Npl = 135.95 kN) moment-rotation
curves involving the bolt variations. 137
Figure 77 - Investigated EE3 (N = +20% Npl = 193.30 kN) moment-rotation
curves involving the bolt variations. 137
Figure 78 - Investigated EE4 (N = +27% Npl = 259.20 kN) moment-rotation
curves involving the bolt variations. 138
Figure 79 - Investigated EE6 (N = -10% Npl = -127.20 kN) moment-rotation
curves involving the bolt variations. 138
Figure 80 - Investigated EE7 (N = -20% Npl = -257.90 kN) moment-rotation
curves involving the bolt variations. 139
Figure 81 - Analytical moment-axial load interaction diagram at different bolts.
139
Figure 82 - Evaluation of the design bending moments (Mint & Mmax) and rotations
(int & max). 145
Figure 83 - Correction Factor strategy method using a three segment division of
the M- curve. 146
Figure 84 - Approximate M- curve using three Correction Factor pairs. 146
Figure 85 - Tri-linear representation of the M- curve methodology. 147
Figure 86 - Flush endplate joint layout, Simões da Silva et al. (2004). 150
Figure 87 - Experimental moment versus rotation curves, Simões da Silva et al.
(2004). 150
Figure 88 - Flush endplate bending moment versus axial force interaction
diagram, Simões da Silva et al. (2004). 151
Figure 89 - Tri-linear strategy used for the experimental M- curves. 152
Figure 90 - Paths used to define the procedure to determine any M- curve present
within these limits. 153
Figure 91 - FE8 M- curve approximation, considering a tensile force of 10% of
the beam’s axial plastic resistance. 154
Figure 92 - FE3 M- curve approximation, considering a compressive force of 4%
of the beam’s axial plastic resistance. 154
Figure 93 - FE4 M- curve approximation, considering a compressive force of 8%
of the beam’s axial plastic resistance. 155
Figure 94 - The whole set of predicted M- curves by using the proposed
methodology. 155
Figure 95 - Baseplate configurations, Guisse et al. (1996). 156
Figure 96 - PC2.15 experimental M- curves and the tri-linear reference M-
curves. 157
Figure 97 - PC2.30 experimental M- curves and the tri-linear reference M-
curves. 157
Figure 98 - PC4.15 experimental M- curves and the tri-linear reference M-
curves. 158
Figure 99 - PC4.30 experimental M- curves and the tri-linear reference M-
curves. 158
Figure 100 - PC2.15.600 M- curve approximation. 159
Figure 101 - PC2.30.600 M- curve approximation. 160
Figure 102 - PC4.15.400 M- curve approximation. 160
Figure 103 - PC4.30.400 M- curve approximation. 161
Figure 104 - First-order approximations error magnitudes versus joint rotation.167
List of Tables
Table 1 - Joint basic components. 42
Table 2 - Design resistance FT,Rd of a T-stub flange (EC3-1-8, 2005). 47
Table 3 – Reduction factor for interaction with shear. 52
Table 4 - Approximate values for the transformation parameter . 53
Table 5 - Effective lengths for an unstiffened column flange. 58
Table 6 - Effective lengths for an endplate. 60
Table 7 - Summary of the mechanical models to predict the joint behaviour. 79
Table 8 - Values adopted for the strain hardening coefficients, μ. 84
Table 9 - Steel mechanical properties. 98
Table 10 - Theoretical values of the resistance and initial stiffness of the extended
endplate joint components, Figure 34, evaluated according to Eurocode 3:1-8
(2005). 99
Table 11 - Characterisation of the extended endplate joint components, Figure 34,
according to the approach given in Chapter 3 - section 3.2. 100
Table 12 - Load situations applied to the joint and their respective mechanical
models. 102
Table 13 - Applicability of each model, Mlim, and evaluation of lever arm d
according to the experimental axial force levels. 102
Table 14 - Values evaluated for the prediction of the moment-rotation curves for
different axial force levels. 104
Table 15 - Comparisons between the experimental and the proposed model initial
stiffness and the experimental and the proposed model design moment. 108
Table 16 - Comparisons between the experimental and analytical points obtained
for the extended endplate joint. 115
Table 17 - Mechanical model row stiffness for the joint ultimate bending moment
resistance. 117
Table 18 - Row-component yield and failure sequence. 118
Table 19 – Main elements of the joint and their respective basic components. 120
Table 20 - Investigated beam profiles and their main dimensions. 120
Table 21 - The weakest component of the mechanical model rows for each
analysed case with N = 0.0. 124
Table 22 - Evaluated ultimate bending moments at different beam profiles. 125
Table 23 - Investigated column profiles and their main dimensions. 126
Table 24 - The weakest component of the mechanical model rows for each
analysed case with N = 0.0. 130
Table 25 - Evaluated ultimate bending moments at different column profiles. 130
Table 26 - Investigated endplate thicknesses and their dimensions. 131
Table 27 - The weakest component of the mechanical model rows for each
analysed case with N = 0.0. 135
Table 28 - Evaluated ultimate bending moments at different endplate thicknesses.
135
Table 29 - Investigated grade 10.9 bolts and their main dimensions. 136
Table 30 - The weakest component of the mechanical model rows for each
analysed case with N = 0.0. 140
Table 31 - Evaluated ultimate bending moments at different bolt diameters. 140
Table 32 - Values evaluated for the reference M- curves. 152
Table 33 - Values evaluated for three tri-linearly approximated M- curves. 153
Table 34 - Nomenclature of the tests and their parameters, Guisse et al. (1996).
156
Table 35 - Values evaluated for the reference M- curves. 157
Table 36 - Values evaluated for three tri-linearly approximated M- curves. 158
Table 37 - Comparisons between the experimental and the proposed methodology
in terms of initial stiffness and design moment capacity for flush endplate
joints. 162
Table 38 - Comparisons between the experimental and the proposed methodology
in terms of initial stiffness and design moment capacity for baseplate joints.
163
Notation
All symbols used in this thesis are defined as they first appear. For the
reader’s convenience, the principal meanings of the commonly used notations are
contained in the list below.
Roman Symbols
a modelling parameter
ba throat thickness of the beam flange-to-column flange weld
ca throat thickness of the column web-to-flange weld
ja least-square curve fitting coefficient
pfp aa ; throat thickness of the weld between the beam flange and the
endplate
1b bar 1: rigid bar representing the beam end
2b bar 2: rigid bar representing the column flange centreline
bb width of the beam cross section
cb width of the column cross section
wcceffb ,, effective width of column web in compression
wbteffb ,, effective width of beam web in tension
wcteffb ,, effective width of column web in tension
jb least-square curve fitting coefficient
)7(bfwc beam flange and web in compression
pb width of the plate welded to an I or H section
)10(bt bolts in tension
)8(bwt beam web in tension
jc modelling parameter
)4(cfb column flange in bending
)2(cwc column web in compression
)1(cws column web in shear
)3(cwt column web in tension
d lever arm: distance from the loading application centre to the
rigid link
bd bolt diameter
hd bolt head diameter
id system displacements, i=1..4: ub1, b1, ub2, b2
nd nut diameter
wd washer diameter; width across points of the bolt head or nut
wcd clear depth of the column web
e distance from the loading application centre to the beam bottom
flange
)5(epb endplate in bending
we 4/wd
yibrf , yield strength of the joint bolt-row i
ycpf joint component yield capacity
ucpf joint component ultimate capacity
if force in spring/row i
yif yield capacity of spring/row i
uif ultimate capacity of spring/row i
bpyf , yield strength of the backing plates
fyf , yield strength of the flange of the I or H section
pyf , yield strength of the plate welded to the I or H section
puf , ultimate strength of the plate welded to the I or H section
wcyf , yield strength of the beam web
wcyf , yield strength of the column web
bh depth of the beam cross section; beam height
ch depth of the column cross section; column height
pep hh ; depth of the plate; endplate height
rh distance of bolt-row r from the compressive centre
th lever arm
k non-dimensional stiffness parameter
1k stiffness coefficient of the column web panel in shear
2k stiffness coefficient of the column web in compression
3k stiffness coefficient of the column web in tension
4k stiffness coefficient of the column flange in bending
5k stiffness coefficient of the endplate in bending
7k stiffness coefficient of the beam flange and web in compression
8k stiffness coefficient of the beam web in tension
10k stiffness coefficient of the bolts in tension
bk factor that depends on the frame type
bbfk elastic stiffness of the bottom flange of the beam
1brk elastic stiffness of bolt-row 1
2brk elastic stiffness of bolt-row 2
3brk elastic stiffness of bolt-row 3
btfk elastic stiffness of the top flange of the beam
ecpk joint component elastic stiffness
pcpk joint component plastic stiffness
ucpk joint component reduced strain hardening stiffness
reffk , effective stiffness coefficient of bolt-row r
eqk equivalent stiffness coefficient
rik , stiffness coefficient representing basic component i in bolt-row r
lcbfk elastic stiffness of the compressive rigid link referred to the
bottom flange of the beam
lctfk elastic stiffness of the compressive rigid link referred to the top
flange of the beam
ltk elastic stiffness of the tensile rigid link referred to the lever arm
1ltk elastic stiffness of tensile rigid link 1 referred to bolt-row 1
2ltk elastic stiffness of tensile rigid link 2 referred to bolt-row 2
3ltk elastic stiffness of tensile rigid link 3 referred to bolt-row 3
wck reduction factor that accounts for the influence of the vertical
normal stress
effl effective length
epl length of the endplate over the beam flange
il distance from joint spring/row i to the beam bottom flange centre
m number of knots (junction of multi-part curve)
m non-dimensional moment resistance parameter
n shape factor
rn total number of bolt-rows in tension
nbr number of bolt-rows
nc row/spring component number
ns system spring/row number
wn number of washers
ar radius of the fillet of the angle legs
cr radius of the fillet of the web-to-flange connection of the column
ir effective stiffness of model spring/row i
eir elastic effective stiffness of spring/row i
pir plastic effective stiffness of spring/row i
uir reduced strain hardening effective stiffness of spring/row i
s length that depends on if the column section is rolled or welded
ps length obtained by dispersion at 45o of the compressive action
through the endplate thickness
at angle thickness
bpt thickness of the backing plates
ept thickness of the endplate
ft thickness of the flange of an I or H section
fbt thickness of the beam flange
fct thickness of the column flange
ht thickness of the bolt head
nt thickness of the nut
pt thickness of the plate (under the bolt or the nut)
wt thickness of the web of an I or H section
wbt thickness of the beam web
wct thickness of the column web
wht thickness of the washer
1bu first bar displacement
2bu second bar displacement
iu absolute displacement of spring/row i (first bar)
iul absolute displacement of spring/row i (second bar)
z lever arm
eqz equivalent lever arm
Capital letter
sA tensile stress area of the bolt
vcA shear area of the column
C constant that controls the curve slope
321 ;; CCC curve-fitting constants
iC spring/row i vertical coordinates
MCF correction factor for the moment axis
CF correction factor for the rotation axis
E elastic modulus of structural steel
F internal loading vector
bbfF row compressive yield capacity (beam bottom flange)
RdwccF ,, design resistance of the column web in compression
brRdwccF ,,, design buckling resistance of the column web in compression
crRdwccF ,,, design crushing resistance of the column web in compression
RdwscF ,, design resistance of the column web in shear
linktF rigid link tensile capacity, which joins the second bar to the
supports
min,RdF smallest design resistance of the basic components
RdtF , design tension resistance of a bolt
RdTF , design tension resistance of a T-stub flange
RdwctF ,, design resistance of the column web in tension
RdtrF , effective tension resistance of bolt-row r
bI second moment of the area of the supported beam section
K model stiffness matrix; parameter that depends on the geometrical
and mechanical properties of the connection details
bK ratio of the relative rigidity of all beams at the top of the storey
cK ratio of the relative rigidity of all columns at the top of the storey
iKK ; initial stiffness
ijK terms of the system stiffness matrix, i=1..4 and j=1..4
PK strain hardening stiffness
bL span of the supported beam; bolt elongation length taken equal to
the grip length (total thickness of material and washers) plus half
the sum of the height bolt head and the height of the nut.
M bending moment applied to the joint
M bending moment versus rotation curve
)0(Mx moment-rotation curve disregarding the axial force effect
)( iNMx moment-rotation curve considering the axial force iN
)(M moment-rotation relationship
fM bending moment referred to a 0.05-radian joint final rotation
uM bending moment that leads the joint to the failure
yM bending moment that leads the joint to the yield
0M initial moment; reference moment
pM ,0 bending moment on the reference M curve disregarding the
axial force at point p
EdbM ,1 joint internal bending moments
EdbM ,2 joint internal bending moments
uibrM , bending moment that leads to the failure of the joint spring/row i,
located between the first and second bars
yibrM , bending moment that leads to the yield of the joint spring/row i,
located between the first and second bars
RdcM , design moment resistance of the beam cross-section
dM design bending moment
uifrM , bending moment that leads to the failure of the joint spring/row i,
located between the second bar and supports
yifrM , bending moment that leads to the yield of the joint spring/row i,
located between the second bar and supports
intM design bending moment considering the axial force iN
jM upper bound moment in the j-th part of the curve
lim,jM limit bending moment of spring/row j, located between the first
and second bars
EdjM , design moment action
RdjM , design moment resistance of the joint, the design plastic moment
resistance of the connected member
maxM design bending moment disregarding the axial force
0NM bending moment referred to )0(Mx curve
pNM , bending moment on the reference M curve considering the
axial force at point p
pM plastic moment; bending moment evaluated for the target M
curve at point p
RdplM , design plastic moment resistance of the connected member
uM ultimate moment; idealised elastic-plastic mechanism moment
N shape parameter obtained through the least square method
uN axial load that leads the joint to the failure
yN axial load that leads the joint to the yield
EdbN ,1 joint internal normal forces
EdbN ,2 joint internal normal forces
iN axial force present in interaction i
plN beam’s axial plastic capacity
P axial load applied to the joint
)(R load-deformation relationship
0R reference load
jS secant stiffness
inijS , initial rotational stiffness of the joint
EdbV ,1 joint internal shear forces
EdbV ,2 joint internal shear forces
RdwpV , design shear force of the column web in shear
Greek Symbols
4,3,2,1 coefficients of Eq. 3.41
transformation parameter which account for the possible influence
of the web panel in shear
0M partial safety factor for resistance of cross-section whatever the
class is
1M partial safety factor for resistance of members to instability
assessed by member checks
2M partial safety factor for resistance of cross-sections in tension to
fracture
1bu first bar virtual displacement
2bu second bar virtual displacement
U internal virtual work
W external virtual work
1b first bar virtual rotation
2b second bar virtual rotation
virtual displacement field
i virtual displacement of spring i
4,3,2,1 coefficients of Eq. 3.41
joint rotation
u joint rotation capacity necessary to develop the joint plastic
bending moment
y joint rotation capacity necessary to develop the joint yield
bending moment
f joint final rotation (assumed to be equal to 0.05 radians)
0 reference rotation
1b first bar rotation
2b second bar rotation
stiffness coefficient (Eq. 3.18)
stiffness coefficient (Eq. 3.18)
p plate slenderness
stiffness ratio jinij SS /, that accounts for the joint non-linear
behaviour
p plastic stiffness strain hardening coefficient
u ultimate stiffness strain hardening coefficient
stiffness coefficient (Eq. 3.23)
reduction factor for plate buckling; stiffness coefficient (Eq. 3.18)
stiffness coefficient (Eq. 3.26)
p,0 rotation on the reference M curve disregarding the axial force
at point p
Cd design rotation capacity
d design rotation
Ed rotation between connected members of the joint
int design rotation considering the axial force iN
max design rotation disregarding the axial force
0N rotation referred to )0(Mx curve
pN , rotation on the reference M curve considering the axial force
at point p
p rotation evaluated for the target M curve at point p
stiffness coefficient (Eq. 3.27)
1 stiffness coefficient (Eq. 3.23)
2 stiffness coefficient (Eq. 3.20)
stiffness coefficient (Eq. 3.20); coefficient that depends on the
connection type
reduction factor to allow for the possible effects of interaction
with shear in the column web panel
1 stiffness coefficient (Eq. 3.23)
2 stiffness coefficient (Eq. 3.20)
Capital letter
relative displacement field
i spring/row i relative displacement
ibr , spring/row i relative displacement located between the first and
second bars
ifr , spring/row i relative displacement located between the second bar
and the supports
yi relative displacement that leads to the yield of the model
spring/row i
ui relative displacement that leads to the failure of the model
spring/row i
stiffness coefficient (Eq. 3.25)
stiffness coefficient (Eq. 3.22)
stiffness coefficient (Eq. 3.22)
1 Introduction 32
1Introduction
1.1.Background
The continuous search for the most accurate representation of structural
behaviour directly depends on a detailed structural modelling, including the
interactions between all the structural elements, linked to the overall structural
analysis procedures, such as material and geometric non-linear analysis. This
strategy enables a more realistic modelling of joints, instead of the usual pinned or
rigid assumptions. This idea is crucial to advance towards a better overall
structural behavioural understanding, since joint response is well-described by the
moment-rotation curve. However, this approach requires a complete knowledge of
semi-rigid joint behaviour, which is, for some situations, beyond the scope of
present knowledge i.e. the influence of axial forces on the joint bending moment
versus rotation characteristic.
In addition to permitting the most accurate structural modelling, the use of
semi-rigid joints has several practical advantages such as those identified in SCI
Publication 183 (1997):
- economy of both design effort and fabrication costs;
- beams may be lighter than in simple construction;
- reduction of mid-span deflection due to the joint inherent stiffness;
- joints are less complicated than in continuous construction;
- frames are more robust than in simple construction; and
- for an unbraced frame, additional benefit may be gained from semi-
continuous joints in resisting wind loading without the extra fabrication
costs incurred when full continuity is adopted.
Under certain circumstances, beam-to-column joints can be subjected to the
simultaneous action of bending moments and axial forces. Although, the axial
force transferred from the beam is usually low, it may, in some situations attain
values that significantly reduce the joint flexural capacity.
1 Introduction 33
These conditions may be found in: structures under fire situations where the
effects of beam thermal expansion and membrane action can induce significant
axial forces in the joint (Ramli-Sulong et al., 2007); Vierendeel girder systems
(Figure 1, widely used in building construction because they take advantage of the
member flexural and compression resistances eliminating the need for extra
diagonal members); regular sway frames under significant horizontal loading
(seismic or extreme wind); irregular frames (especially with incomplete storeys)
under gravity/horizontal loading; and pitched-roof frames (Figure 2).
CLEAR SPAN TRUSS(to support vertical loads& transfer lateral shear)
VIERENDEELPANEL FLOOR SLAB
(also to transferlateral loadshear force)
UNINTERRUPTED FLOOR SPACE
Figure 1 - Schematic illustration of a typical staggered-truss system and the structural
system, Ritchie et al. (1979).
Figure 2 - Pitched-roof portal frame joint, Lima (2003).
Moreover, with the recent escalation of terrorist attacks on buildings, the
investigation of progressive collapse of steel framed buildings has been
highlighted, as can be seen in Vlassis et al. (2006). Examples of these exceptional
conditions are the cases where structural elements, such as central and/or
peripheral columns and/or main beams, are suddenly removed, abruptly increasing
the joint axial forces. In these situations the structural system, mainly the joints,
1 Introduction 34
should be sufficiently robust to prevent the premature failure modes that may lead
to a progressive structural collapse, Figure 3 and Figure 4.
Figure 3 - Sub-structural levels for progressive collapse assessment. (a) Bays adjacent to
the lost column; (b) Floors above de lost column; (c) Single floor system; (d) Individual
beams. Vlassis et al. (2006).
Figure 4 - Structural progressive collapse real example. Nethercot et al. (2007).
Unfortunately, few experiments considering the bending moment versus
axial force interactions have been reported. Additionally, the available
experiments are related to a small number of axial force levels and associated
bending moment versus rotation curves.
Recently, some mechanical models have been developed (Chapter 2,
Literature Review) to deal with the bending moment-axial force interaction.
1 Introduction 35
However these models are not still able to accurately predict the joint moment-
rotation curves, thereby restricting their incorporation into analysis procedures.
Problems in the prediction of the moment-rotation curves are usually related
to the joint initial stiffness evaluation for different axial force levels, as can be
seen in Del Savio et al. (2008a). The magnitude of this problem increases when
joints are subjected to tensile axial forces. This problem relates to the ability of
these models to deal with moment-axial interaction, and consequently changes of
the compressive centre, before the first component yields. If the model is under
the linear-elastic regime, without reaching any component yield (i.e. the
component stiffness response is also linearly), the modification of the joint
stiffness matrix, only due to the geometric stiffness changes, will be insignificant.
From this point upwards to the onset of first component yield, these models are
not able to accurately represent the joint initial stiffness for any level of axial load
and bending moment while still working on the linear-elastic range.
There is, therefore, the need to develop a component-based mechanical
model for semi-rigid beam-to-column joints including the axial force versus
bending moment interaction, which principally aims to overcome this limitation
by allowing modifications of the compressive centre position even before
reaching the first component yield, i.e. in the linear-elastic regime.
1.2.Scope of the Present Work
The main purpose of the present investigation is to develop a generalised
component-based mechanical model for semi-rigid beam-to-column joints
including the axial force versus bending moment as well as to study the influence
of the axial force-bending moment interaction on the overall joint behaviour.
Following a comprehensive literature review on the subject, the component-
based approach is selected as a basis for the development of the new mechanical
model. The relationship of the present development to key prior studies of this
topic is also explained. Detailed formulation of the proposed analytical model is
fully described in this work, as well as all the analytical expressions used to
evaluate the model properties. Detailed examples demonstrate how to use this
model to predict moment-rotation curves for any axial force level. Numerical
results, validated against experimental data, were performed in order to verify the
1 Introduction 36
accuracy and validity of the proposed model. Based on this, a tri-linear approach
to characterise the force-displacement relationship of the joint components was
also proposed.
A series of parametric and sensitivity studies were executed varying several
key parameters that influence on the joint structural behaviour. The axial force-
bending moment interaction was also carefully analysed and the axial force effect
on the joint response was discussed. The proposed model and associated
analytical studies form the basis of important design considerations, involving the
presence of the axial force, which were also suggested to be included in future
improvements of structural design codes.
In addition to the proposed model, a method is also presented herein which
extends the range of application of available experimental data so as to produce
moment-rotation characteristics that implicitly make proper allowance for the
presence of significant levels of either tension or compression in the beam. The
applicably and validity of the proposed methodology is demonstrated through
comparisons against several tests on endplate joints and baseplate arrangements.
1.3.Thesis Layout
This thesis is organised into eight chapters. The present chapter introduces
the background of this study, the scope of the present work and the layout of the
thesis displayed as follows.
A comprehensive review of the literature on the techniques currently
available to predict the joint structural behaviour as well as a discussion of
experimental tests, focusing on the study of joint behaviour under combined
bending moment and axial force using mechanical models, is presented in Chapter
2.
Chapter 3 describes the detailed formulation of the generalised component-
based mechanical model for beam-to-column joints including the axial force-
bending moment interaction. Additionally, a tri-linear characteristic force-
displacement relationship for the joint component characterisation is also
proposed.
The straightforward applicability of the proposed mechanical model is
illustrated in Chapter 4 by means of detailed examples using a set of extended
1 Introduction 37
endplate joints. Moreover, the model validation against experimental tests,
considering or neglecting the axial force effect on the joint behaviour, is also
assessed.
In Chapter 5, parametric and sensitivity studies are carried out in order to
investigate and demonstrate the application scope of the proposed model. Various
scenarios involving the key parameters that influence on the joint structural
behaviour were considered and discussed.
An alternative methodology is presented in Chapter 6 extending the range of
application of available experimental data to produce moment-rotation
characteristics that implicitly make proper allowance for the presence of
significant levels of either tension or compression in the beam. The applicably and
validity of this proposed methodology is demonstrated in Chapter 7 through
comparisons against several tests on endplate joints and baseplate arrangements.
Finally, the most significant conclusions of the present investigation are
summarised in Chapter 8, as well as recommendations for future studies. Based on
the results obtained in this work, design considerations are also suggested aiming
at overcoming the limitations present at the existing code related to the component
method. The current design codes are still not able to suggest procedures to
evaluate the rotational stiffness and moment capacity of semi-rigid joints when, in
addition to the applied moment, an axial force is also presented.
2 Literature Review 38
2Literature Review
2.1.Introduction
This chapter attempts to provide a summary of the techniques currently
available to predict the joint structural behaviour, starting with the conventional
design practice based on the joint flexural response given by Eurocode 3:1-8
(2005), passing by mathematical formulations (empirical models), simplified
analytical models, finite element analysis and culminating in mechanical models
proposed to study of joint behaviour under combined bending moment and axial
force. To conclude, the available experimental tests involving the axial force
effect are brief discussed.
2.2.Conventional Design Practice
Usually, the joints in the design of steel-framed structures are assumed as
either fully rigid or ideally pinned. The first assumption considers the stiff joint,
where the associated small rotations under the transmitted beam end moments
have negligible effect on the distribution of internal forces and moments within
the structure. On the other hand, ideally pinned joint does not transmit bending
moments between the connected members but it can develop significant rotations.
However, it is widely recognized that these two extremes cannot accurately
represent the actual joint behaviour, which in most cases can be described as
semi-rigid, where considerable joint rotations can be developed under transmitted
beam end moments.
2.2.1.Global Analysis
Three analysis methods are currently used to evaluate a joint: elastic
analysis, rigid-plastic analysis and elasto-plastic analysis. For elastic analysis a
linear moment-rotation relationship is sufficient to describe the joint behaviour,
2 Literature Review 39
and thus all joints within the structure should be classified according to their
rotation stiffness. For rigid-plastic analysis, the joints should be classified
according to their strength and the joint rotation capacity should be sufficient to
accommodate the rotations resulting from the analysis. Finally, for elasto-plastic
analysis, the joints should be classified according to both stiffness and strength
and a moment-rotation characteristic of the joints is used to evaluate the
distribution of internal forces and moments. Although the joint response is
generally non-linear, a bi-linear simplification of the design moment-rotation
characteristic is usually adopted.
2.2.2.Classification of the Joints
2.2.2.1.Classification by Stiffness
In line with the joint classification by stiffness, a joint may be classified as
rigid, nominally pinned or semi-rigid according to its rotational stiffness, by
comparing its initial rotational stiffness Sj,ini with the classification boundaries that
can be expressed in terms of a non-dimensional stiffness parameter:
b
binij
EI
LSk , (2.1)
where Sj,ini is the initial joint rotation stiffness corresponding to a bending moment
that does not exceed two-third of the design moment resistance Mj,Rd of the joint,
Lb is the span of the supported beam, E is the elastic modulus of structural steel
and Ib is the second moment of area of the supported beam section. A joint can be
classified as:
5.0ifpinnednominally :3Zone
5.0ifrigid-semi:2Zone
ifrigid:1Zone
k
kk
kk
b
b
(2.2)
where kb is a factor that depends on the frame type. For braced frames where the
bracing system reduces the horizontal displacement by at least 80% kb admits a
value of 8. For all others frames kb can be taken as 25, provided that the ratio of
the relative rigidity Kb=Ib/Lb of all the beams at the top of the storey to the relative
rigidity Kc=Ic/Lc of all the columns of the same storey is greater than or equal to
2 Literature Review 40
0.1, i.e. Kb/Kc 0.1. If the ratio is less than 0.1, the joint should be classified as
semi-rigid irrespective of the non-dimensional stiffness parameter value.
2.2.2.2.Classification by Strength
Regarding the joint classification by strength, a joint may be classified as
full-strength, nominally pinned or partial strength by comparing its design
moment resistance Mj,Rd with the design moment resistances of the members that
it connects. The design resistance of a full strength joint should be not less than
that of the connected members, whilst a nominally pinned joint should be capable
of transmitting the internal force without developing significant moments which
might adversely affect the members or the structure as a whole. On the other hand,
a joint which does not meet the criteria for either a full-strength joint or a
nominally pinned joint should be classified as a partial-strength joint.
The joint classification according to the moment resistance can be also
expressed in terms of a non-dimensional moment resistance parameter:
Rdpl
Rdj
M
Mm
,
, (2.3)
where Mpl,Rd is the design plastic moment resistance of the connected member. For
joints located at the top storey Mpl,Rd is the smallest of the design plastic moment
resistances of the connected beam and column, while for joints at lower storeys
Mpl,Rd should be taken as the smallest of the beam design plastic moment
resistance and twice the column design plastic moment resistance. In this way,
joints can be categorized as:
125.0ifstrength-partial
1ifstrength-full
25.0ifpinnednominally
m
m
m
(2.4)
2.2.3.Design Moment-Rotation Characteristic of Joints
A joint may be represented by a rotational spring connecting the centre lines
of the connected members at the point of intersection, as indicated in Figure 5(a)
and Figure 5(b) for a single-sided beam-to-column joint configuration. The
properties of the spring can be expressed in the form of a design moment-rotation
characteristic that describes the relationship between the bending moment Mj,Ed
2 Literature Review 41
applied to a joint and the corresponding rotation Ed between connected members.
Generally the design moment-rotation characteristic is non-linear as shown in
Figure 5(c).
The key parameters defining the design moment-rotation characteristic are
the moment resistance, the rotational stiffness and rotation capacity, Figure 5. The
joint design moment resistance Mj,Rd is equal to the maximum moment of the
moment-rotation characteristic. The rotation stiffness Sj is the secant stiffness
corresponding to the design moment action Mj,Ed, while the initial rotational
stiffness Sj,ini is the elastic range slope of the design moment-rotation
characteristic. The design rotation capacity Cd of a joint is equal to the maximum
rotation of the design moment-rotation characteristic.
Figure 5 - Design moment-rotation characteristic for a joint.
2.2.4.Component method
The most widely used method for predicting the moment-rotation
characteristic of semi-rigid joint is the component method. The component
method entails the use of relatively simple joint mechanical models, based on a set
of rigid links and spring components. The component method – introduced in
Eurocode 3:1-8 (2005) – can be used to determine the joint’s resistance and initial
stiffness. Its application requires the identification of active components, the
evaluation of the force-deformation response of each component (which depends
on mechanical and geometrical properties of the joint) and the subsequent
assembly of the active components for the evaluation of the joint moment versus
rotation response.
Basically, the standard connection types can be divided into two principal
categories: welded connections and bolted connections. The joint basic
2 Literature Review 42
components associated with these connection types are presented in Table 1 and
identified in Figure 6. Besides this, Figure 6 presents the mechanical model
associated with each connection type.
Table 1 - Joint basic components.
EC3-1-8 (2005) Identification
Basic ComponentAdopted Notation
1 Column web panel in shear cws2 Column web in transverse compression cwc3 Column web in transverse tension cwt4 Column flange in bending cfb5 Endplate in bending epb6 Flange/web cleat in bending:
- top angle in bending ta- web angles in bending wa
7 Beam flange and web in compression bfwc8 Beam web in tension bwt9 Plate in tension or compression
- top angle leg in tension tat- web angle leg in tension wat- seat angle in compression sac
10 Bolts in tension bt11 Bolts in shear bs12 Bolts in bearing bb13 Welds wel
Plate in bearing:- top angle leg in bearing tab- seat angle leg in bearing sab- beam flanges in bearing bfb- beam web in bearing bwb- web angle leg in bearing wab
Regarding welds in particular, they are able to withstand very limited
deformations and they generally exhibit a brittle failure mode. Therefore, as long
as design requirements leading to sufficient weld over strength are satisfied, welds
can be neglected when calculating the moment resistance of the joint. Similarly,
their contribution to the joint rotation stiffness should be taken as equal to infinity.
According to the component method, the design moment resistance Mj,Rd of
any joint may be derived from the distribution of internal forces within the joint
and the resistances of its basic components to these forces. In addition, the
flexibilities of the basic components, each one represented by an elastic stiffness
coefficient ki that has units of length (normalised relative to the elastic modulus of
structural steel), can be combined to determine the joint rotational stiffness Sj.
2 Literature Review 43
cws
cfb cwt
(a) Welded connections.
cwc bfwc
Rigid-plasticcomponent
f
d
Elasto-plasticcomponent
f
d
k
fRd
fRd
cwc cws bfwc
bt bwt
bt bwt
cfb cwt bt epb bwt
cfb cwt bt
cfb cwt
cfb cwt
(b) Endplate connections.
epb
epb
epb
cwc cws bs sab bfwc sac
bt wa bwtbs bwb wab
bt wa bwtbs bwb wab
cfb cwt bt wa bwtbs bwb wab
cfb cwt bt
ta tatbs tab bfb
cfb cwt
cfb cwt
(c) Angle flange cleat connections.
Figure 6 – Joints and their associated mechanical models.
2 Literature Review 44
2.2.4.1.Welded Connections
The evaluation of the welded joint moment resistance is given by:
min,, RdtRdj FhM (2.5)
where FRd,min is the minimum of the design resistances of the five basic
components presented in Figure 6(a) and ht is the lever arm, which can be taken as
the centre-to-centre distance between the two flanges of the supported beam.
Regarding the joint initial stiffness, it can be evaluated as:
i i
tsj
k
hES
1
2
(2.6)
in which ki is the stiffness coefficient for each basic component i that contributes
to the joint stiffness. Usually, the stiffness of the column flange in bending and the
beam flange/web in compression are not considered in the joint rotational
stiffness. is the stiffness ratio Sj,ini/Sj that is used to account for the joint non-
linear behaviour and it is determined from the following relationships:
RdjEdjRdjRdj
Edj
RdjEdj
MMMM
M
MM
,,,
7.2
,
,
,,
3/2if5.1
3/2if1
(2.7)
2.2.4.2.Bolted Connections
The bolted joint moment resistance can be evaluated as:
rn
rRdtrrRdj FhM ,, (2.8)
where Ftr,Rd is the effective tension resistance of bolt-row r, hr is the distance of
bolt-row r from the centre of compression, r is the bolt-row number and nr is the
total number of bolt-rows in tension.
The effective tension resistance of each bolt-row is calculated in sequence,
starting from the bolt-row furthest from the centre of compression and progressing
to the other bolt-rows closer to the centre of compression successively. According
to this procedure, the resistance of bolt-row r is taken as the minimum value of the
resistance of its basic components, considered both individually and as part of all
the possible groups of consecutive bolt-rows consisting of bolt-row r and the
2 Literature Review 45
previous ones. Furthermore, the resistance of any bolt group should not exceed the
resistance of the components which are independent of the bolt-rows, such as the
column web panel in shear, the column web in transverse compression and the
beam flange/web in compression.
Based on the procedure recommended by Eurocode 3:1-8 (2005), the first
step towards the determination of the joint rotational stiffness Sj is the
computation of the effective stiffness coefficient keff,r for each bolt-row r:
i ri
reff
k
k
,
, 11
(2.9)
where ki,r is the stiffness coefficient representing basic component i in bolt-row r.
In the case of joints with endplate connections, keff,r should take into account the
stiffness coefficients ki of the column web in tension, the column flange in
bending, the endplate in bending, and the bolts in tension. The respective
coefficients for angle flange/web cleat connections include the column web in
tension, the column flange in bending, the flange/web cleat in bending, the bolts
in tension, the bolts in shear, and the bolts in bearing.
Evaluated the effective stiffness coefficient for each bolt-row r, the second
step in the procedure is the representation of the basic components corresponding
to all bolt-rows by a single equivalent stiffness coefficient keq. Assuming a linear
rotation profile associated with a rigid rotation of the supported beam web around
the centre of compression, the equivalent stiffness coefficient can be determined
from:
eq
rrreff
eq z
hkk
,
(2.10)
in which zeq is an equivalent lever arm given by:
rrreff
rrreff
eq hk
hkz
,
2,
(2.11)
Finally, considering the contribution of the column web panel in shear and
the column web in compression, which are independent of the bolt-rows, the joint
rotational stiffness Sj can be calculated from:
2 Literature Review 46
eq
eqj
kkk
EzS
111
21
2
(2.12)
where k1 and k2 are the stiffness coefficients corresponding to the column web
panel in shear and the column web in compression, respectively. Similar to
welded connections, the stiffness ratio μ can be obtained from the following
equations (Eurocode 3:1-8 2005):
RdjEdjRdjRdj
Edj
RdjEdj
MMMM
M
MM
,,,,
,
,,
3/2if5.1
3/2if1
(2.13)
where the coefficient ψ depends on the connection type. For endplate connections
ψ should be taken as 2.7, while for angle flange cleats the value of 3.1 is
recommended.
2.2.4.3.Equivalent T-stub
The key components in endplate connections can be analysed using
equivalent T-stub assemblies in tension and compression. This is implemented by
adopting appropriate orientation of the T-stub, which depends on the connection
type and the component row position that is being analysed as shown in Figure 7.
(a) Unstiffened. (b) Stiffened.
Figure 7 - T-stub identification and orientation for bolted extended endplate connections.
2.2.4.3.1.Equivalent T-stub in Tension
In bolted connections an equivalent T-stub in tension may be used to model
the design resistance of the following basic components: column flange in
bending; endplate in bending; flange cleat in bending and baseplate in bending
2 Literature Review 47
under tension. Table 2 presents how to evaluate the design resistance of a T-stub
flange according to its failure modes that may be developed as shown in Figure 8.
Table 2 - Design resistance FT,Rd of a T-stub flange (EC3-1-8, 2005).
Figure 9.
Equation (2.45).
2 Literature Review 48
Figure 8 – Failure modes of a T-stub.
The values for emin, leff and m presented in Table 2 and Figure 9 are given in
section 2.2.4.4 for each joint basic component. The effective length leff is the most
significant parameter since it accounts for the possible yield line mechanisms of
the T-stub flange and varies according to the bolt-row location. For the bolt-row
located outside the tension flange of the beam – extended endplate – the collapse
mechanisms are shown in Figure 10. In the case of multiple bolt-rows three cases
may be identified: yield lines develop separately for each bolt-row - Figure 11(a);
only some bolt-rows constitute a bolt-group - Figure 11(b); bolt-group involves all
the bolt-rows - Figure 11(c).
Finally, in cases where prying forces may be developed, Table 2, the design
tension resistance of a T-stub flange should be taken as the smallest value for the
three possible failure modes 1, 2 and 3. On the other hand, disregarding the prying
force effect, the design tension resistance of a T-stub flange should be assumed as
the smallest value for the two possible failure modes according to Table 2.
Figure 9 – Dimensions of an equivalent T-stub flange (EC3-1-8, 2005).
Mode 1:Complete yielding
of the flanges.
Mode 2:Flange yielding and
bolt failure.
Mode 3:Bolt failure.
2 Literature Review 49
(a) Circular pattern.
(b) Non-circular pattern.
Figure 10 – Collapse mechanisms of the bolt-row outside the beam flange (Faella et al.,
2000).
(a) Single bolt row.
CircularPattern
Non-CircularPattern
(b) Partial bolt group.
CircularPattern
Non-CircularPattern
(c) Global bolt group.
CircularPattern
Non-CircularPattern
Figure 11 – Yield line models of bolt row group (Faella et al., 2000).
Individual Bolt-Rows, Bolt-Groups and Groups of Bolt-Rows
The use of the T-stub approach to model a group of bolt-rows should satisfy
the following conditions:
a) the force at each bolt-row should not exceed the design resistance
determined considering only that individual bolt-row;
b) the total force on each group of bolt-rows, comprising two or more
adjacent bolt-rows within the same bolt-group, should not exceed the design
resistance of that group of bolt-rows.
For evaluation of the design tension resistance of a basic component
represented by an equivalent T-stub flange, the following parameters should be
calculated:
2 Literature Review 50
a) the design resistance of an individual bolt-row, determined considering
only that bolt-row;
b) the contribution of each bolt-row to the design resistance of two or more
adjacent bolt-rows within a bolt-group, determined considering only those
bolt-rows.
In the case of an individual bolt-row leff should be taken as equal to the
effective length leff tabulated in 2.2.4.4 for that bolt-row taken as an individual
bolt-row. On the other hand, for a group of bolt-rows leff should be taken as the
sum of the effective length leff tabulated in 2.2.4.4 for each relevant bolt-row taken
as part of a bolt-group.
2.2.4.3.2.Equivalent T-stub in Compression
In steel-to-concrete joints, the flange of an equivalent T-stub in compression
may be used to model the design resistances for the combination of the following
basic components: the steel baseplate in bending under the bearing pressure on the
foundation; the concrete and/or grout joint material in bearing. As these basic
components are not used in this work, further information about how to evaluate
the design compression resistance of a T-stub flange can be found in EC3-1-8
(2005).
2.2.4.4.Design of the Joint Basic Components
In this section is presented how to evaluate the design resistance and
stiffness of each joint basic component according to Eurocode 3:1-8 (2005).
2.2.4.4.1.Column Web Panel in Shear
Resistance
The design shear resistance including the influence of the distribution of the
internal actions can be evaluated as:
Rdwp
Rdwsc
VF ,
,, (2.14)
where is giving in Table 4 and Vwp,Rd is the design plastic shear resistance of an
unstiffened column web panel obtained using:
2 Literature Review 51
0
,,
3
9.0
M
vcwcyRdwp
AfV
(2.15)
where Avc is the shear area of the column evaluated according to Eurocode 3:1-1
(2005).
The design shear resistance may be increased by the use of stiffeners or
supplementary web plates, however they are not considered in this work.
Stiffness
The stiffness coefficient for the column web panel in shear, for unstiffened
column web panel, is:
z
Ak vc
38.0
1 (2.16)
where z is the lever arm. For a more accurate value z is assumed equal to zeq given
in Eq. 2.11. For stiffened column web panel in shear
1k (2.17)
2.2.4.4.2.Column Web in Transverse Compression
Resistance
The design resistance of an unstiffened column web subject to transverse
compression should be assumed as the smallest between the crushing resistance,
0
,,,,,,
M
wcywcwcceffwccrRdwcc
ftbkF
(2.18)
and the buckling resistance,
1
,,,,,,
M
wcywcwcceffwcbrRdwcc
ftbkF
(2.19)
where is a reduction factor to allow for the possible effects of interaction with
shear in the column web panel according to Table 3.
2 Literature Review 52
Table 3 – Reduction factor for interaction with shear.
Transformation parameter Reduction factor 5.00 1
15.0 11 112
1 1
21 121 1
2 2
2,,
1
/3.11
1
vcwcwcceff Atb
2,,
2
/2.51
1
vcwcwcceff Atb
In Table 3, Avc is the shear area of the column and is a transformation parameter
which account for the possible influence of the web panel in shear. Approximate
values for based on the values of the beam moments Mb1,Ed and Mb2,Ed at
periphery of the web panel, see Figure 12, may be obtained from Table 4.
Figure 12 – Forces and moments acting on the joint. Direction of forces and moments are
considered as positive in relation to equations presented in this section.
2 Literature Review 53
Table 4 - Approximate values for the transformation parameter .
Type of joint configuration Action Value of
EdbM ,1 1
EdbEdb MM ,2,1 00/ ,2,1 EdbEdb MM 10/ ,2,1 EdbEdb MM 20,2,1 EdbEdb MM 2
beff,c,wc is the effective width of column web in compression:
- for a welded connection:
statb fcbfbwcceff 522,, (2.20)
where ac, rc and ab are given in Figure 13.
- for bolted endplate connection:
pfcbfbwcceff sstatb 522,, (2.21)
where sp is the length obtained by dispersion on at 45o of the compressive action
through the endplate thickness and it should be:
ppp tst 2 (2.22)
where tp is the endplate thickness. According to Faella et al. (2000) sp may be
evaluated as:
pfbepepepp ashlhts (2.23)
where tep is the endplate thickness; hep is the endplate depth; lep is the endplate
length over the beam top flange; hb is the beam depth and apf is the throat
thickness of the weld between the beam flange and the endplate.
2 Literature Review 54
Figure 13 - Transverse compression on an unstiffened column.
- for bolted connection with angle flange cleats:
strtb fcaawcceff 56.02,, (2.24)
where ta and ra are illustrated in Figure 13(a); tfc is the column flange thickness
and s is:
c
c
a
rs
2s:columnsection Hor I weldedafor
:columnsection Hor Irolledafor
(2.25)
is the reduction factor for plate buckling:
2/2.0:72.0if
0.1:72.0if
ppp
p
(2.26)
p is the plate slenderness:
2
,,,932.0wc
wcywcwcceffp
Et
fdb (2.27)
where dwc is:
)2(2:columnsection Hor I weldedafor
)(2:columnsection Hor Irolledafor
cfccwc
cfccwc
athd
rthd
(2.28)
kwc is a reduction factor that accounts for the influence of the vertical normal
stress and is generally assumed equal to 1.0 and no reduction is necessary. It can
therefore be omitted in preliminary calculations when the longitudinal stress is
unknown and checked later.
2 Literature Review 55
Stiffeners or supplementary web plates may be used to increase the design
resistance of a column web in transverse compression, however they are not
considered in this work.
Stiffness
The stiffness coefficient for the column web in compression, for unstiffened
column web, is:
wc
wcwcceff
d
tbk ,,
2
7.0 (2.29)
where dwc is the clear depth of the column web given in Eq. 2.28. For stiffened
column web in compression
2k (2.30)
2.2.4.4.3.Column Web in Transverse Tension
Resistance
The design resistance of an unstiffened column web subject to transverse
tension should be determined from,
0
,,,,,
M
wcywcwcteffRdwct
ftbF
(2.31)
where is a reduction factor to allow for the possible effects of interaction with
shear in the column web panel determined from Table 3 using the value of beff,t,wc
given below.
The effective width of a column web in tension should be obtained using:
- for a welded connection:
statb fcbfbwcteff 522,, (2.32)
where ab is given in Figure 13 and s is evaluated in Eq. 2.25.
- for a bolted connection: the effective width beff,t,wc of column web in tension
should be taken as equal to the effective length of equivalent T-stub representing
the column flange in bending, see 2.2.4.4.4.
Stiffeners or supplementary web plates may be used to increase the design
tension resistance of a column web, however they are not considered in this work.
Stiffness
2 Literature Review 56
The stiffness coefficient for the column web in tension, for stiffened or
unstiffened bolted connection with a single bolt-row in tension or unstiffened
welded connection, is:
wc
wcwcteff
d
tbk ,,
3
7.0 (2.33)
where dwc is the clear depth of the column web given in Eq. 2.28. For stiffened
welded connection
3k (2.34)
The effective width of the column web in tension, beff,t,wc, for a joint with a single
bolt-row in tension should be taken as equal to the smallest of the effective
lengths leff (individually or as part of a group of bolt-rows) given for this bolt-row
in Table 5 for an unstiffened column flange.
2.2.4.4.4.Column Flange in Transverse Bending
Resistance
Transverse stiffeners and/or appropriate arrangements of diagonal stiffeners
may be used to increase the design resistance of the column flange in bending,
however only unstiffened column flange is considered.
Bolted connection
The design resistance and failure mode of an unstiffened column flange in
transverse bending, together with the associated bolts in tension, should be taken
as similar to those of an equivalent T-stub flange, see 2.2.4.3, for both each
individual bolt-row required to resist tension and each group of bolt-rows required
to resist tension. The dimensions emin and m used for T-stub flange evaluation
should be determined from Figure 14. The effective length of equivalent T-stub
flange should be determined for the individual bolt-rows and the bolt-group in
accordance with 2.2.4.3.1 from the values given for each bolt-row in Table 5.
2 Literature Review 57
Figure 14 - Definitions of e, emin, rc and m.
2 Literature Review 58
Table 5 - Effective lengths for an unstiffened column flange.
Welded connection
For welded joints, the design resistance of an unstiffened column flange in
bending, due to tension or compression from a beam flange, should be obtained
using:
0
,,,,
M
fbyfbfcbefRdfc
ftbF
(2.35)
where beff,b,fc is the effective breath beff of the beam flange considered as a plate.
For an unstiffened I or H section the effective width should be obtained
from:
fweff ktstb 72 (2.36)
in which:
1but,
,
k
f
f
t
tk
py
fy
p
f (2.37)
where fy,f is the yield strength of the flange of the I or H section; fy,p is the yield
strength of the plate welded to the I or H section; and s should be obtained from
Eq. 2.25. However, the following criterion should be satisfied:
ppu
pyeff b
f
fb
,
, (2.38)
where fu,p and bp are, respectively, the ultimate strength and the width of the plate
welded to the I or H section. Otherwise the joint should be stiffened.
Stiffness
The stiffness coefficient for the column flange in bending, for a single bolt-
row in tension, is:
2 Literature Review 59
3
3
4
9.0
m
tlk fceff (2.39)
where leff is the smallest of the effective lengths (individually or as part of a bolt
group) for this bolt-row given in Table 5 for an unstiffened column flange; and m
is as defined in Figure 14.
2.2.4.4.5.Endplate in Bending
Resistance
The design resistance and failure mode of an endplate in bending, together
with the associated bolts in tension, should be taken as similar to those of an
equivalent T-stub flange, see 2.2.4.3 for both each individual bolt-row and each
group of bolt-rows required to resist tension. The groups of bolt-rows either side
of any stiffener connected to the endplate should be treated as separate equivalent
T-stubs. In extended endplates, the bolt-row in the extended part should also be
treated as a separate equivalent T-stub, see Figure 15. The design resistance and
failure mode should be determined separately for each equivalent T-stub. The
dimension emin required for use in 2.2.4.3 should be obtained from Figure 14 for
that part of the endplate located between the beam flanges. On the other hand, for
the endplate extension emin should be taken as equal to ex, see Figure 15. Finally,
the effective length of an equivalent T-stub flange leff should be determined in
accordance with 2.2.4.3.1 using the values for each bolt-row given in Table 6.
2 Literature Review 60
Figure 15 - Modelling an extended endplate as separate T-stubs.
Table 6 - Effective lengths for an endplate.
2 Literature Review 61
Figure 16 – Values of for stiffened column flanges and endplates.
2 Literature Review 62
Stiffness
The stiffness coefficient for the endplate in bending, for a single bolt-row in
tension, is:
3
3
5
9.0
m
tlk peff (2.40)
where leff is the smallest of the effective lengths (individually or as part of a group
of bolt-rows) given for this bolt-row in Table 6; m is generally as defined in
Figure 16, but for a bolt-row located in the extended part of an extended endplate
m = mx, where mx is as defined in Figure 15.
2.2.4.4.6.Beam Flange and Web in Compression
Resistance
The resultant of the design compression resistance of a beam flange and the
adjacent compression zone of the beam web may be assumed to act at the level of
the centre of compression. In this way, the design compression resistance of the
combined beam flange and web is given by the following expression:
)(,
,,fbb
RdcRdfbc th
MF
(2.41)
where hb is the depth of the connected beam; Mc,Rd is the design moment
resistance of the cross-section, reduced if necessary to allow for shear, see
Eurocode 3:1-1 (2005); and tfb is the flange thickness of the connected beam.
The design moment resistance of the beam could be increased by
reinforcing it with haunches, however this case is not considered in this work.
Stiffness
For beam flange and web in compression, k7 should be taken as equal to
infinity:
7k (2.42)
2 Literature Review 63
2.2.4.4.7.Beam Web in Tension
Resistance
The design tension resistance of the beam web, for a bolted endplate
connection, should be obtained from:
0
,,,,,
M
wbywbwbteffRdwbt
ftbF
(2.43)
where beff,t,wc is the effective width of the beam web in tension and should be taken
as equal to effective length of the equivalent T-stub representing the endplate in
bending obtained from 2.2.4.4.5 for an individual bolt-row or a bolt-group.
Stiffness
For beam web in tension, k8 should be taken as equal to infinity:
8k (2.44)
2.2.4.4.8.Bolts in Tension
Resistance
The design tension resistance of a bolt is evaluated as:
2
,2,
M
sbubRdt
AfkF
(2.45)
where kb2 is equal to 0.63 for countersunk bolt or 0.90 for others cases; fu,b is the
ultimate strength of the bolt and As is the tensile stress area of the bolt.
Stiffness
The stiffness coefficient for the bolts in tension, for a single bolt-row, is:
b
s
L
Ak
6.110 (2.46)
where Lb is the bolt elongation length, Figure 17:
22hn
whwfcepb
tttnttL (2.47)
taken as equal to the grip length (total thickness of material, tep + tfc, and washers,
nwtwh), plus half the sum of the height of the bolt head (th) and the height of the nut
(tn), where nw is the number of washers and twh is the washer thickness.
2 Literature Review 64
db
h
dn
dw
twh
th
tp
tp
twh
tn
Plate
Plate
Plate
Plate
Bolt
Washer
Washer
Nut
Lb
Figure 17 - Bolt elongation length.
2.2.4.5.Axial Force
Nowadays, using the Eurocode 3:1-8 (2005) component method, it is
possible to evaluate the rotational stiffness and moment capacity of semi-rigid
joints when subject to pure bending. However, this component method is not yet
able to calculate these properties when, in addition to the applied moment, an
axial force is also present. Eurocode 3:1-8 (2005) suggests that the axial load may
be disregarded in the analysis when its value is less than 5% of the beam’s axial
plastic resistance, but provides no information for cases involving larger axial
forces.
Although, the component method does not consider the axial force, its
general principles could be used to cover this situation, since it is based on the use
of a series of force versus displacement relationships, which only depend on the
component’s axial force level, to characterize any individual component’s
behaviour.
2.3.Theoretical Models
As an alternative to experimental tests, other methods have been proposed
to predict bending moment versus rotation curves. These procedures range from a
purely empirical curve fitting of test data, passing through ingenious behavioural,
2 Literature Review 65
analogy and semi-empirical techniques, to comprehensive finite element analysis,
Nethercot & Zandonini (1989).
2.3.1.Mathematical Formulations (Empirical Models)
Empirical models are based on formulations that relate the parameters
involved in the mathematical representation of the moment-rotation curve to the
geometrical and mechanical properties of the joints. Although they can be useful
for the prediction of moment-rotation curves, they are limited to the joint
configurations used for calibrating the corresponding formulations.
The first attempt of fitting a mathematical representation to connection
moment-rotation curves dates back to the work of Baker (1934) and Rathbun
(1936), who used a single straight-line tangent to the initial slope, thereby
overestimating connection stiffness at finite rotations.
In the 1970s the use of bilinear representations was introduced by
Lionberger & Weaver (1969) and Romstad & Subramanian (1970). These
recognised the reduced stiffness at higher rotations, however it was only
acceptable for certain joint types and for applications where only small joint
rotations are likely.
Kennedy (1969), Sommer (1969), Frye & Morris (1975) proposed
polynomial representations that recognised the curved nature, but required
mathematical curve fitting and consideration of a family of experimental moment-
rotation curves. The empirical model developed by Frye & Morris (1975), which
is based on an odd-power polynomial representation of the moment-rotation
curve, is given as
53
321 )()()( MKCMKCMKC (2.48)
where C1, C2 and C3 are curve-fitting constants, M is the moment applied to the
connection and the parameter K depends on the geometrical and mechanical
properties of the connection details.
Ang & Morris (1984) replaced the polynomial representation by a Ramberg-
Osgood (1943) type of exponential function that has the advantage of always
yielding a positive slope, but is also dependent on mathematical curve fitting.
Multi-linear representations were proposed by Moncarz & Gerstle (1981)
and Poggi & Zandonini (1985) to overcome the obvious limitation of the bilinear
2 Literature Review 66
model in that it could not deal with continuous changes in stiffness in the knee
region.
B-spline techniques were suggested by Jones et al. (1981) as an alternative
to polynomials as a means of avoiding possible negative slopes:
3
0 1
3
j
m
jjjjj MMbMa (2.49)
where m is the number of knots (junction of multi-part curve) and
00
0
jj
jjj
MMforMM
MMforMMMM(2.50)
where Mj is the upper bound moment in the j-th part of the curve, aj and bj are
coefficients obtained by least-square curve fitting.
Lui & Chen (1986) used an exponential representation that despite being
complex could readily be incorporated in analytical computer programs
(Nethercot et al., 1987). This exponential expression is:
p
n
jj K
jacMM
2exp1
10 (2.51)
where M0 is the initial moment, Kp is the strain hardening type connection
stiffness, a and cj are the modelling parameters.
Although it is possible to closely fit virtually any shape of moment-rotation
curve, purely empirical methods possess the disadvantage that they cannot be
extended outside the range of the calibration data. This is particularly important
for joints such as endplates where the change in geometrical and mechanical
properties of the connection may lead to substantially different behaviours and
collapse mechanisms (Nethercot & Zandonini, 1989).
Aiming to overcome this limitation, Yee & Melchers (1986), Kishi et al.
(1988a,b) and Chen & Kishi (1987) proposed models linking curve fitting
approaches to some form of behavioural model, but these were still dependent on
a mathematical curve fitting. The simplified four-parameter exponential model
proposed by Yee & Melchers (1986) for bolted extended endplate connection is:
p
p
pip K
M
CKKMM
exp1 (2.52)
where Mp is the plastic moment, Ki and Kp are respectively the initial and strain
hardening stiffnesses, and C is a constant controlling the curve slope.
2 Literature Review 67
Focusing on finite element analysis, Richard et al. (1980) used a type of
formula already developed by Richard & Abbott (1975) to represent data
generated by finite element analyses in which the constitutive relations of certain
of the joint components, e.g. bolts in shear, were directly obtained from subsidiary
tests.
Each of the models discussed so far may only be used to describe the joint
behaviour under a single application of a monotonically increasing load.
However, some of them were modified and/or adapted to represent the
performance of certain connection types under cyclic loading, as can be seen in
the work done by Moncarz & Gerstle (1981), Altman et al. (1982) and Mazzolani
(1988).
Aiming to incorporate a limited set of experiments including the axial
versus bending moment interaction into a structural analysis, Del Savio et al.
(2007b, 2008b) developed a consistent and simple approach to determine
moment-rotation curves for any axial force level. This alternative methodology is
also presented in Chapter 6. Basically, this method works by finding moment-
rotation curves through interpolations executed between three required moment-
rotation curves, one disregarding the axial force effect and two considering the
compressive and tensile axial force effects. This approach can be easily
incorporated into a nonlinear joint finite element formulation since it does not
change the finite element basic formulation, only requiring a rotational stiffness
update procedure.
2.3.2.Simplified Analytical Models
Several authors have applied the basic concepts of structural analysis
(equilibrium, compatibility and material constitutive relations) to simplified
models of the key components in various types of beam-to-column connections
(Nethercot & Zandonini, 1989).
Lewitt et al. (1969) provided formulae for the load-deformation behaviour
of double web cleat connections in both the initial and the final plastic phases;
however these models needed to be used in conjunction with knowledge of the
connection rotation centre.
2 Literature Review 68
Chen & Kishi (1987) and Kishi et al. (1988a,b) considered the behaviour of
web cleats, flange cleats and combined web and flange cleat connections where
their resulting values of initial connection stiffness and ultimate moment capacity
were utilized in a Richard type of power expression (Richard & Abbott, 1975) to
represent the resulting moment-rotation curve.
Assuming that the behaviour of the whole joint may be obtained simply by
superimposing the flexibilities of the joint components (member elements,
connecting, elements, fasteners) Johnson & Law (1981) proposed a method for the
prediction of the initial stiffness and plastic moment capacity of flush endplate
connections, however no comparison was conducted against experimental results.
Based on the same philosophy, Yee & Melchers (1986) developed a method
for bolted endplate eaves connections in which an exponential representation was
assumed, which depends on four parameters where only one is dependent on test
data.
Richard et al. (1988) proposed a four-parameter formula to describe the
load-deformation and moment-rotation relationship for bolted double framing
angle connections. These equations are
pNN
p
p
pNN
p
p
K
M
KK
KKM
K
R
KK
KKR
/1
0
/1
0
1
)(
1
)(
(2.53)
where K and Kp are respectively the initial and strain hardening stiffnesses, R0 and
M0 are the reference load and moment respectively, N is the shape parameter
obtained through the least square method. This model is composed of a rigid bar
and a nonlinear spring, representing the angle segments in either tension or
compression. The moment-rotation behaviour of the connections is determined
through an iterative procedure by satisfying equilibrium and compatibility
conditions.
A similar approach was developed and used by Elsati & Richard (1996) in a
computer-based programme to validate the model against the test results of a
variety of connection types for both composite and steel beam connections.
2 Literature Review 69
A three-parameter exponential model was suggested by Wu & Chen (1990)
to model top and seat angles with and without double web angle connection and
due to its simplicity it could be implemented in the analysis of semi-rigid frames.
The moment-rotation relationship is:
0
1ln
nn
M
M
u
(2.54)
where Mu is the idealised elastic-plastic mechanism moment, Ki is the initial
rotational stiffness, 0 is the reference rotation evaluated as Mu/Ki and n is the
shape factor. The key parameters, Ki and Mu are calculated by elastic and plastic
analysis, respectively.
In the same year, Kishi & Chen (1990) proposed a semi-analytical model to
predict moment-rotation curves of angle connections:
nn
ui M
MK
M1
1
(2.55)
where Mu is the ultimate moment, Ki is the initial stiffness and n is the shape
factor. The initial stiffness and ultimate moment capacity are evaluated
analytically using simple failure mechanisms. Later, this model was extended by
Foley & Vinnakota (1994) for unstiffened extended endplate connections.
Although these methods require a few key parameters, the use of test data is
normally necessary to calibrate some of their coefficients. A wider discussion
about some of these methods can be found in Nethercot & Zandonini (1989) and
Faella et al. (2000).
2.3.3.Finite Element Analysis
Numerical simulation started being used as a way to overcome the lack of
experimental results; to understand important local effects that are difficult to be
measured with sufficient accuracy, e.g. prying forces and extension of the contact
zone, contact forces between the bolt and the connection components; and to
generate extensive parametric studies.
The first study into joint behaviour making use of the FEM was executed by
Bose et al. (1972) related to welded beam-to-column connections, where an
incremental analysis was performed, including in the formulation plasticity, with
2 Literature Review 70
strain hardening, and buckling. The comparison with available experimental
results showed satisfactory agreement, but only the critical load levels were
considered.
Since then, several researchers have been using the FEM to investigate joint
behaviour, such as: Lipson & Hague (1978) – single-angle bolted-welded joint;
Krishnamurthy et al. (1979) – extended endplate joints; Richard et al. (1983) –
double-angle joint; Patel & Chen (1984) – welded two-side joints; Patel & Chen
(1985) – bolted moment joint; Kukreti et al. (1987) – flush endplate joints;
Beaulieu & Picard (1988) – bolted moment joint; Atamiaz Sibai & Frey (1988) –
welded one-side unstiffened joint configuration.
More recently, focusing on 3D finite element models the following works
can be mentioned and discussed:
- Sherbourne & Bahaari (1994) developed a methodology based on finite-
element modelling to analytically evaluate the moment-rotation relationships for
steel bolted endplate connections. The endplate, beam and column flanges, webs
and column stiffeners were represented as plate elements, whilst each bolt shank
was modelled using six spar elements and three-dimensional interface elements
were used to model the boundary between column flange and back of the endplate
that may make or break contact. The methodology was demonstrated for an
extended endplate connection and the results were compared against experimental
data. The predicted results were within the range of accuracy of experimental
values.
- Bursi & Jaspart (1997, 1998) studied bolted steel connections by means of
finite elements. They performed the finite element code ABAQUS (1994)
calibration on test data as well as on the simulations of elementary tee stub
connections and then an assemblage of three-dimensional beam finite elements
was proposed to model the bolt behaviour in a simplified fashion. The proposed
3D finite-element model was set with the ABAQUS (1994) code in order to
simulate the stiffness and strength behaviour of isolated extended endplate
connections. Comparison between computed and measured values in each phase
highlighted the effectiveness and degree of accuracy of the proposed finite
element models.
- Yang et al. (2000) investigated double angle connections welded to the
beam web and bolted to the column flange. The connections were subjected to
2 Literature Review 71
axial tensile loads, shear loads, and a combination of these loads. The connections
and the beams were discretized using 3D finite elements in ABAQUS (1994),
including the modelling of the separation of the angle from the column and the
modelling of the contact forces between the bolt heads and the angles. Based on
this study, two mechanical models were proposed, one to be used to approximate
de initial stiffness of the connection under axial loading and another used results
from the 3D analysis to replace the angle by equivalent nonlinear springs. The
proposed mechanical models were able to obtain a simplified prediction of the
connection behaviour.
- Cardoso (2001) worked on numerical and analytical models for extended
endplate and web cleat connections by proposing a finite element based
methodology for the numerical evaluation of moment-rotation relations and load
capacity. A three-dimensional geometrical modelling was used for the main
connection components. Frictionless contact between the connection components
was also considered, by ensuring non-penetration and permitting separation of the
individual parts, together the presence of large deformations, component yielding
and bolt pre-stressing. The numerical results, obtained from the proposed
numerical model, were validated against experimental data and demonstrated
good agreement with the tests.
- Citipitioglu et al. (2002) presented an approach for refined parametric 3D
analysis of partially-restrained bolted steel beam-to-column connections. The
model included the effects of slip by utilizing a general contact scheme and non-
linear 3D continuum elements were used for all parts of the connection. Models
with parameters drawn from a previous experimental study of top and bottom seat
angle connections were generated in order to compare the analysis with test
results, with good prediction shown by the 3D refined models.
Although finite element analysis offers a powerful tool for investigating the
joint behaviour, it can be associated with excessive computational demands,
especially in the case of complex joint configurations, where the interaction of
numerous components, such as bolts, plates and welds, should be taken into
account.
2 Literature Review 72
2.3.4.Mechanical Models
Mechanical models have been developed by several researchers for the
prediction of moment-rotation curves for the whole range of joints, where the
number of physical governing parameters is rather limited. These models have
also been confirmed as an adequate tool for the study of steel joints; however their
accuracy relies on the degree of refinement and accuracy of the assumed load-
deformation laws for the principal components. The determination of such
characteristics requires a complete understanding of the behaviour of single
components, as well as of the way in which they interact, as a function of the
geometrical and mechanical factors of the complete joints, Nethercot & Zandonini
(1989).
Wales & Rossow (1983) effectively introduced the use of mechanical
models, or rather, a component-based method, when they developed a model for
double web cleat connections, Figure 18, in which the joint was idealised as two
rigid bars connected by a homogeneous continuum of independent nonlinear
springs. Each nonlinear spring was defined by a tri-linear load-deformation law
obtained via the analysis of numerical models for the whole connection. Both
bending moment and axial force were considered to act on the connection and
coupling effects between the two stress resultants were then included in the joint
stiffness matrix. Comparisons were made with a single test by Lewitt et al. (1969)
aiming to validate the philosophy. An important feature of this model is to
account for the presence of the axial force. Results obtained by Wales & Rossow
(1983) indicate that greater attention should be given to such axial forces, as a
factor affecting the response of beam-to-column connections.
Figure 18 - Connection and mechanical model for web cleat connections, Wales &
Rossow (1983).
2 Literature Review 73
Kennedy & Hafez (1984) used a technique of connection discretisation to
describe the behaviour of header plate connections. T-stub models were used to
represent the tension and compression parts of the connection. Although this
model had provided good agreement with comparisons done against the author’s
own tests for ultimate moment capacity, the prediction of the corresponding
rotations were not as accurate.
Chmielowiec & Richard (1987) extended the model proposed by Wales &
Rossow (1983) to predict the behaviour of all types of cleated connections only
subjected to bending and shear, Figure 19. Mathematical expressions were
adopted for the force-deformation relationships of the double angle segments and
later calibrated by curve fitting against experimental results obtained by the same
author. Comparisons with experimental data from a different series of connection
tests in general confirmed the accuracy of the method.
Figure 19 - Mechanical model for flange and web cleated connections, Chmielowiec &
Richard (1987).
An extensive investigation into the response of fully welded connections
was conducted by Tschemmernegg (1988), where the mechanical model of Figure
20 was proposed. In this model, springs A are meant to account for the load
introduction effect from the beam to the column, while springs B simulate the
shear flexibility of the column web panel zone. Thirty tests were carried out, using
a wide range of beam and column sections, making possible a calibration of the
mathematical models assumed to describe the spring element properties. The
moment-rotation curves for fully welded connections were determined via the
model for all possible combinations of beams and columns made of European
rolled sections IPE, HEA and HEB. This model was extended by Tschemmernegg
& Humer (1988) for endplate bolted connections by adding new springs (Figure
2 Literature Review 74
21, springs C), to take into account the new sources of deformation. This model
was also calibrated against experimental tests with good results.
Figure 20 - Mechanical model for full welded joints, Tschemmernegg (1988).
Figure 21 - Mechanical model for bolted joints, Tschemmernegg & Humer (1988).
For 10 years, since the proposed model by Wales & Rossow (1983)
considering the bending moment and axial force interaction, nothing had been
done in terms of these coupled effects until Madas (1993) despite the fact that
Wales & Rossow noted that greater attention should be given to such axial forces,
as a factor affecting the response of beam-to-column connections. Madas (1993)
extended the mechanical model proposed by Wales & Rossow (1983) to flexible
endplate, double web angle and top and seat angle connections including both
bare steel and composite connections. Figure 22 shows the idealized beam-to-
column connection used by Madas (1993). This model presented good agreement
with experimental results; however it was not evaluated against experiments
including the axial force versus bending moment interaction.
2 Literature Review 75
Figure 22 - Idealization of beam-to-column connection, Madas (1993).
Based on preliminary studies carried out by Finet (1994), Jaspart et al.
(1999) and Cerfontaine (2003) developed a numerical approach aiming at
analysing the joint behaviour from the first loading steps up to collapse, Figure
23, subjected to bending moment and axial force. This approach is idealised by a
mechanical model comprising extensional springs, Figure 23(b). Each spring
represents a joint component by exhibiting non-linear force-displacement
behaviour, Figure 23(c). Nunes (2006) compared the experimental results
obtained by Lima (2003) for flush and extended endplate joints to the analytical
results using the Cerfontaine (2003) analytical model. This study pointed out
some problems in the joint behaviour prediction using this analytical model, such
as an overestimation of the initial stiffness in the majority of the cases, as well as
variations between over and underestimation of the final moment capacity for
some cases. These discrepancies were more pronounced for the cases in which the
joints were subjected to bending moments and tensile axial forces.
Figure 23 - Mechanical model, Jaspart el al. (1999).
A simplified mechanical model was suggested by Pucinotti (2001) for top-
and-seat and web angle connections as an extension of Eurocode 3:1-8 (1998) to
take into account the web cleats and hardening contributions. Comparisons against
experimental tests showed that this model is able to estimate the initial stiffness
accurately; however the final flexural capacity prediction is slightly erratic.
Using the same general principles, Simões da Silva & Coelho (2001)
formulated analytical expressions for the full non-linear response of a welded
(a) Beam-to-column joint. (b) Mechanical model. (c) Component behaviour.
2 Literature Review 76
beam-to-column joint under combined bending moment and axial force. Each bi-
linear spring of this model was replaced by two equivalent elastic springs using an
energy formulation and a post-buckling stability analysis. A comparison was
made against a welded joint only subjected to bending moments and the results
presented a good agreement with the experiments.
Sokol et al. (2002) developed an analytical model to predict the endplate
joint behaviour subjected to bending moment and axial force interaction. This
model was tested against two sets of experiments with flush endplate beam-to-
beam joints and extended endplate beam-to-column joints carried out by Wald and
Svarc (2001). In general, the results involving moment-rotation comparisons
provided rather close agreement with the experimental tests, however, for all the
analysed cases, the initial stiffness was overestimated whilst the final moment
capacity was underestimated.
Lima (2003) and Simões da Silva et al. (2004) proposed mechanical models
for extended (Figure 24) and flush (Figure 25) endplate joints, respectively.
Following, basically, the same idea and also based on Madas (1993), Ramli-
Sulong (2005) also developed a component-based connection model, Figure 26,
for flush and extended endplate, top-and-seat and/or web angles, and fin-plate
connections. These models basically consist of two rigid bars representing the
column centreline and the beam end, connected by non-linear springs that model
the joint components. Furthermore, these authors included the compressive
components (for instance, cwc - column web in compression, Figure 24, Figure 25
and Figure 26) at the same location as the bolt rows and the tensile components
(for example, cwt - column web in tension, Figure 24 and Figure 25) at the same
location as the flanges (compressive rows). Proposed models by Lima (2003) and
Simões da Silva et al. (2004) were tested against their own experimental tests.
Although these models presented satisfactory results in terms of ultimate flexural
capacity, the prediction of the initial stiffness, for the case of different axial load
levels, was not accurate, predicting almost the same initial stiffness for the whole
set of evaluated cases, Figure 27 and Figure 28. Regarding Ramli-Sulong’s model
(Ramli-Sulong, 2005, and Ramli-Sulong et al., 2007), neither comparison has
been done against experimental moment-rotation curves nor parametric analysis
involving different axial force levels, which are needed to evaluate this model in
terms of quality of moment-rotation curve prediction for moment-axial
2 Literature Review 77
interaction. On the other hand, this model was shown to be able to predict, with a
good accuracy, the experimental moment-rotation curves, disregarding the axial
effect. Comparisons made at elevated temperature with available tests also
presented a good agreement.
cfb cwt bt epb
cfb cwt bt epb bwt
cfb cwt bt epb bwt cwc cws bfwc
cwc cws bfwc cfb cwt bt epb bwt
cwc cws bfwc
cwc cws bfwc
M
N
IPE 240
HEB 240
Figure 24 - Spring model for extended endplate joints, Lima (2003).
cfb cwt bt epb bwt
cfb cwt bt epb bwt cwc cws bfwc
cwc cws bfwc cfb cwt bt epb bwt
cwc cws bfwc
M
N
IPE 240
HEB 240
Figure 25 - Spring model for flush endplate joints, Lima (2003).
Figure 26 - Nonlinear spring connection model, Ramli-Sulong (2005).
Urbonas & Daniunas (2006) proposed a component method extension to
endplate bolted beam-to-beam joints under bending and axial forces, however the
2 Literature Review 78
procedure for joint moment-rotation curve prediction is only applicable and valid
within the elastic regime of structural behaviour. Numerical tests were executed
by the authors with a three-dimensional joint modelling using finite elements with
the goal to validate this model. The results obtained for the beam-to-beam joint
initial stiffness were close to the finite element analysis.
Figure 27 - Numerical simulations of the moment-rotation curves for the extended
endplate joints, Lima (2003).
Figure 28 - Numerical simulations of the moment-rotation curves for the flush endplate
joints, Simões da Silva et al. (2004).
Table 7 presents a summary of the mechanical models for predicting joint
behaviour discussed in this section.
2 Literature Review 79
Table 7 - Summary of the mechanical models to predict the joint behaviour.
Authors (date) Joint Type Forces
Wales & Rossow (1983) Double web cleat connectionsBending moment and axial force
Kennedy & Hafez (1984) Header plate connections T-stub: axial forceChmielowiec & Richard (1987)
All types of cleated connectionsBending moment and shear
Tschemmernegg (1988) Welded connections Bending momentTschemmernegg & Humer (1988)
Endplate bolted connections Bending moment
Madas (1993)Flexible endplate, double web angle and top and seat angle connections
Bending moment and axial force
Jaspart et al. (1999) and Cerfontaine (2003)
Extended and flush endplate connections
Bending moment and axial force
Pucinotti (2001)Top-and-seat and web angle connections
Bending moment
Simões da Silva & Coelho (2001)
Welded beam-to-column jointsBending moment and axial force
Sokol et al. (2002) Endplate jointsBending moment and axial force
Lima (2003) Extended endplate jointsBending moment and axial force
Lima (2003) and Simoes da Silva et al. (2004)
Flush endplate jointsBending moment and axial force
Ramli-Sulong (2005)Flush and extended endplate, top-and-seat and/or web angles, and fin-plate connections
Bending moment and axial force
Urbonas & Daniunas (2006)
Endplate bolted beam-to-beam joints
Bending moment and axial force
Despite the continuous development and improvement of analytical models
to predict the behaviour of joints under bending moment and axial force, there are
still problems in the prediction of the moment-rotation curves, such as the joint
initial stiffness for different axial force levels, as can be seen, for example, in
Figure 27 and Figure 28 or in Nunes (2006). The magnitude of this problem
increases when joints are subjected to tensile axial forces. This problem relates to
the ability of these models to deal with moment-axial interaction, and
consequently changes of the compressive centre, before the first component
yields. If the model is working on the linear-elastic regime, without reaching any
component yield (i.e. the component stiffness is also working linearly), the
modification of the joint stiffness matrix, only due to the geometric stiffness
changes, will be insignificant. From this point upwards to the onset of first
component yield, these models are not able to represent accurately the joint initial
2 Literature Review 80
stiffness for any level of axial load and bending moment while working on the
linear-elastic regime. Aiming to overcome this limitation, a mechanical model is
proposed in Chapter 3, which allows modifications of the compressive centre
position even before reaching the first component yield, i.e. in the linear-elastic
regime.
2.4.Experimental
The study of the semi-rigid characteristics of beam to column connections
and their effects on frame behaviour can be traced back to the 1930s, Li et al.
(1995). Since then, a large amount of experimental and theoretical work has been
conducted both on the behaviour of the connections and on their effects on
complete frame performance. Despite the large number of experiments, few of
them consider the bending moment versus axial force interactions.
A detailed discussion of all available experimental tests is beyond the scope
of this work; a compilation of the experiments is, however, available in Nethercot
(1985); Weynand (SERICON I, 1992) and Cruz et al. (SERICON II, 1998).
Recently, several researchers have paid special attention to joint behaviour
under combined bending moment and axial force. Guisse et al. (1996) carried out
experiments on twelve column bases, six with extended and six with flush
endplates. Wald and Svarc (2001) tested three flush endplate beam-to-beam joints
and two extended endplate beam-to-column joints; however there is no reference
to tests made with only bending moment, which is vital to access the influence of
the axial force in the joint response. Lima et al. (2004) and Simões da Silva et al.
(2004) performed tests on eight flush endplate joints and seven extended endplate
joints.
The investigators concluded that the presence of the axial force in the joints
modifies their structural response and should, therefore, be considered in the joint
structural design.
3 Generalised Mechanical Model for Beam-to-Column Joints Including the Axial-Moment Interaction 81
3Generalised Mechanical Model for Beam-to-Column Joints Including the Axial-Moment Interaction
3.1.Introduction
In this chapter, a generalised mechanical model is developed to describe the
beam-to-column joint behaviour including the axial force versus bending moment
interaction.
This model, Figure 29, based on the component method, contains three rigid
bars representing the column centreline (support bar), the column flange
centreline (second bar, b2) and the beam end (first bar, b1). These rigid bars are
connected by a series of springs that model the joint components. The stiffness of
these springs (rows) are representing by ri whilst ui and uli are the absolute
displacements of springs i referred to the first and second bars, respectively. Ci are
the vertical coordinates of springs i.
Due to the generalised formulation developed in this work, the model is able
to represent any kind of joint, since the joint can be modelled according to the
scheme shown in Figure 29.
ul3
ul4
ul1
ul2
Loadapplication
line
+
-
(+)
Columncentreline
Column flangecentreline
Beamend
M
P
1
r5
r6
r7
r8
r3
r4
r2
1
r5
r6
r7
r8
r3
r4
r2
C2
C3
C4
C1
C6
C7
C8
C5
u1
u5
u6
u2
ub2 ub1
u3
u7
u8
u4
b2 b1
b1b2
(a) Original configuration.(b) Deformed configuration.(c) Coordinates.
Figure 29 - Proposed generalised mechanical model for semi-rigid joints.
3 Generalised Mechanical Model for Beam-to-Column Joints Including the Axial-Moment Interaction 82
The following sections present the adopted behaviour for each joint
component as well as the complete formulation of this generalised mechanical
model.
3.2.Characterisation of the Joint Components
The behaviour of each component of the joint is given by a force-
deformation relationship, which may be characterised, for example, by a bi-linear,
tri-linear or even a non-linear curve. Simões da Silva et al. (2002), based on
Kuhlmann et al. (1998), classified the endplate joint components according to
their ductility:
- Components with high ductility, Figure 30(a): column web in shear
(assuming no occurrence of local buckling), column flange in bending, endplate in
bending and beam web in tension.
- Components with limited ductility, Figure 30(b): column web in
compression, column web in tension and beam flange in compression.
- Components with brittle failure, Figure 30(c): bolts in tension and welds.
However, some comments are necessary regarding this classification:
- Eurocode 3:1-8 (2005) considers a rigid-plastic behaviour for beam web
in tension.
- Lima (2003) verified a ductile behaviour for the beam flange in
compression in his experiments.
- Welds are not considered in the joint rotation stiffness evaluation
according to Eurocode 3:1-8 (2005).
Figure 30 - Constitutive laws of the endplate joint components, Simões da Silva et al.
(2002).
In this work, a tri-linear approach for the force-deformation relationship is
suggested and used for all the joint components as shown in Figure 31. The
component elastic stiffness, kcpe, and the component yield strength, fcp
y, are
3 Generalised Mechanical Model for Beam-to-Column Joints Including the Axial-Moment Interaction 83
calculated according to the Eurocode 3:1-8 (2005) component method. On the
other hand, for the component plastic stiffness, a strain hardening stiffness kcpp is
evaluated as:
ecp
ppcp kk (3.1)
The component reduced strain hardening stiffness, kcpu, referred to the
component material fracture, is:
ecp
uucp kk (3.2)
where μp and μu are the strain hardening coefficients, respectively, for the plastic
and ultimate stiffness, which depend on the component type. Based on the
classification suggested by Simões da Silva et al. (2002), briefly discussed in this
paper, and curve fitting executed on the experimental tests carried out by Lima
(2003), Table 8 presents the values adopted for the strain hardening coefficient for
each joint component.
fcpy
fcpu
kcpp
kcpu
kcpe
f
Tension
ucpy ucp
u u
fcpy
fcpu
kcpp
kcpu
kcpe
f
Compression
ucpy ucp
u
u
Figure 31 - Force-displacement curve for components in tension and compression.
The component ultimate capacity, fcpu, is determined, for each component,
using the ultimate stress instead of the yield stress in equations present in
Eurocode 3:1-8 (2005).
For the case when the component related to the column web panel in shear
is activated, i.e. when unbalanced moments exist in the joint, and the beam top
flange and bottom flange of the joint are in compression, this component will be
divided into two equal springs (one for the beam top flange and another for the
beam bottom flange) characterised by its usual stiffness and yield and ultimate
strengths divided by two.
3 Generalised Mechanical Model for Beam-to-Column Joints Including the Axial-Moment Interaction 84
Table 8 - Values adopted for the strain hardening coefficients, μ.
Designation - ComponentPlastic
μp
Ultimate
μu
1 - Column web in shear 0.500 0.2172 - Column web in compression 0.300 0.1303 - Column web in tension 0.300 0.1304 - Column flange in bending 0.200 0.0875 - Endplate in bending 0.100 0.0437 - Beam or column flange and web in compression ∞ ∞8 - Beam web in tension ∞ ∞10 - Bolt in tension 0.600 0.261
The generalised mechanical model formulation, described in the next
section, uses an effective stiffness for each model row/spring i referred to the bolts
and beam flanges, which is evaluated as:
nc
jucp
uinc
jp
cp
pinc
jecp
eii
k
ror
k
ror
k
rr
111
11
11
11 (3.3)
where nc is the component number that contributes to the stiffness ri of the
row/spring i. The spring/row stiffness depends on the force-deformation
relationship of each joint component that is evaluated according to the proposed
procedure described in this section.
3.3.Generalised Mechanical Model Formulation
The Principle of Virtual Work was used to formulate the model stiffness
matrix and the corresponding equilibrium equations. Assuming the mechanical
system, Figure 29,:in equilibrium under the applied loads P (axial force) and M
(bending moment) and given arbitrary virtual displacements compatible with the
constraints on the system, the virtual work equation becomes:
0 UW (3.4)
where U is the internal virtual work done by the spring forces and W is the
external virtual work done by the applied forces P and M.
The internal virtual work, i.e., the work due to stresses for strains caused by
a virtual displacement field, can be expressed in terms of the tangent stiffness ri
3 Generalised Mechanical Model for Beam-to-Column Joints Including the Axial-Moment Interaction 85
(Eq. 3.3, of the spring i), the relative displacements i and the virtual
displacements i as:
ns
iiii rU
1
(3.5)
where ns is the system spring number. Adopting small displacements, the relative
(i) and absolute (ui and uli) displacements for the system presented in Figure 29
can be evaluated as:
)sin(
)sin()sin(
21
22
2211
bibi
bibibibi
iiiii
Cuul
CuuCuu
uulu
BarBar
(3.6)
where Ci is the spring vertical coordinate i regarding the load application line. The
spring coordinates above the loading application line must have a positive sign
while the springs located below the loading application line should attain a
negative sign. b1 and ub1, b2 and ub2 are the rotations (bi) and displacements
(ubi) of bars 1 and 2, respectively. Similarly to the relative and absolute
displacements, the virtual displacements can be expressed as:
)sin(
)sin()sin(
21
22
2211
bibi
bibibibi
iiiii
Cuul
CuuCuu
uulu
BarBar
(3.7)
On expanding and rearranging the terms, Eq. 3.5 can be put in the form:
KU T (3.8)
where T and are the virtual (Eq. 3.7) and relative (Eq. 3.6) displacement
fields, respectively, and K is the model stiffness matrix.
Approximating the trigonometric expressions in Eq. 3.6 and 3.7 to the first
order, the model stiffness matrix K, Figure 29, for any spring number at any
position can be evaluated as:
3 Generalised Mechanical Model for Beam-to-Column Joints Including the Axial-Moment Interaction 86
2
144
134
133
222412232
122
121411131
121
11
1
11
i
ns
ii
ns
iii
ns
ii
i
ns
ii
ns
iii
ns
ii
CrK
CrKrKSymmetric
KKKKCrK
KKKKCrKrK
K
b
bb
(3.9)
where K11 and K33 are the matrix terms related to the axial deformations of the
beam-to-column joint; K12 and K34 are associated with the interaction between the
axial and the rotational deformations; K22 and K44 are correlated with the
rotational deformations.
The external virtual work, defined as the work performed by the external
forces due to the virtual displacement field, for the spring system of Figure 29 is:
FMPuuW Tbbb 121 )( (3.10)
where P is the axial load, M is the bending moment. b1 and ub1, are the virtual
rotation and displacement of the first bar, respectively. b2 and ub2 are the
virtual rotation and displacement of the second bar, respectively. T is the virtual
displacement field and F is the force vector defined as:
TMPF 0.00.0 (3.11)
According to the Principle of Virtual Work, for a deformable system in
equilibrium, the total internal virtual work is equal to the total external virtual
work for every virtual displacement consistent with the constraints. Thus
FK TT (3.12)
As this holds for every consistent virtual displacement, it is possible to obtain all
the equilibrium relationship for the given mechanical model in Figure 29.
Due to the simplicity of this mechanical model formulation, it can be easily
incorporated into a nonlinear semi-rigid joint finite element formulation, only
requiring a tangent stiffness update procedure of each joint spring.
Regarding the first order approximations for the trigonometric expressions
used on the generalised mechanical model formulation, section 8.1 presents the
error evaluation for these approximations versus joint rotations as well as a
discussion about this.
3 Generalised Mechanical Model for Beam-to-Column Joints Including the Axial-Moment Interaction 87
3.3.1.Analytical Expressions: Displacements and Rotations
This section presents the analytical expressions for the displacements and
the rotations of the proposed generalised mechanical model, Figure 29. The main
goal is to generate equations for the evaluation of these properties without
executing a mechanical model numerical analysis.
Rewriting the equilibrium equations, Eq. 3.12, based on the symmetric
stiffness matrix, Eq. 3.9, provides the complete equilibrium equations as a
function of six stiffness terms, K11, K12, K22, K33, K34 and K44,
PKuKKuK bbbb 212211112111 (3.13)
MKuKKuK bbbb 222212122112 (3.14)
0.0234233112111 bbbb KuKKuK (3.15)
0.0244234122112 bbbb KuKKuK (3.16)
Isolating b2 from the equilibrium Eq. 3.16,
2112112 ,, bbbbbbb uuuu (3.17)
where,
44
34
44
22
44
12 ;;K
K
K
K
K
K (3.18)
Substituting b2, Eq. 3.17, into the equilibrium Eq. 3.15, and isolating ub2,
1212112 , bb
bbb
uuu
(3.19)
where,
;;;14433
2234
33
122
4433
1234
33
112
4433
234
KK
KK
K
K
KK
KK
K
K
KK
K (3.20)
Substituting b2, Eq. 3.17, into the equilibrium Eq. 3.14, and after
substituting ub2, Eq. 3.19, and subsequently isolating ub1,
1
11 , bbb
MMu
(3.21)
where,
2121 ; (3.22)
44
222
221224444
12112
44
3422 ;;K
KKKK
K
KK
K
KK (3.23)
3 Generalised Mechanical Model for Beam-to-Column Joints Including the Axial-Moment Interaction 88
Substituting b2 (Eq. 3.17) into the equilibrium Eq. 3.13, ub2 (Eq. 3.19), then
ub1 (Eq. 3.21), and subsequently isolating b1 generates the expression for the joint
rotation (or first bar rotation), for any axial force and bending moment level:
MP
MPb ,1 (3.24)
where,
(3.25)
1144
34122
344433
12344411
44
212
11 KK
KK
KKK
KKKK
K
KK (3.26)
1144
34122
344433
22344412
44
221212 K
K
KK
KKK
KKKK
K
KKK (3.27)
Substituting b1 (Eq. 3.24) into Eq. 3.21 leads to the joint horizontal
displacement (first bar horizontal displacement):
MP
MPub 1,1 (3.28)
Substituting b1 (Eq. 3.24) and ub1 (Eq. 3.28) into Eq. 3.19 produces the
following expression for the second bar horizontal displacement:
2222
2 1,
MP
MPub (3.29)
Finally, substituting b1 (Eq. 3.24), ub1 (Eq. 3.28) and ub2 (Eq. 3.29) into Eq.
3.17 leads to the expression for the second bar rotation:
22
22
2 ,
M
PMPb
(3.30)
With Eqs. 3.24, 3.28, 3.29 and 3.30 it is possible to evaluate all the joint
displacements and rotations for any interaction level between axial force and
bending moment, as well as, the forces in each spring:
iii rf (3.31)
where ri and i are, respectively, the stiffness (Eq. 3.3) and the relative
displacement (Eq. 3.6) of each spring.
3 Generalised Mechanical Model for Beam-to-Column Joints Including the Axial-Moment Interaction 89
3.3.2.Limit Bending Moments
For the correct use of the component method the prior knowledge of which
model rows (bolts and flanges) are in tension and/or compression is needed due to
their effect on the evaluation of the joint rotation and flexural capacity. In the
usual Eurocode 3:1-8 (2005) mechanical model for joints subjected only to
bending moment actions, a straightforward procedure is used to identify which
rows are in compression and/or tension. However, when additional axial forces act
on the joint, the identification whether each row is in tension or compression is
not known in advance. This fact implies in the determination of the limit bending
moment for the proposed mechanical model, Figure 29, the need to identify when
the row forces change from compression to tension or vice-versa. With these
results in hand, it is possible to adopt a consistent component distribution to be
used following the Eurocode 3:1-8 (2005) principles. The limiting bending
moment, for each j-spring (component) located between the first and second bars,
can be obtained by isolating, ub1 from Eq. 3.6,
)sin()sin( 2211 bjbbjjb CuCu (3.32)
substituting ub1 into the two first equilibrium equations, of Eq. 3.8,
01
bu(3.33)
01
b(3.34)
This is followed by isolating b1 from the first equilibrium equation Eq.
3.33, then substituting it into the second equilibrium equation Eq. 3.34 and
making the relative displacement (j) equal to zero, and finally isolating the
bending moment to generate the following expression for the j-spring limit
bending moment:
1211
1222
11
11
2
lim,11
11
KKC
KCKP
CrrC
CrCCr
PMj
j
ns
iii
ns
iij
ns
iiij
ns
iii
jbb
bb
(3.35)
It is worth noting that Eq. 3.35 depends only on the axial load applied to the
joint, and the stiffness and the vertical coordinates of springs located between the
3 Generalised Mechanical Model for Beam-to-Column Joints Including the Axial-Moment Interaction 90
first and second bars. There is no significant influence of springs located between
the second bar and supports on the limit bending moment evaluation.
According to Eq. 3.35, for instance, for the first spring (j = 1), for M <
M1,lim all rows are compressed; M = M1,lim first spring axial force is equal to zero;
and M > M1,lim there are both tension and compression rows.
3.3.3.Moments that Cause the Joint Rows and the Joint to Yield and Failure
In this section analytical equations are derived, from the analytical
expressions presented in section 2.2, for the evaluation of bending moments that
cause the model springs/rows and the joint to both yield and failure, for any axial
force level.
The displacement Δiy that causes the model spring/row i to yield is obtained
by isolating Δi from Eq. 3.31, and setting fi equal to the weakest component yield
strength of spring/row i, fcpy,
ei
ycpy
i r
f (3.36)
Similarly, the displacement Δiu that causes the model spring/row i to failure is,
pi
ucpu
i r
f (3.37)
where rie and ri
p are the elastic and the plastic stiffness of the spring/row i,
respectively, given in Eq. 3.3. The relative displacement of spring/row i located
between the first and second bars, from Eq. 3.6, is,
))sin(()sin( 2211, bibbibibr CuCu (3.38)
Approximating the trigonometric expressions in Eq. 3.38 to the first order;
then substituting ub1 (Eq. 3.28), b1 (Eq. 3.24), ub2 (Eq. 3.29) and b2 (Eq. 3.30)
into it; and making the relative displacement (br,i) equal to iy (Eq. 3.36) and
subsequently isolating the bending moment generates the expression that causes
the i-spring/row, located between the first and second bars, to yield:
)(
)(
4321
4321,
i
iyiy
ibr C
CPM (3.39)
3 Generalised Mechanical Model for Beam-to-Column Joints Including the Axial-Moment Interaction 91
Similarly, making the relative displacement (br,i) equal to iu (Eq. 3.37), the
expression for the bending moment that causes the i-spring/row, located between
the first and second bars, to fail is produced:
)(
)(
4321
4321,
i
iuiu
ibr C
CPM (3.40)
where the coefficients of Eqs. 3.39 and 3.40 are:
4
223
222
1
4
22
3
222
1
1
11
11
(3.41)
Following the same idea, now, for spring/row i located between the second
bar and supports, the relative displacement, from Eq. 3.6, is,
)sin( 22, bibifr Cu (3.42)
Approximating the trigonometric expressions in Eq. 3.42 to the first order;
then substituting ub2 (Eq. 3.29) and b2 (Eq. 3.30) into it; and making the relative
displacement (fr,i) equal to iy (Eq. 3.36) and subsequently isolating the bending
moment produces the expression that causes the i-spring/row, located between the
second bar and supports, to yield:
32
32,
i
iyiy
ifr C
CPM
(3.43)
Similarly, making the relative displacement (fr,i) equal to iu (Eq. 3.37) leads to
the following expression for the bending moment that causes the i-spring/row,
located between the second bar and supports, to fail:
32
32,
i
iuiu
ifr C
CPM
(3.44)
3 Generalised Mechanical Model for Beam-to-Column Joints Including the Axial-Moment Interaction 92
where the coefficients of Eqs. 3.43 and 3.44 given in Eq. 3.41.
Finally, the joint yield bending moment can be calculated as being the
minimum yield bending moment given in Eqs. 3.39 and 3.43,
}43.3.,39.3.min{ ,, EqMEqMM yifr
yibr
y (3.45)
and the joint plastic bending moment as being the minimum plastic bending
moment evaluated by Eqs. 3.40 and 3.44,
}44.3.,40.3.min{ ,, EqMEqMM uifr
uibr
u (3.46)
The joint rotational capacities, y and u, referred to the joint yield and
plastic bending moments are, respectively,
yy MP
(3.47)
uu MP
(3.48)
For a given joint rotation () and axial force (N), it is also possible to
calculate the corresponding joint bending moment by isolating it from Eq. 3.24,
P
M (3.49)
The analytical expressions developed in this section provide all the
necessary information to predict bending moment versus rotation curves for any
axial force level applied to the joint.
3.4.Prediction of Bending Moment versus Rotation Curve for any Axial Force Level
Based on the equations previously developed, Figure 32 presents an
approach to characterise bending moment versus rotation curves considering the
bending moment versus axial force interaction.
My
Mu
KpKu
Ke
M
y f u
Mf
3 Generalised Mechanical Model for Beam-to-Column Joints Including the Axial-Moment Interaction 93
Figure 32 - Proposed prediction of the bending moment versus rotation curve for any
axial force level.
For each moment-rotation curve, the first point (y, My) defines the joint
initial stiffness corresponding to the attainment of the weakest component yield
while the second point (u, Mu) is obtained when the weakest component reaches
its ultimate strength. The third point (f, Mf) depends on the joint assumed final
rotational capacity for the moment-rotation curve. In this work a 0.05-radian joint
final rotation was adopted based on studies for both frames and individual
restrained member. The joint rotations required at maximum load have shown that
behaviour at rotations beyond 0.05 radians, often much less, has little practical
significance, Nethercot & Zandonini (1989).
Summarising, the points of the moment-rotation curve are:
49.3.;05.03
46.3.;48.3.2
45.3.;47.3.1
EqMradiansPnt
EqMEqPnt
EqMEqPnt
ff
uu
yy
(3.50)
It is worth highlighting that more points could have been used to describe
the bending moment versus rotation curve because, for instance, before reaching
the joint plastic bending moment other joint rows might start yielding by
generating new points between the first and second points (Eq. 3.50) changing the
joint stiffness matrix. However, for simplicity of the approach and examples
described in section 3, three points were adopted.
3.5.Lever Arm d
The lever arm d represents the tensile rigid link position that unites the
second bar to the supports, as can be seen, for instance, in Figure 33. On Figure 33
kbr1, kbr2, kbr3 representing the elastic stiffness of bolt-rows 1, 2 and 3,
respectively; kbbf is the elastic stiffness of the bottom flange of the beam; klcbf is
the compressive rigid link associated with the bottom flange of the beam; and klt is
the elastic stiffness of the tensile rigid link referred to the lever arm.
The evaluation of this lever arm d is needed when a mechanical model is
adopted as in Figure 33, where the first bolt rows are in tension, i.e., the beam top
flange is not under compression. According to Del Savio et al. (2007a), the joint
initial stiffness is strongly influenced by this lever arm d. Based on this fact, an
approach is here presented for evaluation of this lever arm d which is divided into
3 Generalised Mechanical Model for Beam-to-Column Joints Including the Axial-Moment Interaction 94
two equations: one for tensile forces and another for the complimentary cases
disregarding axial forces and/or considering compressive forces applied to the
joint.
M
P
klt
kbbf
kbr3
klcbf
kbr2
b1b2
kbr1
d
Flinkt
Fbbf
e
Column centreline
Column flange centreline
Beam end(endplate centreline)
Column
Beam
l1
l2
l3
Figure 33 - Proposed generalised mechanical model for semi-rigid joints – lever arm d.
3.5.1.Lever Arm Evaluation for the Complementary Cases Disregarding Axial Forces and/or Considering Tensile Forces Applied to the Joint
Considering the support reactions and the applied loads, Figure 33, the
system force equilibrium can be evaluated as:
PFF linktbbf (3.51)
The system moment equilibrium at the beam bottom flange is:
MePedFlinkt (3.52)
where Fbbf is the row compressive yield capacity referred to the beam bottom
flange; Flinkt is the rigid link tensile capacity (assumed to be the greatest tensile
capacity between all the model rows times two) that joins the second bar to the
supports; d and e are, respectively, the distances from the loading application
centre to the rigid link and the beam bottom flange.
Assuming M to be equal to the yield bending moment of the first bolt-row
Mbr,1y given in Eq. 3.39, Fbbf, P and e are already known, the problem variables are
Flinkt and d. Then, isolating Flinkt from Eq. 3.51, substituting it into Eq. 3.52, and
then isolating d leads to the expression for the lever arm position:
3 Generalised Mechanical Model for Beam-to-Column Joints Including the Axial-Moment Interaction 95
eFP
MePd
bbf
ybr
1, (3.53)
which also satisfies the condition where Fbbf and Mbr,1y simultaneously reach the
yield.
3.5.2.Lever Arm Evaluation for Compressive Forces Applied to the Joint
The lever arm d for this case is evaluated as the ratio between the sum of
bending moments referred to the bolt-rows and the axial force at the beam bottom
flange and the sum of forces referred also to the bolt-rows and the axial force
minus the distance from the load application centre to the beam bottom flange:
ePf
ePlfd nbr
i
yibr
nbr
ii
yibr
1,
1,
(3.54)
where nbr is the number of joint bolt-rows, fbr,iy is the yield strength of bolt-row i
and li is the distance from joint bolt-row i to the beam bottom flange centre.
The lever arm d evaluated in either Eq. 3.53 or Eq. 3.54 take into account
the change of the joint compressive centre position according to the axial force
levels and bending moment applied to the joint, before the yield of the first
weakest component is reached.
4 Application of the Proposed Mechanical Model and Its Validation against Experimental Tests 96
4Application of the Proposed Mechanical Model and Its Validation against Experimental Tests
4.1.Introduction
In this chapter, the straightforward applicability of the proposed mechanical
model is illustrated by means of detailed examples using six extended endplate
joints tested by Lima et al. (2004).
The model validation against experimental tests, also carried out by Lima et
al. (2004), considering and disregarding the axial force effect on the joint
behaviour, is assessed as well.
4.2.Application of the Proposed Generalised Mechanical Model
Application of the generalised mechanical model, developed in Chapter 3, to
predict the joint behaviour requires the following steps:
(a) Generation and adoption of a joint model in consonance with the
generalised mechanical model presented in Figure 29.
(b) Joint design according to Eurocode 3:1-8 (2005).
(c) Characterisation of the joint components: force-displacement
relationship of each component according to the approach suggested in Chapter 3,
section 3.2.
(d) Identification of all the possible cases (model for compression,
tension, tension/compression) given that loading may vary from pure bending to
pure compressive/tensile axial force with all intermediate combinations. These
intermediate combinations are derived from the adopted model in step (a).
(e) Evaluation of the limit bending moments for the adopted models
in step (d), with the aid of Eq. 3.35, to define the application domains of each one.
(f) Evaluation of the lever arm d according to the proposed
procedure in Chapter 3, section 3.5, Eq. 3.53 for tensile forces and Eq. 3.54 for
4 Application of the Proposed Mechanical Model and Its Validation against Experimental Tests 97
either compressive forces or without axial forces, considering the change of the
joint compressive centre position.
(g) Prediction of bending moment versus rotation curves for each
axial force level, according to the approach described in Chapter 3, section 3.4.
It is worth highlighting that the incorporation of this approach into a
nonlinear semi-rigid joint finite element formulation does not require steps (d) and
(e), because the complete joint modelling already considers all the possible
situations of loading through each component force-displacement characteristic
curve. In order to explain how each step is evaluated, six extended endplate joints
tested by Lima et al. (2004) were modelled.
4.2.1.Extended Endplate Joints
Starting with the application of step (a) previously described and using the
extended endplate joint properties, Figure 34, the following mechanical model
was adopted, Figure 35. On Figure 35 kbr1, kbr2, kbr3 representing the elastic
stiffness of bolt-rows 1, 2 and 3, respectively. klt1, klt2, klt3 are the elastic stiffness
of the tensile rigid links referred to the bolt-rows 1, 2 and 3, respectively. kbtf and
kbbf are the elastic stiffness of the top and bottom flanges of the beam. klctf and klcbf
are the compressive rigid links associated with the top and bottom flanges of the
beam. klt is the elastic stiffness of the tensile rigid link referred to the lever arm.
62 96 62
32 96 32
160
74
15
65
4
M20 cl10.9
IPE240
HE
B2
40
31
4
tp =
15
mm
62
24
01
2
31
4
74
15
65
4
30
M
N
Figure 34 - Extended endplate joint, Lima et al (2004).
All the dimensions in mm
4 Application of the Proposed Mechanical Model and Its Validation against Experimental Tests 98
Figure 35 - Proposed mechanical model.
Next step (b), with the joint material (Table 9) and geometric (Figure 34)
properties, the theoretical values of the strength and initial stiffness for the
extended endplate joint components are evaluated according to Eurocode 3:1-8
(2005) and are presented in Table 10.
Table 9 - Steel mechanical properties.
SpecimenYield Strength
(MPa)
Ultimate
Strength (MPa)
Young’s
Modulus (MPa)
Ratio
Yield/UltimateBeam Web 363.40 454.30 203713 1.250IPE 240 Flange 340.14 448.23 215222 1.318Column Web 372.02 477.29 206936 1.283HEB 240 Flange 342.95 448.79 220792 1.309Endplate 369,44 503,45 200248 1.363Bolts 900,00 1000,00 210000 1.111Weld - 576.00 210000 -
With the evaluated properties of the joint components, the characterisation
of the force-displacement relationship for each component can be calculated
according to the proposed formulation. Table 11 presents the results of this step
(c).
4 Application of the Proposed Mechanical Model and Its Validation against Experimental Tests 99
Table 10 - Theoretical values of the resistance and initial stiffness of the extended
endplate joint components, Figure 34, evaluated according to Eurocode 3:1-8 (2005).
Componentfcp
y
(kN)
kcpe
(kN/mm)
fiy
(kN)
rie
(kN/mm)
Beam top and bottom flange (compression)
cwc (2) 656.7 2133.6 321.3 464.8bfc (7) 541.6 kbtb/kbbf
cws (1) 321.3 594.3
Beam bottom flange
cwc (2) 656.7 2133.6 541.6 763.4bfwc (7) 541.6 kbbf
cws (1) 642.5 1188.6
First bolt row(h=267.1mm)
cwt (3) 533.2 1476.3 289.8 607.7cfb (4) 311.3 8499.7 kbr1
epb (5) 289.8 4223.1bt (10) 441.0 1630.6
Second bolt row
(h=193.1mm)
Considered individuallycwt (3) 445.4 1476.3 218.6 575.0cfb (4) 218.6 8498.7 kbr2
epb (5) 326.9 3026.1bwt (8) 492.3 bt (10) 441.0 1629.6cwc (2) 366.9 -bfwc (7) 251.6 -cws (1) 352.8 -
Bolt-row belonging to the bolt group: bolt-rows 2 + 1cwt (3) 735.1 -cfb (4) 508.4 -epb 616.7 -
Third bolt row(h=37.1mm)
Considered individuallycwt (3) 410.3 1476.3 33.3 554.7cfb (4) 311.3 8498.7 kbr3
epb (5) 320.3 2538.9bwt (8) 413.2 bt (10) 441.0 1629.6cwc (2) 148.3 -bfwc (7) 33.3 -cws (1) 134.2 -
Bolt row belonging to the bolt group: bolt rows 3 + 2cwt (3) 350.8 -cfb (4) 663.4 -epb (5) 623.9 -bwt (8) 764.7 -Bolt row belonging to the bolt group: bolt rows 3 + 2 + 1cwt (3) 918.7 -cfb (4) 878.8 -cws (1) 898.2
4 Application of the Proposed Mechanical Model and Its Validation against Experimental Tests 100
Table 11 - Characterisation of the extended endplate joint components, Figure 34,
according to the approach given in Chapter 3 - section 3.2.
Componentfcp
u
(kN)
kcpp
(kN/mm)
kcpu
(kN/mm)
fiu
(kN)
rip
(kN/mm)
riu
(kN/mm)
Beam top and bottom flange (compression)
cwc (2) 842.6 640.1 278.3 412.2 202.9 88.2bfc (7) 695.4 kbtb/kbbf
cws (1) 412.2 297.2 129.2
Beam bottom flange
cwc (2) 842.6 640.1 278.3 695.4 308.3 134.0bfc (7) 695.4 kbbf
cws (1) 824.4 594.3 258.4
First bolt row(h=267.1mm)
cwt (3) 684.2 442.9 192.6 394.9 160.3 69.7cfb (4) 407.5 1699.7 739.04 kbr1
epb (5) 394.9 422.3 183.6bt (10) 490.0 977.8 425.1
Second bolt row
(h=193.1mm)
Considered individuallycwt (3) 571.4 442.9 192.6 286.1 139.4 60.6cfb (4) 286.1 1699.7 739.0 kbr2
epb (5) 445.5 302.6 131.6bwt (8) 615.4 bt (10) 490.0 977.8 425.1cwc (2) 466.8 - -bfwc (7) 320.1 - -cws (1) 448.8 - -
Bolt row belonging to the bolt group formed by bolt rows 2 + 1cwt (3) 943.2 - -cfb (4) 665.3 - -epb (5) 840.3
Third bolt row(h=37.1mm)
Considered individuallycwt (3) 526.5 442.9 192.6 42.3 128.1 55.7cfb (4) 407.5 1699.7 739.0 kbr3
epb (5) 436.5 253.9 110.4bwt (8) 516.6 bt (10) 490.0 977.8 425.1cwc (2) 188.7 - -bfwc (7) 42.3 - -cws (1) 170.7 - -
Bolt row belonging to the bolt group formed by bolt rows 3 + 2cwt (3) 1178.7 - -cfb (4) 1150.2 - -epb (5) 1223.9bwt 956.0Bolt row belonging to the bolt group formed by bolt rows 3 + 2 + 1
cwt (3) 1178.7 - -cfb (4) 1150.2 - -epb (5) 1223.9
Note: fcpu is given in Chapter 3 - section 3.2, kcp
p and kcpu are given in Eqs. 3.1 and 3.2,
respectively.
4 Application of the Proposed Mechanical Model and Its Validation against Experimental Tests 101
Based on the adopted mechanical model, step (a), Figure 35, four derived
models are identified and presented in Figure 36. These four models, referred to
step (d), are able to deal with the eight load situations presented in Table 12. For
the experimental tests used in this section, only three load situations depicted in
Table 12 were necessary:
- Number 3, where only bending moment is applied to the joint and the
proposed model presented in Figure 36(c) is sufficient to model the joint.
- Number 5: where a compressive axial force is applied to the joint followed
by a bending moment increase. This situation uses the proposed models depicted
in Figure 36(a) and Figure 36(c).
- Number 6: where a tensile axial force is firstly applied to the joint with a
subsequent bending moment application. In this case, the proposed models in
Figure 36(b) and Figure 36(c) are utilized.
Figure 36 - Proposed mechanical model for each analysis stage.
Before analysing the adopted mechanical models in Figure 36, it is
necessary to identify each model applicability domain, which depends on whether
the joint components are subjected to either compression or tension, for a given
combination of bending moment and axial force. This is done by evaluating the
limit bending moments (Mlim), step (e), for the adopted models in Figure 36 with
the aid of Eq. 3.35, relative to the experimental axial force levels. This step does
not require the knowledge of the lever arm position d since yield of joint bolt-
rows are not affected by this position. In this case, only the joint rotation and the
joint row yield corresponding to the beam flanges are affected. The results of the
limit bending moment evaluations are illustrated in Table 13. For the EE1
4 Application of the Proposed Mechanical Model and Its Validation against Experimental Tests 102
experiment (load situation number 3, Table 12) any bending moment applied to
the joint model, Figure 36(c), induces tension on the joint first bolt row and
compression on the beam bottom flange. For the EE2, EE3 and EE4 experimental
tests (load situation number 5, Table 12), the limit bending moment, which
induces tension on the beam top flange is obtained by using the proposed
mechanical model shown in Figure 36(a). For the EE6 and EE7 tests (load
situation number 6, Table 12), the limit bending moment, which leads the third
bolt row to compression, is calculated by the proposed mechanical model
illustrated in Figure 36(b).
Table 12 - Load situations applied to the joint and their respective mechanical models.
No Load situationsMechanical model(s)
Bending moment Axial Force1 - +P Compressive, Fig. 36(a)2 - -P Tensile, Fig. 36(b)3 +M - Tensile-Compressive, Fig. 36(c)4 -M - Compressive-Tensile, Fig. 36(d)5 +M +P Fig. 36(a) and Fig. 36(c)6 +M -P Fig. 36(b) and Fig. 36(c)7 -M +P Fig. 36(a) and Fig. 36(d)8 -M -P Fig. 36(b) and Fig. 36(d)
Note: +P and -P are compressive and tensile axial forces applied to the joint, respectively. +M is the bending moment that compresses the beam bottom flange and tensions the beam top flange, whilst -M is the bending moment that tensions the beam bottom flange and compresses the beam top flange.
Table 13 - Applicability of each model, Mlim, and evaluation of lever arm d according to the
experimental axial force levels.
Experimental data Mlim (kNm), Eq. 35 Lever Arm (mm)
Test N Compressive Tensile Tensile-Compressive(kN) Fig. 36(a) Fig. 36(b) Fig. 36(c)
EE1 (only M) 0.00 *NA *NA 0.00 to M f Eq. 53: 79.28EE2 (+10% Npl) 135.94 0.0 to 18.12 *NA 18.12 to M f Eq. 54: 86.34
EE3 (+20% Npl) 193.30 0.0 to 25.77 *NA 25.77 to M f Eq. 54: 79.60
EE4 (+27% Npl) 259.20 0.0 to 34.55 *NA 34.55 to M f Eq. 54: 73.05
EE6 (-10% Npl) -127.20 *NA 0.0 to 15.96 13.73 to M f Eq. 53: 46.57
EE7 (-20% Npl) -257.90 *NA 0.0 to 32.36 27.84 to M f Eq. 53: 24.33
Note: “+” indicates compressive axial forces and “-” tensile axial forces. Mf is given in Eq. 3.50.
*NA = not applicable.
Based on these limit bending moments, an appropriate mechanical model
can then be adopted from those shown in Figure 36. For instance, for EE4 test if
the bending moment applied to the joint was smaller than 34.55 kNm, Table 13,
4 Application of the Proposed Mechanical Model and Its Validation against Experimental Tests 103
the compressive model presented in Figure 36(a) should be used. For larger values
the tensile-compressive model should be utilised. On the other hand, if the
proposed mechanical model, Figure 35, was implemented into a nonlinear
structural analysis program, where each component was described by its force-
displacement characteristic curve, these joint components would be automatically
activated or deactivated according to its compressive/tensile characteristic (Figure
31), without the need to previously define a model for each load situation as
shown in Figure 36 and Table 12.
The proposed mechanical models presented in Figure 36(c) and Figure
36(d) require the evaluation of the lever arm d, step (f). Table 13 presents the
lever arm d positions evaluated for the mechanical model shown in Figure 36(c),
where Eq. 3.53 is used for tensile forces applied to the joint and Eq. 3.54 is
utilised for all the other complimentary cases. Regarding the mechanical model in
Figure 36(d), the lever arm d positions were not calculated since they were not
considered in the Lima et al. (2004) experiments.
Finally, with the steps (a) to (f) evaluated for the adopted models in Figure
36, it is possible to predict the bending moment versus rotation curves for each
axial force level, step (g), used in the experimental tests carried out by Lima et al.
(2004). Table 14 presents the values evaluated for each moment-rotation curve,
according to the approach described in Chapter 3 - section 3.4. Point 1 (y, My),
Table 14, defines the onset of the joint yield and is evaluated in Eq. 3.50, by using
the yield strength (Table 10, fiy) and the elastic effective stiffness (Table 10, ki
e)
for rows i. Point 2 (u, Mu) represents the joint ultimate capacity and is obtained
by utilising Eq. 3.50 and the ultimate strength (Table 11, fiu) and the plastic
effective stiffness (Table 11, kip) for rows i. Point 3 (f, Mf), Eq. 3.50, is obtained
by adopting a 0.05-radian final rotation for the joint and the reduced strain
hardening effective stiffness (Table 11, kiu) for rows i. With these results in hand,
the results of each analysis compared to their equivalent experimental tests are
illustrated in Figure 37 to Figure 42. Subsequently, Figure 43 presents the whole
set of numerical results.
4 Application of the Proposed Mechanical Model and Its Validation against Experimental Tests 104
Table 14 - Values evaluated for the prediction of the moment-rotation curves for different
axial force levels.
Poi
nt
EE1
(only M)
EE2
(+10% Npl)
EE3
(+20% Npl)
EE4
(+27% Npl)
EE6
(-10% Npl)
EE7
(-20% Npl)
M M M M M Mmrad kNm mrad kNm mrad kNm mrad kNm mrad kNm mrad kNm
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.01 8.2 105.3 7.2 97.4 7.0 90.1 6.7 83.0 9.6 93.5 11.1 81.4
2 23.6 135.1 21.4 128.2 20.8 119.9 20.1 111.8 26.4 118.3 29.5 102.7
3 50.0 137.3 50.0 143.3 50.0 138.0 50.0 132.8 50.0 107.7 50.0 83.6
Note: Points 1, 2 and 3 defined in Eq. 3.50.
0
20
40
60
80
100
120
140
160
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
EE1, P = 0
Proposed Model, P = 0
Figure 37 - Comparison between experimental EE1 moment-rotation curve (Lima et al.,
2004) and predicted curve by using the proposed mechanical model.
0
20
40
60
80
100
120
140
160
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
EE2, P = +10% Npl
Proposed Model, P = +10% Npl
Figure 38 - Comparison between experimental EE2 moment-rotation curve (Lima et al.,
2004) and predicted curve by using the proposed mechanical model.
4 Application of the Proposed Mechanical Model and Its Validation against Experimental Tests 105
0
20
40
60
80
100
120
140
160
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
EE3, P = +20% Npl
Proposed Model, P = +20% Npl
Figure 39 - Comparison between experimental EE3 moment-rotation curve (Lima et al.,
2004) and predicted curve by using the proposed mechanical model.
0
20
40
60
80
100
120
140
160
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
EE4, P = +27% Npl
Proposed Model, P = +27% Npl
Figure 40 - Comparison between experimental EE4 moment-rotation curve (Lima et al.,
2004) and predicted curve by using the proposed mechanical model.
4 Application of the Proposed Mechanical Model and Its Validation against Experimental Tests 106
0
20
40
60
80
100
120
140
160
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
EE6, P = -10% Npl
Proposed Model, P = -10% Npl
Figure 41 - Comparison between experimental EE6 moment-rotation curve (Lima et al.,
2004) and predicted curve by using the proposed mechanical model.
0
20
40
60
80
100
120
140
160
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
EE7, P = -20% Npl
Proposed Model, P = -20% Npl
Figure 42 - Comparison between experimental EE7 moment-rotation curve (Lima et al.,
2004) and predicted curve by using the proposed mechanical model.
4 Application of the Proposed Mechanical Model and Its Validation against Experimental Tests 107
0
20
40
60
80
100
120
140
160
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
P = 0 P = +10% Npl
P = +20% Npl P = +27% Npl
P = -10% Npl P = -20% Npl
Figure 43 - Prediction of six moment-rotation curves for different axial force levels.
4.3.Results and Discussion
Six experimental moment-rotation curves, of Lima et al. (2004), were used
to validate the proposed mechanical model in Chapter as well as to demonstrate its
application.
Figure 37 illustrates the comparisons between the proposed model and the
EE1 test moment-rotation curve that was only subjected to bending moments. For
this case, the point that characterises the joint initial stiffness was defined by
yielding of the endplate in bending. The initial stiffness is slightly underestimated
by 34 % by the mechanical model whilst the flexural capacity is rather over
predicted by 14 %, Table 15.
Figure 38, Figure 39 and Figure 40 present comparisons between the
proposed model and moment-rotation curves of EE2, EE3 and EE4 tests that
respectively consider compressive forces of 10%, 20% and 27% of the beam axial
plastic capacity. For these three compressive cases, the joint initial stiffness was
defined by yielding of the beam bottom flange in compression. Very good
correlation between the experimental tests and numerical results was obtained,
Table 15.
Figure 41 and Figure 42 illustrate the results for EE6 and EE7 moment-
rotation curves that respectively consider tensile forces of 10% and 20% of the
beam axial plastic resistance. For these last two cases the joint plasticity was
governed by yielding of the endplate in bending, followed by yielding of the beam
bottom flange in compression. An accurate prediction of the initial stiffness and
4 Application of the Proposed Mechanical Model and Its Validation against Experimental Tests 108
flexural capacity are observed, Table 15. However, as the tensile force increases
to 10% of the beam axial plastic resistance (EE6 test), slight difference is
exhibited overestimating the flexural capacity by 6%, Figure 41.
Table 15 - Comparisons between the experimental and the proposed model initial
stiffness and the experimental and the proposed model design moment.
TestsInitial Stiffness (kNm/rad) Design Moment (kNm)
Model Exp Mod/Exp % Model Exp Mod/Exp %
EE1 (only M) 12892 19438 0.66 34 135 119 1.14 -14EE2 (+10% Npl) 13445 13554 0.99 1 128 125 1.02 -2
EE3 (+20% Npl) 12885 15411 0.84 16 120 118 1.02 -2
EE4 (+27% Npl) 12369 12647 0.98 2 112 113 0.99 1
EE6 (-10% Npl) 9771 9087 1.08 -8 118 112 1.06 -6
EE7 (-20% Npl) 7317 6750 1.08 -8 103 101 1.02 -2
Note: Negative percentage means overestimated value of X % whilst positive percentage indicates underestimated value of X %. Joint design moment is determined according to Eurocode 3:1-8 (2005), through the intersection between two straight lines, one parallel to the initial stiffness and another parallel to the moment-rotation curve post-limit stiffness.
Figure 43 illustrates the set of numerical results where it is possible to
observe that the extended endplate joint subjected both to compressive and tensile
forces has its initial stiffness and flexural capacity decreased as either compressive
or tensile force increases. This reduction in the initial stiffness is more pronounced
for tensile forces applied to the joint. Additionally it is worth highlighting that the
joint initial stiffness is strongly influenced by the rigid link lever arm d. Joints
possessing similar rigid link lever arms d exhibited a small variation of the initial
stiffness as can be seen on the compressive force numerical results, Figure 43: P =
+10% Npl, P = +20% Npl and P = +27% Npl.
Generally the global behaviour of the numerical moment-rotation curves,
obtained by using the generalised mechanical model proposed in this work, is in
agreement with the test curves, Lima et al. (2004), producing numerical results
that closely approximate the initial stiffness and flexural resistance, Table 15.
These small discrepancies might be attributed to the simplifications made in the
generalised mechanical model as well as possible inaccuracies in the assumed
material and geometrical properties.
5 Parametric Investigations 109
5Parametric Investigations
5.1.Introduction
In this Chapter, parametric and sensitivity studies were executed to
investigate and demonstrate the application scope of the proposed model. Various
scenarios involving the key parameters that influence on the joint structural
behaviour were considered and carefully discussed. The full axial force-bending
moment interaction was also meticulously analysed and the axial force effect on
the joint response was also discussed in detail.
5.2.Joint Layout
The initial and basic joint layout to be studied is presented in Figure 44. The
mechanical model adopted to model this extended endplate joint is depicted in
Figure 45, whilst the joint steel mechanical properties are shown in Table 9. This
same joint configuration was used in Chapter 4 to demonstrate the application of
the proposed mechanical model and its validation against experimental tests. This
joint configuration was adopted in this chapter as comparison basis to the
parametric investigations.
62 96 62
32 96 32
160
74
15
65
4
M20 cl10.9
IPE240
HE
B2
40
31
4
tp =
15
mm
62
24
01
2
31
4
74
15
65
4
30
M
N
Figure 44 - Extended endplate joint, Lima et al (2004).
All the dimensions in mm
5 Parametric Investigations 110
Figure 45 - Proposed mechanical model.
5.3.Preliminary Studies
Firstly, a initial evaluation of the joint presented in Figure 44 and Figure 45
was performed including the points where the proposed mechanical model
changes its stress state range, i.e., from pure compression to both tension and
compression and vice-versa. The results of each analysis compared to their
equivalent experimental tests are illustrated in Figure 46 to Figure 51, with the
dashed lines representing the initial stiffness without the points considering the
changes in the stress state range. Subsequently, Figure 52 presents the whole set
of numerical results.
0
20
40
60
80
100
120
140
160
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
EE1, P = 0
Proposed Model, P = 0
Figure 46 - Comparison between experimental EE1 moment-rotation curve (Lima et al.,
2004) and predicted curve by using the proposed mechanical model.
5 Parametric Investigations 111
0
20
40
60
80
100
120
140
160
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
EE2, P = +10% Npl
Proposed Model, P = +10% Npl
Beam top and bottom flanges in compression
Beam flanges in compression and first bolt-row in tension
Figure 47 - Comparison between experimental EE2 moment-rotation curve (Lima et al.,
2004) and predicted curve by using the proposed mechanical model.
0
20
40
60
80
100
120
140
160
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
EE3, P = +20% Npl
Proposed Model, P = +20% Npl
Beam top and bottom flanges in compression
Beam flanges in compression and first bolt-row in tension
Figure 48 - Comparison between experimental EE3 moment-rotation curve (Lima et al.,
2004) and predicted curve by using the proposed mechanical model.
5 Parametric Investigations 112
0
20
40
60
80
100
120
140
160
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
EE4, P = +27% Npl
Proposed Model, P = +27% Npl
Beam top and bottom flanges in compression
Beam flanges in compression and first bolt-row in tension
Figure 49 - Comparison between experimental EE4 moment-rotation curve (Lima et al.,
2004) and predicted curve by using the proposed mechanical model.
0
20
40
60
80
100
120
140
160
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
EE6, P = -10% Npl
Proposed Model, P = -10% Npl
All the bolt-rows in tension
Figure 50 - Comparison between experimental EE6 moment-rotation curve (Lima et al.,
2004) and predicted curve by using the proposed mechanical model.
5 Parametric Investigations 113
0
20
40
60
80
100
120
140
160
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
EE7, P = -20% Npl
Proposed Model, P = -20% Npl
All the bolt-rows in tension
Figure 51 - Comparison between experimental EE7 moment-rotation curves (Lima et al.,
2004) and predicted curve by using the proposed mechanical model.
0
20
40
60
80
100
120
140
160
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
P = 0 P = +10% Npl
P = +20% Npl P = +27% Npl
P = -10% Npl P = -20% Npl
Figure 52 - Prediction of six moment-rotation curves for different axial force levels.
Figure 53 illustrates the prediction of the bending moment versus axial force
interaction diagram by EC3-1-8 (2005) and proposed mechanical model for the
joint yield (My) and ultimate (Mu) resistances. Additionally, the experimental
points of six extended endplate joints tested by Lima et al. (2004) are also plotted
together their equivalent points obtained using the proposed mechanical model for
the joint ultimate resistances.
5 Parametric Investigations 114
020406080
100120140160180200
-900 -650 -400 -150 100 350 600 850 1100 1350
Axial Force (kN)
Ben
ding
Mom
ent (
kN.m
) Experimental
Proposed ModelEC3:1-8 (2005)Proposed Model - MyProposed Model - MuEE7 (-20% Npl)
EE6 (-10% Npl)
EE2 (+10% Npl)EE1 (0% Npl)
EE3 (+20% Npl)
EE4 (+27% Npl)
CompressionTension
Figure 53 - Prediction of the bending moment versus axial load interaction diagram using
the proposed mechanical model for the joint yield and ultimate resistances.
5.3.1.Discussion of the Results
The prediction of the bending moment versus axial force interaction
diagram using the proposed mechanical model, Figure 53, demonstrates to be in
agreement with the experimental points of the extended endplate joint tested by
Lima et al. (2004). All the experimental points are within the limits built by the
mechanical model and close to the analytical points. Table 16 presents a
comparison between the experimental and analytical points demonstrating the
accuracy of the points obtained by the mechanical model.
Additionally, the mechanical model is also able to capture an important
characteristic observed in the laboratorial tests performed by Lima et al. (2004). It
was identified that for a compressive force of 135.94 kN (EE2 test) is possible to
obtain a joint bending moment larger than that one obtained disregarding the axial
force. In the bending moment versus axial force interaction diagram, Figure 53,
predicted by the mechanical model a maximum increase in the joint bending
moment for a compressive force of 64.3 kN was also identified confirming the
experimental evidence.
5 Parametric Investigations 115
Table 16 - Comparisons between the experimental and analytical points obtained for the
extended endplate joint.
TestN
(kN)Mu,analytical
(kNm)Mu,experimental
(kNm)Mu,experimental /
Mu,analitical
EE1 0.00 135.06 118.70 0.88EE2 135.94 128.23 125.40 0.98EE3 193.30 119.90 118.10 0.98EE4 259.20 111.80 113.20 1.01EE6 -127.20 118.26 111.50 0.94EE7 -257.90 102.65 101.00 0.98
Table 17 and Table 18 present the mechanical model row stiffness state for
the joint ultimate bending moment resistance and row-component yield and
failure sequence, respectively. These tables aim at assisting to understand deeper
the prediction of the bending moment versus axial load interaction diagram using
the proposed mechanical model depicted in Figure 53. The explanation to follow
is also applicable to the all the analytical moment-axial load interaction diagram
constructed by the proposed mechanical model.
Starting with the compressive side of the analytical moment-axial load
interaction diagram, Figure 53, the first linear segment from the 824.4 kN to 449.8
kN axial loads is characterised by a compressive mechanical model, i.e. the beam
top and bottom flange are in compression, Table 17 (axial loads: 824.4, 642.5,
578.3, 514.0 and 449.8 kN). For the 824.4 kN axial load magnitude the beam top
and bottom flange reach the failure together without any bending moment applied
to the joint. From the 642.5 kN to 514.0 kN axial loads there is a proportional
increasing ultimate bending moment resistance causing firstly the yield of the
beam bottom flange and subsequently its failure, without yielding or failing the
beam top flange, Table 17 and Table 18. At 449.8 kN axial load magnitude the
first bolt-row is activated, Table 17 (449.8 kN axial load), but the joint ultimate
resistance is still controlled by the beam flanges in compression.
The abruptly change in the analytical moment-axial load interaction diagram
from 449.8 kN to 385.5 kN axial loads, Figure 53, was caused due the changes in
the activated mechanical model rows. From 385.5 kN compressive axial load to
the 514.0 kN tensile axial load practically all the bolt-rows are activated in tension
together the beam bottom flange in compression and the lever arm in tension.
5 Parametric Investigations 116
For the compressive axial load of 385.5 kN, 321.3 kN, 257.0 kN and 192.8
kN the beam bottom flange yields and fails in the sequence without reaching any
bolt-row yield, Table 17 and Table 18.
Decreasing the axial load magnitude from 128.5 kN to 0.0 kN, the beam
bottom flange continues controlling the joint ultimate bending moment resistance,
however, before reaching the beam bottom flange failure, the following
mechanical model rows: beam bottom flange, bolt-row 1 and bolt-row 2 yield in
this sequence, Table 18 (128.5 kN, 64.3 kN and 0.0 kN).
In the tensile side of the analytical moment-axial load interaction diagram,
Figure 53, a 64.3 kN tensile axial load leads also the bolt-row 3 to the yield.
Furthermore, from the 128.5 kN tensile load to the 578.3 kN joint tensile
resistance the bolt-row 3 fails without any bending moment applied to the joint,
however the beam bottom flange keeps governing the joint ultimate bending
moment resistance till the 514.0 kN tensile axial load.
Finally, from the 514.0 kN to 578.3 kN tensile axial loads there is a sudden
drop in the joint ultimate bending moment resistance from 75.0 kNm to 0.0 kNm.
This is the effect of the changes in the activated mechanical model rows, where
for the 578.3 kN tensile axial load only the bolt-rows are activated and subjected
to tension. Moreover, without any bending moment applied to the joint, all the
bolt-rows yield and in the sequence the bolt-rows 3 and 2 fail, Table 17 and Table
18.
It is also worth highlighting the changes in the controlling components of
the joint ultimate bending moment resistance in function of the axial load level
magnitudes. For instance, for a 385.5 kN compressive load the weakest
component was beam flange and web in compression for the beam bottom flange
row whilst for a 385.5 kN tensile load the weakest component was the column
flange in bending for the bolt-row 2. This demonstrates how the axial load level
changes the governing components for a same joint layout. For other example, see
Table 18.
5 Parametric Investigations 117
Table 17 - Mechanical model row stiffness for the joint ultimate bending moment
resistance.
Nu Mu BR 1 BR 2 BR 3 BTF BBF LAkN kNm kbr1 klt1 kbr2 klt2 kbr3 klt3 kbtf klctf kbbf klcbf klt d (mm)
824.4 0 - - - - - - p e p e - -642.5 21 - - - - - - e e p e - -578.3 29 - - - - - - e e p e - -514.0 36 - - - - - - e e p e - -449.8 44 e - - - - - e e p e - -385.5 100 e - e - - - - - p e e 63.1321.3 106 e - e - - - - - p e e 67.8257.0 113 e - e - - - - - p e e 73.3192.8 120 e - e - - - - - p e e 79.7128.5 130 p - p - - - - - p e e 87.364.3 141 p - p - - - - - p e e 96.60.0 136 p - p - - - - - p e e 79.3
-64.3 127 p - p - e - - - p e e 61.0-128.5 0 e e e e p e - - - - - -
119 p - p - - - - - p e e 46.3-192.8 0 e e e e p e - - - - - -
111 p - p - - - - - p e e 34.1-257.0 0 e e e e p e - - - - - -
103 p - p - - - - - p e e 23.9-321.3 0 e e e e p e - - - - - -
96 p - p - u - - - p e e 15.3-385.5 0 e e e e e e - - - - - -
89 p - p - u - - - p e e 7.8-449.8 0 e e e e e e - - - - - -
82 p - p - u - - - p e e 1.3-514.0 0 e e e e e e - - - - - -
75 p - p - u - - - p e e -4.5-578.3 0 e e e e e e - - - - - -
0 - - p e u e - - - - - -Note: e is the model row elastic stiffness; p is the model row plastic stiffness; u is the model
row ultimate stiffness; “-” means that the model row is deactivated i.e. is not contributing for the model stiffness; BR 1 is bolt-row 1; BR 2 is bolt-row 2; BR 3 is bolt-row 3; BTF is the beam top flange; BBF is the beam bottom flange; LA is the lever arm; kbr1 is the bolt-row 1 stiffness and its associated tensile rigid link stiffness klt1; kbr2 is the bolt-row 2 stiffness and its associated tensile rigid link stiffness klt2; kbr3 is the bolt-row 3 stiffness and its associated tensile rigid link stiffness klt3; kbtf is the beam top flange stiffness and its associated compressive rigid link stiffness klctf; kbbf
is the beam bottom flange stiffness and its associated compressive rigid link stiffness klcbf; klt is the lever arm rigid link stiffness and its associated position d. All the stiffness and the lever arm used here are presented in Figure 45.
5 Parametric Investigations 118
Table 18 - Row-component yield and failure sequence.
Yield Sequence Failure Sequence
Axial Load(kN)
M y (kNm)
Row: Component
M u (kNm)
Row: Component
824.4 0.0 BTF/BBF: bfwc642.5 0.0 BTF/BBF: bfwc
21.0 BTF/BBF: bfwc578.3 8.0 BBF: bfwc
29.0 BBF: bfwc514.0 15.0 BBF: bfwc
36.0 BBF: bfwc449.8 23.0 BBF: bfwc
44.0 BBF: bfwc385.5 30.0 BBF: bfwc
100.0 BBF: bfwc321.3 78.0 BBF: bfwc
106.0 BBF: bfwc257.0 84.0 BBF: bfwc
113.0 BBF: bfwc192.8 91.0 BBF: bfwc
120.0 BBF: bfwc128.5 99.0 BBF: bfwc
120.0 BR 1: epb120.0 BR 2: cfb
130.0 BBF: bfwc64.3 109.0 BBF: bfwc
114.0 BR 1: epb114.0 BR 2: cfb
141.0 BBF: bfwc0.0 106.0 BBF: bfwc
107.0 BR 1: epb107.0 BR 2: cfb
136.0 BBF: bfwc-64.3 0.0 BR 3: bfwc
100.0 BBF: bfwc101.0 BR 1: epb101.0 BR 2: cfb
127.0 BBF: bfwc-128.5 0.0 BR 3: bfwc 0.0 BR 3: bfwc
94.0 BBF: bfwc95.0 BR 1: epb95.0 BR 2: cfb
119.0 BBF: bfwc
5 Parametric Investigations 119
-192.8 0.0 BR 3: bfwc 0.0 BR 3: bfwc88.0 BBF: bfwc89.0 BR 1: epb89.0 BR 2: cfb
111.0 BBF: bfwc-257.0 0.0 BR 3: bfwc 0.0 BR 3: bfwc
82.0 BR 1: epb82.0 BBF: bfwc82.0 BR 2: cfb
103.0 BBF: bfwc-321.3 0.0 BR 3: bfwc 0.0 BR 3: bfwc
10.0 BR 2: cfb10.0 BR 1: epb76.0 BBF: bfwc
96.0 BBF: bfwc-385.5 0.0 BR 2: cfb 0.0 BR 3: bfwc
43.0 BR 1: epb70.0 BBF: bfwc
89.0 BBF: bfwc-449.8 0.0 BR 2: cfb 0.0 BR 3: bfwc
47.0 BR 1: epb64.0 BBF: bfwc
82.0 BBF: bfwc-514.0 0.0 BR 2: cfb 0.0 BR 3: bfwc
0.0 BR 1: epb58.0 BBF: bfwc
75.0 BBF: bfwc-578.3 0.0 BR 2: cfb 0.0 BR 3: bfwc
0.0 BR 1: epb 0.0 BR 2: cfb
5.4.Joint Key Parameters
Initially, sensitive analyses were carried out, in the next sections, varying all
the basic components that should be in principle considered for an extended
endplate beam-to-column joint design. These sensitive analyses have as the main
goal to identify the most influent basic components that affect the joint behaviour.
These basic components are presented in Figure 6(b) and listed below:
- column web panel in shear (cws, 1);
- column web in transverse compression (cwc, 2);
- column web in transverse tension (cwt, 3);
- column flange in bending (cfb, 4);
5 Parametric Investigations 120
- endplate in bending (epb, 5);
- beam flange/web in compression (bfwc, 7);
- beam web in tension (bwt, 8);
- bolts in tension (bt, 10).
Regarding the load situations, all the load variations depicted in section 5.3,
initial studies, were selected and used in the parametric investigations. They were:
- one without any axial load;
- three considering compressive force magnitudes of 135.94 kN (+10% Npl),
193.30 kN (+20% Npl) and 259.20 kN (+27% Npl);
- two considering the tensile force magnitudes of 127.20 kN (-10% Npl) and
257.90 kN (-20% Npl).
The following sections present the parametric studies, divided into the main
elements that define an extended endplate joint. Table 19 demonstrates this
division and the studied basic components referred to each case.
Table 19 – Main elements of the joint and their respective basic components.
Member Parametric Investigations
SectionVariation Components
Beam Profile bfwc (5) and bwt (8) 5.5Column Profile cws (1), cwc (2), cwt (3) and cfb (4) 5.6Endplate Thickness epb (5) 5.7
Bolts Type bt (10) 5.8
5.5.Beam Profile Investigations
In this section, the variations of the beam profiles are evaluated and their
impact in the related basic components: beam flange and web in compression
(bfwc, 5) and beam web in tension (bwt, 8). Table 20 presents the range of used
profiles in this investigation and their principal dimensions.
Table 20 - Investigated beam profiles and their main dimensions.
Profile hb (mm) bb (mm) twb (mm) tfb (mm) Information
IPE 240 240.0 120.0 6.2 9.8 Reference ProfileIPE 180 180.0 91.0 5.3 8.0 Studied ProfileIPE 200 200.0 100.0 5.6 8.5 Studied ProfileIPE 300 300.0 150.0 7.1 10.7 Studied Profile
5 Parametric Investigations 121
The results of each investigation, for the six axial force levels previously
mentioned in section 5.4 and Table 20, were compared to the reference IPE 240
beam profile illustrated in Figure 54 to Figure 59. Subsequently, Figure 60
illustrates the prediction of the bending moment versus axial force interaction
diagram, for these cases, by the proposed mechanical model for the joint ultimate
resistances.
0
20
40
60
80
100
120
140
160
180
200
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
IPE 240 IPE 180
IPE 200 IPE 300
Figure 54 - Investigated EE1 (N = 0.0 kN) moment-rotation curves involving the beam
profile variations.
0
20
40
60
80
100
120
140
160
180
200
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
IPE 240 IPE 180
IPE 200 IPE 300
Figure 55 - Investigated EE2 (N = +10% Npl = 135.95 kN) moment-rotation curves
involving the beam profile variations.
5 Parametric Investigations 122
0
20
40
60
80
100
120
140
160
180
200
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
IPE 240 IPE 180
IPE 200 IPE 300
Figure 56 - Investigated EE3 (N = +20% Npl = 193.30 kN) moment-rotation curves
involving the beam profile variations.
0
20
40
60
80
100
120
140
160
180
200
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
IPE 240 IPE 180
IPE 200 IPE 300
Figure 57 - Investigated EE4 (N = +27% Npl = 259.20 kN) moment-rotation curves
involving the beam profile variations.
5 Parametric Investigations 123
0
20
40
60
80
100
120
140
160
180
200
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
) IPE 240 IPE 180
IPE 200 IPE 300
Figure 58 - Investigated EE6 (N = -10% Npl = -127.20 kN) moment-rotation curves
involving the beam profile variations.
0
20
40
60
80
100
120
140
160
180
200
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
) IPE 240 IPE 180
IPE 200 IPE 300
Figure 59 - Investigated EE7 (N = -20% Npl = -257.90 kN) moment-rotation curves
involving the beam profile variations.
020406080
100120140160180200
-900 -650 -400 -150 100 350 600 850 1100 1350
Axial Force (kN)
Ben
ding
Mom
ent (
kN.m
)
IPE 240 IPE 180
IPE 200 IPE 300
Figure 60 - Analytical moment-axial load interaction diagram at different beam profiles.
5 Parametric Investigations 124
5.5.1.Discussion of the Results
The joint response (moment-rotation curve and moment-axial force
interaction diagram) under axial forces and bending moments is strongly affected
by different beam profiles. For instance, Table 21, by reducing the reference IPE
240 beam profile to IPE 180 or IPE 200 the capacity of bolt-row 2 changes its
controlling component from the column flange in bending to the beam flange and
web in compression. On the other hand, by increasing the reference profile to IPE
300, the capacities of bolt-row 3 and beam bottom flange change their governing
component from the beam flange and web in compression to the column web in
compression.
Table 21 - The weakest component of the mechanical model rows for each analysed
case with N = 0.0.
Profile Bolt-row 1 Bolt-row 2 Bolt-row3 Beam bottom flange
IPE 240 epb cfb bfwc bfwcIPE 180 epb bfwc bfwc bfwcIPE 200 epb bfwc bfwc bfwcIPE 300 epb cfb cwc cwc
Regarding the joint ultimate bending moment capacities, it is possible to
observe a reduction in the ultimate bending moments in alignment with a profile
reduction. This reduction is more pronounced with increasing compressive force
magnitude levels, see Table 22 (EE2, EE3 and EE4) and Figure 55, Figure 56 and
Figure 57. On the other hand, Table 22 and Figure 54 to Figure 59, the change of
profiles from IPE 240 to IPE 300 increases the ultimate bending moment by
22.7% (minimum EE1: 20.5% - maximum EE4: 24.0%) approximately for all the
analysed cases.
5 Parametric Investigations 125
Table 22 - Evaluated ultimate bending moments at different beam profiles.IP
E P
rofi
le EE1
(only M)
EE2
(+10% Npl)
EE3
(+20% Npl)
EE4
(+27% Npl)
EE6
(-10% Npl)
EE7
(-20% Npl)
Mu Mu Mu Mu Mu Mu
kNm % kNm % kNm % kNm % kNm % kNm %
240 135.1 Ref. 128.2 Ref. 119.9 Ref. 111.8 Ref. 118.3 Ref. 102.7 Ref.180 105.4 -28.2 59.2 -116.5 52.6 -127.9 46.6 -139.9 88.8 -33.2 74.6 -37.7
200 114.4 -18.1 78.8 -62.7 71.5 -67.8 64.6 -73.0 97.7 -21.0 83.0 -23.7
300 169.9 20.5 167.2 23.3 157.0 23.6 147.0 24.0 151.0 21.7 133.4 23.1
Note: The percentage is calculated in function of the Mu of the reference IPE 240 profile.
The joint initial stiffness is slightly reduced by downsizing the beam profiles
for the compressive forces and slightly increased for the tensile forces, Figure 54
to Figure 59. By upsizing the beam profile there is a small increase in the joint
initial stiffness when compared to the reference IPE 240 beam profile for all the
cases.
Finally, from the analytical bending moment versus axial force interaction
diagram, Figure 60, it worth being noted that the joint tensile resistance is
inversely proportional to the downsizing of the beam profile. This fact occurs due
to the reduction of the lever arm defined by the distance from the load application
line to the midpoint between bolt-row 1 and 2. As bolt-row 3 under large tensile
forces is the first row to fail, remaining bolt-row 1 and 2, this lever arm influences
in the failure of these remained bolt-rows. However, with a significant increase in
the beam profile size others factors may become more relevant and the associated
joint tensile resistance might be larger as it is the case of IPE 300 beam profile.
5.6.Column Profile Investigations
In this section, the variations of the column profiles are evaluated and their
effect in the related basic components: column web in shear (cws, 1); column web
in compression (cwc, 2); column web in tension (cwt, 3) and column flange in
bending (cfb, 4). Table 23 presents the range of used profiles in this investigation
and their principal dimensions.
5 Parametric Investigations 126
Table 23 - Investigated column profiles and their main dimensions.
Profile hc (mm) bc (mm) twc (mm) tfc (mm) Information
HE 240 B 240.0 240.0 10.0 17.0 Reference ProfileHE 240 A 230.0 240.0 7.5 12.0 Studied ProfileHE 300 B 300.0 300.0 11.0 19.0 Studied ProfileHE 360 B 360.0 300.0 12.5 22.5 Studied Profile
The results of each investigation, for the six axial force levels previously
mentioned in section 5.4 and Table 23, compared to the reference HE 240 B
column profile are illustrated in Figure 61 to Figure 66. Subsequently, Figure 67
illustrates the prediction of the bending moment versus axial force interaction
diagram, for these cases, by the proposed mechanical model for the joint ultimate
resistances.
0
20
40
60
80
100
120
140
160
180
200
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
HE 240 B HE 240 A
HE 300 B HE 360 B
Figure 61 - Investigated EE1 (N = 0.0 kN) moment-rotation curves involving the column
profile variations.
5 Parametric Investigations 127
0
20
40
60
80
100
120
140
160
180
200
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
HE 240 B HE 240 A
HE 300 B HE 360 B
Figure 62 - Investigated EE2 (N = +10% Npl = 135.95 kN) moment-rotation curves
involving the column profile variations.
0
20
40
60
80
100
120
140
160
180
200
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
HE 240 B HE 240 A
HE 300 B HE 360 B
Figure 63 - Investigated EE3 (N = +20% Npl = 193.30 kN) moment-rotation curves
involving the column profile variations.
5 Parametric Investigations 128
0
20
40
60
80
100
120
140
160
180
200
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
HE 240 B HE 240 A
HE 300 B HE 360 B
Figure 64 - Investigated EE4 (N = +27% Npl = 259.20 kN) moment-rotation curves
involving the column profile variations.
0
20
40
60
80
100
120
140
160
180
200
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
) HE 240 B HE 240 A
HE 300 B HE 360 B
Figure 65 - Investigated EE6 (N = -10% Npl = -127.20 kN) moment-rotation curves
involving the column profile variations.
5 Parametric Investigations 129
0
20
40
60
80
100
120
140
160
180
200
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
) HE 240 B HE 240 A
HE 300 B HE 360 B
Figure 66 - Investigated EE7 (N = -20% Npl = -257.90 kN) moment-rotation curves
involving the column profile variations.
020406080
100120140160180200
-900 -650 -400 -150 100 350 600 850 1100 1350
Axial Force (kN)
Ben
ding
Mom
ent (
kN.m
) HE 240 B
HE 240 AHE 300 BHE 360 B
Figure 67 - Analytical moment-axial load interaction diagram at different column profiles.
5.6.1.Discussion of the Results
The influence of the studied column profile types on the joint response
(moment-rotation curve and moment-axial force interaction diagram) under axial
forces and bending moments is not as pronounced, as expected, as the previous
investigated beam profile cases.
Table 24 presents the joint basic components that govern the capacities of
the mechanical model rows and Table 25 depicts the joint ultimate bending
moment capacities referred to Figure 66. The increase in the column profile sizes
do not significantly affect the joint characteristic curve as can be seen in Figure 61
5 Parametric Investigations 130
to Figure 66 and Table 25. On the other hand, HE 240 A profile causes a
pronounced reduction in the joint ultimate bending moment, around 40.6%, when
coupled with increasing compressive forces, Table 25 (EE2, EE3 and EE4 tests).
Table 24 - The weakest component of the mechanical model rows for each analysed
case with N = 0.0.
Profile Bolt-row 1 Bolt-row 2 Bolt-row3 Beam bottom flange
HE 240 B epb cfb bfwc bfwcHE 240 A epb cwc cwc cwcHE 300 B epb bfwc bfwc bfwcHE 360 B epb bfwc bfwc bfwc
Table 25 - Evaluated ultimate bending moments at different column profiles.
HE
Pro
file
EE1
(only M)
EE2
(+10% Npl)
EE3
(+20% Npl)
EE4
(+27% Npl)
EE6
(-10% Npl)
EE7
(-20% Npl)
Mu Mu Mu Mu Mu Mu
kNm % kNm % kNm % kNm % kNm % kNm %
240B 135.1 Ref. 128.2 Ref. 119.9 Ref. 111.8 Ref. 118.3 Ref. 102.7 Ref.240A 134.1 -0.7 94.0 -36.5 85.3 -40.5 77.3 -44.7 116.0 -1.9 100.1 -2.6
300B 138.9 2.8 132.5 3.2 123.4 2.9 114.6 2.5 121.2 2.4 104.6 1.9
360B 140.7 4.0 132.5 3.2 123.4 2.9 114.6 2.5 122.5 3.5 105.6 2.7
Note: The percentage is calculated in function of the Mu of the reference HE 240 B profile.
The joint initial stiffness presents a slightly reduction by downsizing the
column profiles for the compressive forces, Figure 62, Figure 63 and Figure 64.
However, for the others case the initial stiffness almost remains unchanged.
Regarding the analytical bending moment versus axial force interaction
diagram, Figure 67, it can be highlighted the increase in the joint compressive
resistance with the increase in the analyzed column profile sizes. For tensile forces
applied to the joint the moment-axial force diagram responses are very similar for
the whole range of the investigated column profiles demonstrating small influence
of the column profile variations in the joint tensile resistance.
5.7.Endplate Thickness Investigations
In this section, the variations of the endplate thicknesses are evaluated and
their influence in the related basic component endplate in bending (epb, 5). Table
26 presents the range of used endplate thicknesses in this investigation.
5 Parametric Investigations 131
Table 26 - Investigated endplate thicknesses and their dimensions.
Member tep (mm) Information
Endplate 15.0 Reference ThicknessEndplate 10.0 Studied ThicknessEndplate 12.5 Studied ThicknessEndplate 17.5 Studied Thickness
The results of each investigation, for the six axial force levels previously
mentioned in section 5.4 and Table 26, compared to the reference 15 mm endplate
thickness are illustrated in Figure 68 to Figure 73. Subsequently, Figure 74
illustrates the prediction of the bending moment versus axial force interaction
diagram, for these cases, by the proposed mechanical model for the joint ultimate
resistances.
0
20
40
60
80
100
120
140
160
180
200
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
tep = 15.0 mmtep = 10.0 mmtep = 12.5 mmtep = 17.5 mm
Figure 68 - Investigated EE1 (N = 0.0 kN) moment-rotation curves involving the endplate
thickness variations.
5 Parametric Investigations 132
0
20
40
60
80
100
120
140
160
180
200
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
tep = 15.0 mmtep = 10.0 mmtep = 12.5 mmtep = 17.5 mm
Figure 69 - Investigated EE2 (N = +10% Npl = 135.95 kN) moment-rotation curves
involving the endplate thickness variations.
0
20
40
60
80
100
120
140
160
180
200
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
tep = 15.0 mmtep = 10.0 mmtep = 12.5 mmtep = 17.5 mm
Figure 70 - Investigated EE3 (N = +20% Npl = 193.30 kN) moment-rotation curves
involving the endplate thickness variations.
5 Parametric Investigations 133
0
20
40
60
80
100
120
140
160
180
200
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
tep = 15.0 mmtep = 10.0 mmtep = 12.5 mmtep = 17.5 mm
Figure 71 - Investigated EE4 (N = +27% Npl = 259.20 kN) moment-rotation curves
involving the endplate thickness variations.
0
20
40
60
80
100
120
140
160
180
200
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
tep = 15.0 mmtep = 10.0 mmtep = 12.5 mmtep = 17.5 mm
Figure 72 - Investigated EE6 (N = -10% Npl = -127.20 kN) moment-rotation curves
involving the endplate thickness variations.
5 Parametric Investigations 134
0
20
40
60
80
100
120
140
160
180
200
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)tep = 15.0 mmtep = 10.0 mmtep = 12.5 mmtep = 17.5 mm
Figure 73 - Investigated EE7 (N = -20% Npl = -257.90 kN) moment-rotation curves
involving the endplate thickness variations.
020406080
100120140160180200
-900 -650 -400 -150 100 350 600 850 1100 1350
Axial Force (kN)
Ben
ding
Mom
ent (
kN.m
) tep = 15.0 mm
tep = 10.0 mmtep = 12.5 mmtep = 17.5 mm
Figure 74 - Analytical moment-axial load interaction diagram at different endplate
thicknesses.
5.7.1.Discussion of the Results
The endplate thickness influence over the joint response (moment-rotation
curve and moment-axial force interaction diagram) under axial forces and bending
moments were more significant, as expected, than the previously investigated
cases referred to the beam and column profile variations, sections 5.5 and 5.6,
respectively.
The basic component endplate in bending that has governed the first bolt-
row capacity in the whole previously studied cases (Table 21 and Table 24) just
stops acting when it is assumed a 17.5 mm endplate thickness and then the first
5 Parametric Investigations 135
bolt-row changes its controlling component to be the column flange in bending,
Table 27.
Table 28 presents the enormous impact that the variations of the endplate
thicknesses cause in the joint ultimate bending moment. For example, for a 259.20
kN compressive force and a 10 mm endplate thickness the ultimate bending
moment decreases by 236.9% (EE7 test).
Table 27 - The weakest component of the mechanical model rows for each analysed
case with N = 0.0.
Endplate Bolt-row 1 Bolt-row 2 Bolt-row3 Beam bottom flange
15.0 mm epb cfb bfwc bfwc10.0 mm epb epb epb bfwc12.5 mm epb epb bfwc bfwc17.5 mm cfb cfb bfwc bfwc
Table 28 - Evaluated ultimate bending moments at different endplate thicknesses.
Th
ickn
ess EE1
(only M)
EE2
(+10% Npl)
EE3
(+20% Npl)
EE4
(+27% Npl)
EE6
(-10% Npl)
EE7
(-20% Npl)
Mu Mu Mu Mu Mu Mu
kNm % kNm % kNm % kNm % kNm % kNm %
15.0 135.1 Ref. 128.2 Ref. 119.9 Ref. 111.8 Ref. 118.3 Ref. 102.7 Ref.10.0 59.9 -125.4 81.0 -58.4 87.4 -37.2 94.7 -18.0 45.5 -160.0 30.5 -236.9
12.5 93.6 -44.2 118.5 -8.2 114.8 -4.5 107.7 -3.8 78.3 -51.0 63.6 -61.3
17.5 145.4 7.1 129.5 1.0 121.0 0.9 112.7 0.8 128.3 7.8 112.5 8.8
Note: The percentage is calculated in function of the Mu of the reference 15 mm endplate.
The joint initial stiffness, Figure 73, is strongly dependent on the endplate
thickness, mainly in the studied cases where the adopted endplate thickness is
smaller than the reference 15 mm endplate thickness and there is tensile force
acting on the joint. This fact is also noted in the analytical bending moment versus
axial force interaction diagram, Figure 74, where the joint tensile resistance is
reduced with a simultaneous decrease of the endplate thickness.
It should be also observed in the moment-rotation curves for a 10.0 mm
endplate thickness and tensile loads of 127.2 kN (Figure 72) and 257.9 kN (Figure
73) a strong degradation of the joint ultimate stiffness after yielding. For instance,
the moment-rotation curve for a 10.0 mm endplate thickness and a 257.9 kN
tensile load, Figure 73, shows that after yielding the joint does not have any
additional resistance.
5 Parametric Investigations 136
5.8.Bolt Investigations
In this section, the bolt diameter size influence was evaluated and their
impact in the related basic component bolts in tension (bt, 10). Table 29 presents
the range of used bolts in this investigation and their principal dimensions.
Table 29 - Investigated grade 10.9 bolts and their main dimensions.
Bolt db (mm) dh (mm) th (mm) tn (mm) dw (mm) twh (mm) Information
M 20 20.0 33.27 13.0 16.00 35.03 4 Reference BoltM 12 12.0 22.75 7.5 10.59 23.74 3 Studied BoltM 16 16.0 28.07 10.0 14.45 29.74 4 Studied BoltM 30 30.0 52.44 18.7 24.95 55.40 5 Studied Bolt
The results of each investigation, for the six axial force levels previously
mentioned in section 5.4 and Table 29, compared to the reference M 20 bolt are
illustrated in Figure 80 to Figure 80. Subsequently, Figure 81 illustrates the
prediction of the bending moment versus axial force interaction diagram, for these
cases, by the proposed mechanical model for the joint ultimate resistances.
0
20
40
60
80
100
120
140
160
180
200
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
M 20 M 12
M 16 M 30
Figure 75 - Investigated EE1 (N = 0.0 kN) moment-rotation curves involving the bolt
variations.
5 Parametric Investigations 137
0
20
40
60
80
100
120
140
160
180
200
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
M 20 M 12
M 16 M 30
Figure 76 - Investigated EE2 (N = +10% Npl = 135.95 kN) moment-rotation curves
involving the bolt variations.
0
20
40
60
80
100
120
140
160
180
200
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
M 20 M 12
M 16 M 30
Figure 77 - Investigated EE3 (N = +20% Npl = 193.30 kN) moment-rotation curves
involving the bolt variations.
5 Parametric Investigations 138
0
20
40
60
80
100
120
140
160
180
200
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
M 20 M 12
M 16 M 30
Figure 78 - Investigated EE4 (N = +27% Npl = 259.20 kN) moment-rotation curves
involving the bolt variations.
0
20
40
60
80
100
120
140
160
180
200
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
) M 20 M 12
M 16 M 30
Figure 79 - Investigated EE6 (N = -10% Npl = -127.20 kN) moment-rotation curves
involving the bolt variations.
5 Parametric Investigations 139
0
20
40
60
80
100
120
140
160
180
200
0 10 20 30 40 50Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
) M 20 M 12
M 16 M 30
Figure 80 - Investigated EE7 (N = -20% Npl = -257.90 kN) moment-rotation curves
involving the bolt variations.
020406080
100120140160180200
-900 -650 -400 -150 100 350 600 850 1100 1350
Axial Force (kN)
Ben
ding
Mom
ent (
kN.m
)
M 20 M 12
M 16 M 30
Figure 81 - Analytical moment-axial load interaction diagram at different bolts.
5.8.1.Discussion of the Results
The bolts, similar to the endplate thickness variations, have significant
effect on the joint response as it was again expected. Even though the basic
component bolts in tension has not governed the capacity of any mechanical
model row in the previously executed numeric analysis, it begins to control bolt-
rows 2 and 3 capacities when a M 12 bolt is adopted, Table 30.
Table 30 presents the weakest component of the mechanical model rows for
each studied bolt type, whilst Table 31 draws the joint ultimate bending moments.
The ultimate bending moment variations in functions of the bolt types, Table 31,
5 Parametric Investigations 140
are more evident for decreasing bolt sizes than increasing bolt sizes. For instance,
for a M 12 bolt there is a decrease by 178.5% in the joint ultimate bending
moment while for a M 30 bolt there is an increase of 1.5%, for a 259.20 kN
compressive force (EE7 test).
Table 30 - The weakest component of the mechanical model rows for each analysed
case with N = 0.0.
Bolt Bolt-row 1 Bolt-row 2 Bolt-row3 Beam bottom flange
M 20 epb cfb bfwc bfwcM 12 epb bt (cfb/epb) bt (cfb/epb) bfwcM 16 epb epb epb bfwcM 30 epb bfwc bfwc bfwc
Table 31 - Evaluated ultimate bending moments at different bolt diameters.
Bol
t
EE1
(only M)
EE2
(+10% Npl)
EE3
(+20% Npl)
EE4
(+27% Npl)
EE6
(-10% Npl)
EE7
(-20% Npl)
Mu Mu Mu Mu Mu Mu
kNm % kNm % kNm % kNm % kNm % kNm %
M 20 135.1 Ref. 128.2 Ref. 119.9 Ref. 111.8 Ref. 118.3 Ref. 102.7 Ref.M 12 65.1 -107.6 79.9 -60.4 86.2 -39.1 93.4 -19.7 51.2 -131.2 36.9 -178.5
M 16 106.5 -26.8 121.9 -5.2 114.5 -4.7 107.5 -4.0 90.5 -30.7 75.4 -36.2
M 30 132.4 -2.0 132.5 3.2 123.4 2.9 114.6 2.5 116.2 -1.7 101.2 -1.5
Note: The percentage is calculated in function of the Mu of the reference M 20 bolt.
The joint initial stiffness, Figure 80, is strongly dependent on the bolt type,
similarly to the finding observed in the investigation of the endplate thickness
discussed in section 5.7. The studied cases involving bolts smaller than the
reference M 20 bolt, in general, present significant variation in the joint initial
stiffness what do not happen for bolts larger than the reference M 20 bolt.
Associated also with the reduction in the bolt sizes is the associated reduction of
the joint tensile resistance as can be seen in Figure 81.
5.9.Axial Force Effect
In general, from the parametric investigations, it is possible to note that the
axial force significantly affect the joint structural behaviour. The effect of the
axial force might be more pronounced or not when coupled with variations in the
joint basic components arising from, for instance, different profile sizes, endplate
thickness and bolts.
5 Parametric Investigations 141
Some axial force levels may be also beneficial for the joint ultimate bending
moment as identified in the bending moment versus axial force interaction
diagram (Figure 53, Figure 60, Figure 67, Figure 74 and Figure 81) studied in this
chapter for the majority of the investigated variations.
In bending moment versus axial force interaction diagram for the beam
profile variations, Figure 60, it is observed a beneficial tensile force of 64.25 kN
causing the maximum joint ultimate bending moment for the beam IPE 180 and
IPE 200 profiles, whilst for the reference beam IPE 200 a beneficial compressive
force of 64.25 kN is found. For the upper beam IPE 300 profile, the maximum
ultimate bending moment was reached without axial forces.
A similar situation is noted in moment-axial force diagram for the column
profile variations, Figure 67, where for the column HE 240 A profile larger than
the reference column HE 240 B profile reaches the maximum ultimate bending
moment at a 64.25 kN tensile force and for a upper column HE 360 B profile the
maximum ultimate bending moment was reached without axial forces.
For the endplate thickness variations, a beneficial 64.25 kN compressive
force was detected in Figure 74 for the endplate thicknesses of 10.0, 12.5 and 15.0
mm, where the maximum joint ultimate bending moments were obtained, while
for a 17.5 mm endplate thickness the maximum ultimate bending moment was
reached without axial forces.
Regarding the bolts, the bending moment versus axial force diagram, Figure
81, depicts the maximum joint ultimate bending moments for the M 12 and M 16
bolts for compressive forces of 257.01 kN and 64.25 kN, respectively. The
maximum joint ultimate bending moment for the reference M 20 bolt was also
reached at a beneficial 64.25 kN compressive force, whilst for the upper M 30
bolt, the maximum joint ultimate bending moment was reached without axial
forces.
Based on the investigated situations, it was also possible to conclude that the
positive contribution of the axial force in the maximum joint ultimate bending
moments is more significant with a decrease in the joint stiffness. In general, for
the cases analysed in this chapter where the joint dimensions are reduced when
compared to the joint reference dimensions, the axial force presents a beneficial
contribution for the maximum ultimate bending moment. On the other hand, for
5 Parametric Investigations 142
upper joint dimensions the maximum ultimate bending moments are reached
without axial forces.
5.10.Notes about the Incremental Solution of the Analytical Bending Moment versus Axial Force Interaction Diagram
Some notes must be made about the incremental solution process of the
bending moment versus axial force interaction diagram by using the proposed
mechanical model:
a) The joint yield compressive resistance is used as an input for the starting
point for the incremental solution of the moment-axial force diagram.
Subsequently, this value is decreased until reaching the joint ultimate tensile
resistance. The joint ultimate tensile resistance is identified when the incremental
bending moment is equal to zero.
b) The bending moment applied to the joint is incremented from zero till
reaching the joint ultimate bending moment resistance where the joint does not
have any additional resistance.
c) The yield (Ny, My) and (Nu, Mu) failure points plotted in the bending
moment versus axial force interaction diagram are the maximum reached values
for each axial load level magnitude.
d) The mechanical model stability is checked every time that one of the
model row stiffness changes. For instance, if the mechanical model is only
subjected to a compressive force, only the beam top and bottom flanges are
contributing for the joint resistance. However, if an increasing bending moment is
introduced leading the joint to the tensile-compressive state, the bolt-rows start
being introduced into the mechanical model stiffness matrix together the lever arm
whilst the beam top flange row is removed from the analysis.
6 An Alternative Methodology to Extend the Range of Application of Available Experimental Data so as to Produce Moment-Rotation Characteristics 143
6An Alternative Methodology to Extend the Range of Application of Available Experimental Data so as to Produce Moment-Rotation Characteristics
6.1.Introduction
Few experiments considering the interaction bending moment and axial
force have been reported in the literature. Additionally, the available experiments
are associated with a small number of axial force levels and associated bending
moment versus rotation curves, M-. Thus the, a question still remains for how to
incorporate these effects into a structural analysis. There is, therefore, the need of
M- curves associated with varying axial force levels, which accurately represent
the joint resistance rotational stiffness.
This has led to the development of a relatively simple yet accurate approach
to predict any moment versus rotation curve from tests that include the axial
versus bending moment interaction. This alternative methodology based on the
use of Correction Factors initially proposed by Del Savio et al. (2006), which
extends the range of application of available experimental data, is presented in this
Chapter.
It is worth highlighting that this methodology is not only restricted to the
use of experiments, but can also be applied to results obtained analytically,
empirically, mechanically and numerically. Moreover, since this methodology is
exclusively based on the use of M- curves, the bending moment versus axial
force interactions are intrinsically incorporated, it can be easily implemented into
a nonlinear semi-rigid joint finite element formulation because it does not change
the element formulation, only requiring a rotational stiffness update procedure.
6.2.General Concepts of the Correction Factor
The Correction Factor was initially proposed by Del Savio et al. (2006) to
allow for the bending moment versus axial force interaction, by scaling original
6 An Alternative Methodology to Extend the Range of Application of Available Experimental Data so as to Produce Moment-Rotation Characteristics 144
moment values present in the moment versus rotation curves (disregarding the
axial force effect).
This strategy shifts this curve up or down depending on the axial force level.
However, as it only modifies the bending moment axis, it is not able to fully
describe the bending moment versus rotation associated with different axial force
levels. This fact is highlighted when the joint is subject to a tensile axial force,
where there is a significant difference, mainly, in terms of initial stiffness.
With the aim of improving this basic idea, the Correction Factor has been
divided into two parts: one for the moment axis and another for the rotation axis.
Both corrections are in principle independent, and do not depend on the moment
versus axial force interaction diagram, as was the case for the initial idea
presented by Del Savio et al. (2006). It is now only a function of the moment
versus rotation curves for different axial force levels.
6.3.Extension of the Correction Factors for Both Bending Moment and Rotation Axes
As previously noted, there are two corrections, one to the moment axis and
another to the rotation axis. As a general approach, the Correction Factor for the
moment axis is evaluated in terms of the bending moment versus rotation curves
considering the axial force effect. Using the design bending moment ratio and
considering the axial force effect, the Correction Factor for the moment axis, CFM,
can be evaluated by:
))0.0((max
))((int
max
int
MxfMi
NMxfM
M
M
MCF
(6.1)
where Ni is the axial force present in interaction i; Mx or M- is the bending
moment versus rotation curve; Mint is the design bending moment for the M- (Ni)
curve considering the axial force Ni; and Mmax is the design bending moment for
the M- (0.0) curve without axial forces. Mint and Mmax can be determined
according to Eurocode 3:1-8 (2005), through the intersection between two straight
lines, one parallel to the initial stiffness and another parallel to the M- curve
post-limit stiffness, Figure 82.
6 An Alternative Methodology to Extend the Range of Application of Available Experimental Data so as to Produce Moment-Rotation Characteristics 145
Figure 82 - Evaluation of the design bending moments (Mint & Mmax) and rotations (int &
max).
Similarly, the rotation axis Correction Factor, CF, is evaluated using the
design rotation ratio, i.e.:
))0.0((max
))((int
max
int
Mxfi
NMxfCF
(6.2)
where int and max are the design rotations related respectively to Mint and Mmax.
Both design rotations are found by tracing a horizontal straight line at the design
moment level until it reaches the M- curve. At this point a vertical straight line is
drawn until it intersects the rotation axis, Figure 82.
With the Correction Factors evaluated for both the moment and rotation
axes, Eqs. (6.1) and (6.2) respectively, they are then incorporated into the joint
structural response considering the moment versus axial force interaction by
modifying the M- curve for the zero axial force case, i.e.:
),()(
)()0.0(
00
CFCFMMxNMx
NMxMx
NMNi
i
(6.3)
Basically, the M- curve point coordinates MN=0 and N=0 referred to the
moment and the rotation axis coordinates, respectively, for the case without axial
forces, are multiplied by the Correction Factors CFM and CF, respectively.
However, only using a pair of Correction Factors throughout the whole M-
curve, for the case without axial forces, does not provide a good approximation to
the M- curve considering the axial force, because its response is very sensitive to
the adopted initial and post-limit stiffnesses.
6 An Alternative Methodology to Extend the Range of Application of Available Experimental Data so as to Produce Moment-Rotation Characteristics 146
This prompted the division of the M- curve into three segments with
different pairs of Correction Factors. These divisions were made at two-third, one,
and 1.1 times the design bending moment Md as shown in Figure 83.
Figure 83 - Correction Factor strategy method using a three segment division of the M-
curve.
With this division, the Correction Factors cannot be applied as presented in
Eq. (6.3). This is justified, in fact, because they would provoke two abrupt
variations of stiffness throughout the approximate M- curve at around the point
of intersection of the approximate curve with the vertical lines at the points 2/3d
and d, Figure 84. This is due to the use of three different pairs of Correction
Factors evaluated according to Eqs. (6.1) and (6.2) at two-third, one, and 1.1 times
the design bending moment Md.
Figure 84 - Approximate M- curve using three Correction Factor pairs.
6.4.An alternative methodology
Based on the division of the M- curve into three segments with different
pairs of Correction Factors, previously mentioned, in Figure 85, a tri-linear
representation for the M- curve is proposed. This method overcomes the problem
of abrupt alterations of stiffness presented in Figure 84 as well as guaranteeing an
6 An Alternative Methodology to Extend the Range of Application of Available Experimental Data so as to Produce Moment-Rotation Characteristics 147
accurate approximation of the M- curve at points: (2/3Md, 2/3d); (Md, d) and
(1.1Md, 1.1d).
Figure 85 - Tri-linear representation of the M- curve methodology.
From the tri-linear representation proposed in Figure 85, the bending
moments of the target M- curve, associated with a certain axial force level (Ni),
can be evaluated by:
forceaxialecompressiv0
forceaxialtensile0
1.1;;32
,0,0,
iNN
Ni
N
dM
dM
dMp
pM
Ni
N
pM
pNM
pM
(6.4)
where the subscript p refers to three analysed points: 2/3Md, Md, and 1.1Md; N is
the axial force load level associated with the reference M- curve; Ni is the axial
force load level related to the target M- curve; Mp is the bending moment
evaluated for the target M- curve at point p; MN,p is the bending moment on the
reference M- curve considering the axial force at point p; and M0,p is the bending
moment on the reference M- curve without axial forces at point p.
Likewise, the rotations of the evaluated M- curve, for the associated Ni, can
be calculated by:
forceaxialecompressiv0
forceaxialtensile0
1.1;;32
,0,0,
iNN
Ni
N
dM
dM
dMp
pNi
N
ppNp
(6.5)
6 An Alternative Methodology to Extend the Range of Application of Available Experimental Data so as to Produce Moment-Rotation Characteristics 148
where p is the rotation evaluated for the target M- curve at point p; N,p is the
rotation on the reference M- curve considering the axial force at point p; and 0,p
is the rotation on the reference M- curve without axial force effects at point p.
The evaluations of the bending moments and rotations proposed in Eqs.
(6.4) and (6.5), respectively, for prediction of the target M- curve are, in essence,
linear interpolations between two reference M- curves – considering and
disregarding the axial force – at points: (2/3Md, 2/3d); (Md, d) and (1.1Md, 1.1d).
7 Applicably and Validity of the Proposed Alternative Methodology 149
7Applicably and Validity of the Proposed AlternativeMethodology
7.1.Introduction
This Chapter presents the evaluation and validation of the alternative
methodology developed in Chapter 6 for extending the range of application of
available data so as to produce moment-rotation characteristics that implicitly
make proper allowance for the presence of significant levels of either tension or
compression in the beam. This assessment is executed against a range of available
experimental tests for flush endplate joints (Simões da Silva et al., 2004) and
baseplate joints (Guisse et al., 1996).
7.2.Application of the Alternative Methodology
The main focus of the methodology presented in Chapter 6 is to determine
M- curves for any axial force level from two reference M- curves. The quality
of the obtained approximations depends on the quality of the M- curves used as
input to the method.
This methodology requires, at least, two M- curves, disregarding and
considering either the compressive or tensile axial force effect. However, for a
complete behavioural evaluation of the joint three M- curves are necessary: one
disregarding the axial force effect; another considering the compressive force
effect and finally a third alternative considering the tensile force effect. In this
way, it is possible to study the entirely joint structural response given that loading
applied to the joint may vary from compression to tension.
In order to explain the application of this method to obtain M- curves for
any axial force level, as well as to validate its use, experimental tests carried out
by Lima (2003) and Simões da Silva et al. (2004), and Guisse et al. (1996), on
eight flush endplate joints and twelve column bases have been used.
7 Applicably and Validity of the Proposed Alternative Methodology 150
7.2.1.Flush endplate joints
This section evaluates experimental tests carried out by Simões da Silva et
al. (2004) on eight flush endplate joints. The geometric properties of the flush
endplate, the M- curves describing the experimental behaviour of each test, and
the bending moment versus axial force interaction diagram are shown in Figure 86
to Figure 88.
Figure 86 - Flush endplate joint layout, Simões da Silva et al. (2004).
Figure 87 - Experimental moment versus rotation curves, Simões da Silva et al. (2004).
All the dimensions in mm
7 Applicably and Validity of the Proposed Alternative Methodology 151
Figure 88 - Flush endplate bending moment versus axial force interaction diagram,
Simões da Silva et al. (2004).
The experimental data, Figure 87, provides the necessary input for the
proposed method. The minimum input is composed of two M- curves,
disregarding and considering either the tensile or compressive axial force.
However, the flush endplate joint, tested by Simões da Silva et al. (2004),
exhibited a decrease in the moment resistance for the tensile axial forces whilst
achieving the highest moment resistance for a compressive axial force equal to
20% of the beam’s axial plastic resistance (see Figure 88, FE7). Three reference
M- curves were adopted to demonstrate this joint’s behaviour relative to the type
of axial force: FE1 (N = 0); FE7 (N = -257 kN, -20% Npl, compressive force), and
FE9 (N = 250 kN, +20% Npl, tensile force), where Npl is the beam’s axial plastic
resistance.
These three experimental curves and their tri-linear approximations are
shown in Figure 89. Additionally, Table 32 presents all the values evaluated for
these tri-linear approximations according to Figure 83, where the points for each
tri-linear reference M- curve were obtained from the joint design moment, Md,
which is given by the intersection between two straight lines, one parallel to the
initial stiffness and another parallel to the M- curve post-limit stiffness.
7 Applicably and Validity of the Proposed Alternative Methodology 152
Table 32 - Values evaluated for the reference M- curves.
FE1(N = 0.0)
FE7(N = -257 kN, -20% Npl)
FE9(N = +250 kN, +20% Npl)
Point (mrad) M (kNm) (mrad) M (kNm) (mrad) M (kNm)
0 0.0 0.0 0.0 0.0 0.0 0.0
2/3 Md 6.3 50.6 6.8 56.1 13.0 38.4
Md 27.6 76.0 26.8 84.1 25.8 57.7
1.1 Md 56.1 83.5 67.3 92.2 35.0 63.5
Tri-linear M- curves, Figure 89, are used to define paths between each
curve at points 2/3Md, Md and 1.1Md, Figure 90. These paths were used to guide
the linear interpolators for bending moments, Eq. (6.4), and rotations, Eq. (6.5),
throughout the given range of axial force levels to determine the required set of
M- curves.
Figure 89 - Tri-linear strategy used for the experimental M- curves.
7 Applicably and Validity of the Proposed Alternative Methodology 153
Figure 90 - Paths used to define the procedure to determine any M- curve present within
these limits.
Subsequently, Table 33 depicts the results obtained by using the proposed
methodology to predict three experimental M- curves: FE8 for a 10% tensile
force of the beam’s axial plastic resistance, FE3 and FE4 for compressive forces
of 4% and 8%, respectively, of the beam’s axial plastic resistance.
Following this strategy, as an example, Eq. (7.1) demonstrates how to
calculate point 1.1Md, Table 33, of the FE8 approximated M- curve. Figure 91 to
Figure 93 graphically depict these results. Figure 94 presents the whole set of
predicted M- curves utilising this methodology.
Table 33 - Values evaluated for three tri-linearly approximated M- curves.
FE3(Ni = -53 kN, -4% Npl)
FE4(Ni = -105 kN, -8% Npl)
FE8(Ni = +128 kN, +10% Npl)
Point (mrad) M (kNm) (mrad) M (kNm) (mrad) M (kNm)
0 0.0 0.0 0.0 0.0 0.0 0.0
2/3 Md 6.4 51.8 6.5 52.9 9.7 44.4
Md 27.4 77.6 27.3 79.3 26.7 66.6
1.1 Md 58.4 85.3 60.7 87.1 45.3 73.3
7 Applicably and Validity of the Proposed Alternative Methodology 154
1:2,0
9:2;,
3.451.560.250
0.1281.560.35
,0,0,
3.735.830.250
0.1285.835.63
,0,0,
,0
,
1.1int:8
FETableandp
M
FETableNandpN
M
mradpN
iN
ppNp
kNmp
MN
iN
pM
pNM
pM
p
pN
dMpPoFE
(7.1)
Figure 91 - FE8 M- curve approximation, considering a tensile force of 10% of the
beam’s axial plastic resistance.
Figure 92 - FE3 M- curve approximation, considering a compressive force of 4% of the
beam’s axial plastic resistance.
7 Applicably and Validity of the Proposed Alternative Methodology 155
Figure 93 - FE4 M- curve approximation, considering a compressive force of 8% of the
beam’s axial plastic resistance.
Figure 94 - The whole set of predicted M- curves by using the proposed methodology.
7.2.2.Column bases
This section presents the evaluation of the experiments performed by Guisse
et al. (1996) on twelve column base joints. Test configurations with respectively
four and two anchor bolts, Figure 95(a) and Figure 95(b), were considered. The
steel column profile was a S355 HE160B, whilst the S235 baseplates utilised two
different thicknesses: 15 mm and 30 mm. The baseplates are welded to the
column with 6 mm fillet welds connected with M20 10.9 anchor bolts.
7 Applicably and Validity of the Proposed Alternative Methodology 156
45 mm
45 mm
250 mm340 mm
220 mm
HE 160B
M 20
110 mm
220 mm
50 mm 50 mm120 mm
220 mm
HE 160B
M 20
110 mm
(a) Four anchor bolts. (b) Two anchor bolts.
Figure 95 - Baseplate configurations, Guisse et al. (1996).
Table 34 presents the set of the tested column bases and Figure 96 to Figure
99 show the experimental M- curves obtained by Guisse et al. (1996).
Table 34 - Nomenclature of the tests and their parameters, Guisse et al. (1996).
Name Anchor bolts Plate thickness (mm) Normal force (kN)PC2.15.100 2 15 100PC2.15.600 2 15 600PC2.15.1000 2 15 1000PC2.30.100 2 30 100PC2.30.600 2 30 600PC2.30.1000 2 30 1000PC4.15.100 4 15 100PC4.15.400 4 15 400PC4.15.1000 4 15 1000PC4.30.100 4 30 100PC4.30.400 4 30 400PC4.30.1000 4 30 1000
Since the experiments used only compressive forces, two reference M-
curves were adopted for each set of tests related to the axial forces of 100 and
1000 kN. The experimental M- curves and their tri-linear approximations are
shown in Figure 96 to Figure 99. Additionally, Table 35 presents all the values
evaluated for these tri-linear approximations according to Figure 83.
7 Applicably and Validity of the Proposed Alternative Methodology 157
Table 35 - Values evaluated for the reference M- curves.P
oin
t PC2 PC415.100 15.1000 30.100 30.1000 15.100 15.1000 30.100 30.1000 M M M M M M M M
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 02/3Md 21 21 9 41 25 17 11 46 10 32 16 63 12 46 11 72
Md 40 32 30 62 44 26 29 69 28 48 40 94 33 69 35 1081.1Md 50 35 60 62 51 29 62 75 43 53 60 94 50 76 64 108
Note: M in kNm and in mrad.
0
10
20
30
40
50
60
70
80
0 15 30 45 60 75 90
Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
PC2.15.100: experimentalPC2.15.600: experimentalPC2.15.1000: experimentalPC2.15.100: tri-linearPC2.15.1000: tri-linear
upper compressive limit (N = -1000 kN)
lower compressive limit (N = -100 kN)
Figure 96 - PC2.15 experimental M- curves and the tri-linear reference M- curves.
0
10
20
30
40
50
60
70
80
0 15 30 45 60 75 90
Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
PC2.30.100: experimentalPC2.30.600: experimentalPC2.30.1000: experimentalPC2.30.100: tri-linearPC2.30.1000: tri-linear
upper compressive limit (N = -1000 kN)
lower compressive limit (N = -100 kN)
Figure 97 - PC2.30 experimental M- curves and the tri-linear reference M- curves.
7 Applicably and Validity of the Proposed Alternative Methodology 158
0
20
40
60
80
100
120
0 15 30 45 60 75 90
Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
PC4.15.100: experimentalPC4.15.400: experimentalPC4.15.1000: experimentalPC4.15.100: tri-linearPC4.15.1000: tri-linear
upper compressive limit (N = -1000 kN)
lower compressive limit (N = -100 kN)
Figure 98 - PC4.15 experimental M- curves and the tri-linear reference M- curves.
0
20
40
60
80
100
120
0 15 30 45 60 75 90
Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
PC4.30.100: experimentalPC4.30.400: experimentalPC4.30.1000: experimentalPC4.30.100: tri-linearPC4.30.1000: tri-linear
upper compressive limit (N = -1000 kN)
lower compressive limit (N = -100 kN)
Figure 99 - PC4.30 experimental M- curves and the tri-linear reference M- curves.
Table 36 presents the results obtained by using the proposed method, with
the aid of Eqs. (6.4) and (6.5), to predict four experimental M- curves:
PC2.15.600; PC2.30.600; PC4.15.400 and PC4.30.400.
Table 36 - Values evaluated for three tri-linearly approximated M- curves.
Poi
nt PC2 PC4
15.600 30.600 15.400 30.400 M M M M
0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.02/3Md 13.0 34.7 15.3 36.4 12.7 45.6 11.6 57.6
Md 33.3 52.0 34.0 54.7 33.3 68.4 33.9 86.31.1Md 56.7 53.1 58.3 60.1 50.6 71.1 56.2 90.2
Note: M in kNm and in mrad. Ni is equal to 600 kN for PC2 and 400 kN for PC4.
7 Applicably and Validity of the Proposed Alternative Methodology 159
Since there is no reference to experimental M- curve disregarding the axial
force effect the experimental M- curves related to axial loads of 100 kN are
adopted for the base M- curves. This strategy implies that the axial force load N,
associated with the reference M- curve, used in Eqs. (6.4) and (6.5), was
decreased by 100 kN. Equation (7.2) demonstrates how to calculate point 2/3Md,
Table 36, of the PC2.30.600 approximated M- curve. Finally, Figure 100 to
Figure 103 graphically show these results.
100.30.2:5,0
1000.30.2:5;,
3.150.250.1000.1000
0.6000.255.10
,0,0,
4.363.170.1000.1000
0.6003.170.46
,0,0,
,0
,
3/2int:600.30.2
FCTableandp
M
PCTableNandpN
M
mradpN
iN
ppNp
kNmp
MN
iN
pM
pNM
pM
p
pN
dMpPoPC
(7.2)
0
10
20
30
40
50
60
70
80
0 15 30 45 60 75 90
Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
PC2.15.600: experimental
PC2.15.100: tri-linear
PC2.15.600: approximation
PC2.15.1000: tri-linear
upper compressive limit (N = -1000 kN)
lower compressive limit (N = -100 kN)
Figure 100 - PC2.15.600 M- curve approximation.
7 Applicably and Validity of the Proposed Alternative Methodology 160
0
10
20
30
40
50
60
70
80
0 15 30 45 60 75 90
Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
PC2.30.600: experimental
PC2.30.100: tri-linear
PC2.30.600: approximation
PC2.30.1000: tri-linear
upper compressive limit (N = -1000 kN)
lower compressive limit (N = -100 kN)
Figure 101 - PC2.30.600 M- curve approximation.
0
20
40
60
80
100
120
0 15 30 45 60 75 90
Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
PC4.15.400: experimental
PC4.15.100: tri-linear
PC4.15.400: approximation
PC4.15.1000: tri-linear
upper compressive limit (N = -1000 kN)
lower compressive limit (N = -100 kN)
Figure 102 - PC4.15.400 M- curve approximation.
7 Applicably and Validity of the Proposed Alternative Methodology 161
0
20
40
60
80
100
120
0 15 30 45 60 75 90
Rotation (mrad)
Ben
ding
Mom
ent (
kN.m
)
PC4.30.400: experimental
PC4.30.100: tri-linear
PC4.30.400: approximation
PC4.30.1000: tri-linear
upper compressive limit (N = -1000 kN)
lower compressive limit (N = -100 kN)
Figure 103 - PC4.30.400 M- curve approximation.
7.3.Results and Discussion
7.3.1.Flush Endplate Joints
Three flush endplate joint experimental M- curves, Simões da Silva et al.
(2004), were evaluated and are depicted in Figure 91 to Figure 94. They were
used to validate the proposed methodology presented in Chapter 6 as well as to
demonstrate its application.
Figure 91 illustrates an approximation for the FE8 M- curve that considers
a tensile force equal to 10% of the beam’s axial plastic resistance. This
approximation was obtained from two tri-linear M- curves, disregarding and
considering a tensile force of 20% of the beam’s axial plastic resistance. This
approximation was very close to the FE8 M- test curve, Table 37.
Figure 92 and Figure 93 present approximations for FE3 and FE4 M-
curves that respectively consider compressive forces of 4% and 8% of the beam’s
axial plastic resistance. These approximations were obtained from two tri-linear
M- curves, disregarding and considering a compressive force of 20% of the
beam’s axial plastic resistance. The approximation for FE4 M- curve, Figure 93,
was relatively close to the experimental curve, Table 37. However, for FE3 M-
curve, Figure 92, the obtained response was not as good, underestimating the joint
flexural capacity by 11%, Table 37. This was due to the behaviour of this
7 Applicably and Validity of the Proposed Alternative Methodology 162
experimental curve when compared to the others. It is possible to observe in
Figure 88 that there is an increase in the flush endplate joint moment capacity
from FE1 M- curve (N = 0% Npl) to FE7 M- curve (N = -20% Npl). However,
within this range, with a 4% beam’s compressive plastic resistance the flexural
capacity is larger than the maximum moment obtained with the 8% test.
Following this increasing tendency in the joint flexural capacity registered from
FE1 (N = 0% Npl) to FE7 (N = -20% Npl), the maximum moment obtained with
FE4 (N = -8% Npl) should be larger than FE3 (N = -4% Npl). A possible reason for
this perturbation in the experimental results might be related to problems with the
FE3 experimental test such as measuring errors or assembly eccentricities.
In general, the predictions of the M- curves using the methodology
proposed in Chapter 6 provided accurate correlations with the test curves from
Simões da Silva et al. (2004) as can be seen in Table 37.
Table 37 - Comparisons between the experimental and the proposed methodology in
terms of initial stiffness and design moment capacity for flush endplate joints.
Tests Initial Stiffness (kNm/rad) Design Moment (kNm)Appr Exp Appr/Exp % Appr Exp Appr/Exp %
FE3 (N=-4% Npl) 8097 10132 0.80 20 74 83 0.89 11FE4 (N=-8% Npl) 8147 10903 0.75 25 75 75 1.00 0
FE8 (N=+10% Npl) 4568 5403 0.85 15 64 68 0.94 6
Note: Negative percentage means overestimated value in % whilst positive percentage indicates underestimated value in %. Joint design moment was determined according to Eurocode 3:1-8 (2005), through the intersection between two straight lines, one parallel to the initial stiffness and another parallel to the moment-rotation curve post-limit stiffness.
7.3.2.Column Bases
Regarding the tests performed by Guisse et al. (1996), four baseplate
experimental M- curves were evaluated and are presented in Figure 100 to Figure
103. Figure 100 draws the prediction of PC2.15.600 M- curve for a compressive
force of 600 kN, by using two reference M- curves: PC2.15.100 and
PC2.15.1000. It is possible to note the very close approximation reached at the
evaluated points: 2/3Md, Md and 1.1Md. On the other hand, the initial stiffness was
rather erratic being estimated to be 44% (Table 38) smaller than the experimental
one. This fact occurred because the point 2/3Md, i.e. the first point of the
approximated M- curves, is located above the onset point of physical separation
of the plate and the concrete in the tensile zone. Therefore, the point 2/3Md was
7 Applicably and Validity of the Proposed Alternative Methodology 163
just able to capture the initial stiffness final change not considering the initial
stiffness before the separation of the steel plate and the concrete base.
Figure 101 presents the PC2.30.600 M- curve approximation for a
compressive force of 600 kN, by utilising the reference M- curves: PC2.30.100
and PC2.30.1000. A reasonable approximation was obtained for this M- curve,
however the initial stiffness was underestimated by 32% and the flexural capacity
was slightly under predicted by 5%, Table 38.
Figure 102 demonstrates the PC4.15.400 M- curve prediction for a
compressive force of 400 kN, by employing the base M- curves: PC4.15.100 and
PC4.15.1000. A good correlation between the experimental tests and numerical
results was obtained. Unlike the others results, the initial stiffness and the design
bending moment were over predicted by 26% and 3%, respectively.
Finally, Figure 103 presents the estimation of the PC4.30.400 M- curve for
a compressive force of 400 kN, by having as basis PC4.30.100 and PC4.30.1000
M- curves. This case did not produce an accurate prediction of the M- curve,
Table 38. However, this fact may be justified due to the occurrence of the column
end section yielding as well as column flange local plate buckling. In others
words, the column capacity was reached before achieving the baseplate joint
flexural capacity.
Table 38 - Comparisons between the experimental and the proposed methodology in
terms of initial stiffness and design moment capacity for baseplate joints.
TestsInitial Stiffness (kNm/rad) Design Moment (kNm)
Appr Exp Appr/Exp % Appr Exp Appr/Exp %
PC2.15.600 2667 4800 0.56 44 52 54 0.96 4PC2.30.600 2377 3500 0.68 32 53 56 0.95 5
PC4.15.400 3602 2857 1.26 -26 65 63 1.03 -3
PC4.30.400 4981 9091 0.55 45 85 111 0.77 23
Note: Negative percentage means overestimated value in % whilst positive percentage indicates underestimated value in %. Joint design moment was determined according to Eurocode 3:1-8 (2005), through the intersection between two straight lines, one parallel to the initial stiffness and another parallel to the moment-rotation curve post-limit stiffness.
8 Summary and Conclusion 164
8Summary and Conclusion
8.1.Generalised Mechanical Model
Based on the general principles of the component method, a generalised
mechanical model was proposed, in the present thesis, to estimate the endplate
joint behaviour when both bending moments and axial forces are present. This
mechanical model is able to deal with three basic requirements for the joint
performance: strength, stiffness and deformation capacity.
Application and validation of this mechanical model, using experimental
tests executed by Lima et al. (2004) on six extended endplate joints, was
performed and led to accurate prediction of the experiment’s key variables.
The utilization of this generalised mechanical model is simple and provides
an accurate approach to estimate the bending moment versus rotation curve for
any axial force level acting on the joint. Additionally, bending moment versus
axial force interaction diagrams can also be obtained by using the proposed
mechanical model.
The tri-linear characterisation of the joint components suggested in this
work, is shown to be capable of reasonable approximations for the moment-
rotation curve construction. However, further experimental examination and
numerical analysis using different ranges of joints to check the validity and
application of the proposed strain hardening coefficients beyond the scope of
studied joints in this work is still desirable.
The approach proposed for evaluation of lever arm d, by taking into account
the change of the joint compressive centre position according to the axial force
levels and bending moment applied to the joint, is directly responsible for a
satisfactory estimation for the joint initial stiffness, even before yielding of the
first weakest component was reached.
Parametric and sensitivity investigations demonstrate the application scope
of the proposed mechanical model. Various scenarios involving the key
8 Summary and Conclusion 165
parameters that influence on the joint structural behaviour were considered and
the main conclusions are:
- The prediction of the bending moment versus axial force interaction
diagram using the proposed mechanical model demonstrated to be in agreement
with the experimental points of the extended endplate joint tested by Lima et al.
(2004). Additionally, the mechanical model was able to capture an important
characteristic observed in the experimental tests performed by Lima et al. (2004)
where for certain compressive force levels it was possible to obtain a joint
resistance bending moment larger than that one without axial forces.
- The use of different beam profiles strongly affects the joint response under
axial forces and bending moments. The joint ultimate bending moment resistance
is reduced in alignment with a profile reduction, whilst larger profiles increase the
ultimate bending moment resistance. The joint initial stiffness is slightly reduced
by downsizing the beam profiles for compressive forces and slightly increased for
tensile forces. From the analytical moment-axial load interaction diagram, at
different beam profiles, it was observed that the joint tensile resistance is inversely
proportional to the downsizing of the beam profile. This fact occurs due to the
reduction of the lever arm defined by the distance from the load application line to
the midpoint between the first and second bolt-rows. However, this was not
identified for larger beam profile sizes, where others factors may become more
relevant than the lever arm and, consequently, the joint tensile resistance might be
larger than smaller profiles.
- The influence of the studied column profile types on the joint response was
not as pronounced, as expected, as the previous investigated beam profile cases.
The increase in the column profile sizes does not significantly affect the joint
characteristic curve. On the other hand, the use of smaller profile causes a
pronounced reduction in the joint ultimate bending moment when coupled with
increasing compressive forces. The joint initial stiffness presents a slightly
reduction by downsizing the column profiles for the compressive forces.
However, for the others cases the initial stiffness remains almost unchanged. The
analytical moment-axial load interaction diagram, at different column profiles,
depicts an increasing joint compressive resistance when column profile sizes were
increased. On the other hand, for tensile forces applied to the joint, the results
were very similar for the whole set of the investigated column profiles
8 Summary and Conclusion 166
demonstrating the small influence of the column profile variations in the joint
tensile resistance.
- The endplate thickness influence over the joint structural behaviour was
more significant, as expected, than the previously investigated cases referred to
the beam and column profile variations, causing large variations for the joint
ultimate bending moment resistance mainly for decreasing endplate thickness. It is
interesting to note that this is in line with the experimental observations depicted
by Lima (2003). The joint initial stiffness is strongly dependent on the endplate
thickness, mainly for endplate thickness smaller than the reference 15 mm
endplate thickness when tensile forces are acting on the joint. This fact was also
noted in the analytical moment-axial load interaction diagram at different endplate
thicknesses, where the joint tensile resistance was reduced with a simultaneous
decrease of the endplate thickness.
- The bolts, similar to the endplate thickness variations, had a significant
effect on the joint response, as it was again expected. However, decreasing the
bolt sizes caused a larger joint ultimate bending moment resistance variation than
when the bolt sizes were increased. The joint initial stiffness is strongly dependent
on the bolt type, similarly to the finding observed in the investigation of the
endplate thicknesses. Cases involving bolts smaller than the reference M 20 bolt,
in general, present significant variation in the joint initial stiffness, fact that did
not happen for bolts larger than the reference M 20 bolt. Associated also with the
reduction in the bolt sizes is the associated reduction of the joint tensile resistance
as presented in the analytical moment-axial load interaction diagram at different
bolt sizes.
In general, from the parametric investigations, it is possible to note that the
axial force significantly affects the joint structural behaviour. The effect of the
axial force might be more pronounced or not when coupled with variations in the
joint basic components arising from, for instance, different profile sizes, endplate
thickness and bolts. Some axial force levels may be also beneficial for the joint
ultimate bending moment as identified in the analytical bending moment versus
axial load interaction diagram for the majority of the investigated variations.
Based on the investigation, it was also possible to conclude that the positive
contribution of the axial force in the maximum joint ultimate bending moment
resistances was more significant with a joint stiffness decrease. The joints that had
8 Summary and Conclusion 167
their dimensions reduced when compared to the joint reference dimensions
presented a beneficial contribution in terms of the maximum ultimate bending
moment. On the other hand, for upper joint dimensions the maximum ultimate
bending moments was reached without axial forces.
First order approximations for the trigonometric expressions were used
throughout the generalised mechanical model formulation. Figure 104 presents the
error due to these approximations versus joint rotations. According to Nethercot &
Zandonini (1989), rotations beyond 0.05 radians had little practical significance.
Based on this, this 0.05-radian rotation was adopted for the joint final rotation. For
this rotation value it is possible to observe in Figure 104 an error of 0.0021%.
This indicated that the developed equations in this work were accurate for the
usual problems involving beam-to-column joints.
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
0.00 0.01 0.02 0.03 0.04 0.05
Err
or =
abs
(sin
(q)-
q)x1
00 (
%)
q (rad)
Figure 104 - First-order approximations error magnitudes versus joint rotation.
8.2.Alternative Methodology
A consistent and alternative methodology to determine any moment versus
rotation curve from experimental tests, including the axial versus bending moment
interaction, was also presented. This method extends the application range of
available data so as to produce moment-rotation characteristics that implicitly
make proper allowance for the presence of significant levels of either tension or
compression at the beam.
This methodology can also be applied to results obtained analytically,
empirically, mechanically, and numerically. Due to its simplicity and to the fact
that its basis is M- curves that already consider the moment versus axial force
8 Summary and Conclusion 168
interaction, it can be easily incorporated into a nonlinear semi-rigid joint finite
element formulation. It is also important to observe that the use of the proposed
methodology does not change the basic formulation of the non-linear joint finite
element, only requiring a rotational stiffness update procedure.
This proposed method is a simple and accurate way of introducing semi-
rigid joint experimental test data into structural analysis, through M- curves.
Application and validation of the proposed methodology to obtain M-
curves, for different axial force levels, were performed against experimental tests
executed by Simões da Silva et al. (2004) and Guisse et al. (1996) on eight flush
endplate and on twelve column base joints, respectively.
Finally, it may be suggested that an alternative, though accurate, method to
determine M- curves for endplate and baseplate joints, considering the bending
moment versus axial force interactions, can be made with a simple linear
interpolation between two reference M- curves providing a straightforward
procedure to obtain M- curves for any axial force level.
8.3.Design Considerations
Using the Eurocode 3:1-8 (2005) component method, it is possible to
evaluate the rotational stiffness and moment capacity of semi-rigid joints when
subject to pure bending. However, this component method is not yet able to
calculate these properties when, in addition to the applied moment, an axial force
is also present.
Eurocode 3:1-8 (2005) suggests that the axial load may be disregarded in
the analysis when its value is less than 5% of the beam’s design axial plastic
resistance (Npl,Rd), but provides no information for cases involving larger axial
forces. However, if the applied axial force exceeds the 5% limit, a conservative
approach may be used:
0.1,
,
,
, Rdj
Edj
Rdj
Edj
N
N
M
M(8.1)
where Mj,Ed is the design value of the joint internal moment, Mj,Rd is the joint
moment design resistance, Nj,Ed is the design value of the joint internal axial force
and Nj,Rd is the joint axial force design resistance.
8 Summary and Conclusion 169
Aiming to overcome this limitation in this existing code related to the
component method and based on the results obtained in this work the following
design considerations are suggested, as an extension of the current procedures of
Eurocode 3:1-8 (2005) accounting for the full interaction of the bending moment
and axial forces:
- Rotational stiffness: the generalised mechanical model, developed in this
word, is suggested to estimate the rotational joint stiffness, considering the
influence of the interaction between bending moment and axial loading. The
bending moment versus rotation curve can be readily predicted by evaluating
three main points of the moment-rotation curve: the first point (y, My) defines the
joint initial stiffness corresponding to the attainment of the weakest component
yield while the second point (u, Mu) is obtained when the weakest component
reaches its ultimate strength. The third point (f, Mf) depends on the joint assumed
final rotational capacity for the moment-rotation curve, which is adopted to be
equal to 0.05 radians.
- Strength interaction: the proposed mechanical model can be
straightforwardly used to build bending moment versus axial force interaction
diagrams, where the proposed analytical model is subjected to different levels of
axial load. This is followed by increasing bending moment until the joint ultimate
capacity is reached.
- Deformation capacity: the joint deformation capacity is controlled by the
ductility of its constituent components. In this way a tri-linear characterisation of
the joint basic components is suggested in this thesis.
The bending moment versus axial load interaction diagram, constructed by
using the ideas development in this work, can be used to determine the joint
resistance subjected to any combination of bending moments and axial loads,
supplying an efficient and complete tool for structural joint designs.
8.4.Main Contributions and Developments of the Present Investigation
This section summarises the main contributions and developments of the
present investigation:
- A generalised component-based mechanical model was proposed to
estimate the endplate joint behaviour when both bending moments and axial
8 Summary and Conclusion 170
forces are present. It must be underlined the simplicity of the mechanical model
utilization, given by analytical equations developed in this thesis, and its accurate
prediction of the moment-rotation curves and moment-axial load interaction
diagrams. However, the most important and unprecedented contribution could be
related to the ability that this model has in representing the changes of the joint
compressive centre position according to the axial load levels and bending
moments applied to the joint.
- A tri-linear characterisation of the joint basic components was suggested in
this work, highlighting the novelty of the strain hardening coefficients proposed
for endplate joints that are used to estimate the plastic and ultimate stiffness of the
joint basic components.
- The use of the proposed component-based mechanical model as an
extension of the current procedures of Eurocode 3:1-8 (2005) accounting for the
full interaction of the bending moment and axial forces and dealing with three
basic requirements for the joint performance: strength interaction, stiffness and
deformation capacity.
- A consistent and alternative methodology to determine any moment versus
rotation curve from experimental tests or results obtained analytically,
empirically, mechanically and numerically, including the moment-axial load
interaction, was also presented. From this alternative methodology it may be
underlined its straightforward implementation into nonlinear semi-rigid joint finite
element formulation. However, the most important observation referred to this
alternative methodology is that the prediction of M- curves for endplate and
baseplate joints, considering the bending moment versus axial force interactions,
can be made with a simple linear interpolation between two reference M- curves.
8.5.Future Research Recommendations
This research work has focused on the development of a component-based
mechanical model to describe the beam-to-column joint behaviour including the
full interaction of the bending moment and axial forces. This model is based on a
general idea that permits the model to represent any kind of joint. Moreover, this
model offers practical improvements over current procedure of Eurocode 3:1-8
(2005), because it considers the influence of the axial force effect in the joint
8 Summary and Conclusion 171
behaviour and allows modifications of the compressive centre position even
before reaching the first component yield, i.e. in the linear-elastic regime,
enabling accurate predictions of the moment-rotation curve.
The research topics that have been identified in the process of developing
and applying the proposed mechanical model include the following issues:
- Tri-linear characterisation of the joint components: further experimental
examination and numerical analysis using different ranges of joints to
check the validity and application of the proposed strain hardening
coefficients is still desirable.
- Composite joints: a mechanical model for composite joints may be
formulated from the proposed mechanical model by accounting for the
contribution of the reinforcing bars. A row of reinforcing bars in tension
might be similarly treated as a bolt-row in tension in a steel joint while
the interaction slab-connectors-beam could be considered by adding a
new vertical spring described by the force-displacement characteristic of
this slab-connectors-beam system.
- Lever arm position: further investigation about the lever arm d, which
considers the change of the joint compressive centre position according
to the axial force levels and bending moment applied to the joint. It
would be enviable to aim on determining a single equation for both
tensile and compressive forces and also to prove mathematically if the
suggested lever arm position evaluation accurately represents the
variations in the joint compressive centre position as a function of the
joint loads.
- Experimental investigations: few experiments considering the
interaction bending moment and axial force have been reported in the
literature. Additionally, the available experiments are associated with a
small number of axial force levels and associated bending moment
versus rotation curves. There is, therefore, the need of further tests
associated with various axial force magnitudes and different joint
layouts.
In conclusion, although there is clearly scope for further improvements, it is
believed that the proposed mechanical model offers an effective tool for
assessment of structural joints, considering the axial-moment interaction.
References 172
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