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

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PUC-Rio - Certificação Digital Nº 0410736/CA

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

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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.

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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.

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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.

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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

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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.

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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

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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.

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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

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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


3.2. 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

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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


4.2. 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


5.2. 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


5.6. Column Profile Investigations 125


5.7. Endplate Thickness Investigations 130


5.8. Bolt Investigations 136


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


6.2. General Concepts of the Correction Factor 143

6.3. Extension of the Correction Factors for Both Bending Moment and

Rotation Axes 144

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6.4. An alternative methodology 146

7 Applicably and Validity of the Proposed Alternative Methodology 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

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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

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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



104



105

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105



106



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



110



111



111



112



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

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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 58 - Investigated EE6 (N = -10% Npl = -127.20 kN) moment-rotation




Figure 60 - Analytical moment-axial load interaction diagram at different beam

profiles. 123


column profile variations. 126


curves involving the column profile variations. 127









Figure 67 - Analytical moment-axial load interaction diagram at different column

profiles. 129


endplate thickness variations. 131

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curves involving the endplate thickness variations. 132









Figure 74 - Analytical moment-axial load interaction diagram at different endplate

thicknesses. 134


bolt variations. 136


curves involving the bolt variations. 137









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

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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


curves. 157


curves. 158


curves. 158

Figure 100 - PC2.15.600 M- curve approximation. 159




Figure 104 - First-order approximations error magnitudes versus joint rotation.167

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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

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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 25 - Evaluated ultimate bending moments at different column profiles. 130

Table 26 - Investigated endplate thicknesses and their dimensions. 131



Table 28 - Evaluated ultimate bending moments at different endplate thicknesses.

135

Table 29 - Investigated grade 10.9 bolts and their main dimensions. 136



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

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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

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)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

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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



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

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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



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

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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

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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

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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

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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

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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)


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


reduction factor for plate buckling; stiffness coefficient (Eq. 3.18)


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

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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


1 stiffness coefficient (Eq. 3.23)


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



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




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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.

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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,

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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.

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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

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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


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

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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


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.

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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,

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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

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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

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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

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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.

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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.

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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

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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:

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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

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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).

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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.

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(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:

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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:

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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.

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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.

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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.

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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.

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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

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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.

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Figure 14 - Definitions of e, emin, rc and m.

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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:

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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.

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Figure 15 - Modelling an extended endplate as separate T-stubs.

Table 6 - Effective lengths for an endplate.

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Figure 16 – Values of for stiffened column flanges and endplates.

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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)

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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.

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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,

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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

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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.

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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.

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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.

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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

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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

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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.

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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).

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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

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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.

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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.

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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

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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

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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.

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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


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


Urbonas & Daniunas (2006)

Endplate bolted beam-to-beam joints


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

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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.

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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.

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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

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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.

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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

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(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:

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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.

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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)

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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.

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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

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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)

DBD



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)

DBD



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

DBD



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

DBD



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:

DBD



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.

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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

DBD



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

DBD



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).

DBD



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

DBD



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.

DBD



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

DBD



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,

DBD



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.

DBD



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



DBD



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




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




DBD



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



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




DBD



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

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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.

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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).


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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



DBD



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


Beam top and bottom flanges in compression

Beam flanges in compression and first bolt-row in tension



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






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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






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


All the bolt-rows in tension



DBD



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


All the bolt-rows in tension

Figure 51 - Comparison between experimental EE7 moment-rotation curves (Lima et al.,


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.

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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.

DBD



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.

DBD



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.

DBD



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.

DBD



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







120.0 BR 1: epb120.0 BR 2: cfb


114.0 BR 1: epb114.0 BR 2: cfb


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

DBD



-192.8 0.0 BR 3: bfwc 0.0 BR 3: bfwc88.0 BBF: bfwc89.0 BR 1: epb89.0 BR 2: cfb


82.0 BR 1: epb82.0 BBF: bfwc82.0 BR 2: cfb


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






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);

DBD



- 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

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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.

DBD



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



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



DBD



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


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



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.

DBD




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.

DBD



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.

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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


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



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.

DBD



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


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



DBD



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



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



DBD



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



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.


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

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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).


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


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.

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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


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



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.

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0

20

40

60

80

100

120

140

160

180

200

0 10 20 30 40 50Rotation (mrad)

Ben

ding

Mom

ent (

kN.m

)



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

)




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0

20

40

60

80

100

120

140

160

180

200

0 10 20 30 40 50Rotation (mrad)

Ben

ding

Mom

ent (

kN.m

)




0

20

40

60

80

100

120

140

160

180

200

0 10 20 30 40 50Rotation (mrad)

Ben

ding

Mom

ent (

kN.m

)




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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



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.


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

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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).


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


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.

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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


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.

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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


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



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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



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



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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



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.


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,

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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).


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


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.

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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

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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.

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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

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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.

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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.

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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

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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)

DBD



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).

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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.

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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).


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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.

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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.

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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

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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


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Figure 93 - FE4 M- curve approximation, considering a compressive force of 8% of the


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.

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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.

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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

)





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0

20

40

60

80

100

120

0 15 30 45 60 75 90

Rotation (mrad)

Ben

ding

Mom

ent (

kN.m

)





0

20

40

60

80

100

120

0 15 30 45 60 75 90

Rotation (mrad)

Ben

ding

Mom

ent (

kN.m

)





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.

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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




Figure 100 - PC2.15.600 M- curve approximation.

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0

10

20

30

40

50

60

70

80

0 15 30 45 60 75 90

Rotation (mrad)

Ben

ding

Mom

ent (

kN.m

)








0

20

40

60

80

100

120

0 15 30 45 60 75 90

Rotation (mrad)

Ben

ding

Mom

ent (

kN.m

)








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0

20

40

60

80

100

120

0 15 30 45 60 75 90

Rotation (mrad)

Ben

ding

Mom

ent (

kN.m

)








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

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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

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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.

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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

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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

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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

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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

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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.

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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

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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

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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.

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References 172

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alexandre almeida del savio a component method … forças axiais de compressão ou tração serão...

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