review on the modelling of joint behaviour in steel frames

Journal of Constructional Steel Research 67 (2011) 741–758

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Journal of Constructional Steel Research

journal homepage: www.elsevier.com/locate/jcsr

Review

Review on the modelling of joint behaviour in steel framesConcepción Díaz a, Pascual Martí a, Mariano Victoria a, Osvaldo M. Querin b,∗

a Department of Structures and Construction, Technical University of Cartagena, Campus Muralla del Mar, 30202 Cartagena (Murcia), Spainb School of Mechanical Engineering, University of Leeds, Leeds LS2 9JT, United Kingdom

a r t i c l e i n f o a b s t r a c t

Article history:Received 12 August 2010Accepted 24 December 2010

Keywords:Steel joint analysisSemi-rigid jointsJoint behaviour representationMoment–rotation curveBeam-to-column joints

Steel portal frames were traditionally designed assuming that beam-to-column joints are ideally pinnedor fully rigid. This simplifies the analysis and structural design processes, but at the expense of notobtaining a detailed understanding of the behaviour of the joints, which in reality, have finite stiffnessand are therefore semi-rigid. The last century saw the evolution of analysis methods of semi-rigid joints,from the slope-deflection equation and moment distribution methods, to matrix stiffness methods and,at present, to iterative methods coupling the global and joint structural analyses. Studies agree thatin frame analysis, joint rotational behaviour should be considered. This is usually done by using themoment–rotation curve. Models such as analytical, empirical, experimental, informational, mechanicaland numerical can be used to determine joint mechanical behaviour. The most popular is the mechanicalmodel, with several variances (e.g. Component Method). A summary is given of the advantages anddisadvantages and principal characteristics of each model. Joint behaviour must be modelled whenanalysing semi-rigid frames, which is associated with a mathematical model of the moment–rotationcurve. Depending on the type of structural analysis required, any moment–rotation curve representationcan be used; these include linear, bilinear, multilinear and nonlinear representations. The most accuraterepresentation uses continuous nonlinear functions, although themultilinear representation is commonlyused for mechanical models. This article reviews three areas of steel joint research: (1) analysis methodsof semi-rigid joints; (2) prediction methods for the mechanical behaviour of joints; (3) mathematicalrepresentations of the moment–rotation curve.

© 2011 Elsevier Ltd. All rights reserved.

Contents

1. Introduction........................................................................................................................................................................................................................7432. Analysis methods of semi-rigid joints ..............................................................................................................................................................................7433. Methods for modelling the rotational behaviour of joints ..............................................................................................................................................744

3.1. Experimental testing .............................................................................................................................................................................................7443.2. Empirical models ...................................................................................................................................................................................................745

3.2.1. Frye and Morris model ...........................................................................................................................................................................7453.2.2. Krishnamurthy model ............................................................................................................................................................................7453.2.3. Kukreti model .........................................................................................................................................................................................7463.2.4. Attiogbe and Morris model ....................................................................................................................................................................7463.2.5. Faella, Piluso and Rizzano model...........................................................................................................................................................746

3.3. Analytical models ..................................................................................................................................................................................................7463.3.1. Chen et al. model ....................................................................................................................................................................................7463.3.2. Yee and Melchers model ........................................................................................................................................................................747

3.4. Mechanical models ................................................................................................................................................................................................7473.5. Numerical models..................................................................................................................................................................................................7483.6. Informational models ............................................................................................................................................................................................750

4. Mathematical representation of moment–rotation curve ..............................................................................................................................................751

∗ Corresponding author. Tel.: +44 1133432218, +44 7712532215 (mobile); fax: +44 1133432150.E-mail addresses: [email protected] (C. Díaz), [email protected] (P. Martí), [email protected] (M. Victoria), [email protected] (O.M. Querin).

0143-974X/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.jcsr.2010.12.014

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http://dx.doi.org/10.1016/j.jcsr.2010.12.014

742 C. Díaz et al. / Journal of Constructional Steel Research 67 (2011) 741–758

4.1. Stiffness, resistance and shape factor-based formulations .................................................................................................................................7524.1.1. Linear model ...........................................................................................................................................................................................7524.1.2. Bilinear model.........................................................................................................................................................................................7524.1.3. Multilinear model ...................................................................................................................................................................................7524.1.4. Nonlinear model .....................................................................................................................................................................................752

4.2. Formulation based on curve fitting by regression analysis.................................................................................................................................7545. Conclusions.........................................................................................................................................................................................................................755

Acknowledgements............................................................................................................................................................................................................755References...........................................................................................................................................................................................................................755

Nomenclature

bep end-plate widthbfc beam flange and web in compressionbj least-squares coefficientsbt bolts in tensionbwt beam web in tensioncfb column flange in bendingcwc column web in compressioncws column web in shearcwt column web in tensiond1 distance between the middle lines of the legs

adjacent to the beam flangesd3 distance between the centre of the web angles and

the middle line of the seat angle leg adjacent to thebeam flange

db bolt diameterdg vertical distance between the bolt centrelinesepb end-plate in bendingfy yield stress of the base materialfyb bolt yield stressg gauge of column flange boltsg1 distance between the nut edge and the middle line

of the top angle leg adjacent to the beamg3 distance between the nut edge and the middle line

of the web angle leg adjacent to the beamgb gauge distance between the two bolts in a rowhb beam heightIta inertia moment of the leg adjacent to the column

face of the top angleIwa inertia moment of the leg adjacent to the column

face of the web anglekt distance between the heel of the top angle and the

toe of the filletla angle lengthm number of curve-fitting constants (Eq. (28)); num-

ber of points between two elementary parts of theM–φ curve (Eq. (27))

n shape factor which characterizes the knee of themoment–rotation; shape parameter determined us-ing the method of least squares for differences be-tween the predictedmoments and the experimentaltest data (Eq. (30))

nb number of bolts per angle leg on column flangepf pitch of the bolt (distance from top of the flange to

the centreline of the bolt)ta angle thicknesstep end-plate thicknesstf column flange thicknesstfb beam flange thicknessti angle thicknesstp end-plate thicknesstsa seat angle thickness

tta top angle thicknesstwa web angle thicknesstwb beam web thickness

Capital letters

Ab gross cross-sectional area of the boltC regression parameterC1,2,3 curve-fitting constantsCj curve-fitting parameter obtained from linear re-

gression (Eq. (28)); modelling parameters obtainedby linear regression analysis (Eq. (29))

Dk curve-fitting parameter obtained from linear re-gression

E modulus of elasticityH[φ] Heaviside’s step functionK parameter depending on the geometrical and me-

chanical properties of the structural detail (Eq. (1));rotational stiffness

Kφ joint rotational stiffnessKf joint rotational stiffnessKφ,p joint plastic rotational stiffness; strain-hardening

connection stiffness (Eq. (28))Kφ,y post-yielding rotational stiffnessLsa length of the seat angleLta length of the top angleLwa length of the web angleM joint momentM0 reference bending momentMi initial momentMj joint moment; upper bound moment of the jth part

of th curve (Eq. (27))Mj,p joint plastic momentMj,Rd joint moment resistanceMj,y yielding momentPi joint geometric parameterWb beam section modulus

Greek letters

αi coefficients obtained in such a way as to give a goodfit to the curve (Eq. (3))

α regression parameter (Eq. (25)); scaling factor fornumerical stability (Eq. (28)); shape parameterdetermined using the method of least squares fordifferences between the predictedmoments and theexperimental test data (Eq. (30))

φi initial rotationφk starting rotation of the kth linear component of the

M–φ curveφ joint rotationφ0 joint permanent rotationφCd joint rotational capacity

C. Díaz et al. / Journal of Constructional Steel Research 67 (2011) 741–758 743

1. Introduction

Steel portal frames were traditionally designed assuming thatbeam-to-column joints are ideally pinned or fully rigid. The useof the ideally pinned condition implies that no moment can betransmitted between the beam and the column; this means thatthe connections have no rotational stiffness and cannot transmitmoments although they do transmit axial and shear forces to theattached members (Fig. 1(a)). On the other hand, fully rigid jointshave rotational compatibility and therefore transmit all form ofloads between beam and column (Fig. 1(b)). An important aspect ofthe analysis of these joints is that their behaviour is decoupled fromthe analysis of the structure. Although this simplifies the analysisand structural design processes; it comes at the expense of notbeing able to obtain a detailed understanding of the behaviour ofthe joint. In reality, joints have finite stiffness and are thereforesemi-rigid (Fig. 1(c)). In the last century, analysis methods ofsemi-rigid joints evolved considerably to obtain the true structuralresponse. Starting in the 1930’s with the slope-deflection equationand moment distribution methods, the 1960’s with the matrixstiffness methods, and nowadays, with complex iterative analysismethodswhich couple the structural analysiswith that of the joint.

The true behaviour of a joint can be incorporated within theglobal analysis of the structure by using the moment–rotationcurve (Mj–φ), (Fig. 2). This is achieved by determining themechanical properties of the joint in terms of its rotational stiffness(Kj), moment resistance (Mj,Rd), and rotational capacity (φCd),starting from their geometrical and mechanical properties.

There are several models which can be used to determine themechanical behaviour of joints, these are: analytical, empirical,experimental, informational, mechanical and numerical. The mostpopular of these is the mechanical model, which has severalvariances, the most popular being the Component Method,Eurocode 3 [1]. This method considers a joint as a set of‘‘individual basic components’’, which allows the determinationof the moment resistance and stiffness characteristics of all thedifferent components of the joint.

These joint behaviour models need to be incorporated intostructural analysis packages in order to then be able to analyseand design the joint. To achieve this, mathematical expressionsare requiredwhich allow for the rotational deformation, rotationalstiffness and design moment resistance to be easily incorporatedinto the global structural analysis.

This article provides a state-of-the-art review of three areasof steel joint research: (1) analysis method of semi-rigid joints;(2) predictionmethods for themechanical behaviour of joints; and(3) mathematical representations of the moment–rotation curve.

2. Analysis methods of semi-rigid joints

The first studies on semi-rigid joints were carried out in 1917,when Wilson and Moore [2] investigated the stiffness of rivetedjoints in steel structures. But it was not until the 1930’s thatstudies began into the relationship between the moment androtation of semi-rigid joints and their overall effect on steelstructures. These can be seen in the reports of The Steel StructuresResearch Committee [3–5] (UK), Young and Jackson [6] (Canada)and Rathbun [7] (USA). Since then, there have been numerousexperimental and theoretical studies into the behaviour of semi-rigid steel joints (riveted, bolted and welded) and their effect onthe overall structure.

Batho andRowan [4] proposed a graphicalmethod, called beam-line, which was used to determine the end restraint providedby each joint. To apply this method, requires the use of theexperimentally calculated moment–rotation curve. Baker [4] andRathbun [7], were the first to apply the slope-deflection [8] and

the moment distribution [9] methods to the analysis of semi-rigidjoints.

Between 1936 and 1950, most of the research was focusedon the application of these methods to the analysis of structureswith semi-rigid joints. The most notable publications are those ofBaker andWilliams [5], Johnston andMount [10], Stewart [11] andSourochnikoff [12].

By the 1960s, the matrix stiffness method of structural analysisutilising computers had been established. Monforton and Wu [13]were the first to incorporate the effects of semi-rigid connectionsinto the matrix stiffness method in 1963. This was achieved bymodifying the beam stiffness matrices to take the semi-rigidconnection effects into account in the frame analysis. Similarprocedures were also proposed by Livesley [14], and Gere andWeaver [15], at about the same time. In these analysis methods,a linear Mj–φ relationship was assumed and the linear semi-rigidconnection factor Z = φ/M was used to modify the beam stiffnessmatrices [16].

The dynamic behaviour of semi-rigid frames was investigatedby Lionberger and Weaver [17] in 1969 and by Suko andAdams [18] in 1971. In these analyses the connection elasto-plasticbehaviour was modelled by equivalent springs.

In 1978, the European Convention for Constructional Steelwork(ECCS) published Report 23 on the European recommendationsfor steel construction [19]. This report formed the basis of thecurrent Eurocode 3. These recommendations replaced the methodof allowable stresses by the limit state method, which is based onprobabilistic concepts of safety and the use of enhancement ‘load’factor for the analysis of structural resistance and stability insteadof the traditional reference to allowable stresses.

In 1981, Moncarz and Gerstle [20] proposed a new approxima-tion to the analysis of semi-rigid frames based on modification ofthe basic matrix stiffness technique.

Based on the studies of the ECCS, in 1984 the Commission of theEuropean Community published the first version of the Eurocode3 [21]. In this document, the joints are classified as rigid and semi-rigid for elastic linear analysis and with full- or partial-strengthfor elastic–plastic analysis. However, they neither consider theiruse, nor how to model them. The code was published on a trialbasis (European Pre-Standard, ENV) inviting comments from itsusers as well as professional, scientific, standards and technicalorganisations. Their comments and suggestions were used todevelop the final code (European Standard, EN). In 1989 this workwas transferred to the European Committee for Standardization(CEN).

In 1983, Jones et al. [22] presented a revised review of theanalysis of frames with semi-rigid joints. This work was extendedby Nethercot in [23,24], where he proposed different approachesand improvements for the analysis of semi-rigid frame by adoptingthe basic matrix stiffness technique.

In 1987, Lui and Chen [25], and Goto and Chen [26], proposedmethods for the analysis of semi-rigid frame based on matrixstiffness analysis, using small computers. On the same year,the ECCS [27] created the Working Group TWG 8.2, to studythe influence of semi-rigid connections on the overall framebehaviour. The results of this study helped to establish theTechnical Committee for Structural Connections (TC10) of the ECCSto look at the behaviour of connections.

The Eurocode has evolved [28], and finally in May 2005, theEurocode 3 [1] was published. It was exclusively dedicated to alltypes of joints, including semi-rigid ones, where the response ofa joint is dependent on the geometric and mechanical propertiesof its components, using the component method. This code ofpractice is a collection of decades of research in steel structures.Other international codes of practice which also consider jointbehaviour are those of the USA in AISC-ASD [29], LRFD [30], AISC-ASD/LRFD [31] and China in GB [32].


a b c

Fig. 1. Joint types according to their behaviour, where φ is the angular rotation between the beam and the column: (a) pinned; (b) rigid; and (c) semi-rigid.

There is currently a great range of studies of steel frameswith semi-rigid connections: Jaspart [33], Jaspart andMaquoi [34],Weynand et al. [35], Chen [36], Braham and Jaspart [37], Ashrafet al. [38], Cabrero and Bayo [39], Bayo et al. [40], da S. Vellascoet al. [41], Ashraf et al. [42], Yang and Lee [43], Faella et al. [44],da Silva et al. [45], Daniunas and Urbonas [46], Sekulovic andNefovska-Danilovic [47], Bel Hadj Ali et al. [48], Ihaddoudèneet al. [49], Mehrabian et al. [50], Darío [51], etc. These studies wereconcernedwith twoprincipal themes [52]: (1) the evaluation of themechanical properties of the joints in terms of rotational stiffness,moment resistance and rotation capacity, and (2) the analysis anddesign procedures for frames including rotational joint behaviour.

All studies agree that when carrying out structural analysisof any frame, the rotational behaviour of the joints mustbe considered. It is evident that the prediction of the jointbehaviour by means of one of the above methods has to begenerally accompanied by a mathematical representation of themoment–rotation curve, which is necessary to be used as inputdata in computer programs for the structural analysis of semi-rigid frames. In the next section, all methods for the predictionof the joint rotational behaviour as well as their mathematicalrepresentation will be explained.

3. Methods for modelling the rotational behaviour of joints

To properly model the beam-to-column joint behaviour, themoment–rotation curve for the joints is required. Fortunatelythere are many models which can be used to predict it. The mostcommonly usedmodels are includedhere, grouped into: analytical,empirical, experimental, mechanical, numerical and informationalmodels. The last of which is the most recent. Other classificationscan be found in the work of Nethercot and Zandonini [53], Faellaet al. [54] and Jaspart [52].

3.1. Experimental testing

Themost accurate knowledge of the joint behaviour is obtainedthrough experimental tests, but this technique is too expensivefor everyday design practice and is usually reserved for researchpurposes only [54].

In 1917, Wilson and Moore [2] performed the first experimentto assess the rigidity of steel frame connections. Since then,experimental testing has been continued.

Prior to 1950, most connection tests were focused on rivetedjoints: Batho [3]; Batho and Rowan [4]; Batho and Lash [5]; Youngand Jackson [6]; Rathbun [7]. After 1950, high strength bolts wereused extensively in steel construction.

A large number of tests were made and reported, allowing forthe generation of several data banks. The information requiredfrom each test usually includes: the geometric and mechanical

properties of each component which makes up the joint, themoment–rotation curve, the rotational stiffness (Kj) and momentresistance (Mj,Rd) as well as the name of the researchers.

The four most important data banks are:

1. Goverdhan data bank. The first one to be developed, in1984 [55], has the results of 230 tests from the USA carriedout between 1950 and 1983. It includes tests on the followingconnection typologies: double web angle connections, singleweb angle/plate connections, header-plate connections, end-plate connections and top and seat angle connections with orwithout web angles.

2. Nethercot data bank. The first European data bank on steelconnections was developed in 1985. Nethercot [56,57] exam-ined more than 70 experimental studies collecting more than700 individual tests by other researchers [58]. The connectiontypologies include those examined by Goverdhan as well asT-stub connections with and without web angles.

3. Steel connection data bank. In the USA, the work of Goverd-han [55] was followed by that of Kishi and Chen [59,60] whoprepared a data bank collecting experimental tests from all overthe world carried out from 1936 to 1986. They compiled re-sults from over 303 tests. In addition, they developed the SteelConnection Data Bank (SCDB) program for the recovery ofall the experimental data and the formulation of mathe-matical relationships for the curve fitting of experimentalmoment–rotation behaviour [61,62]. In 1995, Abdalla andChen [63] added the results of 46 additional experimentaltests of steel beam-to-column joints. The tests collected in theprogram SCDB are contained, according to the following con-nection typologies: single angle web1 cleat/plate connections,double angle web cleat connections, top and seat angle cleatsconnections with or without web angles, extended and flushend-plate connections and header-plate connections.

4. SERICON data bank. Developed by Arbed Recherches [64] andAachenUniversity [65], includes only European test results [66].It also contains tests from single joint components and tests oncomposite connections. This data bank was extended into theSERICON II database by Cruz et al. [67].

The use a data bank is mainly devoted to the validation of models,aimed at the prediction of the joint behaviour from its geometricaland mechanical properties, rather than to daily design practice. Infact, the designer has only a low probability of finding in the databank the specific structural detail of the joint studied, due to thegreat variety of connection typologies, geometrical properties andstiffening details of panel zone [54].

1 Also referred to as ‘‘web angle’’.


Fig. 2. Moment–rotation (Mj–φ) curve.

Other experiments include the work of Popov and Takhi-rov [68], who carried out two tests on bolted large seismic steelbeam-to-column joints. Girão et al. [69] evaluated 8 tests to assessthe ductility of extended end-plate connections. Girão et al. [70]carried out 32 tests on bolted T-stub connections made up ofwelded plates. Girão and Bijlaard [71] carried out experimentsto study the behaviour of high strength steel end-plate connec-tions and in [72] the experimental behaviour of high strength steelweb shear panels. Cabrero and Bayo [73], analysed the semi-rigidbehaviour of three-dimensional steel beam-to-column joints sub-jected to proportional loading. Shi et al. [74] carried out 5 ex-periments of beam-to-column bolted extended end-plate joints todevelop an analytical model to obtain the rotational stiffness andthe moment–rotation curve of a joint. Piluso and Rizzano [75] didan experimental analysis and modelling of bolted T-stubs undercyclic loads.

3.2. Empirical models

Empirical models are based on empirical formulations whichrelate the parameters of the mathematical representation ofthe moment–rotation curve to the geometrical and mechanicalproperties of beam-to-column joints. These formulations can beobtained using regression analyses of data which can be derived indifferent ways such as: experimental testing, parametric analysesdeveloped by means of Finite Element (FE) models, analyticalmodels or mechanical models.

The main disadvantage of this type of model is that it is onlyapplicable to joints whose characteristics match those used togenerate the model. It is also not possible to determine how eachparameter of the joint affects its overall performance. Five commonmodels are described next.

3.2.1. Frye and Morris modelThe Frye and Morris model [76] is based on an odd-power

polynomial representation of the moment–rotation curve, Eq. (1).

φ = C1 (KM) + C2 (KM)3 + C3 (KM)5 (1)

where K is a parameter depending on the geometrical andmechanical properties of the structural detail, and C1, C2 and C3 arecurve-fitting constants. For example, for the end-plate connectionswithout column stiffeners of Fig. 3, the curve-fitting constants aregiven by Eq. (2).

C1 = 8.91 × 10−1; C2 = 1.20 × 104

;

C3 = 1.75 × 108; K = d−2.4

g t−0.4p t−1.5

f .(2)

The main drawback of this formulation is that, in some cases,the slope of the moment–rotation curve can become negative forsome values of M [77]. This is physically unrealistic and can cause

Fig. 3. Geometrical parameters for the Frye–Morris polynomial representation ofend-plate connections without column stiffeners.

Fig. 4. Extended end-plate connections with four bolts in the tension zone for theKrishnamurthy model [88].

numerical difficulties in the analysis of semi-rigid frames using thetangent stiffness formulation. To solve this problem, Azizinaminiet al. [78] proposed a different formulation of the parameter K ,Eq. (3).

K = Pα11 Pα2

2 · · · Pαnn (3)

where Pi are geometric parameters of the joint and the αi are thecoefficients obtained to give a good fit to the curve.

This model was used in several studies to investigate the effectof semi-rigid joints on steel frame structures: Picard et al. [79];Altman et al. [80]; Goverdhan [55]; Kameshki and Saka [81];Hadianfard and Razani, [82]; Hayalioglu and Degertekin [83];Prabha et al. [84].

3.2.2. Krishnamurthy modelKrishnamurthy [85,86] carried out a wide parametric study by

means of the FEMethod (FEM) to study the rotational behaviour ofend-plate connections. Experimental tests, limited to 5 prototypes,were used to adjust some of the parameters of the model andconfirm the numerical results.

The two-dimensional (2D) plane stress numerical model wasfor a plane parallel to the beam web. Five experiments were usedto correlate this model [87]. This method was further developed


Fig. 5. Structural detail of a flush end-plate connection analysed [89].

to the case of extended end-plate connections with four boltsin the tension zone (Fig. 4), leading to the development of anempirical model based on the simple power representation of themoment–rotation curve [88], Eq. (4).

φ = CMα (4)

α = 1.58; C =1.4βµp2.03f

A0.36b t1.38ep

; β =0.0056 b0.61ep t1.03fb

h1.30b t0.26wb W 1.58

b;

µ =1.0

f 0.38y f 1.20yb

where Wb is the beam section modulus, fy is the yield stress of thebase material, fyb is the bolt yield stress and Ab is the gross cross-sectional area of the bolt.

These parameters are independent of the geometry of thecolumn as this was considered in the FEmodel. For this reason, themoment–rotation curve is for the connection and not the joint.

3.2.3. Kukreti modelKukreti extended themethod of Krishnamurthy by carrying out

a new parametric study of flush end-plate connections withoutcolumn stiffeners (Fig. 5).

Kukreti et al. [89] also used the FEM to obtain the power modelof Eq. (5).

φ = CMα (5)

α = 1.58; C =359 × 10−6p2.227f

h2.616b t0.501wb t0.038fb d0.849b g0.519

b b0.218ep t1.539ep

where the lengths are in inches and the moments in kip-ft.This method was later applied to a study of the extended end-

plate connection where eight bolts are located in the tensile zoneand the end-plate is stiffened by means of a reinforcing rib [90].

Empirical models, based on the power of the moment–rotationcurve, are able to accurately predict the initial rotational behaviourof the connection, rather than the whole moment–rotationcurve. There is significant scatter between the predicted andexperimental moment–rotation curves for high values of plasticdeformations [91].

3.2.4. Attiogbe and Morris modelAttiogbe and Morris [92] proposed a new model based on lab-

oratory experimental results and the mathematical representa-tion of Goldberg and Richard [93], to predict the moment–rotationcurve for double web angle connections. This model requires four

parameters (ϕ0, M0, n, Kϕ,p) which are related to the geometricalproperties of a connection, Eq. (6).

ϕ0 =t0.595a g−2.817l4.737a h−0.784

b n−5.957b

× 10−3

M0 = t1.136a g−1.515l1.139a h0.258b n0.309

b

n = t0.522a g1.564l−1.073a h−0.737

b n1.704b

Kϕ,p = t0.955a g2.044l−4.445a h0.327

b n7.555b

(6)

where ta is the angle thickness (mm), g the gauge of column flangebolts (mm), la is the angle length (mm), hb is the beam depth andnb is the number of bolts per angle leg on column flange. The unitsϕ0, M0 and Kϕ,p are radians, kN, and kN m/rad, respectively.

3.2.5. Faella, Piluso and Rizzano modelThe empirical model of Faella et al. [94] for the prediction of the

flexural resistance and rotational stiffness of extended end-platebeam-to-column joints was developed by means of a mechanicalmodel [95,96] based on the componentmethod from the Eurocode3 [28].

3.3. Analytical models

Analytical models use the basic concepts of structural analysis:equilibrium, compatibility and material constitutive relations, toobtain the rotational stiffness (Kj) and moment resistance (Mj,Rd)of a joint due to its geometric and mechanical properties.

3.3.1. Chen et al. modelChen and his colleagues worked extensively to predicting the

response of a joint from its geometrical andmechanical properties.The work on joints with the semi-rigid connections with anglesis presented in [97–99]. For top and seat angles with double webangles connections (Fig. 6) the initial stiffness is given by Eq. (7).

Kϕ =3EItad21

g1g21 + 0.78t2ta

+3EIwad23

g3g23 + 0.78t2wa

(7)

Ii =Lit3i12

(8)

where Ita and Iwa are the inertia moments, Eq. (8), of the legadjacent to the column face of the top angle and of the web angle,respectively; ti is the thickness of the angles; g1 and g3 are thedistances between the nut edge and the middle line of the angleleg adjacent to the beam, g1 is referred to the top angle and g3 tothe web angles; d1 is the distance between the middle lines of thelegs adjacent to the beam flanges; d3 is the distance between thecentre of the web angles and the middle line of the seat angle legadjacent to the beam flange.

The ultimate bending moment is given by Eq. (9).

Mj,u = fyLsat2sa4

+Vpt (g1 − kt)

2+ Vptd2 + 2Vpad4 (9)

where Lsa and tsa are the length and thickness of the seat angle, ktis the distance between the heel of the top angle and the toe of thefillet, and d2 and d4 are given by Eqs. (10) and (11).

d2 = d +tsa2

+ kt (10)

d4 =2Vpu +

fytwa2

3Vpu +

fytwa2

Lwa +tsa2

+ LI . (11)


Fig. 6. Top and seat angle connection with double web angles and the geometrical parameters of angles.

The parameters Vpu, Vpt and Vpa are obtained using Eqs. (12)–(14).2Vpu

fytwa

4

+gc − katwa

2Vpu

fytwa

= 1 (12)

2Vpt

fyLtatta

4

+g1 − kt

tta

2Vpt

fyLtatta

= 1 (13)

Vpa =Vpu +

fytwa2

2Lwa (14)

where Lta and Lwa are the lengths of the top angle and of the webangles, respectively.

These relationships were combined non-dimensionally[100,101], to provide the influence of themain geometrical param-eters on the rotational behaviour of connections with angles. Theiruse within a design procedure based on advanced analysis meth-ods has been shown [102].

The main problem with the Chen and Krishnamurthy modelsis that they do not consider the deformation of the column. Theassumption being that the support to a connection is rigid.

3.3.2. Yee and Melchers modelIn 1986, Yee and Melchers [103] proposed a mathematical

model that could predict the moment–rotation relationships ofbolted extended end-plate eave connections, using the connectiondimensions. The model represents a physically based approach tothe prediction of moment–rotation curves, taking into account thepossible failure modes and the deformation characteristics of theconnection elements.

The model included five deformation and six modes offailure. The deformations are: (1) end-plate flexure; (2) columnflange flexure; (3) bolt extension; (4) column web panel sheardeformation; and (5) column web compression. And the failuresare: (1) bolt failure (tension); (2) formation of end-plate plasticmechanism; (3) formation of column flange plastic mechanism;(4) shear yielding of the column web; (5) buckling of the columnweb; and (6) web crippling.

The rotational stiffness of a joint is obtained by combining theelastic displacements of the different components of the joint. Thelimiting moment capacity depends on the strength of the weakeradjoining section.

Johnson and Law [104] developed with a similar approach amethod for predicting the initial stiffness and plastic momentcapacity of flush end-plate connections.

Pirmoz et al. [105] proposed a semi-analytical model ofobtaining themoment–rotation behaviour of bolted top–seat angleconnections under combined axial tension and moment loadingbased on the data bank, created using FE simulation.

3.4. Mechanical models

Mechanical or spring models [54,95,96,103,106] represent thejoint by using a combination of rigid and flexible components,which are modelled by means of stiffness and resistance valuesobtained from empirical relationships. The nonlinearity of theresponse is obtained by means of inelastic constitutive laws usedfor the spring elements. Fig. 7 shows the mechanical model usedby Faella [54] for the extended end-plate beam-to-column joint.

To develop a mechanical model three steps are required:(1) identify the components of the joint that will provide sig-nificant deformation and failure of the joint; (2) determine theconstitutive laws for each component of the joint using analyti-cal, experimental or numerical means, and (3) assemble all of thecomponents together to produce the moment–rotation curve forcomplete joint.

This procedure is very flexible as it can be applied to jointsof any type: bolted or welded, and where specific effects can beintroduced, such as: bolt pretensioning or plastic hardening, etc.This is because all that is required are the constitutive behaviourof the components which make up the joint.

The firsts to introduce this type of model were Wales andRossow [107] in 1983 to simulate the behaviour of a double webangle connection with an applied bending moment and axial load(Fig. 8). The joint was modelled using two rigid bars connectedby a homogeneous continuum of independent nonlinear springs.An important characteristic of this model was that it included anaxial load. Kennedy and Hafez [108] used this model to repre-sent header-plate connections. Chmielowiec and Richard [109] ex-tended this model to predict the behaviour of all types of cleatedconnections subject to bending and shear.

Since then, significant research has been carried out usingmechanical models to study the behaviour of joints and tointroduce their effect in the analysis of structure. Faella et al. [54]developed the program JMRC to evaluate the moment–rotationcurve for welded connections, bolted end-plate connections andbolted connections with angles. Pucinotti [110] proposed a modelfor top and seat andweb angle connection based on a simplificationof the model in part J of the Eurocode 3 [28]. A model for jointsunder bending and axial loadswas proposed by Simões da Silva andGirão [111], Simões da Silva et al. [112], as well as by Urbonas andDaniunas [113], Sokol et al. [114] and del Savio et al. [115]. Bayoet al. [40] proposed an improvement to the Eurocode 3 model byintroducing a component-based finite dimensioned elastic–plastic4-node joint element which takes into account the actual sizeof the joint, its deformation characteristics, including those ofthe panel zone, local phenomena and all the internal forces thatconcur at the joint. Cabrero and Bayo [116] proposed a model to


Fig. 7. Mechanical model for the extended end-plate beam-to-column joint [54].

Fig. 8. Mechanical model for web angle connections [107].

calculate the stiffness in three-dimensional steel beam-to-columnjoints for both major and minor axes. Simões da Silva [117]proposed a generic model for steel joints under generalizedloading. Lemonis and Gantes [118], proposed a model based onthe componentmethod for bolted connectionswith end-plates andwith angles. Simões da Silva et al. [119] proposed a mechanicalmodel to evaluate the behaviour of cruciform flush end-platebeam-to-column steel joints at elevated temperatures.

The component method [1] is a hybrid analytical–mechanicalmethod (Fig. 7). It consists of modelling a joint as an assemblyof extensional springs (components) and rigid links, where eachspring represents a specific part of a joint with its own strengthand rigidity, dependent on the type of loading. The behaviour ofthe joint is obtained by knowing the mechanical and geometricalproperties of each component of the joint. It produces good resultswhen the joint is acting primarily in bending with minimal axialloading.

3.5. Numerical models

Numerical simulation started to be used for several reasons:(1) as a means of overcoming the lack of experimental results;(2) to understand important local effects which are difficult tomeasure with sufficient accuracy, e.g. prying and contact forcesbetween the bolt and the connection components; and (3) togenerate extensive parametric studies.

FE Analysis (FEA) is ideally suited to determine the rotation ofa joint; however such analysis is still computationally expensive.Themoment–rotation curve is the result of the complex interactionbetween the different elements of a joint. The analysis of steel

joints requires the introduction of geometrical and materialnonlinearities of the elementary parts of the connection; boltpreload and its response under a general stress distribution;interaction between bolts and plate components: i.e., shank andhole, head or nut contact; compressive interface stresses andfriction resistance; slip due to bolt-to-hole clearance; variability ofcontact zones; welds; imperfections.

Currently the FEMallows for the introduction into themodel of:large deformations, plasticity, strain-hardening, instability effects,the representation of large strain and/or displacements, contactsbetween plates and pre-stressing of bolts [120,121].

In 1972 Bose et al. [122] carried out the first FEM study ofwelded beam-to-column joints, which included: plasticity, strainhardening and buckling. The results obtained compared favourablywith available experimental results. Since then, several researchershave used the FEM to investigate joint behaviour.

In 1976, Krishnamurthy andGraddy [87]were the first tomodelthree-dimensional (3D) joints. They used an eight-node brickelement to model the end-plate connection. The analysis includedcontact between the different joint elements and preloaded bolts.However due to the limited computational power at the time,the 3D model was only used to develop a correlation factorbetween the two-dimensional (2D) and 3D results to enable theprediction of the more accurate 3D values from the less expensive2D results (Fig. 9). A similar process was proposed by Kukretiet al. [89], to generate the moment–rotation curve for bolted end-plate connection, obtaining very good results.

Kukreti et al. [91] developed a ‘hybrid’ 2D–3D FE model for tee-hanger connections, using 3D FE for the tee-flange, bolt heads andbolt shanks, and 2D FE elsewhere else.


Fig. 9. 2Dmesh of the end-plate connectionwith 581 nodes and 508 elements [87].

Chasten et al. [123] studied large extended unstiffened end-plate connections with eight bolts at the tension flange (four-bolts wide). FEM was used, using shell elements for the end-plateand beam flanges and plane stress elements for the beam web.Contact between the end-plate and column flange was modelledto determine the prying force.

Gebbeken et al. [124] studied extended end-plate connectionsusing shell elements. The characteristics of their model are:bolts with a simplified geometry; plane stress analysis; nonlinearstrain–displacement relationship. For the case of friction betweenend-plate and screw head, only the limit cases when completelystick and frictionless slip were considered; and the frictionbetween the flange and end-plate was neglected.

Sherbourne and Bahaari [125,126] developed a FEM to inves-tigate the behaviour of steel bolted end-plate connections. Wherethe end-plate, beam and column flanges, webs, and column stiff-eners were represented as plate elements with each bolt shankmodelled using six spar elements. Three-dimensional interface el-ements were used to model the boundary between the columnflange and the back of the end-plate that may make or break con-tact.

Bursi and Jaspart modelled T-stub connections [127] andisolated extended end-plate connections [128,129] (Fig. 10). Theycarried out several models using 3D brick elements and contactelements. They considered the effect of: element type, preloading,different constitutive relationships, and friction coefficient. Theirresults compared favourably to test results.

Troup et al. [130] used FEA to create a numerical model of aT-stub and an extended end-plate connection. Simplified bilinearstress–strain curves for the steel sections and bolt shank wereadopted. Material nonlinearity was considered for steel membersand connecting components, togetherwith geometric nonlinearitydue to the changing area of contact between the faces of the end-plate or T-stubs. An encouraging correlation between the modeland experimental tests was observed showing a good comparisonof the stiffness in both thick and thin plate conditions.

Bahaari and Sherbourne [131] developed a detailed 3D FEMto study 8-bolt unstiffened extended end-plate connections usingprimarily shell elements (Fig. 11). Neither the bolt head or nutwere included in the model, instead the end-plate and columnflange thicknesses were increased around the bolt hole. Thebolt shank was represented using truss elements connectingcorresponding nodes between the end-plate and column flange.The contact between the column flange and back of the end-platewas modelled using 3D interface elements.

Sumner et al. [132] also used FE to develop 4- and 8-boltextended unstiffened moment end-plate connections, obtaining

Fig. 10. 3D model of the extended end-plate connection [129].

Fig. 11. 3Dmodel using shell elements of an extended end-plate connection [131].

very good correlation between theirs and test results. Their modelincluded: solid eight-node brick elements for the beam sectionand column flange, which included plasticity effects; solid twenty-node elements for the bolts and end-plate; and contact elementsbetween the end-plate and the rigid column flange.

Swanson et al. [133] presented the results of a FE investigationof the behaviour of T-stub flanges (Fig. 12). Two types of modelswere used; a 3D T-stub model consisting of brick and wedgeelements and several 2D T-stub flange models consisting ofrectangular and triangular elements. All models incorporated


Fig. 12. 3D solid T-stub model [133].

nonlinear material characteristics, nonlinear geometric behaviour,and several contact interactions.

Citipitioglu et al. [134] presented different 3D models of boltedconnections with angles, (Fig. 13), following the recommendationsof [129] on the FE selection. Contact between all parts wasmodelled, including the effect of friction. Their results, similar tothose of [129], confirmed that the effect of friction on the initialstiffness of the joint was negligible, although it was slightly moreon the plastic regions. The effect of bolt pretension was similar tothat of friction, although it could modify the ultimate moment ofthe joint by as much as 25%.

Gantes and Lemonis [135] developed an FE model for boltedT-stub steel connections. Material and geometric nonlinearitiesas well as contact and friction were implemented, which wasvalidated by comparison with experimental data. The impact ofbolt length considered in the model was investigated and shownto be of primary importance.

Ju et al. [136] developed a 3D elasto-plastic FE model to studythe structural behaviour of butt-type steel bolted joints. The resultsshowed that the nominal capacities of the bolted connectioncalculated from the AISC specification and using FEMwere similar.

Maggi et al. [137] carried out parametric analyses on the be-haviour of bolted extended end-plate connections using 3D FEmodels calibrated to experimental results. The models took intoaccount: material nonlinearities, geometrical discontinuities, largedisplacements and contact to account for geometric discontinu-ities. Comparisons between numerical and experimental data forthe moment–rotation curves, displacements of the end-plate, andforces on bolts showed satisfactory agreement.

Xiao and Pernetti [138] proposed several models using shellFE based on [130], where shell elements were shown to giveequivalent results to solid 3D elements but at a fraction of thesolution time. Slip between end-plate and bolt headwas neglected.Contact elements were introduced between the end-plate and thecolumn flange tomodel themovement of end-plate away from thecolumn flange.

Tagawa and Gurel [139] used FE simulations to examine thestrength of steel beam-to-column joints stiffened with boltedchannels. 3D eight-node structural solid elements were used tomodel all components of the joint, with pretensioned bolts.

Abolmaali et al. [120] developed a 3D FE model for flush end-plate connections using 8-noded solid isoparametric elements for

the beam, column, end-plate and bolts. Geometric and materialnonlinearity, contact and pretension in the bolts were considered.

Moreno [140] developed a 3D FE model of flush and extendedend-plate bolted connections. The model included the beam, end-plate, bolts ends and the column. Considering the interactionbetween: end-plate and the column flange; bolts (head andnut) and the column flange; and bolts and the end-plate.The bolt shanks were modelled using truss elements. Theanalysis incorporated material nonlinearity for the plates andbolts. The FE results were compared with numerically predictedmoment–rotation curves,which corresponded to the experimentaltests carried out and with the component method [1].

Cabrero [141] developed two extended end-plate connectionsmodels, following the FEM recommendations of Bursi andJaspar [129]. One of the models used 8-node brick elements withincompatiblemodes,whereas the othermodel used truss elementsfor the bolts and shell elements for the end-plate, beam andcolumn. Different strategies were used for modelling contacts inthe second model, such as gap elements. Both models producedgood results, with a slight underestimation of the rotationalstiffness and a slight overestimation moment resistance whencompared with experimental results.

Pirmoz et al. [142] studied the behaviour of bolted top–seatangle connections with web angles subjected to combined shearforce and moment. Several 3D parametric FE models were usedwith geometric and mechanical properties used as parameters.With all of the connection components, such as beam, column,angles and bolts are modelled using solid elements. The contactsbetween surfaces were simulated by surface-to-surface contactelements. The results were compared with experimental resultswith good agreement.

Mohamadi-shooreh and Mofid [143] presented the resultsof several parametric analyses on the initial rotational stiffnessof bolted flush end-plate beam splice connections using FEMwith 20-noded brick elements, material behaviour, geometricaldiscontinuities and large displacements. The model was verifiedfor three case studies from the literature with the predicted resultscomparing well with reported data.

Lemonis and Gantes [118] proposed amethodology to estimatethe moment–rotation curve of structural beam-to-column jointsbased on the component method. The cases examined in this workincluded bolted connections with end-plates and with angles. Themethodology was found to be very satisfactory compared withexperimental tests and advanced FE models in terms of stiffness,strength and rotational capacity.

Dai et al. [144] made a simulation study of 10 fire testson restrained steel beam–column assemblies using five differenttypes of joints. Three-dimensional solid elements were used inmodelling themain structural members. The results demonstratedgood agreement between numerical simulations and experimentalobservations.

Díaz [145] developed a detailed 3D FE model to study thebehaviour of beam-to-column bolted extended end-plate joints(Fig. 14). The beam, column, extended end-plate, bolts (head, nutand shank)were allmodelled using 8-node brick elementswith fullintegration and incompatible modes. Contact elements were on allcontact surfaces of the joint. The obtained results were in goodagreementwith the real behaviour of joints, found in experimentalrests in the literature. Themodelwas used to develop ametamodelfor use into the design and optimization of semi-rigid connections.

3.6. Informational models

Informational models using Neural Networks (NN), can providean alternative to conventional methods of determining themoment–rotation curve by providing an inside relationship in


Fig. 13. 3D FE models of bolted connections with angles [134].

Fig. 14. 3D FEM of a beam-to-column extended end-plate joint [145].

the form of generalizations between the parameters involved.Thereby obtaining amore approximatemoment–rotation curve byextracting information directly from the experimental results.

Artificial NN (ANN) is an artificial intelligence applicationimplemented by engineers to carry out design tasks. It hasbeen applied to problems of: predicting function approximation;classification; filtering; structural analysis, design, dynamics andcontrol and structural damage assessment [146].

Informational NN formulations are equation-free global repre-sentation. The purpose of curve fitting is to find the parametersfor a mathematical equation, whereas NNmodelling is to learn thebackgroundmechanics. Once this ‘learning’ is done, the neural net-work can be implemented into other structural analysis platformswithout further simplification and calibration challenges [147].

Jadid and Fairbairn [148] investigated the relationship betweenthe behaviour of beam–column joints and the geometrical shape,amount and size of steel reinforcement, fixed beam and columncross-sectional dimensions and concrete strength using ANN.

Anderson et al. [149] used NN to predict the bilinear ap-proximation of the moment–rotation curves of minor axisbeam-to-column flush end-plate joints; Stavroulakis et al. [150]to predict the global moment–rotation curve for single web anglebeam-to-column joints.

Dang and Tan [151] proposed an inner product-based hys-teretic model for the application to piezoceramic actuators;Yun et al. [152] as a model for hysteretic behaviour of materials;

Yun et al. [153] as a hysteretic material model to expedite learningof the cyclic behaviour of connections.

De Lima et al. [154] used NN to predict the flexural resistanceand initial stiffness of beam-to-column steel joints, the resultsof which were consistent with experimental and design codereference values; Guzelbey et al. [155] to estimate the rotationcapacity of wide flange beams. The database used to train the NNwas based on 81 experimental results from the literature.

Pirmoz and Golizadeh [156] and Salajegheh et al. [157] usedNN to estimate the behaviour of bolted top–seat angle connectionswith web angles and Kim et al. [147] to model the nonlinearhysteretic cycle for bolted beam-to-column angle joints in steelframes.

Another methodology to predict the moment–rotation curve isGenetic Programming (GP). Cevik [158] was the first to investigatethe use of GP to determine the rotation capacity of wide flangebeams.

4. Mathematical representation of moment–rotation curve

In order to consider the behaviour of a joint in the globalanalysis of a structure, it is necessary to consider themathematicalrepresentation of the moment–rotation curve.

This representation can be performed by means of differentrelationships and levels of precision. Fig. 15 shows the differentmathematical representations of the moment–rotation curve:linear; (b) bilinear; (c) multilinear (trilinear); (d) nonlinear.


a b

c d

Fig. 15. Different mathematical representations of the (Mj–φ) curve: (a) linear; (b) bilinear; (c) multilinear (trilinear); (d) nonlinear.

The moment–rotation curve can be represented mathemati-cally in one of two ways [54]: (1) depending on parameters withclear physical meaning (e.g. stiffness, resistance) and a shape fac-tor; and (2) based on no clear physicalmeaning as it is derived fromregression analysis, called curve-fitting formulations. For a full re-view of this topic, the reader is referred to Faella et al. [54] andEurocode 3 [1].

4.1. Stiffness, resistance and shape factor-based formulations

The mathematical representation of the moment–rotationcurve depends on parameters with a physical meaning, such as therotational stiffness (K ), moment resistance (M) and a shape factornwhich characterizes the knee of the moment–rotation [54].

4.1.1. Linear modelThe linear model, Eq. (15), is the simplest to use but it is the

least accurate. It overestimates the rigidity of the joint [159] and isonly dependent on the rotational stiffness (Kφ) of the joint. Bathoet al. [3,5], Rathbun [7], Monforton and Wu [13], amongst others,used this model.

Mj = Kφφ. (15)

4.1.2. Bilinear modelThis model depends on three parameters, the: rotational

stiffness (Kφ); plastic moment (Mj,p); and plastic rotationalstiffness (Kφ,p) of the joint, Eq. (16). Used by many [17,160–164]and implemented in FEA programs it has a sharp change in rigidityand the intersection of the two curves (Fig. 15(b)).

Mj =

Kφφ for Mj ≤ Mj,pKφ,pφ for Mj > Mj,p.

(16)

4.1.3. Multilinear modelThis model was proposed to remedy the problem of the bilinear

model. Moncarz and Gerstle [20] use a trilinear representationwith five parameters, Eq. (17), the: rotational stiffness (Kφ); firstyielding moment (Mj,y); post-yielding rotational stiffness (Kφ,y);plastic moment (Mj,p); and plastic rotational stiffness (Kφ,p) of thejoint.

Mj =

Kφφ for Mj ≤ Mj,yKφ,yφ for Mj,y < Mj < Mj,pKφ,pφ for Mj,p ≤ Mj.

(17)

The representation proposed in Eurocode 3 [1] is divided intothree segments (Fig. 16), although for elastic–plastic analysis, asimplified bilinear model is proposed. The first segment of thecurve has the linear behaviour of Eq. (15) up to the momentvalue of 2/3Mj,Rd, where Mj,Rd is the design value of the jointplastic momentMj,p. The second segment is nonlinear according toEq. (18) in the range of 2/3Mj,Rd < Mj < Mj,Rd.

Mj =Kφ

1.5 MjMj,Rd

ξφ (18)

where ξ depends on the [1]:

ξ =

2.7 welded, bolted end-plate and base-plate connections3.1 bolted angle flange cleats.

The last segment is a straight horizontal line representing plasticbehaviour (Mj = Mj,Rd).

Other multilinear models can be can be found in the workof [105,115,165–167].

4.1.4. Nonlinear modelThis is the most accurate model so far. Proposed in 1943 by

Ramberg andOsgood [168], Eq. (19), depends on three parameters:


Fig. 16. Three-segment approximation of the (Mj–φ) curve [1].

n1

n1 <n2 <n3

n2n3

n = ∞

KφKφ

φ0 2φ0 φ

M

M0

Fig. 17. Ramberg–Osgood [168] representation of the (M–φ) curve.

rotational stiffness (Kφ), rotation (φ) of the joint, and the shapefactor (n) which characterizes the knee of the moment–rotationcurve (Fig. 17). The curve becomes bilinear with elastic–perfectlyplastic behaviour as n ⇒ ∞ at which point the plastic moment ofthe joint is equal to the reference momentM0.

φ

φ0=

MM0

1 +

MM0

n−1

where M0 = Kφφ0. (19)

Ang and Morris [169] were the first to use Eq. (19). Abolmaaliet al. [120] compared the moment–rotation curve for flush end-plate connections generated by Eq. (19) with one using FEA, withexcellent results.

Fig. 18 shows the nonlinear representation use by Goldberg andRichard [93]; Richard andAbbot [170] andAttiogbe andMorris [92]which is given by Eq. (20). It depends on four parameters: referencebending moment (M0); rotational stiffness (Kφ); plastic rotationalstiffness (Kφ,p) and a shape factor (n). Which is better than theRamberg–Osgood [168] equation as it allows positive, zero andnegative plastic rotational stiffness (Kφ,p). Negative values arenecessary when the joint fails due to local buckling.

MM0

=

1 − K p

φ

φ01 +

1 − K p

φ

φ0

n 1n

+ K pφ

ϕ0where K p =

Kφ,p

Kφ

. (20)

The exponential equation (21), proposed by Yee and Melch-ers [103] also allows positive, zero and negative plastic rotationalstiffness (Kφ,p) (Fig. 19). A characteristic of this curve is that slopeof the curve at the origin is equal to the initial elastic stiffness ofthe joint.

MM0

= 1 − exp[−

φ

φ0

1 − K p + n′

φ

φ0

]+ K p

φ

φ0

MM0

MM0

1

1

n3

n3

n1

n2

Kp >0

n1 <n2 <n3

φ/φ0

φ/φ0

n2

n3

n1

Kp <0

n1 <n2 <n3

a

b

Fig. 18. Goldberg and Richard [93] nonlinear representation of the mo-ment–rotation curve: (a) positive plastic stiffness; (b) negative plastic stiffness.

MM0

MM0

Kp >0

n1 <n2 <n3

n1

n1

n2

n2

n3

n3

φ/φ0

1

1

Kp <0

n1 <n2 <n3

a

b

Fig. 19. Yee and Melchers [103] exponential representation of the mo-ment–rotation curve: (a) positive plastic stiffness; (b) negative plastic stiffness.


MM0

1

1

n1

n2

n3

n1 <n2 <n3

φ/φ0

Fig. 20. Exponential representation of Pilvin [171].

MM0

1

n1

n2

n3

n1 <n2 <n3

φ/φ01

Fig. 21. Exponential representation of Colson [172].

where

n′

= nφ0

Kφ

K p =Kφ,p

Kφ

.(21)

Two further nonlinear representations are those of Pilvin [171],Eq. (22), (Fig. 20) and Colson [172], Eq. (23), (Fig. 21).

φ

φ0=

MM0

1 +

12n − 1

MM0

1 −MM0

(22)

φ

φ0=

MM0

1

1 −

MM0

n where M0 = Kφφ0. (23)

Wu and Chen [173] proposed the logarithmic representation ofEq. (24) for connections with angles (Fig. 22).

MM0

= n ln1 +

φ

nφ0

. (24)

4.2. Formulation based on curve fitting by regression analysis

An alternative way to determine the moment–rotation curveis using regression analysis. The simplest representation is that ofKrishnamurthy et al. [88], Eq. (25) used for end-plate connections.

φ = CMα (25)

where C and α are regression parameters related to thegeometrical and mechanical properties of the beam-to-columnjoint.

MM0

1n1

n2n3

n1 <n2 <n3

1 φ/φ0

Fig. 22. Logarithmic representation of Wu and Chen [173].

A more accurate representation is that of Kennedy [174];Sommer [175] and Frye andMorris [76], Eq. (26); although its slopecan become negative for some values ofM [77].

φ = C1M + C2M3+ C3M5 (26)

where C1, C2 and C3 are curve-fitting constants depending on thegeometrical and mechanical properties of the joint.

To solve this problem, Jones et al. [176], proposed equation (27)which is given by a segmented cubic B-spline formulation.

φ = φi +

m−j=0

bjM − Mj

3where

M − Mj

=

M − Mj forM > Mj0 forM < Mj

(27)

wherem is the number of points between two elementary parts ofthe moment–rotation curve,Mj is the upper bound moment of thejth part of the curve, while φi is the initial rotation (usually φi = 0)and the coefficients bj are obtained by least-squares curve fitting.

Lui and Chen [177] proposed the exponential relationship ofEq. (28), where Mi is the initial moment, Kφ,p is the strain-hardening connection stiffness and Cj are modelling parametersobtained by linear regression analysis [178]; α is a scaling factorfor numerical stability. It requires (m + 3) parameters, wherem isthe number of curve-fitting constants (Cj); usually, for a sufficientdegree of accuracy, m = 4–6. Although this model provided anexcellent fit, if the slope of the curve changes sharply, the modelcannot capture this adequately [179].

M =

m−j=1

Cj

[1 − exp

− |φ|

2jα

]+ Mi + Kφ,p |φ| . (28)

Kishi and Chen [60] modified Eq. (28) to accommodate linearcomponents of the moment–rotation curve Eq. (29).

M = Mi +

m−j=1

Cj

[1 − exp

− |φ|

2jα

]

+

n−k=1

Dk (|φ| − |φk|)H [|φ| − |φk|] (29)

where Cj and Dk are curve-fitting parameters obtained fromlinear regression [62]; φk is the starting rotation of the kthlinear component of the moment–rotation curve and H[φ] is theHeaviside’s step function (1 for φ ≥ 0 and 0 for φ < 0).

Lee and Moon [180] proposed the 2-parameter log model ofEq. (30) to describe the nonlinearmoment–rotation curve for semi-rigid connections with angles.

M = αln

103nφ + 1

n(30)


Table 1Advantages and disadvantages of models to obtain the rotational joint behaviour.

Model Advantages Disadvantages

Analytical Ease of application Uses simplified modelsLow computational cost Requires verification with experimental results to validate

Empirical Ease of application Requires calibration with other models, e.g. experimentalLow computational cost Its applicability is limited to the connection typologies used to

calibrate itCannot be used to determine the contribution of eachcomponent of a joint to its global behaviour

Experimental Best method to obtain the rotational behaviour of the joints Very expensive to carry out

Informational Can obtain information from experimental data Large data set required to obtain good results

Mechanical Applicable to any type of joint The accuracy of the results depends on the number ofcomponents used and on their mechanical characteristics

Low computational cost

Numerical Can introduce local effects which are difficult to measure, (prying forces,contact, etc.)

High computational cost

Can be used to carry out parametric studies

Table 2Principal characteristics of current models to obtain the rotational behaviour of a joint.

Characteristics ModelAnalytical Empirical Experimental Informational Mechanical Numerical

Advanced analysis available (contact, pretension, etc.) Low Low Medium Low Medium HighLevel of complexity Low Low Medium Medium Medium HighDatabase requirements High High Low High Low MediumCost Low Low High Medium Low MediumReusable for other connection typologies No No No No Yes YesParameterization Low Low Low High Medium HighSolution time Low Low High Low Low MediumUser skills Low Low Medium Medium Medium HighUsability for design optimization Low Low N/A High High HighMatch real behaviour Low Low High Medium Medium MediumProvides extra information No No Yes No Yes Yes

N/A not applicable.

where n and α are shape parameters determined using themethodof least squares for differences between the predicted momentsand the experimental test data [59].

5. Conclusions

Steel portal frames were traditionally designed, assuming thatbeam-to-column joints are ideally pinned or fully rigid, whereasin fact, due to the finite stiffness of the joints, the true behaviouris somewhere between these two extremes. All studies agreethat when carrying out structural analysis of any frame, therotational behaviour of the joint should be considered. Currently,the most common method of accounting for the true behaviour ofa connection is by using themoment–rotation curve in the analysisof the structure.

Several types of models can be used to obtain the mo-ment–rotation curve, these are: analytical, empirical, experimen-tal, informational, mechanical and numerical. The most popular ofthese are the mechanical models, of which the most used is thecomponentmethod.With this method it is possible to evaluate therotational stiffness andmoment capacity of semi-rigid joints whensubjected to only pure bending. The method fails if an axial load isalso present.

A summary of the advantages and disadvantages of each modelis given in Table 1 with Table 2 giving the principal characteristicof each model.

When analysing semi-rigid frames, the behaviour of the jointsneed to be modelled, this is associated with a mathematical modelof the moment–rotation curve. Depending on the type of globalstructural analysis required, one of severalmoment–rotation curve

representations can be used, these are: linear, bilinear, multilinearand nonlinear. The most accurate representation can be obtainedusing continuous nonlinear functions, although the multilinearrepresentation is commonly used for mechanical models.

Acknowledgements

This work was supported by the CARM (Consejería deEducación, Ciencia e Investigación de la Región de Murcia) andthe Technical University of Cartagena. Its support is greatlyappreciated. Travelling funds for the fourth author were providedby the School of Mechanical Engineering at the University of Leeds.

References

[1] European Committee for Standardisation (CEN). Eurocode 3. Design of steelstructures, part 1–8: design of joints (EN 1993-1-8:2005). Brussels; 2005.

[2] Wilson WM, Moore HF. Tests to determine the rigidity of riveted joints insteel structures. In: University of Illinois. Engineering experiment station.Bulletin 104, Urbana (USA): University of Illinois; 1917.

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review on the modelling of joint behaviour in steel frames

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