Liu, Xu & Grierson - Compound-element Modeling Accounting for Semi-rigid Connections and Member Plasticity

Engineering Structures 30 (2008) 1292–1307www.elsevier.com/locate/engstruct

Compound-element modeling accounting for semi-rigid connections andmember plasticity

Yuxin Liua,∗, Lei Xub, Donald E. Griersonb

a ACR Civil & Layout, Atomic Energy of Canada Limited, Ontario, Canada, L5K 1B2b Civil and Environmental Engineering, University of Waterloo, Ontario, Canada, N2L 3G1

Received 18 May 2007; received in revised form 23 July 2007; accepted 24 July 2007Available online 5 September 2007

Abstract

A general model of a compound element is proposed in this article to consider the combined influence of semi-rigid connections and plasticityon the non-linear responses of steel frameworks. The stiffness degradation of semi-rigid connections is modeled by a moment–rotation relationshipwith four parameters, while the plasticity formation of a member end from initial yield to full yield is modeled by an elliptic moment–rotationrelationship. The compound element resulting from the combined influence of member plasticity and semi-rigid connection behaviour is used tofacilitate the derivation of member stiffness coefficients where the effects of geometrical non-linearity and member shear deformation are alsoincluded for the progressive-failure analysis. On the basis of member plasticity behaviour, the categories of semi-rigid connections are investigated.Three semi-rigid steel frameworks are analysed to illustrate the proposed analysis method, and the results are compared with those obtained fromexperiments and the application of other methods.c© 2007 Elsevier Ltd. All rights reserved.

Keywords: Steel framework; Compound element; Semi-rigid connection; Stiffness degradation; Progressive-failure analysis

1. Introduction

Many studies have been devoted to developing practicalmethods of non-linear analysis of frameworks accountingfor semi-rigid behaviour of the connections and/or plasticbehaviour of the members [1–4]. However, little has been doneto investigate the interaction between the behaviour of semi-rigid connections and that of member plasticity, as well astransverse-shear and axial stiffness degradations. This paperfocuses on such interaction by applying a planar-compound-element concept. A member plastic zone forms at the beamend due to internal forces (e.g., moment, shear force, andaxial force). The characteristics of the interaction between aflexural semi-rigid connection and the flexural plastic zone ofits connected member are discussed and illustrated in detail.

Fig. 1(a) exhibits a typical beam-to-column connection joint,where there is member plasticity due to bending. Typically,the connection is semi-rigid, and can include bolts, welds and

∗ Corresponding author. Tel.: +1 905 823 9060x6283; fax: +1 905 403 7307.E-mail address: [email protected] (Y. Liu).

0141-0296/$ - see front matter c© 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2007.07.026

angles. To facilitate a non-linear analysis, the model in Fig. 1(a)is replaced by the analytical model in Fig. 1(b). Here, one of thetwo springs represents the plasticity formed at the member end,while the other spring represents the semi-rigid connection.

An incremental-load method of analysis has beenrecently developed to deal with material and geometric non-linearities [5,6]. The goal of this paper is to extend this methodto account as well for semi-rigid connections. Each stage ofthe analysis accounts for stiffness degradation due to semi-rigidconnection behaviour combined with member plasticity, geo-metric non-linearity and shear deformation when updating thecorresponding tangent stiffness matrix for the structure. Theincremental-load process ends when the specified external loadshave been completely applied to the structure, or a limit failureloading state is reached.

2. Rotational compound element

This section employs an assembly of springs, connectedin series, to develop a compound element representingthe combined rotational stiffness behaviour of a semi-rigidconnection and a member-end plastic hinge. The determination

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Y. Liu et al. / Engineering Structures 30 (2008) 1292–1307 1293

Nomenclature

A, As Cross-sectional area, equivalent shear cross-sectional area

d j , f j Member end displacement, force in j directionDWA Double web-angle connectionE, G Material Young’s modulus, material shear modu-

lusEEP Extended end-plate connectionFEP Flush end-plate connectionH Horizontal loadI Moment of inertia of cross-sectionk, K Member stiffness matrix, structure stiffness

matrixL Member lengthM, M j Bending moment, and moment at member end jMn Nominal moment capacity of connectionMp, My Fully-plastic moment capacity, initial-yield mo-

ment of a sectionM0 Reference moment of connectionN j , n j Axial inelastic stiffness, corresponding degrada-

tion factor at end jP, p Axial force, normalized axial forcePcr , Pe Critical load, Euler buckling loadPy, Pp Initial- and full-yield axial forcesR, r Rotational stiffness, and corresponding degrada-

tion factorRc, Rn , Rotational connection stiffness, and nominal

stiffnessrc, rp Rotational stiffness degradation factors of con-

nection and member plasticityRce, Rcp Initial rotational stiffness, strain-hardening/

-softening stiffness of connectionRe, Rp Elastic and inelastic rotational stiffnessesRi j Rotational stiffness coefficient for member i jT, t Transverse shear stiffness of a cross-section,

corresponding degradation factorV, V j Shear force, shear force at member end jSWA Single web-angle connectionχ j Factors accounting for axial and shear stiffness

degradationδ Axial deformation, or deflectionδp, δs Axial plastic limit deformation, transverse shear

deflectionε Normal strain or axial deformationφ Curvature or rotation of a member end in the

plastic rangeφp Full-yield rotation of a section corresponding to

full plastificationφy Initial flexural yield rotation at which inelasticity

startsγ Shear strain/plastic shear deformation or shape

parameter of connectionsγy, γp Initial- and full-yield shear strainsλ Load factor/multiplier

λ f Load factor used to identify the failure behaviourof the structure

λ j Increment/load scale factor/multiplierθ Rotation at member end, or an angleθc Rotational angle of semi-rigid connectionθe Rotational angle of member endθ j Rotational angle at member end jθn Nominal rotation capacity of semi-rigid connec-

tion

(a) Actual connection with member bendingplasticity.

(b) Analytical model.

Fig. 1. Semi-rigid connection and member-inelasticity model.

of the stiffness of semi-rigid connections is discussed in detail,while that for member-end plasticity is adopted directly fromprevious research [5,6].

2.1. Series element model

The series element model consists of a semi-rigid connectionspring, an inelastic spring and an elastic member, all connectedin series. Herein, an inelastic spring is defined as a springthat characterizes the inelastic behaviour of a cross-sectionfrom initial yield to full yield. The nature of the compoundelement is indicated in Fig. 2, where parameters Rc, Rp, andRe denote the rotational stiffnesses of the semi-rigid connectionspring, the member plasticity spring and the elastic memberend, respectively. Only end 1 of the member is considered (end2 may or may not have the exact same nature as end 1).

The case in Fig. 2(a) is conventionally used in structuralanalysis, where a beam-to-column connection at node 1 isassumed as either a pinned connection (Rc = 0) or afixed connection (Rc = ∞). This assumption simplifies theanalysis for both hand and computer-based analyses. However,if the effect of the actual connections on structural response

1294 Y. Liu et al. / Engineering Structures 30 (2008) 1292–1307

(a) Conventional end. (b) Semi-rigid connection.

(c) Inelastic spring. (d) Series elements.

Fig. 2. Four types of member ends and connection models.

(a) Series elements. (b) Compound element.

Fig. 3. Compound element replaces the elements in series.

is significant, the model including a semi-rigid connectionrepresented by a spring symbol @ in Fig. 2(b) should beaccounted for in the analysis and design of the structure.Another model popular in rigid-plastic analysis assumes that amember plastic hinge abruptly forms, i.e., rather than graduallyforming from initial yield to full yield. To improve theaccuracy in this case, the inelastic-spring model in Fig. 2(c)is suitable for simulating gradual stiffness degradation due toincreasing extent of plastic behaviour. Finally, if both semi-rigid connection and plastic member behaviour occur at thesame time, the series-element model shown in Fig. 2(d) shouldbe introduced in the analysis. Although Yau and Chan [7]previously considered the latter model, the influences ofmember plasticity and semi-rigid connections were consideredseparately.

The rotational deformations of the semi-rigid connection andinelastic member end, indicated in Fig. 2(d), are graphicallyrepresented in Fig. 3(a). It is readily shown that the two series-connected springs can be substituted by the compound elementin Fig. 3(b) having only one spring. It remains to derive theexpression for compound stiffness R representing the combinedeffect of stiffnesses Rc and Rp. To that end, for appliedmoment M in Fig. 3(a), the rotations θc and θp caused bysemi-rigid connection behaviour and member plastic behaviour,respectively, are found as,

θc = M/Rc (1a)

θp = M/Rp. (1b)

Fig. 4. Four-parameter model of semi-rigid connections.

Then, the total rotation θ between the joint and the elasticmember end is, from Eqs. (1),

θ = θc + θp = M/Rc + M/Rp = M/R (2)

from which it is observed that the compound rotational stiffnessaccounting for semi-rigid connection and member plasticbehaviour is,

R =1

1/Rc + 1/Rp=

Rc Rp

Rc + Rp. (3)

2.2. Determining connection stiffness Rc

It remains to determine the stiffness of the compoundelement defined by Eq. (3). To that end, member plasticitystiffness Rp is directly given by Refs. [5,6], and semi-rigidconnection stiffness Rc alone needs to be established in thefollowing.

Several semi-rigid connection models have been investi-gated by Xu [8]. A four-parameter power model, originallyproposed for modeling post-elastic stress–strain behaviour [9],has been commonly adopted in analysis. Recently, experimen-tal data for extended-end-plate and flush-end-plate connectionshas further confirmed this model to be effective and accuratefor predicting the behaviour of end-plate connections [10]. Thefollowing four-parameter model is employed in this study tosimulate the behaviour of semi-rigid connections,

M =(Rce − Rcp)θc

{1 + [(Rce − Rcp)θc/M0]γ }1/γ

+ Rcpθc. (4)

In Eq. (4), θc denotes the rotation of the semi-rigidconnection, and the four parameters Rce, Rcp, M0 and γ are theelastic rotation stiffness, strain-hardening/softening stiffness,reference moment and shape parameter for the connection,respectively. The elastic stiffness Rce = Mcy/θcy , where Mcyand θcy are the initial-yield moment and corresponding rotation.The shape of the moment–rotation curve is defined by theparameter γ , whose magnitude is related to the strain-hardeningand -softening behaviour of the connection (the value γ usedin the model is found by curve fitting experimental data).The four parameters in Eq. (4) can be found for differenttypes of connections from an existing database of experimental


results [8]. The reference moment M0, strain-hardening or -softening stiffness Rcp, and nominal rotation θn determinethe nominal maximum moment or moment capacity of theconnection to be,

Mn = M0 + θn Rcp, (5)

where θn depends on the connection type and is determinedfrom published research results; e.g., Bjorhovde et al. [11]. Itis noteworthy that when the moment–rotation response doesnot have a humped point, the nominal moment capacity isdetermined by the moment at which θn = 0.02, as suggestedin the AISC [12] design specifications.

By differentiating Eq. (4) with respect to rotation θc, thetangent stiffness of the connection is given by [9],

Rc =dM

dθc= Rcp +

Rce − Rcp

{1 + [(Rce − Rcp)θc/M0]γ }1+1/γ

, (6)

where Rce is the elastic rotational stiffness at the initialcondition θc = 0, and Rcp is the strain-hardening and -softeningstiffness when rotation θc tends to infinity. For practical analysisof steel structures, the rotation θc is at most equal to the limitingnominal rotation value when connection fracture occurs [11].

It is seen from Eqs. (4) and (6) that the four-parameter modelreduces to a linear model with Rc = Rce when Rcp tends toRce. A bilinear model is realized when the shape parameterγ approaches infinity; i.e., when θc < M0/(Rce − Rcp), theterm [(Rce − Rcp)θc/M0]

γ tends to zero and Eq. (6) reducesto Rc = Rce, while when θc > M0/(Rce − Rcp), the term[(Rce − Rcp)θc/M0]

γ tends to infinity and Eq. (6) reduces toRc = Rcp. If Rcp is set to zero (i.e., strain-hardening and -softening is ignored), Eq. (4) reduces to the following three-parameter model, suggested by Kishi and Chen [13],

M =Rceθc

[1 + (Rceθc/M0)γ ]1/,γ(7)

where reference moment M0 is equal to nominal momentcapacity Mn . Note that rotation θc can be explicitly obtainedfrom Eq. (7) as,

θc =M

Rce[1 − (M/M0)γ ]1/γ. (8a)

As given in the authors’ studies [5,6], the post-elasticrotation of the connecting member is taken by this study to be,

φ = φp

{1 −

[1 −

(M − My

Mp − My

)e0]1 / e0

}My

Mp≤

M

Mp, ≤ 1 (8b)

where parameters e0 and φp are defined as shape parameter andfull-yield rotation for the section, respectively. Experimentalor analytical results can be used to determine these twoparameters. Based on the previous work [5], values e0 = 4 andφp = 0.0025 radians per unit length are applied in the analyses.

Therefore, from Eqs. (8a) and (8b), the total rotation θ =

θc +φ of the compound element can be explicitly expressed as,

θ =M

Rce[1 − (M/Mu)γ ]1/γ

+ φp

{1 −

[1 −

(M − My

Mp − My

)e0]1 / e0

}My

Mp≤

M

Mp≤ 1 (9)

which represents the moment–rotation relationship of thecompound element. The benefit of using the three-parametermodel Eq. (7) is that rotation θc of the connection is directlyobtained from Eq. (8a) given moment M found by the non-linear analysis; the disadvantage is that the strain-hardeningor -softening nature of the connection is omitted. In contrast,although strain-hardening and -softening is accounted for in thefour-parameter model, an iterative procedure is needed to findthe rotation θc of the connection. Both connection models areconsidered for the verification analysis presented in Section 5.

2.3. Stiffness degradation factors

The flexural stiffness degradation factor associated withsemi-rigid stiffness Rc is given by [14],

rc =1

1 + 3EI/L Rc, (10)

where EI/L is the flexural stiffness of the elastic member. Thefactor rc is interpreted as the ratio of the end rotation of theelastic member to the combined rotation of the elastic memberand the semi-rigid connection due to unit end-moment [8].Similarly, the stiffness degradation factor associated with theinelastic member stiffness Rp is given by [5],

rp =1

1 + 3EI/L Rp, (11)

where the factor rp is interpreted as the ratio of the elasticrotation ML/3EI to the total elastic and inelastic rotationML/3EI + M/Rp due to bending moment M applied at theend connected to the compound element, where the far end ofthe elastic member is simply supported [6].

To evaluate the combined stiffness effect, a stiffnessdegradation factor for the compound stiffness R defined byEq. (3) is introduced and similarly expressed as,

r =1

1 + 3EI/L R(12)

which is the ratio of the rotation of the elastic element to thesum total rotation of the compound element and the rotationof the elastic member, when it is simply supported at thefar end. From Eqs. (3) and (10)–(12), the compound stiffnessdegradation factor is expressed as,

r =1

1 + 3EI/L Rc + 3EI/L Rp=

rcrp

rc + rp − rcrp(13)


Fig. 5. Stiffness degradation relationships at a member end.

which maps R ∈ [0, ∞] to r ∈ [0, 1]. From Eq. (13),the stiffness degradation factor for the compound elementis a function of the degradation factors of the semi-rigidconnection and member inelasticity such that, if any of thesefactors degrades to zero, the stiffness of the compound elementdegrades to zero as well.

3. Characteristics of compound rotational element

The behaviour of the compound rotational element isdependent upon the strength capacities of the connection andthe connected beam members. For the current study, the effectof shear deformation of the panel zone on the behaviour ofthe beam–column connection is ignored. Connection strengthis important in the inelastic analysis of frameworks. Thestrength behaviour of the compound element is analysed in thefollowing.

If only the effect of the member plasticity is considered,the moment–rotation relation in the post-elastic range is thatshown in Fig. 5(a). Alternatively, if the effect of the semi-rigidconnection is accounted for, the moment–rotation relationshipis as shown in Fig. 5(b). In Fig. 5, the nominal maximummoment Mn defined by Eq. (5) is the moment capacity ofthe connection, while My and Mp are the initial-yield andfully-plastic moment capacities of the connected member,respectively. Depending on the interaction between memberinelasticity and semi-rigid connection behaviour, three typesof connections are characterized by the compound element, asdescribed in the following

3.1. Under-strength connections: Mn ≤ My

In this situation, the performance of the compound elementis governed only by the semi-rigid connection, and no plasticityoccurs in the vicinity of the member end. This can occur forSingle Web-Angle (SWA) connections with Mn = MSWA

n .Since the member end does not undergo any plasticity, the non-linear moment–rotation behaviour of the compound elementis determined by the behaviour of the semi-rigid connectionalone; i.e., the moment–rotation relationship defined by thelowest solid curve in Fig. 6 is the same as that given in Fig. 5(b)for an SWA connection. This kind of connection is referredto as under-strength connection, since the strength capacity of

Fig. 6. Combined moment–rotation relationships for a compound element.

the compound element is less than the yield strength capacityof the connected member. If Mn is small enough, this type ofconnection is categorized as a conventional simple or pinnedconnection [12,15]. (Note that the definition of under-strengthconnections in this study is based on there being no plasticityat the member end, whereas the flexible connections defined inAISC-LRFD [12] are based on Mn ≤ 0.2Mp).

3.2. Partial-strength connections: My < Mn < Mp

In this second case, both semi-rigid connection behaviourand member inelasticity govern the behaviour of the compoundelement, but the limit strength is determined by the nature ofthe connection. In other words, the connected member does notreach its moment capacity Mp, while the compound elementachieves nominal moment capacity Mn . Such behaviour for aFlush End-Plate (FEP) connection is illustrated by the middlesolid curve in Fig. 6 (the corresponding dotted curve refersto the middle solid curve in Fig. 5(b)). Although this type ofsemi-rigid connection is here referred to as a partial-strengthconnection, it is somewhat different from the definition in thedesign codes [12], where the inelasticity of the member is notaccounted for.

3.3. Full-strength connections: Mn ≥ Mp

Finally, when the nominal moment capacity of theconnection Mn is equal or greater than the plastic momentcapacity Mp of the connected member, the member inelasticitydominates the behaviour of the compound element (even


though the connection influences the stiffness degradation ofthe compound element due to its non-linear behaviour). Suchbehaviour for an Extended End-Plate (EEP) connection isillustrated in Figs. 5 and 6 (where the dotted moment–rotationcurve refers to the EEP connection alone). It is evident inFig. 6 that the moment–rotation behaviour of the compoundelement (solid curve) is dominated by the plastic behaviour ofthe member. This kind of connection is referred to as a full-strength connection, which is defined the same way as in thedesign codes (e.g., [12]).

It can be concluded from the preceding discussions thatwhen the nominal capacity of a connection is much lower thanthat of the connected member, the connection dominates thebehaviour of the compound element; however, if the nominalcapacity of a connection is much greater than the capacity ofthe connected member, the member plasticity dominates thecompound-element behaviour. In practice, a flexible connectionwith low connection capacity (Mn � My) can be employedin the design of braced frames. In the design of moment-resisting frames, however, excessive deformation can occurif the connection capacity Mn is far less than the capacityMp of the connected member (see Example 3). A satisfactorydesign is achieved if both the connection and the correspondingconnected member have approximately the same strengthcapacity (i.e., Mp ≈ Mn). It is prudent to avoid over-strengthconnections (i.e., Mn � Mp), since this results in over-costlyconstruction because excess connection capacity is not utilized.

Besides the strength of connections, connection stiffnessis another important factor characterizing the behaviour ofconnections, especially in serviceability design concerninginitial elastic stiffness Rce. According to Eurocode 3 [17,18],for example, a beam-to-column connection is assumed to berigid if its elastic stiffness satisfies the following condition,

L Rce

EI≥ 25 (or rc ≥ 0.893 for an unbraced frame) (14)

or

Rce L

EI≥ 8 (or rc ≥ 0.727 for a braced frame), (15)

where the notations are the same as those defined in Eq. (10),except that Rc is replaced by elastic stiffness Rce. Conversely, aconnection is assumed to be flexible if the following conditionis satisfied,

Rce L

EI≤ 0.5

(or rc ≤ 0.143 for either a braced or an unbraced frame).

(16)

When elastic stiffness Rce or corresponding stiffness factorrc is located between the values defined by Eq. (14) or (15)and (16), a semi-rigid connection is attained. Such using initialconnection stiffness Rce and member-end rotation-stiffnessparameter EI/L is referred to as initial-stiffness criterion foridentifying connection behaviour. Note that the stiffness criteriadefined in Eqs. (14) through (16) are related to member lengthL . If only the member length changes, the connection category

Fig. 7. Relationship between the degradation factors for a compound element.

changes according to the stiffness criteria. For instance, ifRce L/EI = 9 for a braced frame, then the connection isrigid; however, when member length L changes to 0.5L , thecorresponding stiffness ratio becomes 0.5Rce L/EI = 4.5,and the same connection becomes semi-rigid. Such a paradoxchallenges the current classification systems for beam-to-column connections and further research is needed. For the timebeing, Eqs. (15) and (16) are used as the criteria in this study tocharacterize connection behaviour.

Contrary to that for connection stiffness, connection strengthcriteria are based on member moment capacities My and Mp, aswell as nominal connection capacity Mn , and are independentof any length variation of the connected member. It isnoteworthy that conventional rigid connections are recognizedas having full strength (Mn ≥ Mp) and rigid stiffness(Rce = ∞). So, if the behaviour of a connection is consideredaccording to the strength and stiffness classification criteria,a rigid connection with full strength is not equivalent to theconventional rigid connection because Rce satisfies Eq. (14) or(15) but is less than infinity. To distinguish from a conventionalrigid connection, a full-strength connection with Rce < ∞ isreferred to as a fully-rigid connection in this study.

The characteristics of stiffness degradation of a compoundelement are further examined in the following. The relation-ships between the compound degradation factor r and theconnection and plasticity factors rc, rp, given in Eq. (13) aregraphed in Fig. 7. It is observed that for the common pinned-connection case when rc = 0, the compound element has zerorotational stiffness for any value of rp and the connected mem-ber exhibits no plasticity. For the other extreme case whenrc = 1, the compound-element behaviour is governed by theplastic behaviour of the member (r = rp). For the two cases,it is evident from Fig. 7 that when the plasticity factor is lessthan unity (e.g., rp = 0.7), the r value of the compound el-ement is approximately the value of rc. This means that evenwhen the member end has undergone some degree of plasticity(e.g., 30% = 1.0 − 0.7), the stiffness of the compound elementis dominated by that of the connection. In other words, the levelof member plasticity has little effect on the stiffness degradationof the compound element.

To numerically demonstrate the interaction between semi-rigid connections and member plasticity, an illustration ispresented in the following for a beam member with three


(a) Beam member. (b) M–θc relations.

Fig. 8. Simple beam structure with semi-rigid connections.

(a) Stiffness for EEP and FEP. (b) Enlarged portion for FEP.

Fig. 9. Stiffness degradation behaviour of different compound elements.

Table 1Parameters for specified connections

Connection M0 (kN m) Rce (kN m/rad) Rcp (kN m/rad) γ

DWA 55.935 20 114 69.608 0.964FEP 95.146 21 470 468.95 1.45EEP 137.86 18 984 1041.86 5.11

different connections. As shown in Fig. 8, the member has spanlength L = 4 m and a W310 × 33 cross-section (with thefollowing properties: elastic and plastic moduli S = 0.415 ×

106 mm3 and Z = 0.48 × 106 mm3, moment of inertiaI = 65 × 106 mm4, Young’s modulus E = 200 000 MPa,yield stress σy = 248 MPa). The residual stress is assumedto be σr = 0.3σy , such that the initial and full-yield momentsMy = 0.7Sσy = 0.7 × 0.415 × 10−3

× 248 × 103= 72 kN m

and Mp = Zσy = 119 kN m, respectively. The parametersfor the three typical connections listed in Table 1 are takenfrom a published databank [8]. Illustrated in Fig. 8(b) are themoment–rotation curves found for the connections using thefour-parameter connection model defined by Eq. (4). It is seenfrom Fig. 8(b) that the Double Web-Angle (DWA) is an under-strength connection, the Flush End-Plate (FEP) is a partial-strength connection, and the Extended End-Plate (EEP) is afull-strength connection.

Since no plasticity occurs at the member end when theDWA connection is employed, only the stiffness degradationbehaviour of the beams with FEP and EEP connections areinvestigated in the following. The variations of stiffnessespredicted by Eqs. (6) for EEP and FEP connections areplotted in Fig. 9 versus applied moment M . The correspondingvalues of the stiffness of these two connections are listed in

Tables 2 and 3. It is seen from Fig. 9 for both the EEP andFEP connections that connection stiffness Rc and compoundstiffness R are equal before member yielding takes place,and that compound stiffness R degrades to zero after memberyielding occurs. The corresponding degradation factors rp, rc,and r in Tables 2 and 3 demonstrate that the member-sectionplastic behaviour dominates the stiffness degradation of thecompound element.

4. Non-linear analysis of frameworks with semi-rigidconnections

Once the stiffness degradation factor of a compoundelement is determined, as discussed in the previous sections,the structural analysis is conducted. This study focuses onplanar semi-rigid steel frameworks comprised of beam–columnmembers with compact sections, for which plastic deformationis not precluded by local buckling [12]. Plastic bending,shearing or axial deformation (φ, γ or δ) of a member underthe action of moment, shear or axial force M, V or P ,respectively, is assumed as concentrated at the member-endsections [6]. Fig. 10(a) represents a general member withYoung’s modulus E , shear modulus G, member length L , cross-section moment of inertia I , sectional area A, and equivalentshear area As . Parameters Rpj , Tpj and Npj are, respectively,the post-elastic rotational bending, transverse shearing andnormal axial stiffness of the member at the two end sectionsj = 1, 2, while Rcj , Tcj and Ncj are, respectively, the rotationalbending, transverse shearing and normal axial stiffness of theconnections at the two end sections. By adopting the compoundelement developed previously in this paper, the simplifiedmember model in Fig. 10(b) is obtained, the correspondingparameters for which are discussed in the following.


Table 2Results for EEP connection

M (kN) Rp (kN m/rad) Rc (kN m/rad) R (kN m/rad) r p rc r

0 18 984 18 984 0.661 0.6619 18 984 18 984 0.661 0.661

19 18 983 18 983 0.661 0.66128 18 979 18 979 0.661 0.66138 18 962 18 962 0.660 0.66047 18 915 18 915 0.660 0.66057 18 810 18 810 0.659 0.65973 4 732 880 719 18 376 18 375 1.000 0.653 0.65375 39 559 835 18 249 18 241 1.000 0.652 0.65284 838 297 17 683 17 318 0.989 0.645 0.64093 167 474 16 857 15 315 0.945 0.634 0.611

101 57 301 15 745 12 351 0.855 0.618 0.559109 22 861 14 366 8 822 0.701 0.596 0.475116 7 139 12 786 4 581 0.423 0.567 0.320119 0 11 861 0 0.000 0.549 0.000

Table 3Results for FEP connection

M (kN) Rp (kN m/rad) Rc (kN m/rad) R (kN m/rad) r p rc r

0 21 470 0.688 0.68810 20 094 0.673 0.67320 18 026 0.649 0.64928 15 873 0.619 0.61936 13 853 0.587 0.58742 12 050 0.553 0.55348 10 483 0.518 0.51856 8 323 0.461 0.46157 8 000 0.451 0.45161 7 033 0.419 0.41964 6 214 0.389 0.38967 5 518 0.361 0.36170 4 926 0.336 0.33672 Infinity 4 420 4420 1.000 0.312 0.31290 274 191 1 524 1516 0.966 0.135 0.13595 125 900 1 082 1073 0.928 0.100 0.099

100 68 670 822 813 0.876 0.078 0.077105 38 651 668 657 0.799 0.064 0.063110 20 013 579 562 0.672 0.056 0.055115 9 206 537 507 0.486 0.052 0.049116 7 157 530 494 0.423 0.052 0.048117 5 187 525 476 0.347 0.051 0.047118 3 213 520 447 0.248 0.051 0.044119 0 514 0 0.000 0.050 0.000

The evaluation of connection and member rotationalstiffnesses Rcj and Rpj in Fig. 10(a), and correspondingstiffness degradation factors rcj and rpj , have been discussedin detail in Section 2. The member transverse shear andnormal axial stiffnesses Tpj and Npj have also been determinedin previous research [5,6], where corresponding stiffnessdegradation factors tpj and n pj for member end j are given by,

tpj =1

1 + 3EI/L3Tpj(17a)

n pj =1

1 + EA/L Npj(17b)

which map Tpj or Npj ∈ [0, ∞] into tpj or n pj ∈ [0, 1]. Sim-ilarly, the transverse and normal stiffness degradation factors

for the connection are expressed as,

tcj =1

1 + 3EI/L3Tcj(18a)

ncj =1

1 + EA/L Ncj, (18b)

where Tcj and Ncj are the transverse shear and normal axialstiffnesses of the connection.

When the connection is in the elastic range, it is assumed thatstiffness Tcj or Ncj is infinite and corresponding degradationfactor tcj or ncj in Eqs. (18) is unity. Conversely, when theconnection is in the plastic range, it is assumed that stiffnessTcj or Ncj is zero and corresponding degradation factor tcj or


(a) Elements in series.

(b) Compound element.

Fig. 10. Compound model of beam–column member.

(a) Transverse stiffness case. (b) Normal axial stiffness case.

Fig. 11. Idealized force–displacement relations for transverse and axialconnections.

ncj is zero. Such idealized elastic–plastic models are depictedin Fig. 11.

For the general planar compound member in Fig. 10(b), thebending stiffness degradation factor r j is found through Eq.(13), while the shearing and axial stiffness degradation factorst j and n j are similarly found as,

t j =tcj tpj

tcj + tpj − tcj tpj(19a)

n j =ncj n pj

ncj + n pj − ncj n pj. (19b)

Also in Fig. 10(b), fi and di (i = 1, 2, . . . , 6) arerespectively the local-axis joint forces and deformationscorresponding to the local stiffness matrix k for the compoundframe element, with account for the effects of sheardeformation and geometrical non-linearity. The local-axisstiffness matrices for all elements are transformed into theglobal coordinate system and then assembled as the structurestiffness matrix Ki , where subscript i refers to the i th stage ofthe incremental-load analysis procedure. If Ki is non-singularat the end of the i th load step, corresponding incremental nodaldisplacements 1ui are solved for and incremental member-endforces 1fi and deformations 1di are found. As well, total nodaldisplacements ui =

∑1ui , member-end forces fi =

∑1fi

and deformations di =∑

1di accumulated over the loadinghistory are found. The initial-yield and full-yield conditionsfor each member-end section are checked to detect plasticbehaviour, and the corresponding bending, shearing and axialstiffness degradation factors are found. Degraded stiffnessesRc, Tc and/or Nc are determined based on the moment, shear

and axial forces found by the analysis at the current loadinglevel. Degradation factors (rp, tp, n p, rc, tc, and nc) are appliedto modify each element stiffness matrices k and, hence, thestructure stiffness matrix K before commencing the next loadstep. The incremental-load analysis procedure continues untileither a specified load level F is reached or the structurestiffness matrix K becomes singular at a lower load level, asa consequence of failure of part or all of the structure. (If thestructure has not failed at load level F, the analysis can becontinued beyond that level until failure of the structure doesoccur.)

The final analysis results include the values of thebending, shearing and axial stiffness degradation factorsr, t and n indicating the extent of the combined memberplasticity and semi-rigid connection deformation in the beam-to-column connection regions of the compound element.Further computational details are provided through the analysisexample presented in the following section.

5. Example studies

Three examples of semi-rigid structural steel frameworksare selected to illustrate the analysis method proposed in theforegoing. The objective of the first example concerning asemi-rigid portal frame is to compare the results obtained bythis study with those obtained from experimental testing [16].The second example illustrates a comparison study of aone-bay by two-storey semi-rigid frame designed by Chenet al. [2]. Finally, the two-bay by two-storey frame describedin Refs. [5,6] is revisited to investigate the influence of semi-rigid connections on the analysis results. In all analyses,Young’s modulus E = 200 000 MPa and shear rigidity G =

77 000 MPa. The residual stress for bending and axial behaviouris σr = 0.3σy , while for shearing behaviour it is τr = 0.05τy ,where σy and τy are respectively the normal yield stress andshearing yield stress of the steel material for each example.

5.1. Example 1: Semi-rigid portal frame

For the semi-rigid portal frame in Fig. 12, for whichexperimental test results are available in the literature [16], theproperties of the beam are: area A = 4740 mm2, moment ofinertia I = 5547 × 104 mm4, plastic modulus Z = 485 ×

103 mm3, normal yield stress σy = 345 MPa, and shear yieldstress τy = 199 MPa (based on the von Mises criterion).The properties of the two columns are: area A = 7600 mm2,moment of inertia I = 6103 × 104 mm4, plastic modulusZ = 654 × 103 mm3, and yield stresses σy = 336 MPa andτy = 194 MPa.

The semi-rigid connections are modeled by the four-parameter model in Fig. 4 [9], for which the parametervalues are obtained from the following pilot-test results.The moment–rotation test results for the beam-to-columnconnection C1 are traced in Fig. 13(a) as the dotted curve.By applying a curve-fitting technique to the model parametersin Eq. (4), the four parameters are determined to be M0 =

79 kN m, Rce = 7202 kN m/rad, Rcp = 144 kN m/rad


Fig. 12. Example 1: Portal frame [16].

(a) Beam-to-column connection C1.

(b) Column-to-base connection C2.

Fig. 13. Example 1: Moment–rotation relations for connections.

and γ = 0.57. Similarly, for the column-to-base connectionC2, whose pilot-test results shown in Fig. 13(b), the modelparameters are determined to be M0 = 148 kN m, Rce =

24 721 kN m/rad, Rcp = 151 kN m/rad and γ = 0.78.To match the experimental test setup, the loads for the

analysis procedure in this study are monotonically increasedup to the collapse load level by incrementally changing thehorizontal load H , while the vertical loads remain fixed at thoseshown in Fig. 12. The beam is divided into three elements,and each column is taken as one element. The analysis resultsconcerning the load–deflection behaviour of joint 6 are given bythe solid line in Fig. 14(a). Also shown are the test results [16]and the computed results from a refined plastic hinge analysismethod called PHINGE [2]. It is obvious at lower loading levels(H < 35 kN) that the load–deflection results found by thisstudy (heavy-solid curve) and the PHINGE method (dashedcurve) are in good agreement with each other and the test results(dotted curve). At higher loading levels (H > 40 kN), theresults of the current study are slightly less than those of the

(a) Force–deflection curves.

(b) Plasticity in semi-rigid frame.

Fig. 14. Example 1: Load–deflection responses and plasticity formation.

PHINGE method, most likely because the latter method doesnot account for the influence of elastic shear deformation.

As is shown in the following, the behaviour of the portalframe is such that semi-rigid connection behaviour rather thanmember inelasticity dominates. The proposed method predictsthat the structure collapses at load level H f = 74 kN, whichis close to the value of 77 kN as predicted by the PHINGEmethod, but both values are considerably less than the 99kN value found as the limit state by the experimental test. Itlikely that this discrepancy between experimental and analyticalresults is as a consequence of the analysis methods usingconnection behaviour data which were determined by separatepilot experiments [16], but which differ from that for thebehaviour of the connections in the actual frame itself.

It is evident in Fig. 14(b) that the development of plasticityat the member ends is not very significant. This occurs becauseconnections C1 and C2 have nominal moment capacities Mn =

82 kN m and Mn = 151 kN m, respectively, which are notmuch greater than yield moment capacities My = 100 kN mand My = 134 kN m of the beam and columns, respectively.Upon referring to the regions defined in Fig. 5 and discussedin Section 3, it is observed that connection C1 is an under-strength connection (Mn < My) while C2 is a partial-strengthconnection (My < Mn < Mp). This is consistent with theplasticity distribution indicated in Fig. 14(b), where 6% and 3%plasticity occurs at joints 4 and 5 of the beam, respectively, andonly 1% plasticity occurs at the bottom end of the right column.

As summarized in Table 4, the C26 column base experiencesrp = 100(1 − 0.994) = 0.6%(≈1%) of plasticity. However,the values of connection stiffness degradation factor rc atthe bases of columns C13 and C26 reduce from their initial


Table 4Example 1: Stiffness degradation factors

Member End Initial Semi-rigid Rigidrc0 rc θc (rad) r p r Case 1: r Case 2: r

C13 E1 0.671 0.085 0.0259 0.998 0.085 0.482 –C26 E2 0.671 0.084 0.0261 0.994 0.084 0.000 –B34 E3 0.168 0.019 0.0236 1.000 0.019 1.000 0.572B56 E6 0.168 0.012 0.0443 1.000 0.012 0.000 0.000

value of 0.671 to approximately 0.085; i.e., the degradationfactors rc decrease by about 87% (0.085/0.671 − 1 = −0.87)

compared with their initial values. For the beam-to-columnconnections, the factor rc varies from 0.168 to 0.012; i.e., adecrease of about 93% (0.012/0.168 − 1 = −0.93). Inessence, then, the stiffness degradation factors of the compoundbeam and column elements are the same as those of theconnections, as shown in Table 4. These results indicate fora framework with low-strength semi-rigid connections thatconnection behaviour rather than member plasticity dominatesthe non-linear response of the structure. Note also fromTable 4 that the rotations of the semi-rigid connections at thecorresponding member ends range from 0.0236 to 0.0443 rad atthe overall structure failure state, and these values are beyondthe nominal rotation capacity of 0.02 rad in AISC [12].

Also shown in Fig. 14(a) are two special cases where it isassumed for the portal frame that some or all of the connectionsare rigid. When both the beam-to-column and beam-to-baseconnections are rigid, it is observed from the correspondingload–deflection behaviour that the deflection at limit load levelH f = 143.3 kN is only about one-fifth of that for semi-rigid connections. When the beam-to-column connections areassumed to be rigid while the column-to-base connections arepinned, a conventional situation in design, the correspondingload–deflection behaviour is close to that when the connectionsare semi-rigid, with frame limit load capacity H f = 82.4 kN.The plasticity behaviour of the frame members at the limit statefor the two cases is exhibited in Fig. 15. From Fig. 15(a), thecase of all rigid connections, four plastic hinges (i.e., 100%plasticity) form in the beam and right column, while the leftcolumn base undergoes 52% plasticity under combined axialforce and bending moment. The formation of the fourth plastichinge at node 4 occurs when the horizontal load H f =

143.3 kN, at which point the frame fails due to inelasticinstability signalled by the horizontal displacement of node6 becoming infinitely large (i.e., the corresponding stiffnesscoefficient tends to zero and causes the structure stiffnessmatrix to become singular). From Fig. 15(b), the case ofbeam-to-column rigid connections and column-to-base pinnedconnections, the beam experiences more plastic deformationthan the columns. The formation of the plastic hinge at theright end of the beam occurs when the horizontal load reachesH = 77.7 kN. At the limit load level H f = 82.4 kN, theframe fails due to inelastic instability signalled by the horizontaldisplacement of node 6 becoming infinitely large (i.e., the samefailure mode as for the rigid frame). Table 4 indicates thedifferent degrees of member-end stiffness degradation for the

(a) Rigid column base.

(b) Pinned column base.

Fig. 15. Plasticity behaviour of rigid frame with different supports.

Fig. 16. Example 2: One-bay by two-storey frame [2].

case where all connections are semi-rigid, and the two caseswhere all or some of the connections are rigid.

5.2. Example 2: One-bay two-storey frame

Consider the one-bay two-storey frame with semi-rigidconnections in Fig. 16. This frame has been analysed and


(a) Bjohovede classification system.

(b) EC3 classification system.

Fig. 17. Example 2: Connection classification.

designed previously by Chen et al. [2] for the loads,member sizes and connections shown in the figure. The leastweight frame design was achieved for the following twocombinations of dead loads (DL), live loads (L L) and windloads (WL): 1.2DL +1.6L L and 1.3WL +1.2DL +0.5L L [12].The latter load combination was found to govern the design [2]and, as such, is alone considered here.

In order to account for the imperfect geometry of the frame,it is assumed that all columns are initially out-of-plumb byh/400, where h is the storey height. Connections C1 andC2 have top and seat angles with double web angles, andare represented by a three-parameter model (i.e., Rcp = 0)using the Mn, Rcp and γ parameters given in Table 5, whichalso reports the values of the plastic moment Mp and initial-yield moment My of the corresponding connected beams. Thenormalized moment–rotation curves of the two connectionsare presented in Fig. 17, where two connection classificationsystems are applied to demonstrate the category of the usedconnections. Both the Bjorhovde classification system [11] inFig. 17(a) and the EC3 classification system [3] in Fig. 17(b)indicate that the two connections are semi-rigid. The rotationaldemands imposed on the connections by the factored gravityloads are smaller than the corresponding rotational capacity

Table 5Example 2: Semi-rigid connection parameters [2]

Connection Mp(kN m)

My(kN m)

Mn(kN m)

Rce(kN m/rad)

Rcp γ

C1 270 164 200 107 804 0 0.81C2 219 134 92 23 269 0 1.27

indicated in Fig. 17 [2]. That is, the connections have adequateductility to allow for the full evolution of plasticity in theconnection. According to the discussion in Section 3 whenthe residual stress of a member section is ignored, and asshown in Fig. 17, the non-dimensional yield stress m y = 0.61and C2 is an under-strength connection while C1 is a partial-strength connection. When the residual stress distribution in themember is taken into account as σr = 0.3σy , the initial-yieldstress m yr = 0.43 and C1 and C2 are both partial-strengthconnections. As a result, in the pure bending case, the roofbeam-ends at connections C2 and the floor member ends atconnections C1 exhibit plasticity that depends on the residualstress levels.

Upon applying the compound-element analysis methodproposed by this study, the lateral load–deflection relationshipat node 8 of the frame up to load-factor level λ f = 1.10is found to be the solid-line curve shown in Fig. 18. Twofully-plastic hinges form at the midspan of the beams. Themember-end plasticity ranges between 9% and 22% at thecolumn ends, and between 1% and 11% at the beam ends. Tofurther see the behaviour of the compound connection-beamelements, corresponding stiffness degradation factors rc, rp,and r as well as initial connection stiffness factors rc0 are listedin Table 6. From this table it is observed that the connectionstiffness degrades significantly; e.g., the connection stiffnessfactor for end E5 of beam B45 drops 94% (0.045/0.766 − 1 =

−0.94) from its initial value rc0 = 0.766 to its final valuerc = 0.045. It is also observed from the fourth and sixthcolumns in Table 6 that although a beam end such as E5undergoes 10.9% (rp = 0.891) plasticity, the member stiffnessdegradation behaviour does not affect the stiffness degradationof the compound elements (i.e., rc = r ). This confirms thatsemi-rigid connections dominate compound-element behaviourwhen the connection moment capacity Mn is significantly lessthan the moment capacity Mp of the connected member, as isthe case for this framework (see Table 5).

The formation of the second plastic hinge at node 7 in Fig. 18occurs when the load factor λ = 1.04. When λ f = 1.101,the frame fails due to inelastic instability instigated by thehorizontal displacement of node 8 becoming extremely large.For the purpose of comparison, the dashed-curve in Fig. 18is obtained by the PHINGE analysis method [2], which findsthe load factor λ f = 1.096. Obviously, the results from thisstudy and PHINGE are in good agreement. It is also observedfrom Table 6 that rotations 0.0374 and 0.0415 rad for the roof-beam connections are greater than 0.0133 and 0.0153 rad for thefloor-beam connections, and only the rotations of roof memberconnections exceed the nominal maximum capacity of 0.02 radper AISC [12].


Fig. 18. Example 2: Comparison with PHINGE [16].


Beam End Initial Semi-rigid Rigidrc0 rc θc (rad) r p r r

B34 E3 0.766 0.055 0.0133 0.901 0.054 0.798B45 E5 0.766 0.045 0.0153 0.891 0.044 0.737B67 E6 0.490 0.005 0.0374 0.991 0.005 0.982B78 E8 0.490 0.004 0.0415 0.991 0.004 0.983

To consider the difference between a semi-rigid connectiondesign and a conventional rigid connection design, Fig. 19 alsoincludes the analysis results found by the method proposedin this study for the rigidly-connected frame. Note that theload factor λ f at collapse increases 6.8% from 1.10 forthe frame with semi-rigid connections to 1.18 for the framewith rigid connections. As well, the plasticity formation inthe rigid frame is much different than that in the semi-rigidframe. Specifically, five plastic hinges form at the columnends in the rigid connected frame. This demonstrates thatthe rigid connections transmit substantial bending moments tothe columns. Moreover, plasticity increases from about 10%to 100% at the upper column ends, while the plastic hingesection at node 7 in the semi-rigid frame experiences only 73%plasticity in the rigid frame. Similar to that for the semi-rigidframe, the rigid frame fails due to inelastic instability signalledby the horizontal displacement of node 8 becoming extremelylarge.

5.3. Example 3: Two-bay two-storey frame

The third example illustrated in Fig. 20 is a two-bay by two-storey frame with semi-rigid connections, which was previouslyanalysed in [5,6] without rigid connections. The loads shownin Fig. 20 are at the design load level for the frame. Theframe is investigated here to demonstrate the effect of semi-rigid connections on structural response up to failure. Twoconnection cases are analysed: (1) under-strength semi-rigidconnections, and (2) fully-rigid connections.

In the first case, the two connection curves from Example2 are applied to the frame in Fig. 20. The connection modelparameters for all floor beam-to-column connections C1 and

Fig. 19. Example 2: Comparison with rigid connection analysis.

Fig. 20. Example 3: Frame and service-level design gravity loading.

C2 assume the values in the second row of Table 5, while theparameters for all roof beam-to-column connections C3 and C4are those in the third row of Table 5. According to the memberMp moment values given in the second column of Table 7, andthe connection Mn moment values given in the fourth columnof Table 5, the frame has under-strength connections becauseMn < My (= 0.7 Mp/1.15, assuming residual stress 0.3σy andshape factor 1.15 for the W-section). From the stiffness criteriadefined in Eqs. (14) and (16), and the rc0 values given in thethird column of Table 8, all the connections are categorizedas being semi-rigid because 0.143 < rc0 < 0.893. This isconsistent with the classification results shown in Fig. 17.

Upon applying the compound-element analysis method,the semi-rigid frame was found to collapse when the loadfactor reached λ f = 0.688 (i.e., at 68.8% of the specifiednominal design load level), as indicated by the heavy solid-line curve in Fig. 22. To consider the difference between thesemi-rigid connection design and conventional rigid connectiondesign, the analysis results obtained by other methods for rigidconnections are also indicated in Fig. 22. It is seen from thefigure that the collapse-load factor λ f decreases by 36.3% from1.08 for the rigid frame to 0.688 for the semi-rigid frame, andthat large lateral translation occurs for the semi-rigid frame. Inaddition to the significant changes in the loading capacity, theplasticity formation in Fig. 23 for the semi-rigid frame variessubstantially from that for the rigid frame [6]. Because of theunder-strength semi-rigid connections, all of the member ends


Table 7Example 3: Semi-rigid connection parameters [10]

Connection Mp (kN m) Mn (kN m) Rce (kN m/rad) Rcp (kN m/rad) γ

C1 (CF6 − U12 × 96) 995 1736 1 240 000 56 900 1.39C2 (EP8 with shim) 2773 3252 15 300 000 81 600 1.20C3 (CF5-U10x49) 387 867 893 000 30 300 1.18C4 (CF5-U10x68) 1240 1494 1 020 000 46 100 1.69


Beam End Semi-rigid Fully-rigid Rigidrc0 rc θc (rad) r p r rc0 rc θc (rad) r p r r

B45 E4 0.315 0.131 0.0011 1.000 0.131 0.841 0.837 0.0000 1.000 0.837 1.000B45 E5 0.315 0.117 0.0013 1.000 0.117 0.841 0.221 0.0036 0.472 0.168 0.000B56 E5 0.231 0.002 0.0257 1.000 0.002 0.977 0.190 0.0056 0.693 0.173 0.847B56 E6 0.231 0.013 0.0074 1.000 0.013 0.977 0.857 0.0002 0.992 0.842 1.000B78 E7 0.252 0.134 0.0025 1.000 0.134 0.928 0.919 0.0000 1.000 0.919 1.000B78 E8 0.252 0.192 0.0012 1.000 0.192 0.928 0.361 0.0023 0.521 0.255 0.000B89 E8 0.158 0.001 0.0380 1.000 0.001 0.892 0.274 0.0089 0.798 0.256 0.891B89 E9 0.158 0.009 0.0128 1.000 0.009 0.892 0.299 0.0036 0.970 0.288 0.952

(a) Bjohovede classification system.

(b) EC3 classification system.

Fig. 21. Example 3: Connection classification.

at the connections do not undergo any plasticity, as indicated inthe fifth column of Table 8. However, the connection stiffnessfactors associated with the beams in the right large-span baydrop almost to zero, as shown in column four of Table 8.The factors relevant to the left short-span bay drop by about24%–63% when the rc values in column four are comparedwith the r values in column six of Table 8, it is noted that

Fig. 22. Example 3: Comparison with rigid-connection analysis.

the compound-element behaviour is dominated by that of theunder-strength connections. Similar to the failure mode of theframes in the previous two examples, the two-bay by two-storeyframe fails at load-factor level λ f = 0.688 due to inelasticinstability signalled by the horizontal displacement of node 9becoming extremely large.

From the load–deflection response designated by the heavysolid-line curve in Fig. 22, the flexibility of the frame increasesconsiderably when the connections are semi-rigid. As such,serviceability design requirements might not be satisfied dueto excessive deflections. To enhance the stiffness and strengthof the frame, stiffer connections are now selected whilemaintaining all the same member properties. Specifically, theparameters of the moment–rotation connections in the last threecolumns of Table 7 are chosen for the structural analysis.(The four parameters for each connection in Table 7 areobtained from the research of Kishi et al. [10] concerningextended end-plate connections). According to the memberMp and connection Mn values in the second and thirdcolumns of Table 7, the connections C1, C2, C3 and C4are categorized as being full strength. Also, from the rc0


Fig. 23. Example 3: Plasticity at failure load-factor level λ f = 0.688.

values in column seven of Table 8, and the criteria in Eqs.(13), all of the connections are categorized as being fully-rigid per initial-stiffness criterion. This is consistent with theresults from the Bjorhovde classification system as indicatedin Fig. 21(a) except that limited portions of connections C2and C4 curves pass through the semi-rigid region. If theEC3 classification system is employed, however, as shown inFig. 21(b), only connection C3 curve reaches the boundaryof rigid-connection region, while the other three connectioncurves are close to the rigid boundary but remain in the semi-rigid region.

After conducting the non-linear analysis for the fully-rigidframe, the plasticity distribution for the members, the lateralload–deflection curve at joint 9, and the degradation factorscorresponding to the connections, are found to be as givenin Figs. 24 and 22 and Table 8, respectively. These results revealthe following structural behaviour. First, the loading capacityof the fully-rigid frame is the same λ f = 1.08 value as forthe conventional rigid frame, as indicated in Fig. 22. Secondly,the structural stiffness of the fully-rigid frame is considerablygreater than that of the semi-rigid frame, and approaches thestiffness of the conventional rigid frame. From Fig. 22, theheavy dotted-line load–deflection curve at joint 9 of the fully-rigid frame almost coincides with the curve of the rigid framewhen the load factor λ is below 0.6. Unlike the load–deflectionbehaviour of the conventional rigid frame, however, there is nosudden kink transition as the external loading approaches thelimit state. Thirdly, the plasticity distribution of the fully-rigidframe is much different from that of the semi-rigid frame inFig. 23, but close to that of the conventional rigid frame. For thefully-rigid frame, plastic hinges appear at the top end of columnC25, the top and bottom ends of column C69, and the midspanof beam B56, as indicated in Fig. 24, while the three plastichinges shown in Fig. 23 for the semi-rigid frame no longerappear. More importantly, as indicated in Fig. 24, plasticitynow appears at the beam member ends linked to the stifferfully-rigid connections. Upon comparing Fig. 24 with Fig. 14in Ref. [6], it is observed that all the columns have similarplasticity behaviour except for the different order of the plastichinge formation. However, a significant difference betweenthe plasticity formation in these two figures is observed forthe compound elements where plasticity occurs; more detailedinformation in this regard is shown by the rp and r values in the

Fig. 24. Example 3: Plasticity at failure load-factor level λ f = 1.08.

ninth and last columns of Table 8. For instance, end E8 of beamB78 has a considerable difference in stiffness factors, rangingfrom rp = 0.521 for the fully-rigid connection to r = 0.00 forthe conventional rigid connection. It is noted that large rotationsof semi-rigid connections at failure state are reached when low-strength connections are selected, as shown in column 5 ofTable 8. Rotation 0.0257 rad for E5 of beam B56 and rotation0.0380 rad for E8 of beam B89 exceed 0.02 rad of the nominalrotation capacity. As expected for the frame with fully-rigidconnections, the connection rotations are considerably small asshown in column 10 of Table 8. Finally, after the formation ofthe fourth plastic hinge at the midspan of beam B56 occurs atload factor λ f = 1.08, the frame fails due to inelastic instabilitywithout a sudden change of the force–deflection relation likethat for the rigid frame in Fig. 22.

6. Summary and conclusions

A hybrid member model is introduced to include theeffects of both member plasticity and semi-rigid connections,while a four-parameter model is employed to simulatethe non-linear moment–rotation relationship of semi-rigidconnections. A compound stiffness degradation factor isdeveloped as a function of semi-rigid connection and inelasticmember stiffness degradation factors. The interactive effect ofconnection semi-rigidity and member inelasticity is illustratedfor several semi-rigid frames.

The proposed compound-element approach can be effec-tively used to model the interactive action of member andconnection behaviour. Member plasticity dominates the failurebehaviour of a compound element if the connection is rigid,while non-linear connection behaviour dominates the failurebehaviour of a compound element if the connection is semi-rigid or flexural. The proposed non-linear analysis method us-ing compound elements is effective in predicting the responseof steel frameworks involving semi-rigid connection behaviourand member plasticity up to the collapse state. Connection be-haviour is a pivotal factor that affects structural response. Themember-connection compound element developed in this arti-cle will be used to investigate the influence that both damagedconnections and semi-rigid connections have on the capacity offramework structures to resist progressive collapse failure un-der abnormal loading.


Acknowledgement

This work was funded by the grants from the NationalScience and Engineering Research Council of Canada.

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Liu, Xu & Grierson - Compound-element Modeling Accounting for Semi-rigid Connections and Member Plasticity