experimental and theoretical analysis of the moment

Journal of Constructional Steel Research 63 (2007) 1279–1293www.elsevier.com/locate/jcsr

http://www.paper.edu.cn中国科技论文在线

Experimental and theoretical analysis of the moment–rotation behaviour ofstiffened extended end-plate connections

Yongjiu Shi∗, Gang Shi, Yuanqing Wang

Department of Civil Engineering, Tsinghua University, Beijing 100084, PR China

Received 19 January 2006; accepted 1 November 2006

Abstract

A new theoretical model to evaluate the moment–rotation (M–φ) relationship for stiffened and extended steel beam–column end-plateconnections has been derived in this paper. Based on a specific definition of the end-plate connection rotation, the end-plate connection isdecomposed into several components, including the panel zone, bolt, end-plate and column flange. The complete loading–deformation process ofeach component is then analysed. Finally the loading–deformation process for the whole connection is obtained by superimposing the behaviour ofeach component. In addition, 5 joint tests have been conducted to verify the proposed analytical model. By comparing it with the test results, it hasbeen concluded that this analytical model can evaluate the rotational behaviour of end-plate connections, as well as the moment–rotation (M–φ)curve and the initial rotational stiffness accurately. Furthermore, it can analyse every contribution to the joint’s rotational deformation, such as theshear deformation of the panel zone, the bolt extension, the bending deformation of the end-plate and column flange, etc. This analytical modelalso provides moment–shear rotation (M–φs ) and moment–gap rotation (M–φep) curves, which establish a reliable foundation for analysing thedetailed rotational behaviour of end-plate connections.c© 2006 Elsevier Ltd. All rights reserved.

Keywords: End-plate connection; Semi-rigid; Rotational stiffness; Moment–rotation

1. Introduction

Conventional analysis and design of steel frames are usuallycarried out under the assumption that the connections joiningthe beams to the columns are either fully rigid or ideallypinned. In fact, as is evident from experimental observations,all connections used in current engineering practice possessrigidities which fall between the extreme cases of fully rigidand ideally pinned [1], i.e., the connection is actually semi-rigid. The behaviour of semi-rigid connections significantlyinfluences not only the internal force distribution, but alsothe deformation of steel structures [2]. For most connections,the axial and shearing deformations are usually low comparedto the rotational deformation. Consequently, for practicaldesign, it is essential to determine the connection’s rotationaldeformation. Therefore, almost all the steel design codes ofdifferent countries from all over the world [3–6] require that theeffect of connection deformations should be taken into account

∗ Corresponding author. Tel.: +86 10 6278 2012; fax: +86 10 6278 8623.E-mail address: [email protected] (Y. Shi).

0143-974X/$ - see front matter c© 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.jcsr.2006.11.008

in the global analysis and design of steel frames with semi-rigidconnections. Hence, much effort has been focused in recentyears toward determining connection moment–rotation (M–φ)relationships [7].

Certainly, full-scale and carefully conducted joint experi-ments are the most reliable sources and direct method of ob-taining M–φ relationships. While more than 800 tests of beam-to-column connections have been performed around the worldtoday, only about 300 of them have provided currently usefulmoment–rotation data [2]. Since the connection details consistof a number of components, any changes in these connectiondetails may lead to significant variations in the connection char-acteristics [8]. In addition, many other variable parameters, forexample, details of the fabrication and assembly of the connec-tions, also vary enormously and can affect their behaviour [9].It is impossible to test all of the connections that might be usedin steel construction. Some researchers such as Goverdhan,Nethercot, Kishi and Chen, have collected the available experi-mental results and constructed steel connection data banks thatprovide the user with not only the test data, but also some pre-dictive equations [1]. Where the connections detailing, beam,

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1280 Y. Shi et al. / Journal of Constructional Steel Research 63 (2007) 1279–1293

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Notation

Ae effective area of bolt shank threaded section;E elastic modulus of steel;Eb elastic modulus of bolt;Eh hardening modulus of steel;Ebh hardening modulus of bolt;G shear modulus;Ic f,l moment of inertia of the column panel zone left

flange;Ic f,r moment of inertia of the column panel zone right

flange;M the joint moment;MRd,s the design moment resistance of the panel zone;N1 the transferred tension force of a single bolt in the

first bolt-row;Ni the transferred tension force of a single bolt in the

i th bolt-row;1Ni the actual increase of the bolt tension force in the

i th bolt-row;Vp volume of the panel zone;bep end-plate width;e f distance from the bolt centre to the beam flange

surface;ew distance from the bolt centre to the beam web

surface;fv the steel shear strength;fy the steel yield strength;fby the bolt yield strength;fbu the bolt ultimate tension strength;hbw beam web height;hcw column web height;ht the distance between the centre lines of the beam

top and bottom flanges;1l deformation at the beam tension flange centreline

caused by the bolt extension;1l1 the extension of a single bolt in the first bolt-row;1l2 the extension of a single bolt in the second bolt-

row;lb bolt shank length, equal to the thickness of

plates which the bolt clamped plus the washersthickness;

tc f,l thickness of the column panel zone left flange;tc f,r thickness of the column panel zone right flange;tcw column web thickness;tep end-plate thickness;yi distance from the i th bolt-row centreline to the

neutral axis;y1 distance from the first bolt-row centreline to the

neutral axis;y2 distance from the second bolt-row centreline to

the neutral axis;α the ratio of the compression distribution area

between the contacted surfaces around a bolt onthe bolt shank area;

εi the bolt tension strain of the i th bolt-row.

and column sizes used in frame analysis are significantly dif-ferent from the available experiments, however, the connectionbehaviour retrieved from a database may not correctly repre-sent the actual connections. Furthermore, not every structuralengineer has access to the database of experimental results.

The most widely used semi-rigid connections are the anglecleat and end-plate connections. For end-plate connections,the common approaches for predicting the M–φ relationship,besides experiments, are the T-stub analogy, the yield linetheory and the finite element analysis [10].

Early attempts to develop a design methodology for end-plate connections were based on the T-stub analogy developedby Douty and McGuire [11], Nair et al. [12], Kato andMcGuire [13], Agerskov [14] and others. More recently,methods based on refined yield line analysis have beensuggested, in which the widely accepted design proceduresof end-plate connections are derived from [15–19]. Eurocode3 [3] has formally recommended an M–φ curve for end-plateconnections based on the T-stub yield line theory.

The finite element analysis of end-plate connections wasfirst developed by Krishnamurthy [20–22]. An exhaustiveanalytical study of four-bolt, unstiffened, extended end-plates,along with a series of experimental investigations, leads tothe development of the design procedure found in Ref. [23].Tarpy and Cardinal carried out an elastic finite elementstudy and experimental verification for unstiffened end-plateconnections and also proposed a design methodology [24].Maxwell et al. [25] developed a prediction equation for theultimate moment of the connection and the M–φ relationshipsbased on the finite element method and experimental as well.Sherbourne and Bahaari used 3-D finite elements to analyzeend-plate connections. In addition to the overall behaviour,the contribution of the bolt, end-plate and column flangeflexibility to the connection rotation was singled out [26].With the finite element method, they also studied the structuralproperties of an extended end-plate connected to an unstiffenedcolumn flange [27]. Based on 34 stiffened, extended end-plateconnections and 19 end-plate connections without stiffeners inthe tension region, they produced a single standardized M–φ

function for each of these two connection types by curvefitting [28,29]. Shi et al. used many new functions of the finiteelement method and simulated the mechanical behaviour ofend-plate connections and each component more accurately [9,30,31].

Currently the widely accepted nonlinear M–φ relationshipformulae are the polynomial model, the power model and theexponential model. From a survey of these existing models,it was found that the coefficient and parameters involved inthese modeling formulae, by and large, were calibrated fromthe relevant test results or finite element analysis results.But curve fitting will not give any indications as to howconnection components deform or fail, and provides less helpto designers for improving connection design. In the designM–φ curve recommended by Eurocode 3, the nonlinear partis also determined by curve fitting, and the correspondingresearch is mostly based on the end-plate connections with

Y. Shi et al. / Journal of Constructional Steel Research 63 (2007) 1279–1293 1281

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2

tbwcpsucctsey

rcrotebfla

ota

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emis5af

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nstiffened columns, which may not be suitable for all the end-late connection types. Furthermore, the last part of this curves the horizontal line and this cannot consider the hardeningffect, which leads to great discrepancies with the actualehaviour of the connection [32]. Finite element analysis isrecise and reliable, and can analyze complicated connectionrofiles that may be difficult to investigate by experiment.owever, such finite element analysis generally requires a large

ommercial package and is not therefore feasible for manyractical applications. Most importantly, the effects of torque,ack of fit, construction imperfections and defects, etc., may beifficult to include in a finite element model. A well-definedheoretical method that can be easily carried out and is onlyased on the connection’s details is needed indeed for thenalysis of the end-plate connection M–φ relationship.

. End-plate connection standard details

Many experiments have been performed to investigatehe influences of the connection details on the connection’sehaviour. Ghobarah et al. [33–35] concluded that connectionsith unstiffened columns showed very poor behaviour as

ompared with those that were stiffened. Tsai and Popov [36]ointed out that an end-plate rib stiffener and stronger bolts canignificantly improve the behaviour of end-plate connectionsnder large cyclic loading, and the extended end-plate momentonnections can be designed to develop the full plastic momentapacity of the beam under cyclic loading, and the effect ofhe prying force was reduced by the use of the end-plate ribtiffener. Adey et al. [37] proposed that the application ofxtension stiffeners increases the connection flexural strength,ield rotation as well as energy dissipation capacity.

Summarising the available test results and other relevantesearch results, standard details of end-plate momentonnections for multistorey steel frames, especially in seismicegions, can be proposed as follows: The end-plate extendsn both sides; the column flange and end-plate are stiffened;he thickness of the column flange stiffener and the end-platextension stiffener should be no less than the thickness of theeam flange and web respectively; the thickness of the columnange is equal to the end-plate within the range of 100 mmbove and below the extension edge of the end-plate.

In this paper, an analytical model for the M–φ relationshipf this type of end-plate connection has been proposed andhe corresponding joint tests have been conducted to verify thenalytical results.

. Test specimens

Five specimens of stiffened and extended beam-to-columnnd-plate connections with various details are tested underonotonic loads. A sketch of a typical connection specimen

s shown in Figs. 1 and 2. The out-of-plane deformation ofpecimens was restrained during tests. The details of these

specimens are shown in Table 1 and Fig. 3. The beamnd column sizes with welded I-shaped cross-sections usedor all these 5 specimens were identical. The section depths,

Fig. 1. Test specimen and loading arrangement.

Fig. 2. Testing set.

able 1ypes and details of specimens

pecimen number End-plate thickness (mm) Bolt diameter (mm)

PC-1 20 20PC-2 25 20PC-3 20 24PC-4 25 24PC-5 16 20

eb thicknesses and flange thicknesses of the columns andeams are 300 mm, 8 mm and 12 mm, and the flange widthsre 250 mm and 200 mm, respectively. The thickness of theolumn flange is equal to that of the end-plate within the rangef 100 mm above and below the extension edge of the end-late. The thicknesses of the column stiffener and end-plate ribtiffeners are 12 mm and 10 mm respectively.

Full penetration welds are applied between the end-plate andeam flanges as well as the column flange splices, and the otherelds, including the welds between the flanges and webs ofeams and columns, end-plates and beam webs, are fillet weldsith 8 mm leg size.



Table 2Material properties

Material Measured yieldstrength (MPa)

Measured tensilestrength (MPa)

Measured elasticmodulus (MPa)

Design value of boltpre-tension force (kN)

Measured bolt averagepre-tension force (kN)

Steel (thickness ≤ 16 mm) 391 559 190 707 – –Steel (thickness > 16 mm) 363 537 204 228 – –Bolts (M20) 995 1160 – 155 185Bolts (M24) 975 1188 – 225 251

Table 3Test results

Specimennumber

Loadingcapacity (kN)

Momentresistance (kN m)

Moment resistance changecompared with EPC-1 (%)

Initial rotationalstiffnessS j,ini(kN m/rad)

Initial rotationalstiffness changecompared withEPC-1 (%)

Failure mode

EPC-1 286.4 343.7 0.0 52 276 – Bolt fractureEPC-2 268.4 322.1 −6.3 46 094 −11.8 Bolt fractureEPC-3 325.3 390.3 13.6 46 066 −11.9 Buckling of beam flange and

web in compressionEPC-4 342.3 410.8 19.5 47 469 −9.2 Buckling of beam flange and

web in compressionEPC-5 296.1 355.4 3.4 41 634 −20.4 Bolt fracture and buckling of

end-plate rib stiffener incompression

Fig. 3. Details of connections.

The steel is grade Q345 (nominal yielding stress fy =

345 MPa), and the bolts are high strength friction-grip bolts(Grade 10.9). The material properties of the steel and boltsare obtained from tensile tests on coupons and from the bolts’certificate of quality, as shown in Table 2. The proof elasticmodulus of the bolts is taken as 206 000 N/mm2.

One line of bolts on each connection was instrumented usingstrain gauges, and these bolts are numbered in Fig. 3. Twoshallow slots were grooved symmetrically on the unthreadedportion of the bolt shank, and in each slot a strain gaugewas fixed and covered with resin for protection (Fig. 4).The alignment of the two strain gauges was secured to beperpendicular to the beam flange during the tightening of thebolts, so that the maximum and minimum strain of the boltin the connection moment plane could be measured. The boltaxial force can be calculated by taking the average of thetwo strain gauges’ measured values. All bolts were tightenedby the calibrated wrench method. The design values of bolt

Fig. 4. A gauged bolt.

pre-tension forces and the actually applied bolt pre-tensionforces are listed in Table 2. The contact surface between theend-plate and column flange was prepared by blasting, with aslip coefficient 0.44.

Fig. 1 displays all the displacement transducers installedto measure the joint deformation. No. 1 was used to monitorthe displacement at the loading point. Nos. 2–10 measure therelative deformation between the end-plate and column flange.Nos. 11 and 12 measure the inner shearing deformation of thepanel zone. Nos. 13 and 14 were arranged next to the columnstiffeners to measure the shearing deformation of the panelzone. No. 15 measures the slippage between the end-plate andcolumn flange.

4. Test results and discussion

4.1. Rotational stiffness and M–φ, M–φep, M–φs curves

The test results of the specimens are summarised in Table 3.The failure modes of specimens are shown in Fig. 5. In thesetests, the fracture and necking positions of all the failed boltsappear on the threaded portion and not at the strain gauge slots,as shown in Fig. 5, which indicates that these grooved slots donot damage the bolt capacity.

With regard to the joint rotation φ, the conventionaldefinition is suggested as the angle change between the beam


atibiptflcTr∆lcδ

tdt2Mts

pmbcl


Fig. 5. Failure modes of the specimens.

Ss3b2isthsass41c

pEs21in

Fig. 6. Definition of joint rotation.

nd the column from its original configuration [1]. In this paper,he joint rotation φ of the beam-to-column end-plate connections defined as the relative rotation between the centre lines of theeam top and bottom flanges at the beam end, and it usuallyncludes two parts: the shearing rotation φs , contributed by theanel zone of the column, and the gap rotation φep, caused byhe relative deformation between the end-plate and the columnange, including the bending deformation of the end-plate andolumn flange as well as the extension of the bolts (Fig. 6).he shearing rotation φs is calculated by ∆/ht , and the gap

otation φep is calculated by δ/ht , giving φ = φs + φep, whereis the displacement difference of the panel zone at the centre

ines of the top and bottom beam flanges at the beam end whichan be measured by displacement transducer Nos. 13 and 14;is the gap between the end-plate and the column flange at

he beam tension flange centre line, which can be measured byisplacement transducer No. 4; and ht is the distance betweenhe centre lines of the top and bottom beam flanges, and is88 mm. M–φ curves of all the specimens are shown in Fig. 7.–φs and M–φep curves of each specimen are also measured

o verify the analytical model, and the comparison results arehown in Figs. 12 and 13.

The loading capacity presented in Table 3 is the maximumushing load applied. The moment resistance is calculated byultiplying the load with the arm of the loading(1.2 m). It can

e seen from Fig. 7 that the M–φ curves of all the end-plateonnection specimens are almost linear when the moment isess than 60 kN m; therefore the initial rotational stiffnesses,

Fig. 7. M–φ curves of the specimens.

j,ini, of the connections are defined as the secant rotationaltiffnesses up to this bending moment. According to Eurocode[3], S j,ini is compared to the flexural stiffness of the connectedeam EIb/Lb. The connection is rigid when S j,ini is larger than5EIb/Lb for unbraced frames, nominally pinned when S j,ini

less than 0.5EIb/Lb, and semi-rigid when S j,ini is betweenese two values. If we assume that the natural beam length of

pecimens is 1.2 m × 2 = 2.4 m, then EIb/Lb is 9751 kN m,nd S j,ini for all the specimens are about 4.3–5.4 times EIb/Lb,o all the tested connections are semi-rigid. If the tested beamection sizes are applied in 4–5 m span frames and EIb/Lb is681–5851 kN m, the Sj,ini of all the connections are about 7.1–1.2 times EIb/Lb, and therefore all the connections are stilllassified as semi-rigid.

The influence of connection details on the behaviour of end-late connections can be analyzed from Table 3 and Fig. 7.PC-1 is a reference specimen, and the other connectionpecimens alter only one or two parameters from EPC-1. EPC-has increased the end-plate thickness compared with EPC-

, but its moment resistance decreases remarkably and itsitial rotational stiffness changes less; its rotational stiffness



Fig. 8. Bolt tension force generated by applied moment.

.t

,

is much larger than EPC-1 when nonlinearity of the M–φ

curve occurs. EPC-3 has increased the bolt diameter, which alsomeans a larger bolt pre-tension force; its moment resistanceis much higher than EPC-1, which fails by bolt fractureEPC-4 has increased both the end-plate thickness and boldiameter; its moment resistance is the largest among all thespecimens; higher rotational stiffness is observed during itsloading process. EPC-5 has reduced the end-plate thicknessbut its moment resistance increases on the contrary, because the

end-plate is thin and its bending stiffness is smaller, leadingto the bolt tension force distribution among the four bolts intension being more uniform. Meanwhile, its initial rotationalstiffness decreases significantly, with excellent ductility androtation capacity.

4.2. Bolt force and distribution

The bolt tension forces generated by the applied momentsare shown in Fig. 8. In this paper, the bolt tension force


pγ

at

t

zc

wom

op4

flt

5

edt


Table 4Moment values of curve of Specimens 1, 2 and 3 (kN m)

Specimen number Curve 1 Curve 2 Curve 3

EPC-1 80 199 248EPC-2 80 210 248EPC-3 100 224 300EPC-4 100 226 289EPC-5 70 209 230

indicates the tension force increment caused by the jointmoment, excluding the pre-tension force. The loading stagescorresponding to curve 1, 2 and 3 in Fig. 8 are listed inTable 4. The applied moment at curve 1 is around the designmoment resistance of the connection from the Chinese codes [6,38,39], where the connection is in the elastic stage. Theapplied moment at curve 2 is around the knee point of theM–φ curves of each specimen, where the nonlinearity andpartial plasticity occur. Curve 3 corresponds to the momentwhere the nonlinearity is obvious and plasticity has developedsignificantly.

From Fig. 8, it can be noted that, for all these connections,generally the neutral axis of the connection is around the centreline of the bolts group, which is on the centre line of theconnection when the bolts are arranged symmetrically aboutthe beam section centre line, and the bolt force generated bythe bending moment is approximately distributed linearly. Thenegative values in the compression zone means that the bendingmoment will reduce the resultant bolt force to levels below thepre-tension level. For EPC-1, EPC-2, EPC-4 and EPC-5, themaximal tension force appears on the first bolt-row, and thetension force of the second bolt-row develops gradually. ForEPC-3, the tension force of the second bolt-row is maximalat the initial loading stage. With the moment increasing, thetension force of the first bolt-row exceeds the second bolt-rowand remains maximal up to failure.

5. Analytical model

As specified above, the end-plate joint rotation φ consistsof the shearing rotation φs and the gap rotation φep. The gaprotation φep is contributed by the flexural deformation of theend-plate and column flange as well as the extension of thebolts. This analytical model will evaluate the M–φs and M–φepcurve respectively, and then superpose these two curves to getthe final M–φ curve.

5.1. Analytical model for M–φs curve

From the relevant research findings [32,40], the M–φs curveadopting a trilinear model is recommended as shown in Fig. 9.Some research [40] has indicated that the restraints around thecolumn panel zone, especially the column flange, will increasethe panel zone loading capacity and its stiffness. The Chinesesteel structural design code [6] simplifies this contributionand presents the calculation equation for the design momentresistance of the panel zone as:

MRd,s =43

fvVp =43

fvhbwhcwtcw =43

·fy

√3

· hbwhcwtcw.(1)

Fig. 9. M–φs model.

When the joint applied moment M is equal to MRd,s , theanel zone yields, and the panel zone shear strain reaches= γy = fv/G. The shear rotation actually is the shear strain,

nd is also the shearing rotation φsy = γy . When M ≤ MRd ,he joint shearing rotation initial stiffness Ks,ini is given by:

Ks,ini =MRd,s

φsy=

MRd,s

γy

=

43 fvhbwhcwtcw

fv/G=

43

· G · hbwhcwtcw. (2)

From Eqs. (1) and (2), it is noted that the restraints aroundhe panel zone increase its resistance and stiffness by 1/3.

According to Krawinkler et al. [40], after yielding, the panelone rotational stiffness can be attributed to the bending of theolumn flanges, and it can be computed as:

K p =125

EIc f,l

tc f,lβ+

125

EIc f,r

tc f,rβ(3)

here β is a factor intended to account for the beneficial effectf column shear above and below the joint, whose calculationethod can be obtained from Ref. [40].This post-yielding stiffness is developed up to the yielding

f the column flanges occurring for a shear deformation of theanel zone, which can be approximately assumed to be equal toγy . Therefore, the corresponding moment is given by:

Mp = K s,ini · γy + K p · 3γy . (4)

After the development of this moment, i.e., the columnange yielding, the panel zone rotational stiffness attributed to

he strain hardening can be computed as:

Ks =Eh

EKs,ini. (5)

.2. Analytical model for M–φep curve

(i) Analysis assumptionsFrom the test results obtained by this paper, and the typical

nd-plate connections meeting the abovementioned standardetails and requirements, we can simplify and assume thathe bolt tension strain generated by the applied moment is


t

)

)

)

ls

)

s

)

)

y

,l

)

)

l,

)

)

,

f


Fig. 10. Bolt tension strain and force generated by the applied moment.

approximately a linear distribution, and the neutral axis of theconnection is located approximately at the beam mid-height,and the bolt tension strains in each of the two rows of boltsare considered as equal. Here the specified bolt tension strainis the increment caused by the applied joint moment, excludingthe pretension strain. Until the end-plate and the column flangeseparate, the bolt force contributed by the bending momentcan be assumed to be linearly distributed as shown in Fig. 10.The tension force indicates an increment in the bolt force andcompression force indicates a decrement in the bolt force.

(ii) Joint rotation contributed by bolt extensionBefore the moment is applied to the connection, the bolts

are tightened to the bolt pre-tension force P . From the staticequilibrium, the contact force C between the end-plate and thecolumn flange around this bolt is also equal to P .

With the joint moment developing, the bolts in tension canexperience the following three stages:

Stage 1: The first stage lasts until the end-plate and thecolumn flange separate around bolt location, i.e., εi ≤ ε0.During this stage, with the joint moment developing, the contactforce between the end-plate and the column flange decreaseswhile the bolt tension force increases. Assuming the appliedmoment is M , and the contact force becomes C ′

= C − 1C ,and the total bolt tension force is increased to Ni = P + 1Ni ,where C is the initial contact force between the end-plate andcolumn flange under pre-tension P; 1C, 1N are the contactforce decrement and bolt tension increment caused by appliedmoment. From the static equilibrium, it can be obtained that1C = α1Ni , and α is the ratio of the contact area around a boltto the bolt shank area, according to the analysis results [31], thevalue of α can be taken as 10 conservatively.

The actual increment in bolt tension force can be expressedas

1Ni = Eb Aeεi (6)

and the tension force on this bolt is given by

Ni = (α + 1)1Ni (7)

and the extension deformation of this bolt is given by

1li = εi · lb. (8)

Whenever the contact force between the end-plate and thecolumn flange around this bolt is reduced to zero, i.e., the end-plate and the column flange separate around bolt, the decrementof this contact force is given by 1C = α1Ni = P , so that

1Ni =Pα

. (9)

Substituting Eq. (9) into Eqs. (6)–(8), it can be obtained thaat this moment, the bolt tension strain is given by

ε0 =1Ni

Eb Ae=

PαEb Ae

. (10

The total bolt force is

Ni =α + 1

αP (11

and the extension deformation of this bolt is

1li = ε0 · lb. (12

Stage 2: The second stage starts from the separation of theend-plate and the column flange around the bolt and lasts untibolt yielding in tension, i.e., ε0 < εi ≤ εby − εp. During thistage, the actual increase of the bolt tension force can still becalculated by Eq. (6), and the total tension force on this bolt is

Ni = P + Eb Aeεi (13

and when yielding, the total bolt strain is equal to the yieldingstrain εby =

fbyEb

, and the actual bolt tension strain increment igiven by

εi = εby − εp =fby

Eb−

PEb Ae

=fby Ae − P

Eb Ae. (14

The total tension force of this bolt is given by

Ni = fby Ae (15

where εp is the bolt pre-tension strain taken as εp =P

Eb Ae; fb

is the bolt yield strength.Stage 3: The third stage starts from the bolt yielding in

tension until the bolt reaches its ultimate tension strengthi.e., εby − εp < εi ≤ εbu − εp. During this stage, the actuaincrement of the bolt tension force is given by

1Ni = fby Ae + Ebh Ae(εi − εby) − P. (16

The total tension force of this bolt is evaluated as

Ni = fby Ae + Ebh Ae(εi − εby). (17

When the bolt reaches its ultimate tension strength, its totastrain is equal to the ultimate tension strain εbu =

fbyEb

+fbu− fby

Ebhand this bolt tension strain increment is

εi = εbu − εp. (18

The total tension force of this bolt is given by

Ni = fbu Ae. (19

The joint rotation contributed by bolt extension deformationwhich is also called bolt extension rotation in the followingtext, can be obtained by the following pivotal points oprocedure (a)–(e). Linking these points with straight lines, themoment–bolt extension rotation curve will be obtained:

(a) The contact force between the end-plate and the columnflange at the first bolt-row is reduced to zero, i.e., at thestage where the end-plate separates from the column flange byapplying moment at the first bolt-row.



Fig. 11. End-plate stiffness.

Here the actual increments on the bolt tension force, the bolttension strain, the total tension force and the extension of thebolt in the first bolt-row can be calculated by Eqs. (9)–(12)respectively.

From the linear distribution assumption, the bolt tensionstrain in tension zone εi can be calculated from

εi =ε1

y1· yi . (20)

The tension force of this bolt can be calculated from Eqs. (6)and (7), and can also be calculated as

Ni =N1

y1· yi . (21)

(b) The contact force between the end-plate and the columnflange at the second bolt-row is reduced to zero, i.e., at themoment the end-plate is separated from the column flange byapplying moment at the second bolt-row.

Here the actual increment on the bolt tension force, the bolttension strain, the total tension force and the extension of thebolt in the second bolt-row can be calculated by Eqs. (9)–(12)respectively.

Similarly, bolt tension strain in tension zone εi can becalculated as

εi =ε2

y2· yi . (22)

(c) The first bolt-row yields.Here the bolt tension strain and the total tension force of the

bolt in the first bolt-row can be calculated by Eqs. (14) and (15)respectively. The bolt tension strain in tension zone εi can becalculated by Eq. (20).

(d) The second bolt-row yields.Here the bolt tension strain and the total tension force of the

bolt in the second bolt-row can be calculated by Eqs. (14) and(15) respectively. The bolt tension strain in tension zone εi canbe calculated by Eq. (22).

(e) The bolt of the first bolt-row reaching ultimate tensionstrength.

Here the bolt tension strain and the total tension force of thebolt in the first bolt-row can be calculated by Eqs. (18) and (19)respectively. The bolt tension strain in tension zone εi can becalculated by Eq. (20).

After the tension strain of the bolts(Fig. 10) in the tensionzone has been obtained, the actual increment on the bolt tensionforce, the total tension force and the extension deformation ofthis bolt can be calculated. The corresponding joint moment andthe bolt extension rotation can be calculated as:

M = 2∑

(Ni · yi ) (23)

φb =1lht

=1l12y1

=1l22y2

=1li2yi

(24)

where,∑

is summation for all the bolts in the tension zone.From the above calculation procedure (a)–(e), we can get

some pivotal points for the joint moment and the correspondingbolt extension rotation. Linking these pivotal points by thesequence of the magnitudes of the corresponding bolt extensionrotation, the moment–bolt extension rotation curve can beobtained, extending the straight line segment by its own slopebeyond the point (e).

(iii) Joint rotation contributed by the bending deformation ofthe end-plate and column flange

The joint rotations contributed by the bending deformationof the end-plate and column flange are called the end-platerotation and the column flange rotation respectively. Thebending deformation of the end-plate and column flange can becalculated according to the bolt tension force obtained by theabove calculation procedure (a)–(e), taking into account onlythe deformation of the end-plate and column flange around onebolt-row on both sides of the beam tension flange.

Since the calculation procedure of the column flangedeformation is the same for the end-plate, the end-plate rotationis given to introduce the calculation method.

For the end-plate connection meeting the above standarddetails requirements, the beam flange, the beam web and theend-plate extension rib stiffener can be considered as the fixedrestraints of the end-plate segment. The end-plate segmentaround the bolt in tension can be separated and simplified intoa two-edge fixed plate [41] as shown in Fig. 11. The arrow isthe bolt clamp force. As the pre-tension bolt clamps the plateseffectively, it can be assumed that the end-plate segment is fixedat the bolt centre line. Furthermore, this end-plate segment canbe decomposed into two plates with opposite ends fixed. Thestiffness of this end-plate segment kI or kII is equal to the



Fig. 12. Comparison of the M–φs curve from the analytical models and tests.

)

dr

s

eel

summation of stiffness of these two plates:

kI or kII = k1 + k2 (25

where kI or kII is the stiffness of the end-plate segment arounone bolt of the first and second bolt-rows respectively. If b1 o

b2 is larger than ew + e f , b1 or b2 is set equal to ew + e f , ashown in Fig. 11.

In particular, when the plate bending spans are of thsame order of magnitude as the plate thickness itself [42], thplate’s shear deformation cannot be neglected. From materia



Fig. 13. Comparison of the M–φep curve from analytical model and tests.


)

)

r

1

-

s.,ts

es

)

e

)

e

.

)

tee

t

see

e

rteese

,

,e.ee

-

s.e-e

-e

e

ee,

sf

ee


mechanics, the k1 and k2 can be expressed as

k1 = β11

e3w

12EI +αs ew

GA

= β11

e3w

Eb1t3ep

+αs ew

Gb1tep

(26

k2 = β21

e3f

12EI +αs e fGA

= β21

e3f

Eb2t3ep

+αs e f

Gb2tep

(27

where αs = 1.5, is a parameter taking into account the sheadeformation; β1 = 1− A2/(b1ew) and β2 = 1− A1/(b2e f ) arethe reduction coefficients, because the stiffness of the areas Aand A2 in Fig. 11 have been calculated repeatedly.

After yielding occurs at the section edge of the endplate segment and it reaches its bending resistance, accordingto references [3,32], the end-plate segment stiffness iapproximate to 1/7 of its initial stiffness, i.e., kI /7 or kII/7After the total cross-section of the end-plate segment yieldsit reaches its ultimate bending resistance. Taking into accounthe strain hardening, the end-plate segment stiffness is taken aEhE kI or Eh

E kII .From Ref. [41], the bolt tension force corresponding to th

bending resistance of a two-edge fixed end-plate segment igiven by

Ny =bept2

ep fy

6e f+

(e f + ew)t2ep fy

3ew

. (28

When this segment reaches its ultimate bending resistance, thbolt tension force is taken as

Nu = 1.5Ny . (29

Based on the bolt tension force calculated from the abovcalculation procedure (a)–(e), the end-plate deformation at thefirst bolt-row ∆ep1 and second bolt-row ∆ep2 can be calculatedThe deformation at the centre line of the beam tension flange∆ep, is taken as the average of ∆ep1 and ∆ep2, i.e., ∆ep =

(∆ep1 +∆ep2)/2. The corresponding end-plate rotation is givenby

φep =∆ep

ht. (30

From the above calculation procedure (a)–(e), the joinmoment and the corresponding end-plate rotation can bcalculated. The moment–end-plate rotation curve can bobtained. After point (e), the line slope is taken as Eh

E Ki , whereKi is the initial stiffness of this curve, i.e. the slope of the firsstraight line segment of this curve.

For the actual behaviour of all the end-plate connectionmoment–rotation (M–φ) curves, its tangent stiffness ialways decreasing. Accordingly, it is required to revise thmoment–end plate rotation curve obtained above. Whenever thslope of any straight line segment is larger than its precedingstraight line segment, its slope is taken equal to its precedingstraight line segment, and the moments of the pivotal points donot change.

For the calculation procedure (e), if the edge of the end-platsegment does not yield, i.e. the bolt tension force corresponding

to the bending resistance of the end-plate segment is largethan the bolt’s ultimate tension resistance, this indicates thathe end-plate is rather thick and it does not yield. Here, thslope of the last straight line segment, i.e. the straight linsegment after point (e), is taken to be equal to the slope of itpreceding straight line segment. This circumstance should bavoided during the design. According to the relevant tests andinvestigation results [3,32,43], the failure of the end-plate inbending can provide the joint with higher deformation capacitywhile the failure of bolts in tension is much less dissipative andless deformable. The seismic design rule of “strong connectionweak plate” should be adopted. It also suggested applying thend-plate with moderate thickness and adequately stiffeningThe bolt diameter should be adequate so that the bolt ultimattension resistance is higher than the end-plate segment ultimatbending resistance, in order to assure the joint ductility andenergy dissipation capacity.

For the end-plate connection with a strong bolt and weakend-plate, it may happen that the bending resistance of the endplate segment at the first bolt-row is below the bolt tensionforce of the first bolt-row when the end-plate and columnflange separate at this position, i.e. the end-plate segment yieldprior to the separation of the end-plate and column flangeHere a pivotal point should be added to the moment–end-platrotation curve, which is the bolt tension force of the first boltrow corresponding to the yielding resistance of the end-platsegment at this position. The relevant bolt transferred tensionforce, the bolt extension, the bending deformation of the endplate and the corresponding end-plate rotation can then bcalculated.

(iii) Moment–gap rotation (M–φep) curveThe moment–gap rotation (M–φep) curve can be obtained

by superposing the moment–bolt extension rotation curve, thmoment–end plate rotation curve and the moment–columnflange rotation curve.

The calculation method for the moment–column flangrotation curve is the same as that for the moment–end platrotation curve. If the column flange is wider than the end-platethe values of b1 and b2 for the column flange should be takenas the values of the end-plate.

5.3. Moment–rotation (M–φ) curve

The moment–rotation (M–φ) curve can be obtained bysuperposing the moment-shearing rotation (M–φs) curve andthe moment–gap rotation (M–φep) curve. During the procesof superposing, for the moment value of each pivotal point othe M–φs curve and M–φep curve, adding the correspondingshearing rotation φs and gap rotation φep, the joint rotation φ

is equal to φs + φep. Linking these pairs of M and φ, the M–φ

curve of the end-plate connection can be obtained.

6. Comparison of analytical and test results and discussion

The M–φ curves for test specimens EPC-1–EPC-5 havbeen calculated using the above analytical model, and thcomparison between the M–φs curves, the M–φep curves and



Fig. 14. Comparison of the M–φ curves from the analytical model and tests.

the M–φ curves of the analytical models and test results areshown in Figs. 12–14. The comparison between the joint initialrotational stiffness, S j,ini, of the analytical models and testresults are listed in Table 5. The value of S j,ini from the

analytical model is taken as the slope of the first straight linesegment of its M–φ curve. In Figs. 12–14 and Table 5, Model1 denotes that the analytical results are calculated according tothe material property values specified in the current Chinese


2/


Table 5Comparison of S j,ini of specimens from analytical and test results

Specimen number Model 1 Model 2 Test Model 1/ Model(kN m/rad) (kN m/rad) (kN m/rad) Test Test

EPC-1 49 690 46 811 52 276 0.95 0.90EPC-2 53 911 50 577 46 093 1.17 1.10EPC-3 49 845 39 712 46 066 1.08 0.86EPC-4 54 136 50 775 47 469 1.14 1.07EPC-5 43 441 40 331 41 634 1.04 0.97

Average 1.08 0.98

dtef

ed’s’s’sis

2

,,

d

telalyeiss

tse.es.ye---nctyoye

s

nessonernen

nso-ffhdll,h,

dopne

she

):

steel design code; the yield strength of Q345 steel is fy =

345 MPa; the ultimate strength is fu = 470 MPa; the yielstrength of grade 10.9 bolt is fby = 940 MPa and its ultimatension strength is fbu = 1040 MPa; the elastic modulus osteel is E = 2.06 × 105 N/mm2 and its shear modulus G =

79 × 103 MPa. Model 2 denotes that the analytical results arcalculated according to the material actual properties obtainefrom coupon tests which are listed in Table 2. The steelhardening modulus is assumed to be Eh = 0.04E ; the bolthardening modulus is assumed to be Ebh = 0.1Eb; Poissonratio is taken as µ = 0.3; and the elastic modulus of bolttaken as Eb = 2.06 × 105 N/mm2.

The other connection parameters are taken as hbw =

276 mm, hcw = 276 mm, tcw = 8 mm, Ae = 244.8 mm(M20) or 352.5 mm2 (M24); lb = 48 mm (EPC-1), or 58 mm(EPC-2), or 50 mm (EPC-3), or 60 mm (EPC-4), or 40 mm(EPC-5); α = 10, d f = 288 mm, y1 = 200 mm, y2 = 88 mmew = 49 mm, e f = 50 mm, b1 = 100 mm, b2 = 95 mmβ = 0.856.

From the comparison of the results in Figs. 12–14 anTable 5, it can be concluded that:

(1) The M–φ curves, the M–φs curves and the joininitial rotational stiffnesses obtained from the analytical modcoincide well with the test results if the actual materiproperties are applied. The comparison verifies the accuracof this analytical model. In practical structural design, wherthe nominal material property specified in current codeapplied, the analytical models also give satisfactory solutionwith adequate accuracy.

(2) The M–φep curves obtained from the analytical resulcoincide well with the tests results at the initial loading stagAfter nonlinearity occurs in the loading curves, there are somdiscrepancies between the analytical results and the test result

The possible reasons behind these discrepancies mabe explained as follows: The end-plate and column flangdeformation is evaluated under the assumption that the endplate segment is fixed at the bolt centre line. The endplate and column flange contact closely due to the bolt pretension force at the initial loading stage, and this assumptiois reasonable. With the joint moment increasing, the contaforce between the end-plate and column flange is significantlreduced and even if they separate, this assumption will lead tsome discrepancies, but the discrepancies do not considerablinfluence the evaluation of joint total rotation (Fig. 14). Th

final results of the M–φ curve are in accordance with the testgenerally.

7. Conclusions

(1) In this paper, the stiffened and extended beam–columend-plate connection has been recommended for end-platmoment connections in multistory steel frames, and itstandards details have been proposed. Five full-scale joint testof this type of end-plate connection have been conducted tinvestigate the influences of bolt size and end-plate thickness othe joint behaviour. The rotational stiffness, moment resistancand moment–rotation (M–φ) curves are obtained. A cleadefinition for the end-plate connection rotation has beeproposed. With a special method to measure the bolt strain, thdistribution and the development of bolt tension force have beeobtained during the tests.

(2) A new analytical model to evaluate the moment–rotatio(M–φ) relationship of this type of end-plate connection habeen proposed. The end-plate connection is decomposed intseveral components, including the panel zone, bolt, endplate and column flange. The complete loading process oeach component is analysed. The moment–rotation curve othe whole connection is obtained by superimposing eaccomponent. Comparing with the test results, it has been verifiethat this analytical model can sufficiently predict the rotationabehaviour of end-plate connections, such as the initial rotationastiffness and the moment–rotation (M–φ) curve. Furthermorethe contributions to the joint rotational deformation of eaccomponent, such as shear deformation of the panel zonethe bolt extension, bending deformation of the end-plate ancolumn flange etc. are provided. This analytical model can alsprovide the moment–shear rotation (M–φs) and moment–garotation (M–φep) curves, thus provide a reliable foundatiofor analysing the detailed rotational behaviour of end-platconnections.

Acknowledgements

The writers gratefully acknowledge the support for thiwork, which was funded by the Tsinghua Basic ResearcFoundation (Grant No. JCqn2005006) and the Natural SciencFoundation of China (No. 50578083).

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experimental and theoretical analysis of the moment

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