Journal of Constructional Steel Research 108 (2015) 60–69

Contents lists available at ScienceDirect

Journal of Constructional Steel Research

Shear behaviour of trapezoidal column panels. I: Experiments and finiteelement modelling

Eduardo Bayo a,⁎, Alfonso Loureiro b, Manuel Lopez b

a University of Navarra, Spainb University of A Coruña, Spain

⁎ Corresponding author. Tel.: +34 948425600; fax: +3E-mail address: ebayo@unav.es (E. Bayo).

http://dx.doi.org/10.1016/j.jcsr.2014.10.0260143-974X/© 2014 Elsevier Ltd. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 12 April 2014Accepted 18 October 2014Available online 15 November 2014

Keywords:Column shear panelBeams of unequal depthsExperimental validationFinite element modelling

The behaviour of joints exerts an important influence in the overall performance of steel and compositestructures, both in deformations and in ultimate resistance. As a consequence, a considerable effort has beencarried out in recent years to better characterize the behaviour of joints in these kinds of structures. Moderncodes, including Eurocodes 3 (EC3) and 4 (EC4), have included these research advances so that they may beused in common practice.The method adopted in EC3 and EC4 to characterize the connections is the so-called component method. One ofthe most important components is the column web panel in shear, and its behaviour has been investigatedthoroughly for the case of rectangular shear panels arising when the beams have equal depths. However, thecase of trapezoidal column panels, formed by beams of different depths at each side of the column, has notbeen researched as much. Formulae or design recommendations for the shear behaviour of steel beam-to-column joints with beams of unequal size are not currently included in design codes.The aim of this paper is the characterization of the shear behaviour of trapezoidal column panels. In order toisolate the shear behaviour from other components, stiffened beam to column connections with commercialsections are experimentally tested. Also, finite elementmodelling and analysis are carried out to compare results,and a mechanical model is proposed. Current modelling procedures are tested and the results compared withthose coming from the experiments and numerical simulations.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The characterization of steel joint behaviour and properties has beena matter of research for a number of years, and all the accumulatedknowledge has been compiled to a large extent in currently availablestructural steel design codes. One important aspect is the behaviour ofthe column panel subjected to the shear forces arising from themoments of the adjacent beams as well as the shear forces acting onthe columns. A correct definition of the column panel zone deforma-tions under static conditions is of great importance due to its influenceon the overall sway behaviour of the frame. An increase in frame driftdue to panel zone shear deformation may render the frame unservice-able. This may even happen for commonly considered rigid joints.Modelling of the column panel is also important for the avoidance oflocal failure under ultimate limit state conditions.

Krawinkler et al. [1] reported the importance that panel sheardeformations have on the frame behaviour under lateral loads, and pro-posed a formulation for the stiffness and resistance of shear panels ofbeam to columnconnectionswith beams of equal depths. An alternative

4 948425629.

formulation has been proposed in the Eurocode 3, part 1.8 [2] followingthe componentmethod [3–5]. These and other proposedmethods allowintroducing the flexibility of joints in both the elastic and inelastic rangein order to assess the frame response [6,7] and also including highstrength steel [8].

The lines connecting the flanges of beams of unequal depthsdefine a trapezoid within the panel column zone. Hoogenbroom andBlaauwendraad [9], and Curtis and Greiner [10] have proposed analyti-cal and computational methods to characterize the shear behaviour ofisolated quadrilateral panels. More recently, Hashemi and Jazany [11]have investigated the connection detailing of joints of unequal beamdepths under seismic loads. One of their conclusions is that inclinedstiffeners connecting the lower flanges of the beams perform betterthan the horizontal ones. Jordao et al. [12] have studied the perfor-mance of this type of joints for high strength steel without the use ofweb stiffeners. As a consequence, the stress field at the panel zonebecomes rather complex due to the fact that the compression, tensionand shear components are all coupled together within the panel zone.The common failure mode of the experiments carried out in [12] wasweb buckling due to compression.Within the context of the componentmethod, they propose a joint model based on two subpanels delimitedby 3 levels of load introduction coming from the beams. A suitable

15

COLUMN

SHALLOW BEAMDEEP BEAM

860

1000 1000120 120

15

15

1515

B A1930

Fig. 1. Experimental setup.

61E. Bayo et al. / Journal of Constructional Steel Research 108 (2015) 60–69

modelling of the joint for global analysis can be achieved by consideringthe cruciform element proposed by Bayo et al. [13,14].

Work remains to be done to better characterize the complex behav-iour of steel jointswith beamsof unequal depths and trapezoidal panels.The study involves a large number of variables and intervening factors.In this paper, experimental work and numerical (finite element) resultsare presented that provide important information to characterize andmodel this type of joints. The moment rotation diagrams depend on alarge number of material, geometrical and loading variables.

The research presented concentrates on the shear stiffness and resis-tance of the trapezoidal panel zone and the additional contribution pro-vided by the elements surrounding the panel such as the columnflangesadjacent to the panel. These elements play amajor role in the joint post-elastic reserve shear strength and stiffness. Since the aim is to study theshear behaviour and component of the panel zone, stiffeners are includ-ed to avoid the effects of other components such as the column web intension and compression, and to prevent their interaction with theshear component as much as possible. Furthermore, an inclined stiffen-er in the lower part of the panel assures trapezoidal behaviour. Conse-quently, only stiffened fully welded connections were considered inthis investigation.

Themoments acting at both sides of the joint are themost importantcontribution to the panel shear force, and they are usually considered asthe load variables characterizing the strength and stiffness of the joint.

Table 1Characteristics of the tests: beam and column configurations.

Test Column Deep beam Shallow beam

E1A HEA 240 HEB 300 HEB 160E1B HEA 240 HEB 300 HEB 160E2A HEA 240 HEB 300 HEB 180E2B HEA 240 HEB 300 HEB 180

However, the beneficial effect of the column shear should also be con-sidered (particularly in the case of short columns) in the joint stiffnessand strength as proposed byKrawinkler et al. [1]. The deformation is de-fined in terms of the average shear distortion of the panel zone (PZ), andit ismeasured in this investigation bymeans of two inclinometers locat-ed on themiddle section of the panel. The deformations on both sides ofthe panel (the shallow and deep sides) will be considered as well. Theaim is to understand the mechanisms of deformation and characterizethemoment rotation curves for the shear component of the trapezoidalpanel.

2. Experimental work and finite element models

The experimental work has been carried out in two different proto-types with beams of unequal depth. The overall scheme is illustrated inFig. 1. The column is pinned at both ends and actuation is applied atpoints A and B on the attached beams. The distance from these pointsand the column flanges is exactly 1000 mm. Table 1 shows the beamsand column sizes in both tests, and Fig. 2 depicts a picture of the exper-imental setup.

The top horizontal and the inclined stiffeners were welded asdepicted in Figs. 1 and 2 to avoid, as mentioned above, any type of fail-ure other than that produced by shear. The stiffeners were 15mm thickand in all caseswere rigid enough to provide the necessary resistance to

Depth ratio Loading point Type of loading

1.88 A Elastic1.88 B Failure1.67 A Elastic1.67 B Failure

Fig. 2. Experiment E1 during the loading process.

Table 2Steel properties obtained from coupons.

Part σy

(MPa)σu

(MPa)E(MPa)

HEA 240 330 493 207,000HEA 240 328 494 211,000HEA 240 326 496 209,000HEA 240 329 490 210,000Stiffeners 300 446 213,000Stiffeners 309 449 211,000

Fig. 4. Detail of the finite element model.

62 E. Bayo et al. / Journal of Constructional Steel Research 108 (2015) 60–69

prevent either tension or compression failure of the column webs, aswell as out of plane bending of the column flanges. The column webwas 8 mm thick. The type of steel used for all the parts was S275.Couponswere extracted from both the columns and stiffeners to obtainthe true properties of the material. Table 2 shows those properties thatwere subsequently used in the finite element analyses for comparisonpurposes.

The panel was instrumented with 4 strain gauges placed at thecorners of the panel as shown in Fig. 3. They served tomonitor the yield-ing sequence and the levels of shear strain. Five inclinometers wereused as shown in Fig. 3 to obtain the rotations. Two were placed verti-cally at the beams adjacent to the joint. Twomorewere placed verticallyat the top and bottom of the web panel to capture the possible differentrotations at those levels. A fifth one was placed in the middle of thepanel to capture the rotation of the columndue to bending. This rotation

3

4

1

5

1

34

2

Fig. 3. Placement of strain ga

is subtracted from those at the beams to obtain the rotation due to theshear deformations.

The finite element analysis was performed using Abacus®. Solidelements with reduced integration (to avoid shear locking) and hour-glass control (C3D8R) were adopted. Fig. 4 illustrates the part of themodel that corresponds to the column panel. The material behaviourwas introduced by means of the true stress–strain data obtained fromthe coupons. Static nonlinear material and geometric analyses withforce controlwere performed. TheVonMisses yield criterionwas select-ed to define the inelastic response.

For all the tests, a previous simulation had been done in order todetect the load at which plastic behaviour begins. These values havebeen used in tests E1A and E2A as the upper limit of the applied forces,with the objective of avoiding damage in the web panels. The experi-ments were simulated with the exact dimensions of the elements. Inall cases, the analysis was extended until the complete plastic behaviourof the panel was reached. The finite element models were refined nearthe most stressed zones (the panel and adjacent zones) with the aimof assessing correctly the strain and stress fields. The convergence ofthe model was evaluated to reach good results with a suitable level ofmeshing, and to allow parametric studies with a reasonable computa-tional cost.

3. Description of the results

3.1. Test E1A

The first test E1A consisted in loading the prototype E1 at the tip ofthe shallow beam (point A). The aim was to obtain the elastic stiffness

2Inclinometer

Strain gauge

uges and inclinometers.

Fig. 5. Elastic shear stress contours at the end of the elastic range in experiment E1A.Fig. 7. Elastic shear stress levels in test E1B.

63E. Bayo et al. / Journal of Constructional Steel Research 108 (2015) 60–69

of the trapezoidal panel under the shear coming from loading theshallow beam. The maximum load applied was 30 kN, and afterwardsit was unloaded. A previous finite element analysis had predicted pureelastic behaviour up to that loading level. The shear stress distributionwithin the panel obtained from the finite element analysis is illustratedin Fig. 5, which corresponds to the level of strains just at the end of theelastic range. Fig. 6 shows the strain gauge data for this test.

It may be seen that the stress distribution is uniform over an upperrectangular part of the panel. The readings of the inclinometers 3 and4 were similar, and the inclinometers 1 and 2 showed that the rotationat the shallow beam was approximately 2.4 times the rotation at thedeep beam.

3.2. Test E1B

The following test, E1B, consisted in loading the prototype E1 atthe tip of the deep beam (point B) until failure, which occurred at aload of 230 kN. The shear stress distributions within the panel comingfrom the finite element analysis during the elastic and inelastic partsof the response are illustrated in Figs. 7 and 8, respectively. It may beseen how even in the elastic range the shear stress field now extendsdeeper towards the bottom of the panel becoming nonrectangularin shape.

Fig. 6. Results from strain gauges in experiment E1A.

During the elastic part of the response, the readings of the inclinom-eters 1 and 2 also showed a rotation at the shallow side 1.2 times largerthan the rotation at the deep side. However, within the inelasticresponse the angles at the shallow beam became much larger thanthose of the deep beam reaching 40% more right before failure. Thefinal failure was due to the cracking of the panel at the upper rightcorner (right next to strain gauge 1)

3.3. Test E2A

The test E2A consisted in loading the prototype E2 at the tip of theshallow beam. The qualitative behaviour of this test was similar tothat of E1A. The maximum load applied was 37 kN, and afterwards itwas unloaded. Again, a previous finite element analysis had predictedpure elastic behaviour up to that loading level. The shear stress distribu-tion within the panel obtained from the finite element analysis is illus-trated in Fig. 9, which corresponds to the level of strains just at the endof the elastic range. Fig. 10 shows the strain gauge data for this test. Thereadings of inclinometers 3 and 4 were similar, and the rotation at theshallow beam was approximately double the size of the rotation of thedeep beam.

Fig. 8. Inelastic shear stress levels in test E1B.

Fig. 9. Elastic shear stress levels at the end of the elastic range in test E2A. Fig. 11. Elastic shear stress levels in test E2B.

64 E. Bayo et al. / Journal of Constructional Steel Research 108 (2015) 60–69

3.4. Test E2B

The final test E2B consisted in loading the experiment E2 at the tip ofthe deep beam (point B) until failure, which occurred at a load of230 kN. The shear stress distributions within the panel coming fromthe finite element analysis during the elastic and inelastic parts of theresponse are illustrated in Figs. 11 and 12 respectively. The rotation ofthe lower part of the panel was similar to that of the higher part rightbefore failure. The angles at the shallow beam were 1.15 times thoseof the deep beam during the elastic part of the response. However,within the inelastic response the angles at the shallow beam becamemuch larger than those of the deep beam reaching a difference of 40%right before failure.

Figs. 13 and 14 show the readings of the strain gauges correspondingto tests E1B and E2B, respectively. The pictures of the deformed paneland deformed shape of the experiment before failure are shown inFigs. 15 and 16, respectively. The plastic shear stress levels of the columnpanel as well as the final deformed shape of the specimen obtained byfinite element simulation can be seen in Fig. 17. It may be observedhow the finite element model replicates the actual deformed shape ofthe experiment. It is also worth noting how the shallow beam gets

Fig. 10. Results from strain gauges in experiment E2A.

much more inclined than the deep beam due to the deformation ofthe trapezoidal panel as an articulated quadrilateral mechanism.

4. Kinematics of the deformation and mechanical model

As mentioned above and as observed from tests and finite elementmodels, three different degrees of freedom can be identified in theshear panel during the elastic range loading. Fig. 18 shows an enlargedpicture of the finite element panel deformation when the shallowbeam is loaded (Test E1A), and it evidences the fact that the rotationof the shallow side is larger than those of the centre and deeper partsof the panel, respectively. Fig. 19 shows the panel deformation whenthe deep beam is loaded (Test E1B); again the rotation of the shallowside is larger than those of the centre and deeper parts of the panel.Finally, Fig. 20 shows the deformed shape due to shear forces, comingfrom the column, acting in the panel; and again the same pattern isobserved. As explained below, these deformations do not correspondto those of a single degree of freedom articulated quadrilateral, andtherefore they are independent. The departure from the kinematics ofan articulated quadrilateral in the elastic range is due to the bendingdeformation produced in the column flanges in both sides of the panel;

Fig. 12. Inelastic shear stress levels in test E2B.

Fig. 13. Results from strain gauges in test E1B.Fig. 15. Panel inelastic deformation.

65E. Bayo et al. / Journal of Constructional Steel Research 108 (2015) 60–69

this can be seen by observing the deformation of the shallow sidein Fig. 19.

In regard to the initial stiffness and deformations, Figs. 21 and 22show the moment–rotation curves corresponding to the deep beam,the centre of the panel and the shallow beam for tests E1A and E2A,respectively. Figs. 23 and 24 show those corresponding to the testsE2A and E2B, respectively. In all cases, by observing the slopes of themoment–rotation curves one can identify different stiffness and seehow the rotation of the shallow beam side is larger than the rotationof the centre of the panel, and this one is in turn larger than the rotationat the deep beam side.

As mentioned above the joint shows three independent degrees offreedom in the elastic range, which do not follow the deformations ofan articulated quadrilateral. However, once the panel becomes plastic,the incremental rotations follow the kinematics of a one-degree of free-dom articulated quadrilateral. The data appearing in Table 3 explainsthis effect. It can be seen in this table how the ratios between the heightsof the deep and shallow beams (the depth ratio) are very different fromthose of the corresponding incremental rotations in the elastic range.However, when the panel deforms within the plastic range the ratio ofthe incremental rotations and the depth ratio become quite similar,

Fig. 14. Results from strain gauges in test E2B.

thus corroborating the kinematics of a one-degree of freedomarticulatedquadrilateral mechanism.

From the previous discussion it was possible to observe that thereare three independent rotations and therefore three degrees of freedom(DOF) during the elastic range: left connection, right connection andcentre of panel; and only one DOF in the inelastic range correspondingto an articulated quadrilateral mechanism. In view of this, a mechanicalmodel is presented in Fig. 25. This model is made up of rigid bars con-nected by springs. The model illustrates the three possible rotations:ϕ1 on the right part of the panel, ϕ2 on the left part of the panel, andϕp on the panel itself. When the panel becomes plastic the value of Kp

becomes significantly lower than K1 and K2, and the model resemblesa one-degree of freedom system. The corresponding dual internal forcesareM1,M2 and V.M1 is the right sidemoment,M2 is the left side one, andV is the shear coming from the column. The spring Kp models the stiff-ness of the panel under shear; the springs K1 and K2 take into accountthe relative rotation between the centre of the panel and the right andleft connections, respectively. The parameters h1, h2 and hp representthe equivalent height of the shallow, centre and deep parts of the joint,respectively. The values of the springs and parameters are defined aftera numerical parametric analysis in the companion (Part II) paper [15].

Fig. 16. Test deformed shape before failure.

Fig. 17. Final stage of stress and deformation in test E2B.

66 E. Bayo et al. / Journal of Constructional Steel Research 108 (2015) 60–69

5. Moment rotation curves: stiffness and resistance

This section presents a comparison of the moment rotation curvesobtained from the experiments, the finite element analysis, theEurocode 3 [2] and the method proposed by Krawinkler et al. [1].Since the last two methods do not include trapezoidal panels theyare applied to each connection (left and right) as if the panel was

Fig. 18. Enlarged panel deformed shape in test E1A (load in shallow beam).

Fig. 19. Enlarged panel deformed shape in test E1B (load in deep beam).

rectangular for the corresponding beam depth. It is important to notethat the EC3 and Krawinkler's formulaewere developed for rectangularpanels that correspond to beams of equal depths, and as such do not fitin this study. However, they are included in this comparison to observetheir results and assess the possible errors that may derive from theiruse to trapezoidal panels. The intention is to inform and warn thosedesigners whomay use these formulae for cases other than rectangularpanels.

The comparison is established in terms of the moment at theconnection versus the average shear rotation of the panel, as well asthe other possible rotations at each side of the joint (additional DOFs).The three experimental rotations of the panel: right, centre and lefthave been measured using the following relations:

Rotation at the right of the panel = inclin(2)− inclin(5)Rotation at the centre of the panel = (inclin(3) + inclin(4)) /2 −inclin(5)Rotation at the left of the panel = inclin(1) − inclin(5)

Inclinometers 1 and 2 measure the rotations at the left (deep) andright (shallow) parts of the panel, respectively. Inclinometers 3 and 4measure the total rotation at the top and bottom parts on the panelalong the centre line (see Fig. 3). The reading of inclinometer 5 issubtracted from all the measures to take into account the flexuralrotation of the column at the joint level.

Figs. 26 and 27 show themoment–rotation curves corresponding tothe tests E1A and E2A, respectively. The results from the experiments

Fig. 20. Enlarged panel deformed shape (shear load in panel).

Fig. 21. Shallow, centre and deep side rotations in the FEM for test E1A.

Fig. 22. Shallow, centre and deep side rotations in the FEM for test E2A.

Fig. 23. Shallow, centre and deep side rotations in the FEM for test E1B.

Fig. 24. Shallow, centre and deep side rotations in the FEM for test E2B.

67E. Bayo et al. / Journal of Constructional Steel Research 108 (2015) 60–69

only provide the initial stiffness since, asmentioned above, they are onlyloaded in the linear elastic range (themaximumapplied loadwas 30 kNfor test E1A and 37 kN for test E2A), and their plots are hidden under-neath the finite element ones. Figs. 28 and 29 show the moment rota-tion curves corresponding to the tests E1B and E2B, respectively. Itmay be seen that the predictions of the finite element model in termsof stiffness and resistance are good when considering the rotation atthe centre of the panel. The differences at the knee level are due touncertainties in the modelling of the welding material properties.

EC3 provides a good prediction of the stiffness but a poor predictionof the resistance, even when including the additional resistance provid-ed by the column flanges. Krawinkler's model underestimates both thestiffness and resistance, although the latter is better approximated thanEC3.

Table 4 compares the values of the stiffness obtained by the differentmethods, as well as the relative errors, when considering the rotation ofthe central section of the panel as the reference for the rotations (thirddegree of freedom). It may be observed that the finite element modelprovides quite good results. The EC3 formulation also gives good results(around 5% error) and Krawinkler's model gives larger errors of upto 20%.

The results are similar when considering the deep section of thepanel as the reference for the rotations (second degree of freedom).Table 5 shows similar differences for the EC3 case although it providesstiffness values that are non-conservative (note that the cases E1A andE2A do not appear since there is no moment applied at that side).

Amuchworse situation arises when considering the shallow section(first degree of freedom) of the panel as the reference for the rotations.Table 6 shows that the errors arising from the finite element model arewithin 10%, however the errors introduced by both the EC3 andKrawinkler's models are considerable, 97% and 57% respectively, andalso non-conservative. Therefore the application of the formula provid-ed by EC3 and Krawinkler's model is not recommended for the shallowbeam side.

Table 3Incremental shallow/deep rotations in the elastic and plastic ranges.

FEM Depth ratio Incremental shallow/deep rotation

Elastic range Plastic range

E1A 1.87 2.37 1.78E1B 1.87 1.19 1.62E2A 1.66 2.05 1.61E2B 1.66 1.15 1.48

Fig. 25. Proposed mechanical model of the shear panel.

68 E. Bayo et al. / Journal of Constructional Steel Research 108 (2015) 60–69

6. Conclusions

In this paperwe have investigated the shear behaviour (component)of stiffened trapezoidal shear panels appearing in joints with unequalbeam depths. Tests and FE simulations have been performed. A

Fig. 26. Moment–rotation at the centre of panel for test E1A (the test plot is underneaththe FEM curve).

Fig. 27. Moment–rotation at the centre of panel for test E2A (the test plot is underneaththe FEM curve).

mechanical model of rigid bars and springs has also been proposed tocapture the kinematics and internal forces acting on the joint. Themain conclusions can be summarized as follows:

1. The shear deformation zone corresponds to a part of the upper rect-angle when loading the shallow beam, and the whole trapezoidwhen loading the deep beam. Consequently the mechanics of thedeformation and the initial stiffness values of the left and rightconnections are different.

2. Three independent degrees of freedom are identified in the elasticrange: the rotations corresponding to the left, right and centralsections of the panel. The inelastic range is well characterized byone degree of freedom, which corresponds to an articulated quadri-lateral mechanism. These degrees of freedom should be consideredat the time of characterizing the joint stiffness for frame analyses.

3. The finite element analysis predicts the stiffness and resistance withgood accuracy. EC3 approximates the initial stiffness reasonably wellwhenusing the dimensions of the beamattached to the correspondingconnection, and themiddle section and deep beam side of the panel asthe references for the rotations (degrees of freedom 2 and 3). Howev-er, the initial stiffness estimated by EC3 becomes very poor whenconsidering the rotation of the shallow beam (degree of freedom 1).Krawinkler's model provides slightly worse results than the EC3when using the middle section and deep beam side of the panel asthe reference for the rotations. However, the stiffness predicted byKrawinkler's model becomes very poorwhen considering the rotationat the shallow beam side.

Fig. 28.Moment–rotation at the centre of panel for test E1B.

Fig. 29.Moment–rotation at the centre of panel for test E2B.

Table 4Comparison of rotational stiffness (kNm/mrad) considering the centre of the panel.

Test FEM Error (%) EC3 Error (%) Krawinkler Error (%)

E1A 29.1 28.7 −1.4 31.9 −2.0 25.4 −22.0E1B 59.0 57.5 −2.5 60.9 −1.3 51.7 −16.2E2A 31.8 31.4 −1.3 36.0 1.8 28.9 −18.3E2B 56.7 55.5 −2.2 60.9 1.2 51.7 −14.1

Table 5Comparison of rotational stiffness (kNm/mrad) considering the deep side of the panel.

Test FEM Error (%) EC3 Error (%) Krawinkler Error (%)

E1B 59.8 61.3 2.5 60.9 −1.9 51.7 −16.7E2B 57.5 59.2 3.0 60.9 8.9 51.7 −7.5

Table 6Comparison of rotational stiffness (kNm/mrad) considering the shallow side of the panel.

Test FEM Error (%) EC3 Error (%) Krawinkler Error (%)

E1A 16.7 18.3 9.6 31.9 96.9 25.4 56.8E2A 19.7 21.6 9.7 36.0 88.5 28.9 51.3

69E. Bayo et al. / Journal of Constructional Steel Research 108 (2015) 60–69

4. Amechanicalmodel of rigid bars and springs has been proposed. Thismodel includes the three degrees of freedomof the trapezoidal shearpanel. The variables that characterize this model will be obtainedafter a detailed parametric analysis in the companion (Part II)paper [15].

In the context of advanced global analysis allowed by modern codes[16,17] and optimized solutions [18] the use of the proposed jointmodel could be particularly useful since it allows the joint to be accu-rately modelled within the structure.

Acknowledgements

Thefinancial support provided by the SpanishMinisterio de Ciencia eInnovación under contract BIA2010-20839-02-C01-C02 is gratefullyacknowledged.

References

[1] Krawinkler H, Bertero VV, Popov EP. Shear behaviour of steel frame joints. J StructDiv ASCE 1975;101(11):2317–36.

[2] CEN. Eurocode 3, EN 1993-1-8:2005: Design of steel structures. Part 1.8: Design ofjoints. Brussels, Belgium: European Committee for Standardization; 2005.

[3] Zoetemeijer P. Summary of the research on bolted beam-to-column connections(period 1978–1983). Technical Report 6-85-M. Delft: Steven Laboratory;1983.

[4] Yee YL, Melchers RE. Moment–rotation curves for bolted connections. J Struct EngASCE 1986;112(3):615–35.

[5] Faella C, Piluso V, Rizzano G. Structural steel semi-rigid connections. Boca Raton:CRC Press; 2000.

[6] Castro JM, Elghazouli AY, Izzuddin BA. Modeling of the panel zone in steel andcomposite frames. Eng Struct 2005;27:129–44.

[7] Charney FA, Downs WM. Modeling procedures for panel zone deformations inmoment resisting frames. Connections in Steel Structures V. Delft, the Netherlands:Bouwen met Staal; 2005 121–30.

[8] Girão-Coelho A, Bijlaard F, Kolstein H. Numerical modelling of high strength steelcolumn web shear panel behaviour. Eurosteel 2008. Brussels: ECCS EuropeanConvention for Constructional Steelwork; 2008 1125–30.

[9] Hoogenbroom P, Blaauwendraad J. Quadrilateral shear panel. Eng Struct 2000;22:1690–8.

[10] Curtis H, Greiner G. A stress-based quadrilateral shear panel. Finite Elem Anal Des1996;21:159–78.

[11] Hashemi BH, Jazany RA. Study of connection detailing on SMRF seismic behaviourfor unequal beam depths. J Constr Steel Res 2012;68:150–64.

[12] Jordao S, Simoes da Silva L, Simoes R. Behaviour of welded beam-to-column jointswith beams of unequal depth. J Constr Steel Res 2013;91:42–59.

[13] Bayo E, Cabrero JM, Gil B. An effective component basedmethod tomodel semi-rigidconnections for the global analysis of steel and composite structures. Eng Struct2006;28:97–108.

[14] Bayo E, Gracia J, Gil B, Goñi R. An efficient cruciform element to model semi-rigidcomposite connections for frame analysis. J Constr Steel Res 2012;72:97–104.

[15] Lopez M, Loureiro A, Bayo E. Shear behaviour of trapezoidal column panels. II:parametric study and cruciform element. J Constr Steel Res 2015;108:70–81.

[16] CEN. Eurocode3, EN1993-1-1-2005. Design of steel structures. Part 1.1: General rulesand rules for buildings; 2005.

[17] AISC. Specification for structural steel buildings. Chicago, IL: American Institute ofSteel Construction; 2005.

[18] Carbonell-Marquez JF, Gil-Martin LM, Hernandez-Montes E. Strength designoptimization of structural steel members according to Eurocode 3. J Constr SteelRes 2013;80:213–23.

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