Finite Element Instability Analysis of the Steel Joist of Continuous Composite Beams with Flexible Shear Connectors

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1 Journal of Solid Mechanics Vol. 9, No. (07) pp Finite Element Instability Analysis of the Steel Joist of Continuous Composite Beams ith Fleible Shear Connectors H. Bakhshi,*, H.R. Ronagh Engineering Faculty, Hakim Sabzevari University, Sabzevar, Iran Instiutute for Infrastructure Engineering, Western Sydney University, Penrith, NSW 75, Australia Received 5 March 07; accepted 4 May 07 ABSRAC Composite steel/concrete beams may buckle in hogging bending regions. As the top flange of I-beam in that arrangement is restricted from any translational deformation and tist, the eb ill distort during buckling presenting a phenomenon often described as restricted distortional buckling. here are limited studies available in the literature of restricted distortional buckling of composite steel/concrete I-beams subjected to negative or hogging bending. here is none hoever that includes the effect of partial shear interaction. In this paper, finite element models for in-plane analysis and out-of-plane buckling of continuous composite I-beams including the effects of partial shear interaction are presented. 07 IAU, Arak Branch.All rights reserved. Keyords : Finite element model; In-plane analysis; Out-of-plane buckling; Restricted distortional buckling; Partial shear interaction. INRODUCION HIS paper discusses the instability of the steel joist of a composite beam composed of a steel I-beam and a concrete slab connected together by shear connectors hich is subjected to negative or hogging bending. his is a relatively complicated problem. he instability analysis becomes even more comple hen partial shear interaction beteen steel joist and concrete slab is taken into account. As the top flange of I-beam in that arrangement is restricted from any translational deformation and tist, the eb ill distort during buckling presenting a phenomenon often described as restricted distortional buckling. If this buckling is prevented, large rotational capacities can be achieved beyond the plastic moment of resistance and advantageous plastic design is feasible. Consequently, the moment redistribution in the hogging region that is possible due to the ductility of such beams is achievable prior to failure and significant economies can be obtained due to that distribution. he very early ork on the concept of imperfect composite action beteen concrete and steel for composite structures ere tests on si simply supported composite steel and concrete -beams performed by Nemark et al. []. Unlike simply-supported composite beams, continuous composite beams are more comple due to concrete cracking in hogging regions. heoretical studies ere published in order to address the problem of the buckling of steel joist of composite beams ith both continuously varying aial force and bending moment (Bradford and Gao []; Dekker et al. []; ehami [4]; Jasim and Atalla [5]; Nie and Cai [6]; Nie et al. [7]; Ranzi et al. [8]). Bradford and Gao [] presented a simple method to eamine the stability of fied-ended composite steel-concrete beams taking into account the difference beteen its sagging and hogging bending rigidities, due to the concrete cracking in * Corresponding author. el.: ; Fa: address: h.bakhshi@hsu.ac.ir (H. Bakhshi). 07 IAU, Arak Branch. All rights reserved.

2 5 H. Bakhshi and H.R. Ronagh tension. Bradford [9] developed a rational model for predicting the elastic buckling load of thin-alled I-section columns, restrained fully against translation and elastically against tist at one flange and subjected to a uniformly distributed aial force. Bradford and Ronagh [0] etended the degrees of freedom element developed by Bradford and rahair [] to the 6 degrees of freedom element for predicting the distortional buckling of restrained I-section members more accurately. Bradford and Ronagh [] considered the elastic lateral-distortional buckling of composite cantilevers, hose steel portion as tapered, under moment gradient. Bradford and Ge [] demonstrated an in-plane analysis of elastic distortional buckling of a to-span continuous I-beam ith a concentrated load in each span. It as concluded that the distortional buckling resistance of to-span continuous beams decreased hen the concentrated loads ithin each span ere located toard the centre of each span, since the bending moment distribution as more uniform. Bradford and Kemp [4] revieed the research to date into local and lateraldistortional buckling in the negative moment region of continuous composite beams, and provided design proposals. Vrcelj [5] presented the development of a rational generic model to investigate the elastic restrained distortional buckling of I-section members in both half-through girder bridges and composite tee-beams in hogging regions. Her study, hoever, only addressed the issue in the light of full interaction analysis. Xu and Wu [6] presented a ne plane stress model for the composite beams ith interlayer slips using the state space method. Wang and Chung [7] proved that the deformation characteristics of shear connectors are very important in predicting the structural behavior of composite beams ith large rectangular eb openings. Zona and Ranzi [8] considered the shear deformability of the steel and slab components at various load levels. Chakrabarti et al. [9] proposed a ne finite element model based on a higher order beam theory for the analysis of composite beams. he proposed model takes into account the effect of partial shear interaction beteen the adjacent layers as ell as transverse shear deformation of the beams. ezgy-nazargah [0] presented an isogeometric approach based on non-uniform rational B-spline (NURBS) basis functions for the analysis of composite steel concrete beams. ezgy-nazargah and Kafi [] proposed a finite element model for the analysis of composite steel-concrete beams based on a refined highorder theory. he employed theory of these researchers satisfies all the kinematic and stress continuity conditions at the layer interfaces and considers effects of the transverse normal stress and transverse fleibility. He and Yang [] made comparisons among the classical, Reddy s higher order and plane stress models. hey also presented a higher order model for composite beams and verified it through comparison via other models. Chen et al. [] presented an eperimental study of the fleural behavior and shear transfer mechanisms of shallo cellular composite beams the concrete passing through the steel eb opening is combined ith a steel tie-bar element to form the shear connection. In this paper, finite element models for in-plane analysis and out-of-plane buckling of continuous composite I- beams including the effects of partial shear interaction are presented. he first finite element model is developed to investigate the in-plane behavior of composite beams. he model provides the bending moment distribution along the length alloing for the variation in the rigidities of the cross-section in hogging and sagging regions. he stress resultants that act in the steel joist are then used as the input data for the out-of-plane finite element. he out-ofplane finite element model undertakes the out-of-plane distortional buckling analysis of the steel joist. Both of the models and some results are presented in the folloing. HE IN-PANE MODE In the in-plane finite element formulation, a composite beam is modelled as to one-dimensional beam elements, one for concrete and the other for steel. Each beam has to nodes ith three degrees of freedom per node totaling to telve degrees of freedom for the element (Fig.(a)). Based on the assumption that the curvature is the same in both elements at the same point along the length due to the lack of transverse separation at the contact interface, displacements and tists can be reduced to eight degrees of freedom as is shon in Fig. (b). A ne model ith to nodes and four degrees of freedom per node is then proposed. In the first model ith telve degrees of freedom (Fig. (a)), the generalized displacements containing these freedoms are presented by the vector qf as: q q q q q q q q q q q q q f f f f f 4 f 5 f 6 f 7 f 8 f 9 f 0 f f () 07 IAU, Arak Branch

3 Finite Element Instability Analysis of the Steel Joist of Continuo 5 Fig. In-plane composite beam finite element. In the second model ith eight degrees of freedom (Fig. (b)), the vector q p can be ritten as: q q q q q q q q q p p p p p 4 p5 p6 p7 p8 () Introducing the shape functions N X () N N N N N X (4) X X (5) (6) 4 (7) 5 (8) 6 he displacements along the length of the concrete and the steel can be ritten as: Aial displacement: u N q N q (9) c f f 4 u N q N q (0) s f 7 f 0 Vertical displacement: v N q N q N q N q () c f 4 f 5 f 5 6 f 6 v N q N q N q N q () s f 8 4 f 9 5 f 6 f Rotation: 07 IAU, Arak Branch

4 54 H. Bakhshi and H.R. Ronagh dv c c v c N qf N 4 qf 4 N 5 qf 5 N 6 qf 6 d () dv s s v s N qf 8 N 4 qf 9 N 5 qf N 6 qf d (4) Having this, the displacements at the shear connector level usc and u sc shon in Fig. can be calculated as: u u h (5) sc c c c u u h (6) sc s s s he slip is then defined as: s u u (7) sc sc s u u ( h h ) (8) c s c c s s Fig. Slip relations of in-plane composite element. Replacing Eq.(9), (0), () and (4) into Eq.(7), e get s N h N h N N h N h N N h N h N N h N h N q c c 4 c 5 c 6 s s 4 s 5 s 6 f (9) or as: S D q s f q q q q q q q q q q q q q f f f f f 4 f 5 f 6 f 7 f 8 f 9 f 0 f f D N h N h N N h N h N N h N h N N h N h N S c c 4 c 5 c 6 s s 4 s 5 s 6 and D S is given in the Appendi. Stiffness matri of the composite element is the sum of the stiffness matrices of the concrete, the steel and the shear connector elements. Using the energy approach, the strain-energy of shear connector element is defined as: 07 IAU, Arak Branch

5 Finite Element Instability Analysis of the Steel Joist of Continuo 55 U shear k s s d 0 k s U shear qf Ds Ds qf d (0) U q K q 0 shear f shear f shear s s s K k D D d () 0 Is identified as the stiffness matri of the shear connector element. he stiffness matri K cs of the concrete and the steel is developed from the basic stiffness matri of beam element. he total stiffness of composite element is thus obtained from U qf K f qf () K f Kshear Kcs () On the other hand, a relation beteen the displacements of the element ith telve degrees of freedom and the element ith eight degrees of freedom in Fig. can be epressed as: q f H q (4) p In hich H is given in the Appendi. Eq. () can be reritten as: U qf K f qf q p K p q p (5) p (6) f K H k H Is identified as the total stiffness matri of the composite element ith eight degrees of freedom. A finite element program called PARIASHEAR is then compiled to produce the slip, the lateral displacements and tist of the concrete and the steel, and the stress resultants of steel joist. hese ill be used as input for the out-of-plane buckling investigation performed by the out-of-plane finite element. As the elements ork ith fied properties along the length, the sagging and hogging properties may need to be defined based on the position of the contra fleure point. Since this position is unknon in prior, trial and error based analysis needs to be carried out to find its location. he analysis therefore starts ith a set number of elements at set lengths. he lengths of elements neighboring the contra fleure point are then adjusted to allo the zero bending point to occur eactly at their junction. he process of adjustment continues until the point of contra fleure remains at the assumed position ithin an acceptable error range. 07 IAU, Arak Branch

6 56 H. Bakhshi and H.R. Ronagh HE OU-OF-PANE MODE he geometry of the beam-column element and the -y-z Cartesian ais system are defined as shon in Fig.. he element is a prismatic I-girder ith flange idth b f, flange thickness t f, eb depth h, and eb thickness t. ateral displacement u and tist of the top and the bottom of the eb along the length z are used to define the shape of the beam during buckling. he subscripts and B refer to the top and bottom flanges, respectively. Due to the restraint provided through the shear connectors to the concrete slab, the displacements at the top flange of I- girder are assumed to be zero. Fig. he beam element and its displacements. he element is considered to be subjected to end moments M 0 and M, end aial forces N 0 and shear forces V 0 and V in the manner shon in Fig. 4. N, and end Fig.4 Configuration of element in loading. he bending moment and aial force are assumed to vary linearly along the length of the element, so that 0 z z M M M 0 z z N N N (7) (8) he shears are equal and can be obtained from static s equilibrium as: V M M 0 0 V (9) In the z-direction, the buckled shape is represented by the lateral and tist displacements of the flanges, ub, and B. All these deflections and tists are defined as cubic polynomials along the length. he flanges displacements are therefore given as: z z ub z 4 (0) 07 IAU, Arak Branch

7 Finite Element Instability Analysis of the Steel Joist of Continuo 57 z z z () z z z B 9 0 () hich can be ritten in the matri form as: B B u u () M M z z z z z z (4) (5) M M M M (6) or as: u M (7),,..., (8) In the y-direction, the buckling deformation of the eb is also defined by cubic polynomials linking from the top to the bottom of the eb. he deflection of any eb point in the cross-section can then be epressed as: or y y u h y (9) u H (40) y y H y h h (4) (4) 07 IAU, Arak Branch

8 58 H. Bakhshi and H.R. Ronagh Fig.5 Web element shape function. he displacement variation of eb element shon in Fig. 5 can also be epressed in natural coordinates as: u N u N N (4) B B N y y (44) N h 8 y y y 4 h (45) N h 8 y 4 y y h (46) Eq.(4) can be ritten as: u N N N u (47) B B u u (48) and from Eq.(7): u M Hence, Eq. (47) becomes u N N N M (49) or u N (50) N N N N M (5) 07 IAU, Arak Branch

9 Finite Element Instability Analysis of the Steel Joist of Continuo 59 Fig.6 Buckling displacements q. Generalized buckling displacements used are shon in Fig. 6. A prime ( ) denotes differentiation ith respect to z. he vector q containing these freedoms is ritten as: B, B,,, B, B, B, B,,, B, B q u u u u (5) he generalized buckling displacements q are related to the natural coordinate system. he relation can be easily obtained by looking at the shape functions at the boundaries. his relation can be epressed as: q C (5) the matri C is a matri of order and is given in Appendi. Eq.(5) can then be reritten ith the inverse of the matri C as: C q (54) In a similar manner, buckling displacements u are related to the generalized displacement system q by substituting Eq.(54) into Eq.(7). his gives: u M C q (55) C is given in Appendi.. Derivation of the stiffness matri In order to derive the stiffness matri using an energy approach, it is necessary to calculate the total energy stored in the beam during buckling. he beam elements representing the flanges and the plate element representing the eb ould deflect sideays and tist. he strain energy stored can be epressed as: U U U U fu f (56) U fu is the energy associated ith the bending of flanges about their major ais, U f is the energy associated ith the tist of flanges and bending about their minor ais, and is the strain energy associated ith the bending and tisting of the eb as a plate. hese energy components can be calculated as: U U fu Es I yf ub dz 0 (57) bf v v U D v v v ddz z z B ( ) (58) f f B 0 b f 07 IAU, Arak Branch

10 60 H. Bakhshi and H.R. Ronagh U 0 dy dz (59) U f is composed of to components, the first component bf f f B 0 bf U D v v ddz (60) Is associated ith the bending of flanges about their minor ais.he second component of U f is equal to bf v v B f ( ) z z 0 b f U f D v ddz (6) and is associated ith the tist of the flanges. Since the flanges are assumed to remain rigid, the vertical (y-direction) displacement of the flanges can be epressed as: v v B B (6) (6) Substituting Eqs.(6-6) into Eq.(60) and then integrating over the idth of the flanges shos that this component contains the usual tisting terms of the Saint Venant linear torsion as: 0 GJ f B dz (64) G is the shear modulus equal to: E s G ( v ) (65) and J f is the torsional rigidity of each flange, hich is equal to: bt f J f f (66) Hence, U f can be reritten as: bf U D v v ddz GJ dz (67) f f B f B 0 bf 0 07 IAU, Arak Branch

11 Finite Element Instability Analysis of the Steel Joist of Continuo 6 he term I yf is the second moment of area of each flange about the y-ais, hich is equal to: f f bt I yf (68) he parameter Df represents the plate rigidity of flanges and is given by D f s f Et ( v ) (69) As the eb is considered a plate, the generalized strain and stress vector of Eq.(59) can be ritten as: u u u y z yz (70) v 0 v 0 Et s D v 0 v 0 ( v ) v v (7) D s Et ( v ) (7) From Eqs. (54): C q (7) herefore: q C (74) Using Eqs. (57-59), the strain energy equation can be mathematically manipulated to produce U K a (75) Substituting and from Eqs.(54, 74) into Eq.(75) e obtain U q C K C q q K q (76) From hich K C K C (77) 07 IAU, Arak Branch

12 6 H. Bakhshi and H.R. Ronagh Is identified as the stiffness matri of the element.. Derivation of the stability matri he stability matri relates to the ork done by the surface tractions or the loss of the potential energy of the stresses. his loss of potential energy can be ritten as: W W W W fu f (78) W fu the ork is associated ith the bending of flanges about their major ais, W f is the ork associated ith the tist of flanges and bending about their minor ais, and is the ork associated ith the bending of the eb as a plate. W W fu Af B ub dz 0 (79) bf W t v v ddz (80) f f B B 0 bf u / z u / z W t dydz u / y 0 u / y (8) 0 In hich Af is the area of each flange and I yf is the second moment of area of each flange about the y-ais. he terms and represent the stresses along the eb obtained in the usual ay as: N My A I v h t (8) (8) as: V M 0 M from Eq.(9). he terms and B the stresses at the top and bottom flanges, are calculated N Mh A I N Mh B A I (84) (85) In the matri form, the ork done by the surface tractions or the loss of the potential energy of the stresses becomes 07 IAU, Arak Branch

13 Finite Element Instability Analysis of the Steel Joist of Continuo 6 W S a (86) Similar to Eq.(75), Eq.(86) has to be derived in terms of the generalized displacement q. Using Eqs. (54, 74), this transformation can be performed as: W q C S C q q S q (87) S C S C (88) Is identified as the stability matri of the element.. Buckling solution he global stiffness and stability matrices of the structure are calculated by assembling the stiffness and stability matrices of all members. he buckling load factor can be found using an eigen technique that can solve the linear non-standard eigen problem of 0 K S q g (89) K and S are the assembled stiffness and stability matrices and he solution of Eq. (89) for nontrivial q g q g is the global buckling degrees of freedom. is a standard eigen problem, and is solved herein for the buckling load factor and buckling mode shape or eigenvector q g. In order to minimise the numerical efforts, the stiffness and stability matrices, K and S, are derived symbolically as much as possible. Here, the stiffness and stability matrices are calculated symbolically using MAHEMAICA. he elements of these matrices are then input into a finite element program called FED ritten in FORRAN. his program processes the input of the elements, calculates the matrices, solves for the eigenvalue and provides the associated eigen modes. 4 EXAMPES A standard to-span continuous composite I-beam, 0m in span-length subjected to 50 KN/m uniformly distributed load is considered. Cross-sectional dimensions of the beam are E, 000 N mm, E 00, 000 N mm, h,000 mm, b 400 mm, t 8 mm, t 6 mm. he concrete slab dimensions are,500mm 50mm and the f f reinforcement ratio ( Ar Ac) is set at.8%. Figs. 7, 8 and Fig. 9 contain the results of the in-plane analysis for various degrees of interaction. he stress resultants of steel joist are then used as the input data for the out-of-plane buckling investigation. he in-plane bending moment of steel joist in hogging region is larger for beams ith a loer degree of interaction. hese beams are therefore more susceptible to out-of-plane buckling. As is seen in able., the partial interaction design has affected the buckling resistance of the beam. he buckling mode shapes have minor influences due to partial shear interaction and are illustrated in Fig. 0 and Fig.. c s 07 IAU, Arak Branch

14 64 H. Bakhshi and H.R. Ronagh Fig.7 Slip of continuous composite beam. Fig.8 In-plane bending moment of steel joist. Fig.9 In-plane aial force of steel joist. Fig.0 ateral displacement of bottom flange along the length. 07 IAU, Arak Branch

15 Finite Element Instability Analysis of the Steel Joist of Continuo 65 Fig. Buckling tist of bottom flange along the length. able Partial interaction (PI) effects on critical buckling load. Degree of Interaction ( ) Critical Buckling oad cr ( KN m ) PI Effects cr, PI cr, FI 00 cr, FI (%) 0., , , ,59.7.0, CONCUSIONS he finite element models for analyzing the in-plane analysis and out-of-plane restricted distortional buckling of continuous composite I-beams including the effects of partial shear interaction are presented. he proposed models can be used effectively to account for the differences in the beam s fleural rigidities in hogging and sagging regions due to the concrete cracking in tension. hese versatile models are capable of accommodating different types of boundary and loading conditions. hey can be used to develop more economical and efficient designs. he eample presented shos that the to-span continuous composite I-beam is more prone to buckling at loer degrees of interaction. Since the degree of interaction affects the buckling resistance of steel joist, the partial interaction design is found to be critical in terms of strength and stability of continuous composite I-beams subjected to uniformly distributed loads. APPENDIX he degree of interaction,, is defined in this paper as minus the ratio of the slip at the steel/concrete interface in a shear span ith partial interaction to the slip in case of no-interaction. Varies from 0 (no-interaction) to (fullinteraction). he relation for the degree of interaction can be ritten as: S S S ni, is the slip at position of the steel/concrete interface of composite beam ith partial interaction (ith any k s as the stiffness of the shear connectors); S ni, is the slip at position of the steel/concrete interface of composite beam ith no-interaction (ith k s 0 ). Often is chosen at the location of maimum slip. he matrices developed in the finite element models can be presented in the folloing. D d d d d d d d d d d d d s s s s s 4 s 5 s 6 s 7 s8 s 9 s0 s s 07 IAU, Arak Branch

16 66 H. Bakhshi and H.R. Ronagh 6hc 6hc 4hc hc s s s c d d d h 6hc 6hc hc hc s 4 s 5 s 6 d d d 6hs 6hs 4hs hs s 7 s 8 s 9 s d d d h 6hs 6hs hs hs s0 s s d d d he matrices H and H are defined as: H H he transformation matrices C and C are given as follos: C C u 0 C C 0 Cu C C : C C C u 0 0 u C IAU, Arak Branch

17 Finite Element Instability Analysis of the Steel Joist of Continuo 67 C C he eb displacement matri is given as follos: 07 IAU, Arak Branch

18 68 H. Bakhshi and H.R. Ronagh N y y y y z y y z y y z y y y 8 4 y y y z 8 4 h y y y z 8 4 y y y 8 4 y y y z 8 4 y y y z 8 4 y y y z 8 4 y y y z 8 4 REFERENCES [] Nemark N.M., 95, est and analysis of composite beams ith incomplete interaction, Proceedings of Society for Eperimental Stress Analysis 9(): [] Bradford M.A., Gao Z.,99, Distortional buckling solutions for continuous composite beams, Journal of Structural Engineering 8(): [] Dekker N.W., 995, Factors influencing the strength of continuous composite beams in negative bending, Journal of Constructional Steel Research 4: [4] ehami M.,997, ocal buckling in class continuous composite beams, Journal of Constructional Steel Research 4(-): [5] Jasim N.A., Atalla A., 999, Deflections of partially composite continuous beams: A simple approach, Journal of Constructional Steel Research 49: 9-0. [6] Nie J., Cai C.S., 00, Steel-concrete composite beams considering shear slip effects, Journal of Structural Engineering 9(4): [7] Nie J., Fan J., Cai C. S., 004, Stiffness and deflection of steel-concrete composite beams under negative bending, Journal of Structural Engineering 0(): [8] Ranzi G., Bradford M.A., Uy B., 004, A direct stiffness analysis of a composite beam ith partial interaction, International Journal for Numerical Methods in Engineering 6: [9] Bradford M.A., 997, ateral-distortional buckling of continuously restrained columns, Journal of Constructional Steel Research 4(): -9. [0] Bradford M.A., Ronagh H.R., 997, Generalized elastic buckling of restrained I-beams by FEM, Journal of Structural Engineering (): [] Bradford M.A., rahair N.S., 98, Distortional buckling of I-beams, Journal of he Structural Division 07: IAU, Arak Branch

19 Finite Element Instability Analysis of the Steel Joist of Continuo 69 [] Bradford M.A., Ronagh H.R., 997, Elastic distortional buckling of tapered composite beams, Journal of Structural Engineering and Mechanics 5(): [] Bradford M.A., Ge X.P., 997, Elastic distortional buckling of continuous I-beams, Journal of Constructional Steel Research 4(-): [4] Bradford M.A., Kemp A.R., 000, Buckling in continuous composite beams, Progress in Structural Engineering and Materials : [5] Vrcelj Z., 004, Buckling Modes in Continuous Composite Beams, PhD hesis, he University of Ne South Wales, Sydney, Australia. [6] Xu R., Wu Y.F.,007, o-dimensional analytical solutions of simply supported composite beams ith interlayer slips, International Journal of Solids and Structures 44: [7] Wang A.J., Chung K.F., 008, Advanced finite element modelling of perforated composite beams ith fleible shear connectors, Engineering Structures 0(0): [8] Zona A., Ranzi G., 0, Finite element models for non-linear analysis of steel concrete composite beams ith partial interaction in combined bending and shear, Finite Element in Analysis and Design 47(): [9] Chakrabarti A., Sheikh A.H., Griffith M., Oehlers D.J., 0, Analysis of composite beams ith partial shear interactions using a higher order beam theory, Engineering Structur 6: 8-9. [0] ezgy-nazargah M., 04, An isogeometric approach for the analysis of composite steel concrete beams, hin-walled Structures 84: [] ezgy-nazargah M., Kafi., 05, Analysis of composite steel-concrete beams using a refined high-order beam theory, Steel and Composite Structures 8(6): [] He G., Yang X., 05, Dynamic analysis of to-layer composite beams ith partial interaction using a higher order beam theory, International Journal of Mechanical Sciences 90: 0-. [] Chen S., imazie., an J., 05, Fleural behavior of shallo cellular composite floor beams ith innovative shear connections, Journal of Constructional Steel Research 06: IAU, Arak Branch