Chemrouk, M.

Ihaddoudène, A.N.T.

Saidani, Messaoud

Coventry University Pure Portal

Modeling of steel frames with semi rigid joints  Ihaddoudène, A.N.T. , Chemrouk, M. and Saidani, M.  Published version deposited in CURVE October 2012  Original citation & hyperlink:  Ihaddoudène, A.N.T. , Chemrouk, M. and Saidani, M. (2008) Modeling of steel frames with semi rigid joints. International Journal of Applied Engineering Research, volume 3 (7): 955-967. http://www.ripublication.com/Volume/ijaerv3n7.htm     Copyright © and Moral Rights are retained by the author(s) and/ or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This item cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder(s). The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders.         CURVE is the Institutional Repository for Coventry University http://curve.coventry.ac.uk/open  International Journal of Applied Engineering Research ISSN 0973-4562 Volume 3, Number 7 (2008), pp. 955–967 © Research India Publications http://www.ripublication.com/ijaer.htm  Modeling of steel frames with semi rigid Joints   A.N.T.Ihaddoudène 1 , M. Saidani2,a and M.Chemrouk3  1, 3 Built Environment Research Laboratory, Faculty of Civil Engineering, U.S.T.H.B., Algiers, Algeria 2 Faculty of Engineering and Computing, Department of Built Environment, Coventry University, Coventry, England, UK. Emails: nacer_i@yahoo.fr, m.saidani@coventry.ac.uk, epau_chemrouk@yahoo.fr   Abstract Beam to column connections are usually considered either as perfectly rigid or as nominally pinned. However, as is now well established, this does not represent the actual behaviour of the connection, and hence the impact it has on the behaviour of the structure itself, as connections are often semi-rigid. The present paper presents a mechanical model of the connections, where the rigidity of the joint is represented by means of rotational and translational springs introducing the concept of non deformable element of nodes, describing relative displacements and rotations between the nodes and the elements of the structure. Examples are provided to illustrate the simplicity and efficiency of the method and comparisons to recently published results have proved the generalization, the simplicity and the efficiency of the proposed model.  Keywords: Rigid, Semi-rigid, Connection, Mechanical model, Frames, Buckling  Notations: k k1 2,    :   Elastic constants of the springs in rotation at nodes " "i and " "j , respectively vv CC 21 , :   Elastic constants of vertical springs at nodes " "i and " "j , respectively.                         956  A.N.T. Ihaddoudene, M. Saidani and M. Chemrouk  uu CC 21 ,  :  Elastic constants of axial springs at nodes " "i and " "j , respectively. Ψ :     Area of bending moment diagram for a simply supported beam. nl ml, :   Distance to left support and right support respectively, from gravity centreΨ  ω :     Flexural rigidity per unit length,lEI iΔ :       Relative vertical displacement between nodes " "i and " "j  jjii MVMV ,,, : Reactions at nodes " "i and" "j  ,   in local reference.  Fe  :     Vector force in local reference.   Introduction  Conventional analysis of steel frames assumes either perfectly rigid or pinned connections. The rigidity of most connections behaves, in fact, somewhat between these two extremes and incorporating the effect of connection flexibility of the frame becomes necessary.      To assess the real behaviour of the frame, it is therefore necessary to incorporate the effect of connection flexibility of the frame (Bjorhovde et. al. [1]; Gerstle [2]; Kishi and Chen [3]; Saidani [4]; Ihaddoudène  and Chemrouk [5]). In the past few decades, extensive research works on static loading tests for different types have been carried out and conducted on the commonly used connections from which large collections of test data have been reported (Jaspart [6]; Azizinamini et al [7]; Nethercot and Zandonini [8]).       Some researchers such as Kishi and Chen [3], have collected available experimental results and constructed steel connection data banks that provided the user with not only the test data, but also some predictive equations, however, not every structural engineer has access to the data base of experimental results.      For the beam-moment connections, it is clearly shown (Kishi and Chen [3]; Jaspart [6]; Azizinamini et al [7]; Nethercot and Zandonini [8]; Atamaz Sibai [9]; Aribert et al [10]) that the moment-rotation relationship is non-linear for all types of connections and varies depending on connection flexibility.      Several mathematical models have been proposed to fit the moment-rotation curves from experimental data. These models vary widely in their complexity to describe ( )θ−M  curves of connections. It is presented in its exponential form (Ihaddoudène and Chemrouk [5]) by the equation: αkM=Θ  (1)      Because of the high number of the parameters influencing the behaviour of connections, accurate modeling of such behaviour becomes complex. Globally, initial rigidity and the ultimate moment of the connection are the two most significant characteristics to define the behaviour of a joint (Bjorhovde et. al. [1]; Ihaddoudène and Chemrouk [5].  Modeling of Steel Frames With Semi Rigid Joints 957  Mechanical model      To incorporate the effect of semi rigid joints on steel frames (Fig.1a), the concept of a non-deformable element of node (Fig.1b) describing relative displacements and rotations between the nodes and the elements of the structure is introduced. The adopted model (Ihaddoudène and Chemrouk [5]) is based on the analogy of three springs with two translational and one rotational (Fig.1c) and the effect of each spring is considered separately (Fig.2).                 (a): Semi rigid joints; (b): Non deformable node; (c): Bar element and non deformable node.  Figure 1: Mechanical model adopted          Equilibrium equations and rotational deformations  2.1.1. Bar element subjected to axial force:   N N Nij ji= − =  (2.a) (c) Non deformable          nodes  (b) Semi rigid       joints (a)    Bar Rotational spring Translational spring Non deformable node ii)(θfM =NCu u=≡i  VCv v=+(a): Bar element and non deformable node;(b): Axial spring; (c):Translational and rotational springs   Figure 2 :Analogy of three springs 958  A.N.T. Ihaddoudene, M. Saidani and M. Chemrouk  Δ ju uNlEAC N C N,1 1 2= + +  (2.b)  2.1.2. Bar element subjected to transversal forces: When considering the actions of the vertical and rotational springs, the equilibrium equations are expressed as: 0=−+ RVV ji  (3.a) 0=−++ lVRZMM jji  (3.b) The rotational deformations are for the node ""i  (see Figure.3)  ωωω 63)(1 121 jiijviviiMMkMlmVCVCl−++Ψ++−Δ=Θ  (3.c) and for the node "" j ,  ωωω 63)(1 221 ijjjvivijMMkMlnVCVCl−++Ψ−+−Δ=Θ  (3.d) In a simplified form, these equations can be written as: ( ) )(lmlMBMBA ijiΨ−Δ−=−−+ 666212 1 ωωθ      (3.e) )( ( )lnlMBAMB jjiΨ+Δ−=++−− 666221 2 ωωθ  (3.f)  Where   )31( 11 ωkA += , )31( 22 ωkA +=    and       2 21)(3lCCBvv +=ω         where  )( 21* jvivi VCVC +−Δ=Δ   3 5i j 1l*Δ iMk1  iM  1=iθ  i  vC1  iV  1k  ⎟⎟⎠⎞⎜⎜⎝⎛−wMwM ji63Figure 3: Unit rotation of node ""i  Modeling of Steel Frames With Semi Rigid Joints 959  Rigidity of the bar element 2.2.1. Rigidity of the bar element for the displacement Δ j ,1 in direction 1:  By imposing the unit displacement to the node "" j , the axial effort in the bar element is expressed as: ) 1211, =++⎜⎝⎛=Δ uuj CCEAlN    (4.a) lEAaCCEAlEANuu 1121 )(=++=   (4.b) where:    a EAlC Cu u111 211=+ +( ) 2.2.2. Rigidity of the bar element for the displacement of the node " "i   in the transversal direction: By imposing to node " "i , a unit displacement Δ i ,3 1=  in the direction 3( see Figure.4)      Replacing the limit conditions Δ Δi i,3 1= =  and Θ Θ Ψi j R, ,5 5 0= = = =  in the equations (3.e) and (3.f) leads to:  lBBABAAM iω6)21())((4)21(2212−−+++−= (5.a) lBBABAAM jω6)21())((4)21(2211−−+++−=  (5.b)  222121 12)21())((41lBBABAAAV iω−−++++=   (5.c)  2.2.3. Rigidity of the bar element to the rotation around the axe 5  *Δ  jMji2k1kvC2vC1jViVFigure 4: Unit displacement of node ""i  3 5i j 1 M960  A.N.T. Ihaddoudene, M. Saidani and M. Chemrouk      Imposing to the node " "i of the structure the unit rotation 15, == ii θθ around the axe 5, replacing the limit conditions 15, == ii θθ , 0==Ψ==Δ Rjθ  in the equations (3.e) and (3.f), one obtain: ω4)21())((4)(32212BBABABAM i−−+++=  (6.a) ω2)21())((4)21(3221 BBABABM j−−++−=  ( 6.b)  lBBABAAViω6)21())((4212212−−+++−=   (6.c)  2.2.4. Reactions of the bar element for fixed ends: If the nodes " "i  and " "j are fixed in their initial positions, the internal efforts at each end of the bar element can be determined from the equations (3.e) and (3.f) by imposing the limit conditions: 0===Δ ji θθ . [ ][ ]2212)21())((4)21()(26BBABAlBnBAmM i−−++−−+Ψ−=   (7.a) [ ][ ]2211)21())((4)21()(26BBABAlBmBAnM j−−++−−+Ψ=   (7.b)    Stiffness matrix To establish the modified stiffness matrix considering the effect of connection flexibility, the direct method is used, i.e. the rigidity kij  of an element is the reaction in the direction " "j  due to a unit displacement in the direction" "i .      The nodes of the beam are represented by non deformable frames at each ends. Considering in the mechanical model adopted (Ihaddoudène and Chemrouk [5]), the Figure 5: Unit rotation of node 1i =θ  iMivC1  iV1k  jMj2kvC2jV3 5i j 1 1=iθModeling of Steel Frames With Semi Rigid Joints 961  beam with different rotational flexibilities 1k  and  2k  at ends i  and j  respectively. In order to establish the different elements of the stiffness matrix eK in local reference, equilibrium equations and rotational deformations are considered for each element kij .      The stiffness matrix, eK , in local coordinates is given by: ⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=44434241343332312423222114131211kkkkkkkkkkkkkkkkKe                                                                          (8.a) Terms jk2  of the stiffness matrix (8.a), for instance in bending, are obtained by letting 1=iθ , 0=Δ i ; 0=jθ ; 0==Ψ R  ; 0=B  in the Eq. (3.e) and (3.f). For linear behaviour αθ kM=  where 1=α  Which gives:      [ ]1)31)(31(4)21(1821221−+++−=ωωωωkklkk                     (8.b)  1)31)(31(4)31(1221222−+++=ωωωωkkkk  (8.c) 2123 kk −=  (8.d) 1)31)(31(462124−++=ωωωkkk                                                                      (8.e) The same procedure is followed in deriving all the terms of the local stiffness matrix eK .  Nodal load vector When the beam with different flexibilities 1k  and 2k at both fixed ends, is subjected to an external load ""q , the nodal load vector is established in local reference as: [ ] [ ]TjjiiTjjjiiie MVMVMYXMYXF −−−−== ,,0,,,0,,,,,     (9) In global reference, the stiffness matrix and the vector force are obtained respectively as:  eeTee T.K.TK =                                                                                             (10.a) [ ]TjjjiiieTee MVVMVVFTF −−−−== ,cos,sin,,cos,sin. ββββ   (10.b) In which eT is the transformation stiffness matrix given by: ⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡−−=1000000cossin0000sincos0000001000000cossin0000sincosββββββββeT 962  A.N.T. Ihaddoudene, M. Saidani and M. Chemrouk  The angle β defines the orientation of the element with respect to the global reference system. Finally, the internal forces are calculated by using the well known equation:  [ ]{ } { }eee FUK =  (11)  Buckling consideration The system of equilibrium equation is now expressed in the deformed state of the bar element, which connections are semi rigid. Here, the effects of semi rigid joints on the buckling load of steel frames are considered. In order to express the terms jk2 , for instance, of the stiffness matrix, let us considering a column with semi rigid joints (Fig.6), the equilibrium equations are expressed as: HHH ji ==     (12.a) jx MHxNyM −+=  (12.b) ji MHlM −=  (12.c) The governing differential equations is given as follows: jMHxNyxEIy +−−=)("      (13.a) EIMHxyxy j+−=+ 2)(" α      (13.b) Where EIN=2α     The general solution of Eq.(12) is: pyxBxAxy ++= αα sincos)(                   (13.c)      Where py , A and B are respectively the particular solution and integration constants to be determined using the boundary conditions at the ends supports: iM1=iθFigure 6: Bar element with semi rigid joints iHjH  1k2kP  l  xyjMiMk1  PModeling of Steel Frames With Semi Rigid Joints 963  0)0( =y   leads to EIMB j2α−=  (13.d) 0)( =ly    leads to    [ ]HllMlEIA j +−= )1(cossin12 ααα (13.e) Hence, [ ]EIMHxEIxMHllMlEIxxy jjj 222cos)1(cossinsin)(ααααααα +−−+−=     (14) The derivative: [ ]EIHEIxMHllMlEIxxy jj 222sin)1(cossincos)('ααααααααα−++−= (15) The reactions iMH , and jM  are determined employing the boundary conditions jMky 2)0(' =    (16.a) iMkly 11)(' +−=    (16.b) We obtain : )(1 vlwH ξ=  (17.a) )(1 vwM j φ=  (17.b) )(2 vwM i φ=   (17.c) Where lv α=       From which the terms of jk2  are derived (see Table 1)  Table 1: jk2  expressions in several situations  Both axial force and semi rigid connections are accounted Only axial force is accounted [ ]DvvwkvvlEIk sincos1 22221+−=  [ ])sincos22(cos12221 vvvvvlEIk−−−=  [ ]DvwvkvvvvlEIk sincossin2222+−=  [ ])sincos22(cossin22vvvvvvvlEIk−−−=  [ ]DvvwkvvlEIk sincos1 22223+−−=  [ ])sincos22(cos12223 vvvvvlEIk−−−−=  [ ]DvvvlEIk sin24−=  [ ])sincos22(sin24vvvvvvlEIk−−−=   Where ),,()sincos22( 21 kkvvvvD ξ+−−=  The same procedure is employed to derive all the components of the matrix. The buckling load for the frame is obtained when the determinant is equal to zero, i.e: 964  A.N.T. Ihaddoudene, M. Saidani and M. Chemrouk  [ ] 0det =K                        (18) Example The frame below [11] which has a single span mL 20= , a story height mh 201 =  and mh 152 =  is subjected to concentrated and distributed load kNP 12= , mlkNq /4.2=  respectively. Different flexibilities have been considered; as for the columns AB and DC, they are 21,kk  and 56,kk  respectively, for the beam they are 43,kk  as shown in Figure.7.         In bending, the rotational spring is the essential component and hence the equations of rotational deformations (7.a) and (7.b) are reduced and expressed as:  [ ][ ]1)31)(31(4)31(26212−++−+Ψ−=ωωωkklnkmMi     [ ][ ]1)31)(31(4)31(26211−++−+Ψ=ωωωkklmknM j   lRZMMV jii−+=  Table 2 below summarises the calculation data for each element: AB CD6k  m15 m20  kN12  1k  2k  3k 4k 5kcI5  cI2  cI3  Figure 7: Example of the reference [11] ml/kN4.2q =m6m12 Modeling of Steel Frames With Semi Rigid Joints 965  Table 2: Example data [11]  Elements 1 2 3  (No charge) Ψ [m2] 2432m  21600m  - m  444.0  5.0  - n  555.0  0.5  wki31+  5.2  666.1  333.3  wk j31+  2.0 5.2  25.1  iM [kN.m] 262.9−  28.61−  - jM [kN.m] 684.17  744.35  -     The load vector force is then:    ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−+−=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=468.8744.35592.43468.8744.3528.61684.1721HMMF        The matrix stiffness of the structure is obtained by assembling the elements and the internal forces are calculated by using the equation with six degree of freedom:     [ ]{ } { }eee FUK =  3k 4kFigure 7 : b. Beam element lqB C  B  2k  1k  2Ψ 10=nl 8=ml481Ψ8  4 G A kNR 12= mZ 12= m6Figure 7: a Area of bending moment diagram for a simply supported beam. 966  A.N.T. Ihaddoudene, M. Saidani and M. Chemrouk  The final bending moment diagram is:   The proposed method is in excellent agreement with the numerical results obtained by the author of reference [11].   Conclusions The influence of the flexibility of connections in steel frames is investigated and a simple method of analysis and design is provided through a mechanical model for the joints.     The significant role of semi rigid joints on the buckling load on the frame is also included in the analysis of steel frames for linear buckling effect.      Illustrative examples of simple frame are examined and comparisons between the results obtained are similar, suggesting the proposed model is adequate and may be a useful tool in the analysis of steel frames with semi-rigid joints.   References  [1] Bjorhovde, R., Colson, A., and Brozzetti, J., 1990, “Classification system for beam-to column connections,” J.  Str. Eng., 116(11), pp. 3059-3077. [2] Gerstle, K.H., 1988, “Effect of connections on frames,” J. Constr. Steel Research., 10, pp. 241-267. [3] Kishi, N., and Chen, W.F., 1989, “Semi-rigid steel beam to column connections: Data base and modeling,” J. Str. Eng., 115(1), pp. 105-119. [4] Saidani, M., 1998, “The effect of eccentricity connection on the distribution of axial force and bending moments in RHS lattice girders,” J. Constr. Steel Research., 47(3), pp. 211-221. Figure 8: Bending moment diagram (kN.m), present study and [11] 2467.33(33.25) 247.33(33.25) 888.62(62.89) 961.36(36.95) 3216.8(8.32) 3123.8(8.32) 1.30  Modeling of Steel Frames With Semi Rigid Joints 967  [5] Ihaddoudène, A.N.T., and Chemrouk, M., 2004, “Influence of Semi-Rigid Joints on the Behaviour of Steel Beam-Column Structures,” Proc. of the International Conference on Computational & Experimental Engineering & Science., Paper No.249, Madeira. [6] Jaspart, J.P., 1991, “Etude de la semi-rigidité des nœuds poutre-colonne et son influence sur la résistance et la stabilité des ossatures en acier,” Ph.D. thesis, University of Liège.  [7] Azizinamini, A., Bradburn, J.H., and Radziminski, J.B, 1987, “Initial stiffness of semi-rigid beam-to-column connections,” J. Constr. Steel Research., 8, pp. 71-90. [8] Nethercot, D.A., and Zandonini, R., 1989, Methods of prediction of joints behaviour: beam-to-column connections, Structural Connection: stability and strength, edited by Narayanan, R., Elsevier Applied Science,  London and NY, Chap. 2. [9] Atamaz, S.W., 1991, “Semi-rigid joint modelling for nonlinear analysis of flexibly connected frames,” Ph.D. thesis, Swiss Federal Polytechnic School, Lausanne. [10] Aribert, J.M., Braham, M., and Lachal, A., 2004, “Testing of simple joints and their characterisation for structural analysis,” J. Constr. Steel Research., 60, pp. 659-681. [11] Wang, C.K., 1983, Intermediate Structural Analysis, McGraw-Hill, Chap.18. 

Modeling of steel frames with semi rigid joints

https://pure.coventry.ac.uk/ws/files/3995948/Modeling%20of%20steel%20frames.pdf

Modeling of steel frames with semi rigid joints

Abstract

Similar works

Full text

Available Versions

Coventry University Pure Portal