3D beam finite element including nonuniform torsion

Transcription

1 Avalable onlne at Proceda Engneerng 48 (212 ) as 212 3D beam fnte element ncludng nonunform torson Justín urín a *, Vladmír Kutš a, Vtor Králov a, bor Sedlár a a Department of Appled echancs and echatroncs of IPAEE, FEI SU n Bratslava, Ilovova 3, Bratslava, Slovaa Abstract New 3D beam fnte element ncludng non-unform torson wll be presented n ths contrbuton whch s sutable for analyss of beam structures of open and closed cross-sectons. he secondary torson moment deformaton effect wll be ncluded nto the stffness matrx. Results of the numercal experments wll be dscussed and evaluated. 212 he Authors. Publshed by Elsever Ltd. 212 Publshed by Elsever Ltd.Selecton and/or peer-revew under responsblty of the Branch Offce of Slova etallurgcal Socety at Faculty Selecton of etallurgy and/or peer-revew and Faculty under of echancal responsblty Engneerng, of the Branch echncal Offce Unversty of Slova of Košce etallurgcal Open access Socety under CC at Faculty BY-NC-ND of lcense. etallurgy and Faculty of echancal Engneerng, echncal Unversty of Košce. Keywords: Non-unform torson, 3D beam fnte element, secondary torson moment deformaton effect, closed thn walled cross Nomenclature e u nodal dsplacement vector e F nodal load vector u, v, w dsplacement (m) N, y, normal and shear forces (N) z, torson moment (Nm) x, bendng moments (Nm) y z bmoment (Nm 2 ), j stffness coeffcent I, I y z quadratc area moments of nerta (m 4 ) I torson constant (m 4 ) I warpng constant (m 6 ) I s secondary torson constant (m 4 ) E Young s modulus (Pa) G shear modulus (Pa) Gree symbols ν Posson rato * Correspondng author. el.: E-mal address: justn.murn@stuba.s Publshed by Elsever Ltd.Selecton and/or peer-revew under responsblty of the Branch Offce of Slova etallurgcal Socety at Faculty of etallurgy and Faculty of echancal Engneerng, echncal Unversty of Košce Open access under CC BY-NC-ND lcense. do:1.116/j.proeng

2 Justín urín et al. / Proceda Engneerng 48 ( 212 ) ϕ angular dsplacement x, y, z ϑ warpng part of the twst angle frst dervatve (m -1 ) secton ordnate (m 2 ) σ normal stress caused by bmoment (Pa) σ normal bendng stress (Pa) τ p z prmary shear stress (Pa) τ s prmary shear stress (Pa) Subscrpts,j, nodal pont R at corners an analytcal 1. Introducton he effect of warpng must be assumed n stress and deformaton analyses of structures above all wth thn-walled crosssectons loaded by torson. Warpng effects occur manly at the ponts of acton of the concentrated torson moments (except for free beam ends) and at sectons wth free-warpng restrctons. Specal theores of torson wth warpng non-unform torson have been used to solve such problems analytcally (e.g. [1]). he analogy between the 2nd order beam theory (wth tensonal axal force) and torson wth warpng s also very often used (e.g. [2], [3]). One has to pont out that n the lterature and practce, as well as n the EC-3 [4] and EC-9 [5] gudelnes, strong warpng s assumed to occur n open crosssectons only. Warpng-based stresses and deformatons n closed sectons, however, are assumed to be nsgnfcant and have been therefore neglected. Accordng to the above mentoned theory of torson of open cross-sectons wth warpng and the analogy, specal 3Dbeam fnte elements have been desgned and mplemented nto the fnte element codes (e.g. [6], [7]). he warpng effect s ncluded through an addtonal degree of freedom at each nodal pont - the frst dervatve of the angle of twstng of the beam cross-secton. Important progress n the soluton of torson wth warpng has been reached n papers [8] and [9] where a combnaton of boundary and fnte element method was used allowng a warpng analyss for composte beams wth longtudnally varyng cross-secton. However, latest theoretcal results have shown that the effect of warpng must be consdered n the case of non-unform torson of closed-secton beams [1]. For prsmatc beams, the analogy between the torson wth warpng (ncludng the secondary torson moment deformaton effect) and the 2nd order beam theory (ncludng the shear force deformaton effect) has to be used. hs approach was mplemented nto the computer code IQ-1 [13]. hs analogy does not hold for nonprsmatc beams [11]. Accordng to the last research results n ths area ([1], [11], [12], [14]), the local stffness relaton of a new two-node fnte element for torson wth warpng of straght beam structures s presented n contrbuton [15]; agan based on the above-mentoned analogy. he warpng part of the frst dervatve of the twst angle has been consdered as the addtonal degree of freedom n each node at the element ends whch can be regarded as part of the twst angle curvature caused by the warpng moment. hs new fnte element can be used n non-unform torson analyses of open and closed cross-secton beams. Fnally n [16], the boundary element method has been appled n the non-unform torson analyss of smply or multples connected bars of doubly symmetrcal arbtrary constant cross-secton, tang nto account secondary torson moment deformaton effects. Necessty of the non-unform torson effect n the analyss of close shaped crosssecton has been confrmed. In ths paper, a new 3D beam fnte element wll be presented whch s sutable for structural analyss of 3D beam structures. he classc 12x12 local stffness matrx of the 3D beam fnte element wll be enhanced to 14x14 stffness matrx. he warpng part of the frst dervatve of the twst angle has been consdered as the addtonal degree of freedom n each node at the element ends whch can be regarded as part of the twst angle curvature caused by the warpng moment. he transformaton of the local fnte element equaton to the global fnte element equaton wll be done. he derved fnte element equatons wll be mplemented nto the computer program and the numercal experments wll be done. Results of the numercal analyss concernng the spatal combned loadng of chosen beam structures of open and closed cross-secton wll be presented and dscussed.

3 438 Justín urín et al. / Proceda Engneerng 48 ( 212 ) he local and global fnte element equatons 2.1. Local fnte element equaton Fgure 1 shows a prsmatc beam element of length L wth two nodes and, and wth approprated geometrc, statc, nematcs and materal quanttes: A [m 2 ] s the double symmetrc cross-sectonal area; I y [m 4 ] and I z [m 4 ] are the quadratc area moments of nerta; I [m4 ] s the torson constant; I [m6 ] s the warpng constant, I s [m 4 ] s the secondary torson constant; E s the Young s modulus; G s the shear modulus. In order to nclude the warpng, a new degree of freedom s added to the classcal nodal varables n the stffness matrx n each element nodal pont. he warpng part of the frst dervatve of the twst angle ( ϑ ) has been consdered as the addtonal degree of freedom n each node at the element ends [15]. hs can be regarded as part of the twst angle curvature whch s caused by the warpng moment. hs choce brngs advantages when applyng the boundary condtons. If the secondary torson moment deformaton effect s not consdered ϑ ( x) = ϑ ( x). he nodal dsplacement vector n the local coordnate system, as shown n Fg. 1, s and the respectve nodal load vector s e { u } { u v w ϕ ϕ ϕ ϑ u v w ϕ ϕ ϕ ϑ } = (1) x y z e { F } { N N } y z x y z = (2) where and are the torson moments, and are the warpng moments at the nodal ponts. he geometrcal meanng of all other nematcs and statc varables s evdent from Fgure 1. y z x x y y z z v y y ϕ y N u x ϕ x x L ϑ y, ϕ w z z z v y y ϕ y N u z ϕ ϕ w x x z z z ϑ Fg. 1. 3D beam element ncludng non-unform torson n local coordnate system. By enhancng the classc 3D beam fnte element about the local stffness matrx of non-unform torson of straght rod [15] we get the local element equaton of the 3D beam fnte element ncludng the non-unform torson wth effect of

4 Justín urín et al. / Proceda Engneerng 48 ( 212 ) secondary torson moment deformaton effect: N N y z x z y z y x y z = 1,1 2,2 S 3,3 Y 4,4 u u v w ϕx ϕ y ϕz ϑ v w ϕx ϕ y ϕz ϑ 7,7 7,11 7,14 3,5 5,5 2,6 6,6 E 4,7 1,8 8,8 R 2,9 6,9 9,9 Y 3,1 5,1 1,1 4,11 11,11 3,12 5,12 1,12 12,12 2,13 6,13 9,13 13,13 4,14 11,14 14,14 (4) he stffness terms n (14) are: 3,3 1,1 = 3,1 = EI y / = 12 L ; 3 = = = EA/ L ; 1,1 8,8 1,8 3,5 3,12 = 1,12 = 1,12 = EI y / 2,2 9,9 = 2,9 = EIz / = 12 L ; 2 3 2,6 2,13 = EI z / = 6 L ; = 6 L ; 4,4 = 11,11 = 4,11 = c1 / κ2 ; 2 ( ) = κ ; = 4EI y / L ; = 4 EI z / L ; 4,7 4,14 = 7,11 = 11,14 = c = 2 / 2 13 ( b b ) 7 / 7 = / ;,7 = 14,14 = κ ;, ,9 9,13 = EI z / 5,5 12,12 = 2 6,6 13,13 = = 6 L ; 6,13 = 2 EI z / L ; 5,12 = 2EI y / L. GI Wth K = κ, f = K and x = L beng the transfer constants b j calculated as: EI snh( fx) b = cosh( fx), b1 = f, for b j 2 a j 2 j 2 : b j j = and wth 1 K a =, for x j 1: a j = j!. b1 he stffness constants are: 1 = EI ; b 2 = 2 EI ; b 3 b1 3 =. EI GI s he arbtrary cross-sectonal characterstcs are descrbed by: A cross-sectonal area [m 2 ]; I torson constant [m 4 ]; I s secondary torson constant [m 4 ]; I - warpng constant [m 6 ]. ateral propertes: E elastcty modulus; G shear modulus. I he secondary torson moment deformaton effect s encountered through constant = 1 + κ and the transfer I s constants b j, j, 3. Parameter κ = 1 f ths effect s neglected. hs s usually made n the case of open form crosssectons where the effect of the secondary torson moment has been assumed nsgnfcant. Deformaton effect of the secondary torson moment must be consdered frst of all n the case of the closed form cross-sectons as demonstrated n [3]. he expressons for calculaton of the secondary torson constant (denoted as I s ) depend on the cross-secton type. hese can be found n [8] and [12], for example. When the nematc and statc varables n Eq. (4) are nown at the nodal ponts, the nodal forces, bendng moments, prmary and secondary torson moment, normal and shear stress can be calculated n a usual way [19]. he nodal ponts secondary torson moments are: s = κ ( GI ϑ ); κ ( GI ϑ ) s =. (5) 1 2

5 44 Justín urín et al. / Proceda Engneerng 48 (212) he prmary torson moments at the nodal ponts are: p = s ; p = s. (6) Expressons for the shear and normal stress calculaton depend on the cross-sectonal area type. hs problem has been descrbed n detal n [1]. hese expressons wll be used for stress calculatons n our numercal examples. For the straght beam structures the local relaton (4) concdes wth the global one. hs new fnte element can be used for analyss of torson wth warpng of constant both open and closed-shaped cross-sectons. he mplementaton of the expresson (4) nto the local equaton of the general 3D-beam fnte element s straghtforward, and t wll be done n the next chapter Global fnte element equaton Local element stffness matrx (4) after formaton has to be transformed to global coordnate system. he transformaton s performed by extended transformaton matrx and can be formally expressed as K G = K L ; (7) where KL s local element stffness matrx, s transformaton matrx and KG s element stffness matrx transformed to global coordnate system. ransformaton matrx has dmenson 14x14 and can be expressed as ªa «««=«««««a 1 a a º»»»»;»»» 1 ¼» (8) where submatrx a wth dmenson 3x3 has classcal form [ANSYS]. 3. Numercal experments In ths sub-chapter, the authors present a numercal study of a closed-secton prsmatc beam loaded by non-unform torson and a concentrated force (Fg. 8). In the analyss, the bendng and non-unform torson parameters wll be calculated by the new 3D beam fnte element. he materal propertes of the prsmatc cantlever alumnum beam (Fgure 8) are as follows: Young modulus E = 79 GPa, shear modulus G = E/(2(1+ν )) = 31.1 GPa and Posson raton ν =.27. he appled concentrated force F e = 1 [N] results n beam bendng. Restrctons at the clamped end and the applcaton of an concentrated torson moment e = 1 [Nm] result n non-unform torson. Fg. 2. Geometrcal parameters, loads and boundary condtons.

6 Justín urín et al. / Proceda Engneerng 48 (212) Accordng to Fg. 8, the followng geometrcal parameters have been chosen and calculated [12], [19]: L1 =.8 m, L2 =.15 m, h =.58 m, b =.18 m, t = s =.2 m, A = 2As + 2AG = 2(hs+ bt) =.34 m2 - the cross-sectonal area hb ht bs contanng the area of the webs As and flanges AG, warpng ordnate at the corners R = = m2, 4 ht + bs 2(hb) 2 2A = m6, torson constant I = = m4, secondary torson warpng constant I = R2 h/ s + b/t 3 2(h / s + b / t ) I A constant I s = = m4. ( Ahb) 2 2(h 2 + b 2 ) 2 + AS AG 3 Fg. 3. Normal and shear stress dstrbuton caused by non-unform torson. Normal stress caused by bendng can be easly calculated usng [1]. he authors consder outputs obtaned usng ANSYS Beam188 and Shell181 elements, the IQ1 software and the new 3D beam element n comparson wth the hn ube heory calculatons. In the numercal analyses, outputs beng the bmoment Ȧ, torson moment x, normal stress (from bendng ız and bmoment ıȧ), vertcal deformaton v, bendng angle ijz twst angle ijx. Fg. 4. Bmoment flow as calculated analytcally usng and numercally va ANSYS Beam188 and va the new 3D beam element.

7 442 Justín urín et al. / Proceda Engneerng 48 ( 212 ) Bendng and warpng deformatons are calculated separately to pont out the advantages of the new 3D beam element n comparson wth tradtonal commercal FE approach. Normal and shear stresses (Fg. 9) at any pont of the beam can be calculated accordng to [1], [12], [13] and [18], respectvely. For ths partcular beam and the torson moment = 1 Nm, the calculated parameters are as follows: maxmal bmoment =.4143 Nm 2, normal stress σ = R, an = = Pa, maxmal prmary shear stress I p s R τ = = p, max Pa, maxmal secondary shear stress τ s, max = = Pa. he total torson 2 hbt 2 2 I t A h + b G + 4 6( h / s + b / t) o angle s ϕ x = e able 1: Comparson of obtaned analytcal and numercal results (nought values are omtted). Please, note the strong msscalculaton n ANSYS Beam188 of non-unform parameters. value unt poston 1: x = mm fxed (clamped) support SHELL 181 BEA 188 3D beam F y [N] z [Nm] z [Pa] x [Nm] [Nm 2 ].4143 n/a [Pa] poston 2: x = 8 mm load applcaton F y [N] v [mm] z [ ] x [Nm] x [ ] [Nm 2 ] n/a [Pa] poston 3: x = 95 mm free end v [mm] z [ ] x [ ]

8 Justín urín et al. / Proceda Engneerng 48 (212) Fg. 5. axmum normal stress dstrbuton along the beam s longtudnal drecton: analytcal, ANSYS Beam 188 and Shell 181 element. In support of the new 3D beam element, the authors wsh to pont out that usng only two new 3D beam elements for ths analyss one obtans satsfactory results (see able 1 and Fg. 11) even for non-unform torson parameters. oreover, the presented results are n very good agreements wth the recently obtaned expermental data [17]. 4. Conclusons New 3D beam fnte element ncludng non-unform torson s presented n ths contrbuton whch s sutable for analyss of beam structures of open and closed cross-sectons. he secondary torson moment deformaton effect has been ncluded nto the stffness matrx. Results of the numercal experments show very hgh effectveness and accuracy of our new beam fnte element.

9 444 Justín urín et al. / Proceda Engneerng 48 ( 212 ) Acnowledgements hs paper has been supported by Grant Agency VEGA (grant No. 1/534/12). References [1] Vlasov, V.Z., enostnné pružné prúty, SNL. [2] Ro, K., Sedlace G., heore der Wölbrafttorson unter Berücschtgung der seundären Schubverformungen - Analogebetrachtung zur Berechnung des querbelasteten Zugstabes, Stahlbau, 35, p. 43. [3] Rubn, H., 25. Wölbrafttorson von Durchlaufträgern mt onstantem Querschntt unter Berücschtgung seundärer Schubverformung, Stahlbau, 74, Heft 11, p [4] EC-3 Eurocode 3. Desgn of steel structures. [5] EC-9 Eurocode 9. Desgn of alumnum structures. [6] ANSYS Swanson Analyss System, Inc., Johnson Road, Houston, PA 15342/13, USA. [7] RSAB, 26. Ingeneur - Software Dlubal GmbH, efenbach. [8] Sapountzas, E.J., oos, V.G., D beam element of composte cross-secton ncludng warpng and shear deformaton effect, Computers and Structures, 85, p. 12. [9] oos, V.G., Sapountzas, E.J., D beam element of varable composte cross-secton ncludng warpng effect, Acta echanca, 171 (3-4), p. 73. [1]Rubn, H., 27. orsons-querschnttswerte für rechtecge Hohlprofle nach EN 121-2: 26, Stahlbau, 76, Heft 1. [11]Rubn, H., 27. Zur Wölbrafttorson geschlossener Querschntte und hrer Irrtümer - Grundlagen, 29th Stahlbausemnar, Band 14, ISSN: [12]Rubn, H., Amnbagha,., 27. Wölbrafttorson be veränderlchen, offenem Querschntt - hat de Begezugstabanaloge noch Gültget?, Stahlbau, 76, p [13]Rubn, H., Amnbagha,., Weer, H., 26. IQ-1. he cvl engneerng structures program, U Venna, Bautabellen für Ingeneure, Werner Verlag, 17. Auflage. [14]Rubn, H., 27. Baustat 2, anusrpt der Vorlesung. Insttute of Structural Analyss, Venna Unversty of echnology. [15]urn, J., Kuts, V., 28. An Effectve Fnte Element for orson of Constant Cross-Sectons Includng Warpng wth Secondary orson oment Deformaton Effect, Engneerng Structures, ISSN , Vol. 3, Iss. 1, [16]oos, V.G., Sapountzas, E.J., 211. Secondary torsonal moment deformaton effect by BE, Internatonal Journal of echancal Scence. IJS- 1323R1. Accepted for publcaton. [17]urín, J., Sedlár,., Králov, V., Goga, V., Kalaš, A., Amnbagha,., 212. Numercal Analyss and easurement of Non-unform orson. Proceedngs of the Eleventh Internatonal Conference on Computatonal Structures echnology. Accepted for publcaton.