MODELLING OF ELASTO-STATICS OF POWER LINES BY NEW COMPOSITE BEAM FINITE ELEMENT Bratislava

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1 ODING OF ASTO-STATICS OF POW INS BY NW COPOSIT BA FINIT NT urín Justín 1 rabovský Jura 1 Gogola oman 1 utš Vladmír 1 Paulech Jura 1 1 Insttute of Automotve echatroncs FI STU n Bratslava Ilkovčova Bratslava -mal: ustn.murn@stuba.sk eceved 7 a 215; accepted 15 a Introducton The power lne s from mechancal pont of vew a 3D sstem. It can be loaded n longtudnal horontal and vertcal drecton. so the torsonal loadng s possble as well. But n the techncal calculatons t s mostl smplfed to the one dmensonal sstem. In the lterature e. g [1] an analtcal method s presented for the power lne elasto-statc analss n ts vertcal plane. The analtcal methods are not much effectve for the general spatal analss. The more effectve are the numercal methods over all the fnte element method. For the smple elasto-statc analss the geometrcall nonlnear lnk fnte element can be used that s able to anale the tensonal forces and stresses and the elongaton of the lne e.g. [2]. For dnamc analss the beam fnte element s preferable [3]. The heterogeneous cross-secton are of several constructon e.g. as shown n Fg.1 [4]. Because the materal of the power lne s nhomogeneous the homogenaton of materal propertes s needed. Therefore the aal fleural and torsonal stffness must be stated. For the mechancal analss also the sold fnte elements are avalable but modelng of the complcated geometr and heterogenet s a ver demandng procedure. In ths paper the results of elasto-statc analss of the sngle and bundle power lnes are presented. Fg. 1. Constructon of power lne cross-secton [4]. For modelng and smulaton of the problem a new 3D composte beam fnte element s used whch was developed at our nsttute [3]. The second order beam theor has been used for the fnte element stffness matr formulaton. The effectve tensonal fleural and torsonal stffness of chosen power lnes are consdered [3]. The results are calculated evaluated and compared wth the ones obtaned b the standard fnte element software [2]. 2. Composte fnte beam element equatons et us consder a 3D straght fnte beam element (Tmoshenko beam theor and Sant-Venant torson theor) of doubl smmetrc cross-secton Fg. 2. The nodal degrees 375

2 of freedom at node are: the dsplacements u v w n the local as drecton and the cross-sectonal area rotatons -. The degrees of freedom at the node are denoted n a smlar manner. The nternal forces at node are: the aal force N the transversal forces and the bendng moments and and the torson moment. The frst dervatve wth respect to of the relevant varable s denoted wth an apostrophe. Fg. 2: The local nternal varables and loads. Furthermore n n s the aal force dstrbuton q q and q q transversal and lateral force dstrbutons m m m m and m m are the are the dstrbuted moments A s the cross-sectonal area I and I are the second area moments I p I I s the area polar moment. The effectve homogened and longtudnall varng N N N stffness reads: A A s the aal stffness ( s the effectve elastct modulus for aal loadng) I I s the effectve elastct modulus for bendng about as ) I I s the fleural stffness about the as ( sm bendng about as ) GA Gk A G G s the effectve shear modulus and s the fleural stffness about the -as ( s the effectve elastct modulus for s the reduced shear stffness n drecton ( sm k s the average shear correcton factor n sm drecton [5]) GA G k A s the reduced shear stffness n drecton ( s the effectve shear modulus and sm k G G s the average shear correcton factor n drecton [5]) G I T s the effectve torsonal stffness ( G G s the torsonal elastct modulus and I T s the torson constant). Detaled dervaton of the effectve materal propertes s presented n [3] and [4]. stablshng of the local composte beam fnte element equatons s presented n [6]: 376

3 377 w v u w v u Y T Y S N N (1) In (1) the terms contan the lnear and the lneared geometrc non-lnear stffness terms contanng the aal force effect on the fleural beam stffness. Shear correcton s accounted as well. The global stffness matr of the beam structures can be done b classcal methods. stablshng of the local and global stffness matrces as well as the whole soluton procedure were coded b the software ATATICA [7]. 3. Numercal eperments The smmetrc power lne marked as 456 (3+9 steel and alumnum wres) whch s loaded n ts ntal state b the self-weght n -drecton has been consdered. Span of the power lne s = 3 m the mamal deflecton s m ma 3966 and the average nternal aal force s N II = kn. The power lne s subsequentl loaded at ts mdpont b forces 1 F N and 1 F N. The dameter of the alumnum wres s d = 45 mm and the dameter of the steel wres s d = 28 mm. The effectve crosssectons of the power lne parts are: A = 7389 mm 2 A = mm 2 and the effectve cross-sectonal area of the power lne s A = mm 2. Fg. 3: power lne n ntal state consequentl loaded b concentrated forces. ateral propertes of the components are constant and ther values are: alumnum the elastct modulus GPa 7 the Posson s rato 32 the mass denst 3 27 kgm ; steel the elastct modulus GPa 21 the Posson s rato 28 the mass denst kgm. For the elasto-statc analss of the sngle power lnes the followng effectve materal propertes have been used [3]: the effectve elastct modulus for aal loadng - Pa N the fleural stffness - Pa

4 the effectve shear modulus - G G Pa the torsonal elastct modulus - G Pa The followng calculatons were done wth our 3D FG beam fnte element (NF). The same problem has been solved usng a commercal F program ANSYS wth 3 of beam fnte elements (BA188). The global dsplacements v [m] n ( drecton) and w [m] n ( drecton) at dstances of 5 1 and 15 m from the left lne end calculated wth ANSYS as well as wth the NF are presented n Table 1. The average relatve dfference [%] between dsplacements calculated b our NF-method and the ANSYS soluton has been evaluated as well. Tab. 1. Spatal dsplacements at the selected ponts (DSP) of the power lne DSP NF ANSYS [%] v w v w v v The comparson of total dsplacement of the power lne calculated b our new approach and commercal F program ANSYS s shown n Fgure 4. Fg. 4: Total dsplacement of the power lne. The local nternal forces and moments n the power lne can be calculated from the local dsplacements at the fnte elements nodal ponts. Accordng to the gven loads and the low power lne stffness for bendng the relevant nternal force s the aal tensonal force N. Its mamal value s at the left and rght lne end: N ma 5324 kn. The average normal N stress s N/ A and the normal stran / at the -poston. Because of dfferent elastct modulus of the alumnum and the steel a dfferent normal stress n the materal components wll be arse. Poston of the crtcal cross secton (were the mamal normal force arses) s at the left and rght end of the power lne. In our case the mamal 75 Pa and for steel stress for the alumna s ma Pa. ma 378

5 4. Conclusons New composte beam fnte element was used for the elasto-statc analss of the power lne. omogenaton of the heterogeneous materal propertes was made b the reference volume method (V). For comparson of the effectveness and accurac of the new fnte element the same problem was solved b the commercal F software ANSYS. A ver good agreement of both results has been obtaned. The most advantage of the new beam fnte element s that the stffness matr contans the effect of aal force and all the stffness (tensonal fleural and torsonal) are calculated b a consstent manner. Acknowledgement Ths paper has been supported b the Slovak Grant Agences VGA No. 1/453/15 and APVV and VGA No. 1/228/14. Ths contrbuton s the result of the proect mplementaton: Industral esearch Centre for operatng lfetme of selected components of power plants (ITS: ) supported b the esearch & Development Operatonal Programme funded b the DF. eferences: [1] Š. cko et. al.: Vonkaše elektrcké vedena enesans s.r.o. Bratslava ISBN (21). [2] ANSYS Swanson Analss Sstem Inc. 21 Johnson oad ouston PA 15342/13 USA. [3] J. urín J. rabovský. Gogola G. Gálk: odal analss of the power lnes b fnte element methods. Časops Vol. 2. NO 5/S (214). [4] Ocelovo-hlníkové vodče ( lana) pro venkovní elektrcké vedení. Avalable on web: [5] J. urín J. rabovský V. utš J. Paulech: Shear correcton functon dervaton for the FG beams. In: 2nd Internatonal Conference on ult-scale Computatonal ethods for sold and Fluds Saraevo Bosna and ercegovna (215). [6] J. urín. Amnbagha J. rabovský V. utš J. Pulech: 3D beam fnte element for elasto-statc analss of the FG structures. In: 8ICC Opata Croata (215). [7] S. Wolfram ATATICA 5 Wolfram research Inc. (