6 th INTERNATIONAL CONFERENCE

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1 6 th INTERNATIONAL CONFERENCE Contempoay achievements in civil engineeing 20. Apil Subotica, SERBIA DUALITY OF BUCKLING AND VIBRATION PHENOMENA USING FINITE STRIP METHOD Dagan D. Milašinović 1 Peta Maić 2 Žako Živanov 3 Mioslav Hajduković 4 UDK: :519.6 DOI: /konfeencijaGFS Summay: It is well ecognized that when using exact theoy, as opposed to the moe usual appoximate finite element method, thee exists a duality between buckling and natual vibation of an elastic stuctue, even though the fome is a static poblem wheeas the latte is a dynamic one. The solution pocedues fo the both poblems ae analogous because both can be essentially epesented by a tanscendental eigenvalue poblem, the eigenvalues being natual fequencies in vibation poblems and load factos in buckling poblems. Thus the elastic buckling of a stuctue is often egaded as the degeneated case of its natual vibation at zeo fequency. In this pape duality is poved fo the both elastic and viscoelastic (o damage) stuctue using semi-analytical finite stip method (FSM) and heological-dynamical analogy (RDA). The govening dynamic RDA modulus has been deived in [1]. This pape pesents an investigation of composite thin-walled wide-flange columns. Numeical examples showing the theoetical consideations ae pesented and agee with the expeimental data. Two efeence Open Souce Softwae implementations ae povided: fo appoximating the natual fequency fom stess via physical duality (and vice vesa) and fo the visual analysis of the physical duality appoximation effectiveness. Keywods: Duality, FSM, RDA, dynamic RDA modulus, Composite thin-walled wideflange column, numeical analysis, accuacy of numeical evaluation 1. INTRODUCTION The conventional FSM is based on the eigen-functions fo the vibating beam, and poved to be efficient tool fo analyzing a geat deal of stuctues fo which both geomety and mateial popeties can be consideed as constants along a main diection, 1 Pof. d Dagan D. Milašinović, dipl. inž. gađ., Gađevinski fakultet Subotica, Kozaačka 2a, tel: 024/ , ddmil@gf.uns.ac.s 2 Doc. d Peta Maić, dipl. el. inž., Fakultet tehničkih nauka u Novom Sadu, Tg Dositeja Obadovića 6, Novi Sad, tel: 021/ , petamaic@uns.ac.s 3 Doc. d Žako Živanov, dipl. el. inž., Fakultet tehničkih nauka u Novom Sadu, Tg Dositeja Obadovića 6, Novi Sad, tel: 021/ , zzako@uns.ac.s 4 Pof. d Mioslav Hajduković, dipl. el. inž., Fakultet tehničkih nauka u Novom Sadu, Tg Dositeja Obadovića 6, Novi Sad, tel: 021/ , hajduk@uns.ac.s CONFERENCE PROCEEDINGS INTERNATIONAL CONFERENCE (2018) 133

2 6. МЕЂУНАРОДНА КОНФЕРЕНЦИЈА Савремена достигнућа у грађевинарству 20. април Суботица, СРБИЈА staight o cuved, while only the loading distibution may vay. As fa as linea analysis is concened, it takes advantage of the othogonality popeties of eigen-functions in the stiffness matices fomulation. The analysis of a vaiety of elastic stuctues subjected to static loads lead to the same matix equation of Kq = Q (1) in which K is the stiffness matix of stuctue, q the vecto containing all nodal displacement paametes, and Q the vecto containing all nodal foces. If the stuctue is moving then it is also possible to educe the dynamic poblem to a static one by applying D'Alembet's pinciple of dynamic equilibium in which an inetial foce equal to the poduct of the masss and the acceleation is assumed to act on the stuctue in the diection of negative acceleation. Thus at any instant ot time the equibium equation fo a stuctue in which both damping and extenal excitation foces ae assumed to be non-existent is Kq( t) = ( M + m) q ( t) (2) 2 2 whee q(t) is now a function of time and.. epents t. In the above equation M is a diagonal matix of concentated o line masses at the nodal lines, and is simply equal to zeo when no such concentated o line masses ae acting on the stuctue, and m is an oveall mass matix of the stuctue assembled fom individual stip consistent mass matices m s. The assembly pocess fo mass matices and fo stiffness matices ae identical [2]. Fo fee vibation, the system is vibating in a nomal mode, and it is possible to make the substitutions 2 q( t) = q sinωt, q ( t) = ω qsinωt (3) into (2) to obtain 2 ( K ω m) q = 0 (4) whee ω is the natual fequencies of the modes and the common tem sinωt has been concelled out. It is possible to tansfom (4) into 1 1 ( K m) q = q (5) 2 ω which becomes, theefoe, a tanscendental eigenvalue poblem with the fom Ax = λx. (6) Compaatively little wok has been done on the application of the finite stip to stability poblems, although vibation and stability shae many simila featues and both equie the detemination of eigenvalues and eigenvectos. Two types of column failue (buckling) ae well known fo wide-flange sections: local buckling and oveal (Eule) column buckling. In local modes, only non-linea tems such as squae deivatives of tansvese displacement w need to be included (von Kaman appoach). It is obvious that if the total potential enegy of the stip, i.e. the sum of stain enegy due to bending, potential enegy due to nodal line foces, and the additional potential enegy due to the initial stess, is now minimized with espect to the nodal displacement paametes, the following elationship would be obtained: 134 ЗБОРНИК РАДОВА МЕЂУНАРОДНЕ КОНФЕРЕНЦИЈЕ (2018)

3 6 th INTERNATIONAL CONFERENCE Contempoay achievements in civil engineeing 20. Apil Subotica, SERBIA Sq + SGq = Q (7) in which S G is efeed to as the geometic stiffness matix of stip o initial stess matix and takes up the same sign as the stesses. Upon assembly of the contibutions fom all the stips an oveall set of equilibium equations is established, Kq + KGq = Q (8) Fo linea stability, the nodal foces ae zeo and it is theefoe possible to aive at eigenvalue equations simila to the ones given in (4), ( K+ λkg ) q = 0 (9) with λ being scaling facto elated to the citical load λmin σ = σc = (10) 2 t whee t is the web (flange) thickness. 2. DUALITY OF BUCKLING AND VIBRATION The popagation of mechanical waves (o stess waves) with tansition fom the shottime modulus of elasticity (E D) to the long-time one (E H) epesents a physical basis fo the analogy between two diffeent physical phenomena, the heological and the dynamical. Geneally speaking, the RDA is deived in ode to solve dynamic poblems, but it can be used in the analysis of quasi-static loading consideing the coesponding limit values of the deived analytical expessions. The govening dynamic RDA modulus has been deived in [1] ϕ+ δ E ER = E H ω 2 2 H, ER = ER =, δ = σ. (11) ( 1+ ϕ) + δ limδ ϕ ω ϕ is the stuctual-mateial ceep coefficient and ω σ is the fequency of excitation. Elastic buckling analysis by the FSM was pioneeed by Yoshida [3] and Pzemieniecki [4]. Plank and Wittick [5] genealized the appoach to include shea loading, by using complex hamonic functions. They pointed out that duality exists between the feevibation behavio and buckling unde unifom end compession. 2 σ im m π ωim = (12) ρ a whee i = numbe of degees of feedom (DOF), ρ = mass density, m = numbe of seies tem and a = length of the column. In this pape duality is poved fo the both elastic and viscoelastic stuctue using the FSM and RDA. 3. SOFTWARE IMPLEMENTATION The fsm_eigenvalue poject [6] povides a efeence Open Souce Softwae implementation fo paametic modeling of buckling and fee vibation in pismatic CONFERENCE PROCEEDINGS INTERNATIONAL CONFERENCE (2018) 135

4 6. МЕЂУНАРОДНА КОНФЕРЕНЦИЈА Савремена достигнућа у грађевинарству 20. април Суботица, СРБИЈА stuctues, pefomed by solving the eigenvalue poblem in the hamonic coupled finite stip method (HCFSM). The fsm_eigenvalue poject takes the semi-analytical finite stip model data file (geomety, mateials, loading) as its input and then pefoms its computations as a paamete sweep ove 4 sepaate dimensions: D1: Pefoms the buckling and fee vibation analysis D2: Iteates ove all stip lengths, in the ange specified by the input D3: Iteates ove all stip thicknesses, in the ange specified by the input D4: Iteates ove all modes, in the ange specified by the input. The HDF5 (Hieachical Data Fomat 5) [7] fomat has been selected fo stoing the computed esults, because of its innate ability to efficiently stoe and oganize lage amounts of numeical data. HDF5 is an extensible and open standad, compised of platfom independent technologies which ae available unde Open Souce licenses. HDF5 fomat has been designed with the objective of ceating data stoages that ae selfdesciptive, flexible, and have extemely fast and efficient access pattens to the stoed data. Fo each combination of mode, stip length and thickness the fsm_eigenvalue poject outputs a lage amount of numeical data: citical stess, natual fequency, thei appoximations via physical duality, appoximation eos, minimal citical stess vecto, minimal natual fequencies vecto, etc. To appoximate the natual fequency fom stess via physical duality (and vice vesa) a sepaate Open Souce pogamming libay has been ceated, named physical_dualism [8]. When lage finite stip models ae analyzed, such as the example data files povided by the fsm_eigenvalue poject that have 51 modes, 7800 stip lengths ( mm with a 0.5 mm iteation step) and 140 stip thicknesses (2-9 mm with a 0.05 mm iteation step), computed esults can amount to ove 50 GB of compessed data pe HDF5 file. Such data can t be inspected manually, so it was necessay to build a new Open Souce poject fo automated visualization and modal analysis of the paametic model of buckling and fee vibation in pismatic shell stuctues, named fsm_modal_analysis [9]. This softwae has been used to geneated Figs. 1, 2, 3 and 4 within this pape. 4. APPLICATIONS The simply suppoted ideally staight thin-walled wide-flange H-section column of length a = 2310 mm consists of a web and two flanges of side b = 152 mm [10]. The thickness t is 6.35 mm. The column is compessed axially. The bending stiffness EI=89.6 knm 2 about the weak axis of the analysed column was computed fom the infomation povided by the manufactue fo each section following the methodology developed by Babeo and Tomblin [11]. This infomation includes the type of fibes and matix mateial, the local oientation of the fibes and the fibe content in the cosssection. Tab. 2 shows the flange and web bending stiffness components fo 152 mm x 152 mm x 6.35 mm pultuded WF H-section column. 136 ЗБОРНИК РАДОВА МЕЂУНАРОДНЕ КОНФЕРЕНЦИЈЕ (2018)

5 6 th INTERNATIONAL CONFERENCE Contempoay achievements in civil engineeing 20. Apil Subotica, SERBIA Table 1. Weak axis bending stiffness components of flanges and web: test data [10] Stiffness component (knm) Flanges Web D 11 D 22 D 12 D The buckling behaviou of ideally staight column to axial loading and small tansvese concentated load has aleady been investigated by the HCFSM [12]. Milašinović [2] developed a hamonic coupled FSM by including the Geen-Lagange stain tems in the fomulations, thus bending and membane ae coupled in the geometic stiffness to give moe accuate buckling behaviou pediction, especially in lage-deflection and elastic post-buckling analysis. The following appoximate functions ae used fo the simply flat shell stip. 0 0 u u u u u u u u u um u um um [ 1 ] um m= 1 m= 1 u = Aq = Y Nq = Y xb xb q, a a v = Aq = Y Nq = Y xb xb q, v v v v v v v u u um u um um [ 1 ] um m= 1mπ m= 1mπ [ ] w= Aq = Y Nq = Y N N N N q, w w wm w wm wm wm m= 1 m= ( ) = ( ) + ( ), N ( x) = b x b ( xb) + ( xb) N1 x 1 3 xb 2 xb 2 3, ( ) = ( ) ( ), N ( x) = b ( xb) + ( xb) N3 x 3 xb 2 xb v um u Yum ( y) sin( mπ y a) Ywm ( y) u ( ) ( π ) cos( π ) um = =, 4 2 3, Y y = dy dy = m a m y a, m = 1,2,3,... (13) The mateial popeties in Table 3 fo the implementation of HCFSM ae obtained as detailed in [12], and thee ae used in this pape. CONFERENCE PROCEEDINGS INTERNATIONAL CONFERENCE (2018) 137

6 6. МЕЂУНАРОДНА КОНФЕРЕНЦИЈА Савремена достигнућа у грађевинарству 20. април Суботица, СРБИЈА Mateial Popeties Table 2. Mateial popeties of flange and web Elastic Viscoelastic Flanges Web Flanges Web E x (N/mm 2 ) E y µ x µ y G Figue 1. Elastic natual fequencies 138 ЗБОРНИК РАДОВА МЕЂУНАРОДНЕ КОНФЕРЕНЦИЈЕ (2018)

7 6 th INTERNATIONAL CONFERENCE Contempoay achievements in civil engineeing 20. Apil Subotica, SERBIA Figue 2. Elastic citical buckling stesses Figue 3. Viscoelastic natual fequencies CONFERENCE PROCEEDINGS INTERNATIONAL CONFERENCE (2018) 139

8 6. МЕЂУНАРОДНА КОНФЕРЕНЦИЈА Савремена достигнућа у грађевинарству 20. април Суботица, СРБИЈА Figue 4. Viscoelastic citical buckling stesses Fig. 1 shows the esults of natual fequencies fo the elastic solutions of the poblem. The column length has been successively inceased fom 2000 to 4000 mm, with a step of 0.5 mm. Thee is a noticeable lag between the viscoelastic, Fig. 3 and the elastic natual fequencies. Fig. 2 pesents the elastic buckling cuve fo all lengths of columns fom 2002 to 4000 mm, with a step of 0.5 mm. Viscoelastic buckling stess, Fig. 4 lags behind the elastic buckling stess acoss all modes, which is a consequence of the visoelastic behaviou of mateials. The viscoelastic behaviou is chaacteized by the delay time T D [1]. As the length of the column is lage, the obseved lag inceases. At the column length of 2310 mm, the computed elastic citical stess of the fist global mode coespond to test esults [10]. Also, the computed viscoelastic stess at 2310 mm of the fist global mode has been peviously confimed in [12]. 5. CONCLUSIONS Numeical examples wee computed by application of extensive hybid paallelization. Results fom the numeical studies fo all lengths of columns fom 2000 to 4000 mm, show that columns have all elastic and inelastic chaacteistics in the same (fist) mode. Because of that the mode inteaction is not occued. Numeical examples showing the theoetical consideations ae pesented and agee with the expeimental data at the column length of 2310 mm. Two efeence Open Souce Softwae implementations ae povided: fo appoximating the natual fequency fom stess via physical duality (and 140 ЗБОРНИК РАДОВА МЕЂУНАРОДНЕ КОНФЕРЕНЦИЈЕ (2018)

9 6 th INTERNATIONAL CONFERENCE Contempoay achievements in civil engineeing 20. Apil Subotica, SERBIA vice vesa). In this pape duality is poved fo the both elastic and fist time fo viscoelastic columns using the RDA. REFERENCES [1] Milašinović DD. Rheological-Dynamical Theoy of Visco-Elasto-Plasticity and Fatigue: Pat 2. Multidiscipline Modelling in Mateials and Stuctues, 2(2), 2006, pp [2] Milašinović DD. The Finite Stip Method in Computational Mechanics. Faculties of Civil Engineeing: Univesity of Novi Sad, Technical Univesity of Budapest and Univesity of Belgade: Subotica, Budapest, Belgade, [3] Yoshida K. Buckling analysis of plate stuctues by stip elements, Poc. Japan Soc. Nav. Achit., 130, 1971, pp [4] Pzemieniecki JS. Matix analysis of local instability in plates, stiffened panels and columns. Int. J. Nume. Meth. Eng., 5, 1972, pp [5] Plank RJ. and Wittick WH. Buckling unde combined loading of thin flat-walled stuctues by a complex finite stip method. Int. J. Nume. Meth. Eng., 8, 1974, pp [6] P. Maić, D.D. Milašinović, fsm_eigenvalue public code epositoy, 2017, URL [7] The HDF Goup, HDF5, 2015, URL [8] P. Maić, D.D. Milašinović, physical_dualism public code epositoy, 2017, URL [9] P. Maić, fsm_modal_analysis public code epositoy, 2018, URL [10] Babeo EJ, Dede EK and Jones D. Expeimental veification of buckling-mode inteaction in intemediate-length composite columns. Intenational Jounal of Solids and Stuctues, 37(29), 2000, pp [11] Babeo E and J. Tomblin. A Phenomenological Design Equation fo FRP Columns with Inteaction between Local and Global Buckling. Thin-Walled Stuctues, 18, 1994, pp [12] Milašinović DD. Hamonic coupled finite stip method applied on buckling-mode inteaction analysis of composite thin-walled wide-flange columns. Thin-Walled Stuctues, 50(1), 2012, pp ДУАЛИТЕТ ПОЈАВА ИЗВИЈАЊА И ВИБРАЦИЈА КОРИШТЕЊЕМ МЕТОДА КОНАЧНИХ ТРAКА Резиме: Добро је познато, када се користи аналитичка теорија за разлику од чешће кориштеног апроксимативног метода коначних елемената, да постоји дуалитет између извијања и сопствених вибрација еластичне конструкције, иако је први статички проблем, док је други динамички. Процедуре решавања за оба проблема су аналогне јер се оба могу представити трансценденталним CONFERENCE PROCEEDINGS INTERNATIONAL CONFERENCE (2018) 141

10 6. МЕЂУНАРОДНА КОНФЕРЕНЦИЈА Савремена достигнућа у грађевинарству 20. април Суботица, СРБИЈА проблемом сопствених вредности, сопствене вредности су природне фреквенције у проблемима вибрација, а фактори оптерећења у проблемима извијања. Због овога се еластично извијање конструкције често посматра као дегенерисани случај свог природног вибрационог стања са нутом сопственом фреквенцијом. У овом раду дуалитет је доказан за обе, еластичну и вискоеластичну (оштећену) конструкцију кориштењем пулуаналитичког метода коначних трака (МКТ) и реолошко динамичке аналогије (РДА). Владајући динамички модул је изведен у [1]. Овај рад представља истраживање композитних танкозидних стубова са широким фланшама. Нумерички примери приказују теоријска разматрања, а у сагласности су са експерименталним подацима. Два референтна 'Open Souce Softwae' су обезбеђена за прорачун сопствених фреквенција преко критичних напона према физичком дуалитету (и обрнуто) као и за визуелну анализу ефикасности апроксимације физичким дуалитетом. Кључне речи: Дуалитет, МКТ, РДА, динамичи РДА модул, Композитни танкозидни стуб са широким фланшама, нумеричка анализа, тачност нумеричке евалуације 142 ЗБОРНИК РАДОВА МЕЂУНАРОДНЕ КОНФЕРЕНЦИЈЕ (2018)