Geometrically Non-Linear Dynamic Analysis of Thin-Walled Beams

Transcription

1 roceedings o he World Congress on Engineering 9 Vol WCE 9, July -, 9, London, U.K. eomerically Non-Linear Dynamic Analysis o Thin-Walled Beams Kuo Mo Hsiao, Wen Yi Lin, and Ren Haw Chen Absrac A co-roaional inie elemen ormulaion or he geomerically nonlinear dynamic analysis o hin-walled beam wih large roaions bu small srain is presened. The elemen developed here has wo nodes wih seven degrees o reedom per node. The elemen nodes are chosen o be locaed a he cenroid o he end cross secions o he beam elemen and he cenroid ais is chosen o be he reerence ais. The kinemaics o he beam elemen is described in he curren elemen coordinae sysem consruced a he curren coniguraion o he beam elemen. The elemen nodal orces are convenional orces, momens and bimomens. Boh he elemen deormaion nodal orces and ineria nodal orces are sysemaically derived by consisen linearizaion o he ully geomerically non-linear beam heory, he d'alember principle and he virual work principle in he curren elemen coordinaes. An incremenal-ieraive mehod based on he Newmark direc inegraion mehod and he Newon-Raphson mehod is employed here or he soluion o he nonlinear equaions o moion. Numerical eamples are presened o demonsrae he accuracy and eiciency o he proposed mehod. nde Terms Co-roaional ormulaion, Dynamics, eomerical nonlineariy, Thin-walled beam.. NTRODUCTON The nonlinear dynamic behavior o beam srucures, e.g., ramed srucures, leible mechanisms, and robo arms, has been he subec o considerable research. However, he applicaion o co-roaional ormulaion in he nonlinear dynamic analysis o hree-dimensional beams has been raher limied (e.g. [-]). n [] a consisen co-roaional inie elemen ormulaion or he nonlinear dynamic analysis o hree-dimensional elasic Euler beam using consisen linearizaion o he ully geomerically non-linear beam heory was presened. The ormulaion was proven o be very eecive by numerical eamples sudied in []. However, he eec o warping resrain was negleced in []. To he auhors knowledge, he applicaion o co-roaional ormulaion in he geomeric nonlinear dynamic analysis or hin-walled beams wih he consideraion o he warping Manuscrip received February, 9. This work was suppored in par by he Naional cience Council, Taiwan under ran NC96--E9-76. Kuo Mo Hsiao is wih Deparmen o Mechanical Engineering, Naional Chiao Tung Universiy, Ta Hsueh Road, Hsinchu,, Taiwan (phone: ; a: ; kmhsiao@mail.ncu.edu.w). Wen Yi Lin is wih Deparmen o Mechanical Engineering, De Lin nsiue o Technology, Alley 8, Ching Yun Road, Tucheng, Taiwan ( wylin@dli.edu.w). Ren Haw Chen is wih Deparmen o Mechanical Engineering, Naional Chiao Tung Universiy, Ta Hsueh Road, Hsinchu,, Taiwan ( chenrh@mail.ncu.edu.w). rigidiy has no been repored in he lieraure. The obec o his paper is o presen a co-roaional inie elemen ormulaion or he geomeric nonlinear dynamic analysis o hin-walled beams wih open secion. n order o capure correcly all coupling among bending, wising, and sreching deormaions o he beam elemens, he ormulaion o beam elemens migh be derived by he ully geomerically non-linear beam heory. The eac epressions or he elemen nodal orces, which are required in a oal Lagrangian ormulaion or large displacemen/small srain problems, are highly nonlinear uncions o elemen nodal parameers. However, he dominan acors in he geomerical nonlineariies o beam srucures are aribuable o inie roaions, he srains remaining small. For a beam srucure discreized by inie elemens, his implies ha he moion o he individual elemens o a large een will consis o rigid body moion. he rigid body moion par is eliminaed rom he oal displacemens and he elemen size is properly chosen, he deormaional par o he moion is always small relaive o he local elemen aes. Thus in conuncion wih he co-roaional ormulaion, he higher order erms o nodal parameers in he elemen nodal orces may be negleced by consisen linearizaion. The elemen deormaion and ineria nodal orces are sysemaically derived by using he d'alember principle and he virual work principle. An incremenal-ieraive mehod based on he Newmark direc inegraion mehod and he Newon-Raphson mehod is employed here or he soluion o he nonlinear equaions o moion. Numerical eamples are presened and compared wih he resuls repored in he lieraure o demonsrae he accuracy and eiciency o he proposed mehod.. FNTE ELEMENT FORMULATON The kinemaics o he beam elemen presened in [4] and he co-roaional inie elemen ormulaion proposed in [] are employed here. n he ollowing only a brie descripion o he beam elemen is given. A. Basic Assumpions The ollowing assumpions are made in derivaion o he beam elemen behavior: () The beam is prismaic and slender, and he Euler-Bernoulli hypohesis is valid. () The cross secion o he beam is doubly symmeric. () The uni eension o he cenroid ais o he beam elemen is uniorm. (4) The cross secion o he beam elemen does no deorm in is own plane and srains wihin his cross secion can be negleced. BN: WCE 9

2 roceedings o he World Congress on Engineering 9 Vol WCE 9, July -, 9, London, U.K. B. Coordinae ysems n his paper, a co-roaional ormulaion is adoped. n order o describe he sysem, we deine hree ses o righ handed recangular Caresian coordinae sysems:. A ied global se o coordinaes, i (i =,, ) (see Fig. ); he nodal coordinaes, displacemens, roaions, velociies, and acceleraions, and he equaions o moions o he sysem are deined in his coordinaes.. Elemen cross secion coordinaes, i (i =,, ) (see Fig. ); a se o elemen cross secion coordinaes is associaed wih each cross secion o he beam elemen. The origin o his coordinae sysem is rigidly ied o he cenroid o he cross secion. The aes are chosen o coincide wih he normal o he unwrapped cross secion and he and aes are chosen o be he principal direcions o he cross secion.. Elemen coordinaes, i (i =,, ) (see Fig. ); a se o elemen coordinaes is associaed wih each elemen, which is consruced a he curren coniguraion o he beam elemen. The origin o his coordinae sysem is locaed a node, and he ais is chosen o pass hrough wo end nodes o he elemen; he and aes are deermined by he mehod proposed in [5]. Noe ha his coordinae sysem is us a local coordinae sysem no a moving coordinae sysem. The deormaions, deormaion nodal orces, ineria nodal orces, siness mari and mass mari o he elemens are deined in erms o hese coordinaes. n his paper he elemen deormaions are deermined by he roaion o elemen cross secion coordinae sysems relaive o his coordinae sysem. C. Kinemaics o Beam Elemen n his sudy only he doubly symmeric cross secion is considered. Le Q (Fig. ) be an arbirary poin in he beam elemen, and be he poin corresponding o Q on he cenroid ais. The posiion vecor o poin Q in he undeormed and deormed coniguraions may be epressed as [4]: s u p v w Fig.. Coordinae sysems (,,) l Q z y r e + ye + ze = () p ( e + v( e + w( e +, e + ye ze r = θ ω + () where p (, v( and w ( are he, and coordinaes o poin, respecively, in he deormed coniguraion, θ, = θ, ( is he wis rae o he deormed cenroid ais, ω ( y, z) is he ain Venan warping uncion or a prismaic beam o he same cross secion, and e i and e i (i =,, ) denoe he uni vecors associaed wih he i and i aes, respecively. Noe ha e i and e i are coinciden in he undeormed sae. The relaionship beween e i and e i is given in [4] and no repeaed here. Here, he laeral delecions o he cenroid ais, v( and w (, and he roaion abou he cenroid ais, θ,, are assumed o be he Hermiian polynomials o. The relaionship among p (, v (, and w (, and may be given as [4] c,, ] ( = u + [( + ε ) v w d () p where u is he displacemen o node in he direcion. Noe ha due o he deiniion o he elemen coordinae sysem, he value o u is equal o zero. However, he variaion and ime derivaives o u are no zero. Making use o he assumpion o uniorm uni eension, ε c and he aial displacemens o he cenroid ais may be calculaed using () and he curren chord lengh o he beam elemen. D. Elemen Nodal Force Vecor, iness Mari and Mass Mari The elemen proposed here has wo nodes wih seven degrees o reedom per node. The nodal parameers are chosen o be u i ( u = u, u = v, u = w ), he i (i =,, ) componens o he ranslaion vecors u a node ( =, ), φ i, he i (i =,, ) componens o he roaion vecors φ a node ( =, ), and β, he wis rae o he cenroid ais a node. Here, he values o φ are rese o zero a curren coniguraion. Thus, δφ i, he variaion o φ i, represens ininiesimal roaions abou he i aes [5], and he generalized nodal orces corresponding o δφ i are m i, he convenional momens abou he i aes. The generalized nodal orces corresponding o δ ui, he variaions o u i, are i, he orces in he i direcions. The generalized nodal orces corresponding o δβ, he variaions o β, are bimomen B. The elemen nodal orce vecor is obained rom he virual work principle and he d Alember principle in he curren BN: WCE 9

3 roceedings o he World Congress on Engineering 9 Vol WCE 9, July -, 9, London, U.K. elemen coordinaes. The virual work principle requires ha δwe = δq = δwin = ( σ δε + σ δε + σ δε + ρδr & r ) dv = δq V D = + =, m,, m, } { B D θ θ θ θ θ = θ + θ = {, m,, m, B * * q { δu, δθ, δu, δθ, δβ} q = { δu, δφ, δu, δφ, δβ} } θθ δ θ = (6) δ ε = (,, ) r r, ε = r, r, y.. ε =,, z r r (7) where δ u = δu, δv, δw }, δ φ = δφ, δφ, δφ }, * { { δθ = δθ, δw, δv }, =,, }, θ { { θ θ θ { m, m, m θ { θ θ θ {,, m = m, m, m }, = }, m = } ( =, ), δ β = { δβ,δβ } and D B = { B, B}. and are elemen deormaion nodal orce vecor and ineria nodal orce vecor, respecively. V is he volume o he undeormed beam elemen, δε i (i =,, ) are he variaion o ε i in (7) corresponding o δq θ. Noe ha because δε i are uncion o δq θ, δ Win may be epressed by δq D θ θ. θ and θ are generalized deormaion nodal orce vecor and ineria nodal orce vecor corresponding o δq θ. σ i (i =,, ) are he second iola-kirchho sress. For linear elasic maerial, σ = Eε, σ = ε, and σ = ε, where E is Young s modulus and is he shear modulus. ρ is he densiy, δ r and & r& are he variaion and he second ime derivaive o r in (), respecively. Noe ha because he elemen coordinae sysem is us a local coordinae sysem no a moving coordinae sysem, & r& is he absolue acceleraion. The higher order erms o nodal parameers in he elemen nodal orces are negleced by consisen second order linearizaion in his sudy. The relaion beween δq and δq θ, and he relaion beween and θ may be epressed as [4] δqθ = Tθφδq, = T θφ θ (8) where θ may be calculaed using (-7). The elemen siness mari and mass mari may be epressed as k =, q (4) (5) m = (9) && q R where F is he unbalanced orce among he ineria nodal orce F, deormaion nodal orce F D, and he eernal D nodal orce. F and F are assembled rom he elemen nodal orce vecors, which are calculaed using (4) and (8) irs in he curren elemen coordinaes and hen ransormed rom elemen coordinae sysem o global coordinae sysem beore assemblage using sandard procedure.. NUMERCAL TUDE An incremenal ieraive mehod based on he Newmark direc inegraion mehod and he Newon-Raphson mehod [] is employed here. The irs eample considered is a simply suppored beam subeced o uniorm load as shown in Fig.. This eample was analyzed by [6]. Tweny elemens are used or discreizaion. A ime sep size o Δ =.sec is used. The ime hisories o displacemens a poin C are shown in Fig. ogeher wih he soluion given in [6]. As can be seen, he discrepancy beween hese wo soluions is disinc. The discrepancy may be aribued a leas in par o ha he momen o ineria o he beam cross secion is no considered in [6].,W A,V C Cross secion d ( b w L C B L = m =. m w=. m d =.56 m b =. m E = a = 8.77 a m= 88.4 kg/m = m C = kn / m,u E. Equaions o Moion The nonlinear equaions o moion may be epressed by R D F = F + F = (). (sec) Fig.. imply suppored beam subeced o uniorm load. BN: WCE 9

4 roceedings o he World Congress on Engineering 9 Vol WCE 9, July -, 9, London, U.K...5 resen [6] W C U V W φ Displacemens (m) U C Time (sec) ω =.78 (rad/s) ω = ω 4 =.44 Fig.. Time hisories o displacemens a poin C.,W A,V C B,U Modal delecion ω = L/ L/.5in End cross secion 5in Loading poin Time Hisory o Load (. (sec) Fig. 4. imply suppored beam subeced o eccenric aial orce. The second eample considered is a simply suppored W4 4 beam subeced o an eccenric aial sep loading wih magniude = 5 kip as shown in Fig. 4. The ends o he beam are ree o warp and ree o roae abou and aes, bu resrained rom roaion abou ais. The ranslaion is resrained a end poin A, and is ree only in he direcion o ais a poins B. The geomerical and maerial properies are L = in, b = in, =. 5 in, d =. 66 in, w =. 5 in, Young's modulus E = 9 ksi, and he shear modulus = ksi, / 4 ρ =.8 lb s in. The irs aial naural requency corresponding o he π undeormed sae may be given by E ρ = L 6.95 rad / sec.the saic buckling load or his eample given in [4] is cr = 9. kip. Tweny elemens are used or discreizaion. The irs ive naural requencies and vibraion Displacemens (in) Figure 5. Vibraion modes or simply suppored beam subeced o eccenric aial orce ω 5 =.67 /L V C Fig. 6. Time hisory or simply suppored beam subeced o eccenric aial orce. modes corresponding o he saic equilibrium coniguraion a = 5 kip are calculaed and given in Fig. 5. can be seen ha he irs and ourh vibraion modes are dominaed by he laeral vibraion in he direcion, he second and ih vibraion mode is a coupled laeral-orsional vibraion, and he hird vibraion mode is dominaed by he laeral vibraion in he direcion. A ime sep size o U B W C Time (sec) = 5 kip Δ =.sec is used. The ime hisories o laeral displacemens a poin C and aial displacemen a poin B are shown in Fig. 6. can be seen rom Figs. 5 and 6 ha he BN: WCE 9

5 roceedings o he World Congress on Engineering 9 Vol WCE 9, July -, 9, London, U.K. ime hisories given in Fig. 6 are dominaed by he irs aial vibraion mode, and he irs and he hird vibraion modes. V. CONCLUON A consisen co-roaional oal Lagrangian inie elemen ormulaion or he geomerically nonlinear vibraion analysis o doubly symmeric hin-walled beams wih open secion is presened. The nodal coordinaes, displacemens, roaions, velociies, acceleraions, and he equaions o moion o he srucure are deined in a ied global se o coordinaes. The beam elemen has wo nodes wih seven degrees o reedom per node. The elemen nodal orces are convenional orces and momens. The kinemaics o beam elemen is deined in erms o elemen coordinaes which are consruced a he curren coniguraion o he beam elemen. Boh he elemen ineria and deormaion nodal orces are sysemaically derived by using consisen second order linearizaion o he ully geomerically nonlinear beam heory, he d'alember principle and he virual work principle. n conuncion wih he co-roaional ormulaion, he higher order erms o nodal parameers in elemen nodal orces are consisenly negleced. However, in order o include he nonlinear coupling among he bending, wising, and sreching deormaions, erms up o he second order o nodal parameers are reained in elemen deormaion nodal orces. should be noed ha he elemen coordinae sysem is us a local coordinae sysem, which is updaed a each ieraion, no a moving coordinae sysem. Thus, he velociy and acceleraion described in he elemen coordinaes are he absolue velociy and acceleraion. The elemen equaions are consruced irs in he elemen coordinae sysem and hen ransormed o he global coordinae sysem by using sandard procedure. From he numerical eamples sudied, he accuracy and eiciency o he proposed mehod are well demonsraed. is believed ha he consisen co-roaional ormulaion or beam elemen and numerical procedure presened here may represen a valuable engineering ool or he dynamic analysis o hree dimensional hin-walled beam srucures. REFERENCE [] T. Belyschko and B. J. Hsieh, Nonlinear ransien inie elemen analysis wih conveced coordinaes, nerna. J. Numer. Mehs. Engrg, Vol. 7, 97, pp [] T. Belyschko and L. chwer, Large displacemen, ransien analysis o space rames, nerna. J. Numer. Mehs. Engrg., Vol., 977, pp [] K. M. Hsiao, J. Y. Lin and W. Y. Lin, A Consisen Co-Roaional Finie Elemen Formulaion or eomerically Nonlinear Dynamic Analysis o -D Beams, Compu. Mehods Appl. Mech. Engrg., Vol. 69, 999, pp. -8. [4] W. Y. Lin and K. M. Hsiao, Co-Roaional Formulaion or eomeric Nonlinear analysis o Doubly ymmeric Thin-Walled Beams, Compu. Mehods Appl. Mech. Engrg., Vol. 9,, pp [5] K. M. Hsiao, A Co-roaional Toal Lagrangian Formulaion or Three Dimensional Beam Elemen, AAA Journal, Vol., 99, pp [6] J. T. Kasikadelies and. C. Tsiaas, Nonlinear Dynamic Analysis o Beams wih Variable iness, J. ound Vib., Vol. 7, 4, pp BN: WCE 9