A New Approach for Deriving the Instability Potential for Plates Based on Rigid Body and Force Equilibrium Considerations

Transcription

1 Avalable onlne at Poceda Engneeng 4 (20) 4 22 The Twelfth East Asa-Pacfc Confeence on Stuctual Engneeng and Constucton A New Appoach fo Devng the Instablty Potental fo Plates Based on Rgd Body and Foce Equlbum Consdeatons Y. B. Yang a, S. R. Kuo b a Dept. of Constucton Eng., Natonal Yunln Unv. of Scence and Technology, Yunln, Tawan b Dept. of Habo and Rve Engneeng, Natonal Tawan Ocean Unv., Keelung, Tawan Abstact A new method s poposed fo devng the nstablty potental of plates based on the gd body and foce equlbum consdeatons usng the updated Lagangan fomulaton. By statng fom the gd body ule, the vtual nstablty potental s deved fo an ntally stessed thn plate unde eal gd dsplacements. Next, by utlzng the equlbum equatons fo the bounday foces actng on the plate at the C and C 2 states, anothe vtual nstablty potental s deved fo the plate unde vtual gd dsplacements. Fnally, by compang the two sets of vtual nstablty potental fo eal and vtual gd dsplacements, the total nstablty potental s deved. The pesent appoach s advantageous n that only smple ntegatons and analogcal compason of elated vtual wok tems ae equed, theeby avodng the physcally unappaent, complcated mathematcal devatons nvolved n conventonal pocedues. Examples ae pepaed to demonstate the applcablty of the theoy to the postbucklng analyss of vaous stuctues. 20 Publshed by Elseve Ltd. Open access unde CC BY-NC-ND lcense. Keywods: eometc nonlneaty; Instablty potental; Plate; Postbucklng; Rgd body ule. Intoducton In an ncemental nonlnea analyss of sold stuctues, thee typcal can be dentfed,.e., the ntal state C 0, last state C, and cuent state C 2. Assumng that the behavo of a stuctue, ncludng defomatons and loadngs, fom C 0 to C s known, one s nteested n ts behavo dung the ncemental step fom C to C 2 unde a load ncement (Yang and Kuo 994). In ths pape, the updated Lagangan (UL) fomulaton s adopted to deve the ncemental nonlnea theoy fo the plate, by whch C s selected as the confguaton of efeence. One advantage wth the vtual wok methods fo devng the stablty theoy of a sold element s that the pocedue can be caed out n a systematc and ountne manne. Namely, by substtutng the stan Publshed by Elseve Ltd. Open access unde CC BY-NC-ND lcense. do:0.06/j.poeng

2 Y.B. Yang and S.R. Kuo / Poceda Engneeng 4 (20) dsplacement elatons, stess-stan elatons, and foce-stess elatons fo the element nto the vtual wok equaton, the govenng dffeental equatons, along wth the natual and geometc bounday condtons, can be deved by utlzng the abtay natue of vaatons. Howeve, fo the plate element of concen hee, the above pocedue s not foolpoof. Thee ae seveal ssues that must be adequately addessed befoe a successful esult can be obtaned. Fst, when dealng wth the vtual wok elated to the sx nonlnea vtual stans, thee always appea some tems that ae not physcally meanngful, as a esult of the adopton of dffeent hypotheses fo plane sectons afte defomaton. The othe ssue s elated to the devaton of equvalent shea foces on the boundaes of the plate at C 2, whch may dffe fo dffeent hypotheses of plate sectons. It s ealzed that only when the bounday shea foces ae coectly specfed at C 2 can Kchhoff s foces be accuately computed fo the plate element, and that the latte s essental to devaton of the vtual woks done by suface tactons at C 2 (Kuo et al. 2009). A fequently encounteed stuaton s that even f a nonlnea plate theoy was delbeately deved, thee s no guaantee that the theoy passes the gd body test (Yang and Chou 987; Yang and Kuo 994), a fundamental popety to be obeyed by all ntally stessed elements. A new, smple method, based on the fundamental pncples of gd body ule and foce equlbum, s pesented heen fo devng the vtual nstablty potental of a thn plate, unde the condtons of ethe gven eal o vtual gd dsplacements. By an analogcal compason of the two sets of vtual nstablty potental deved, the total nstablty potental can be deved. 2. Method of fomulaton By the UL fomulaton, the vtual wok equaton fo the plate can be wtten wth efeence to C as follows (Yang and Kuo 994): V o C klj 2 e e dv dv t u ds t u ds () kl j j V j S S UE U W2 W (2) whee U E = vtual lnea stan enegy, U = vtual nstablty potental of the plate, W2 = 2 vtual wok done by the suface tactons t actng at C 2, and W = vtual wok done by the suface tactons t actng at C. Othe symbols n Eq. () ae defned as follows: = vaaton; C klj = mateal coeffcents, e kl = lnea stan components, j = Cauchy stesses; j = nonlnea stan components; u = dsplacement ncements; V = volume; and S = suface aea of the element. The vtual lnea stan enegy U E s a functon of the dsplacement ncements u and vtual dsplacements u of the plate element,.e., UE E( u, u) ; u uxex uyey uzez (3) Snce the ntenal foces exstng at C ae known, one can expess the vtual nstablty potental U and extenal vtual wok W at C as follows: U u, u W u (4) (5)

3 6 Y.B. Yang and S.R. Kuo / Poceda Engneeng 4 (20) 4 22 Snce the extenal foce ncements at C 2 ae elated to the dsplacement ncements u, one can expess the extenal vtual wok W 2 at C 2 as a functon of the dsplacement ncements u and vtual dsplacements u as W2 2 u, u In ths pape, we shall use the ncemental vtual wok equaton n Eq. (2), togethe wth the gd body ule and equlbum condtons, to deve the vtual nstablty potental u, u gven eal gd dsplacements and u, u of the plate fo fo gven vtual gd dsplacements. By compang these two expessons fo the plate unde the same actons, one can deve the total vtual nstablty potental, u u of the plate. In the followng, one wll llustate how the followng thee tems can be deved: the vtual nstablty potental u, u of the plate fo gven eal gd dsplacements, the one u, u vtual gd dsplacements, and the total nstablty potental, (6) fo gven u u. Due to estcton of pape length, all the statc and knematc descptons of the plate (Fgue ) n the pe-bucklng and buckng stages wll be omtted. The eades should be efeed to Kuo and Yang (200) fo detaled pesentaton of the theoy. Fgue : Thn plate at C and elated coodnates 3. Devaton of u, u based on gd body ule The pncple of objectvty mples that usng dffeent efeences to descbe the behavo of a sold unde the same actons, the same esult should be obtaned. The gd body ule poposed by Yang and Chou (987) fo nonlnea fnte elements can be egaded as an example of objectvty, whch means that when a fnte element equlbated by a set of foces at C s subjected to a gd otaton, the foces actng on the element at C 2 should otate followng the gd otaton, whle the magntudes eman unchanged. Hee we shall deve fom the gd body ule the extenal vtual wok W( u, u) done by the bounday tactons and the coespondng vtual nstablty potental U ( u, u) fo a plate unde gven eal gd dsplacements. Fst, we can wte down the vtual wok W done by the bounday foces at C. Fo a plate undegong a gd dsplacement, we can also wte out the vtual wok W 2 done by the suface tactons at C 2. Subtactng W fom W2 yelds the vtual wok ncement done by the bounday foces dung the gd dsplacement fom C to C 2. W

4 Y.B. Yang and S.R. Kuo / Poceda Engneeng 4 (20) Fo an ncemental step contanng only gd dsplacement, the lnea vtual stan enegy vanshes,.e., U E 0. Consequently, we can obtan fom Eq. (2) the followng elaton fo the vtual nstablty potental U and the vtual wok ncement W done by the bounday foces as U ( u, u) W Nzy Nyzu ds (7) N N v ds x z z x y x x y N N w ds The pecedng equaton s exactly the condton fo the vtual nstablty potental of the plate to satsfy dung a gd dsplacement. The vtual nstablty potental ( u, u) n Eq. (7) s a bounday ntegal fo all the foces actng on the bounday of the plate, whch can be tansfomed by een s theoem nto an aea ntegal n tems of the stess esultants and genealzed stans exstng ove the plate. The vtual wok tems n Eq. (7) ae done by the n-plane actons N x, N y and bounday actons N z (ncludng the equvalent sheas). Fo the pesent puposes, we shall dvde the vtual wok n Eq. (7) nto two pats elated to the n-plane and out-of-plane actons as U U U (8) I O whee afte some lengthy opeatons, the two vtual wok tems on the ght sde can be deved as follows: w w w w U x Nyy Nxy da y Nxx Nxy da y x x y v v u u z Nxx Nxy da z Nyy Nxy da x y y x I (9) w v w u w v w u U M O xx M 2 2 yy 2 2 y x x x y y x y Mxy M n w v w u da y xy x xy w v w u ds y n x n (0)

5 8 Y.B. Yang and S.R. Kuo / Poceda Engneeng 4 (20) Devaton of u, u based on foce equlbum Equlbum s a popety that must be obeyed by a sold element at any step of an ncemental nonlnea analyss. Wth the UL fomulaton, the lnea vtual stan enegy U E s ndependent of the nodal foces at C, but the vtual nstablty potental U s a functon of the ntal nodal foces at C. Specfcally, the vtual nstablty potental U should be equal to the vtual wok done by the foce ncements n connecton wth the dsplacement ncements dung the ncemental step fom C to C 2. The followng s the pocedue fo devng U ( u, u) gven the vtual gd dsplacements. Fst, sx foce equlbum equatons can be wtten fo the plate each at C and C 2. Fo a vtual dsplacement feld consstng of thee vtual gd dsplacements x, y, z and thee vtual otatons x, y, z, one can wte down the vtual woks W and W 2 done by the bounday foces at C and C 2. By takng nto account the equlbum condtons at C, one fnds that the vtual wok W of the plate at C vanshes,.e., W 0. Smlaly, by takng nto account the equlbum condtons at C 2, one can deve the vtual wok W 2. Fo gven vtual gd dsplacements, the lnea vtual stan enegy educes to zeo,.e., U E E( u, u) 0. Fom Eq. (2), one fnds that the vtual nstablty potental U (, ) u u can be elated to the vtual wok ncement Wb done by the bounday foces of the plate as ( u, u) U Wb W 2 W * x w Ny v Nz z nx Mn ds, () * w N u N n M ds y x z z y n v N u N M ds z x y n n The pecedng bounday ntegals can be tansfomed by een s theoem nto aea ntegals and dvded nto two pats elated to the n-plane and out-of-plane actons as follows: U U U (2) I O whee, afte lengthy opeatons, the two tems on the ght sde can be wtten as w w w w w w w w U N I xx Nyy Nxy da x x y y y x x y v v u u v v u u Nxx Nyy Nxy da x x y y y x x y v N C e N x xx xy yy u C2 exy da y v u NxyC3 exy NxyC4 exy da y x (3)

6 Y.B. Yang and S.R. Kuo / Poceda Engneeng 4 (20) v w u w v w u w U M O xx M 2 2 yy 2 2 x y x x y y y x Mxy M M n n v w u w da xy y xy x w v w u ds y n x n v w u w ds n y n x w w y x * * Mn z nx z ny ds As can be seen, all knds of actons of the plate have been duly taken nto account n the potental expessons gven n Eqs. (3) and (4). 5. Total vtual nstablty potental u, u U( u, u) In the pevous sectons, one has deved the vtual nstablty potental u, u gd dsplacements u and anothe potental u, u These two potentals can be egaded as the specal cases of the total potental, wth the followng elatons: u, u u, u uu u, u u, u uu (4) fo gven eal fo gven vtual gd dsplacements u u u. of the plate The pecedng two equatons mply that the total vtual nstablty potental, (5) (6) u u satsfes u, u, one can fnd the coespondng both the gd body ule and foce equlbum equements. By an analogcal compason of the ntegal tems contanng the same actons n u, u and tems fo the total potental, u u that satsfes the gd body ule and foce equlbum equements. The fnal esult s expessed sepaately n tems of the n-plane and out-of-plane actons of the plate as follows:

7 20 Y.B. Yang and S.R. Kuo / Poceda Engneeng 4 (20) 4 22 U I u v w Nxx 2 x x x u v w N yy 2 N y y y v v u u w w da x y y x y x xy (7) v w u w v w u w U M O xx M 2 2 yy 2 2 x y x x y y y x 2 2 v w u w 2 Mxy da xyy xyx v w u w * Mn ds Mnz n ds n y n x The last two ntegals on the ght sde of Eq. (8) ae the two tems that ae new compaed wth the exstng theoes fo plates. Because of the ncluson of these two tems, the vtual nstablty potental deved, as gven n Eqs. (7) and (8), s featued by the fact that t s gd body-qualfed and confoms to all equlbum condtons. (8) y 4 L 300 mm 2 (a) R 20 mm 8 (b) Fgue 2: Cylndcal shell: (a) geomety, (b) loadng 6. Applcatons of the pesent theoy The vtual nstablty potental of the plate deved heen plus the conventonal vtual stan enegy, can be used to deal wth the gd body otatonal behavo of the plates n the postbucklng ange. By the UL-based fnte element pocedue, a lnea stffness matx [ k e ] can be deved fom the vtual stan enegy, and a geometc stffness matx [ k g ] fom the vtual nstablty potental fo the plate. The two matces can be used n combnaton wth a elable path-tacng scheme, e.g., the genealzed

8 Y.B. Yang and S.R. Kuo / Poceda Engneeng 4 (20) dsplacement contol method poposed by Yang and Sheh (990), to solve the postbucklng esponse of plate stuctues. Fgue 2 shows a cylndcal shell fxed at one end and subjected to hozontal sheas at the fee end, wth the followng mateal popetes: elastc modulus E = kn/mm 2, thckness t = 5 mm, and Posson s ato = 0.3. Fgue 3(a) shows the convegence chaactestcs of analyss upon mesh efnement usng the pesent theoy. As ndcated n Fgue 3(b), usng the conventonal appoach,.e., neglectng the two last tems n Eq. (8), does not yeld a convegent soluton upon mesh efnement. Ths s manly due to the fact that the conventonal appoach fals to cope wth gd otatons n the post bucklng ange. Fgue 3: Cylndcal shell: (a) mesh test, (b) compason 7. Conclusons A new appoach s pesented fo devng the vtual nstablty potental of plates based on the fundamental pncples of gd body ule and foce equlbum usng the updated Lagangan fomulaton. Compaed wth the conventonal pocedues fo fomulatng the plate theoy, no knematc assumpton s equed heen concenng the plane sectons of the plate afte defomaton. Futhemoe, the pesent theoy s gd-body qualfed, whle all knds of actons on the nstablty of the plate ae duly taken nto account. In the numecal example, the pesent theoy has been demonstated to be supeo to the conventonal ones n that t can adequately cope wth the gd otatons of shells n the postbucklng ange. Acknowledgments The eseach epoted heen was sponsoed lagely by the Natonal Scence Councl though the Tawan-Austa Jont Reseach Poject wth ant No E MY3. Ths pape has been pesented at the Russa-Austa Jont Wokshop on Advanced Dynamcs and Model Based Contol of Stuctues and Machnes held at Johannes Keple Unvesty Lnz, Austa on Apl 25-28, 200. Refeences [] Kuo SR, Ch CC, and Yang YB. A complete stablty theoy fo Kchhoff thn plate unde all knds of actons. J Mane Sc. & Tech.; 2009, 7(3), [2] Kuo SR, and Yang YB. A new method fo devng the nstablty potental of plates based on gd body and equlbum consdeatons. To be publshed; 200

9 22 Y.B. Yang and S.R. Kuo / Poceda Engneeng 4 (20) 4 22 [3] Yang YB, and Kuo SR. Theoy and Analyss of Nonlnea Famed Stuctues. Pentce-Hall, Sngapoe; 994 [4] Yang YB, and Chou HT. Rgd body moton test fo nonlnea analyss wth beam elements. J Eng. Mech.; 987, ASCE, 3(9), [5] Yang YB, and Sheh MS. Soluton method fo nonlnea poblems wth multple ctcal ponts. AIAA J. ; 990, 28(2),