Buckling and Initial Post-Buckling of Generally Stiffened Conical Shells

Transcription

1 7th AIAA/AE/ACE/AH/AC tuctues, tuctual Dynamics, and ateials Cnfee - ay, ept, Rhde Island AIAA -7 Buckling and Initial P-Buckling f Geneally tiffened Cnical hells Y. Gldfeld * Technin-Isael Initute f Technlgy, Haifa, Isael The sensitivity f ened cnical shells t impefectin is cnsideed, via the initial puckling analysis. Unlike ened cylindical shells, in the case f geneally ened cnical shells the enes inclinatin and the diance eteen the enes vay ith the shell cdinates, hich cmplicates the plem cnsidealy. The main jective f the udy is t inveigate the influence f the enes n the uckling lad and n the impefectin sensitivity. Thus, y finding the paametes that influence the shell s impefectin sensitivity, it is pssile t impve the ehavi f the hle uctue. I. Intductin HE uctues ae idely used in aenautic, maine and civil engineeing uctues, they elng t the thinall uctue family, hich ae vey sensitive t impefectin, and thei sensitivity depends n the gemety f the shell and its mechanical ppeties. One f the main gals, in this field, is t find the vaius paametes that influence the shell s sensitivity, thus t impve the ehavi f the hle uctue. One f the ays t incease the uckling lad and t decease the sensitivity t impefectin as ell is t add enes t the shell uctue. Heve, adding enes t the shell uctues mu e dne ith ict attentin, ecause adding enes incectly uld, in sme cases, nt change the ehavi f the shell at all. Cnical shells ae usually used as a cnnectin eteen t cylindical shells ith diffeent diametes. Unlike ened cylindical shells, in the case f cnical shells ith enes in a geneal psitin the diance eteen the enes and the angle f inclinatin f the enes vay ith the shell cdinates, hich ultimately esults in cdinate dependence f the cntiutin t the ness cefficients f the A, B and D matices. This effect cmplicates the plem cnsidealy. The fi level f cmpleity is attiuted t the need t find an analytical epesentatin f thse functins. An ehauive udy f the ness functins and thei dependence n these facts has een pefmed in Ref.. The secnd level f cmpleity is assciated ith the intductin f cdinate dependent eness matices int the mathematical mdel and the slutin f the syem f nnlinea gvening patial diffeential equatins ith vaiale cefficients. In ealie inveigatins - the enes ee cnsideed nly as inges (psitined in the aial diectin, the diance eteen the inges vay linealy) as ings (psitined in cicumfeential diectin, equally diiuted). Othe inveigats - used finite elements cmmecial cmpute cde t calculate the uckling and p-uckling ates f inge-ened cnical shells. As fa as the auth kns, the uckling and the initial p-uckling ehavi and the impefectin sensitivity f geneally ened cnical shells taking int accunt the vaiatin f the mateial ppeties ith the shell s cdinates has nt een inveigated, as yet. This k pesents a quantitative udy f the impefectin-sensitivity f eccentically ened cnical shells in a geneal diectin, y cnsideatin f the vaiatin f the ness cefficients and thei dependence n the shell s cdinate. Inveigatin is made ithin the famek f Kite s 7 geneal they f p-uckling ehavi. The calculatins pvide a measue f the etent t hich the shells ae sensitive insensitive t impefectins in thei shape and thus indicate t hat etent the classical uckling esults can e eliale. The nn-linea equiliium diffeential equatins ae deived fm the asis f thei kinematic appach, ug the displacement cmpnents (aial (u), cicumfeential (v) and nmal ()) as the unknn dependent vaiales. The asympttic technique is used t cnvet the nn-linea equatins int thee linea sets. These equatins ae slved thugh epansin f the dependent vaiales in Fuie seies in the cicumfeential diectin and in finite diffeences in the aial diectin. Afteads the Galekin pcedue is used t minimize the e due t the tuncated fm f the seies. A special cmpute cde as develped and used f a ide ange f paametic udy f uckling and sensitivity ehavi f geneally ened cnical shells. * eni lectue, Faculty f Civil and Envinmental Engineeing. Cpyight y Yiska Gldfeld, Technin- Isael Initute f Technlgy. Pulished y the, Inc., ith pemissin.

2 II. Gvening Equatin This slutin pcedue has een used efe -, heve, f the case f ened cnical shells sme additinal cmplicatins ccu. A. Kinematic Relatin Dnnell s they The ain-displacement elatin can e itten as: { ε} { ε } z{ χ} () hee {ε } and {χ} ae, espectively, the ain and change-f-cuvatue vects f the efeence suface, cmpsed as flls, see Ref. 9: ε u' v' ε cs γ u' ( ) ( ) v v ' ( ) ( ) u ( ) ( ) ( ) χ χ χ ' ' ( ) ( ) ' ' ' ( ) ( ) (-) hee (), and (), dente the deivatives ith espect t the aial () and cicumfeential () cdinate, espectively; () is the adius at the cdinate, and α is the cne semi-vete angle (see Fig. ). B. Cnitutive equatins tiffened Cnical hells The pesent k alls f th intenally and etenally eccentic ene in a geneal psitin. The shell is assumed t e clsely ened. Theefe, thee is n pssiility f lcal uckling eteen enes and it is pssile t smea the enes, s the ess esultants and mment pe unit length ae given y: shell () shell shell hee and shell ae the ess esultants and cuples pe unit length f the shells sheet. and ae the enes cntiutins t the lad caying capacity f the shell (pe unit length). These cntiutins ae ased n the flling assumptins egading the cnitutive elatins, : ) The enes ae made f ipic mateial ith linea elaic ehavi. ) The enes can e psitined in any diectin and Detail f enes () ae diiuted mathematically, f calculatin pupses, ve the hle suface f the shell. ) The ene-laminate cnnectin is mnlithic, hence the nmal ains (ε zz ) vay linealy in the ene as ell as in the sheet and equal at thei pint f cntact. ) The enes d nt tansmit shea pependicula t thei aes. ) The ene esis tsin due t its tsinal igidity. As a esult f these assumptins the intenal fces and mments aut the suface f efeence, due t the enes nly, ae tained as functins f the ains and the cuvatues f the efeence suface in the diectin f the enes (, ). e R R α β( ) Figue : Gemety, sign cnventins f the cdinates, and the psitin f the ene n a cnical shell.

3 a ε χ χ ε d d χ hee a E A ( ) E Ae d d E ( ) ( ) ( I A e ) G I ( ) t (-) E is the mdulus f elaicity f the ene, G is the shea mdulus f elaicity f the ene, A is the csssectinal aea f the ene, e is the diance f the centid f the ene css-sectin fm the suface f efeence, I is the mment f inetia f the ene css-sectin aut itself, I t is the tsin cnant f the ene css-sectin. is the diance eteen the enes and f a cnical shell it depends n the shell s cdinates, (, ). F ptimizatin pupse the gemetical chaacteiics f the css-sectin f the enes can e chsen t e als sme functin f and. Theefe, a,, d and ae ng functins f the shell s cdinate. Unde the classical laminate they, the cntiutin f the enes t ness matices epessed in the asic cdinates syem f the shell (, ) is given y: A B D a d d (7) Whee C C C C C C C C C C( C ) ( ) ( ) ( ) ( ) C C C C C C C C C (-9) C hee Ccs(β(,)) and (β(,)), β is the angle f inclinatin f the enes and is a functin f and hich depends n the chsen psitin f the enes. In this k a gedesic path is chsen t epesents the vaiatin f the enes in the shell s cdinate, theefe the enes inclinatin β and the diance eteen the enes change nly in the aial diectin and ae cnant in the cicumfeential diectin. The enes inclinatin β() is given y: R α β () ( ) ac β R α hee β is the ene inclinatin at the small end f the shell, is the lngitudinal cdinate in the cne suface, see Fig.. The diance eteen the enes () is given y: ( ) ( R α ) csβ () R α csβ hee is the diance eteen the fies at the nae edge f the cne. Theefe, the ness matices A, B and D, ae ng functins f the lngitudinal cdinate and have t e β( ) ( )/ β β 7 β... [ m] β β 7 β... [ m] Figue : Vaiatin f the enes inclinatin and the diance eteen the enes alng the cne slant length f cnical shell α.

4 added t the ness matices given y the shell s sheet. te that f β ne tains the case f inges (cntiutin t the ness cefficients A, B and D, hich ae vay linealy alng the aial diectin). In the case f a ene ientatin f βπ/ the assumptin f a gedesic path is nt valid. The ene inclinatin and the diance eteen enes emain cnant alng the aial diectin and the uckling lad and the initial p uckling ehavi can e calculated n the asis f cnant ness cefficients. In this case (βπ/), ne tains the case f ing enes (cnant cntiutin t the ness cefficients A, B and D ). In Fig. the change f the ene inclinatin and the diance eteen the enes ae pltted vs. the lngitudinal cdinate f cnical shell (ith cne semi-vete angle f α, a slant length f. m and a shte adius f the tuncated cne R.7m) ith vaius initial enes inclinatin (β ). The ene inclinatin deceases alng the aial diectin and the diance eteen enes inceases alng the aial diectin. The highe the initial ene inclinatin the apid the change alng the aial diectin. F a geneal angle f inclinatin β, the ening f the enes ill e unalanced. T have alanced ening ne mu add the same enes ith an angle f inclinatin -β. F this case ne tains: A B D a d d () hee the alanced matices and ae given y: C C C C C C ( ) C C (-) C It mu e emphasized that f cylindical shell, the ness cefficients, due t the enes ith cnant gemetical chaacteiics, ill ecme autmatically cnants. C. Equiliium equatins The nnlinea equiliium equatins and the apppiate unday cnditins ae deived n the asis f the atinay ptential enegy citein. The flling thee nnlinea equiliium equatins ae tained as:,,,, ( ), ( ), ( ) [ ] qv ( ), ( ) ( ) ( ) [ ( ) ]' ( ) cs q u ' ( ) ( ),,, ( ) ( ) ' ' q () ith the flling unday cnditins:,, ' ' ( ) ( ) u u v v ( ) ( ) ( ) Q α ()

5 Ug Eqs. (-) and the cnitutive elatins the equiliium equatins and the unday cnditins can e tained in tems f the displacement functin 9. D. Impefectin sensitivity In Kite s geneal they 7 f elaic aility the impefectin sensitivity f a uctue is clsely elated t its initial puckling ehavi and the they is eact in the asympttic sense. This detemines hethe the lad initialy inceases deceases afte uckling. The classical uckling lad f the pefect uctue is dented y λ c, and in all cases cnsideed hee it is the lad at hich a nnaisymmetic ifucatin fm the peuckling ate ccus. Assuming that the eigenvalue plem f the uckling lad λ c ill yield a unique uckling mde u, a slutin t e valid in the initial puckling egime, is sught in the fm f the flling asympttic epansin. λ λ c aξ ξ u u v v ( ) ( ) ( ) u ξ v () () () u ξ v ( ) ( ) ( )... (7) Whee λ is the lad paamete deviating fm the ifucatin uckling lad λ c and ξ is the amplitude f the uckling mde. The supescipts (), () and () dente the peuckling, uckling and puckling ate, espectively. A fmal suitutin f this epansin int the nnlinea gvening equatins (Eqs. ) geneates a sequence f equatins f the functins appeaing in the epansins: The ze-th de tems yield the patial diffeential equatins f the peuckling ate: [ e] ( ) ( ) [ e] ( ) ( ) ( ) [ e] ( ) ( T ) (, T, ) P [ e] ( ), e,, () The fi de tems yield the patial diffeential equatins f the uckling ate: [ ] ( ) ( ) [ e] ( ) ( ) ( ) [ ( ) (, T )] [ ] ( ) ( ) ( ) [ e] ( ) ( ) ( ) [ ( ) (, T, ) (, T, )] [ e] () e, T [ e] ( ) ( ) ( ) e, T, e,, (9) The secnd de tems yield the patial diffeential equatins f the puckling ate: [ e] ( ) ( ) [ e] ( ) ( ) [ e] ( ) ( ) ( ) ( ) (, T ) (, T ) (, T, ) [ e] ( ) ( ) ( ) [ e] ( ) ( ) ( ) (, T, ) (, T, ) [ e] () () [ e] () () ( ) [ e] ( ) () () [ e] () ( ) () (, T ) (, T, ) (, T, ) (, T, ) [ e] ( ) e,, () [e], [e] and [e], [e],,, ae espectively linea, quadatic and cuic diffeential peats having vaiale cefficients. They ae given y Ref. 9. tice that the sum f the indices f the peuckling peats is ze, thse f the uckling peats is ne, and that f the puckling peats is t. The nn-linea slutin f the peuckling ate, Eq. (), shuld e pefmed y an adequate numeical pcedue hich ultimately yields a limit-pint. In de t simplify the plem the slutin pcedue f the peuckling ate is pefmed y igning the nn-linea tems and slving nly the linea pat f Eq. (), that is [ e] ( ) [ e] [ e] ( ) P. The applied lading P cnsis f aial cmpessin, intenal etenal pessue and clckise cunte-clckise tque. It is assumed t have a unifm spatial diiutin and is divided int a fied pat and a vaiale pat. The magnitude f the vaiale pat is alled t vay in pptin t a lad paamete λ. This leads t an eigenvalue plem f the citical lad λ c. These equatins admit sepaale slutins, in the cicumfeential diectin y ug Fuie seies and the - dependence is eliminated y applying Galekin s pcedue. A cental finite diffeence scheme is used in the aial diectin t educe the dinay diffeential equatins t the flling algeaic nes:

6 F the peuckling ate, { } { P} ( ) [ ] Z K () and f the uckling ate, { } {[ ] [ ]} ( G Z ) K λ () hee K and G ae the ness and gemety matices espectively, Z () and Z () ae unknn vects cnsiing f u, v,, u,, v, and, f the peuckling ate and the uckling ate, espectively. Eq. () is an eigenvalue plem in hich λ epesents the uckling lad paametes and Z the uckling mde. Ug a linea peuckling analysis the equatins gvening the p-uckling ate ecme [ e] ( ) ( ) [ e] ( ) ( ) ( ) λc [ (, T ) (, T )] [ e] ( ) ( ) ( ) [ e] ( ) ( ) ( ) [ e] ( ) ( ) ( ) c [ (, T, ) (, T, ) (, T, )] [ e] () () [ e] () () ( ) [ e] ( ) () () [ e] () ( ) () (, T ) λ [ (, T, ) (, T, ) (, T, )] [ e] ( ) λ c e,, () The puckling-ate (nd-de tems) (), v () and u () ae tained fm the slutin f the set f thee inhmgeneus linea patial diffeential equatin linea ith the assciated unday cnditins. ince the ight hand side f Eq. () epesents the quadatic level f the knn uckling mde, the puckling ate viusly depends n c, c eing the scala y hich the uckling mde is nmalized. lving the p-uckling ate ug the same sepaale slutins, the flling algeaic equatins tained f the puckling ate: { } { F} ( ) [ ] Z A () hee A and F ae the p-uckling mati and the ight hand side vect, espectively. Z () is unknn vects cnsiing f u (), v (), () ( ) ( ) ( ), u ', v ', ' f the p-uckling ate. F pefect shells ne is inteeed in the vaiatin f λ(ξ) ith ξ in the vicinity f λλ c. nea the ifucatin pint λ c the asympttic epansin given in Eq. (7) is valid. Due t the peidicity f the uckling mde in the cicumfeential diectin the fi puckling cefficient a vanishes. The ell-knn fmula f the case f a memane peuckling ate deived in Ref., f the case f linea peuckling the flling fmula tained: σ ( u) σ ( u, u) λ [σ ( u, u ) σ ( u )] c () When has a psitive value the uctue can develp cnsideale puckling ength, and lss f aility f the pimay path des nt esult in uctual cllapse. Heve, hen has a negative value then in m cases uckling ill ccu vilently and the magnitude f the citical lad is suject t the degading influence f initial impefectins. F cnical shell, ith the vaiales u, v, the peat ill e: ( j ) ( k ) ' ( j ) ( k ) ( k ) ( j ) ' π () ( ) ( ) () () i j k i i σ i ( u j, uk ) [ ],,,, dd i j k () ( ) ( ) ( ) a Hee, the supescipts (i), (j) and (k) dente the apppiate ate (()-peuckling, ()-uckling and ()- initial puckling).

7 III. Results and Discussin F the pcedue utlined ave simply-suppted (-) ened cnical shells unde aial cmpessin ae eamined. The gemetic ppeties ae: slant length.m, the adius at the small end R.7m, the thickness f the shell s sheet t.7m (R /t). The gemetic ppeties f the enes ae: the diance eteen the enes at the small end f the shell πr / ( enes in the cicumfeential diectin assue that n lcal uckling ill ccu), the css-sectinal aea f the ene A.7 t, the mment f inetia f the ene css-sectin aut itself I t //(-ν ), the tsinal igidity f the ene css-sectin is neglected (I t ), the diance f the centid f the ene css-sectin fm the suface f efeence e±.t. F alanced alanced alanced enes cnfiguatin A A / and I I /, thus alanced and unalanced cnfiguatins have the same amunt f mateial. The sheet and the enes ae made f ipic mateial ith mateial ppeties as fll: mdulus f elaicity E. /m, Pissn s ati ν.. In Fig. the uckling lad and the Kite- paamete ae pltted again the enes inclinatin at the small end f the shell (β ) f cnical shell ith α having diffeent enes cnfiguatins. As as epected alanced ene cnfiguatins give highe uckling lad than unalanced ene cnfiguatins, heve usually cnical shell ith alanced ene cnfiguatin ae me sensitive t impefectin than unalanced cnfiguatin. The psitin f the enes has a significant influence n the uckling lad and n the impefectin sensitivity; etenal enes give highe uckling lad ut ae me sensitive t impefectin than intenal enes. Intenal enes give elatively le uckling lad ut usually assue insensitive shell. Centic enes have n significant advantage cmpae t the the cnfiguatin, especially in the case f alanced cnfiguatin (it gives le uckling lad and als vey sensitive t impefectin) ce it has kind f thtpic cnfiguatin. F ptimizatin pupse, it seems that the enes cnfiguatin f β ~ f alanced enes and β ~ f unalanced cnfiguatin give the highe uckling lad ut ae sensitive t impefectin. * k/m 7 β - ene inclinatin Kite -paamete etenal enes - unalanced etenal enes - alanced centic enes - unalanced centic enes - alanced intenal enes - unalanced intenal enes - alanced In Fig. the uckling lad and the Kite- paamete ae pltted again the cne semi-vete angle f cnical shells ith enes having β at the small end f the shell. Hee again, the unalanced ene cnfiguatins give le uckling lad ut insensitive shells. Etenal enes give highe uckling lad ut me sensitive shell. Centic enes is the cnfiguatin f alanced enes, it gives le uckling lad and vey sensitive shell. F th eamples the cicumfeential ave numes f the alanced cnfiguatins ae le than thse f the unalanced cnfiguatin n n n 7 7 n n7 n7 ncicumfeantial ave nume β - ene inclinatin Figue : Aial cmpessive uckling lad and Kite paamete vesus ene inclinatin at the small end f the shell f cnical shell ith α. n n n n n 7

8 * k/m 7 α - cne semi-vete angle Kite -paamete n 9 n n n 7 n (intenal enes-unalanced)>> n α - cne semi-vete angle 7 Figue : Aial cmpessive uckling lad and Kite paamete vesus the cne semi vete angle f the shell f cnical shell ith ene inclinatin at the small end β. IV. Cnclusin In this udy, the uckling lad and the impefectin sensitivity f a ened cnical shell ee inveigated unde the assumptin that enes ientatin changes as a gedesic path. The impvement in this eseach is achieved y the adptin f a suitale analytical epesentatin t descie the cdinate dependent ness and, especially, y the udy f the influence f the vaiatin f the ness cefficients n the impefectin sensitivity f the ened cnical shells. The main cnclusin fm this inveigatin is that the psitin f the enes has a significant influence n the uckling lad and n the impefectin sensitivity. Etenal enes give highe uckling lad ut ae me sensitive t impefectin than intenal enes cmpae t intenal enes. Centic enes have n significant impvement f the shells ehavi. Balanced enes cnfiguatins give highe uckling lad ut me sensitivity t impefectin. Refeences Bauch,., Acz,. and Zhang, G.Q. "aminated Cnical hells- Cnsideatins f the Vaiatins f tiffness Cefficients". Rept R-7, Faculty f Aespace Engineeing, Delft Univesity f Technlgy, The ethelands, Apil 99. Bauch,., inge,. and Haai, O., Geneally Inaility f Cnical hells ith n-unifmly paced tiffenes unde Hydatic Pessue, Isael unal f Technlgy, Vl.,., 9, pp. -7. Bauch,. and inge,., Geneally Inaility f tiffened Cicula Cnical hells unde Hydatic Pessue, The Aenautical Quately, ay 9. halev, A., Bauch,. and issim, E., Buckling Analysis f Elaically Cnained tiffened Cnical hells unde Hydatic Pessue y The Cllcatin ethd, Isael unal f Technlgy, Vl.,. -, 97, pp Chyssanthpuls,. K. and pagnali, A., The Influence f Radial Edge Cnaint n the taility f tiffened Cnical hells in Cmpessin, Thin-Walled tuctues, Vl. 7,., 997, pp. 7-. pagnli, A. and Chyssanthpuls,. K., Buckling Design f tinge tiffened Cnical hells in Cmpessin. unal f tuctual Engineeing, anuay Kite, W. T., The taility f Elaic Equiliium, Thesis, Delft, H.. Pais, Amedam, 9; U.. Ai Fce Flight Dynamics a. Ai Fce yem Cmmand, AFFD-TR-7-, Wight-Pattesn AFB, OH, Fed. 97 (English Tanslatin). Gldfeld, Y., heinman, I. and Bauch,. "Impefectin ensitivity f Cnical hell." AIAA unal,., Vl.,, pp Gldfeld, Y. and Acz., Elaic Buckling f aminated Cnical hells ug a Hieachical High-Fidelity Analysis Pcedue, sumitted f pulicatin,. Gldfeld, Y., Impefectin ensitivity f aminated Cnical hells, sumitted f pulicatin,. Bauch,. and inge,., Effect f Eccenticity f tiffenes n the Geneal Inaility f tiffened Cylindical hells unde Hydatic Pessue, unal f echanical Engineeing cience, Vl.,., 9, pp. -7. Budiansky, B. and Hutchinsn,.W., Dynamic Buckling f Impefectin ensitive tuctues. Pceeding XI Intenatinal Cngess n Applied echanics, unich, 9, pp. -.