A Preliminary Study on Material Utilization of Stiffened Cylindrical Shells

Transcription

1 Reearch Journal of Appled Scence, Engneerng and echnology 6(5): , 03 ISSN: ; e-issn: Maxwell Scentfc Organzaton, 03 Submtted: December 8, 0 Accepted: February 08, 03 Publhed: Augut 0, 03 A Prelmnary Study on Materal Utlzaton of Stffened Cylndrcal Shell, L Zhu and Xu Ba College of Shpbuldng Engneerng, Harbn Engneerng Unverty, Harbn 5000, Chna Equpment Procurement Center, Equpment Department of Navy, Bejng 0007, Chna Abtract: he tffened cylndrcal hell ued for leg of offhore platform and preure hull of ubmerble frequently. he extng degn method only focu on the trength and tablty of tructural component. he am of th tudy to preent a new crteron for degn of tffened cylndrcal hell. A a reference, an un-tffened cylndrcal hell taken nto conderaton. By fxng the man cale value of tructure, changng the hell thckne, the dmenon and number of tffener, cylndrcal hell wth crcumferental rb are created. hen, the bucklng analy on the famly of hell, whch the revoluton of contant ma, carred out by FEM. Baed on movng leat quare method wth nterpolaton condton, the fttng urface of polynomal functon got by MALAB mulaton. Reult of polynomal how the relatonhp between the materal utlzaton of tructure and the crtcal load under unform external preure. Keyword: Bucklng analy, materal utlzaton, hell tructural degn, tffened cylndrcal INRODUCION Stffened cylndrcal hell are wdely ued n offhore engneerng, uch a leg of offhore platform and ubmarne preure hull. he degn problem of thee knd of tructure have been a ubject over the lat decade and are tll vvdly and broadly nvetgated today. It becaue there are tll many unolved problem. Nowaday, ue have concernng the feld of cylndrcal hell tructure, a n the book of Chen (00) and the paper of Young-Shn (009). here are many paper condered the optmzaton degn of tffened cylndrcal hell, for example, the optmum degn problem of cylndrcal hell under arbtrary aymmetrcal boundary condton and wth unform dtrbuted radal preure reported by Lang and Yue (00). hen, n order to decreae the tructural weght and ncreae the effectve payload, the tffened cylndrcal hell of workng platform n deep ea wa optmzed by Gao-feng et al. (00). Relatvely new reult baed on mult-objectve optmzaton algorthm for rng tffened cylndrcal hell were contaned n the paper of Bagher et al. (0). he man requrement of modern engneerng tructure are that they hould be afe for load-carryng capacty, ft for producton and be economcal. However, very few paper related to that. he reearch on mnmum cot of a welded orthogonally tffened cylndrcal hell wa preented by Jarma et al. (006), n whch the cot functon and the functon pecfyng the contrant are hown. And mnmzaton of the weght of rbbed cylndrcal hell condered by Sryu (0). Accordng to thee achevement, the deal degn way of tffened cylndrcal hell eem to be a method whch cont wth materal utlzaton and tructure utlzaton bede trength and tablty. Fg. : he geometry of a cylndrcal hell he am of th tudy to expound materal utlzaton n tructural degn of tffened cylndrcal hell. A procedure for a whole famly of hell whch evolved from keepng contant ma workng out. he un-tffened cylndrcal hell taken nto conderaton a reference. Fgure preent the geometry of a cylndrcal hell, the value of the man cale of tructure, radu of hell R and axal length L, are fxed. hen, a famly of tffened cylndrcal hell wth crcumferental rb created by changng the hell thckne, the dmenon and number of tffener. Furthermore, comparng to other factor, the tablty of cylndrcal hell epecally mportant n cylndrcal hell of hp and offhore engneerng. he effect of materal utlzaton on crtcal load the manly conderaton n th tudy. ANSYS code, one of the Fnte Element Method (FEM) oftware, utlzed to calculate crtcal load by bucklng analy. Fnally, materal utlzaton wll be got from analyzng the FEM reult. Correpondng Author: Xu Ba, College of Shpbuldng Engneerng, Harbn Engneerng Unverty, Harbn 5000, Chna 757

2 SIFFENED CYLINDRICAL SHELLS WIH CONSAN MASS Frtly, create a famly of tffened cylndrcal hell whch evolved from keepng contant ma. A a reference, a column of end both fxed contructed and loaded by unform external preure. hen, quare croecton rb are welded nde of the hell by crcumferental fllet weld. Hence, ma of the tffened cylndrcal hell hown n Fg. approxmately gven by: Re. J. Appl. Sc. Eng. echnol., 6(5): , 03 m = πrρ tl + na () he followng varable wll be ntroduced when tt 0 repreent the thckne of untffened cylndrcal hell: t A x =, y t = () tl 0 0 Introducng varable () nto Eq. () gve, after reorderng: x + ny = 0 (3) Solvng th quadratc equaton the followng formula for varable n are: Fg. : Cylndrcal hell wth rng tffened Y n = 0.4 n = 0.3 n = n = X Fg. 3: A famly of contant ma lne for dfferent rb n= x / y (4) A t wa mentoned above the axal length L of the hell and the radu R are contant. And then, t clear that changng parameter x and y, a ere lne repreented hell wth contant ma can be got. An example hown n Fg. 3. he value of x and y obtaned from the Eq. (3) and (4) provded the tffened cylndrcal hell wth the condered ma. In Fg. 3, the tffened cylndrcal hell are the nterecton pont of the contant ma lne wth the coordnate ax. Summng up, t clear now that by aumng the value of x and y, the other parameter of the hell obtaned. From the Fg. 3, the pont (x = and y = 0) repreent an un-tffened cylndrcal hell. he nterecton pont of each lne wth the ax of coordnate repreent tffened cylndrcal hell. Bede thee pont, other are correpondng to tffened cylndrcal hell wth dfferent geometry. Conderng the longtudnal coordnate, all pont n th lne repreent llegal parameter of hell. So t not need to be condered here. An example wll be hown now. A a reference an un-tffened cylndrcal hell of the ma choen. he 758 able : A famly of hell wth contant ma mm = 83. kg L = 000 mm tt 0 = 0 mm R = 600 mm Contant value Parameter n = n = n = 3 n = 4 a A (mm ) t (mm) b A (mm ) t (mm) c A (mm ) t (mm) d A (mm ) t (mm) e A (mm ) t (mm) f A (mm ) t (mm) g A (mm ) t (mm) radu of the hell R = 600 mm and the thckne tt 0 = 0 mm. h determne the value L = 000 mm. And the properte of the materal are: E =.05 0 MPa, υ = 0.3, ρ = 7.85 g / cm 5 3 Now by choong arbtrary value of the number of rb n and calculatng value of parameter A and t by expreon () and (4), t poble to got a famly of tffened cylndrcal hell whch evolved from keepng contant ma. hee hell are hown n able.

3 FEM ANALYSES Re. J. Appl. Sc. Eng. echnol., 6(5): , 03 A bucklng analy ha been carred out n order to how how the materal of the cylndrcal hell nfluence the crtcal load under unform external preure. So a famly of tffened cylndrcal hell wth contant ma ha been condered. he analy wa carred out by ANSYS procedure. A Sold 45 fnte old element ha been choen wth eght node and x degree of freedom n each node. he boundary condton mpoed n the analy are hown n Fg. 4. he movement of a normal edge drecton and a rotaton allowed. A hell loaded wth unform external preure only. Becaue the appled preure value equal unty (p = MPa) and no preload (dead load) wa appled, the egenvalue obtaned from the analy correpond to the bucklng load drectly. he materal pecfcaton the ame a n the example above, t : Fg. 4: Boundary condton E =.05 0 MPa, υ = 0.3, ρ = 7.85 g / cm 5 3 he frt tep, a meh convergence analy wa made by un-tffened cylndrcal hell under the ame condton. Fgure 5 how the bucklng hape of the untffened cylndrcal hell a an example. he reult are Fg. 5: Bucklng hape of untffened cylndrcal hell (a) Change of load (b) Relatve change of load Fg. 6: Meh convergence analy for untffened cylndrcal hell Pcr (MPa) Relatve change of Pcr (%) Number of element (a) Change of load Number of element (b) Relatve change of load Fg. 7: Meh convergence analy for tffened cylndrcal hell 759

4 Re. J. Appl. Sc. Eng. echnol., 6(5): , 03 Crtcal load (MPa) 90 Dfferent hell thckne Number of rb t = 8 mm Fg. 8: Part of the FEM analy reult hown n Fg. 6. It wa decded that 30,000 fnte element of the hell would be enough for the analy. And the analy reult of f8 of the tffened cylndrcal hell alo preent 30,000 fnte element howed n Fg. 7. he econd tep, the famly of tffened cylndrcal hell wth contant ma wa examned. he FEM analy reult are hown n Fg. 8. he a 4 of tffened cylndrcal hell correpond to the crtcal load obtaned from ANSYS, whch equal p cr = MPa. Smlar value, that, p cr = MPa, gve by comprehenve conderaton of the Eq. (5) and (6). he crtcal load of local bucklng: 0.6 Et p = (5) u 0.37 R And the crtcal preure of general bucklng gven by: 4 α + E ( α + n ) t p = n + 0.5α.68 βt R 3 ( n ) αu ( + β) R (6) he non-dmenonal parameter relatng to the degn varable and contant are defned accordng to theory and experment a follow: 0.647l u = Rt β = lt / F t = 6 mm t = 4 mm t = mm t = 0 mm 8 POLYNOMIAL FUNCIONS he bucklng load one of the mot mportant objectvely degn varable of an externally preurzed tffened cylndrcal hell. It ntegrate tructural parameter whch react the bearng capacty. he relatonhp between the tructure parameter and bucklng load nonlnear trongly. he mulaton reult are not dentcal wth the real value and the relatonhp between materal dtrbuton and bucklng load cannot be got from Eq. (5) and (6) drectly. So, a mlar polynomal functon whch reflect the changng trend of thee dcrete data pont need to be got by urface fttng. Surface fttng a data proceng method, whch ue a contnuou urface to depct the functonal relaton among pace dcrete pont. It can be contructed by the leat quare method whch doe not need to go through all the data pont. Recently, movng leat quare method propoed for fttng the data. Compared wth the tradtonal leat quare method, t ha obvou advantage and ued broad n the data fttng and analy. In th tudy, the fttng urface need to go through one key pont, o the nterpolaton condton of the movng leat quare method hould be condered. Accordng to the paper of Feng-Me et al. (0), n the movng leat quare method, the objectve polynomal functon can be expreed a: m (7) = = α = α f x x p x p x x he coeffcent are: ( x) ( x), ( x),..., ( x) α = α α αm he bac functon :,,..., p x = p x p x pm x Conderng the complete polynomal ba, uch a: Lnear bae: = (,, ) ( = 3) px xy m Quadratc bae: α = πr/ L ( ) 3 U = Lt / I R t + F / l / l And t can verfy that parameter m =, n = 6 n expreon (6) gve the mallet value. 760 = (,,,,, ) ( = 6) p x x y x xy y m In order to obtan a more accurately local approxmaton, the dfference of the weghted quare between local approxmaton f (x ) and pont value hould be mnmum. In the paper of Xao-Hong et al.

5 Re. J. Appl. Sc. Eng. echnol., 6(5): , 03 (005) and N et al. (00), the dcretely weghted norm of redual : n α (8) J = w x x p x x y = w (x-xx ) the weght functon of the pont x whle n the number of pont n the olvng doman and f (x) the fttng functon. Uually, the crcle choen a the upport doman the plne functon contructed to be the weght functon. Let ' = x-x, = '/r, then the cubc plne functon : , w = + < < 3 3 0, > 3 4 4, (9) when J take the mnmum value, matrx form of Eq. (8) a follow: ( ) ( ) J = Pa x Y W x Pa x Y (0) (,,..., ) Y = y y y n = (,,..., n ) W x dag w x w x w x w = w x x P... m ( x ) p x p x p x p = p x p x... n m n a (x) obtaned by the leat quare method: o, the objectve functon ubttuted by Eq. () through (): m () = k k = φ = ψ f x y xy Ψ kk (x) the hape functon and k the order of bac functon: ( x) =,,..., = p ( x) A ( x) B( x) k k k k ψ φ φ φn (3) Whle the fttng functon hould go through ome key pont, the movng leat quare method wth nterpolaton condton appled to contruct the fttng urface formula. Suppong there are a et of cattered node (xx, yy, zz ), =,,.., n and the nterpolaton condton are (xx, yy, zz ), =,,.., t, t n. P (x, y) the fttng urface contructed by the movng leat quare method, then, the fttng urface wth nterpolaton condton can be wrtten a: t z= Pxy, l xyδ, (4) l ( x) = t ( j) = ( ) ( j) ( ) x x y y = x x y y j j j j =,,..., t δ = P x y z (, ) h formula can alo be condered a the modfcaton of the movng leat quare fttng formula where the correcton functon tt = ll (x) δδ xx. So, n th tudy, the man procee of fttng functon are: Obtan the movng leat quare fttng urface wthout nterpolaton condton. Calculate the devaton of the key pont. Obtan the movng leat quare fttng urface wth nterpolaton condton. ax = A ( xbxy ) () Accordng to the mulaton reult, the et of cattered node are got. In order to analy the materal utlzaton, the dmenonle value hould be ndependent varable, the bucklng load value hould be dependent varable. he detaled data hown n A( x) = PW( x) P able. he MALAB mulaton tool wa ued to wrte the B( x) = PW( x) fttng procedure and pant the urface graph. he fttng 76

6 Re. J. Appl. Sc. Eng. echnol., 6(5): , 03 Fg. 9: he fttng urface Fg. 0: Surface of X partal dervatve functon able : he orgnal data of urface fttng z x y z x y able 3: he relatve error of fttng urface he maxmal error he mnmal error he average error 3% 0.6%.6% urface wa contructed by the nterpolaton node whch are (.648,, 0), whch hown n Fg. 9 and the formula of fttng expreed a: (, ) = 0 = 0 = 0 = 0 f xy = axy + bxy + cxy + dxy + e a0 a a a3 a b0 b b b c0 c c = d0 d e 3770 Accordng to the fttng functon, the relatve error among fttng urface and orgnal data are hown n able 3. he error ft for the accuracy requrement. MAERIAL UILIZAION Maxmzng materal utlzaton n tructure degn paramount mportant. Materal typcally repreent 75% and even more of the total cot n manufacture, o 76 Fg. : Surface of Y partal dervatve functon a poor degn can gnfcantly ncreae cot of tructure over t lfe. he concept of materal utlzaton a common vocabulary n energy ndutry, materal cence and etc., but relatvely new n tructure degn. he defnton alo dfferent from dfferent tructure degn. In th tudy, Materal utlzaton defned a the effectve degree of materal ue n tffened cylndrcal hell tructure. Wth the ame materal amount the hgher the materal utlzaton rato, the tronger the carryng capacty of tructure. So, the materal utlzaton the change rate of the functon of tructure parameter and target varable. In the fttng functon, the ndependent varable are the hell and rb proporton of the total materal. herefore, the effectve degree of materal uage the man factor that the ndependent varable affect the dependent varable. Partal dervatve functon for X and Y coordnate ax are got repectvely. A hown n Fg. 0 and, the change of the parameter x hgher than the change of the parameter y. hat to ay, n th cae, the utlzaton of the hell hgher than the utlzaton of the rb. Materal utlzaton wll be the new crtera and method of tructural degn. hrough the procedure, the

7 Re. J. Appl. Sc. Eng. echnol., 6(5): , 03 utlzaton of the hell and rb n other cae can be got. How to meaure materal utlzaton and t uage n tructure degn are the key content of the author reearch n the future. CONCLUSION A mple procedure to create a famly of tffened cylndrcal hell, whch the revoluton of contant ma, ha been hown. A lot of data pont about bucklng load and materal hare of the hell and rb are got form bucklng analy ung ANSYS code. A concluon can be drawn from the fttng functon: n the cae of unform external preure load tffened cylndrcal hell, the utlzaton of the hell are hgher than the rb. Materal utlzaton n tructural ytem ha pecal gnfcance to degn reaonable tructure and enure the afety of the overall tructure. In the future tudy, the crtera of materal utlzaton n tructural degn wll be tuded baed on the reult of th tudy. h reearch project nnovatve n t approach of tffened cylndrcal hell degn. It expected that th approach wll make mmene contrbuton to the tructural degn. ACKNOWLEDGMEN he author would lke to expre ther grattude and ncere apprecaton to the anonymou revewer for contructve uggeton to mprove the qualty and readablty of th tudy. h tudy wa conducted wth upported of the Defene Pre-reearch Foundaton of Shpbuldng Indutry (No. J.3.). REFERENCES Bagher, M., A.A. Jafar and M. Sadeghfar, 0. Mult-objectve optmzaton of rng tffened cylndrcal hell ung a genetc algorthm. J. Sound Vb., 330: Chen, J., 00. Stablty heory and Degn of Steel Structure. Scence Pre, Bejng, Chna, pp: -0. Feng-Me, Y., R. Shu-Y and Z. Rong-X, 0. Emprcal comparon term tructure of nteret rate model baed on polynomal plne functon wth penalty term. Syt. Eng. heory Pract., 3(4): Gao-Feng, S., Z. A-Feng and W. Zheng-Quan, 00. Optmum degn of cylndrcal hell under external hydrotatc preure. J. Shp Mech., 4(): Jarma, K., J.A. Snyman and J. Farka, 006. Mnmum cot degn of a welded orthogonally tffened cylndrcal hell. Comput. Struct., 84: Lang, B. and J. Yue, 00. Optmum degn of cylndrcal hell on tablty. J. Mech. Strength, 4(3): N, H., Z. L and H. Song, 00. Movng leat quare curve and urface fttng wth nterpolaton condton. Proceedng of ICCASM Internatonal Conference on Computer Applcaton and Sytem Modelng. ayuan, pp: Sryu, V., 0. Mnmzaton of the weght of rbbed cylndrcal hell made of a vcoelatc compote. Mech. Comput. Mater., 46(6): Xao-Hong, S., Z. Gang and F. Zhao-Hua, 005. A homogeneou hgh prece drect ntegraton baed on legendre polynomal ere. Chnee J. Comput. Mech., (3): Young-Shn, L., 009. Revew on the cylndrcal hell reearch. J. Mech. Sc. echnol., 33: