THE DEFORMATION OF CYLINDRICAL SHELLS SUBJECTED TO RADIAL LOADS

Transcription

1 Proceedngs of PACAM XI Copyrght 009 by ABCM 11th Pan-Amercan Congress of Appled Mechancs January 04-08, 010, Foz do Iguaçu, PR, Brazl THE DEFORMATION OF CYLINDRICAL SHELLS SUBJECTED TO RADIAL LOADS Madurera, Luísa R, Departamento de Engenhara Mecânca e Gestão Industral, Faculdade de Engenhara da Unversdade do Porto R. Dr Roberto Fras Porto, Portugal Fonseca, Elza M. M Departamento de Engenhara Mecânca, Insttuto Poltécnco de Bragança, Portugal Melo, F. J. Querós Departamento de Engenhara Mecânca, Unversdade de Avero, Campus de Santago, 3810 Avero Portugal Abstract. Cylndrcal shells have a smple geometry and applcaton n pressure vessels and ppng engneerng. The development of calculaton algorthms n structural project s mpelled by a constant challenge n the search of more accurate and fast desgn tools n engneerng. The objectve of ths work s to contrbute wth a smple and relable numercal tool for the stress analyss of cylndrcal vessels subjected to generalzed forces. A hybrd formulaton n the defnton of forces and dsplacements s proposed for cylndrcal shells subjected to radal loads. Varatonal technques coupled wth functonal analyss are used to obtan an optmzed soluton for the shell dsplacement and further stress feld evaluaton. As t s not possble to obtan exact solutons for the dsplacements or deformaton feld whenever the external loads are ether concentrate or locally dstrbuted, the soluton here proposed deals wth the combnaton of unknown analytc functons combned wth Fourer expansons, where the former depend on the axal shell coordnate and the trgonometrc terms are dependent upon the cylnder crcumferental polar angle. These functons are expanded n Fourer seres where dsplacement ampltudes are combned wth trgonometrc terms. The result s a system of ordnary dfferental equatons where the soluton s analytc after evaluaton of egenvalues and egenvectors. The boundary condtons are then used to reach the fnal soluton. As an example a large cylndrcal shell subjected to pnchng loads s consdered. The results for the radal dsplacement and secton ovalzaton are analyzed where the soluton was obtaned wth three terms (n θ =6 for the accuracy s acceptable n ths case. The transverse dsplacement presents mportant dependence on the shell thckness vs radus, as the shell can be a thn-walled one (ths case s ncluded n the presented example up to a moderately thck one, where the surface dsplacement ranges untl the extreme edges, whch s not the case analyzed. The proposed method leads to accurate results wth a relatvely low complexty nput data. For conclusons of ths work t s remarked that the defntons of the load system and boundary condtons are easly processed as the method has pre-defned possbltes for each load case or edge boundary condtons. An analytc soluton s obtaned and a low number of terms n the Fourer seres show good accuracy. A comparson wth fnte element methods s presented. Keywords: ppng engneerng, Fourer seres, system of dfferental equatons, boundary condtons. 1. INTRODUCTION The dsplacement feld s defned as follows: u( x, n = θ θ b ( xcos( θ = n = n = θ a v x, θ ( xsn( θ ( (1 w( x, θ = θ a ( xcos( θ = where u, v and w are respectvely, the axal, the merdonal and the transverse shell dsplacements. The cylndrcal shell deformatons clude axal, crcumferental and shear membrane terms ε xx, ε θθ, γ xθ, also as bendng curvatures and twst. k xx k θθ and k xθ. These terms together wth nternal forces N xx N xθ, N θθ complete the hybrd soluton unknowns.

2 Proceedngs of PACAM XI Copyrght 009 by ABCM 11th Pan-Amercan Congress of Appled Mechancs January 04-08, 010, Foz do Iguaçu, PR, Brazl The deformaton energy U s a functonal U(u, v, w, N xx N xθ, N θθ for whch the statonarty s obtaned performng a weak formulaton wth Fourer seres ampltude parameter varatons. The result s a system of ordnary dfferental equatons where the soluton s analytc after evaluaton of egenvalues and egenvectors. The boundary condtons are then used to reach the fnal soluton. As an example a large cylndrcal shell subjected to pnchng loads s consdered. The results for the radal dsplacement and secton ovalzaton are as depcted n the Fgures n secton where the soluton was obtaned wth sx terms (n θ =6 for the accuracy s acceptable n ths case. The transverse dsplacement w presents mportant dependence on the shell thckness vs radus, as the shell can be a thnwalled one (ths case s ncluded n the presented example up to a moderately thck one, where the surface dsplacement ranges untl the extreme edges, whch s not the case analyzed. In the expansons n Eq. (1 the functons a, b are the ampltudes for the ovalzaton and warpng dsplacements. These terms are ncluded n the Fourer θ-expansons, but they are only x-dependent functons. In ths problem the terms used for evaluaton of the total energy are, followng Mllard and Roche (1984: 1 D(1 υ U= Aε xx + Sγ xθ + Dk xx + Dk θθ + k xθ dω = ( 1 a = [ ( + ( 1 = A b cos θ S b ( sn θ + D( + 1 ( a cos θ +,3,4,5,6,7 r r 4 1 ( + D(1 υ + a sn θ + Da cos θ ] dω r (3 where d Ω = rdθdx and k xx = ( a cos θ (the shell axal curvature 1 1 k xθ = + r a sn θ (the shell surface twst ( + 1 a θ 1 kθθ = cos (the crcumferental curvature (4 r ε xx = ( b cos θ (the axal membrane stran ε θθ = 0 (the crcumferental membrane stran; s zero as the ppe s crcumferentally nextensble a γ x θ = b sn θ (the membrane shear stran r The statonarty condton for U, (correspondng to a mnmum, as n fact t s a natural varatonal prncple, s obtaned wth separate varatons of functonal U, for each of the unknowns a, b (Mllard and Roche, U δb b U δa a U + δb b U + δa a = 0 = 0 (5 The soluton s now wrtten wth totally assumed dsplacements and ts dervatves. Followng ths procedure, a system of dfferental equatons s obtaned. The dfferental equatons for the straght ppe, assumng that the deformaton s caused by a force F appled on the surface of the ppe are descrbed n Eq. (6 and solved by evaluaton of egenvalues and egenvectors (Wyle and Barret, 1995: D v ( 1 π rs D rda π 1 π (1 ν π = a + πsb + + a + F r + 3 r (6.1

3 Proceedngs of PACAM XI Copyrght 009 by ABCM 11th Pan-Amercan Congress of Appled Mechancs January 04-08, 010, Foz do Iguaçu, PR, Brazl S ra b π π = b + πsa r (6.. EXAMPLE A pnched cylnder wth rgd end daphragms s presented. The cylnder s subjected to dametrcal load as shown n Fg 1. Ths s a benchmark case study frequently analyzed by several authors. The example of a pnched cylnder wth the present methodology s compared wth the work by Crak, Ortz and Schroder (000 and Fonseca (007. Smultaneously, numercal smulaton usng ANSYS fnte element program wll be presented. The materal propertes and the geometrc parameters are the followng (unts are accordng to the references for comparson: 6 E = 3 10 ps ν = 0. 3 r=300 nch L=600 nch h=3 nch In ths example lengths are n nches and load n pounds as publshed by Fonseca (007 and Crak, Ortz and Schroder (000. Fgure 1 shows the case study ncludng all consdered parameters. Geometry L=600 nch R=300 nch H=3 nch Materal propertes E= ν =0.3 Concentrate forces F=1.0 lb Fg.1 - Geometry of a two-radal loads pnched cylnder.1 The varatonal method applcaton Equaton (7 represents the load submtted for ths type of analyss and ntroduced n dfferental Eq. (6. 59 F 86 nπ 314 nπ nπx F(x = cos + cos sn n= 1 nπ (7 Fgure shows the load consdered on the surface as a Fourer expanson where F=-0.5pound Fg. Appled load

4 Proceedngs of PACAM XI Copyrght 009 by ABCM 11th Pan-Amercan Congress of Appled Mechancs January 04-08, 010, Foz do Iguaçu, PR, Brazl Fg.3 Separate solutons ocorrespondng to each ovalzaton mode a x, a ( x,..., a ( ( 3 7 x Superposng the sx solutons gves the optmal soluton for the deformaton, the vertcal dsplacement along length of cylnder obtaned wth ths hybrd formulaton s shown n Fg. 4. nch E-05 nch Fg.4 - a x + a ( x a ( for ppe deformaton ( 3 7 x The next graph shows that these results match the ones obtaned wth fnte element methods by Fonseca (007, Crak et al ( E+00 Length E-06 Vertcal dsplacement -8.0E-06-1.E E E-05 Rng element (8elem Rng element (4elem Shell8 element Fg.5 Radal dsplacement wth fnte element formulaton (Rng and COSMOS/M (shell, (Fonseca, 007

5 Proceedngs of PACAM XI Copyrght 009 by ABCM 11th Pan-Amercan Congress of Appled Mechancs January 04-08, 010, Foz do Iguaçu, PR, Brazl Fg.6 Ppe ovalzaton at md length resultng from dametrcally opposed par of forces.. The fnte element method applcaton The pnched cylnder problem tests the ablty of smulaton to model complex bendng and membrane states. Researchers use the pnched cylnder problem as a benchmark test to assess the performance of curved beam or shells elements as dd Fonseca (007 and Crak et al (000. Results of the fnte element method usng ANSYS program are shown n the next Fgures to compare the obtaned soluton wth the functonal analyss. In ANSYS program a fnte Shell93 element was used, wth 8 nodes and 6 degrees of freedom per each node: translatons n the nodal x, y, and z drectons and rotatons about the nodal x, y, and z-axes. Shell93 s partcularly well suted to model curved shells. The deformaton shapes are quadratc n both n-plane drectons. Dfferent analyses are presented. Due to the geometry and load symmetry a one-eghth of the cylnder s modelled and all complete cylnder, beng the fnte element mesh used n Fg. 7. For the one-eghth symmetry model, one-fourth of the load s appled. (a (b Fg. 7 Fnte element mesh used for one-eghth (a and total cylnder length (b. Fgure 8 shows the postprocessor obtaned for the vertcal dsplacement results usng ANSYS program. Uymax=-0.18E-04 [n] (a Uymax=-0.180E-04 [n] (b Fg. 8 Vertcal dsplacement for eghth (a and total cylnder length (b. The reference dsplacement, as reported by Dvornk and Bathe (1984 and Kouha (000 and the exact deflecton at the load pont are equal (0.1848E-4[n], usng wexact = 164.4( F Eh. The same value was obtaned wth ANSYS program and the varatonal method presented.

6 Proceedngs of PACAM XI Copyrght 009 by ABCM 11th Pan-Amercan Congress of Appled Mechancs January 04-08, 010, Foz do Iguaçu, PR, Brazl Ths s known as a benchmark result where there s a domnant bendng behavour n the thn shell lmt. The vertcal dsplacement along length of cylnder obtaned wth two dfferent meshes was compared n the followng Fg. 9, smlar to the results obtaned by Fonseca (007 n Fg. 5..0E-0 (nch -4.0E-06 Length (nch Vertcal dsplacement -8.0E-06-1.E E E-05 Shell93 [1/8] Shell93 [all] Fg. 9 Vertcal dsplacement along the cylnder length (ANSYS soluton 3. CONCLUSIONS The presented numercal tools have shown a good agreement altogether when analyzng a recurrent problem n ppng engneerng, the structural behavour of thn walled cylndrcal ppes subjected to radal loads. The varatonal soluton based on functonal analyss, though ntally demandng some work wth closed form analytc methods, has the advantage of leadng to a qute smple computatonal soluton by computer mplementaton. Moreover, the results match consstently wth correspondng work run wth fnte element technques.and referenced authors. The soluton accuracy can be mproved by easly expandng the number of trgonometrc elements consdered n the Fourer expansons ncluded n the presented soluton. 4. REFERENCES Crak, F; Ortz, M and Schroder, P, Subdvson surfaces: a paradgm for thn shell fnte element analyss, Int Journ for Numercal Methods n Engneerng, (000; 47: Dvorkn, E., Bathe, K.J. (1984. A contnuum mechancs based four-node shell element for general nonlnear analyss. Engneerng Computaton, 1, Fonseca, Elza M.M., F.J.M.Q. de Melo and C.A.M. Olvera, Trgonometrc functon used to formulate a mult-nodal fnte tubular element, Mechancs Research Communcatons, Volume 34:1 (007 pp 54-6 Kouha, R., Stenberg, R. (000. A smple lnear nonconformng shell element. IASS-IACM 000, Fourth Internatonal Colloquum on Computatonal of Shell Spatal Structures, Greece. Mllard, A., Roche, R. "Elementary Solutons for the Propagaton of Ovalzaton along Straght Ppes and Elbows", Int. Journ. of Press. Vessels & Ppng, Volume 16:1 (1984 pp Wyle C and Barret L, Advanced Engneerng Mathematcs 6 th ed, McGraw-Hll ( RESPONSIBILITY NOTICE The authors are the only responsble for the prnted materal ncluded n ths paper. 6. AKNOWLEDGEMENTS The authors wsh to thank Fundação para a Cênca & Tecnologa the support for ths research.