Ielastic spherical caps with defects JAAN LLLP RNST TUNGL Istitute of athematics Uiversity of Tartu Liivi str Tartu 549 STONIA aalellep@utee Tartu College Talli Uiversity of Techology Puiestee 78 58 Tartu STONIA ersttugel@ttuee Abstract : Limit aalysis miimum weight desig of stepped spherical shells is studied The caps have piece wise costat thicess are subected to the uiform exteral pressure The shells are made of a ielastic material obeyig a approximatio of the Tresca yield surface The aim of the paper is to develop a procedure for miimum weight desig for give limit load Necessary optimality coditios are derived with the aid of variatioal methods of the theory of optimal cotrol Numerical results are preseted for a simply supported spherical cap Keywords: optimizatio spherical shell ielastic material crac Itroductio Various approaches to the limit aalysis of axisymmetric plates shells solutios of problems of load carryig capacity of spherical caps ca be fid i boos by [] [4]; [] Ielastic spherical cap with a cetral hole was studied by [7] assumig that the thicess was piece wise costat the material obeyed geeralised square yield coditio iimum weight desigs for shallow shells are obtaied by [5] The shell uder cosideratio is pierced with a cetral hole it is subected to the iitial impact loadig Optimal desigs of shells of piece wise costat thicess are established uder the coditio that the maximal residual deflectio attais the miimum value for give total weight Spherical shells of ises material were studied by [8] whereas coical shells were cosidered by [6] I [8] a optimizatio procedure is developed for spherical shells of piece wise costat thicess made of a ielastic material obeyig the ises material associated flow law The desigs of spherical shells correspodig to maximal load carryig capacity are established for give material volume or weight of the shell I the preset paper stepped spherical caps with cracs at re-etrat corers of steps are cosidered maig use of a approximatio of the Tresca yield coditio The aim of the paper is to establish miimum weight desigs of the shell for give load carryig capacity correspodig author ISBN: 978--684-8- 88
Formulatio of the problem Let us cosider a spherical cap of radius A simply supported at the edge with cetral agle β (Fig ) The shell is subected to the uiform exteral pressure of itesity P The pressure loadig is assumed to be quasistatic iertial effects will be eglected Let the thicess of the shell be piece wise costat eg h h for ; ) where ( β Thicesses h ( ) agles ( ) will be treated as desig parameters to be defied so that a cost fuctio attais its miimal value It is wellow that sharp corers i structures geerate stress cocetratio which etails cracs It is assumed herei that at ( ) circular cracs are located If the problem is to maximize the ultimate load to be sustaied by the cap the the quatity V is cosidered as a give costat Basic equatios I the case of rotatioal symmetry the equilibrium equatios of a shell elemet ca be preseted as (see [] [7] [8]) ( N ( N ( I () si)' N N PA)si ( S si)' si)' N cos S si cos AS si () N st for membrae forces for bedig momets i the two pricipal directios respectively S is the shear force Here heceforth prims deote the differetiatio with respect to We shall use a simple approximatio of the exact yield surface is obtaied assumig that the stress state of the shell correspods to the ridge N ± ( ( ) ) () N Fig : Geometry of the shell We are looig for the miimum weight desig of the spherical cap for the fixed limit load The other problem we are dealig with cosists i the maximizatio of the limit load for give material cosumptio Sice the middle surface of the cap is a sphere of radius A the cost criterio for the problem of miimum weight ca be preseted as V h (cos cos ) () of the exact yield surface ([] [4]) whereas bedig momets satisfy the yield coditio (hexago) o the plae of momets Here N st for the limit momet limit force for a portio of the shell with thicess h eg σ h / 4 N σ h σ (4) beig the yield stress of the material I the limit aalysis as well as dyamic plasticity of axisymmetric shells it is usual that the bedig momet attais its limit values Thus it is reasoable to expect that where V V/ πa volume of the shell V beig the material (5) ISBN: 978--684-8- 89
accordig to () N for [ ] ; The latter meas that N throughout the shell Let h be the thicess of a referece shell of costat thicess N * - yield force yield momet for the referece shell It seems to be reasoable to itroduce followig o-dimesioal quatities h γ h AN N N PA p N m S s N (6) Variables (6) admit to preset the equilibrium equatios () with (5) as ' cot s s' scot p m ' m cot γ cot s / ; for [ ] Boudary coditios for equatios (7) are m () m () () γ () s() m ( β ) (7) (8) I the case of simply supported spherical caps it is expected that the optimal shape of the shell is such that > γ γ (9) for each Sice we are looig for statically admissible solutio of the problem we have to chec if m ( ) γ () for ) ; if ( It ca be show that the momet m membrae force are mootoic fuctios of the agle Thus the admissible values of m are exceeded at boudary poits itervals [ ] ; if ay of This meas that we have to chec the admissibility of stress compoets at ( ) We must bear i mid that the sectios of the cap are weaeed by cracs of depth c These sectios are able to sustai bedig momets with maximal value ( ) m σ h c at Similarly the 4 maximal admissible value of the membrae force N is N m σ ( h c ) for Therefore the costraits () () ca be replaced by equalities m ( ) ν γ () ( ) ν γ () provided (9) holds good I () () st for so-called slac variables c ν (4) h for The problem posed above will be treated as a problem of the theory of optimal cotrol (see Bryso []) 4 Optimality coditios I order to derive ecessary coditios of optimality for the problem with cost fuctio () state equatios (7) with boudary coditios (8) costraits ()- () we compile a exteded fuctioal (see [] [5] [6] [7] [8]) ( ) γ () ISBN: 978--684-8- 9
J [ ( ' ( s' ( m ' m { γ (cos cos cot s) s cot p) cot { ( m ( ) ν γ ) ( ( ) ν γ ) γ cot)] d} )} ( m () γ ) () s() m ( β ) s (5) I (5) st for co-state (cougate) variables ( ); are uow Lagrage multipliers It is worthwhile to emphasize that the co-state variables are certai fuctios of whereas Lagrage multipliers are treated as uow costats Whe calculatig the total variatio of the exteded fuctioal (5) oe has to tae ito accout the distictios betwee ordiary (wea) variatios of state variables total variatios at boudary poits of itervals ; ) We call z( ± ) the total variatio ( of a variable z at δ z( ± ) the value of the ordiary variatio δ z at It is ow that (see [] [6] [7] [8]) z( ± ) δz( ± ) z'( ± ) It meas that at the boudary poits of itervals ; ) oe has ( ( ) ( ) m ( ) m ( ) s( ) s( ) Note that whe deducig the last relatios the cotiuity of state variables s m e g ( ) ( ) s ( ) s ( ) m ( ) m ( ) is tae ito accout Guidig by the cosideratios give above it ca be recheced that the equatio J yields the co-state system ' cot ' cot ' cot (6) also equatios for determiatio of parameters cos γ for cos cotd ν γ ν (7) cos cos γ γ cotd for (8) (9) for The trasversality coditios at boudary poits are ( β ) ( β ) ( β ) () ( ) () ( () ) whereas ump coditios for co-state variables have the form ( ) ( ) ± ( ) ( ) ( ) ( ) ± where Fially variatio of (5) yields () ISBN: 978--684-8- 9
si ( γ γ ( ) s' ( ) ( ) m '( ) ( ) '( ) ( ) s' ( ) ( ) m '( ) ) ( ) '( ) () for each aig use of (7) () the equatios () ca be put ito the form si ( γ γ ) ± ( s( ) ( )cot ) s( ) [ ( )]( m ( ) cot ) cot ( γ ( ) γ ( )) (4) for I (4) square bracets deote fiite umps [ ( )] ( ) ( ) It is easy to rechec that the geeral solutio of the system (6) is s m at coditios (8) maig use of boudary 5 Numerical results discussio The detailed aalysis shows that fially we obtai equatios for determiatio of arbitrary costats A B C ( ) m C ( A si B cos )si ( A cos B si)si C si (5) Fig : Bedig momet m for ; ); ( I order to solve the posed problem up to the ed oe has to itegrate (7) maig use of (8) to solve equatios (7)-(4) maig use of (5) It appears that equatios (7) ca be itegrated i each regio ; ) ; The ( result is p cot D cot p s (cot ) D cot p F m γ ( cot D cot ) si (6) for ( ) ; where D F are arbitrary costats Uow costats D F i (6) ca be defied from the cotiuity of state variables Fig : embrae force This set of equatios is solved umerically Results of calculatios are preseted i Fig Fig Table for the spherical cap with sigle step I Fig the distributios of the bedig momet m the membrae force are preseted Solid lies i Fig correspod to the optimized stepped shell whereas dashed lies are associated with the referece shell of costat thicess It ca be see from Fig ISBN: 978--684-8- 9
that bedig momet m mootoically decreases from its limit value γ at the pole util zero at the supported edge It is somewhat surprisig that the distributios of the bedig m correspodig to the optimized momet shell to the referece shell of costat thicess respectively are quite close to each other Calculatios showed that the stress state of the shell is statically admissible I Table optimal values of parameters γ are preseted for differet values of the itesity of the exteral pressure Here V sts for the optimal value of the material volume whereas V is the material volume of the referece shell of costat thicess P is the collapse pressure of the shell of costat thicess Table correspods to shells with ; ν 9 Note that the material volume of the referece shell with thicess h * ca be expressed as V (cos h* β ) For the sae of simplicity i the Table are accommodated the data correspodig to the case whe h * h Table : Optimal desig for β P γ e V /V 99P 979 95 98 98P 956 78 9687 97P 9 44 9557 96P 9 47 948 95P 87 455 98 9P 666 675 97 Calculatios carried out showed that the efficiecy of the desig e V /V depeds o the shell parameters β o the loadig p o the crac legth ν For istace if β ; ν 9; P9P oe has 666 γ 675 e 97 It meas that i this case the material savig is 99% 6 Cocludig remars The behaviour of ielastic spherical caps uder uiformly distributed exteral pressure loadig was studied The material of the cap obeys a approximatio of the Tresca yield surface i the space of membrae forces bedig momets Necessary optimality coditios for the posed problem are derived with the aid of variatioal methods of the theory of optimal cotrol Numerical results are preseted for a simply supported cap with uique step I the study it was assumed that the stepped caps had circular cracs at re-etrat corers of steps It was established that the depth of a crac had relatively wea ifluece o the optimal desig of the shell Acowledgemets: The partial support from the target fiaced proect SF 88S8 odels of applied mathematics mechaics from the Grat of stoia Sciece Foudatio TF 9 Optimizatio of structural elemets is acowledged Refereces [] A Bryso Ho Yu-Chi Applied Optimal Cotrol Wiley New Yor975 [] J Charabarty Applied Plasticity Spriger Berli New Yor [] PG Hodge Plastic Aalysis of Structures Krieger New Yor 98 [4] PG Hodge Limit Aalysis of Rotatioally Symmetric Plates Shells Pretice Hall New Jersey 96 [5] J Lellep H Hei Optimizatio of clamped plastic shallow shells subected to iitial impulsive loadig g Optim Vol 4 No 5 pp 545-556 [6] J Lellep Puma Optimizatio of coical shells of ises material Struct ultidisc Optim Vol No pp 49-56 [7] J Lellep Tugel Optimizatio of plastic spherical shells with a cetral hole Struct ultidisc Optim Vol No pp - 4 [8] J Lellep Tugel Optimizatio of plastic spherical shells of vo ises material Struct ultidisc Optim Vol No 5 5 pp 8-87 ISBN: 978--684-8- 9