Geometrically Non-linear Analysis of Axisymmetric Plates and Shells

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1 Iteratioal Joural of Sciece & echolog olume, No, 33-4, 7 Geometricall No-liear Aalsis of Aismmetric Plates ad Shells Cegiz POLA ad Zülfü Çıar ULUCAN Fırat Uiersit, Facult of Egieerig, Departmet of Ciil Egieerig, 379, Elazig, URKIYE (Receied: 7..7; Accepted: 9.3.7) Abstract: A geometricall oliear formulatio is preseted for the aismmetric plate ad shell structures. he formulatio is based o the total Lagragia approach. he oliear equilibrium equatios are soled usig the Newto-Raphso method. Differet umerical eamples are performed to obtai the geometricall oliear behaiour of aismmetric plates ad shells. Kewords: Aismmetric plate ad shell, geometric oliearit, Newto-Raphso method. Eseel Simetri Pla e Kabuları Geometri Baımda Lieer Olmaa Aalizi Özet: Pla e abu apılar içi geometri baımda lieer olmaa bir formülaso erilmetedir. Formülasoda toplam Lagrage alaşımı esas alımatadır. Lieer olmaa dege delemleri Newto- Raphso metodu ullaılara çözülmetedir. Eseel simetri pla e abuları geometri baımda lieer olmaa daraışıı elde etme içi değişi ümeri öreler gerçeleştirilmiştir. Aahtar elimeler: Eseel simetri pla e abu, geometri baımda lieer olmaa daraış, Newto-Raphso metodu.. Itroductio I the liear aalsis, the displacemets ad strais deeloped i the structure are small. hat is, the geometr of the structure assumed remais uchaged durig the loadig process ad liear strai approimatios ca be used. Howeer, the geometr of the structure chages cotiuousl durig the loadig process, ad this fact is tae ito accout i the geometricall oliear aalsis. I the liear aalsis, the load carrig capacit of the structure caot be predicted correctl. herefore, it is ecessar to use the o-liear equilibrium equatios to describe the structural behaiour [,]. I this paper, the geometricall oliear behaior of aismmetric plates ad shells subjected to differet aismmetric loads for arious shell parameters is studied. he total Lagragia approach, i which all static ad iematic ariables at the curret state are referred to the iitial cofiguratio, is adopted. he odal displacemets are oliear fuctios of odal rotatios i the aismmetric shell formulatio, thus restrictios of small odal rotatios durig deformatio process is remoed. I the umerical applicatio, the isoparametric aismmetric shell elemets are used. For such elemets the elemet geometr is described i terms of the coordiates of middle surface odes ad the mid-surface odal poit ormals. he icremetal equatios of equilibrium are soled b usig the Newto-Raphso method. he oliear aalsis of aismmetric plates ad shells is performed b a computer program which is writte b the authors i MALAB software.. Formulatio.. Kiematics of Deformatio For the isoparametric aismmetric shell elemets, mid-surface is defied b ol two coordiates ξ ad. Figure shows the odal ariables ad the defiitios of agles for a

2 C. Polat ad Z. Ç. Uluca tpical ode i the elemet. he elemet geometr is defied b N + N t cos ϕ () N + N t si ϕ () u N u + N t F (3) ad [3] N + N t F (4) F is defied b the followig epressio where t is the elemet thicess at the ode, ad is the umber of odes i the elemet. For the fiite elemet model, displacemet field is approimated b usig odal displacemets u ad, odal rotatio α, ad the shape fuctio N at the ode as follows; F F F cosϕ (cosα si ϕ (cosα ) si ϕ si α ) + cosϕ si α (5) ( ξ, ) p( ξ, ) α u( ξ, ) u ϕ ξ Figure. Defiitios of agles ad local aes of aismmetric shell elemet.. Noliear Fiite Elemet Formulatio I the total Lagragia approach, secod Piola-Kirchhoff stresses ad correspodig Gree-Lagrage strais are cosidered [4]. It is assumed that shell thicess does ot chage durig deformatio that is, σ ad ε ca be eglected. For a liear elastic material, stresses ad strais i the local coordiate sstem ( z) ca be writte as σ D ε (6) [ σ σ τ ] σ (7) σz [ ε ε γ ] ε (8) εz where D is the elasticit matri i the local coordiate sstem ad it ca be obtaied usig these two assumptios σ ad ε. Similarl i the global coordiate sstem ( z) stresses ad strais are defied as σ D ε (9) [ σ σ τ σ ] σ () z 34 [ ε ε γ ε ] ε () z

3 Geometricall No-liear Aalsis of Aismmetric Plates ad Shells where D is the elasticit matri i the global coordiate sstem ad ca be defied b D () D where is the trasformatio matri which is defied as c s sc s c sc (3) sc sc c s where c cos(a) ad s si(a) ; a is the agle betwee the global ais ad the local ais i the udeformed cofiguratio. Gree-Lagrage strais i the global coordiate sstem ca be writte as ε ε u + ε L u + + u [( u ) + ( ) ] [( u ) + ( ) ] [ u u + ] u (4) where ε ad ε L are the liear ad oliear strais, respectiel. If we defie [ u u u / ] θ (5) thus the liear ad oliear strais of Eq. (4) ca be writte coeietl as ε θ H θ (6) u u εl θ A θ (7) u u u / aig the ariatio of the liear strai dε H dθ (8) ad let d θ G du (9) the, we ca obtai the liear straidisplacemet matri B as ε HG du B du ; B HG () d where the ariatio of the odal parameters du is gie b du [ du d dα K du d α ] d,,.. () aig the ariatio of the oliear strai usig Eq. (7) dε L da θ + Adθ () ad we hae da θ A dθ (3) the dε L A dθ (4) If Eq. (4) is rewritte usig Eq. (9), the oliear strai-displacemet matri B L ca be obtaied as dε L A G du BL du ; B L A G (5) hus usig Eq. () ad Eq. (5), we ca write 35

4 C. Polat ad Z. Ç. Uluca [ H A] G du dε B du ( B + B ) du (6) L + Hece, the Eq. (3) ca be gie as where B is called as the strai-displacemet matri [5,6]. he equilibrium equatio of the oliear sstem ca be writte as (i) R( u) B σ d P F λ P (7) where R, F ad P represet the out-of-balace force ector, the iteral force ector ad the eterall applied load ector respectiel, ad (i) λ is the load-leel parameter. aig the ariatio of Eq. (7) we hae db σ d + B dσ d dr K du (8) ad from Eq. (9) we ca write d σ Ddε DBdu (9) the Eq. (8) becomes dr db σ d + K du (3) he matri B is defied b the Eq. (6), ad the matri K is defied as K B DB d K + K L (3) Substitutig B ad B L matrices ito the first part i the right had of the Eq. (3) istead of B, that part ca be writte as or db σ d (dg H K du (db + dg A + db L ) σ d + G da ) σd db σ d ( K σ + K σ3 + K σ)du σ (3) (33) 36 dr K du + K du ( K + K ) du K du σ σ (34) where K is the taget stiffess matri ad the eplicit defiitio of this matri ca be foud i the thesis of Polat [7]..3. Solutio Method I order to compute the odal displacemets, it is ecessar to sole the sstem of oliear equilibrium equatios usig a icremetal/iteratie method. he load cotrolled Newto Raphso method is the earliest method that is used to trace the equilibrium path [8]. his method is based o the liearizatio of the equilibrium equatios at a (i) prescribed load leel, that is, λ i Eq.(7) is ept costat durig iteratios. he iteratios are performed util the residual is smaller tha a prescribed tolerace. his method ca trace the load displacemet cure before the occurrece of a limit poit, but geerall it will fail to coerge beod this poit. We adopted two coergece criterios together i the algorithm. Oe of them is the displacemet-based coergece criterio ad it ca be writte as u < u (35) βu where u are the iteratie displacemet chages, u the total displacemets ad β u prescribed tolerace. he orm of the iteratie displacemet chage ca be er small while the out-of-balace force orm is er large. o use ol this criterio ma result i some errors. herefore, we used a additioal coergece criterio ow as eerg-based coergece criterio of the form [9] (i) u R < β u ( λ P) (36) r

5 Geometricall No-liear Aalsis of Aismmetric Plates ad Shells.4. Numerical Eamples.4.. Geometric o-liear aalsis of a clamped circular plate subjected to a uiforml distributed load: he large deformatio aalsis of a clamped circular plate subjected to a uiforml distributed load q with Poisso s ratio ν. 3, Youg s modulus 7 E MPa, radius R mm, ad thicess t mm, is cosidered i this eample. Liear (L), quadratic (L3) ad cubic (L4) elemets are used to model the circular plate (Fig.). he elemet matrices ad the load ectors are formed usig Gauss quadrature., 3 ad 4 itegratio rules are emploed for the liear, quadratic ad cubic elemets, respectiel. able I shows the ceter deflectio of the plate for differet elemets ad the aaltical solutio. I additio, i the Figure 3, the L3 elemet ad the aaltical solutio results are compared. he results of the higher order elemets are er close to the aaltical solutio. ( a) ( b) (c) Figure. Meshes of a circular plate uder uiform loadig; a) eight liear elemet, b) four quadratic elemet, c) three cubic elemet. able I. No-dimesioal cetral deflectio (/t) of the circular plate with clamped boudaries uder uiform load. 4 4 qr / Et L L3 L4 Aaltical [] Figure 3. Cetral deflectio of the circular plate. 37

6 C. Polat ad Z. Ç. Uluca.4.. Geometric o-liear aalsis of a clamped circular plate subjected to a cocetrated load at the cetre of the plate: he circular plate with the same geometric, material propert ad the same meshes as the oe i Sectio.4. is subjected to a cocetrated load at the cetre. he odimesioal cetral deflectio of the circular plate from the preset elemets b total Lagragia formulatio is compared with the aaltical solutio b Ref. [] i able II. he o-liear relatioships betwee load ad deflectio are also show i Figure 4. Also, the solutio of the higher order elemets is er close to the aaltical solutio for this problem. able II. No-dimesioal cetral deflectio (/t) of the circular plate with clamped boudaries uder cocetrated load. Yü ( 4 N) L L3 L4 Aaltical [] Figure 4. Cetral deflectio of the circular plate.4.3. Geometric o-liear aalsis of a clamped spherical cap uder uiform eteral pressure: I Figure 5, the material ad geometric properties of the spherical cap uder uiform eteral pressure p are gie. I the figure, λ is the shell parameter ad p the classical buclig pressure. Cubic shell elemets are used to model the clamped spherical cap. he elemet matrices ad the load ectors are formed usig Gauss quadrature with 4 itegratio rule. For differet alues of λ, dimesioless aismmetric sap-through pressures p / p of clamped spherical cap was iestigated. Results were show i Figure 6 with a referece solutio. he obtaied results are er close to the aaltical solutio. 38

7 Iteratioal Joural of Sciece & echolog olume, No, 33-4, 7 p H R λ [3 ( ν )] p / 4 E t R [3( ν )] / ( Η/ t ) / R mm E 5 MPa t mm ν / 3 Figure 5. Geometric ad material properties of the aismmetric shell uder uiform pressure. Figure 6. Compariso of preset alues of p/p o with those of Luo ad eg []. 3. Coclusio A geometricall oliear formulatio based o total Lagragia approach is gie for the aismmetric shell elemets. Geometricall oliear behaiour of aismmetric plates subjected to differet loads are iestigated. I additio, aismmetric sap-through pressures of clamped spherical caps also are studied uder uiform pressure. he obtaied results are similar to the aaltical solutio. Especiall, those of the higher order elemets gie more accurate results. Refereces Zieiewicz, O. C. (977). he Fiite-Elemet Method. 3rd editio, McGraw-Hill, Lodo. Parete, E., az., L. E. (3). O Ealuatio of Shape Sesitiities of No-Liear Critical Loads. Iteratioal Joural for Numerical Methods i Egieerig, 56(6), Suraa, K. S. (98). Geometricall Noliear Formulatio for the Aismmetric Shell Elemets. Iteratioal Joural for Numerical Methods i Egieerig, 8, Redd, J. N. (997). Mechaics of Lamiated Composite Plates ad Shells: heor ad Aalsis. d editio, CRC Press, New Yor. 5 Polat, C., Calaır, Y. (3). Sabit Dıs Basıca Maruz Eseel Simetri Kabuları Geometri

8 C. Polat ad Z. Ç. Uluca Olara Lieer Olmaa Aalizi. XIII. Ulusal Meai Kogresi, , Gaziatep. 6 Polat, C., Calair, Y., Uluca, Z. Ç. ( 6). Postbuclig Behaior of Geometricall No-liear Aismmetric Shells, Seeth Iteratioal Cogress o Adaces i Ciil Egieerig, October -3, Yildiz echical Uiersit, Istabul, ure 7 Polat, C. (6). Geometri Baımda Lieer Olmaa Kabu Yapıları Stati e Diami Daraışı, Dotora ezi. 8 Memo, B. A., X. Z. Su. (4). Arc-Legth echique for Noliear Fiite Elemet Aalsis. Joural of Zhejiag Uiersit, 5(5), Warre, J.E. (997). Noliear Stabilit Aalsis of Frame-pe Structures with Radom Geometric Imperfectios Usig a otal-lagragia Fiite Elemet Formulatio, Ph. D. hesis, Facult of the irgiia Poltechic Istitute ad State Uiersit, irgiia, p. Zhag, Y. X., Cheug, Y. K. (3). A Refied No-Liear No-Coformig riagular Plate/Shell Elemet, Iteratioal Joural for Numerical Methods i Egieerig, 56, Luo, Y. F., eg, J.G. (998). Stabilit Aalsis of Shells of Reolutio o Noliear Elastic Foudatios. Computers ad Structures, 69,