A SIMPLE CORRECTION TO THE FIRST ORDER SHEAR DEFORMATION SHELL FINITE ELEMENT FORMULATIONS

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1 A SIMPLE CORRECTION TO THE FIRST ORDER SHEAR DEFORMATION SHELL FINITE ELEMENT FORMULATIONS Romil Tano and Ala Tabiei Center of Ecellence in DYNAD Analsis Department of Aerospace Engineering and Engineering Mecanics Uniersit of Cincinnati, OH 45-7, USA ABSTRACT Te present ork concentrates on te deelopment of correct representation of te transerse sear strains and stresses in Mindlin tpe displacement based sell finite elements. Te formlation tilies te robst standard first order sear deformation sell finite element for implementation of te proposed representation of te transerse sear stresses and strains. In tis manner te need for te sear correction factor is eliminated. In addition, modification to an eisting sell finite element for te correct representation of transerse sear qantities is minimal. Some modifications to correct Mindlin tpe elements are presented in te literatre. Tese modifications correct te distribtion of te transerse sear stresses onl and se te constant transerse sear strains trog te tickness. As compared to te aboe, te present formlation ses te correct distribtion and is consistent for bot transerse sear stresses as ell as transerse sear strains. Keord: sell finite element, transerse sear stresses and strains, iger order sell teor, sear correction factor. INTRODUCTION One of te major disadantages of te first order sear deformation sell teories is tat altog te accont for te transerse sear te cannot correctl represent its trogtickness distribtion. Neerteless, teir abilit to accratel predict te oerall sell beaior and teir relatie simplicit makes tem te basis for most sell elements tilied in te finite element codes noadas. Te first order sell elements are sall capable of also prodcing good reslts for te in-plane strain and stress distribtion bt teir formlation reslts in constant transerse sear strains as opposed to te realistic parabolic distribtion. As a reslt te traction conditions at te sell srfaces are iolated. Te also reqire sear correction coefficients to correct te corresponding strain energ terms and tese coefficients are problem dependent and are not alas eas to determine. Nmeros efforts ae been made to oercome tis disadantage of te first order formlation most of ic reslt in a iger order sear deformation teor (e.g. see Panda and Kant, Redd,Ha 4, Noor et al. 5 ). An efficient remed for te transerse sear inconsistenc is implemented in te finite element codes ABAQUS and MSC/NASTRAN and described in Capter.6.8 of 6 and Capter 6.5 of 7. It is based on te stress and moment eqilibrim eqations and reslts in a parabolic trog tickness distribtion for te transerse sear stresses. Hoeer, in tis formlation te transerse sear strains are still constant trog te sell tickness and Gradate Res. Assistant Assist. Prof. and Director, ator to om correspondence sold be addressed 8-

2 8- terefore in its implementation te sear correction factors are still reqired in te strain energ terms. Te erein-presented approac for treating te transerse sear strains and stresses in omogeneos sells reslts in parabolic distribtion for bot strains and stresses and, terefore, it eliminates te need of an sear correction factors. It reqires onl minor canges in te first order sear deformation formlation and totall preseres its efficienc. It is applicable to an displacement-based formlation and etremel eas to implement in an first order sell finite element. THEORETICAL FORMULATION Te present formlation starts it te tird order displacement field, () ere and are te in-plane, and is te transerse displacement component. Here,, and are te reference srface linear displacements along te coordinate aes,, and respectiel. i, i, are te reference srface rotations, and i and i are te iger order terms in te displacement polnomial epansion. is te coordinate along te -ais normal to te sell reference srface. Using te strain-displacement relations i j j i ij () te components of te strain ector corresponding to te displacement field () are: (). Vanising of te transerse sear stresses at te top and bottom sell srfaces, ( ) ( ) ± ± σ σ, makes te corresponding strains tere ero, ic ields: 4 ; 4. (4)

3 8- Here is te sell tickness. Using tese relationsips te displacement field, Eq. (), and te strain epressions, Eq. (), simplif into: (5) and (6) 4 4. Let s no assme tat te first to terms in te,,and epressions in Eq. (6) represent te trog-tickness distribtion of te in-plane strains it enog accrac. Tis means tat e can neglect te contribtion of te deriaties of and it respect to and and simplif te strain relations as follo: (7) 4 4. Tese epressions are identical to te strain epressions from te first order sear deformation displacement field, (8)

4 ecept for te transerse strain epressions. Te are different from te sal relations,,, and reslt in parabolic trog-tickness distribtion for te transerse sear strains. Te strain epressions, Eq. (7), and teir corresponding displacement field, Eq. (8), define te present formlation, ic is actall a first order sear deformation sell formlation it corrected transerse sear. It is eident tat it reslts in a parabolic distribtion of te transerse sear strains and satisfies te ero transerse sear stresses reqirements at te sell srfaces. As seen, it also reqires insignificant modifications to be implemented in eisting displacement based first order sell elements. No, let s compare te aboe formlation it te approac in 6 and 7. Te latter approac is based on te eqilibrim eqations: τ σ M ; V. (9) Te in-plane normal stress, σ, can be epressed trog te bending moment M M σ M V σ ;. () I In te last epression it is assmed tat te sell tickness does not ar (or ar slol) it position along te sell. Note tat in te erein-presented formlation tis assmption is not reqired. Sbstitting te second eqation in () into te first relation in (9) and integrating ields 6V τ C. () V At ±, τ andc. Ten for te transerse sear stress e get V 4 τ. () Comparing tis epression it te last relation in (7) e see tat te are er similar. We V kno tat for a omogeneos sell gies te maimm ale of te transerse sear stress τ. Obiosl as similar meaning for te transerse sear strain. Terefore, it is epected tat bot approaces old gie te same reslts for te transerse sear stresses. Note tat if te transerse strains in 6 and 7 are calclated from te strain displacement relations, Eq. (), tere ill be an inconsistenc beteen te transerse strains and te transerse stresses, and te approac ill still reqire a sear correction factor in te strain energ epression. Tis is not te case it te present formlation. EXAMPLE PROBLEMS To illstrate te performance of te present formlation and compare it it reslts from oter approaces it is implemented in te eplicit finite element code DYNAD. Te formlation of te Beltscko-Lin-Tsa 8 sell element is canged to reflect te present approac. To models are inestigated and te reslts acqired sing te present approac 8-4

5 are compared it reslts from oter soltion approaces and it preiosl pblised reslts. First, a simple model is constrcted and soled: a strip of lengt mm, idt 5 mm, and eigt mm is clamped at bot ends and sbjected to niform distribted ertical load of magnitde kpa. Te material is steel it E 7 GPa, ν., and densit ρ 7.8 kg/m. Reslts for transerse sear stress and strain, and normal stress are presented in Figres - corresponding to to different sections of te strip considered. Stresses and strains are collected at section it coordinate.5 mm and 5.5 mm. Figre presents predictions for te transerse sear stresses of te present approac. Te predictions are compared it reslts obtained from te closed form elasticit soltion, reslts from te finite element code ABAQUS based on te approac described in 6, and reslts from te original first order sear deformable formlation (FOSDT). As seen te FOSDT reslts in constant transerse sear stresses trog te tickness and te rest of te reslts coincide er ell it eac oter. Figre presents te reslts for te transerse sear strains, γ. As seen bot ABAQUS and te FOSDT reslt in constant transerse sear strains trog te tickness, ic are incorrect. Hoeer, te transerse sear strains predicted it te present approac agree er ell it tat obtained from te elasticit soltion. Te difference beteen te ales from ABAQUS and FOSDT is de to te fact tat a sear correction factor of is sed in te FOSDT calclations ile it is determined atomaticallin ABAQUS. From Figres and te inconsistenc beteen te distribtion of transerse sear stresses and strains can be obsered (parabolic for stresses, oeer, linear for strains). Finall, Figre sos te inplane normal stresses, σ, distribtion at different sections along te strip. As seen te corrected transerse sear does not inflence te in-plane normal strains, ic proide frter confidence in te deeloped corrections. Second, a model inestigating te aial bckling of a clindrical composite sell is taken from te std of Anastasiadis et al. 9. Reslts are acqired sing tree different approaces: a standard approac for laered sells based on constant transerse sear strains trog te sell tickness (Approac ); an approac for laered sells based on te differential eqations of eqilibrim (as described in 6 and 7 and denoted it Approac ); and te present approac. Te model consists of a composite clinder fied at bot ends, ic as a radis of.95 m and tickness of.7 mm. Te sell consists of ortotropic boron/epo laers it te folloing material properties: E 6.8 GPa, E E 8.6 GPa, ν ν., ν.45, G G 4.48 GPa, G.55 GPa. It is aiall loaded and te bckling ale of te load is reported. For L/R seeral stacking seqences are inestigated and te reslts are presented in table. Stacking Seqence Table. Critical aial compression in N/m 6 HOSD from 9 Approac Approac Present (,9, ) s (9,,9 ) s ( 45,45, 45 ) s (45,45, 45 ) s As seen te present reslts are er good compared to te ales in 9 calclated sing a iger order sear deformable sell formlation. Fig. 4 sos te force s. displacement relation 8-5

6 acqired in te analsis for a standard FOSDT sell and for te same sell it te ereinproposed modification. CONCLUSIONS A correction to te first order sear deformable sell teor is proposed, ic reslts in parabolic trog-tickness distribtion of te transerse sear strains and stresses. It eliminates te need for sear correction factors in te first order teor. Te approac is applicable to all displacement based first order formlations and is simple and etremel eas to implement in an standard sell finite element. As compared to te oter modifications presented in te literatre, in ic correction is made in te distribtion trog te tickness of te transerse sear stresses onl, te proposed modification is more consistent. Te present formlation ses te correct distribtion and is consistent for bot transerse sear stresses as ell as transerse sear strains. REFERENCES. Tabiei, A., and Simitses, G. J., ''Torsional Instabilit of Moderatel Tick Composite Clindrical Sells, b Varios Sell Teories'', AIAA Jornal, Vol. 5, No. 7, pp. 4-46, B. N. Panda and T. Kant, Higer-Order Sear Deformable Teories for Flere of Sandic Plates Finite Element Ealations, International Jornal of Solids and Strctres, 4 (), (988).. J. N. Redd, A General Nonlinear rd-order Teor of Plates it Moderate Tickness, International Jornal of Non-Linear Mecanics, 5 (6), (99). 4. K. H. Ha, Finite Element Analsis of Sandic Plates: An Oerie, Compters & Strctres, 7 (4), 97 4 (99). 5. A. K. Noor, W. S. Brton and C. W. Bert, Comptational Models for Sandic Panels and Sells, Applied Mecanics Reies, 49 (), (996). 6. ABAQUS Teor Manal, Version 5.5, Hibbitt, Karlsson &Sorensen, Inc., Patcket, RI, (995). 7. MSC/NASTRAN V68 Reference Manal, MacNeal-Scendler Corporation, Los Angeles, CA, (99). 8. T. Beltscko, J.I. Lin, and C.-S. Tsa, Eplicit Algoritms for te Nonlinear Dnamics of Sells, Compter Metods in Applied Mecanics and Engineering, 4, 5 5 (984). 9. J.S. Anastasiadis, A. Tabiei, and G.J. Simitses, Instabilit of Moderatel Tick, Laminated, Clindrical Sells nder Combined Aial Compression and Pressre, Composite Strctres, 7, (994). 8-6

7 .8 Tickness coordinate, [mm] Teor ABAQUS FOSDT Present 5.5 mm.5mm Sear stress [MPa] Figre Transerse Sear Stress, σ, Distribtion trog te Sell Tickness in Eample Problem.8 Tickness coordinate, [mm] Teor ABAQUS FOSDT Present 5.5mm.5 mm Sear strain [/] Figre Transerse Sear Strain, γ, Distribtion trog te Sell Tickness in Eample Problem 8-7

8 .8.6 Teor ABAQUS Present Tickness coordinate, [mm] mm 5.5mm.5mm Normal stress [GPa] Figre In-plane Normal Stress, σ, Distribtion trog te Sell Tickness in Eample Problem.. HOSD from reference 9 A B AB A B P/Pcr.6.5 AB.4 AB... A Original FOSDT sell B Present FOSDT sell. End Displacement Figre 4 Force s. End Displacement Cre Normalied it Respect to te Critical Force Vale in Eample Problem 8-8