Axisymmetric Buckling Analysis of Porous Truncated Conical Shell Subjected to Axial Load

Transcription

1 Jornal of Solid Mecanics Vol. 9 No. (7) pp. 8-5 Aisymmetric Bcling Analysis of Poros Trncated Conical Sell Sbjected to Aial Load M. Zargami Deagani M.Jabbari * Department of Mecanical Engineering Sot Teran Branc Islamic Azad University Teran Iran Received Marc 7; accepted 5 April 7 ABSTRACT Tis paper stdied Bcling analysis of poros trncated conical sell sbjected to aial load. It is considered tat a flid ndrained between poros material and te Poros material properties vary across te ticness of sell wit a specific fnction also assmed tat te edge of te sell is simply spported. Te governing eqations are based on te Sanders inematics eqations and te first-order sell teory and by sing of variational formlations. Te general mecanical non-linear eqilibrim and linear stability eqations are derived. At te end te reslt of dimensionless bcling critical load ratio dimensionless ticness in different condition sc as variation in ticness porosity and angle of conical sell is investigated. Te mecanical load reslts are verified by te nown reslts in te literatre. 7 IAU Ara Branc.All rigts reserved. Keywords : Aisymmetric; Poros material; Bcling analysis; Conical sell; Aial load. INTRODUCTION T HE biot model of a flid-filled poros material is composed on te conceptal model of a coerent solid strctre and a freely moving pore flid (in oter words bot solid and flid or gas pases are flly connected). Poros material freqently fond in natre sc as wood stone and layers of dst. Detornay and Ceng [] stdied fndamentals of poroelasticity. Seide [] stdied te bcling of conical sells nder te aial loading. Singer [4] considered te bcling of conical sells nder te aisymmetrical eternal pressre. Bcling of te stiffened conical sells nder ydrostatic pressre is stdied by Barc and Singer [5]. Barc et al. [6] stdied te bcling of conical sells nder te ydrostatic pressre and te bcling loads of aially compressed conical sells for different sets of bondary conditions [7]. Singer [89] analyzed te bcling of ortotropic conical sells. Weigarten and Seide [] stdied te stability of conical sells nder te aial compression and eternal pressre. Te same ators considered te stability of conical sells nder te combined aial compression and internal pressre []. Ari-Gr et al. [] analyzed te bcling of cylindrical sell nder combined loading. Tey considered a cylindrical sell nder aial preload nonniform eating and torqe. L and Cang [] stdied te termal bcling of conical sells. Tey sed te Galerin metod for integrating te eqilibrim eqations in teir analysis. Tani [4] presented a soltion on te bcling of conical sell nder combined pressre and eating loads. Free vibration and bcling beavior of FG trncated conical sells sbjected to termal loads is investigated by Bangale et al. [5] based on te first order sear deformation teory. Te effects of initial stresses on te freqencies of te FG sells are stdied. Termoelastic stability analysis of FG trncated conical sells is presented by Sofiyev [6]. Modified Donnell type stability and compatibility eqations are derived and te Galerin metod is * Corresponding ator. Tel.: address: : M_Jabbari@azad.ac.ir (M.Jabbari). 7 IAU Ara Branc. All rigts reserved.

2 M. Zargami Deagani and M.Jabbari 9 applied to obtain te closed-form soltion. Te instability of FG trncated conical sells sbjected to termal and mecanical loads is analyzed based on te classical sell teory and te Sanders nonlinear inematics relations by Naj et al. [7]. In a series of wors Sofiyev stdied te bcling of FG trncated conical sells nder varios mecanical loads inclding ydrostatic pressre eternal pressre and combined aial tension and ydrostatic pressre (Sofiyev et al. [8] and Sofiyev [9]). By applying te Galerin metod to stability and compatibility eqations te critical bcling load of te sell is obtained. Recently Sofiyev [ ] presented te termal and mecanical bcling analysis of FG circlar sells resting on a two-parameters elastic fondation by solving te eigen-vale problem. Te critical bcling loads wit and witot elastic fondation are obtained sing te Galerin metod. Bic et al. [] stdied te instability of FG conical panels based on te classical tin sell teory nder aial compression eternal pressre and combination of tem. Te effects of te initial imperfections on te bcling beavior of FG trncated conical sells are illstrated by Sofiyev [4]. Sperposition and Galerin metods are applied to te modified nonlinear Donnell type stability and compatibility eqations and te pper and lower critical aial loads are obtained. Applying te Galerin metod to te governing eqations wic are based on iger order sear deformation sell teory reslts in a closed form soltion for te critical bcling temperatre difference. In anoter stdy Mirzavand and Eslami [5] Using te variational approac eqilibrim and stability eqations are obtained based on te classical sell teory and te Sanders nonlinear inematics relations. Jabbari et al. [6] stdied te bcling analysis of soft ferromagnetic FG circlar plates made of poro material. Jabbari et al. [7] stdied te termal bcling of radially solid circlar plate made of poros material wit piezoelectric actator layers presented. Jabbari et al. [8] stdied te bcling analysis of radially solid circlar plate made of poros material bonded wit te layers of piezoelectric actators. Stdied ai-symmetrical deflection and bcling of circlar poros-celllar plate by Magnca-Blandzi [9]. Stdied failre by bcling mode of te pore-strt for isotropic tree-dimensional reticlated poros metal foams nder different compressive loads []. Presented mecanical and termal stability of eccentrically stiffened fnctionally graded conical sell panels resting on elastic fondations and in termal environment [] te sell is a poros material tat a flid ndrained between Poros and simply spported at bot ends and it is assmed to be sbjected to a niform aial load. Te prebcling forces are obtained by considering membrane soltions of linear eqilibrim eqations. Applying te Galerin metod to te stability eqations reslts in an eigen-vale problem wic provides te critical bcling aial load and finally reslts are presented. DERIVATION OF THE GOVERNING EQUATIONS Consider a trncated conical sell of ticness and alf ape angle β made of poros material were te crvilinear coordinate system is defined as (θz) and te displacement components of te middle srface are v and w along te meridian tangential and lateral directions respectively. Te geometry of sell is sown in Fig. te relationsip between E and z for poros sell is assmed fnctionally. ( z ) E ( z ) E [ ecos( )] ( z ) G ( z ) G [ ecos( )] E G e ( ) ( ) E G () were e is te coefficient of sell porosity e E and E Yong s modls of elasticity at z and z respectively and G and G are te sear modls at z and z respectively. Te relationsip between te modls of elasticity and sear modls for j and is E G ( v ) and ν is Poisson s ratio wic is assmed to be constant across te sell ticness. Mecanical properties of te poros material vary across te ticness of te sell ( G G). j j 7 IAU Ara Branc

3 4 Aisymmetric Bcling Analysis of Poros Trncated Fig. Coordinate and Geometry system of a poros conical sell.. Basic eqations Te linear poroelasticity teory of Biot as two featres. An increase of pore pressre indces a dilation of pore.. Compression of te pore cases a rise of pore pressre. Te stress strain relation for poros material of te sell are written as following: G ij G ij ij ppij p K K p () () were p M[ jj ] (4) p M G (5) Here p p is pore flid pressre M is Biot s modls v is ndrained Poisson s ratio v v.5 is te Biot coefficient of effective stress K is bl modls ( K G( v / ( v ))) is variation of flid volme content and is te volmetric strain. Te two dimensional stress-strain law for plane-stress condition in te crvilinear coordinates for te ndrained condition ( ) is given by p M[ jj ] (6) p And by simplification Eq. () in crvilinear coordinates can be written as: A B A B G (7) were te constants are 7 IAU Ara Branc

4 M. Zargami Deagani and M.Jabbari 4 E 4 G( z) ( ) 4 E A G( z) ( ) n ( ) ( ) ( )( ) n G( z) ( ) B ( )( ) G( z) G( z) n G ( z ) n G z G( z) ( )( )( ) n ( ) G( z) n (8) Te sell is assmed to be tin and according to te Love Kircoff assmptions planes normal to te median srface are assmed to remain plane after deformation. Ts sear deformations normal to te sell are disregarded. Using first-order sell teory strain components at distance z from te middle plane are given in matri form as: z (9) were and are te strain components along te θ and z-directions respectively. Here and are te middle plane strain components and and are bending and twisting crvatres wit respect to te - and θ-aes respectively. Te relations between te middle plane strains and crvatres wit te displacement components according to te Sanders assmption strain displacement relations are obtained as: w v w w v sin tan sin tan v w v w sin sin tan w sin w w cos v cos v sin w cos v sin w () were (vw) represent te corresponding components of te displacement of a point on te middle sell srface and () indicates partial differentiation. Te force and moment resltants of conical sell according to te first order sell teory are 7 IAU Ara Branc

5 4 Aisymmetric Bcling Analysis of Poros Trncated / / () N dz M z dz ij ij ij ij / / Sbstitting Eqs. (7) in Eqs. () yield N A A B B N A A B B N D D M A A B B 4 4 M A A B B 4 4 M D D 4 () Te coefficients are given in te Appendi.. Strain energy Te total potential energy for poros conical sell can be written as: E U ij d ij dv () E were ij is elastic strain. Elastic strain energy for poros materials is comprised of elastic strain energy for solid body and flid in pores. Sbstitting Eq. (4) into Eq. () in te ndrained condition strain energy is obtained as: ii ii (4) K Here ( B ) is relation between drained bl modls and ndrained bl modls B is copling between pore flid effects and macroscopic deformation and B is septon coefficient. By sbstitting K (4) gives E in Eq. E ii ii ( ) (5) Te total potential energy for poros conical sell can be written as: U sin d d dz (6) And by sbstitting strain-stress relations U A B A B G sin d d dz In wic (7) 7 IAU Ara Branc

6 M. Zargami Deagani and M.Jabbari 4 A B D A U d d sin (8) B D A4 B 4 D. Stability eqations Te eqilibrim eqations are derived sing te fnctional of potential energy eqation of U Eq. (8) and employing te Eler eqations. In te fnctional of total potential energy te membrane and bending are inclded. For a conical sell te eqilibrim eqations sing te variational principle are derived as: N N N sin N N N cot N N sin M cot M M tan sin M M M M N cot sin sin N N N N sin (9) And w w v sin tan () In Eqs. () and are te rotations of te normal to te middle srface abot te and θ-aes respectively. Te stability eqations of te conical sell are derived sing te adjacent eqilibrim criterion. We assme v and w as te displacement components of te eqilibrim state and v and w as te virtal displacements corresponding to a neigboring state. Te displacement components of te neigboring state are v v v w w w () According to te adjacent eqilibrim criterion in te neigboring state of eqilibrim te stability eqations are fond. Similar to Eqs. () te stress and moment resltants are fond to be te sm of tose related to te eqilibrim and neigboring states as: N M N M N M i () i i i i i i Sbstitting relations () and () in Eqs. (9) collecting te second order terms te stability eqations are obtained as: 7 IAU Ara Branc

7 44 Aisymmetric Bcling Analysis of Poros Trncated N N N sin N N N cot N N sin M cot M M tan sin M M M M N cot sin sin N N N N sin () Te sbscript refers to te eqilibrim state and sbscript refers to te stability state. Te terms wit te sbscript are te soltion of te eqilibrim eqations for te given load. In wic te linear forms of strains and crvatres are given as: v w sin tan v sin w w cos v sin w sin w w cos v cos v sin (4) For simplicity te membrane soltion of te eqilibrim eqations are considered []. For tis aim all te moment and rotation terms mst be set eqal to zero in te eqilibrim eqations. By solving te membrane form of Eqilibrim eqations it is fond tat Pcr N N N sin (5) Sbstitting Eqs. (4) () into Eqs. () and () (5) into Eqs. () te stability eqations in terms of te displacement components are derived..4 Aisymmetric bcling Up to tis point it was assmed tat te bcling mode is not aisymmetric. In tis section we will consider te aisymmetric mode in wic v te displacements of mid-srface in circmferential direction is eqated to zero..5 Bondary conditions Consider a conical sell wit te simply spported bondary edges. Te bondary conditions are assmed as [6] 7 IAU Ara Branc

8 M. Zargami Deagani and M.Jabbari 45 v w M at L m ln L ln ln A sin ln A sin L v w B sin ln v v C sin ln C sin (6) were m... is te nmbers of te meridional wave and A and C are constant coefficients. Te approimate soltions (6) are sbstitted in Eqs. () sing te Galerin minimization tecniqe to yield a a A a a C (7) were te coefficients aij are fnctions of geometric parameters of te conical sell. In order to obtain te critical aisymmetric bcling aial load one sold set te determinant of te coefficient matri to zero and solve te reslting eqation. As seen from te definitions of constants a ij only a contains te aial load resltant tat te coefficients a ij are in Appendi. a a a a (8) Terefor te critical aisymmetric bcling aial load is sin aa ' aa Pcr a sin Q sin Q 8 (9) RESULTS AND DISCUSSION Consider a poros trncated conical sell. Te geometry is sown in Fig. te Yong s Modls and Poisson s ratio are E 7.6 Gpa v. respectively. Simply spported bondary conditions are assmed. Sow te dimensionless bcling aial compressive load verss te dimensionless parameter L / R. Te classical bcling aial compressive load sggested by Seide and E E [] F cos ( ) E () For te problem nder consideration te aisymmetric stability for mecanical bcling load of a poros conical sells presented. Fig. and Fig. sow te variation of te bcling aial compressive load verss te dimensionless 7 IAU Ara Branc

9 46 Aisymmetric Bcling Analysis of Poros Trncated parameter L / R for different semi-verte angles and coefficient porosity respectively. Te crves in Fig. sow te critical aial loads increase as te semi-verte angle decreases also te crves in Fig. sow te critical aial loads increase wen te coefficient of sell porosity decreases. From tese figres tey are fond tat by increasing vales of L / R te critical aial compressive load is decreased. Fig.4 indicated tat increasing te ticness of te poros conical sell increase stability and bcling aial compressive load of poros conical sell. Fig. Variation of te dimensionless critical aial compressive load verss dimensionless parameter L / R for stability of te poros conical sells in different semi-verte angles (beta) and 6. Fig. Variation of te dimensionless critical aial compressive load verss dimensionless parameter L / R for stability of te poros conical sells in different coefficient porosity ( e ). Fig.4 Variation of te dimensionless critical aial compressive load verss dimensionless parameter L / R for stability of te poros conical sells in different dimensionless parameter radis verss ticness ( R / ). Fig.5 sows variation of te dimensionless critical aial compressive load verss dimensionless parameter L / R for stability of te poros conical sells tat increasing nmbers of te meridional wave decrease te stability and 7 IAU Ara Branc

10 M. Zargami Deagani and M.Jabbari 47 critical aial compressive load. Fig.6 illstrated variation of te dimensionless critical aial compressive load verss dimensionless parameter L / R for stability of te poros conical sells in different ndrained Poisson s ratio ( v ) as decreasing ndrained Poisson s ratio decreased te stability and critical aial compressive load. Fig.5 Variation of te dimensionless critical aial compressive load verss dimensionless parameter L / R for stability of te poros conical sells in different nmbers of te meridional wave (m). Fig.6 Variation of te dimensionless critical aial compressive load verss dimensionless parameter L / R for stability of te poros conical sells in different ndrained Poisson s ratio ( v ). 4 CONCLUSIONS In te present article te energy metod is sed for te mecanical bcling analysis of sell made of pore material and derivation is based on te first-order sell teory Sanders inematics eqations and by sing of variational formlations. Te bondary condition of te sell are assmed to be clamped. Te effects of coefficient porosity semi-verte angle te meridional wave ndrained Poisson s ratio ticness on critical load of trncated conical sell are presented and te conclsions are:. Te critical aial load decreases and te sell will be nstable by increasing te porosity ( e ).. By increasing semi-verte angles ( ) critical aial load will be redced.. Te critical aial load increases by increasing te sell ticness (). 4. Te critical aial load decreases as te ndrained Poisson s ratio decreased ( v ). 5. As te meridional wave (m) increase te critical aial load will be decreases. 6. By increasing dimensionless parameter L / RTe critical aial load decreases. 7 IAU Ara Branc

11 48 Aisymmetric Bcling Analysis of Poros Trncated APPENDIX A A A A 4 dz A z dz A z dz B B B B dz B z dz B 4 z dz D G dz D Z G z dz D Z 4 G Z z dz a sin ( B A ) Q sin (4 A ) Q B A a sin B B B Q sin A 4A B 4A 4A A B Q 4 tan B A sin B B Q sin A 4 A A A B B Q tan 5 6 cot sin sin sin cot sin 4 sin 4 a A Q B A Q B A B Q A Q A A Q a a' A B A B B B 4 B 4 a ' 6B 4 A4 sin Q 4A tan tan 4 sin Q sin Q tan B 4 A4 A4 A4 4 4 A4 A4 A4 A4 4A4 6A4 A4 8A4 6A4 sin Q 6B 4 4B 4 6B 4 6A4 6B 4 A4 B 6B A sin Q 4A Q B B 4A A sin Q tan A 8A A 8A A 8A sinq N o sin Q7 sin Q L L L L 4 sin cos sin sin 4 sin Q d Q d Q d Q d L L L L 5 cos sin 6 cos sin 7 8 Q d Q d Q L L L L 9 cos sin sin sin sin sin L Q sin d 4 sin d Q sin sind Q d Q d Q d Q d L L 7 8 L L L Q sin d Q sin cosd Q sin cosd Q sin cosd Q sin cosd 7 IAU Ara Branc

12 M. Zargami Deagani and M.Jabbari 49 REFERENCES [] Detornay E. Ceng A.H.D. 99 Fndamentals of Poroelasticity Capter 5 in Compreensive Roc Engineering: Principles Practice and Projects Vol. II Analysis and Design Metod ed. C. Fairrst Pergamon Press. [] Seide P. 956 Aisymmetric bcling of circlar cones nder aial compression Jornal of Applied Mecanics : [] Seide P. 96 Bcling of circlar cones nder aial compression Jornal of Applied Mecanics 8: 5-6. [4] Singer J. 96 Bcling of circlar conical sells nder aisymmetrical eternal pressre Jornal of Mecanical Engineering Science : -9. [5] Barc M. Singer J. 965 General instability of stiffened conical sells nder ydrostatic pressre Aeronatical Qarterly 6: [6] Barc M. Harari O. Singer J. 967 Inflence of in-plane bondary conditions on te stability of conical sells nder ydrostatic pressre Israel Jornal of Tecnology 5(-): -4. [7] Barc M. Harari O. Singer J. 97 Low bcling loads of aially compressed conical sells Jornal of Applied Mecanics 7: [8] Singer J. 96 Bcling of ortotropic and stiffened conical sells NASA TN D-5: [9] Singer J. 96 Donnell-type eqations for bending and bcling of ortotropic conical sells Jornal of Applied Mecanics : -5. [] Weigarten V.I. Seide P. 965 Elastic stability of tin walled cylindrical and conical sells nder combined eternal pressre and aial compression AIAA Jornal : 9-9. [] Weigarten V.I. Seide P. 965 Elastic stability of tin walled cylindrical and conical sells nder combined internal pressre and aial compression AIAA Jornal : 8-5. [] Ari-Gr J. Barc M. Singer J. 979 Bcling of cylindrical sells nder combined aial preload nonniform eating and torqe Eperimental Mecanics 9: [] L S.Y. Cang L.K. 967 Termal bcling of conical sells AIAA Jornal 5(): [4] Tani J. 984 Bcling of trncated conical sells nder combined pressre and eating Jornal of Termal Stresses 7: 7-6. [5] Bangale R. Ganesan N. Padmanaban C.. 6 Linear termoelastic bcling and free vibration beavior of fnctionally graded trncated conical sells Jornal of Sond and Vibration 9: 4-7. [6] Sofiyev A.H. 7 Termo-elastic stability of fnctionally graded trncated conical sells Composite Strctres 77: [7] Naj R. Sabziar M. Eslami M.R. 8 Termal and mecanical instability of fnctionally graded trncated conical sells Tin-Walled Strctres 46: [8] Sofiyev A.H. Krogl N. Trmen M. 9 Bcling of FGM ybrid trncated conical sells sbjected to ydrostatic pressre Tin-Walled Strctres 47: 6-7. [9] Sofiyev A.H.9 Te vibration and stability beavior of freely spported FGM conical sells sbjected to eternal pressre Composite Strctres 89: [] Sofiyev A.H. Te bcling of FGM trncated conical sells sbjected to combined aial tension and ydrostatic pressre Composite Strctres 9: [] Sofiyev A.H. Bcling analysis of FGM circlar sells nder combined loads and resting on te Pasterna type elastic fondation Mecanics Researc Commnications 7: [] Sofiyev A.H. Termal bcling of FGM sells resting on a two-parameter elastic fondation Tin-Walled Strctres 49: 4-. [] Bic D. Pong N. Tng H. Bcling of fnctionally graded conical panels nder mecanical loads Composite Strctres 94: [4] Sofiyev A.H. Inflence of te initial imperfection on te non-linear bcling response of FGM trncated conical sells International Jornal of Mecanical Sciences 5: [5] Mirzavand B. Eslami M.R. A closed-form soltion for termal bcling of piezoelectric FGM ybrid cylindrical sells wit temperatre dependent properties Mecanics of Advanced Materials and Strctres 8: [6] Jabbari M. Mojaedin A. Hagi M. 4Bcling analysis of tin circlar FG plates made of satrated poros-soft ferromagnetic materials in transverse magnetic field Tin-Walled Strctres 85: [7] Jabbari M. Farzane Jobane E. Mojaedin A.4Termal bcling analysis of poros circlar plate wit piezoelectric actators based on first order sear deformation teory International Jornal of Mecanical Sciences 8: [8] Jabbari M. Farzane Jobane E. Korsidvand A.R. Eslami M.R. Bcling analysis of poros circlar plate wit piezoelectric actator layers nder niform radial compression International Jornal of Mecanical Sciences 7: [9] Magnca-Blandzi E. 8 Ai-symmetrical deflection and bcling of circlar poros-celllar plate Tin-Walled Strctres 46(): -7. [] Li P.S. Failre by bcling mode of te pore-strt for isotropic tree-dimensional reticlated poros metal foams nder different compressive loads Materials & Design (6): IAU Ara Branc

13 5 Aisymmetric Bcling Analysis of Poros Trncated [] Ngyen D.D. Pam H.C. V M.A. V D.Q. Pong T. Ngo D.T. Ngyen H.T. 5Mecanical and termal stability of eccentrically stiffened fnctionally graded conical sell panels resting on elastic fondations and in termal environment Composite Strctres : [] Meyers C.A. Hyer M.W. 99 Termal bcling and postbcling of symmetrically laminated composite plates Jornal of Termal Stresses 4: IAU Ara Branc