A Finite Element Formulation for Analysis of Functionally Graded Plates and Shells

Transcription

1 J Basic ppl Sci es 96- Textoad Pblication SS 9-X Jornal of Basic and pplied Scientific esearc wwwtextroadcom Finite lement Formlation for nalysis of Fnctionally Graded Plates and Sells oammad Setare and oammad eza svandzibaei Department of ecanical ngineering ndimes Branc slamic zad niversity ndimes ran BSTT Stdy of te vibration cylindrical sells made of a fnctionally gradient material FG composed of stainless steel and nicel is important aterial properties are graded in te ticness direction of te sell according to volme fraction power law distribtion Te objective is to stdy te natral freqencies te inflence of constitent volme fractions and te effects of bondary conditions on te natral freqencies of te FG cylindrical sell Te stdy is carried ot sing tird order sear deformation sell teory Te governing eqations of motion of FG cylindrical sells are derived based on sear deformation teory eslts are presented on te freqency caracteristics inflence of constitent volme fractions KY WODS: FG ylindrical sell Hamilton's principle TODTO ylindrical sells ave fond many applications in te indstry Tey are often sed as load bearing strctres for aircrafts sips and bildings nderstanding of vibration beavior of cylindrical sells is an important aspect for te sccessfl applications of cylindrical sells esearces on free vibrations of cylindrical sells ave been carried ot extensively [-5] ecently te present ators presented stdies on te inflence of bondary conditions on te freqencies of a mlti layered cylindrical sell [6] n all te above wors different tin sell teories based on Love ypotesis were sed Vibration of cylindrical sells wit ring spport is considered by Loy and Lam [7] Te concept of fnctionally graded materials FGs was first introdced in 98 by a grop of materials scientists in Japan [8-9] as a means of preparing termal barrier materials Since ten FGs ave attracted mc interest as eat-sielding materials FGs are made by combining different materials sing power metallrgy metods [] Tey possess variations in constitent volme fractions tat lead to continos cange in te composition microstrctre porosity etc reslting in gradients in te mecanical and termal properties [-] Vibration stdy of FG sell strctres is important However stdy of te vibration of FG sells wit ring spports is limited Te FGs considered are composed of stainless steel and nicel were te volme fractions follow a power-law distribtion Te stdy is carried ot based on tird order sear deformation sell teory eslts presented inclde te freqency caracteristics of cylindrical sells and te inflence of bondary conditions Te present analysis is validated by comparing reslts wit oters in te literatre - FTOLLY GDD TL Fnctionally gradient materials FG are obtained by combining two or more materials ost of te fnctionally gradient materials are employed in ig- temperatre environments and many of te constitent materials may possess temperatre-dependent properties Te material P can be expressed as a fnction of temperatre [] as P P P T P T PT P T were P P P P and P are te coefficients of temperatre T K expressed in Kelvin and are niqe to te constitent materials Te material properties P of FGs are a fnction of te material properties and volme fractions of te constitent material and are expressed as P P V j fj j *orresponding tor: oammad eza svandzibaei is PD Stdent in ecanical ngineering mail: esvandzebaei@yaoocom 6

2 Setare and svandzibaei were j P and fj V are respectively te material property and volme fraction of te constitent material j Te volme fractions of all te constitent materials sold add p to one ie j fj V For a cylindrical sell wit a niform ticness and a reference srface at its middle srface te volme fraction can be written as f z V / were is te power- law exponent For a fnctionally gradient material wit two constitent materials te yong's modls Poisson ratio and te mass density can be expressed as From tese eqations wen / z and and wen / z and Te material properties vary continosly from material at te inner srface of te cylindrical sell to material at te oter srface of te cylindrical sell - STS-DSPLT LTOSHPS Te strain-displacement relationsips for a tin sell [] 8 9 r r were and are te fndamental form parameters or Lame parameters and are te displacement at any point and are te radis of crvatre related to and respectively Te tird- order teory of eddy sed in te present stdy is based on te following displacement field: 5 7

3 J Basic ppl Sci es 96- Tese eqations can be redced by satisfying te stress-free conditions on te top and bottom faces of te laminates wic are eqivalent to at Ts 6 Were Sbstitting q 6 into nonlinear strain-displacement relation 8 - it can be obtained for te tird-order teory of eddy 7 8 were 9 8

4 Setare and svandzibaei Were are te membranes strains and are te bending strains nown as te crvatres - FOLTO For a tin cylindrical sell te stress -strain relationsip are defined as For a isotropic cylindrical sell te redced stiffness were i j= and 6 are defined as were is te Yong's modls and is Poisson's ratio Defining stiffness / 5 6 B D F G H d / are fnctions of z for fnctionally gradient materials Here 8 denote te extensional D te bending stiffness B te bending-extensional copling stiffness and F G H are te extensional bending copling and iger-order stiffness For a tin cylindrical sell te force and moment reslts are defined as P P P d d d P d P 9 d d - TH TOS OF OTO FO VBTO OF G SHLL Te eqations of motion for vibration of a generic sell can be derived by sing Hamilton's principle wic is described by t K dt V t Were K and V are te total inetic potential strain and loading energies t and t are arbitrary time Te inetic strain and loading energies of a cylindrical sell can be written as: K dv dv V q q q d d 5 9

5 J Basic ppl Sci es 96- Te infinitesimal volme is given by d d d dv 6 wit te se of qs 7- and sbstitting into q we get te eqations of motions a generic sell P P 6 P P 7 6 c P P P P 8 / / P P P P P P P P P P P P 6 9 P P P P 6 P P P P 6 For qs 7 are defining as i i d - qations of motion for vibration of cylindrical sell Te crvilinear coordinates and fndamental from parameters for a cylindrical sell are: x a a Sbstitting relationsip into qs 7- te eqations of motions for vibration of cylindrical sell wit te tird-order teory of eddy are converted to

6 Setare and svandzibaei a x x P P 6 x a a a a a a P P P a a a x a x x x P 6 a x a x a a a x a P P a a a a x x x P P a a a a x x 8 a a Te displacement fields for a FG cylindrical sell and te displacement fields wic satisfy tese bondary conditions can be written as x cos n cos t x 9 B x sin n cos t x cos n cos t x D cos n cos t x x sin n cos t were B D and are te constants denoting te amplitdes of te vibrations in te x and z directions and are te displacement fields for iger order deformation teories for a cylindrical sell x is te axial fnction tat satisfies te geometric bondary conditions Te axial fnction x is cosen as te beam fnction as mx mx mx mx x cos cos m sin sin 5 L L L L Sbstitting q 9 into qs - 8 for tird order teory we can be expressed det 5 xpanding tis determinant a polynomial in even powers of is obtained were i i 5 are some constants q 5 is solved five positive and five negative roots are obtained Te five positive roots obtained are te natral anglar freqencies of te cylindrical sell based tird-order teory Te smallest of te five roots is te natral anglar freqency stdied in te present stdy 5- SLTS D DSSSO To validate te present analysis reslts for cylindrical sells are compared wit Loy and Lam [5] and wit svandzibaei [6] Te comparisons sow tat te present reslts agreed well wit tose in te literatre

7 J Basic ppl Sci es 96- Table omparison of natral freqency Hz for a clamped isotropic cylindrical sell L cm 58cm 5cm 7788* m 866gm 7756 n m Loy[5] svandzibaei [6] Present From te comparisons presented in Table it can be seen tat te present reslts agree well wit tose in te literatre n tis paper stdies are presented for a FG cylindrical sell wit are considered Te effects of te FG configration are stdying te freqencies of two FG cylindrical sells: Type FG cylindrical sell and Type FG cylindrical sell Type FG cylindrical sell as icel on its inner srface and stainless steel on its oter srface and Type FG cylindrical sell as stainless steel on its inner srface and nicel on its oter srface Figs and sow te variations of te volme fractions V of icel and Stainless Steel respectively in te ticness direction z for f a Type FG cylindrical sell n Fig te volme fraction for icel V f decreased from at z=- 5 to at z=5 n Fig te volme fraction of Stainless Steel V fss increased from at z=-5 to at z=5 Vf 8 6 =5 =7 = = =5 =5 Fig Variation of volme fraction of icel 6 8 V f wit radial distance z in te ticness direction Vfss =5 =7 = = =5 =5 Fig Variation of volme fraction of stainless steel V fss wit radial distance z in te ticness direction

8 Setare and svandzibaei 6- OLSOS Te stdy was carried ot for two types of fnctionally graded cylindrical sells were te configrations of te constitent materials in te fnctionally graded cylindrical sells are different One is termed as a Type FG cylindrical sell and as properties tat vary continosly from nicel on its inner srface to stainless steel on its oter srface Te oter is termed as a Type FG cylindrical sell and as properties tat vary continosly from stainless on its inner srface to nicel on its oter srface Te inflence of te constitent volme fraction on te freqencies for Type and FG cylindrical sells as been fond to be different For te Type FG cylindrical sells te natral freqencies decreased wen increased and for te Type FG cylindrical sells te natral freqencies increased wen decreased Te present analysis is validated by comparing reslts wit tose available in te literatre FS [] rnold Warbrton GB 98 Flexral vibrations of te walls of tin cylindrical sells Proceedings of te oyal Society of London ; 97:8-56 [] Ldwig Krieg 98n analysis qasi-exact metod for calclating eigen vibrations of tin circlar sells J Sond vibration; [] ng H 98 Free vibration analysis of circlar cylindrical sells J Sond vibration; 7-59 [] Soedel W 98 new freqency formla for closed circlar cylindrical sells for a large variety of bondary conditions J Sond vibration; 79-7 [5] Forsberg K 96 nflence of bondary conditions on modal caracteristics of cylindrical sells J; 8-89 [6] Lam KL Loy T 995 ffects of bondary conditions on freqencies caracteristics for a mlti- layered cylindrical sell J Sond vibration; [7] Loy T Lam KY 996Vibration of cylindrical sells wit ring spport Jomal of mpact ngineering; 996; 5:55 [8] Koizmi 99 Te concept of FG eramic Transactions Fnctionally Gradient aterials [9] aino rai Kitajima H Oasi K Transient temperatre response of fnctionally gradient material sbjected to partial stepwise eating Transactions of te Japan Society of ecanical ngineers Part B 99; 6:-699 [] non 996FG components: P meets te callenge etal powder eport 5:8- [] ang XD Li D Ge 99 Termal stress analysis of axial symmetry fnctionally gradient materials nder steady temperatre field Jornal of Fnctional aterials; 5:5-5 [] Weterold Seelman S Wang J 996 se of fnctionally graded materials to eliminate or control termal deformation omposites Science and Tecnology; 56:99- [] ajafizade Hedayati B efined Teory for Termoelastic Stability of Fnctionally Graded irclar Plates Jornal of termal stresses; 7: [] Soedel W 98 Vibration of sells and plates L DKK ew Yor [5] Loy T Lam KY eddy J 999Vibration of fnctionally graded cylindrical sells; :9- [6] ajafizade svandzibaei 7 Vibration of fnctionally graded cylindrical sells based on iger order sear deformation plate teory wit ring spport cta ecanica; 9:75-9