39. Vibration Analysis of Curved Composite Beams using Hierarchical FEMRajamohan Ganesan and Wasim ArshadConcordia Centre for CompositesDepartment of Mechanical and Industrial EngineeringConcordia University, Montreal, Quebec, CanadaE-mail: ganesan@encs.concordia.caAbstractCurved beams made of polymer-matrix fiber-reinforced composite materials are increasingly beingusedinmechanical,aerospace,automotive, and civil engineering applications.The conventional finite element formulation has limitations in performing the dynamic analysis ofcurved composite beams. The requirement of finer mesh leads to discontinuities of stress and strain distributionsatelementinterfaces. Therecentlydevelopedhierarchicalfiniteelementformulation provides us withtheadvantages of usingfewer elements andobtainingbetter accuracyinthe calculation of dynamic response. In the present work, the hierarchical finite element formulation foruniform-thickness curved beams made of composite material is developed. The approximating cubic polynomial functions for both tangential and radial displacements are modified by adding suitable trigonometric functions. To generate the finite element model, different combinations of hierarchicalterms are used. The stiffness and mass matrices of curved composite beams are determined using these combinations, the reduced composite shell theory and the weak (variational) formulation. The combinationof hierarchical terms that gives the most accurate andfast convergingresults is determined. The free vibration analysis of curved beams made of cross-ply [ ]s 890 / 0 laminate, angle-ply[ ]s 845 tlaminate, and quasi-isotropic[ ]s 445 / 45 / 90 / 0 +laminate is performed. The relative performances of these curved laminates are assessed. Keywords: Vibration, hierarchical finite element, laminated composites6. INTRODUCTIONCurvedbeam structures madeofpolymer-matrix composite materialsareoften encountered inpractice. Examples of applications of curved beams can be found infuselage rings, reinforcement rings for cylindrical and conical shells and arches. In the present work, beams curved and bent only in the plane of curvature so that no torsion is involved are considered. There are various procedures that exist for the refinement of the finite element solutions. Broadly these fall into two categories: The first, and the most common, involves refining the mesh while keeping the degree of the elements fixed. This is termed as h-method. The second method, called p-method, involves keeping the mesh size constant and letting the degree of the approximating polynomial totend toinfinity. Hierarchical Finite Element Method (HFEM) belongs to the category of p-method. Hierarchical functions were initially introduced by Zienkiewicz et al. [1] with the objective of introduction of p-graded meshes in ana priorichosen manner. Polynomial functions are morecommon in thefiniteelementanalysis.The HFEM has been applied to linear analysis of plates [2]. Ganesan and Nigam [3] applied the hierarchicalfinite element formulationfor theanalysis of thickandthincomposite beams. Theyalsostudiedthe dynamic response of variable-thickness composite beams using HFEM. The HFEM provides us with critical advantages of using fewer elements and obtaining better accuracy in thecalculationof displacements, stressesandbucklingloadsof metallicandcompositecurvedbeams. Ganesan and Arshad [4] applied hierarchical FEM for the static analysis of curved composite beams. In the present study, the free vibration analysis of curved composite beams using HFEM is conducted. 7. HIERARCHICAL CURVED COMPOSITE BEAM FINITE ELEMENT Thecubic-cubiccircularlycurvedelement hasbeenusedforthepresent workandisdescribedbythe following displacement functions for tangential (v) and radial (w) displacements.3423 2 1) ( s a s a s a a s v + + + (1a)3423 2 1) ( s c s c s c c s w + + + (1b)where s is the tangential distance variable measured from nodal point 1 as shown in Figure 1. The element has constant radius of curvature R and subtending angle . Length L is equal to R. The angular variable and distance variable sthat is equal to R are measured from nodal point 1. The element possesses four physicaldegrees offreedomat eachnodal point: a tangentialdisplacementv,aderivativeoftangential displacement (v/s) or vs, a radial displacement w, and a derivative of radial displacement (w/s) or ws, or slope .Figure 1: Eight Degrees-of-Freedom Circular Beam ElementIn the hierarchical formulation, the approximating cubic polynomial functions (Equations (1a) and (1b)) for bothtangential (v) andradial (w) displacements aremodifiedbyaddingtrigonometric functions. The tangential displacement (v) and radial displacement (w) are expressed as:( ) [ ]++ + + + Nrr rs a s a s a s a a s v143423 2 1sin (2)( ) [ ]++ + + + Nrr rs c s c s c s c c s w143423 2 1sin (3)where Lrr ,r = 1, 2, 3N, andai and ci are coefficients to be determined.The polynomial terms in the assumed displacement field are used to define the element nodal degrees of freedomandthe trigonometric terms are usedtoprovide additional degrees-of-freedomthat are not physical to the interior of the element. The above equation can be written in the matrix form as( ) [ ] [ ] v s g a (4) ( ) [ ] [ ] w s g c (5) where[ ] { }2 31, , , , sinrg s s s s 1 ](6){ }1 2 3 4 4, , , ,Tra a a a a a+1 ](7){ }1 2 3 4 4, , , ,Trc c c c c c+1 ] (8)where{ }4 + racontains terms suchas8 7 , 6 5, , a a a aandsoon. Inasimilar manner[ ] { } sr sinand { }4 + rc are defined. Upon evaluating and w v vs, , ,at node 1 (i.e.ats = 0)and at node 2 (ats = L) and evaluating the hierarchical terms, e.g. when r = 2, the interpolation functions are obtained as follows.2 311 3 2 N + (9)( )2 322 N L +(10)2 333 2 N (11)( )2 34N L + (12)where RRLs(13)and the trigonometric hierarchical shape functions are( )24) 1 ( 2 rrr r rN + + +

( ) rrr) 1 ( + (14)whereLrr ,r = 1, 2, 3, ..NThedisplacement fieldfortheelement, intermsofthenodal degreesoffreedomandthehierarchical degrees of freedom, can now be written as+ ++ + + + Nrr v r s sv N v N v N v N v N v14 4 2 4 2 3 1 2 1 1(15)for tangential displacement (v) and similarly+ ++ + + + Nrr w rw N N w N N w N w14 4 2 4 2 3 1 2 1 1 (16)for radial displacement (w).To generate the finite element model using the HFEM, different combinations of hierarchical terms were tried to get the most accurate results. The best combination is figured out which gives the most converged andaccurateresultsfordifferent laminateconfigurationsandboundaryconditions. Thesecombinations involve one to four hierarchical terms for each of the tangential (v) and radial (w) displacements. First for symmetric combinations of hierarchical terms, same number of hierarchical terms are used with tangential (v) and radial (w) displacement functions e.g. 3 3 2 2 1 1, , w v w v w v and 4 4w v , where 1 1w v means that one hierarchical termis used with tangential displacement function (v) and one with radial displacement function (w). A similar representation applies for 3 3 2 2, w v w v and 4 4w v . The non-symmetric combinations of hierarchical terms were applied in the following way.(i) nw v 0 wheren = 1, 2, 3, 4which means that tangentialdisplacementfunction (v) is provided with no hierarchicalterm and radial displacement function (w) is provided with 1, 2, 3, and4 hierarchical terms successively. If a group uses 1 hierarchical term with tangential displacement function (v) then only 0, 2, 3 and 4 hierarchical terms will be used with radial displacement function (w) for all the non-symmetric combinations. The use of 1 hierarchical termis intentionally avoided to make it non-symmetric. However, it has already been considered for the case of symmetric combinations.Similarly, the following cases are considered.(ii)nw v 1 where n = 0, 2, 3, 4( n = 1 intentionallyavoided for non-symmetric combination)(iii) nw v 2wheren = 0, 1, 3, 4( n = 2 intentionallyavoided for non-symmetric combination)(iv) nw v 3wheren = 0, 1, 2, 4( n = 3 intentionallyavoided for non-symmetric combination)(v)nw v 4wheren = 0, 1, 2, 3 ( n = 4 intentionallyavoided for non-symmetric combination)3.STIFFNESS AND MASS MATRICESConsider acylindrical plateof constant radiusRasillustratedinFigure2. Thethicknessandin-plane dimensions are denoted by h, and a and b respectively. The displacements in the x, s, and z directions are denoted by u, v, and w, respectively. Figure 2: Nomenclature for Curved Laminated PlateThe strain-displacement relations for general thin shells are [5]:22 0xwzxux (17)22 0 0) / 1 ( swR zzsvRzRwsvs++ + (18) s xwRzzsusvRzsx + ++ 2 0 0) 2 ( ) 1 ( (19)where 0uand 0vare the axial and tangential displacements of the mid plane, respectively. Since the plate is shallow ( R > > h ), z/R is small compared to unity. Thus, 1 ) / 1 ( + R z (20)Therefore, Equations (17-19) can be written in the formxxxz + 0(21)sssz + 0 (22)sxsxsxz + 0(23)The ply constitutive relations are as follows

111]1

111]1

111]1

sxxsk k kk k kk k kksxkxksQ Q QQ Q QQ Q Q) (66) (26) (16) (26) (22) (21) (16) (12) (11) () () ((24)where ijQ are the reduced stiffnesses for the plane stress case.The force and moment resultants are defined as follows. dz N N Nksxkshhkx sx s x) , , ( ) , , () ( ) (2 /2 /) ( (25) dz z M M Mksxkshhkx sx s x) , , ( ) , , () ( ) (2 /2 /) ( (26)Combining Equations (21-23) and substituting the results into Equations (25) and (26) and performing the integrations, the laminate constitutive relations are obtainedas 11111111]1

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sxxssxxssxxssxxsD D D B B BD D D B B BD D D B B BB B B A A AB B B A A AB B B A A AMMMNNN00066 26 16 66 26 1626 22 12 26 22 1216 12 11 16 12 1166 26 16 66 26 1626 22 12 26 22 1216 12 11 16 12 11 (27)whereij ijB A ,andijDare the stiffnesses defined as2 /2 /2 ) () , , 1 ( ) , , (hhkij ij ij ijdz z z Q D B A (28)The strain energy, U, for curved plate in terms of an x ,s ,z coordinate system while taking into account the basic assumption of laminated plate theory i.e., 0 sz xz z , is given by the relationship. + + xs xks xkskQ Q Q U ) (16) (122 ) (112 2 (21 dz ds dx Q Q Qxskxkxs sk) 22 ) (662 ) (22) (26 + + + (29)For a1-Dproblemall theterms associatedwithxdirectionareneglected. Substitutingthekinematic relations, Equations (21-23), into Equation (29) and integrating with respect to z, the following equation is obtained.2221 11 12 21 10221

,_

+ ,_

'+11]1

,_

+ ,_

swDsvwRARwsvA UL ss}dsswsvRDsvRD1]1

,_

,_

,_

+221122112(30)The kinetic energy for the hierarchical curved beam element with two symmetric hierarchical terms with respect to tangential (v) and radial (w) displacement functions is20[ ( )]2LvvAT v s dx& (31)20[ ( )]2LwwAT w s dx&(32)where 2 4 2 3 1 2 1 1) (s sv N v N v N v N s v + + + 6 6 5 5 v vv N v N + + (33)2 4 2 3 1 2 1 1) ( N w N N w N s w + + +

6 6 5 5 w ww N w N + +(34)The strain energy expression (30) can be divided into three expressions asww vw vvU U U U + + (35)The energy expressions Uvv, Uvw,and Uww are associated with axial, axial-flexural coupling, and flexural behaviors, respectively. Substitutingthetangential (v) andradial (w) displacement functions withtwo hierarchicalterms,intothe energy expressionsand thenperforming partial differentiationsofthe strain energy with respect to each of the eight degrees of freedom,the 1212 stiffness matrix equations for the element are obtained. ;'11111111111111111]1

;' 6522116522116 6 6 66 6 6 6212211212211] [ ] [] [ ] [wwvvssww wvvw vvwwvvwwwwvvvvvvk kk kFFMYMYFFXXXX(36)where 1X,2X, vF1, vF2, wF1, wF2 are the counterpart generalized forces associated with the degrees of freedom 1 sv,2 sv,5 vv,6 vv,5 ww, and6 wwrespectively. Thecoefficientsinthe66sub-matricesare obtained as ds N NRA kj iLvvij

,_

+ 02111(37)ds N N N NRAk kj i j iLwv vwji ij

,_

011(38) ds N NRN NA kLj ij iwwij

,_

+ 0211(39)The equations of motion are obtained by the use of the Lagranges equation as follows:vii id T UFdt v v _ + ,& (40) wii id T UFdt w w _ + ,& (41)where iv and iwrepresent the ith degree of freedom, and vv wwT T T +. Substituting the displacement functions (33) and (34) into the kinetic energy expressions (31) and (32) and then performing the differentiations as indicated in Equations (40) and (41), one can obtain the mass matrix as 111]1

] [ ] [] [ ] [] [6 6 6 66 6 6 6wwvvm Matrix NullMatrix Null mm (42)The coefficients in the 66 sub-matrices of mass matrix are given by: Lj i wwij vvijds s N s N A m m0) ( ) ( (43)4.ANALYSIS OF CURVED COMPOSITE BEAM USING RITZ METHOD In this section approximate solution based on Ritz method is developed. The approximate solution is given by single summation series: ( ) ( )Mmm ms S A s w10 (44)( ) ( )Nnn ns S B s v10 (45)The functions ( ) s Sm and ( ) s Sn are chosen so as to satisfy the boundary conditions. The coefficients mAand nBare determined using the stationary conditions.mwwmATAU~ ~ (46)nvvnBTBU~ ~(47) In Equations (46-47) U~ is strain energy,andwwT~andvvT~are kinetic energies associated with radialand tangentialdisplacements respectively, obtained by substituting the approximate expressions for the deflections into Equations (31-32).5. FIXED-FREE CURVED BEAM EXAMPLEA uniform thickness composite curved beam with fixed-free boundary condition as shown in Figure 3 is considered. It ismadeof NCT-301Graphite-Epoxycompositematerial. Thematerial propertiesofthe NCT-301 material are given as:, 99 . 7 , 43 . 1292 1GPa E GPa E GPa G 28 . 4 , 021 . 012 21 The geometric properties of the beam are: length L =2 / 381 2 / R; individual ply thickness (ke) = 0.125 mm. There are 32 plies in the laminate. Figure 3: Fixed-Free Curved Composite BeamThree different laminate configurations are chosen to see the effect of fiber orientations on the values of natural frequencies for the fixed-free beam.Natural frequencies are calculated for the following types of laminates that are, angle-ply[ ]s 845 tlaminate, cross-ply[ ]80/ 90slaminate and quasi-isotropic [ ]s 445 / 45 / 90 / 0 + laminate. Table 1 summarizes the values of the 1st natural frequency. It took about 8 termsfortheRitzsolutiontoconvergetoasinglevalue. Angle-plylaminate[ ]s 845 tgivesthelowest natural frequency (3.596 Hz) value among all the three laminate configurations. Cross-ply laminate [ ]80 / 90sgives the highest value of natural frequency while quasi-isotropic laminate gives a natural frequency value in between that of the other two laminates. Table 1: 1st Natural Frequency (Hz) values Calculated using Ritz Method for Fixed-Free Curved Composite Beam made of[ ]s 845 t,[ ]80/ 90s, and [ ]s 445 / 45 / 90 / 0 + Laminates[ ]s 845 t [ ]80/ 90s[ ]s 445 / 45 / 90 / 0 +1 38460 66126 522922 179.12 308.55 244.143 7.46 13.863 11.2174 3.626 6.747 5.4615 3.611 6.722 5.4426 3.597 6.696 5.4217 3.596 6.693 5.4188 3.592 6.686 5.412The curved beam example problem is solved by using hierarchical finite element method and the results will be compared with the solutions obtained using the Ritz method. The approximate Ritz solution for angle-ply laminate[ ]s 845 tis matched using only 4-elements mesh for 4 symmetric hierarchical terms. It was observed that the results keep on improving as we increase the number ofhierarchical termswithtangential displacement function. Theresultsforthegroupofcombination(4 nv w ) for angle-ply laminate[ ]s 845 tare shown in Table 2. The combination (4 3v w ) gives the best convergence among all the 24 combinations tried for angle-ply laminate.It took just 4-elements mesh to reach the approximate Ritz solution compared to 9-elements mesh for the combination (1 0w v ). Table 2: 1st Natural Frequency Calculated by using Non-Symmetric Trigonometric Hierarchical Terms (4 nv w ) for [ ]s 845 tFixed-Free LaminateNumberof ElementsNumber of D.O.F.1st NaturalFrequency (Hz)Numberof D.O.F.1st NaturalFrequency (Hz)Number of D.O.F.1st NaturalFrequency (Hz)Number of D.O.F.1st NaturalFrequency (Hz) v4-w0 hierarchical terms v4-w1 hierarchical terms v4-w2 hierarchical terms v4-w3 hierarchical terms1 8 66.891 9 5.220 10 3.679 11 3.6732 16 3.709 17 3.689 18 3.689 19 3.6903 24 3.674 25 3.674 26 3.674 27 3.6744 32 3.665 33 3.665 34 3.665 35 3.661Approximate Solution by Ritz Method: 1st Natural Frequency = 3.592 HzSimilar trends are observed for cross-ply [ ]80/ 90slaminate and quasi-isotropic [ ]s 445 / 45 / 90 / 0 +laminate.The best combination (4 nv w ) has been used for plotting the first three natural frequencies for the three considered laminate configurations and the result is shown in Figure 4. As evident, cross-ply laminate gives the highest values of the natural frequencies, followed by the quasi-isotropic and the angle-ply laminate configurations in the case of fixed-free curved beam.1 2 305101520253035404550ModeNumberNatural FrequencyAngle-ply[+45/-45]8sCross-ply[0/90]8sQuasi-Isotropic[0/90/+45/-45]4sFigure 4: First Three Natural Frequencies obtained for Different Laminate Configurations for the Fixed-Free Curved Composite Beam8. CONCLUSIONInthepresent work, thehierarchical finiteelement formulationforthedynamicanalysisofthecurved composite beams has been developed.It is shown that best results in terms of computationaleffort and convergencerateforfreevibrationanalysisofcurvedbeamcannot beachievedbyjust increasingthe number of hierarchical terms but is achieved by applying the particular combination of hierarchical terms which gives the best convergence rate and accuracy.From the free vibration analysis of the fixed-free curved beam, it is observed that the cross-ply[ ]s 890 / 0laminate has the highest fundamental natural frequency value whereas angle-ply [ ]s 845 tlaminate has the lowest natural frequency value. This is because the angle-ply laminate has the lowest stiffness alongsdirection. Quasi-isotropic[ ]s 445 / 45 / 90 / 0 + laminate has its natural frequencies in between the values of the other two laminates. The results for fixed-free boundary condition show that hierarchical terms used with tangential displacement function have more effect on convergence rate than that of terms used with radial displacementfunction.The combination (4 3v w ) proved to be the best combination in terms of convergence rate and accuracy.7. REFERENCES[1] Zienkiewicz, O.C., Irons, B.M., Scott, F.C., and Campbell, J., (1971) Three Dimensional Stress Analysis,University of Liege Press, Proc. IUTAM Symp. On High Speed Computing of Elastic Structures, pp. 413-433.[2] Han, W., and Petyt, M., (1996) Linear Vibration Analysis of Laminated Rectangular PlatesusingtheHierarchical FiniteElement Method I: FreeVibrationAnalysis, Computers and Structures, Vol. 61, pp. 705-712.[3] Ganesan, R., andNigam, A., (2002) VibrationAnalysisof Variable-Thickness Composite Beams using Hierarchical Finite Elements,Seventeenth Annual Technical Conference of American Society for Composites, West Lafayette.[4] Ganesan, R., andArshad, W., (2006) Hierarchical FiniteElement Analysis of CurvedComposite Beams, American Society for Composites, 21st Annual Technical Conference, Dearborn, MI, USA. [5] Whitney, J. M., (1987) Structural Analysis of Laminated Anisotropic Plates, Lancaster, PA., Technomic Publishing Company.