Free Vibration Analysis of Rectangular Plates Using a New Triangular Shear Flexible Finite Element

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1 International Journal of Emerging Engineering Research and echnolog Volume Issue August PP - ISSN - (Print) & ISSN - (Online) Free Vibration Analsis of Rectangular Plates Using a New riangular Shear Fleible Finite Element Mihir Chandra Manna Department of Aerospace Engineering and Applied Mechanics Indian Institute of Engineering Science and echnolog Shibpur Howrah India mcmbecdu@gmail.com Abstract: In this paper free vibration analses of isotropic and composite rectangular plates with different thickness ratios aspect ratios boundar conditions and stacking sequences as applicable have been investigated using a new triangular shear fleible finite element. First-order shear deformation theor (FOSD) is used to include the effect of transverse shear deformation. he element has si nodes on the sides and fort five degrees of freedom.he geometr of the element is epressed b shape function in terms of area coordinates. he finite element formulation is displacement-based. Numerical eamples are presented to show the accurac and eleganc of the proposed element. he non-dimensional frequenc parameters of isotropic and composite plates under different thickness ratios aspect ratios etc. are presented in tabular form along with the available published results. Kewords: Rectangular plates FOSD transverse shear degrees of freedom.. INRODUCION Free vibration analsis of plate is ver important in the field of structural engineering because of its wide application in practical life. he finite element method is regarded as one of the most versatile analsis tools specificall in structural mechanics problems. he analsis of plates and shells are the first problems where the finite element method was first applied. Kirchoff s hpothesis was utilized where a number of problems were faced. he major problem concerned the satisfaction of normal slope continuit at the element edges which could not be solved satisfactoril. In the subsequent stud the above problem has been avoided b adopting Mindlin s hpothesis where the effect of shear deformation has been considered. Man eact solutions for elastic thin isotropic plate for bending have been well documented in imoshenko s monographs. However the analtical solution of plates with higher thickness ratio is not sufficient in literatures. Analsis of rectangular and triangular plates with different thickness ratios is quite vast. But there are limited numbers of literatures on triangular plates particularl for higher thickness ratios. Some papers have been discussed below which related to present work. Zienkiewiczet al. have proposed a discrete shear triangular element which is formulated based on Discrete Kirchhoff heor. Chen et al. proposed two refined triangular thin/thick plate elements the conforming displacement element DKM with one point quadrature for the part of shear strain and the element RDKM with the re-constitution of the shear strain based on the Mindlin/Reissner plate theor. In the formulations the eact displacement function of the imoshenko's beam is used to derive the element displacements of the refined elements.hankamet al. described the buckling behavior of rectangular laminate plates subject to thermal loads. he element is based on the transverse displacement field.gauss Quadrature formula is used to compute element matrices. Shufrinet al. have been investigated the free vibration of rectangular thick plates with variable thickness and different boundar conditions b using the etended Kantorovich method. wo shear deformation theories which included the effect of both transverse shear stresses and rotar inertia have been applied to the analsis. Sang Wook Kang et al. proposed a practical analtical method for the free vibration analsis of a simpl supported rectangular plate with unidirectional arbitrar thickness variation. Huang et al. presented a method which is developed for analzing the free vibration problem of orthotropic rectangular plates with variable thickness. IJEER

2 Mihir Chandra Manna Cai et al. proposed a new three node triangular plate element labeled as DS-S (Discrete Shear riangular element with etra Shear degrees of freedom) for the analses of plate/shell structures comprising of thin or thick members. he formulation is based on the DK (Discrete Kirchhoff echnique) and an appropriate use of the independent shear DOF. Liewet. Al. presented the comprehensive sets of accurate vibration frequencies for thick rectangular plates subjected to boundar conditions involving all possible combination of clamped simpl supported and free edges. In this stud sets of mathematicall complete two-dimensional polnomials are assumed in the displacement and rotational functions to approimate mode shapes. he energ function has been derived using the Raleigh-Ritz procedure which leads to the governing eigen value equations. Sets of reasonabl accurate vibration frequencies are presented for a wide range of aspect ratios a/b and relative thickness t/b for each boundar condition. Liew presented an analsis of free fleural vibration of thick smmetric rectangular laminates in which the Ritz method with a set of admissible beam characteristic orthogonal polnomials is used. A simple first order Reissner/Mindlin shear deformation theor was emploed to account for the transverse shear effects. Choo et al. developed two plate bending element one with DOF triangular and other one with DOF quadrilateral based on the hbrid-refftz method. Among the two independent displacement fields i.e. the internal and the boundar displacements the used the Mindlin Reissner s thick plate solution with a particular solution under pressure load as the internal displacement field. Boundar displacement fields are approimated as transverse displacement and rotations and b cubic and quadratic hierarchical shape functions respectivel. ransverse shear strains are derived from constitutive equations and equilibrium equations respectivel and additional degrees of freedom of hierarchical shape functions are removed using the relations between these two shear strains. Brasile presented a new assumed stress triangular element for Reissner Mindlin plates called IP with three nodes and three degrees of freedom per node. he kinematics is constructed b means of the so-called linked interpolation ruled b technicall significant degrees of freedom (i.e. one transversal displacement and two rotations per node) without using additional bubble modes. he static representation starts from a moment shear uncoupled polnomial approimation and is constrained to satisf some equilibrium conditions in order to reduce the stress parameters to a minimum number. Zhuang et al. have developed a new locking-free triangular thick plate element with standard kinematic degrees of freedom and additional degrees of freedom for shear strains for analzing plate/shell structures of thin or thick members. In the present work free vibration of isotropic and composite rectangular plates with different thickness ratios aspect ratios boundar conditions and stacking sequences has been analzed using a new triangular plate element. o include the effect of transverse shear deformation first-order shear deformation theor (FOSD) is used. he element has si nodes on the sides and fort five degrees of freedom. Numerical eamples are solved to show the accurac and eleganc of the proposed element. he non-dimensional frequenc parameters of isotropic and composite plates under different thickness ratios aspect ratios etc. are presented in tabular form along with the available published results.. FINIE ELEMEN FORMULAION Fig.. Element configuration International Journal of Emerging Engineering Research and echnolog

3 Free Vibration Analsis of Rectangular Plates Using a New riangular Shear Fleible Finite Element he formulation is based on the Reissner-Mindlin plate theor. In this theor it is assumed that the transverse deflection of the plate is small compared to the plate thickness and the normal to the plate mid surface which is taken as the reference plane remains straight but ma not remain normal to the deformed mid surface. he proposed element is shown in Fig.. he element has eight degrees of freedom ( u v w θ θ θ γ γ ) degrees of freedom ( u v w ) midpoints of the sides of the element. he natural co-ordinates of the nodes are ( ) ( ) ( ) (. ) (.. ) (.) at nodes - and seven θ θ γ γ at nodes -. Nodes - are at the vertices and - at and. he co-ordinates of an point within the element with respect to the global co-ordinate sstem are given b = ( ξ η) + ξ + η ( ) where a = = a = a = ξ η + ξ + η + b + c a ξ = + b + c η = Δ Δ a () c = b = b = b = and c = c = ( a a a ) Δ= + +. he field variables i.e. the independent displacement components at the reference plane ma be epressed as follows: = [ ]{ α } = [ ]{ α } [ ]{ α } u N u v N v and γ = [ N ]{ αγ } where = γ [ N ]{ αγ } w N w = { α } = [ α α α α α α ] { α } = [ α α α α α α ] u { α } [ α α α α α α... α ] w = { α } [ α α α α α α ] v γ = { α } [ α α α α α α ] [ N ] = ξ η ξ ξη η and N = ξ η ξ ξη η ξ ξ η ξη η ξ ξ η ξ η ξ η η ξ ξ η ξ η ξ η ξη η. Using Eqs. () and () θ and θ ma be written as γ = () { } w w ξ w η θ = + γ = + + N ξ η αγ b = + + Δξ Δη b [ N] [ N] { αw} [ N]{ αγ } { αw} N { αγ } = N + + () International Journal of Emerging Engineering Research and echnolog

4 Mihir Chandra Manna ξ η N ξ η { } c [ N] [ N] { αw} [ N]{ αγ Δη } { αw} N { αγ } w w w θ = + γ = + + αγ c = + + Δξ = N + () Again using Eqs. () and() θ ma be epressed as θ θ θ ξ θ η θ ξ θ η θ = + = ξ η ξ η bc bc + bc bc = N + N + N w Δ ξ Δ ξη Δ η [ ] [ ] [ ] { α } c c b b Δξ Δη Δξ Δη [ N] [ N] { αγ } [ N] [ N ] { αγ } { α } = N w + N αγ + N αγ { } { } ( b b Δξ Δη [ N ] = [ N ] + [ N ] [ N ] = [ N ] + [ ] ) c c N Δξ Δη bc bc + bc bc Δ ξ Δ ξη Δ η [ N ] = [ N ] + [ N ] + [ ] c c Δξ Δη N b b N Δξ Δη [ N ] = [ N ] + [ N ] and [ N ] = [ N ] + [ ]. Eqs. () () ma be assembled in matri form as [ ] [ ] [ ] [ ] [ ] u N N N N N v [ ] [ ] [ ] [ ] [ ] N N αu αu w [ N] [ N] [ N] [ N] [ N] α v α v θ [ N] [ N] [ N] [ N] [ N] αw [ ] w θ = N α [ N] [ N] [ N] [ N] [ N] = A αγ α γ θ [ N] [ N] [ N] [ N] [ N] αγ [ N] [ N] [ N] [ N] [ N] αγ γ γ [ N] [ N] [ N] [ N] [ N] and [ ] [ ] [ ] [ ] [ ] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ ] [ ] [ ] [ ] [ ] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N ] [ N ] [ N ] [ N ] [ N ] u N N N N N v αu αu w α v α v θ N N N N N αw N α = B θ αγ α γ γ αγ α γ γ where () for nodes () = [ ] w for nodes () where [ N A ] and [ N B ] are matrices of order and respectivel and [ N] and [ N ] are null matrices of order and respectivel. Substituting Eqs. () at nodes and () at nodes the coefficients ( α i i=...) in the displacement functions () can be epressed in terms of nodal unknowns as { δ} = [ A]{ α} or { α} = [ A] { δ} () International Journal of Emerging Engineering Research and echnolog

5 Free Vibration Analsis of Rectangular Plates Using a New riangular Shear Fleible Finite Element where () () () () () () A A A B B B [ A] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] = {} α = { α α α α α α α α α }... and the matri [ A ] having an order of {} { δ = u v w θ θ θ γ γ u v w θ θ γ γ can be formed with ( i i ) and ( ξi η i) of si nodes.... }... According to first order shear deformation theor (FOSD) the displacement components of a point at a distance of z from the reference plane ma be epressed in terms of field variables ( ) uvwγ γ at the reference plane as w ( ) u( z) = u( ) z + γ w ( ) v( z) = v( ) z + γ and wz ( ) w ( ) =. he generalized stress-strain relationship ma be epressed as { σ} = [ D]{ ε}. () In the above equation the generalized stress vector is { } { N N N M M M Q Q} σ =. () he generalized strain vector {} ε { ε} = { ε ε ε κ κ κ εz εz} in terms of displacement fields is u v u v θ θ = + θ γ γ () and the rigidit matri [ D ] is given b A A A B B B A A B B sm. A sm. B D D D D = D D sm. D A. A sm sm. A [ ] () he rigidit matri of a laminate [ D ] is constituted with the contributions of its individual laminae and the material properties ( E E ν G G G) and the fiber orientation of each lamina. Now substituting Eqs. () () and () in Eq. () the strain-displacement relationship ma be epressed as { ε} = B { α} () where B is a ( ) matri and is given b International Journal of Emerging Engineering Research and echnolog

6 Mihir Chandra Manna [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] () [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N ] [ N ] [ N ] [ N ] [ N ] B = where =. [ N ] = [ N ] and [ N ] [ N ] Finall the strain vector { ε } can be epressed as {} ε = B [ A] { δ} or { ε} = [ B]{ δ} () Once the strain displacement matri [ B ] and the rigidit matri [ D ] are obtained the stiffness matri can be obtained following the virtual work technique and it ma be epressed as K e B = D B [ ] [ ] [ ][ ]d d In a similar manner the consistent mass matri for an element can be epressed as e = () [ M ] [ A] [ C ] [ ρ ][ C ][ A] dd and the equation of motion of an element ma be epressed as e e ω M = [ K] [ ]. () where [ ] and [ ρ ] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ N] [ ] [ ] [ ] [ ] [ ] [ N] [ N] [ N] [ N] [ N] [ N ] [ N ] [ N ] [ N ] [ N ] C = N N N N N ρh ρh ρh = ρh. ρh In the above equation matrices [ C ] and [ ρ ] are of order and respectivel and ρ and h are the densit of the plate/laminate material and thickness of the plate/laminate respectivel. Finall the element stiffness and mass matrices are assembled together to form the overall stiffness matri [ K ] and mass matri [ M ] respectivel. Integration in the above equations () and () is carried out using Gauss quadrature method.. RESULS AND DISCUSSIONS Isotropic uniform thickness rectangular plates having different thickness ratios aspect ratios and different boundar conditions have been considered. he required Fortran program is developed based on the formulation discussed in the previous section for free vibration analsis of isotropic plates as well as composite plates. Analses have been performed both on free vibration of isotropic and laminated rectangular plates having different thickness ratios aspect ratios boundar conditions fiber () International Journal of Emerging Engineering Research and echnolog

7 Free Vibration Analsis of Rectangular Plates Using a New riangular Shear Fleible Finite Element orientation angles and number of laers as applicable. he results of the analses have been compared with the available published results. he boundar conditions used in the stud are: a. Boundar line parallel to -ais: Simpl supported condition: = = θ = θ = γ = u w Clamped condition: = = = θ = θ = θ = γ = γ = u v w Smmetric condition: θ θ γ b. Boundar line parallel to -ais: v= = = =. Simpl supported condition: = = θ = θ = γ = v w Clamped condition: = = = θ = θ = θ = γ = γ = u v w Smmetric condition: θ θ γ. Isotropic Rectangular Plates u= = = =. ( ) he Poisson s ratio ν and the shear correction factor ( κ ) of the plate material are taken as. and / respectivel unless specified. he eigenvalue tests of the stiffness matri of a single element having different configurations have been performed and it has been found that the element is free from an spurious modes. Net the free vibration analsis of rectangular isotropic plates having different boundar conditions is performed. he boundar condition is designated as SCFC if the boundar line parallel to -ais and = is simpl supported the boundar line parallel to -ais and =a is clamped the boundar line parallel to -ais and = is free and the boundar line parallel to - ais and =b is clamped respectivel. Mesh division is designated as ( m n) if the plate is divided into m equal divisions along the -direction and n equal divisions along the -direction as shown in Fig.. An isotropic rectangular plate is first studied to validate the programme for different mesh divisions boundar conditions and thickness ratios ( t/ b=.and.) and aspect ratios ( a / b =. and.). he mesh arrangement used is shown in Fig. which is also used in the subsequent problems. he plate is analzed for different boundar conditions i.e. CFFFCSCS and CSFF. hefirstfournon-dimensional frequenc parameters able. Frequenc parameters conditions ( λ= ωb / π ρh / D) for isotropic rectangular plates with different boundar a/b h/b Source Frequenc modes CFFF CSCS CSFF PS PS PS PS Liew et al. [] PS PS PS PS Liew et al. [] International Journal of Emerging Engineering Research and echnolog

8 Mihir Chandra Manna... ( λ= ωb / π ρt/ D PS PS PS PS Liew et al. [] PS PS PS PS Liew et al. [] ) obtained b the present model are presented in able with the results b Liew et al.. Liewet al. has used mathematicall complete two-dimensional polnomials in the displacement and rotation functions to approimate the appropriate mode shapes with the Raleigh- Ritz procedure. he consistent mass matri is used for the present analsis. able shows a close agreement between the results. Fig.. Isotropic plate having mesh division of m n.. Composite Rectangular Plates he problem of free vibration analsis of composite rectangular plates having different lamina laers is considered. he laminate consists of laers of equal thickness. he plate is analzed b using four different mess sizes i.e. and. he rectangular plate is analzed numericall with different boundar conditions namel SSSS CCCC and CSFF. with different thickness ratios ( tb=. and.) and aspect ratios ( a/ b=. and.). A shear correction factor k= π / is used for all the numerical computations. he material properties of each lamina are taken as E/ E = G=.E G= G=. E ν =. and ν =.. From the eigenvalue tests of the stiffness matri of a single element it has also been found that the element does not possess an spurious mode. λ= ωb / π ρt/ D obtained b the present he first four non-dimensional frequenc parameters ( ) analsis are presented in able with the results b Liew where D = Eh /( νν). Liew performed the analsis of free fleural vibration of thick smmetric rectangular laminates using the Ritz method with a set of admissible beam characteristic orthogonal polnomials. hree-pl five-pl and eight-pl smmetric laminates are studied to investigate the effects of different parameters as mentioned above.hree pl laminated plates having stacking sequence are used for the all ( ) the numerical computations. he first four non-dimensional frequenc parameters are shown in able.able shows that the present results are ver close to the results obtained b Liew. International Journal of Emerging Engineering Research and echnolog

9 Free Vibration Analsis of Rectangular Plates Using a New riangular Shear Fleible Finite Element able. Frequenc parameters ( λ ωb / π ρh/ D ) boundar conditions a/ b.. = for three-pl laminated plates ( ) with different h/b Source Frequenc modes.... PS-. PS-. PS-. PS-. Liew[ ] SSSS CCCC CFFF. PS-. PS-. PS-. PS-. Liew []. PS-. PS-. PS-. PS-. Liew []. PS-. PS-. PS-. PS-. Liew []. CONCLUSIONS A si-node triangular plate bending element with fort-five degrees of freedom has been used to investigate free vibration of isotropic and composite rectangular plates with different thickness ratios aspect ratios boundar conditions and stacking sequences. Considering different mesh divisions ( and ) a comparative stud of present results with those of earlier investigators shows the convergence characteristics and accurac of the present element for thin (thickness ratio of.) to thick (thickness ratio of.) plates. It has also been found that the element is free of shear locking and does not ehibit an spurious modes. REFERENCES [] Zienkiewics O. C. and ailor R. L. he finite element (two volumes) New York McGraw- Hill International Journal of Emerging Engineering Research and echnolog

10 Mihir Chandra Manna [] imoshenko S. P. and Gere J. M. heor of elastic stabilit McGraw-Hill International New York nd Edition. [] imoshenko S. P. and Woinowsk-Kriegers S. heor of plates and Shell McGrraw-Hill International New York nd Edition. [] Zienkiewics O. C. ailor R. L. Papadopoulos P. and Onate E. Plate bending elements with discrete constraints: new triangular elements Comput. Struct.. [] Chen W.J. and Cheung Y.K. Refined -dof triangular Mindlin plate elements Int. J. Numer. Methods Eng.. [] Sitahankam V. Gajbir Singh VenkateswaraRaoG. and Rath A. K. hermal post-buckling behaviour of laminated plates using a shear fleible element based on coupled-displacement field Composite Structure -. [] Shufrin I. and Eisenberger M. Vibration of shear deformable plates with variable thickness first-order and higher-order analses Journal of Sound and Vibration -. [] Sang Wook Kang and Sang-Hun Kim Variation analsis of simpl supported rectangular plates with unidirectionall arbitraril varing thickness Journal of Sound and Vibration -. [] Huang M. Ma X. Q. Sakiama. Matuda H. and Morita C. Free vibration analsis of orthotropic rectangular plates with variable thickness and general boundar conditions Journal of Sound and Vibration -. [] Cai Y.C. ian. L.G. Atluri S.N. A simple locking-free discrete shear triangular plate element Comput.Model. Eng. Sci [] Liew K. M. Xiang Y. and Kitipornchai S. ranseverse vibration of thick rectangular plates I. Comprehensive set of boundar condition Comput. Struct. -. [] Liew K. M. Solving the variation thick smmetric laminates b reissner / mindlin plate theor and P-Ritz method J. Sound Vib. () -. [] Choo Y.S. Choi N. and Lee B.C. A new hbrid refftz triangular and quadrilateral plate elements Appl. Math. Model.. [] Brasile S. An isostatic assumed stress triangular element for Reissner-Mindlin plate-bending problem Int. J. Numer. Methods Eng.. [] Zhuang X.Y. Huang R.Q. Zhu H.H. Askes H. and Mathisen K. A new and simple locking-free triangular thick plate element using independent shear degrees of freedom Finite Elements in Analsis and Design. International Journal of Emerging Engineering Research and echnolog

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