American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 © Global Society of Scientific Research and Researchers http://asrjetsjournal.org/ Free Vibration Analysis of Isotropic Plates by Alternative Hierarchical Finite Element Method Based on Reddy’s C1 HSDT S. M. N. Serdouna*, S. M. Hamza-Cherifb a,bFaculty of Engineering, Department of Mechanical Engineering, University of Tlemcen, B.P. 230, Tlemcen 13000, Algeria a Serdounn2006 @hotmail.com bsmhamzacherif@yahoo.fr Abstract This paper presents the free vibration analysis of isotropic thick rectangular plates, based on higher order shear deformation theory (HSDT). The plate theory ensures a zero shear-stress condition at the top and bottom surfaces of the plate, and do not requires a shear correction factor. The model requires inter-element C1 continuity for the transverse displacement. To overcome this hindrance, a new hierarchical p-element with six degrees of freedom per node is developed and used to find natural frequencies of thick plates. Convergence studies and comparison have been carried out for with different boundaries conditions. It is shown that the present element enables rapid convergence. Keywords: Free vibration; Thick isotropic plates; hierarchical finite element method; third order C1 HSDT. 1. Introduction Thick plates are extensively used in many fields of engineering, including aerospace, civil structures, hydraulic structures, etc. For plates analysis different theories exists, the classical plates theory (CPT) is adopted for thin plates, where the effect of shear deformation is neglected [1]. The Reissner. Mindlin plate theory is used for moderately thick plates, known as the first order shear deformation theory (FSDT), in which the effect of shear deformation is considered by using a proper choice of a shear correction factor which depend on loading and boundary conditions [2]. The simplifying assumptions made in CPT and FSDT are reflected by the high percentage errors in the results of thick plates analysis. For these plates, higher-order shear deformation theories (HSDT) are required. The HSDT ensure a zero shear-stress condition on the top and bottom surfaces of the plate, and do not require a shear correction factor, which is a major feature of these theories. * Corresponding author. 1 http://asrjetsjournal.org/ American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 9, No 1, pp 1-19 Nelson and Lorch [3], the authors in [4] presented a HSDT for laminated plates however the displacement field does satisfy the shear-stress free condition on the top and bottom surfaces of the plate. Lewinson [5], Murthy [6], and Reddy [7] presented a new higher order shear deformation theories considered as an extension of hencky’s theory, which include a realistic displacement field satisfying the conditions of zero transverse shear- stress and/or strains, known as Reddy’s third-order theory. This model requires C1 inter element continuity requirement. Phan and Reddy developed a non conforming rectangular element with seven degrees of freedom per node, based on C1 Reddy’s third order theory to analyze laminated composites plates. Kant and his colleagues [8] investigate the free and transient vibration analysis of composites and sandwich plates based on a refined theory by using the finite element method and analytical solution. The authors in [9,10] investigate the free vibration and transient response of composite sandwich plates by using two C0 assumed strain finite element based on Reddy’s third-order theory. Sheikh and Chakrabarti [11] used a triangular element based on Reddy’s higher order shear deformation plate theory. Batra and his colleagues . [12] used a HSDT and the finite element method to analyze free vibrations and stress distribution in tick isotropic plate. Kulkarni and Kapuria [13] used a discrete Kirchoff quadrilateral element based on the third order theory for composite plates. Because Reddy’s third-order theory requires inter element C1 continuity on the transverse displacement. The conclusion can be made from the literature review, that a very few conforming element based on this plate theory are developed. To overcome this hindrance, the hierarchical finite element method can be used. In the hierarchical finite element method the mesh keeps unchanged and the polynomial degree of the shape functions is increased. See for instance Szabo and Sahrmann [14], Szabo and Babuska [15] and Hamza-Cherif [16]. In this paper we address these above-mentioned points. The new approach with hierarchical finite element method is formulated for thick plates vibration analysis. A new hierarchical p-element with six degrees of freedom per node is developed, based on the C1 higher order shear deformation theory. The continuity along the inter-element boundary is not required in the model. To demonstrate the convergence and accuracy of the proposed method, present results are compared with existing data available from other analytical and numerical methods. Then, natural frequencies of rectangular plates under different boundary conditions are tabulated for a wide range of aspect ratios and thickness to length ratios. 2. Formulation 1.1. Energy formulation Consider an homogeneous, isotropic, thick plate bounded by 0 ≤ x ≤ a, 0 ≤ y ≤ b , and −h/2 ≤ z ≤ h/2, as show in Fig. 1. The displacement of the plate are decomposed into three orthogonal components, u, v and w parallel to the x- axis ,y-axis and z-axis, respectively. In accordance with the higher-order shear deformable theory [10,11], the displacements can be expressed as 2 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 9, No 1, pp 1-19 ( ) 0 x x wu z θ   θ x f z ∂ = − + ∂  ( ) 0 y wv z θ   θyy f z  ∂ = − + ∂  0w= w In which ( ) 3 2 4 3 zf z h = (2) Where w0 is the transverse displacement of middle plate components and θx , θy are the rotations of the normal to the middle plane about the x-axis and y-axis respectively. The linear strain-displacement relationships. { } ( ) ( ) 2 0 2 2 0 20 0 0 0 0 0 0 0 0 00 0 x x xx y yy y y yz xz x x xy y x θ w θ x x xε θ w ε w wy yθ θγε z  y y zγ w wθ θ γ x xθ θ f x y z f z ∂ ∂ ∂  +    ∂ ∂ ∂      ∂    ∂      ∂ ∂   ∂ ∂∂+ +       = = + − −∂ ∂       ∂       ∂ ∂       + + ∂ ∂         ∂ ∂     +    ∂ ∂  2 0 0 0 2y x yθ y θ θ w x y xy       ∂  + ∂          ∂ ∂ ∂ + + ∂ ∂ ∂  (3) The constitutive equations for linear elastic isotropic material are {σ} = [C]{ε} (4) In the case of plane stress the stress vector can be written as {σ} = {σxx σxx τyz τxz τxy } (5) Where [C] = ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ E 1−ν2 ν E 1−ν2 0 0 0 ν E 1−ν2 E 1−ν2 0 0 0 0 0 E 2(1+ν) 0 0 0 0 0 E 2(1+ν) 0 0 0 0 0 E 2(1+ν) ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ (6) (1) 3 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 9, No 1, pp 1-19 Here Ci,j are the material property coefficients, in which E, and v are the Young’s modulus and, Poisson’s ratio, respectively. The kinetic energy of a bending vibrating thick plate is given by 21 1 20 0 2 0 0 2 2 20 0 02 1 1 2 2 1 2 D D x F x D D y F y A w wEc a b a a w w w d d a a ρ ρ θ ρ θ ξ ξ ρ ρ θ ρ θ ρ ξ η η η   ∂  ∂ = + +  ∂ ∂   ∂  ∂ + + + +  ∂ ∂   ∫ ∫          (7) Where ρ is the mass density per unit volume. / 2 / 2 , h A h dzρ ρ − = ∫ ( ) / 2 2 / 2 , h D h f z dzρ ρ − = ∫ ( ) ( )( ) / 2 2 2 / 2 2 h F h f z z f z z dzρ ρ − = − +∫ The strain energy of a thick plate is expressed as 2 21 1 2 2 2 2 0 0 0 011 11 11 11 4 2 4 2 3 2 3 2 0 0 22 2 2 2 0 0 011 11 12 12 2 2 2 2 2 2 2 2 2 21 2 2 2 yx yx x w w w wD D E EEd ab a b a a w w wF F D E a b a b ab θθ ξ η ξ ξ η η θθ θ ξ η ξ η η ξ  ∂       ∂ ∂ ∂ ∂ ∂ = + + +       ∂ ∂ ∂ ∂ ∂ ∂         ∂       ∂  ∂ ∂ ∂ ∂ + + + +      ∂ ∂ ∂ ∂ ∂ ∂         ∫ ∫    2 22 2 2 0 0 012 12 12 122 2 4 2 4 2 222 2 0 012 12 12 12 3 2 3 2 2 2 2 2 33 0 0 2 2 2 2 2 2 2 2 4 y yx y yx x w w wE D DF a b a b w wE E F F a b b b D w w a b θ θθ ξ η ξ η ξ η θ θθ θ ξ ξ η η ξ η ξ η ∂ ∂      ∂ ∂ ∂ ∂ + + + +      ∂ ∂ ∂ ∂ ∂ ∂       ∂ ∂     ∂ ∂ ∂  ∂  + + + +      ∂ ∂ ∂ ∂ ∂ ∂        ∂ ∂ +  ∂ ∂ 2 2 33 0 33 0 2 2 22 2 33 33 33 33 0 33 0 2 2 2 2 2 0 044 44 33 442 4 4 2 2 2 yx y yx x x x y H w H w ab a b G G G I w I w b a ab a a w wI II I b b θθ ξ η η ξ η ξ θ θθ θ θ η ξ η ξ ξ ξ θ θ η η ∂   ∂ ∂ ∂ + +    ∂ ∂ ∂ ∂ ∂ ∂     ∂ ∂    ∂  ∂  ∂   ∂  + + + + +        ∂ ∂ ∂ ∂ ∂ ∂          ∂   ∂  + + + +   ∂ ∂    2 y d dθ ξ η    (11) where ( ) / 2 2 , , / 2 , h i j i j h D C f z dz − = ∫ ( ) ( )( ) /2 2 , , /2 2 h i j i j h E C f z z f z dz − = −∫ (8, 9, 10) (12, 13) 4 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 9, No 1, pp 1-19 ( ) ( )( ) /2 2 2 , , /2 2 h i j i j h F C f z z f z z dz − = − +∫ , ( ) ( )( ) /2 2 , , /2 2 1 h i j i j h G C f z f z dz − = − +∫ ( ) ( )( ) /2 2 , , /2 h i j i j h H C f z f z dz − = −∫ , ( ) ( )2/2 , , /2 2 1 h i j i j h f z f z I C dz z z−  ∂ ∂   = − +  ∂ ∂   ∫ Where ξ(= x/a) and η(= y/b) are the non-dimensional coordinates. 2.1. Hierarchical finite element formulation A fournode rectangular hierarchical finite element with six degrees of freedom per node (𝑤𝑤0, 𝜕𝜕𝑤𝑤0/𝜕𝜕𝜕𝜕, 𝜕𝜕𝑤𝑤0/𝜕𝜕𝜕𝜕, 𝜕𝜕𝑤𝑤0/𝜕𝜕𝜕𝜕𝜕𝜕, 𝜃𝜃𝜕𝜕 , 𝜃𝜃𝜕𝜕 ) is developed on the basis of a third-order plate theory (See Fig. 2). Trigonometric hierarchical functions are used as shape functions. The model requires C0 continuity for 𝜃𝜃𝜕𝜕 and 𝜃𝜃𝜕𝜕 and C1continuity for 𝑤𝑤0. The displacement and rotations of the rectangular plate p-element are expressed as 𝑤𝑤0(𝜉𝜉, 𝜂𝜂, 𝑡𝑡) = � �𝑊𝑊𝑚𝑚𝑚𝑚 (𝑡𝑡)𝑔𝑔𝑚𝑚(𝜉𝜉)𝑔𝑔𝑚𝑚(𝜂𝜂) 𝑃𝑃𝑤𝑤 𝑚𝑚=1 𝑃𝑃𝑤𝑤 𝑚𝑚=1 𝜃𝜃𝜕𝜕(𝜉𝜉, 𝜂𝜂, 𝑡𝑡) = � �𝜃𝜃𝜕𝜕𝑚𝑚𝑚𝑚 (𝑡𝑡)𝑓𝑓𝑚𝑚(𝜉𝜉)𝑓𝑓𝑚𝑚(𝜂𝜂) 𝑃𝑃𝜃𝜃 𝑚𝑚=1 𝑃𝑃𝜃𝜃 𝑚𝑚=1 (18) 𝜃𝜃𝜕𝜕(𝜉𝜉, 𝜂𝜂, 𝑡𝑡) = � �𝜃𝜃𝜕𝜕𝑚𝑚𝑚𝑚 (𝑡𝑡)𝑓𝑓𝑚𝑚(𝜉𝜉)𝑓𝑓𝑚𝑚(𝜂𝜂) 𝑃𝑃𝜃𝜃 𝑚𝑚=1 𝑃𝑃𝜃𝜃 𝑚𝑚=1 Where 𝑃𝑃𝑤𝑤 and 𝑃𝑃𝜃𝜃 are the number of shape functions used in the model. The firsts shape functions (𝑓𝑓1 𝑓𝑓2 and 𝑔𝑔1to𝑔𝑔4) are commonly used in the finite element method. The functions (𝑓𝑓𝑚𝑚+2 and 𝑔𝑔𝑚𝑚+4 )are the trigonometric shape functions and lead to zero transverse displacement, and zero slope at each node. This feature is highly significant since these functions only give additional freedom to the edges and the interior of the element. The trigonometric hierarchical shape functions 𝑓𝑓𝑖𝑖(𝜉𝜉) for C0 continuity and 𝑔𝑔𝑖𝑖(𝜉𝜉)for C1 continuity are given by [24]         = = = = −= + ,...3,2,1 )(sin 1 2 2 1 r rr rf f f n πδ ξδ ξ ξ (19) (14, 15) (16, 17) 5 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 9, No 1, pp 1-19 and ( )( ) ( )( )[ ] ( )            = = +−+−−++−= +−= −= +−= +−= + ,...3,2,1 sin1112 23 2 231 32 4 32 4 32 3 32 2 32 1 r rr rrg g g g g rr n πδ ξδξξξδ ξξ ξξ ξξξ ξξ (20) The displacement and rotationscan be expressed in matrix form � 𝑤𝑤0 𝜃𝜃𝜕𝜕 𝜃𝜃𝜕𝜕 � = [𝑁𝑁]{𝑞𝑞} (21) [N] is the matrix of the shape functions, given by [𝑁𝑁] = � [𝑁𝑁𝑤𝑤 ] 0 0 0 [𝑁𝑁𝜃𝜃 ] 0 0 0 [𝑁𝑁𝜃𝜃 ] � (22) where {𝑞𝑞} = � 𝑞𝑞𝑤𝑤 𝑞𝑞𝜃𝜃𝜕𝜕 𝑞𝑞𝜃𝜃𝜕𝜕 � (23) In which 𝑞𝑞𝑤𝑤 , 𝑞𝑞𝜃𝜃𝜕𝜕 and 𝑞𝑞𝜃𝜃𝜕𝜕 are the generalized displacements. The matrices of theshape functions are given by [ ] ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( )1 1 1 2 P P1 2 P P , ,... ,... w w w w w k l r N g g g g g g g gξ η ξ η ξ η ξ η =    (24) where 1 wk ,...,P= , 1 wl ,...,P= , and ( )1 wr j i P= + − and 6 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 9, No 1, pp 1-19 [ ] ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( )1 1 1 2 P P1 2 P P , ,... ,...i j m N f f f f f f f f θ θ θ θ θ ξ η ξ η ξ η ξ η =    (25) where 1i ,...,Pθ= , 1j ,...,Pθ= ,and ( )1m j i Pθ= + − . The discretized system of equations of bending of free vibration of isotropic plate can be expressed as [ ]{ } [ ]{ } 0=+ qKqM  (26) Here [K] is the stiffness matrix of the p-element, determined from the strain energy [𝐾𝐾] = � [𝐾𝐾𝑤𝑤𝑤𝑤 ] [𝐾𝐾wθx] �𝐾𝐾wθy� [𝐾𝐾wθx]𝑇𝑇 [𝐾𝐾θxθx] �𝐾𝐾θxθy� �𝐾𝐾wθy� 𝑇𝑇 �𝐾𝐾θxθy� 𝑇𝑇 �𝐾𝐾θyθy� � (27) Where [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 1 1 2 2 2 2 2 2 11 12 22 4 2 2 2 2 2 2 4 2 2 0 0 33 44 44 2 2 2 2 2 2 2 4 T T T w w w w w w ww T T T w w w w w w N N N N N ND D D K ab a a b b N N N N N ND F F d d a b a b a b ξ ξ ξ η η η ξ η ξ η ξ η ξ ξ η η  ∂ ∂ ∂ ∂ ∂ ∂ = + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + +  ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂  ∫ ∫ (28) [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 1 1 2 2 2 3311 12 2 2 2 2 2 0 0 44 wθx 22 2 2 T T T w w w T w N N N N N NEE E ab a a b ab NF N d d a K θ θ θ θ ξ ξ ξ η ηξ η ξ η ξ  ∂ ∂ ∂ ∂ ∂ ∂ = + + ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ +  ∂  ∫ ∫ (29) [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 1 1 2 2 2 3312 22 2 2 3 2 2 0 0 55 wθy 42 2 2 T T T w w w T w N N N N N NEE E a b a b b a b NF N d K d a θ θ θ θ η η ξ η ξξ η ξ η η  ∂ ∂ ∂ ∂ ∂ ∂   = + +  ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ +  ∂  ∫ ∫ (30) 7 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 9, No 1, pp 1-19 [ ] [ ] [ ] [ ] [ ] [ ] [ ] 1 1 3311 442 0 θxθ 2 0 x T T TN N N NGG ab F N N d d a b K θ θ θ θ θ θ ξ η ξ ξ η η  ∂ ∂ ∂ ∂ = + +  ∂ ∂ ∂ ∂  ∫ ∫ (31) [ ] [ ] [ ] [ ] [ ] [ ] 1 1 33 22 44θyθy 2 2 0 0 T T TN N N NG G ab F N N d d a b K θ θ θ θ θ θ ξ η ξ ξ η η  ∂ ∂ ∂ ∂   = + +   ∂ ∂ ∂ ∂  ∫ ∫ (32) [ ] [ ] [ ] [ ]1 1 3312 0 θ 0 xθy 22 T TN N N NGG ab d d ab ab K θ θ θ θ ξ η ξ η η ξ  ∂ ∂ ∂ ∂   = +   ∂ ∂ ∂ ∂  ∫ ∫ (33) [M]is themass matrix of the p-element,given by the following relation [𝑀𝑀] = � [𝑀𝑀𝑤𝑤𝑤𝑤 ] [𝑀𝑀wθx] [𝑀𝑀wθy] [𝑀𝑀wθx]𝑇𝑇 [𝑀𝑀θxθx] [𝑀𝑀θxθy] [𝑀𝑀wθy]𝑇𝑇 [𝑀𝑀θxθy]𝑇𝑇 [𝑀𝑀θyθy] � (34) Where [ ] [ ] [ ] [ ] [ ] [ ] [ ]1 1 2 2 0 0 T T T w w w wE E ww A w w N N N N M a b N N d d a b ρ ρρ ξ η ξ ξ η η  ∂ ∂ ∂ ∂ = + +  ∂ ∂ ∂ ∂  ∫ ∫ (35) [ ] [ ] [ ] 1 1 0 0 T w w x G N M a b N d dθ θρ ξ η ξ ∂ = ∂∫ ∫ (36) [ ] [ ] 1 1 0 0 T w w y G N M ab N d dθ θρ ξ η η ∂   =  ∂∫ ∫ (37) [ ] [ ] [ ]θxθx θy 1 1 0 θy 0 T HM M ab N N d dθ θρ ξ η = =  ∫ ∫ (38) 3. Numerical results and discussion 3.1. Convergence study Tables 1 to 5 show that good convergence and accuracy of the solutions are obtained by increasing the number of trigonometric shape functions, for all cases. It is seen that good results from thick plates are obtained by 8 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 9, No 1, pp 1-19 using, only six shape function in the case of SSSS plates, 14 shape functions in the case CCCC plates,12 shape functions in the case of FFFF plates, and 17 shape functions in others cases. Table 1.Convergence of frequency parameters Ω = ω b2 π2 � ρ h D for a thick plates (ν = 0.3, a/b = 1, h/a = 0.5) with SSSS boundary condition. Pθ = Pw 1 2 3 4 5 4 1.263 2.47 2.47 2.919 2.919 5 1.246 2.414 2.414 2.919 2.919 6 1.245 2.308 2.308 2.919 2.919 Converged solution 1.245 2.308 2.308 2.919 2.919 Table 2.Convergence of frequency parameters Ω = ω b2 π2 � ρ h D for a thick plates (ν = 0.3, a/b = 1, h/a = 0.5) with CCCC boundary condition. Pθ = Pw 1 2 3 4 5 4 3.623 3.623 4.026 4.557 5.069 5 1.662 3.617 3.617 3.965 4.515 6 1.659 2.639 2.639 3.466 3.695 7 1.618 2.613 2.613 3.448 3.691 8 1.615 2.586 2.586 3.405 3.687 9 1.604 2.575 2.575 3.392 3.686 10 1.601 2.566 2.566 3.379 3.686 11 1.598 2.559 2.559 3.369 3.685 12 1.595 2.556 2.556 3.366 3.685 13 1.595 2.552 2.552 3.358 3.685 14 1.593 2.55 2.55 3.358 3.685 Converged solution 1.593 2.55 2.55 3.358 3.685 Table 3.Convergence of frequency parameters Ω = ω b2 π2 � ρ h D for a thick plates (ν = 0.3, a/b = 1, h/a = 0.5) with FFFF boundary condition. Pθ = Pw 1 2 3 4 5 4 0.905 1.354 1.693 1.939 1.939 5 0.905 1.266 1.514 1.805 1.805 9 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 9, No 1, pp 1-19 6 0.895 1.266 1.514 1.798 1.798 7 0.895 1.265 1.511 1.793 1.793 8 0.893 1.265 1.511 1.792 1.792 9 0.893 1.265 1.511 1.79 1.790 10 0.892 1.265 1.511 1.79 1.790 11 0.892 1.265 1.511 1.789 1.789 12 0.892 1.264 1.511 1.789 1.789 Converged solution 0.892 1.264 1.511 1.789 1.789 Table 4. Convergence of frequency parameters Ω = ω b2 π2 � ρ h D for a thick plates(ν = 0.3, a/b = 1, h/a = 0.5) with CFCF boundary condition. Pθ = Pw 1 2 3 4 5 4 2.883 3.378 3.45 3.56 3.792 5 1.139 1.248 1.947 2.721 3.369 6 1.137 1.241 1.944 2.261 2.450 7 1.114 1.213 1.913 2.252 2.439 8 1.112 1.209 1.91 2.226 2.411 9 1.106 1.202 1.903 2.218 2.403 10 1.103 1.199 1.900 2.210 2.395 11 1.102 1.197 1.898 2.204 2.389 12 1.100 1.195 1.897 2.202 2.387 13 1.100 1.195 1.896 2.198 2.382 14 1.099 1.193 1.895 2.198 2.382 15 1.099 1.193 1.895 2.194 2.379 16 1.098 1.192 1.894 2.194 2.378 17 1.098 1.192 1.894 2.192 2.376 Converged solution 1.098 1.192 1.894 2.192 2.376 Table 5.Convergence of frequency parameters Ω = ω b2 π2 � ρ h D for a thick plates (ν = 0.3, a/b = 1, h/a = 0.5) with CSCS boundary condition. Pθ = Pw 1 2 3 4 5 4 3.418 3.499 3.592 4.331 4.432 5 1.656 3.418 3.485 3.585 4.073 6 1.654 2.611 2.635 3.351 3.521 10 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 9, No 1, pp 1-19 7 1.612 2.580 2.610 3.343 3.517 8 1.609 2.554 2.583 3.304 3.514 9 1.599 2.542 2.572 3.295 3.513 10 1.595 2.533 2.563 3.284 3.513 11 1.593 2.526 2.556 3.276 3.512 12 1.590 2.523 2.553 3.273 3.512 13 1.590 2.519 2.549 3.267 3.512 14 1.587 2.517 2.548 3.267 3.512 15 1.587 2.514 2.545 3.263 3.512 16 1.586 2.514 2.544 3.263 3.512 17 1.586 2.512 2.542 3.260 3.512 Converged solution 1.586 2.512 2.542 3.260 3.512 3.2. Discussion The results obtained for an isotropic plate by applying Higher-order using rectangular p-element, are compared with those available in the literature. The linear natural frequencies of plates with free edges (FFFF), simply supported (SSSS), fully clamped plates (CCCC), and combined boundary condition (CFCF), (SFSF) are considered. The frequency parameter of the plate is expressed as Ω = 𝜔𝜔 𝑏𝑏2 𝜋𝜋2 � 𝜌𝜌 ℎ 𝐷𝐷 , (36) Where D = E h3 12 (1−ν2) is the flexural rigidity of the plate. The first five frequency parameters Ω for simply-supported (SSSS) square plates with different thickness-side ratio h/b = 0.001, 0.1, 0.2, 0.5 computed using the present method are given in Table 6 and compared with other published solutions, Leissa [17], Houmat [24], 3-D exact solution [18], Nayak [9], The authors in 1956 [2], Lim and his colleagues [18], Srinivas and his colleagues [19], Malik and Bert, 1998 [27], Zhou and his colleagues [20], very good agreement can be observed, in the results with concealment compare of Nayak [9] are obtained with 1245 DOF, or ours are obtained with 480 DOF Pθ = Pw = 19, in the example illustrated in table 6, a very good accuracy is observed. In table 7, good agreement is achieved by comparing the present results with those obtained by Wang [21], Leissa [17], Lim and his colleagues [1], Liew and his colleagues [23], Wang [22], The first five frequency parameters Ω for fully clamped (CCCC) square plates with different thickness-side ratio h/b = 0.001, 0.1, 0.2, 0.5. 11 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 9, No 1, pp 1-19 Table 6.Comparison of frequency parameters Ω = ω b2 π2 � ρ h D of a SSSS square plate (ν = 0.3). h/a Solution methods 1 2 3 4 5 0.001 Present (Pθ=Pw=19) 2,000 5,003 5,003 8,004 10,017 CPT-exact [17] 2,000 5,000 5,000 8,000 10,000 0.1 Present (480 dofs) 1,932 4,609 4,609 7.074 8.622 HSDT FEM [9] (1245 dofs) 1,931 4,614 4,614 7.085 8.657 MindlinTheory [2] 1,931 4,605 4,605 7.064 8.607 HSDT [18] 1,932 4,609 4,609 7.073 8.617 3-D exact [19] 1,934 4,622 4,622 7.103 8.662 0.2 Present (Pθ=Pw=12) 1,768 3.870 3.870 5.599 6.619 MindlinTheory [2] 1,766 3.858 3.858 5.573 - 3-D exact [19] 1,756 3.899 3.899 5.653 - 0.5 Present (Pθ=Pw=6) 1,245 2.308 2.308 2.919 2.919 HSDT [18] 1,245 2.308 2.308 2.917 2.917 3-D exact [19] 1,259 - - - - 3-D Ritz [20] 1,259 2.331 2.331 - - Table 7.Comparison of frequency parameters Ω = ω b2 π2 � ρ h D of a CCCC square plate (ν = 0.3). h/a Solution methods 1 2 3 4 5 0.001 Present (Pθ=Pw=24) 3.674 7.506 7.506 11.162 13.488 CPT Ritz (S wang) [21] 3,646 7,436 7,436 10,945 13,332 CPT Ritz [17] 3,647 7,438 7,438 10,970 13,338 0.1 Present (Pθ=Pw=22) 3.306 6.323 6.323 8.879 10.477 HSDT [18] 3,303 6,311 6,311 8,858 10,446 3-D Ritz [22] 3,322 6,346 6,346 8,903 10,498 0.2 Present (Pθ=Pw=22) 2.720 4.786 4.786 6.451 7.384 MindlinTheory [23] 2,681 4,675 4,675 - - 0.5 Present (Pθ=Pw=14) 1.593 2.550 2.550 3.358 3.685 12 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 9, No 1, pp 1-19 HSDT [18] 1,587 2.536 2.536 3.337 3.684 3-D Ritz [22] 1,550 2,515 2,515 3,193 3.654 In table 8, good agreement is achieved by comparing the present results with those obtained by, Liew and his colleagues . [23] and Lim and his colleagues [18], The first five frequency parameters Ω for combined boundary condition (CFCF) square plates with tree different thickness-side ratio h/b = 0.01, 0.1, 0.5. Table 8.Comparison of frequency parameters Ω = ω b2 π2 � ρ h D of a CFCF square plate (ν = 0.3). h/b Solution methods 1 2 3 4 5 0.01 Present (Pθ=Pw=22) 2.245 2.673 4.410 6.189 6.795 3-D Ritz [22] 2.248 2.674 4.408 6.197 6.799 0.1 Present (Pθ=Pw=22) 2.095 2.441 3.920 5.366 5.813 HSDT [18] 2.093 2.438 3.913 5.357 5.802 3-D Ritz [22] 2.105 2.449 3.923 5.386 5.827 0.5 Present (Pθ=Pw=17) 1.098 1.192 1.894 2.192 2.376 HSDT [18] 1.095 1.189 1.891 2.185 2.369 3-D Ritz [22] 1.072 1.193 1.871 2.193 2.325 In table 9, shows a comparison of results for square thin and thick plates by, Houmat [24], and Lim and his colleagues [18], The first five frequency parameters Ω for free edges (FFFF) square plates with different thickness-side ratio h/b = 0.001, 0.1, 0.5. Table 9.Comparison of frequency parameters Ω = ω b2 π2 � ρ h D of a FFFF square plate (ν = 0.3). h/a Solution methods 1 2 3 4 5 0.001 Present (Pθ=Pw=20) 1,365 1,986 2,460 3,529 6,199 HFEM [24] 1,365 1,986 2,459 3,526 6,190 0.1 Present (Pθ=Pw=20) 1,291 1,919 2,363 3,239 3,239 HSDT [18] 1,289 1,919 2,363 3,235 3,235 0.5 Present (Pθ=Pw=12) 0,892 1,264 1,511 1.789 1.789 HSDT [18] 0,891 1,264 1,511 1.788 1.788 13 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 9, No 1, pp 1-19 In table 10, good agreement has been achieved. The attained accuracy is also confirmed by comparing our results, Leissa [17], Hosseini and his colleagues [25,26], Malik [27], Liew and his colleagues [23], The first five frequency parameters Ω = ω a2�ρh/D for combined boundary condition (SFSF) square plates with different thickness-side ratio h/b = 0.001, 0.1,0.2, 0.5. Table 10.Comparison of frequency parameters Ω = ω a2�ρh/D of a SFSF square plate (ν = 0.3). h/a Solution methods 1 2 3 4 5 0.001 Present (Pθ=Pw=17) 9.638 16.142 36.730 39.027 46.860 CPT-exact [17] 9,631 16,135 36,726 38,945 46,738 HSDT Exact [25] 9,631 16,131 36,716 38,943 46,732 0.1 Present (Pθ=Pw=20) 9,442 15,401 33,896 36,367 42,821 Exact FSDT [26] 9,446 15,405 33,916 36,425 42,887 3-D DQM [27] 9,446 15,400 33,911 36,437 42,887 3-D Ritz [22] 9,446 15,400 33,913 36,438 42,887 HSDT Exact [25] 9,446 15,392 33,868 36,349 42,801 0.2 Present (Pθ=Pw=20) 8,984 14,104 29,170 31,296 36,006 Exact FSDT [26] 9,000 14,134 29,256 31,434 36,165 3-D DQM [27] 9,001 14,123 29,263 31,472 36,173 3-D Ritz [22] 9,001 14,122 29,263 31,472 36,173 HSDT Exact [25] 8,984 14,101 29,162 31,293 35,999 0.5 Present (Pθ=Pw=20) 7,121 10,070 18,196 19,470 21,628 3-D Ritz [22] 7,166 10,100 18,210 19,613 21,652 HSDT Exact [25] 7,121 10,069 18,194 19,469 21,626 In table 11 and 12, in has to study the effect of the boundary conditions, thickness ratio and rapport a/b on frequency parameters Ω, These tabulated frequency parameters should be useful as benchmark solutions for researchers who are developing numerical techniques and software for solving isotropic HSDT plate vibration problems. 14 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 9, No 1, pp 1-19 Table 11.Frequency parameter Ω = ω b2 π2 � ρ h D (ν = 0.3) of FFFF, CCCC and CFCF plates. B.C a/b h/a 1 2 3 4 5 FFFF 0.5 0.1 2.096 2.439 5.172 5.466 7.850 0.2 1.917 2.090 4.219 4.477 6.220 0.5 1.201 1.365 2.278 2.408 2.772 1 0.1 1.291 1.919 2.363 3.239 3.239 0.2 1.187 1.763 2.148 2.798 2.798 0.5 0.892 1.264 1.511 1.789 1.789 1.5 0.1 0.865 0.948 1.963 2.166 2.463 0.2 0.809 0.906 1.766 1.978 2.196 0.5 0.630 0.731 1.243 1.402 1.451 2 0.1 0.538 0.646 1.409 1.467 2.149 0.2 0.524 0.609 1.293 1.367 1.962 0.5 0.538 0.646 1.409 1.467 2.149 0.5 0.1 7.872 9.913 13.323 17.15 17.739 0.2 5.569 6.925 9.039 10.885 11.631 0.5 2.863 3.510 4.331 4.541 4.959 1 0.1 3.306 6.323 6.323 8.879 10.477 0.2 2.720 4.786 4.786 6.451 7.384 0.5 1.593 2.550 2.550 3.358 3.685 CFCF 1.5 0.1 2.531 3.811 5.767 5.858 6.831 0.2 2.140 3.111 4.407 4.553 5.149 0.5 1.303 1.804 2.363 2.508 2.755 2 0.1 2.314 2.960 4.086 5.599 5.632 0.2 1.968 2.478 3.331 4.287 4.435 0.5 1.203 1.493 1.939 2.298 2.476 0.5 0.1 2.080 3.152 5.328 6.852 9.214 0.2 1.782 2.473 4.098 5.074 6.851 0.5 1.095 1.253 2.186 2.513 3.434 1 0.1 2.095 2.441 3.920 5.366 5.813 0.2 1.793 2.036 3.194 4.120 4.430 0.5 1.098 1.192 1.894 2.192 2.376 15 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 9, No 1, pp 1-19 1.5 0.1 2.101 2.264 2.907 4.225 5.381 0.2 1.798 1.915 2.41 3.446 4.13 0.5 1.099 1.150 1.443 1.990 2.198 2 0.1 2.105 2.198 2.555 3.265 4.400 0.2 1.800 1.867 2.140 2.703 3.589 0.5 1.100 1.131 1.285 1.619 2.084 Table 12.Frequency parameter Ω = ω b2 π2 � ρ h D (ν = 0.3) of CFFF, CSCS and SFSF plates. B.C a/b h/a 1 2 3 4 5 CFFF 0.5 0.1 1.367 1.997 3.658 6.611 7.332 0.2 1.257 1.730 3.018 5.120 5.321 0.5 0.908 1.120 1.834 2.339 2.499 1 0.1 0.348 0.818 2.038 2.586 2.868 0.2 0.339 0.747 1.790 2.282 2.435 0.5 0.296 0.531 1.125 1.502 1.523 1.5 0.1 0.155 0.502 0.941 1.666 2.294 0.2 0.152 0.467 0.879 1.490 2.057 0.5 0.142 0.346 0.649 1.023 1.380 2 0.1 0.087 0.360 0.534 1.161 1.471 0.2 0.086 0.337 0.513 1.062 1.349 0.5 0.082 0.256 0.417 0.766 0.958 CSCS 0.5 0.1 7.867 9.898 13.297 17.142 17.706 0.2 5.565 6.913 9.023 10.87 11.616 0.5 2.859 3.504 4.319 4.535 4.898 1 0.1 3.302 6.312 6.314 8.854 10.461 0.2 2.710 4.769 4.770 6.412 7.358 0.5 1.586 2.512 2.542 3.260 3.512 1.5 0.1 2.528 3.804 5.762 5.848 6.814 0.2 2.132 3.093 4.395 4.533 5.115 0.5 1.295 1.779 2.359 2.491 2.728 2 0.1 2.313 2.955 4.078 5.596 5.622 0.2 1.962 2.462 3.310 4.279 4.413 0.5 1.194 1.470 1.917 2.294 2.453 SFSF 0.5 0.1 0.945 2.527 3.641 5.716 7.698 0.2 0.900 2.155 3.137 4.575 6.006 0.5 0.718 1.202 1.963 2.414 2.611 1 0.1 0.957 1.560 3.434 3.685 4.338 0.2 0.910 1.429 2.955 3.171 3.648 0.5 0.722 1.020 1.844 1.973 2.191 1.5 0.1 0.964 1.274 2.204 3.702 3.777 0.2 0.917 1.188 1.977 3.185 3.225 0.5 0.726 0.892 1.343 1.945 1.98 2 0.1 0.968 1.153 1.721 2.655 3.711 0.2 0.921 1.085 1.576 2.349 3.193 0.5 0.729 0.831 1.124 1.540 1.984 16 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 9, No 1, pp 1-19 4. Conclusion An hierarchical finite element formulation is presented based on an Higher-order shear plates theory for homogeneous thick plates, a computer program is developed, A p-version, hierarchical finite element was presented and applied to thick plate with trigonometric hierarchical shape functions. 1- Based on the Reddy’s higher-order theory, these elements have been implemented with a very simple and understandable mathematical framework and are easily programmed. 2- High accuracy, stable numerical computation and rapid convergence have been observed in the analysis. 3- The results are compared with other shear deformation theories and exact elasticity solutions. Good agreement may be noted. Several examples are solved. The effect of boundary condition and ratio a/b and h/a the analyzes is also studied. References [1] A. E. H. Love. (1888) On the small free vibrations and deformations of elastic shells, Philosophical trans. of the Royal Society (London), 17 491–549. Available: http://www.jstor.org/stable/90527 [2] Mindlin, R.D. Schacknow, A. Deresiewicz (1955, Jun), Flexural vibrations of rectangular plates, ASME J. Appl. Mec, 23 (1956) 430–436. [3] R.B. Nelson, DR. Lorch. (1974, Mar), A refined theory for laminated orthotropic plates. ASME, J. Appl. Mech, 41 177-183.Available:http://appliedmechanics.asmedigitalcollection.asme.org.sci- hub.org/article.aspx?articleid=1401537 [4] K.H Lo, R.M. Christen, E.m Wu. (1977, Dec), A higher order rheory of plate deformation – Part 1: Homogeneous plates, ASME, J. Appl. Mech, 44 663-668. Available:http://appliedmechanics.asmedigitalcollection.asme.org/article.aspx?articleid=1403567 [5] M. Levinson. (1980), An accurate simple theory of the statics and dynamics of elastic plates, Mech. Res. Commun. 7 343-350. [6] M.V.V. Murty. (1981), An improved transverse shear deformation theory for laminated anistropic plates. NASA Technical Paper 1903. [7] J. N. Reddy.(1984, Dec), A simple higher-order theory for laminated composite plates, ASME J. Appl. Mech. 51 745-752.Available:http://appliedmechanics.asmedigitalcollection.asme.org /article.aspx?articleid=1407769 [8] T. Kant, J.H. Varaiya, C.P. Arora, Finite element transient analysis of composite and sandwich plates based a refined theory and implicit time integration shemes, Comput. Struct, 36 (1990) 401- 420.Available:http://www.sciencedirect.com/science/article/pii/004579499090279B 17 http://appliedmechanics.asmedigitalcollection.asme.org.sci-hub.org/article.aspx?articleid=1401537 http://appliedmechanics.asmedigitalcollection.asme.org.sci-hub.org/article.aspx?articleid=1401537 http://appliedmechanics.asmedigitalcollection.asme.org/article.aspx?articleid=1403567 http://www.sciencedirect.com/science/article/pii/004579499090279B American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 9, No 1, pp 1-19 [9] A.K. Nayak, S.S.J. Moy, R.A. Shenoi , Free vibration analysis of composite sandwich plates based on reddy's higherorder theory, Composites Part B: Engineering. 33 (7) (2002) 505-519. Available: http://www.sciencedirect.com/science/article/pii/S1359836802000355 [10] A.K. Nayak, R.A. Shenoi ,S.S.J Moy , Transient response of composite sandwich plates, Comput. Struct, 64 (2004) 249-267. Available: http://www.sciencedirect.com/science/article/pii/S0263822303001351 [11] A.H. Sheikh, A. Chakrabarti, A new plate bending element based on higher order shear deformation theory for the analysis of composite plates, Finite. Elem. Anal. Des, 39 (2003) 137-155. vailable:http://www.sciencedirect.com/science/article/pii/S0168874X02001373 [12] R.C. Batra, S. Aimmanee ,Vibrations of thick isotropic plates with higher order shear and normal deformable plate theories, Comput. Struct, 83 (2005) 934-955. Available: http://www.sciencedirect.com/science/article/pii/S0045794905000507?np=y [13] S. Kapuria, SD Kulkarni, An improved discrete Kirchhoff quadrilateral element based on third-order zigzag theory for static analysis of composite and sandwich plates, J. Numer. method. Eng, 69 (2007)1948-1981. Available: http://onlinelibrary.wiley.com/doi/10.1002/nme.1836/abstract [14] B.A. Szabo, and G.J. Sahrmann, Hierarchical plate and shells models based on p extension, Int. J. Numer.method. Eng, 26 (1988) 1855-1881. Available:http://onlinelibrary.wiley.com/doi/10.1002/nme.1620260812/abstract;jsessionid=3DCB6A45626B110 6E11AB1A7F34B4282.f04t04 [15] B.A. Szabo , I. Babuska., Finite Element Analysis, (1990) Wiley-lnterscience, New York, 1991. [On-line]. http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0471502731.html [16] S. M. Hamza-Cherif, Free vibration analysis of rotating cantilever plates using the p-version of the finite element method, Structural Engineering and Mechanics, 22 (2006) 151-167. [17] A.W. Leissa , The free vibration of rectangular plates, J. Sound Vib. 31 (1973) 257–293. Available: http://www.sciencedirect.com/science/article/pii/S0022460X73803712 [18] C. W. Lim a, K.M. Liew b, S. Kitipornchai, a Numerical aspects for free vibration of thick Part I: Formulation and verification plates, Comput. Methods Appl. Mech. Eng, 156 (1998) 15-29. Available: http://www.sciencedirect.com/science/article/pii/S0045782597001977 [19] S. Srinivas, C.V. JogaRao, A.K. Rao, An exact analysis for vibration of simply-supported homogeneous and laminated thick rectangular plates, J. Sound Vib. 12(2) (1970) 187-199. Available: http://www.sciencedirect.com/science/article/pii/0022460X70900891 18 http://www.sciencedirect.com/science/article/pii/S1359836802000355 http://www.sciencedirect.com/science/article/pii/S1359836802000355 http://www.sciencedirect.com/science/article/pii/S1359836802000355 http://www.sciencedirect.com/science/article/pii/S1359836802000355 http://www.sciencedirect.com/science/article/pii/S0263822303001351 http://www.sciencedirect.com/science/article/pii/S0168874X02001373 http://www.sciencedirect.com/science/article/pii/S0045794905000507 http://www.sciencedirect.com/science/article/pii/S0045794905000507 http://www.sciencedirect.com/science/article/pii/S0045794905000507?np=y http://onlinelibrary.wiley.com/doi/10.1002/nme.1836/abstract http://onlinelibrary.wiley.com/doi/10.1002/nme.1836/abstract http://onlinelibrary.wiley.com/doi/10.1002/nme.1836/abstract http://www.sciencedirect.com/science/article/pii/S0022460X73803712 http://www.sciencedirect.com/science/article/pii/S0045782597001977 http://www.sciencedirect.com/science/article/pii/0022460X70900891 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 9, No 1, pp 1-19 [20] D. Zhou a, Y.K. Cheung, F.T.K. Au, S.H. Lo,Three-dimensional vibration analysis of thick rectangular plates using Chebyshev polynomial and Ritz method, Int. J. Solids Struct, 63(2002) 39– 6353.Available:http://www.sciencedirect.com/science/article/pii/S0020768302004602 [21] S. Wang, Vibration of thin skew fiber reinforced composite laminates, J. Sound. Vib. 201(3) (1997) 335– 352.Available:http://www.sciencedirect.com/science/article/pii/S0022460X96907452 [22] S. Wang, Free vibration analysis of skew fiber-reinforced composite laminates based on first-order shear deformation plate theory, Comput. Struct, 63 (3) (1997) 525-538. Available:http://www.sciencedirect.com/science/article/pii/S0045794996003574 [23] K.M. Liew, K.C. Hung, M.K. Lim, A continuum three-dimensional vibration analysis of thick rectangular plates, Int. J. Solids Struct. 30 (24) (1993) 3357-3379. Available:http://www.sciencedirect.com/science/article/pii/002076839390089P [24] A. Houmat, An alternative hierarchical finite element formulation applied to plate vibrations, J. Sound Vib. (1997) 206 (2) 201-215. Available: http://www.sciencedirect.com/science/article/pii/S0022460X97910762 [25] S. Hosseini-Hashemi , M. Fadaee , H. RokniDamavandiTaher, Exact solutions for free flexural vibration of Lévy-type rectangular thick plates via third-order shear deformation plate theory, Appl. math. Model.35 (2011) 708–727. Available: http://www.sciencedirect.com/science/article/pii/S0307904X10002775 [26] S. Hosseini-Hashemi, M. Arsanjani, Exact characteristic equations for some of classical boundary conditions of vibrating moderately thick rectangular plates, Int. J. Solids. Struct. 47 (2005) 819–853. Available:http://www.sciencedirect.com/science/article/pii/S0020768304003671 [27] M. Malik, C.W. Bert, Three-dimensional elasticity solutions for free vibrations of rectangular plates by the differential quadrature method, Int. J. Solids. Struct. 35 (1998) 299–318. Available: http://www.sciencedirect.com/science/article/pii/S0020768397000735 19 http://www.sciencedirect.com/science/article/pii/S0020768302004602 http://www.sciencedirect.com/science/article/pii/S0022460X96907452 http://www.sciencedirect.com/science/article/pii/S0045794996003574 http://www.sciencedirect.com/science/article/pii/002076839390089P http://www.sciencedirect.com/science/article/pii/S0022460X97910762 http://www.sciencedirect.com/science/article/pii/S0307904X10002775 http://www.sciencedirect.com/science/article/pii/S0020768304003671 http://www.sciencedirect.com/science/article/pii/S0020768397000735