WCCM V Fifth World Congress on Computational Mechanics July 7-12, 2002, Vienna, Austria Eds.: H.A Mang, F.G Rammerstorfer, J Eberhardsteiner Transverse Vibration of Symmetrically Laminated Elastically Restrained Plates of Variable Thickness Ahmed S Ashour International Islamic University Malaysia, Department of Science in Engineering, 53100 Kuala Lumpur, Malaysia e-mail: ashour@iiu.edu.my Key words: Finite strip method, vibration, symmetrically laminated, elastic restrained Abstract The natural frequencies of symmetrically laminated plates of variable thickness are analyzed using the finite strip transition matrix technique In this paper, the natural frequencies of such plates are determined for edges with elastic restrained against rotation, transition or both A successive conjunction of the classical finite strip method and the transition matrix method is applied to develop a new modification of the finite strip method to reduce the complexity of the problem The displacement function is expressed as the product of a basic trigonometric series function in the longitudinal direction and an unknown function that has to be determined in the other direction Using the new transition matrix, after necessary simplification and the satisfaction of the boundary conditions, yields a set of simultaneous equations that leads to the characteristic matrix of vibration Numerical results for different combinations of elastic rotational or transitional edges have been presented and compared with those available from other methods in the literature Also, the effect of the tapered ratio and the aspect ratio on the natural frequencies of the plates is presented The good agreement with other methods demonstrates the validity and the reliability of the proposed method Ahmed S Ashour Introduction Laminated plates are widely used in many engineering application In some applications, the designer has to construct variable thickness plates to save material or to meet certain criteria The vibration of uniform plates with elastic restrained boundary conditions has been investigated by many authors [18] On the other hand, the vibration of elastically restrained plates of variable thickness has been studied by relatively fewer authors Kobayashi and Sonoda [9] presented an exact method for analyzing the free vibration and buckling of isotropic plate with linearly tapered thickness in one direction with two opposite edges are simply supported and the other two edges are elastically restrained against rotation Grossi and Bahat [10] used Rayleigh-Ritz method with the boundary characteristics orthogonal polynomials as shape functions and Rayleigh Schmidt method to find the natural frequencies of isotropic tapered rectangular plates with edges restrained against rotation and translation Gutierrez and Laura [11] used the differential quadrature method to determine the fundamental frequencies of rectangular plates with linearly varying thickness and non-uniform boundary conditions Filipich et al [12] used the Galerkin method to obtain an approximate solution to the vibration of isotropic rectangular plates of variable thickness with two opposite edges simply supported and very general boundary conditions on the other two edges Rais-Rohani and Marcellier [13] presented approximate analytical solutions for the free vibration and buckling of rectangular anisotropic plates as well as rectangular sandwich plates with edge restrained against rotation To the best of the author’s knowledge there is no publication available in the open literature on this problem The main objective of this paper is to determine the natural frequencies of cross play symmetrically laminated plates of variable thickness subjected to elastically restrained boundary conditions against both rotation and translation in the variable thickness direction and any combination of clamped or simply supported boundary conditions in the other direction Governing Equations of Elastic Restrained plates Consider a cross play symmetrically laminated rectangular plate of variable thickness h(y), length a, width b, density ρ and with elastic restrained boundary conditions at y = and y = b as shown in Fig The governing equation can be written as: h3 ( y ) h3 ( y ) ∂ h3 ( y ) ∂ h3 ( y ) D11 Wxxxx + ( D12 + D66 ) Wxxyy + Wxxy + ν 12 D22 Wxx h0 h0 ∂ y h0 ∂ y h0 + D22 ∂ h3 ( y ) ∂ h3 ( y ) h(y ) h ( y ) Wyyyyy + Wyyy + Wyy = −m0 Wtt h0 ∂y ∂y h0 where W is the flexural displacement, Dij = Dij (1) h3 are the bending rigidities, ho is the plate height at y ho3 = 0, N Dij = ∑ h/2 ∫ k =1 − h / Qij( k ) z dz i, j = 1, 2,6 (2) Qij( k ) are the material constants of the kth lamina , N no of layers, zk is the distance from the (k ) midplane of the plate to the bottom of the k th lamina and Qij are the plane stress reduced stiffness coefficient of the lamina k given by WCCM V, July 7-12, 2002, Vienna, Austria Fig 1: Laminated plate of variable thickness E yy ν yx Exx Exx Q11 = , Q22 = , Q12 = , Q21 = Q12 , Q66 = Gxy (1 − ν xyν yx ) (1 − ν yxν xy ) (1 − ν xyν yx ) (3) Here Exx , E yy are the longitudinal and transverse plate moduli, respectively and Gxy is the in plane shear modulus, and ν xy and ν yx are the Poisson’s ratios 2.1 Boundary Conditions The considered boundary conditions along the y-direction are elastically restrained against both rotation and translation At y = , the boundary condition for this case are, ∂ 2W ∂W ∂ 2W (4) − D220 + ν xy =0 ∂y ∂ x2 ∂y ∂ 3W ∂ 3W T0W + D220 + D − D =0 (5) ν 30 xy 220 ∂ y3 ∂ x2 y where D3 = D12 + D66 , the suffixes “0, b” means the rigidities are calculated at y = and y = b respectively The boundary conditions on the other elastically restrained end (y = b) are ∂ 2W ∂W ∂ 2W (6) Rb + D22b + ν xy =0, ∂y ∂ x2 ∂y R0 ( ) ∂ 3W ∂ 3W (7) + − ν = 0, D D ( ) xy 22 ∂ y3 ∂ x2 y where suffixes “b, 0” means rigidities are calculated at y = 0, y = b respectively The boundary conditions at the other two edges x = and x = a can be any combination of the classical boundary conditions (simple, clamped, free) TbW − D22 b 2.2 b b Method of solution and the eigen value problem Assuming a solution of the form M W = ∑ X m ( x ) ym ( y ) , (8) m =1 where Xm (x) are the beam functions that satisfy the boundary conditions at x = and x = a The governing equation can be transformed into 4N number of first order differential equations in terms of the normalized coordinates which can be solved as in [14] In many papers, e.g [1, 2], the boundary Ahmed S Ashour conditions are solved approximately by neglecting some of the terms in Eqs (4-7) In this paper, the elastically restrained boundary conditions are solved exactly Using Eq (8), the boundary conditions at the normalized coordinates η = are d 3Y j dη + M cij dYi D3 − ν = − Yi xy ∑ β D22 φT0 i =1 a jj dη (9) ν xy M cij dYi Yi = ∑ β i =1 aij φ R dη (10) d 2Y j dη where φT0 = D220 bT0 D220 and φR0 = bR0 dη + M cij dYi D3b − ν xy ∑ = Yi β D22b i =1 a jj dη ϕTb (11) ν xy M cij dY ∑ Yi = − φ dηi β i =1 aij R (12) d 2Y j dη D22b and φRb = and at η = d 3Y j where φTb = + + b D22b bTb bRb In the next sections, the symbol S-C-S-ER (for example) means that the edges x = 0, y = 0, x = a, y = b are simply supported, clamped, simply supported and elastic restrained respectively A linearly tapered plate is used to illustrate the above technique with the following non-dimensional variable thickness h(η ) = + δη , where δ is the taper ratio Convergence and Comparison Investigation 3.1 Convergence analysis Since no solution exists for the above problem, one has to carry out several convergence studies First, a convergence investigation is carried out to examine the effect of number of terms of the power series M used in the solution The results are shown in table for uniform three-layer cross-ply symmetrically laminated plates The material properties used in this case are given by E11 / E22 = 40 , G12 / E22 = 0.6 , ν 12 = 0.25 , and the frequency parameter λ used is given by λ = λ (E 22 /E11 )/β Form Table 1, it is very clear that the method converge very rapidly Other convergence studies are carried out for linearly tapered plate but they are not shown here due to the limitation of the space Table 1: Convergence investigation of frequency parameters λ for three-ply (0,90,0) fully clamped plates (CCCC) CCCC [16] β M λ1 λ2 λ3 λ4 1 14.6196 14.6196 14.6195 14.6195 14.6195 14.6195 14.666 17.5211 17.5211 17.5203 17.5203 17.5202 17.5202 17.614 24.3951 24.3951 24.3928 24.3928 24.3925 24.3925 24.511 35.4739 35.4739 35.4694 35.4694 35.469 35.4689 35.532 WCCM V, July 7-12, 2002, Vienna, Austria 3.2 Comparison analysis Table shows the frequency parameters for clamped unidirectional laminated uniform plates compared with some available results in the literature The material properties used in this case are given by E11 / E22 = 15.4 , G12 / E22 = 79 , ν 12 = 30 , the results agree very well with other results in the literature Table 2: Comparison of frequency parameters of five-layer laminated plate (CCCC) CCCC β Q PRESENT Han and Petyt Han and Dickinson λ1 23.8516 λ2 29.7192 λ3 41.7636 λ4 59.3016 23.852 23.852 29.715 29.715 41.721 41.721 60.327 59.963 Numerical Results The frequency parameter λ for uniform laminated plates subject to classical boundary conditions are presented and compared with results from ref [16] in table It can be seen that the results agree very well with ref [16] A parametric investigation is carried out to study the effect of elastic restrained boundary conditions on the frequency parameters for uniform and non-uniform laminated plates The results are shown in Fig 2a and Fig 2b for uniform plates and in Figs 3-5 for non uniform plates First, we consider(S-ER-S-ER) uniform laminated plates Figure shows the frequency parameter surfaces of the fundamental (a) and the second modes (b) surfaces versus the elastic restrained against rotation coefficient φr and against translation coefficient φt for three layer laminated plate (90, 0,90) Table 3: The frequencies parameters of three-ply uniform laminated plates (0,90,0) for different combinations of boundary conditions with some comparison Boundary Condition λ1 λ2 λ3 λ4 CSCS CSCC CSCF CCCF CCCC CCCC [16] CCCF CFCS CFCC CFCF[16] CFCF SSSS SSSS [16] SSSC SSSF SCSS SCSC SCSC[16] SFSS SFSC SFSF SFSF[16] 14.2449 14.3842 14.0621 14.3842 14.6195 14.666 14.0769 14.0634 14.0784 14.072 14.0429 6.5828 625 6.8726 6.2241 6.8726 7.349 7.396 6.2241 6.2563 6.1886 6.208 15.7717 16.531 14.4702 16.531 17.5202 17.614 14.7177 14.4711 14.7185 14.199 14.2275 9.3456 9.447 10.5672 7.0176 10.5672 12.0486 12.144 7.0176 7.502 6.3161 6.436 20.563 22.343 16.6666 22.343 24.3925 24.511 17.6643 16.6667 17.6642 15.037 14.8231 16.0759 16.205 18.2894 10.738 18.2894 20.7366 20.841 10.738 12.2179 7.6705 7.975 29.5577 32.3872 22.4882 32.3872 35.4553 35.532 24.5361 22.4875 24.5352 18.136 17.8036 26.3315 25.115 25.1361 18.433 25.1361 25.2829 25.365 18.433 20.8733 12.3799 12.752 Ahmed S Ashour O r t O ✁ ✁ t r Fig 2: The frequency parameters l for laminated plate with edges elastically restrained against both rotation and translation (a) Fundamental mode (b) Second mode WCCM V, July 7-12, 2002, Vienna, Austria 10 Frequency Paramter O δ=0 δ = 0.2 δ = 0.4 1.0E-08 1.0E-06 1.0E-04 1.0E-02 1.0E+00 1.0E+02 Elastic Restrained Against Rotation )r (a) Frequency Paramter O 15 δ=0 14 δ = 0.2 13 δ = 0.4 12 11 10 1.0E-05 1.0E-03 1.0E-01 1.0E+01 1.0E+03 Elastic Restrained Against Rotation )r (b) Fig 3: The Frequency parameters λ of the fundamental (a) and the second mode (b) versus the elastic restrained against rotation coefficient φr with different taper thickness ration δ for S-ER-S-ER plate (Φt = 0.000001) Ahmed S Ashour 18 Frequency Paramter O δ=0 δ = 0.2 δ = 0.4 17 16 15 14 13 1.0E-05 1.0E-03 1.0E-01 1.0E+01 1.0E+03 Elastic Restrained Against Rotation )r (a) Frequency Paramter O 22 δ=0 δ = 0.2 δ = 0.4 21 20 19 18 17 16 15 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 Elastic Restrained Against Rotation )r (b) Fig 4: The Frequency parameters λ of the fundamental (a) and the second mode (b) versus the elastic restrained against rotation coefficient φr with different taper thickness ration δ for C-ER-C-ER plate (Φt = 0.000001) WCCM V, July 7-12, 2002, Vienna, Austria Frequency Paramter O 17 δ= δ = 0.2 16 δ = 0.4 15 14 13 1.0E-04 1.0E-02 1.0E+00 1.0E+02 Elastic Restrained Against Rotation ) r Fig 4: The fundamental frequency parameters λ versus the elastic restrained against rotation coefficient φr with different taper thickness ration δ for C-ER-C-ER plate (Φt = 0.1) The relation between the frequency parameter λ and the elastic restrained against rotation coefficient φr while the elastic restrained against translation Φt is kept equal 0.000001, almost zero, ( for different taper ratios δ = 0., 0.2 and 0.4) is shown in Fig for S-ER-S-ER plate and in Fig for C-ER-C-ER plate In both figures, the fundamental mode is presented in (a) while the second mode is presented in (b) In Figure 5, the fundamental frequency parameters is plotted versus the elastic restrained against translation coefficient φr while the coefficient of elastic restrained against translation is considered equal (Φt = 0.1) It can be observed that in the case of S-ER-S-ER, the elastic restrained against rotation has more effect on the fundamental frequency than in the case of C-ER-C-ER In all cases, the effect is significant only in the range (.001