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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2011, Article ID 403684, 13 pages doi:10.1155/2011/403684 Research Article Free Vibration of Laminated Composite Hypar Shell Roofs with Cutouts Sarmila Sahoo Department of Civil Engineering, Meghnad Saha Institute of Technology, Kolkata 700107, India Correspondence should be addressed to Sarmila Sahoo, sarmila ju@yahoo.com Received 6 May 2011; Revised 20 September 2011; Accepted 20 September 2011 Academic Editor: K. M. Liew Copyright © 2011 Sarmila Sahoo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Use of laminated composites in civil engineering structural components including shell roofs is increasing day by day due to their light weight, high speciﬁc strength, and stiﬀness properties. In the present paper, laminated composite hypar shell (hyperbolic paraboloidal shells bounded by straight edges) roofs with cutouts are analyzed for their free vibration characteristics using ﬁnite element method. An eight-noded curved shell element is used for modeling the shell. Speciﬁc numerical problems of earlier investigators are solved to compare their results with the present formulation. A number of problems are further solved where the size of the cutouts and their positions with respect to the shell centre are varied for diﬀerent edge constraints. The results are furnished in the form of ﬁgures and tables. The results are examined thoroughly to arrive at some meaningful conclusions useful to designers. 1. Introduction Malhotra et al. [3] studied free vibration of composite plate with cutout for diﬀerent boundary conditions. One Diﬀerent shell forms are common in civil engineering of the early reports on free vibration of curved panels with applications. Among the shell forms used as rooﬁng units, cutout was due to Sivasubramonian et al. [4]. They analysed the skewed hypars (hyperbolic paraboloidal shells bounded the eﬀect of cutouts on the natural frequencies of plates by straight edges) have a special position because these with some classical boundary conditions. The plate had a architecturally pleasant forms may be cast and fabricated curvature in one direction and was straight in the other. conveniently being doubly ruled surfaces. Hypar shell is The eﬀect of ﬁbre orientation and size of cutout on natural preferred in many places, particularly in medical, chemical frequency on orthotropic square plates with square cutout and food processing industries where entry of north light was studied using Rayleigh-Ritz method. Later Sivakumar is desirable. Application of hypars in these industries often et al. [5], Rossi [6], Huang and Sakiyama [7], and Hota necessitates provision of cutouts for the passage of light, and Padhi [8] studied free vibration of plate with various to provide accessibility to other parts of the structure, for cutout geometries. Chakravorty et al. [1] analyzed the eﬀect venting and also sometimes for alteration of the resonant of concentric cutout on diﬀerent shell options. Again in 1999, frequency. A comprehensive idea about the static and free Sivasubramonian et al. [9] studied the eﬀect of curvature and vibration characteristics of such shell roofs with cutouts cutouts on square panels with diﬀerent boundary conditions. is essential for a designer for successfully applying these The size of the cutout (symmetrically located) as well as forms. Moreover, nowadays researchers are emphasizing curvature of the panels is varied. Hota and Chakravorty more on laminated composite shells realizing the strength [10] published useful information about free vibration of and stiﬀness potentials of this advanced material. stiﬀened conoidal shell roofs with cutout. Later Nanda and Bandyopadhyay [11] studied the eﬀect of diﬀerent paramet- The free vibration of composite as well as isotropic plate with cutout was studied by diﬀerent researchers from ric variation on free vibration of cylindrical shell with cutout. time to time. Reddy [2] investigated large amplitude ﬂexural A scrutiny of the literature on vibration of shell panels with cutout indicates that there is enough scope of research vibration of composite plate with cutout. Later in 1989, 2 Advances in Acoustics and Vibration where {F}= N N N M M M Q Q , x y xy x y xy x y ⎡ ⎤ A A A B B B 00 11 12 16 11 12 16 ⎢ ⎥ A A A B B B 00 ⎢ 12 22 26 12 22 26 ⎥ c ⎢ ⎥ ⎢ ⎥ A A A B B B 00 16 26 66 16 26 66 ⎢ ⎥ ⎢ ⎥ B B B D D D 00 11 12 16 11 12 16 ⎢ ⎥ [D] = , (2) ⎢ ⎥ B B B D D D 00 12 22 26 12 22 26 ⎢ ⎥ ⎢ ⎥ B B B D D D 00 ⎢ 16 26 66 16 26 66 ⎥ ⎢ ⎥ ⎣ 0 0 0 000 S S ⎦ 11 12 0 0 0 000 S S 12 22 0 0 0 0 0 ε ε γ k k k γ γ {ε}= x y xy . x y xy xz yz The force and moment resultants are expressed as Figure 1: Surface of a hypar shell with cutout. Surface equation z = ⎧ ⎫ ⎧ ⎫ ⎪ N ⎪ ⎪ σ dz ⎪ x x (4c/ab)(x − a/2)(y − b/2) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ N σ dz y y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ N ⎪ ⎪ τ dz ⎪ xy xy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ h/2 ⎨ ⎬ M σ zdz x x = . (3) ⎪ M ⎪ ⎪ σ zdz⎪ y y ⎪ ⎪ −h/2⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ M τ zdz ⎪ ⎪ ⎪ ⎪ xy xy b ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Q τ dz ⎪ x ⎪ ⎪ xz ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ Q τ dz 2 y yz The stiﬀness coeﬃcients are functions of Young’s moduli, k−1 shear moduli, and the Poisson’s ratio of the laminates. They also depend on the angle which the individual lamina of a laminate makes with the global x-axis. The detailed expressions of the elements of the elasticity matrix are available in several references including Qatu [12]and Vasiliev et al. [13]. The stiﬀness coeﬃcients are deﬁned as np A = Q (z − z ), ij ij k k−1 k=1 Figure 2: Laminations in skewed hypar shell. np 2 2 B = Q z − z , ij ij k k−1 k=1 (4) np to be carried out on the hypar shell. Although Chakravorty 3 3 D = Q z − z i, j = 1, 2, 6, ij ij k k−1 et al. [1] did a few studies about hypar shell with con- k=1 centric cutout, many practically important aspects are yet np to be addressed. In the present paper the free vibration of S = F F G (z − z ) i, j = 1, 2, ij i j ij k k−1 hypar shell with cutouts (Figure 1) is studied considering k=1 diﬀerent boundary conditions. The variation of fundamental where Q are elements of the oﬀ-axis elastic constant matrix ij frequency due to change in eccentricity of cutout along x and which is given by y direction is also considered. Q = [T] Q [T] (5) ij ij oﬀ on 2. Governing Equations in which ⎡ ⎤ Q Q 0 A laminated composite hypar shell (Figure 2) of uniform 11 12 ⎢ ⎥ Q Q 0 thickness h and twist radius of curvature R is considered. Q = ⎣ ⎦ , xy ij 12 22 on Keeping the total thickness same, the thickness may consist of 00 Q ⎡ ⎤ (6) any number of thin laminae each of which may be arbitrarily 2 2 m n mn ⎢ ⎥ oriented at an angle θ with reference to the x-axis of the co- 2 2 [T] = ⎣ n m −mn ⎦ ordinate system. The constitutive equations for the shell are 2 2 −2mn 2mn m − n given by (a list of notations is already given) with {F}= [D]{ε},(1) m = cos θ, n = sin θ. (7) Advances in Acoustics and Vibration 3 The elements of the [Q ] matrix [13]are ij on −1 −1 Q = (1 − ν ν ) E , Q = (1 − ν ν ) E , 11 12 21 11 22 12 21 22 (8) −1 6 Q = (1 − ν ν ) E ν , Q = G . 12 12 21 11 21 66 12 F and F of (4) are two shear correction factors presently i j taken as unity for thin shells and the elements of the G ij 7 matrix are given by 2 2 G = G cos θ + G sin θ, xx 13 23 ( ) G = G − G cos θ sin θ, xy 13 23 (9) G = G sin θ + G cos θ. yy 13 23 The G matrix has the form 8 ij Figure 3: Eight noded shell element with isoparametric coordi- G G xx xy G = . (10) ij nates. G G xy yy The strain-displacement relations on the basis of improved ﬁrst-order approximation theory for thin shell [14]are where the shape functions derived from a cubic interpolation established as polynomial [14]are T T 0 0 0 0 0 (1+ ξξ ) 1+ ηη ξξ + ηη − 1 ε , ε , γ , γ , γ = ε , ε , γ , γ , γ i i i i x y xy xz yz x y xy xz yz N = ,for i = 1, 2, 3, 4, , 2 +z k , k , k , k , k x y xy xz yz (1+ ξξ ) 1 − η N = ,for i = 5, 7, (11) 1+ ηη 1 − ξ where, the ﬁrst vector is the midsurface strain for a hypar N = ,for i = 6, 8. shell and the second vector is the curvature. These are given, (14) respectively, by ⎧ ⎫ The generalized displacement vector of an element is ∂u ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ expressed in terms of the shape functions and nodal degrees ⎪ ⎪ ⎧ ⎫ ⎪ ⎪ ∂x ⎪ ⎪ ∂α ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ of freedom as ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂v ⎪ ⎪ ⎪ ∂x ⎪ ⎧ ⎫ ⎧ ⎫ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [u] = [N]{d }, (15) ∂y e ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ∂β ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k ⎪ ⎪ ⎪ ⎨ y ⎬ ⎨ ⎬ ⎨ y ⎬ ⎨ ⎬ ∂u ∂v 2w ∂y that is, = , k = . xy xy + − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂α ∂β⎪ 0 ∂y ∂x R ⎧ ⎫ ⎡ ⎤ ⎧ ⎫ xy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ γ k ⎪ ⎪ ⎪ ⎪ ⎪ xz⎪ ⎪ ⎪ xz u N u ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i i ⎩ ⎭ ⎪ ⎪ ⎩ ⎭ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂w ⎪ ⎪ ⎪ ∂y ∂x⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ yz yz ⎪ α + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ v 8 N v ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎢ i ⎥ ⎨ i⎬ ⎪ ⎪ ⎪ ⎪ ∂x ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ {u}= w = ⎢ N ⎥ w . (16) i i ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ∂w ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ α i=1⎣ N ⎦ α ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i i β + ⎩ ⎭ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ ∂y β N β i i (12) 2.2. Element Stiﬀness Matrix. The strain-displacement rela- Theradiusofcross curvaturemay be evaluatedbydiﬀerenti- tion is given by ating the surface equation of shell in the form z = f (x, y)and for shallow shells which are taken up for the present study the {ε}= [B]{d }, (17) same may be expressed as 1/R = d z/dx dy. xy where ⎡ ⎤ 2.1. Finite Element Formulation. An eight-noded curved N 000 0 i,x ⎢ ⎥ quadratic isoparametric ﬁnite element (as shown in Figure 3) 0 N 00 0 ⎢ i,y ⎥ ⎢ ⎥ is used for hypar shell analysis. The ﬁve degrees of freedom −2N ⎢ i ⎥ ⎢ N N 00 ⎥ i,y i,x taken into consideration at each node are u, v, w, α, β. The ⎢ ⎥ xy ⎢ ⎥ ⎢ ⎥ following expressions establish the relations between the 00 0 N 0 [B] = . (18) i,x ⎢ ⎥ ⎢ ⎥ displacement at any point with respect to the coordinates ξ i=1 0 000 N i,y ⎢ ⎥ ⎢ ⎥ and η and the nodal degrees of freedom: 00 0 N N ⎢ i,y i,x⎥ ⎢ ⎥ ⎣ 00 N N 0 ⎦ 8 8 8 i,x i 00 N 0 N i,y i u = N u , v = N v , w = N w , i i i i i i i=1 i=1 i=1 (13) The element stiﬀness matrix is 8 8 α = N α , β = N β , i i i i [K ] = [B] [E][B]dx dy. (19) i=1 i=1 4 Advances in Acoustics and Vibration y 120 0 0.1 0.2 0.3 0.4 Figure 4: Typical 10 × 10 nonuniform mesh arrangements drawn a /a to scale. CSCS CCCC SCSC CSCC CSSS CCSC 2.3. Element Mass Matrix. The element mass matrix is SSSC CCCS obtained from the integral CSSC SCSS CCSS SSSS [M ] = [N] [P][N]dx dy, (20) Figure 5: Values of non-dimensional fundamental frequency (ω)of 0/90/0/90 hypar shell with cutout for diﬀerent sizes of central cutout where and boundary conditions. a/b = 1, a/h = 100, a /a = 0.1, a /b = 1, ⎡ ⎤ c/a = 0.2; E /E = 25, G = 0.2E , G = G = 0.5E , ν = 11 22 23 22 13 12 22 12 N 0000 ⎢ ⎥ ν = 0.25. 0 N 000 8 i ⎢ ⎥ ⎢ ⎥ [N] = ⎢ 00 N 00 ⎥ , ⎢ ⎥ i=1⎣ ⎦ 000 N 0 the integrals in the local natural coordinates ξ and η of the 0000 N (21) ⎡ ⎤ 2 × 2 Gauss quadrature because the shape functions are P 00 0 0 derived from a cubic interpolation polynomial (Dey et al. ⎢ ⎥ 8 0 P 00 0 ⎢ ⎥ [14]) and it is an established fact that a polynomial of degree ⎢ ⎥ 00 P 00 [P] = ⎢ ⎥ , 2n− 1 is integrated exactly by n point Gauss quadrature [15]. ⎢ ⎥ i=1⎣ ⎦ 000 I 0 Then the element matrices are assembled after performing 000 0 I appropriate transformations due to the curved shell surface to obtain the respective global matrices [K]and [M]. in which The free vibration analysis involves determination of np np z z k k natural frequencies from the condition P = ρdz, I = zρdz. (22) z z k−1 k−1 2 k=1 k=1 [K] − ω [M] = 0. (23) This is a generalized eigen value problem and is solved by the 2.4. Modeling the Cutout. The code developed can take the subspace iteration algorithm. position and size of cutout as input. The program is capable of generating nonuniform ﬁnite element mesh all over the shell surface. So the element size is gradually decreased near 3. Validation Study the cutout margins. One such typical mesh arrangement is shown in Figure 4. Such ﬁnite element mesh is redeﬁned The accuracy of the present formulation is ﬁrst validated by in steps, and a particular grid is chosen to obtain the comparing the results of the following problem available in fundamental frequency when the result does not improve by the existing literature. more than one percent on further reﬁning. Convergence of Free vibration of simply supported and clamped hypar results is ensured in all the problems taken up here. shell with (0/90) shell of aspect ratio a/b = 1, c/a = 0.2, and side to thickness ratio a/h = 100 with cutouts [1]is considered as the benchmark problem. The nondimensional 2.5. Solution Procedure for Dynamic Analysis. The element 1/2 2 2 stiﬀness and mass matrices are evaluated ﬁrst by expressing frequency parameter is ω = ωa (ρ/E h ) .Material Non-dimensional fundamental frequency Advances in Acoustics and Vibration 5 Table 1: Non-dimensional fundamental frequencies (ω) for hypar shells (lamination (0/90) ) with concentric cutouts. Chakravorty et al. [1] Present ﬁnite element model a /a Simply supported Clamped Simply supported Clamped 8 ×810 × 10 12 × 12 8 ×810 × 10 12 × 12 0.0 50.829 111.600 50.573 50.821 50.825 111.445 111.592 111.612 0.1 50.769 110.166 50.679 50.758 50.779 109.987 110.057 110.233 0.2 50.434 105.464 50.323 50.421 50.400 105.265 105.444 105.443 0.3 49.165 101.350 49.045 49.157 49.178 101.110 101.340 101.490 0.4 47.244 97.987 47.132 47.242 47.141 97.670 97.985 97.991 a/b = 1, a/h = 100, a /b = 1, c/a = 0.2; E /E = 25, G = 0.2E , G = G = 0.5E , ν = ν = 0.25. 11 22 23 22 13 12 22 12 21 4. Results and Discussion In order to study the eﬀectofcutoutsizeonthe free vibration response additional problems for hypar shells with 0/90/0/90 and +45/−45/+45/−45 lamination and diﬀerent boundary conditions have been solved. The selection of the 0/90/0/90 and +45/−45/+45/−45 lamination is based on an earlierstudy by Sahooand Chakravorty[16]which revealed that repeating 0/90 unit and +45/−45 unit more than once and keeping the total shell thickness constant do not improve the fundamental frequency to an appreciable extent. The positions of the cutouts are varied along both of the plan directions of the shell for diﬀerent practical boundary conditions to study the eﬀect of eccentricity of cutout on the fundamental frequency. 4.1. Free Vibration Behaviour of Shells with Concentric Cutouts. Figures 5 and 6 furnish the results of non-dimen- 0 0.1 0.2 0.3 0.4 sional frequency (ω) of 0/90/0/90 and +45/−45/+45/−45 a /a hypar shells. The shells considered are of square plan form CSCS CCCC (a = b), and the cutouts are also taken to be square in plan SCSC CSCC (a = b ). The cutouts are placed concentrically on the shell CSSS CCSC surface. The cutout sizes (i.e., a /a)are varied from 0to0.4, CCCS SSSC and boundary conditions are varied along the four edges. CSSC SSCS CCSS SSSS The boundary conditions are designated by describing the support clamped or simply supported as C or S taken in Figure 6: Values of non-dimensional fundamental frequency (ω) an anticlockwise order from the edge x = 0. This means a of +45/−45/+45/−45 hypar shell with cutout for diﬀerent sizes of shell with CSCS boundary is clamped along x = 0, simply central cutout and boundary conditions. a/b = 1, a/h = 100, a /a = supported along y = 0 and clamped along x = a and 0.1, a /b = 1, c/a = 0.2; E /E = 25, G = 0.2E , G = G = 11 22 23 22 13 12 simply supported along y = b. The material and geometric 0.5E , ν = ν = 0.25. 22 12 21 properties of shells and cutouts are mentioned along with the ﬁgures. properties are as follows: E /E = 25, G = 0.2E , G = 11 22 23 22 13 G = 0.5E , ν = ν = 0.25. 12 22 12 21 The fundamental frequencies of hypar shell with cutout 4.1.1. EﬀectofCutoutSizes. From the ﬁgures it is seen that obtained by the present method agree well with those when a cutout is introduced to a shell the fundamental reported by Chakravorty et al. [1]asevident from Table 1, frequency increases in 6 out of 12 cases in case of symmetric establishing the correctness of the present results. The crossply shell. But in case of symmetric angle ply one the fact that the cutouts are properly modeled in the present fundamental frequency increases in 10 out of 12 cases. In formulation is thus also established. The present approach order to study the eﬀect of cutout size, in more detail, uses the improved ﬁrst-order approximation theory for thin the ratio of the fundamental frequency of a concentric shells [13] considering the radius of cross curvature. For this punctured shell to that of a shell without cutout is expressed class of thin shells a shear correction factor of unity is found in percentage. The increase or decrease in percentage of to yield good results. It is observed that the results remain fundamental frequency from the full shell is denoted by the same when analysis is repeated with the commonly used p. Tables 2 and 3 contain such p values for 0/90/0/90 shear correction factor of π/ 12. and +45/−45/+45/−45 shells, respectively. Negative sign Nondimensional fundamental frequency 6 Advances in Acoustics and Vibration Table 2: Values of “p” for 0/90/0/90 hypar shell. Cutout size (a /a) Boundary conditions 0 0.1 0.2 0.3 0.4 CCCC 0 −1.56 −5.48 −10.01 −12.94 CSCC 0 −0.9 −3.87 −14.97 −22.62 CCSC 0 −0.99 −5.88 −16.04 −23.3 CCCS 0 −0.9 −4.33 −15.42 −22.76 CSSC 0 −1.12 −8.24 −20.47 −30.27 CCSS 0 −1.12 −8.24 −20.47 −30.27 CSCS 0 0.38 −1.67 −4.16 −4.21 SCSC 0 0.38 −1.71 −4.29 −4.43 CSSS 0 0.3 −1.05 −3.35 −5.53 SSSC 0 0.3 −1.09 −3.45 −5.74 SSCS 0 0.33 −1.06 −3.35 −5.54 SSSS 0 0.23 −0.63 −2.92 −6.08 a/b = 1, a/h = 100, a /b = 1, c/a = 0.2; E /E = 25, G = 0.2E , G = G = 0.5E , ν = ν = 0.25. 11 22 23 22 13 12 22 12 21 Table 3: Values of “p” for +45/−45/+45/−45 hypar shell. Cutout size (a /a) Boundary conditions 0 0.1 0.2 0.3 0.4 CCCC 0 0.08 −1.2 −7.4 −16.29 CSCC 0 1.13 −6.11 −21.59 −29.25 CCSC 0 0.67 −5.84 −21.9 −28.99 CCCS 0 0.75 −5.78 −21.96 −29.22 CSSC 0 −0.82 −5.14 −12.44 −24.09 CCSS 0 −0.97 −5.18 −12.77 −24.19 CSCS 0 0.99 0.53 1.53 1.52 SCSC 0 0.93 0.53 −0.01 1.87 CSSS 0 0.77 0.64 1.36 −0.29 SSSC 0 0.74 0.64 1.34 −0.07 SSCS 0 0.58 0.64 1.35 −0.3 SSSS 0 0.44 0.8 1.66 0.36 a/b = 1, a/h = 100, a /b = 1, c/a = 0.2; E /E = 25, G = 0.2E , G = G = 0.5E , ν = ν = 0.25. 11 22 23 22 13 12 22 12 21 indicates decrease in frequency. It is evident from Tables 2 combinations in a particular group have equal number of and 3 that in all the cases with the introduction of cutout with boundary reactions. The groups are of the following forms: a /a = 0.1 the increase or decrease in frequency is not more Group I: contains CCCC shells. than 1%. But with further increase in cutout size, that is, when a /a = 0.2, fundamental frequency decreases in all the Group II: contains CSCC, CCSC, and CCCS shells. cases, except for simply supported symmetric angle ply shells. Group III: contains CSSC, CCSS, CSCS, and SCSC Because in such cases loss of stiﬀness is more signiﬁcant shells. than loss of mass. Here also the decrease in fundamental Group IV: contains SSSS shells. frequency is between 1 and 10%. But with further increase in cutout sizes a /a = 0.3 and 0.4 fundamental frequency As evident from Figures 5 and 6, fundamental frequen- decreases to an appreciable extent (up to 30%). For some cies of members belonging to diﬀerent boundary combina- cases of symmetric angle ply shell no such uniﬁed trend tions may be grouped according to performance. is observed. This leads to the engineering conclusion that According to the values of (ω), the group III may concentric cutouts may be provided safely on shell surfaces be subdivided into Group IIIa and Group IIIb for both for functional requirements up to a /a = 0.2. 0/90/0/90 and +45/−45/+45/−45 shells. Group I: contains CCCC shells. 4.1.2. Eﬀect of Boundary Conditions. The boundary con- ditions have been divided into four groups, so that the Group II: contains CSCC, CCSC, and CCCS shells. Advances in Acoustics and Vibration 7 Table 4: Clamping options for 0/90/0/90 hypar shells with central cutouts having a /a ratio 0.2. Number of sides to Improvement of frequencies with Marks indicating the Clamped edges be clamped respect to simply supported shells eﬃciencies of clamping Simply supported no edges 0 —0 clamped (SSSS) (a) Along x = 0 (CSSS) Slight improvement 13 (b) along x = a (SSCS) Slight improvement 13 (c) along y = b (SSSC) Slight improvement 13 (a) Two alternate edges Good improvement 28 2 (CSCS, SCSC) (b) Two adjacent edges Marked improvement 82 (CSSC, CCSS) 3 edges excluding y = 0 Remarkable improvement and CSCC 3 frequency becomes almost equal to 3 edges excluding x = a that of fully clamped shells CCSC 3 edges excluding y = b CCCS 4 All sides (CCCC) Frequency attains a maximum value 100 Table 5: Clamping options for +45/−45/+45/−45 hypar shells with central cutouts having a /a ratio 0.2. Number of sides Improvement of frequencies with Marks indicating the Clamped edges to be clamped respect to simply supported shells eﬃciencies of clamping Simply supported no edges clamped 0 —0 (SSSS) (a) Along x = 0 (CSSS) Slight improvement 9 (b) along x = a (SSCS) Slight improvement 9 (c) along y = b (SSSC) Slight improvement 10 (a) Two alternate edges (CSCS, SCSC) Good improvement 22 (b) Two adjacent edges (CSSC, CCSS) Marked improvement 52 3 edges excluding y = 0 CSCC Remarkable improvement 72 3 edges excluding x = a CCSC Remarkable improvement 72 3 edges excluding y = b CCCS Remarkable improvement 72 4 All sides (CCCC) Frequency attains a maximum value 100 Group IIIa: contains CSSC, CCSS shells. 4.2. Eﬀect of Eccentricity of Cutout Position on Fundamen- tal Frequency. To study the eﬀect of cutout positions on Group IIIb: contains CSCS and SCSC shells. fundamental frequencies, results are obtained for diﬀerent Group IV: contains SSSS shells. locations of a cutout with a /a = 0.2. As with the This observation indicates that the impact of arrange- introduction of cutout with a /a = 0.2, the change in ment of boundary constraints is far more important than fundamental frequency with that of an unpunctured shell their actual number in determining the free vibration is within 1–10%, so a /a = 0.2 is chosen for the further characteristics. study. Each of the non-dimensional coordinates of the cutout The frequencies are further studied, and marks are given centre (x = x/a, y = y/a) is varied from 0.2 to 0.8 along both to the options of clamping the edges of a simply supported the plan directions so that the distance of a cutout margin shell in order to gradually improve performances. Tables 4 from the shell boundary is not less than one-tenth of the and 5 furnish such clamping options for crossply and angle plan dimension of the shell. The study is carried out for ply shells, respectively. The scale is chosen like this: 0 is all the twelve boundary conditions for both 0/90/0/90 and assigned to a simply supported shell and 100 to a clamped +45/−45/+45/−45 hypar shells. The ratio of the fundamental frequency of a shell with an eccentric puncture to that of a shell. These marks are furnished for cutouts with a /a = 0.2. These tables will help a practicing engineer. If one takes the shell with concentric puncture (obtainable from Figures 5 frequency of a clamped shell as upper limit and that of the and 6)expressedinpercentageisdenoted by r. Tables 6 and simply supported as lower limit, one can easily realize the 7 contain the value of r for 0/90/0/90 and +45/−45/+45/−45 eﬃciency of a particular boundary condition. hypar shells. 8 Advances in Acoustics and Vibration Table 6: Values of “r” for 0/90/0/90 hypar shells. Edge condition y 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 98.34 98.04 97.82 97.7 97.77 97.98 98.29 0.3 98.13 97.49 96.74 96.4 96.71 97.46 98.1 0.4 98.02 96.88 95.55 95.04 95.54 96.88 98.02 CCCC 0.5 97.98 96.62 95.08 94.52 95.08 96.62 97.98 0.6 98.02 96.88 95.54 95.04 95.55 96.88 98.02 0.7 98.1 97.46 96.71 96.4 96.74 97.49 98.13 0.8 98.29 97.98 97.77 97.71 97.82 98.04 98.34 0.2 96.11 96.68 96.37 96.31 96.38 96.67 97.25 0.3 96.11 95.46 95.31 95.45 95.38 95.50 96.1 0.4 95.35 94.72 94.81 95.51 94.88 94.75 95.36 CSCC 0.5 95.27 94.51 94.49 96.13 94.49 94.52 95.27 0.6 95.49 94.58 94.05 93.43 94.03 94.59 95.51 0.7 95.83 94.81 93.71 92.57 93.68 94.83 95.87 0.8 102.93 95.19 93.75 92.41 93.74 95.22 96.33 0.2 96.19 95.74 95.42 95.26 95.36 95.95 97.05 0.3 95.08 94.73 94.51 94.50 94.77 95.49 96.76 0.4 93.26 93.26 93.65 94.21 94.77 95.44 96.73 CCSC 0.5 91.67 91.87 92.74 94.12 95.51 95.61 96.81 0.6 93.21 93.21 93.58 94.15 94.77 95.48 96.76 0.7 95.05 94.69 94.47 94.48 94.80 95.57 96.84 0.8 96.19 95.73 95.41 95.28 95.42 96.08 97.17 0.2 96.32 95.22 93.73 92.33 93.65 95.16 95.16 0.3 95.86 94.83 93.65 92.44 93.59 94.78 95.83 0.4 95.50 94.59 94.02 93.38 93.97 94.56 95.48 CCCS 0.5 95.28 94.52 94.47 95.68 94.43 94.49 95.26 0.6 95.34 94.49 94.85 95.51 94.79 94.71 95.33 0.7 96.08 95.50 95.37 95.45 95.31 95.46 96.09 0.8 97.23 96.67 96.38 96.31 96.37 96.68 97.28 0.2 96.08 95.31 94.74 94.43 94.54 95.05 95.64 0.3 95.38 95.13 95.25 95.43 95.50 95.44 95.10 0.4 91.75 91.74 92.23 93.10 94.94 95.44 94.57 CSSC 0.5 90.85 90.57 90.76 91.46 93.56 95.23 94.56 0.6 92.37 91.62 91.32 91.40 92.88 94.92 94.98 0.7 94.10 93.04 92.27 91.43 92.24 94.84 95.64 0.8 95.61 94.49 93.33 91.74 92.13 95.18 96.46 0.2 95.54 94.41 93.26 91.63 91.98 94.95 96.31 0.3 94.05 92.98 92.21 91.34 92.10 94.66 95.50 0.4 92.36 91.61 91.32 91.34 92.80 94.77 94.87 CCSS 0.5 90.91 90.66 90.80 91.76 93.68 95.11 94.50 0.6 91.71 91.75 92.27 93.17 95.24 95.40 94.55 0.7 95.28 95.03 95.14 95.36 95.47 95.43 95.11 0.8 95.96 95.22 94.69 94.44 94.62 95.18 95.77 0.2 99.07 100.14 98.96 98.08 98.93 100.14 99.10 0.3 99.27 100.79 99.18 97.98 99.14 100.76 99.28 0.4 99.56 101.15 99.46 98.20 99.44 101.13 99.55 CSCS 0.5 99.67 101.24 99.57 98.33 99.56 101.24 99.67 0.6 99.54 101.13 99.44 98.20 99.45 101.15 99.55 0.7 99.28 100.76 99.14 97.98 99.17 100.78 99.27 0.8 99.10 100.14 98.92 98.08 98.95 100.14 99.07 Advances in Acoustics and Vibration 9 Table 6: Continued. Edge condition y 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 99.04 99.30 99.56 99.67 99.56 99.27 98.99 0.3 100.18 100.79 101.13 101.23 101.14 100.81 100.18 0.4 99.04 99.22 99.43 99.54 99.44 99.23 99.04 SCSC 0.5 98.17 98.07 98.20 98.29 98.20 98.07 98.17 0.6 99.04 99.23 99.44 99.54 99.43 99.22 99.04 0.7 100.18 100.82 101.14 101.23 101.13 100.80 100.18 0.8 98.99 99.27 99.56 99.82 99.56 99.30 99.04 0.2 98.49 98.97 98.19 97.42 97.62 98.61 98.94 0.3 98.82 99.92 99.27 98.39 98.59 99.53 99.48 0.4 99.28 100.60 99.91 98.83 98.95 99.95 99.89 CSSS 0.5 99.44 100.78 100.07 98.95 99.07 100.12 100.07 0.6 99.26 100.59 99.89 98.82 98.97 99.98 99.89 0.7 98.83 99.91 99.22 98.37 98.62 99.58 99.49 0.8 98.54 99.02 98.17 97.43 97.67 98.65 98.93 0.2 98.92 99.40 99.86 100.08 99.87 99.41 98.90 0.3 98.71 99.58 99.94 100.10 99.95 99.59 98.70 0.4 97.69 98.68 98.96 99.05 98.96 98.67 97.67 SSSC 0.5 97.44 98.46 98.82 98.91 98.82 98.44 97.42 0.6 98.23 99.29 99.86 100.01 99.85 99.27 98.22 0.7 99.00 99.94 100.56 100.74 100.55 99.93 99.01 0.8 98.39 98.81 99.26 99.43 99.26 98.83 98.45 0.2 98.93 98.65 97.68 97.42 98.18 99.02 98.55 0.3 99.48 99.58 98.62 98.37 99.22 99.91 98.83 0.4 99.89 99.98 98.96 98.82 99.88 100.59 99.27 SSCS 0.5 100.08 100.12 99.06 98.94 100.06 100.78 99.45 0.6 99.88 99.95 98.94 98.82 99.90 100.60 99.27 0.7 99.48 99.53 98.58 98.39 99.26 99.91 98.81 0.8 98.94 98.60 97.61 97.42 98.17 98.96 98.49 0.2 97.78 97.55 96.93 96.64 96.85 97.51 97.94 0.3 98.22 99.14 98.94 98.81 98.80 98.99 98.06 0.4 97.52 99.41 99.42 99.28 99.24 99.20 97.43 SSSS 0.5 97.22 99.41 99.58 99.37 99.59 99.41 97.22 0.6 97.44 99.20 99.23 99.27 99.43 99.41 97.52 0.7 98.06 98.99 98.80 98.81 98.93 99.14 98.22 0.8 97.94 97.50 96.84 96.66 96.93 97.55 97.78 a/b = 1, a/h = 100, a /b = 1, c/a = 0.2; E /E = 25, G = 0.2E , G = G = 0.5E , ν = ν = 0.25. 11 22 23 22 13 12 22 12 21 10 Advances in Acoustics and Vibration Table 7: Values of “r” for +45/−45/+45/−45 hypar shells. Edge condition y 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 97.79 94.67 95.97 102.56 95.97 94.82 97.78 0.3 94.87 94.55 96.55 101.00 96.53 94.55 94.87 0.4 95.96 96.43 97.07 98.73 97.07 96.43 95.95 CCCC 0.5 102.07 100.76 98.61 98.80 98.61 100.76 101.92 0.6 95.95 96.43 97.10 98.77 97.07 96.43 95.95 0.7 94.87 94.55 96.52 100.99 96.52 94.55 94.87 0.8 97.79 94.82 95.97 102.30 95.97 94.82 97.78 0.2 97.57 94.48 97.00 104.29 97.32 94.75 97.73 0.3 99.55 97.12 98.28 101.72 98.39 97.16 99.54 0.4 100.05 97.96 97.40 99.02 97.28 97.79 101.05 CSCC 0.5 100.24 97.51 94.53 93.90 94.31 97.28 100.18 0.6 97.85 95.01 91.14 90.09 90.91 94.74 97.89 0.7 97.55 92.19 87.69 86.65 87.45 91.90 97.32 0.8 97.21 90.03 85.03 84.15 84.80 89.75 97.08 0.2 97.20 97.40 98.19 100.24 99.74 99.45 97.77 0.3 89.82 91.97 94.82 97.28 97.77 97.17 94.74 0.4 84.74 87.48 90.99 94.48 97.44 98.49 97.24 CCSC 0.5 84.02 86.63 90.17 94.16 99.51 102.03 104.53 0.6 84.74 87.47 90.99 94.48 97.44 98.49 97.24 0.7 89.82 91.97 94.82 97.28 97.77 97.17 94.74 0.8 97.20 97.40 98.19 100.24 99.74 99.48 97.77 0.2 97.12 89.92 85.08 84.42 85.08 89.92 97.12 0.3 97.36 92.03 87.73 86.93 87.73 92.03 97.36 0.4 98.00 94.81 91.17 90.40 91.17 94.81 98.00 CCCS 0.5 100.15 97.25 90.40 94.22 94.51 97.25 100.15 0.6 99.96 97.83 97.38 99.37 97.38 97.83 100.67 0.7 99.58 97.23 98.44 101.84 98.44 97.23 99.95 0.8 97.74 94.71 97.22 104.30 97.22 94.71 97.76 0.2 98.26 99.11 101.00 99.30 96.94 95.60 95.03 0.3 90.66 92.23 94.92 97.24 97.39 96.69 95.59 0.4 86.66 88.94 92.14 95.13 96.60 97.47 96.95 CSSC 0.5 89.02 91.38 93.92 94.86 94.78 97.23 99.41 0.6 95.40 97.29 97.18 93.64 91.91 94.83 100.96 0.7 98.69 98.73 96.98 91.14 88.82 92.16 99.05 0.8 99.02 98.80 95.27 88.93 86.67 90.63 98.26 0.2 99.03 98.82 95.03 88.84 86.67 90.65 98.33 0.3 98.70 98.68 96.83 91.07 88.82 92.17 99.06 0.4 95.40 97.30 97.09 93.59 91.92 94.84 100.99 CCSS 0.5 89.01 91.37 93.87 94.82 94.96 97.25 99.42 0.6 86.65 88.93 92.10 95.10 96.61 97.47 96.93 0.7 90.63 92.21 94.89 97.22 97.39 96.69 95.57 0.8 98.26 99.07 100.96 99.30 96.95 95.63 95.03 0.2 98.68 99.70 96.60 94.59 96.60 99.70 98.68 0.3 100.35 100.95 99.60 98.53 99.60 100.95 100.35 0.4 100.63 101.43 100.62 99.97 100.62 101.43 100.62 CSCS 0.5 100.75 101.79 101.07 100.45 101.07 101.79 100.75 0.6 100.63 101.43 100.62 99.97 100.62 101.43 100.62 0.7 100.35 100.95 99.60 98.53 99.60 100.95 100.35 0.8 98.68 99.70 96.60 94.59 96.60 99.70 98.68 Advances in Acoustics and Vibration 11 Table 7: Continued. Edge condition y 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 98.88 100.37 100.64 100.83 100.64 100.37 98.88 0.3 99.76 101.04 101.52 101.86 101.53 101.04 99.75 0.4 96.65 99.74 100.713 101.14 100.76 99.74 96.64 SCSC 0.5 94.69 98.67 100.09 100.53 100.09 98.67 94.69 0.6 96.64 99.74 100.76 101.14 100.76 99.74 96.64 0.7 99.76 101.04 101.53 101.86 101.52 101.04 99.75 0.8 98.88 100.37 100.64 100.83 100.64 100.37 98.88 0.2 97.88 98.69 96.54 93.98 94.57 97.61 98.39 0.3 99.43 99.86 99.39 98.38 98.64 99.79 99.86 0.4 99.76 100.55 100.55 100.23 100.53 101.37 101.32 CSSS 0.5 100.33 101.13 100.94 100.64 101.07 102.09 101.97 0.6 99.76 100.55 100.55 100.23 100.52 101.54 101.32 0.7 99.43 99.86 99.39 98.38 98.64 99.80 99.86 0.8 97.88 98.68 96.54 93.98 94.57 97.61 98.39 0.2 98.24 99.78 101.36 102.07 101.36 99.77 98.24 0.3 97.52 99.76 101.40 102.14 101.40 99.76 97.51 0.4 94.58 98.68 100.53 101.03 100.53 98.68 94.58 SSSC 0.5 94.08 98.44 100.24 100.63 100.24 98.44 94.08 0.6 96.58 99.47 100.62 100.96 100.62 99.47 96.57 0.7 98.72 99.90 100.61 101.15 100.61 99.90 98.72 0.8 98.03 99.41 99.73 100.34 99.72 99.41 98.03 0.2 98.39 97.61 94.57 93.99 96.55 98.69 97.88 0.3 99.87 99.80 98.65 98.39 99.40 99.87 99.43 0.4 101.32 101.38 100.53 100.23 100.55 100.56 99.76 SSCS 0.5 101.98 102.10 101.07 100.64 100.95 101.13 100.33 0.6 101.33 101.38 100.53 100.23 100.55 100.56 99.76 0.7 99.87 99.80 98.65 98.39 99.40 99.87 99.43 0.8 98.39 97.62 94.58 93.99 96.54 98.69 97.88 0.2 97.32 97.03 111.45 93.16 94.72 97.03 97.32 0.3 96.96 98.53 98.61 98.09 98.62 98.53 96.96 0.4 94.66 98.58 100.57 100.43 100.57 98.58 94.65 SSSS 0.5 93.19 98.05 100.34 100.80 100.34 98.05 93.18 0.6 94.65 98.58 100.57 100.43 100.57 98.58 94.65 0.7 96.95 98.53 98.61 98.09 98.62 98.53 96.96 0.8 97.31 97.03 94.72 93.16 94.72 97.03 97.32 a/b = 1, a/h = 100, a /b = 1, c/a = 0.2; E /E = 25, G = 0.2E , G = G = 0.5E , ν = ν = 0.25. 11 22 23 22 13 12 22 12 21 12 Advances in Acoustics and Vibration From the r values it is observed that a shell with eccentric roofs with cutouts as this approach produces results cutout has a fundamental frequency above 90% to that of in close agreement with those of the benchmark a shell having concentric cutout in almost all the cases of problems. 0/90/0/90 shells. So cutout centre may be moved along any (2) Concentric cutouts may be provided safely on hypar directions, resulting in loss of frequency of not more than shell surfaces for functional requirements up to 5–10% with respect to a shell with concentric cutout. But in a /a = 0.2. case of a +45/−45/+45/−45 shell, the r value is between 84.02 and 102.10. (3) The arrangement of boundary constraint along the For 0/90/0/90 shell when the four edges are clamped four edges is far more important than their actual the stiﬀness of shell decreases more when the cutout centre number so far the free vibration stiﬀness is con- is nearer to the shell centre line whereas when the cutout cerned. centre goes towards the boundary shell stiﬀness increases (4) Fundamental frequency undergoes marked improve- resulting in increased frequency. For such a shell r value ment when the edge is converted to clamped from is minimum along centre line and increases towards the simply supported condition. boundary. But such increase or decrease is marginal in each case. When one simply supported edge is introduced along (5) Tables 4 and 5 provide a clear picture about the the shell boundary r value is maximum along the simply relative free vibration performances of hypar shells supported boundary in almost all the cases. Similarly when for diﬀerent combinations of edge conditions along two adjacent edges are simply supported the values of r have the four sides and are expected to be very useful in comparatively greater value towards simply supported edges. decision making for practicing engineers. But exactly reverse trend is observed when two opposite edges are simply supported. In such cases the maximum r (6) Tables 6 and 7 provide the information regarding behaviour of hypar shell with eccentric cutouts for values are obtained along the centre line of the shell which is equidistant from each simply supported edge and r value wide spectrum of eccentricity and boundary condi- decreases towards the simply supported edges. It is further tions and may be used as design aids by structural engineers. noticed that when three edges are simply supported the r value is maximum along the line which is equidistant from two simply supported edges and r value is minimum along Notations the centre which is equidistant from one simply supported edge and another clamped edge. When four edges are simply a, b: Length and width of shell in plane supported r value is maximum along the diagonal. a , b : Length and width of cutout in plane In case of +45/−45/+45/−45 shells when the four edges c: Rise of hypar shell are clamped the r values are maximum along the centre line {d }: Element displacement of shell in both directions but towards the boundary no E , E : Elastic moduli 11 22 uniﬁed trend is observed. Also along each clamped edge r G , G , G : Shear moduli of a lamina with respect to 12 13 23 value is maximum at the middle of each edge. When one of 1, 2and 3axesofﬁbre the edges is simply supported, the r value decreases towards h: Shell thickness the edges opposite to the simply supported edges and are M , M : Moment resultants x y minimum at the centre of that edge. But along the simply M : Torsion resultant xy supported edge and other two clamped edges fundamental np: Number of plies in a laminate frequency is maximum at the middle of each edge. When the N –N : Shape functions 1 8 two adjacent edges are simply supported and other two are N , N : Inplane force resultants x y clamped minimum r values are obtained when cutout centre N : Inplane shear resultant xy is near the middle of the clamped edges and maximum when Q , Q : Transverse shear resultant x y the same is near the middle of the simply supported edges. R : Radii of cross curvature of hypar shell xy When two opposite edges are simply supported in more than u, v, w: Translational degrees of freedom 50% cases r value is greater than 100. In rest of the cases r x, y, z: Local co-ordinate axes values are more than 95 except one or two cases. When three X, Y , Z: Global co-ordinate axes edges are simply supported in a central rectangular zone the z : Distance of bottom of the kth ply from r value is greater than 100. But when four edges are simply midsurface of a laminate supported the zone where r value is greater than 100 gets α, β: Rotational degrees of freedom reduced. ε , ε : Inplane strain component x y φ: Angle of twist γ , γ , γ : Shearing strain components xy xz yz 5. Conclusions ν , ν : Poisson’s ratios 12 21 ξ, η, τ: Isoparametric coordinates From the present study the following conclusions are drawn. ρ:Densityofmaterial (1) The ﬁnite element code used here is suitable for σ , σ : Inplane stress components x y analyzing free vibration problems of hypar shell τ , τ , τ : Shearing stress components xy xz yz Advances in Acoustics and Vibration 13 ω: Natural frequency ω: Non-dimensional natural frequency 1/2 2 2 = ωa (ρ/E h ) . References [1] D. Chakravorty, P. K. Sinha, and J. N. Bandyopadhyay, “Appli- cations of FEM on free and forced vibration of laminated shells,” Journal of Engineering Mechanics, vol. 124, no. 1, pp. 1–8, 1998. [2] J. N. Reddy, “Large amplitude ﬂexural vibration of layered composite plates with cutouts,” Journal of Sound and Vibra- tion, vol. 83, no. 1, pp. 1–10, 1982. [3] S. K. Malhotra, N. Ganesan, and M. A. Veluswami, “Vibration of composite plates with cut-outs,” Journal of Aeronautical Society of India, vol. 41, pp. 61–64, 1989. [4] B. Sivasubramonian, A. M. Kulkarni, G. Venkateswara Rao, and A. Krishnan, “Free vibration of curved panels with cutouts,” Journal of Sound and Vibration, vol. 200, no. 2, pp. 227–234, 1997. [5] K.Sivakumar,N.G.R.Iyengar,and K. Deb, “Freevibration of laminated composite plates with cutout,” Journal of Sound and Vibration, vol. 221, no. 3, pp. 443–465, 1999. [6] R. E. Rossi, “Transverse vibrations of thin, orthotropic rectan- gular plates with rectangular cutouts with ﬁxed boundaries,” Journal of Sound and Vibration, vol. 221, no. 4, pp. 733–736, [7] M. Huang and T. Sakiyama, “Free vibration analysis of rectangular plates with variously-shaped holes,” Journal of Sound and Vibration, vol. 226, no. 4, pp. 769–786, 1999. [8] S. S. Hota and P. Padhi, “Vibration of plates with arbitrary shapes of cutouts,” JournalofSound andVibration, vol. 302, no. 4-5, pp. 1030–1036, 2007. [9] B. Sivasubramonian, G. V. Rao, and A. Krishnan, “Free vibra- tion of longitudinally stiﬀened curved panels with cutout,” Journal of Sound and Vibration, vol. 226, no. 1, pp. 41–55, [10] S. S. Hota and D. Chakravorty, “Free vibration of stiﬀened conoidal shell roofs with cutouts,” JVC/Journal of Vibration and Control, vol. 13, no. 3, pp. 221–240, 2007. [11] N. Nanda and J. N. Bandyopadhyay, “Nonlinear free vibra- tion analysis of laminated composite cylindrical shells with cutouts,” Journal of Reinforced Plastics and Composites, vol. 26, no. 14, pp. 1413–1427, 2007. [12] M. S. Qatu, Vibration of Laminated Shells and Plates,Elsevier, London, UK, 2004. [13] V. V. Vasiliev, R. M. Jones, and L. I. Man, Mechanics of Composite Structures, Taylor & Francis, Boca Raton, Fla, USA, [14] A. Dey, J. N. Bandyopadhyay, and P. K. Sinha, “Finite element analysis of laminated composite paraboloid of revolution shells,” Computers and Structures, vol. 44, no. 3, pp. 675–682, [15] R. D. Cook,D.S.Malkus, andM.E.Plesha, Concepts and Applications of Finite Element Analysis,JohnWiley &Sons, New York, NY, USA, 1989. [16] S. Sahoo and D. Chakravorty, “Finite element vibration characteristics of composite hypar shallow shells with various edge supports,” JVC/Journal of Vibration and Control, vol. 11, no. 10, pp. 1291–1309, 2005. 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Published: Dec 25, 2011
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