Mode frequency analysis of antisymmetric angle-ply laminated composite stiffened hypar shell with cutout

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Abstract

This article deals with finite element method for the analysis of antisymmetric angle-ply laminated composite hypar shells (hyperbolic paraboloid bounded by straight edges) that applies an eight-noded isoparametric shell element and a three-noded beam element to study the mode-frequency analysis of stiffened shell with cutout. Two-, 4-, and 10-layered antisymmetric angle-ply laminations with different lamination angles are considered. Among these, 10-layer antisymmetric angle-ply shells are considered for elaborate study. The shells have different boundary conditions along its four edges. The formulation is based on the first-order shear deformation theory. The reduced method of eigen value solution is chosen for the undamped free vibration analysis. The first five modes of natural frequency are presented. The numerical studies are conducted to determine the effects of width-to-thickness ratio (b/h), degree of orthotropy (E11/E22), and fiber orientation angle (θ) on the nondimensional natural frequency. The results reveal that free vibration behavior mainly depends on the number of boundary constraints rather than other parametric variations such as change in fiber orientation angle and increase in degree of orthotropy and width-to-thickness ratio.

1 Introduction

Laminated composite shells now constitute a large percentage of structures including aerospace, marine, and automotive structural components. Structural engineers have already picked up laminated composite hypar shells (hyperbolic paraboloid bounded by straight edges) as roofing units because these can cover large column-free areas with reduced dead weight. This class of shells has only the radius of cross curvature that is unique to this shell form. Roof structures are sometimes provided with cutout to allow the entry of light, venting and to provide the accessibility of parts of the structures, and also to alter the resonant frequency. Shells with cutout stiffened along the margin are an efficient way to enhance the stiffness of the structure without adding much mass. These stiffeners slightly increase the overall weight of the structure but have positive effect on the structural strength and stability. So to apprehend the laminated composite stiffened hypar shells with cutout and to use this shell form efficiently, its characteristics under vibration need to be explored comprehensively.

The subject of laminated shells has attracted several researchers during the past decade. Considerable attention has been paid to dynamic analyses, including free vibration, impact, transient, shock, etc. From the review of literature, it is observed that shell research has been conducted with emphasis on complicating effects in material such as damping and piezoelectric behavior and complicated structures such as stiffened shells with cutout with various boundary conditions. Applications of various shell theories such as classical, shear deformation, 3D, and nonlinear for various shell geometries have received extensive attention from scholars round the globe [1, 2, 3, 4, 5, 6, 7, 8]. Some studies used higher order shell theories [9, 10, 11, 12, 13, 14, 15, 16], whereas others [17, 18, 19, 20, 21, 22, 23] considered finite element approach based on the first-order shear deformation theory to study the free vibration aspects of stiffened shell panels of different forms, that is, cylindrical, elliptic paraboloid, hyperbolic paraboloid, hypar, conoid, and spherical shells in the presence of cutout. But the analysis of stiffened shells with cutout for higher modes is scanty in the literature. Some studies [24, 25] also reported the mode frequency analysis of laminated spherical and cylindrical shells. However, the analysis of stiffened hypar shells with cutout for higher modes is yet to be reported in literature. Hence, an attempt has been made in the present article to provide some information about the effects of width-to-thickness ratio (b/h), degree of orthotropy (E11/E22), and fiber orientation (θ) on the nondimensional natural frequency for antisymmetric angle-ply hypar shells with 10-layer laminates in the presence of cutout and stiffeners for different practical boundary conditions.

2 Mathematical Formulation

A laminated composite hypar shell of uniform thickness h (Figure 1) and twist radius of curvature Rxy is considered. The shell thickness may consist of any number of thin laminae each of which may be arbitrarily oriented at an angle θ with reference to the X-axis of the coordinate system. An eight-noded, curved, quadratic, isoparametric, finite element (Figure 2a) is used. The five degrees of freedom taken into consideration at each node include two in-plane and one transverse displacement and two rotations about the X- and Y-axes. The strain–displacement and constitutive relationships together with the systematic development of stiffness matrix for the shell element has been reported earlier [7] and the same is used in the present case as well.

Figure 1
Figure 1

Surface of a skewed hypar shell with cutout

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0022

Three-noded isoparametric beam elements (Figure 2b) are used to model the stiffeners, which are taken to run only along the boundaries of the shell elements. In the stiffener element, each node has four degrees of freedom, that is, usx, wsx, αsx, and βax for X-stiffener and vsy, wsy, αsy, and βsy for Y-stiffener. The generalized forcedisplacement relation of stiffeners can be expressed as follows (the notations have been defined in Appendix A):

Figure 2
Figure 2

(a) Eight-noded shell element. (b) Three-noded stiffener element: (i)X-stiffener and (ii) Y-stiffener

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0022

X stiffener:{Fsx}=[Dsx]{εsx}=[Dsx][Bsx]{δsxi};Y stiffener:{Fsy}=[Dsy]{εsy}=[Dsy][Bsy]{δsyi}

where

{Fsx}=[NsxxMsxxTsxxQsxxz]T;{εsx}=[usx.xαsx.xβsx.x(αsx+wsx.x)]T

and

{Fsy}=[NsyyMsyyTsyyQsyyz]T;{εsy}=[vsy.yβsy.yαsy.y(βsy+wsy.y)]T.

Elasticity matrices are expressed as follows:

[Dsx]=A11bsxB11bsxB12bsx0B11bSXD11bSXD12bSX0B12bSXD12bSX16(Q44+Q66)dSXbSX30000bSXS11[Dsy]=A22bsyB22bsyB12bsy0B22bsy16(Q44+Q66)bsyD12bsy0B12bsyD12bsyD11dsybsy30000bsyS22,

where

Dij=Dij+2eBij+e2Aij

and Aij, Bij, Dij, and Sij are explained in an earlier paper [7].

Here the shear correction factor is taken as 5/6. The sectional parameters are calculated with respect to the midsurface of the shell by which the effect of eccentricities of stiffeners is automatically included. The element stiffness matrices are of the following forms.

forXstiffener:[Kxe]=[Bsx]T[Dsx][Bsx]dx;forYstiffener:[Kye]=[Bsy]T[Dsy][Bsy]dy.

The integrals are converted to isoparametric coordinates and are carried out by 2 point Gauss quadrature. Finally, the element stiffness matrix of the stiffened shell is obtained by appropriate matching of the nodes of the stiffener and shell elements through the connectivity matrix and is written as follows:

[Ke]=[Kshe]+[Kxe]+[Kye]

The element stiffness matrices are assembled to get the global matrices.

The element mass matrix for shell is obtained from the following integral equation:

[Me]=T[P][N]dxdy

where

[N]=i=18Ni00000Ni00000Ni00000Ni00000Ni,[P]=i=18P00000P00000P00000I00000I,

in which

P=k=1Znpk1zkρdzandI=k=1Znpk1zkzρdz

Element mass matrix for the stiffener element is

[Msx]=T[P][N]dxforXstiffenerand[Msy]=T[P][N]dyforYstiffener.

Here, [N] is a 3 × 3 diagonal matrix.

[P]=i=13ρ.bsxdsx0000ρ.bsxdsx0000ρ.bsxdsx2/120000ρ(bsx.dsx3+bsx3.dsx)/12

for X – stiffener,

[P]=i=13ρ.bsydsy0000ρ.bsydsy0000ρ.bsydsy2/120000ρ(bsy.dsy3+bsy3.dsy)/12

for Y-stiffener.

The mass matrix of the stiffened shell element is the sum of the matrices of the shell and the stiffeners matched at the appropriate nodes.

[Me]=[Mshe]+[Mxe]+[Mye]

The element mass matrices are assembled to get the global matrices.

In regards to the modeling of the cutout, the code developed can take the position and size of cutout as input. The program is capable of generating nonuniformal finite element mesh all over the shell surface. So the element size is gradually decreased near the cutout margins. One such typical mesh arrangement is shown in Figure 3. Such finite element mesh is redefined in steps and a particular grid is chosen to obtain the fundamental frequency when the result does not improve by more than 1% on further refining. Convergence of results is ensured in all the problems taken up here.

Figure 3
Figure 3

Typical 10 × 10 nonuniformal mesh arrangement drawn to scale

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0022

Finally, the free vibration analysis involves the determination of natural frequencies from the condition

|[K]ω2[M]|=0

This is a generalized eigen value problem and is solved by the subspace iteration algorithm.

3 Numerical Examples

Problems are solved with two different objectives. Benchmark problems are used to check the suitability of the present approach and a number of authors’ own problems are taken up to assess the mode–frequency behavior of antisymmetric angle-ply composite stiffened hypar shells with cutout.

The accuracy of the present formulation is first validated by comparing the results of the following problems available in the existing literature. The results presented in Table 1 show that the agreement of the present results with the earlier ones [4, 26] is excellent and the correctness of the stiffener formulation is established. Free vibration of simply supported and clamped hypar shell having (0/90)4 lamination and with cutouts is also considered. The fundamental frequencies of hypar shell with cutout obtained by the present method agree well with those reported by Chakravorty et al. [1], which is evident from Table 2, establishing the correctness of the cutout formulation. Thus it is evident that the finite element model proposed here can successfully analyze vibration problems of stiffened skewed hypar composite shells with cutout that is reflected by a close agreement of the present results with benchmark ones.

Table 1

Natural frequencies (Hz) of centrally stiffened clamped square plate

Mode no.Mukherjee and Mukhopadhyay [26]Nayakand Bandyopadhyay [4]Present method
N8 (FEM)N9 (FEM)
1711.8725.2725.1733
a = b = 0.2032 m; thickness = 0.0013716 m; stiffener depth = 0.0127 m; stiffener width = 0.00635 m; stiffener eccentric at bottom. Material property: E = 6.87 × 1010 N/m2, ν = 0.29, ρ = 2,823 kg/m3

Table 2

Nondimensional fundamental frequencies (ω) for hypar shells (lamination (0/90)4) with concentric cutouts

a′/aChakravorty et al. [1])Present finite element model

SimplyClampedSimply supportedClamped

supported8 × 810 × 1012 × 128 × 810 × 1012 × 12
0.050.829111.60050.57350.82150.825111.445111.592111.612
0.150.769110.16650.67950.75850.779109.987110.057110.233
0.250.434105.46450.32350.42150.400105.265105.444105.443
0.349.165101.35049.04549.15749.178101.110101.340101.490
0.447.24497.98747.13247.24247.14197.67097.98597.991
a/b = 1; a/h = 100; a′/b′ = 1; c/a = 0.2; E11/E22 = 25; G23 = 0.2 E22; G13 = G12 = 0.5 E22; V12 = V21 = 0.25.

Antisymmetric angle-ply laminated composite stiffened hypar shells with cutout are analyzed to study the behavior of the shell under free vibration at higher mode for different parametric variation. The cutouts are placed concentrically on the shell surface. The stiffeners are placed along the cutout periphery and extended up to the edge of the shell. The material and geometric properties of the shells are a/b = 1, a/h = 100, a′/b′ = 1, a′/a = 0.2, c/a = 0.2, E11/E22 = 25, G23 = 0.2 E22, G13 = G12 = 0.5 E22, ν12 = ν21 = 0.25, and p = 100 N-s2/m4 unless otherwise specified.

Seven laminated stacking sequences, namely, antisymmetric angle-ply (0/−0)10, (15/−15)10, (30/−30)10, (45/−45)10, (60/−60)10, (75/−75)10, and (90/−90)10 are considered. Numerical analyses are performed to determine the effect of fiber orientation angle (θ = 0°, 15°, 30°, 45°, 60°, 75°, and 90°), degree of orthotropy (E11/E22 = 5, 10, 20, 25, 30, 40, and 50), and width-to-thickness ratio (b/h = 10, 20, 50, 100) on nondimensional natural frequency.

The different boundary conditions that are used in the present analysis are CSCS, CSSC, FCCF, FCFC, FSFS and FSSF. The boundary conditions are designated as: C for clamped, S for simply supported, and F for free edges. The four edges are considered in an anticlockwise order starting from the edge x = 0. For example, a shell with CSCS boundary is clamped along x = 0, simply supported along y = 0, clamped along x = a, and simply supported along y = b.

4 Results and Discussion

Table 3 presents the nondimensional fundamental frequency (the first-mode frequencies) for 2-, 4-, and 10-layered antisymmetric angle-ply laminated composite stiffened hypar shells with cutout with the fiber orientation angle varying between 0° and 90°. In each column, the maximum value is indicated by an asterisk. It is observed that for the two-layered hypar shell, maximum fundamental frequency occurs at the lamination angle of either 0° or 90° except for CSSC shell. For CSSC shell, the maximum fundamental frequency occurs at the lamination angle of 45°. It is also observed that for all the boundary conditions considered here, 4- and 10-layered laminates exhibit the maximum value of frequency parameter for the same lamination angle., The maximum fundamental frequency occurs at the lamination angle, θ, of 75° and 45°, respectively, for CSCS and CSSC shells; at θ = 0° for FCFC shells; and at θ = 30° for FCCF, FSFS and FSSF shells. This observation is valid for both 4- and 10-layered hypar shells. According to the number of boundary constraints, boundary conditions can be grouped as CSCS & CSSC; FCCF & FCFC; and FSFS & FSSF. For all the layers considered here, as the number of boundary constraint increases, the fundamental frequency increases. Thus CSCS & CSSC perform better than FCCF & FCFC, which in turn perform better than FSFS & FSSF shells. It is also observed from Table 3 that with the increase in layer, the frequency parameter increases. The increments are sharper from 2 to 4 layers compared to 4 to 10 layers, in which a mild increase in the frequency parameter is observed. As the 10-layered laminates exhibit best performance, so far the fundamental frequency is concerned, they are considered for further studies.

Table 3

Nondimensional fundamental frequency of antisymmetric angle-ply multilayered laminated composite stiffened hypar shell with cutout

Angle-plyBoundary ConditionTwo LayersFour LayersTen Layers
CSCS6.3566.3566.356
15°6.0567.4677.892
30°6.7879.39910.121
45°7.82511.35612.324
60°8.91412.62213.631
75°10.19612.91913.642*
90°13.254*13.254*13.254
CSSC13.01713.01613.016
15°12.44913.97914.307
30°12.60714.95415.547
45°13.356*15.369*15.799*
60°12.52614.92815.549
75°12.40213.99814.362
90°13.00813.00813.008
FCCF6.5396.5396.539
15°6.0116.8557.082
30°6.2637.327*7.555*
45°6.2337.1617.377
60°6.1967.2127.436
75°6.0006.8067.024
90°6.548*6.5486.548
FCFC11.707*11.706*11.708*
15°8.03110.74611.394
30°5.8758.6289.246
45°4.2836.0336.435
60°3.1473.7953.958
75°2.6742.7412.759
90°2.6042.6042.604
FSFS4.367*4.3674.367
15°4.0414.7324.882
30°3.9744.984*5.204*
45°2.9414.0384.293
60°2.1942.5832.679
75°1.8601.9051.918
90°1.8061.8061.806
FSSF3.628*3.6283.628
15°3.3103.9454.127
30°3.3494.190*4.403*
45°3.2904.1664.380
60°3.3044.1044.307
75°3.2683.8604.027
90°3.5603.5603.560

4.1 Effect of fiber orientation

The total thickness of the laminate was maintained constant, and the number of layers is 10. Figure 4 shows the variation of nondimensional frequency with boundary conditions and lamination angle. Seven laminate stacking sequences, namely, antisymmetric angle-ply (0/−0)10, (15/−15)10, (30/−30) 10, (45/−45)10, (60/−60)10, (75/−75)10, and (90/−90)10 are considered. The nondimensional frequency parameter for the first, second, third, fourth, and fifth mode increases with an increase in the fiber orientation angle from 0° to 45° for CSSC, FCCF, and FSFS shells, but further increase in the lamination angle decreases the fundamental frequency. Similarly, for CSCS shell, the fundamental frequency increases up to 60° and then decreases. Again for FCFC shell, fundamental frequency decreases with an increase in the lamination angle. On the other hand for FSSF shell, the change in the frequency with change in lamination angle is very insignificant. For all the laminations and boundary conditions considered here, frequency parameter increases from the first mode to the fifth mode except in few cases in which frequency parameter is almost the same in two consecutive modes. The lamination and boundary conditions interact in a complex manner so that no unified conclusion can be reached. The reason behind this is that the frequencies depend on the contribution made by extensional stiffness, coupling stiffness, and bending stiffness terms in addition to the boundary conditions and panel geometry, among others. However, for all the lamination angles considered here, CSCS & CSSC perform better than FCCF & FCFC, which in turn perform better than FSFS & FSSF. So it can be concluded that the number of boundary constraints plays a great role for free vibration. CSSC performs better than CSCS for lower lamination angles, but for higher lamination angle, CSCS performs better than CSSC. But reverse trend is observed when free edges are involved. FCFC and FSFS perform better in lower lamination angle but FCCF and FSSF perform better in higher lamination angle.

Figure 4
Figure 4

Variation of nondimensional fundamental frequency with the fiber orientation angle

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0022

4.2 Effect of material anisotropy

The effects of material anisotropy on the frequencies of 10-layer antisymmetric angle-ply square shells with fiber orientation angles of 0°, 15°, 30°, 45°, 60°, 75°, and 90° for CSCS, CSSC, FCCF, FCFC, FSFS, and FSSF edge boundary conditions, respectively, are demonstrated in Figures 511.

Figure 5
Figure 5

Variation of nondimensional fundamental frequency with material anisotropy for (0/−0)10 lamination

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0022

Figure 6
Figure 6

Variation of nondimensional fundamental frequency with material anisotropy for (15/−15)10 lamination

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0022

Figure 7
Figure 7

Variation of nondimensional fundamental frequency with material anisotropy for (30/−30)10 lamination

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0022

Figure 8
Figure 8

Variation of nondimensional fundamental frequency with material anisotropy for (45/−45)10 lamination

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0022

Figure 9
Figure 9

Variation of non-dimensional fundamental frequency with material anisotropy for (60/−60)10 lamination

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0022

Figure 10
Figure 10

Variation of nondimensional fundamental frequency with material anisotropy for (75/−75)10 lamination

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0022

Figure 11
Figure 11

Variation of nondimensional fundamental frequency with material anisotropy for (90/−90)10 lamination

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0022

These results are obtained by keeping the material properties as constant, that is, G12/E22 = 0.5 and ν12 = 0.25, and changing the E11/E22 ratio. As observed from these figures, as the degree of orthotropy increases, the first, second, third, fourth, and fifth frequency parameter increases monotonically for all the laminations and boundary conditions considered here. These increments are sharper for CSSC shell. FCFC shell shows better performance for lower lamination angle, but with the increase in the lamination angle, the performance of CSCS shell is better. For other boundary conditions, mild increase in the frequency parameter is observed. It is also observed that for almost all the cases, the frequency parameter increases from the first mode to the fifth mode. In a few cases, the frequency remains almost same between two consecutive modes.

4.3 Effect of width-to-thickness ratio

If the width-to-thickness ratio is increased while maintaining the width of the laminate a constant and the number of layers being fixed at 10, the thickness of the shell is decreased. Figures 1218 show the variation of the nondimensional frequency for the first, second, third, fourth and fifth mode with variation of width-to-thickness ratio and boundary conditions for various values of lamination angles. Ten-layer antisymmetric angle-ply laminates with varying fiber orientation angle, such as 0°, 15°, 30°, 45°, 60°, 75°, and 90°, having different width-to-thickness ratios (b/h = 10, 20, 50, 100) are analyzed. It is evident from Figures 1218 that with the increase in width-to-thickness ratio, the dimensionless frequencies decrease. This decrease in frequency is very much significant in case of CSSC, CSCS, and FCFC shells. For other boundary conditions, this decrease in dimensionless frequency is significant at higher value of width-to-thickness ratio. These are true for all the five modes. Here also for all the combination of stacking sequences, boundary conditions, and width-to-thickness ratios, the nondimensional frequency increases from the first mode to the fifth mode except few cases.

Figure 12
Figure 12

Variation of nondimensional fundamental frequency with b/h ratio for (0/−0)10 lamination

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0022

Figure 13
Figure 13

Variation of non-dimensional fundamental frequency with b/h ratio for (15/−15)10 lamination

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0022

Figure 14
Figure 14

Variation of nondimensional fundamental frequency with b/h ratio for (30/−30)10 lamination

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0022

Figure 15
Figure 15

Variation of nondimensional fundamental frequency with b/h ratio for (45/−45)10 lamination

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0022

Figure 16
Figure 16

Variation of nondimensional fundamental frequency with b/h ratio for (60/−60)10 lamination

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0022

Figure 17
Figure 17

Variation of nondimensional fundamental frequency with b/h ratio for (75/−75)10 lamination

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0022

Figure 18
Figure 18

Variation of nondimensional fundamental frequency with b/h ratio for (90/−90)10 lamination

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0022

5 Conclusions

In this article, analysis of antisymmetric angle-ply laminated composite stiffened hypar shell with cutout is presented for different modes of vibration using finite element method based on the first-order shear deformation theory. It can be concluded by analyzing the results that as the number of layers increase, the fundamental frequency increases. With the increase in lamination angle, nondimensional natural frequency may increase or decrease, but from the first mode to the fifth mode, natural frequency always increase or remain same in very few cases. The first, second, third, fourth, and fifth nondimensional frequency parameter increases monotonically for all the laminations and boundary conditions as the degree of orthotropy increases. With the increase in width-to-thickness ratios, dimensionless frequencies decrease from the first mode to the fifth mode. Free vibration behavior mainly depends on the number of boundary constraints whatever may be the other parametric variations such as change in fiber orientation angle and increase in the degree of orthotropy and width-to-thickness ratio.

Nomenclatureν12, ν21

Poisson’s ratios

ω

natural frequency

ω

nondimensional natural frequency = ωa2(ρ / E22 h2)1/2

ρ

density of material

E11, E22

elastic moduli

G12, G13, G23

shear moduli of a lamina with respect to 1, 2 and 3 axes of fiber

αsx, βsx

rotational degrees of freedom at each node of X-stifiener element

αsy, βsy

rotational degrees of freedom at each node of Y-stiffener element

δsxi, δsyi

nodal displacement of stiffener element

a′, b

length and width of cutout in plan

Bsx, Bsy

strain displacement matrix of stiffener element

Msxx, Msyy

moment resultants of stiffeners

Nsxx, Nsyy

axial force resultants of stiffeners

Qsxxz, Qsyyz

transverse shear resultants of stiffeners

Tsxx, Tsyy

torsion resultants of stiffeners

a, b

length and width of shell in plan

bsx, bsy

width of X and Y stiffeners, respectively

c

rise of hypar shell

dsx, dsy

depth of X and Y stiffeners, respectively

esx, esy

eccentricities of X and Y stiffeners with respect to mid surface of shell

e

eccentricity of stiffeners with respect to mid surface of shell

h

shell thickness

np

number of plies in a laminate

Rxy

radii of cross curvature of hypar shell

usx, wsx

axial and transverse translational degrees of freedom at each node of X-stiffener element

vsy, wsy

axial and transverse translational degrees of freedom at each node of Y-stiffener element

X, Y, Z

global coordinate axes

x, y, z

local coordinate axes

zk

distance of bottom of the k-th ply from mid surface of a laminate

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    Sahoo S.: Dynamic characters of stiffened composite conoidal shell roofs with cutouts: Design aids and selection guidelines J. Engg. ID 230120 1-18 2013.

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    Sahoo S.: Free vibration of laminated composite stiffened saddle shell roofs with cutouts IOSR J. Mech. Civil Engg. ICAET 30-34 2014.

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    Sahoo S.: Laminated composite stiffened shallow spherical panels with cutouts under free vibration – a finite element approach Engg. Sci. Tech. An Int. J.17(4) 247-259 2014.

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    Sahoo S.: Free vibration behavior of laminated composite stiffened elliptic paraboloidal shell panel with cutout Curved and Layered Struct.2(1) 162-182 2015.

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    Sahoo S.: Laminated composite stiffened cylindrical shell panels with cutouts under free vibration Int. J. Manuf. Mat. Mech.Engg.5(3) 37-63 2015.

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    Sahoo S.: Performance evaluation of free vibration of laminated composite stiffened hyperbolic paraboloid shell panel with cutout Int. J. Engg. Technol.71-24 2016.

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    Topal U.: Mode-frequency analysis of laminated spherical shell Proceedings of the 2006 IJME. INTERTECH International Conference-Session ENG P501-001 Kean University NJ; 19-21 October 2006.

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    Srinivasa C.V. Suresh Y.J. Prema Kumar W.P.: Experimental and finite element studies on free vibration of cylindrical skew panels Int. J. Advanced Struct. Engg. 6(1) 1-11 2014.

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    Mukherjee A. Mukhopadhyay M.: Finite element free vibration of eccentrically stiffened plates Comp. Struct. 30 1303-1317 1988.

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    Chakravorty D. Bandyopadhyay J.N. Sinha P.K.: Applications of FEM on free and forced vibrations of laminated shells ASCE J. Engg. Mech.124(1) 1–8 1998.

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    Rikards R. Chate A. Ozolinsh O.: Analysis for buckling and vibrations of composite stiffened shells and plates Composite Struct.51 361–370 2001.

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    Nayak A.N. Bandyopadhyay J.N.: Free vibration analysis and design aids of stiffened conoidal shells ASCE J. Engg. Mech.128(4) 419–427 2002.

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    Nayak A.N. Bandyopadhyay J.N.: On the free vibration of stiffened shallow shells J. Sound and Vibr.255(2) 357–382 2002.

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    Nayak A.N. Bandyopadhyay J.N.: Free vibration analysis of laminated stiffened shells. ASCE J. Engg. Mech. 131(1) 100–105 2005.

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    Nayak A.N. Bandyopadhyay J.N.: Dynamic response analysis of stiffened conoidal shells J. Sound and Vibr. 291 1288-1297 2006.

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    Sahoo S. Chakravorty D.: Finite element vibration characteristics of composite hypar shallow shells with various edge supports J. Vibr. Control 11 1291-1309 2005.

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    Sahoo S. Chakravorty D.: Stiffened composite hyparshell roofs under free vibration: behaviour and optimization aids. J. Sound and Vibr. 295 362-377 2006.

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    Pradyumna S. Bandyopadhyay J.N.: Free vibration analysis of functionally graded curved panels using a higher order finite element formulation J. Sound and Vibr. 318 176-192 2008.

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    Kumar A. Bhargava P. Chakrabarti A.: Natural Frequencies and mode shape of laminated composite skew hypar shells with complicated boundary conditions using finite element method Adv. Mat. Res. 585 44-48 2012.

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    Kumar A. Bhargava P. Chakrabarti A.: Vibration of laminated composite skew hypar shells using higher order theory Thin Walled Structures63 82-90 2013.

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    Kumar A. Chakrabarti A. Bhargava P.: Vibration analysis of laminated composite skew cylindrical shells using higher order shear deformation theory J. Vib. Control21(4) 725-735 2015.

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    GulshanTaj M.N.A. Chakraborty A.: Dynamic response of functionally graded skew shell panel Latin Amer. J. Solids Struct. 10 1243-1266 2013.

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    Tornabene F. Fantuzzi N. Bacciocchi M. Dimitri R.: Dynamic analysis of thick and thin elliptic shell structures made of laminated composite materials Compos. Struct.133 278-299 2015.

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  • [17]

    Sahoo S.: Behaviour and optimization aids of composite stiffened hypar shell roofs with cutout under free vibration ISRN Civil Engg. ID 989785 1-14 2012.

  • [18]

    Sahoo S.: Dynamic characters of stiffened composite conoidal shell roofs with cutouts: Design aids and selection guidelines J. Engg. ID 230120 1-18 2013.

  • [19]

    Sahoo S.: Free vibration of laminated composite stiffened saddle shell roofs with cutouts IOSR J. Mech. Civil Engg. ICAET 30-34 2014.

  • [20]

    Sahoo S.: Laminated composite stiffened shallow spherical panels with cutouts under free vibration – a finite element approach Engg. Sci. Tech. An Int. J.17(4) 247-259 2014.

    • Crossref
    • Zitation exportieren
  • [21]

    Sahoo S.: Free vibration behavior of laminated composite stiffened elliptic paraboloidal shell panel with cutout Curved and Layered Struct.2(1) 162-182 2015.

  • [22]

    Sahoo S.: Laminated composite stiffened cylindrical shell panels with cutouts under free vibration Int. J. Manuf. Mat. Mech.Engg.5(3) 37-63 2015.

  • [23]

    Sahoo S.: Performance evaluation of free vibration of laminated composite stiffened hyperbolic paraboloid shell panel with cutout Int. J. Engg. Technol.71-24 2016.

  • [24]

    Topal U.: Mode-frequency analysis of laminated spherical shell Proceedings of the 2006 IJME. INTERTECH International Conference-Session ENG P501-001 Kean University NJ; 19-21 October 2006.

  • [25]

    Srinivasa C.V. Suresh Y.J. Prema Kumar W.P.: Experimental and finite element studies on free vibration of cylindrical skew panels Int. J. Advanced Struct. Engg. 6(1) 1-11 2014.

  • [26]

    Mukherjee A. Mukhopadhyay M.: Finite element free vibration of eccentrically stiffened plates Comp. Struct. 30 1303-1317 1988.

    • Crossref
    • Zitation exportieren
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Zeitschrifteninformation
Impact Factor


CiteScore 2018: 0.90

SCImago Journal Rank (SJR) 2018: 0.191
Source Normalized Impact per Paper (SNIP) 2018: 0.698

Abbildungen
  • View in gallery

    Surface of a skewed hypar shell with cutout

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    (a) Eight-noded shell element. (b) Three-noded stiffener element: (i)X-stiffener and (ii) Y-stiffener

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    Typical 10 × 10 nonuniformal mesh arrangement drawn to scale

  • View in gallery

    Variation of nondimensional fundamental frequency with the fiber orientation angle

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    Variation of nondimensional fundamental frequency with material anisotropy for (0/−0)10 lamination

  • View in gallery

    Variation of nondimensional fundamental frequency with material anisotropy for (15/−15)10 lamination

  • View in gallery

    Variation of nondimensional fundamental frequency with material anisotropy for (30/−30)10 lamination

  • View in gallery

    Variation of nondimensional fundamental frequency with material anisotropy for (45/−45)10 lamination

  • View in gallery

    Variation of non-dimensional fundamental frequency with material anisotropy for (60/−60)10 lamination

  • View in gallery

    Variation of nondimensional fundamental frequency with material anisotropy for (75/−75)10 lamination

  • View in gallery

    Variation of nondimensional fundamental frequency with material anisotropy for (90/−90)10 lamination

  • View in gallery

    Variation of nondimensional fundamental frequency with b/h ratio for (0/−0)10 lamination

  • View in gallery

    Variation of non-dimensional fundamental frequency with b/h ratio for (15/−15)10 lamination

  • View in gallery

    Variation of nondimensional fundamental frequency with b/h ratio for (30/−30)10 lamination

  • View in gallery

    Variation of nondimensional fundamental frequency with b/h ratio for (45/−45)10 lamination

  • View in gallery

    Variation of nondimensional fundamental frequency with b/h ratio for (60/−60)10 lamination

  • View in gallery

    Variation of nondimensional fundamental frequency with b/h ratio for (75/−75)10 lamination

  • View in gallery

    Variation of nondimensional fundamental frequency with b/h ratio for (90/−90)10 lamination

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