Document Type : Research Paper
Authors
^{1} Department of Civil Engineering, Sanjivani College of Engineering, Savitribai Phule Pune University, Kopargaon423603, Maharashtra, India
^{2} Department of Structural Engineering, Sanjivani College of Engineering, Savitribai Phule Pune University, Kopargaon423603, Maharashtra, India
Abstract
Keywords
Main Subjects
A New HigherOrder Shear and Normal Deformation Theory for the Free Vibration Analysis of Laminated Shells
^{a}Department of Civil Engineering, Sanjivani College of Engineering, Savitribai Phule Pune University,
Kopargaon423603, Maharashtra, India
^{b}Department of Structural Engineering, Sanjivani College of Engineering, Savitribai Phule Pune University,
Kopargaon423603, Maharashtra, India
KEYWORDS 

ABSTRACT 
Fifthorder theory; Shear deformation; Normal deformation; Free vibration; Laminated shells. 
In this paper, the free vibration analysis of laminated composite and sandwich, cylindrical and spherical shells is presented using a new higherorder shear and normal deformation theory. The novelty of the present theory is that it includes the effects of both transverse shear and normal deformations along with higher order expansions of displacement field. A fifthorder polynomial type shape function is used in the inplane displacements to represent the effect of transverse shear deformation for the first time whereas transverse displacement is a function of x, y, and z coordinates to account for the effect of transverse normal deformation. The equations of motion are derived using Hamilton’s principle. Navier’s solution technique is employed to obtain the nondimensional fundamental frequencies. To validate the accuracy of the present theory, the present results are compared with other higherorder theories available in the literature. It is observed that the values of fundamental frequencies obtained using the present theory are in close agreement with those available in the literature. 
Laminated composite and sandwich shells are having a wide application in the area of aircraft, spacecraft, undermining, marine constructions, etc. due to their attractive features such as high load carrying capacity, large spantodepth ratio, high strengthtoweight ratio, high stiffnesstoweight ratio, etc.
In the case of mechanical and structural industries, the structural components get subjected to extreme loads and deformations due to vibration and resonance, which leads to catastrophic failure. Also, in the case of aircraft, to avoid the severe consequences during inflight conditions the wings need to be designed to eliminate the resonance, in the case of civil structures it needs to be designed considering the windinduced vibration.
Therefore, static and vibration analysis of laminated composite shells becomes an active area of research among researchers. 169 years ago Kirchoff [1] has developed a classical shell theory (CST) for the analysis of thin shells which neglect the effect of shear deformation. However, this theory is not useful for the analysis of thick shells. Therefore, Mindlin [2] has developed a firstorder shear deformation theory (FSDT) which considers the effect of shear deformation for the first time. However, this theory fails to satisfy the zero transverse shear stress condition at top and bottom surfaces of the shell. These drawbacks of the CST and FSDT lead to the development of higherorder shear deformation theories (HSDT). Qatu [35], Qatu et al. [6], Mallikarjuna and Kant [7], Thai and Kim [8], Sayyad and Ghugal [911] have published several review articles on free vibration analysis of laminated composite beams, plates, and shells.
Reddy [12] has developed a wellknown parabolic shear deformation theory (PSDT) for the analysis of laminated composite plates and shells. Bhimaraddi [13] has presented a free vibration analysis of doubly curved shells using three dimensional elasticity theory assuming that the ratio of shell thickness to its middle surface radius is negligible. Timaraci and Soldatos [14] studied the dynamic behavior of symmetric crossply circular shells using various shear deformation theories. Khare et al. [15] have developed a finite element formulation using a higherorder facet shell element for the free vibration analysis of laminated composite and sandwich cylindrical and spherical shells. A layerwise shear deformation theory is developed by Ferreira [16] for the static analysis of laminated composite plates. Static and free vibration analysis of laminated composite shells is presented by Ferreira et al. [17] by developing a meshless solution of Reddy’s higherorder shell theory. Garg et al. [18] presented a closedform solution for the free vibration analysis of doubly curved laminated composite and sandwich shells. Pradyumna and Bandopadhyay [19] investigated a C0 finite element formulation based on higherorder shear deformation theory for the static and free vibration analysis of laminated composite shells. Matsunga [20] presented free vibration and stability analysis of crossply laminated composite shells, considering the effect of transverse shear and normal deformations. Carrera and Brischetto [21] have presented an analysis of laminated composite shells using refined and mixed shear deformation theories. Brischetto et al. [22] studied a free vibration analysis of sandwich plates and shells by introducing a zigzag function in the displacement field of classical and higherorder theories. Zhao et al. [23] have applied the Ritz method for the static and dynamic analysis of functionally graded cylindrical shells. Noh and Lee [24] have presented the free vibration analysis of laminated composite shells by developing a finite element formulation based on a thirdorder shear deformation theory. Bending and free vibration analysis of laminated composite plates and shells is presented by Mantari and Soares [25] using higherorder shear deformation theory. Tornabene [26, 27], Tornabene et al. [28, 29] have proposed a GDQ method for the free vibration analysis of laminated composite and functionally graded shells. Qatu and Asadi [30] presented a comprehensive study on free vibration analysis of spherical, cylindrical, and hyperbolic paraboloidal shells using a Ritz method for various boundary conditions. Asadi et al. [31] have presented a 3D solution for static and vibration analysis of thick deep laminated cylindrical shells. The hierarchical trigonometric Ritz formulation is used by Fazzolari and Carrera [32] for the free vibration analysis of doubly curved laminated composite shells. Taj and Chakrabarti [33] have studied the bending analysis of functionally graded ceramicmetal skew shell panels using a C0 finite element. Dai et al. [34] have obtained an exact series solution for the free vibration analysis of cylindrical shells for various boundary conditions. Wang et al. [35] have predicted the free vibration response of laminated composite circular panels and shells of revolution using a FourierRitz method for various boundary conditions. Rawat et al. [36] have developed a finite element model for the free vibration analysis of thin circular cylindrical shells. Pandey and Pradyumna [37] have presented a thermally induced vibration analysis of functionally graded sandwich plates and shell panels. Biswal et al. [38] studied free vibration and stability analysis of doubly curved laminated shell panels based on Sander’s approximation. Fares et al. [39] have presented the bending and free vibration analysis of functionally graded doubly curved shells using a layerwise theory. Monge et al. [40] have carried out an asymptotic evaluation of the best theories for the free vibration analysis of laminated composite and sandwich shells. Cong et al. [41] have extended a third order shear deformation theory of Reddy by developing a new approach to investigate the nonlinear dynamic response of doubly curved sandwich shells. A semianalytical method is used by Li et al. [42] and Pang et al. [43] to study the free vibration analysis for laminated composite doubly curved shells of revolution. Kiani et al. [44] investigated the free vibration characteristics of functionally graded carbon nanotube skew cylindrical shells based on ChebyshevRitz formulation. Sayyad and Ghugal [45] developed a generalized shell theory to investigate the static and dynamic response of laminated composite and sandwich spherical shells. Draiche et al. [46], Allam et al. [47], Zine et al. [48] presented a higher order shear deformation theory for the bending and free vibration analysis of laminated composite, sandwich plates, and shells. Arefi [4953], Arefi and Rabczuk [54], Arefi and Elyas [55], Arefi and Amabili [56] have presented an electroelastic and free vibration analysis of piezoelectric doubly curved nanoshells. Arefi and Zenkour [5758] have presented the bending and free vibration analysis of functionally graded nanobeams whereas Arefi et al. [59] presented the bending response of FG composite doubly curved nanoshells considering the thickness stretching effects.
Draiche et al. [60] presented a static analysis of laminated reinforced plates using first order shear deformation theory. Belbachir et al. [6162], Abualnour et al. [63], Sahla et al. [64] have applied a four variable refined theory for the static and free vibration analysis of laminated composite plates and shells under mechanical and thermal load.
Carrera [65] reported in his research that for the accurate structural analysis of composite laminates, higherorder theories must be expanded up to minimum fifthorder polynomial. It is also recommended by Carrera that it is important to consider the effect of transverse normal deformation for the analysis of composite laminates. However, limited literature is available on refined theories considering the effects of transverse normal strain. Also, refined theories representing higher order (minimum fifthorder) expansion of displacement field are limited. Based on these observations Sayyad and his coauthors Sayyad and Naik, [66], Naik and Sayyad, [6769], Ghumare and Sayyad [7072], Shinde and Sayyad [73] have given due consideration to these recommendations of Carrera and developed a fifthorder shear and normal deformation theory for the analysis of laminated composite, sandwich, and functionally graded beams, plates, and shells. In the present work, this theory is extended for the free vibration analysis of laminated shells. The theory considers the effects of transverse shear and normal deformations which is neglected by classical theories as well as many other higherorder shell theories including Reddy’s theory. A fifthorder polynomial type shear strain function is used in the inplane displacements to consider the effect of transverse shear deformation, whereas a fourthorder function is considered in the transverse displacement to account for the effect of normal deformation i.e. transverse normal stress. The equations of motion are derived using Hamilton’s principle. Navier’s solution technique is employed to obtain the nondimensional fundamental frequencies. The numerical results obtained for the natural frequencies using the present theory are compared with other higherorder theories presented by Asadi et al. [31], Bhimaraddi [13], Sayyad, and Ghugal [45], etc. In overall, the numerical results predicted by the present theory are in excellent agreement with the 3D elasticity solution.
A simply supported laminated composite shell on rectangular planform with a width a in the xdirection, breadth b in the ydirection, and thickness h in the zdirection as shown in Fig. 1 is considered. The shell has N number of orthotropic layers made up of fibrous composite materials. For spherical shell R_{1}=R_{2}=R and for cylindrical shell R_{1}=R, R_{2}=∞.
Fig. 1. Geometry and coordinate system of a spherical shell.
The present theory is built upon classical shell theory. Extension and bending components associated with the present theory are analogous to the classical shell theory. Fifthorder polynomial type shape functions are used in the inplane displacements to account for the effect of transverse shear deformation. Whereas, fourthorder shape function in terms of thickness coordinates is used in the transverse displacement to account for the effect of transverse normal deformations. The theory is presented by Sayyad and Naik, [65], Naik and Sayyad, [6769], Ghumare and Sayyad [7072] and, Shinde and Sayyad [73]. The displacement field of the present theory is as follows.

(1) 
where

(2) 
where u, v, and w are the displacements in the x, y and z directions, respectively; u_{0}, v_{0}, and w_{0} are the unknown displacements of the midplane of the shell in x, y and z directions respectively; are the shear slopes about the yaxis, ^{ }are the shear slopes about the xaxis, are the shear slopes about the zaxis. All these shear slopes are unknowns that need to be determined. (‘) represents the derivative of the function with respect to the z coordinate. The present theory has nine unknowns. The following general straindisplacement relationships are used to determine nonzero strain components, Bhimaraddi [13].

(3) 
Substituting displacement expressions from the displacement field, stated in Eq. (1) one can obtain the following expressions for nonzero strains.

(4) 
The threedimensional Hooke’s law is used to derive expressions for stresses in the shell domain.

(5) 
where,

(6) 
where

(7) 
E_{i}’s (i=1, 2, 3) are the modulus of elasticity; G_{ij}’s are the modulus of rigidity, and ’s are the Poisson’s ratio. The equations of motion for the free vibration analysis of the shell are obtained using Hamilton’s principle.

(8) 
where, dK represents the kinetic energy due to inertia forces, dU represents the strain energy due to stresses, and dV represents the potential energy due to external load. d is the variational operator. t_{1} and t_{2} are the initial and final times respectively. Substituting values of these energies in Eq. (8), one can rewrite the Eq. (8) as

(9) 
where r represents the mass density. Substituting the expressions of stresses and strains from Eqs. (3)(7) into the Eq. (9), integrating by parts, collecting the coefficients of and setting them equal to zero, one can derive the following equations of motion.

(10) 
Boundary conditions of the present theory are expressed in the following form
Along the edges x =0 and x =a
Either or is prescribed
Either or is prescribed
Either or is prescribed
Either or is prescribed
Either or is prescribed
Either or is prescribed
Either or is prescribed
Either or is prescribed
Either or is prescribed
Either or is prescribed
Along with the edges y =0 and y =b,
Either or is prescribed
Either or is prescribed
Either or is prescribed
Either or is prescribed
Either or is prescribed
Either or is prescribed
Either or is prescribed
Either or is prescribed
Either or is prescribed
Either or is prescribed
where force and moment resultants can be derived from the following relations

(11) 
The inertia constants are derived as,

(12) 
The Navier’s solution technique is used to obtain the analytical solutions for the free vibration analysis of simply supported crossply laminated composite shells. The boundary conditions at the simply supported edges of the shell are as follows.
Along the edges x =0 and x =a

(13) 

Along the edges y =0 and y =b

(14) 

The following trigonometric form of unknown variables is assumed which satisfies the simply supported boundary conditions exactly.

(15) 
where ; ; is the fundamental frequency and are the unknown parameters to be determined. Substitution of Eq. (15) into Eq. (10) by setting q = 0 leads to the following eigenvalue problem.

(16) 
where [K] is stiffness matrix, [M] is the mass matrix, and is the vector of amplitudes. Elements of these matrices are mentioned in Appendix.
In the present study, natural frequencies for a homogeneous twolayer (0^{0}/90^{0}) and threelayer (0^{0}/90^{0}/0^{0}) laminated composite cylindrical and spherical shells are obtained using MATLAB 2015a. In laminated shells, all layers are considered of equal thickness. The numerical results are obtained for different values of the R/a ratio and aspect (a/h) ratio. The following material properties for the laminated composite shells are used, Bhimaraddi [13]

(17) 

The natural frequency is presented in the following nondimensional form unless and until specified,

(18) 
The comparison of the first five natural frequencies of twolayer (0^{0}/90^{0}) and threelayer (0^{0}/90^{0}/0^{0}) laminated composite cylindrical shells is shown in Tables 12 respectively. Natural frequencies are presented for a/h=10 and 20 at R/a=2, 1, 0.5. Material properties of laminated shells are mentioned in Eq. (17). The comparison shows that the present results are in good agreement with those results presented by Asadi et al. [31] by different models. It is observed that as the a/h ratio and R/a ratio increases the natural frequency increases. It is found that the natural frequencies are decreasing with respect to the increase in radius of curvature (R/a).
Also, from Tables 1 and 2 it is clearly observed that the natural frequency is less in deep shells and more in shallow shells. The variation of fundamental frequency with respect to the aspect ratio (a/h) is also shown in Tables 1 and 2, which predicts that the frequency is found more in thick shells.
Table 3 shows a comparison of natural frequencies for varying modes of vibration of a twolayer 0^{0}/90^{0} laminated cylindrical shells at a/h=10 and R/a=1. The numerical results are obtained for m=n=1, 2, 3, 4, 5, 6 to change modes of vibration. The comparison shows that the present theory predicts natural frequencies in good agreement with those presented by Bhimaraddi [13] using various theories.
Table 4 shows the effect of R/a and h/a ratios on the natural frequencies of orthotropic and laminated composite cylindrical shells. Variations of natural frequencies in laminated composite cylindrical shells are presented with the help of Fig. 2.
Table 5 through 7 and Figures.3 through 6 deal with the natural frequencies of orthotropic and laminated composite spherical shells using the present theory. The present results are obtained for different modes of vibrations, a/h ratios, and R/a ratios. The comparison of all tables shows that the present results are compared with those presented by Bhimaraddi [13] and, Sayyad and Ghugal [45] using various theories. The present results are found in good agreement with previously published results for parameters.
From Tables 5 through 7 it is seen that the fundamental frequency of laminated shells decreases with an increase in the radii of curvature (R/a ratio). These results are also plotted in Figures 3 through 6.
Figures 5 and 6 show the variation of fundamental frequency with respect to modes of vibration (m, n) in a single layer (0^{0}) and a doublelayer (0^{0}/90^{0}) laminated shell.
From Figures 5 and 6 it is observed that as the mode (m, n) increases the value of frequency also increases.
Table 8 shows the fundamental frequencies of three layered (0^{0}/core/0^{0}) sandwich spherical shells. The thickness of each face sheet is 0.1h and the middle core is 0.8h, where h is the total thickness of the shell. Following are the material properties used for the middle core of sandwich shell, Sayyad and Ghugal [46]:












The elastic properties of the face sheets are assumed as ‘C’ times the elastic properties of the core and the value of C are taken as 1, 2, 5, 10, and 15. From Table 8 it is observed that the result obtained by using the present theory are in good agreement with other theories available in the literature. The fundamental frequency of sandwich shells decreases with an increase in the radii of curvature (R/a).
Also, the results quoted in Table 8 show that the values of fundamental frequency increase with an increase in softness of the core.
Table 1. First five nondimensional natural frequencies in twolayer (0^{0}/90^{0}) laminated composite cylindrical shell
for varying a/R and a/h ratio (R_{1}=R and R_{2}=∞).
a/h 
R/a 
Theory 





20 
2.0 
Present 
11.579 
25.514 
28.015 
36.485 
50.747 
FSDTQ [31] 
11.530 
25.357 
27.913 
36.324 
50.210 

3DFEM [31] 
11.537 
25.378 
27.951 
36.434 
50.253 

1.0 
Present 
15.967 
26.053 
34.840 
38.115 
50.925 

FSDTQ [31] 
15.859 
25.648 
34.867 
37.831 
50.263 

3DFEM [31] 
15.861 
25.658 
34.890 
37.942 
50.297 

0.5 
Present 
24.855 
27.686 
43.444 
50.689 
51.091 

FSDTQ [31] 
24.809 
26.193 
42.664 
49.382 
51.170 

3DFEM [31] 
24.805 
26.162 
42.743 
49.359 
51.167 

10 
2.0 
Present 
9.5271 
21.995 
22.464 
30.415 
39.651 
FSDTQ [31] 
9.4577 
21.676 
22.150 
29.959 
38.608 

3DFEM [31] 
9.4855 
21.743 
22.246 
30.193 
38.745 

1.0 
Present 
10.859 
22.190 
24.319 
30.899 
39.820 

FSDTQ [31] 
10.666 
21.705 
24.090 
30.368 
38.722 

3DFEM [31] 
10.686 
21.767 
24.191 
30.614 
38.896 

0.5 
Present 
14.099 
22.496 
29.446 
32.266 
39.788 

FSDTQ [31] 
13.771 
21.037 
29.574 
31.200 
38.073 

3DFEM [31] 
13.772 
21.040 
29.639 
31.411 
38.266 
Table 2. First five nondimensional natural frequencies in threelayer (0^{0}/90^{0}/0^{0}) laminated composite cylindrical shell
for varying a/R and a/h ratio (R_{1}=R and R_{2}=∞).
a/h 
R/a 
Theory 





20 
2.0 
Present 
15.353 
24.850 
41.093 
42.808 
46.932 
FSDTQ [31] 
15.551 
21.646 
37.022 
46.309 
48.938 

3DFEM [31] 
15.245 
21.370 
36.803 
43.529 
46.148 

1.0 
Present 
18.596 
32.641 
42.761 
47.928 
50.801 

FSDTQ [31] 
18.710 
21.974 
36.794 
49.770 
49.852 

3DFEM [31] 
18.471 
21.703 
36.567 
47.074 
47.416 

0.5 
Present 
25.896 
42.591 
49.308 
51.416 
66.365 

FSDTQ [31] 
23.178 
25.978 
35.923 
52.746 
57.077 

3DFEM [31] 
22.924 
25.840 
35.668 
50.360 
56.448 

10 
2.0 
Present 
11.982 
19.324 
28.921 
31.916 
32.717 
FSDTQ [31] 
12.443 
18.677 
30.839 
31.323 
34.456 

3DFEM [31] 
11.769 
18.159 
28.600 
30.471 
31.928 

1.0 
Present 
12.854 
21.665 
28.786 
32.931 
34.688 

FSDTQ [31] 
13.187 
18.524 
30.564 
32.232 
34.523 

3DFEM [31] 
12.590 
18.005 
29.732 
30.189 
32.037 

0.5 
Present 
15.134 
27.579 
28.281 
33.701 
42.195 

FSDTQ [31] 
15.250 
17.989 
29.491 
34.795 
34.913 

3DFEM [31] 
14.840 
17.468 
29.094 
32.464 
33.046 
Table 3. Nondimensional natural frequencies in twolayer (0^{0}/90^{0}) laminated composite cylindrical shell
for varying modes of vibration (R_{1}=R, R_{2}=∞, R/a=1 and a/h=10).
n 
Source 
m =1 
m =2 
m =3 
m =4 
m =5 
m =6 
1 
Present 
1.0859 
2.2190 
3.9820 
5.8774 
7.7433 
9.4882 

3DElasticity [13] 
1.0408 
2.4127 
4.1157 
5.9337 
7.7818 
9.6281 
2 
Present 
2.4319 
3.0899 
4.554 
6.3011 
8.0866 
9.7855 

3DElasticity [13] 
2.0956 
3.0069 
4.4760 
6.1778 
7.9611 
9.7672 
3 
Present 
4.1705 
4.5649 
5.6724 
7.1589 
8.7769 
10.367 

3DElasticity [13] 
3.7949 
4.4010 
5.5338 
6.9958 
8.6193 
10.316 
4 
Present 
6.0501 
6.3079 
7.1502 
8.3748 
9.7868 
11.225 

3DElasticity [13] 
5.6331 
6.0816 
6.9643 
8.1881 
9.6232 
11.177 
5 
Present 
7.9595 
8.1370 
8.8001 
9.8109 
11.025 
12.300 

3DElasticity [13] 
7.4876 
7.8550 
8.5704 
9.6035 
10.864 
12.274 
6 
Present 
9.8524 
9.9762 
10.514 
11.360 
12.402 
13.520 

3DElasticity [13] 
8.6842 
9.4979 
10.254 
11.143 
12.258 
13.536 
Table 4. Nondimensional natural frequencies in laminated composite cylindrical shell (R_{1}=R, R_{2}=∞)
for different R/a and a/h ratios.
R/a 
Source 
Orthotropic 

0^{0}/90^{0} 

h/a=0.05 
h/a =0.1 
h/a=0.15 

h/a =0.05 
h/a =0.1 
h/a =0.15 

1 
Present 
0.8727 
1.2919 
1.5740 

0.7983 
1.0859 
1.3714 

3DElasticity [13] 
0.8917 
1.3241 
1.6169 

0.7868 
1.0408 
1.2909 
2 
Present 
0.7602 
1.2495 
1.5609 

0.5789 
0.9527 
1.2815 

3DElasticity [13] 
0.7663 
1.2674 
1.5924 

0.5725 
0.9362 
1.2537 
3 
Present 
0.7354 
1.2407 
1.5582 

0.5243 
0.9231 
1.2609 

3DElasticity [13] 
0.7396 
1.2562 
1.5878 

0.5207 
0.9144 
1.2450 
4 
Present 
0.7263 
1.2376 
1.5573 

0.5034 
0.9120 
1.2529 

3DElasticity [13] 
0.7304 
1.2522 
1.5452 

0.5011 
0.9061 
1.2409 
5 
Present 
0.7220 
1.2361 
1.5568 

0.4933 
0.9067 
1.2487 

3DElasticity [13] 
0.7255 
1.2503 
1.5842 

0.4916 
0.9020 
1.2384 
10 
Present 
0.7163 
1.2342 
1.5563 

0.4793 
0.8989 
1.2423 

3DElasticity [13] 
0.7194 
1.2473 
1.5825 

0.4785 
0.8956 
1.2337 
20 
Present 
0.7149 
1.2337 
1.5562 

0.4757 
0.8965 
1.2399 

3DElasticity [13] 
0.7179 
1.2463 
1.5821 

0.4750 
0.8934 
1.2314 
∞ 
Present 
0.7144 
1.2336 
1.5562 

0.4743 
0.8952 
1.2381 

3DElasticity [13] 
0.7173 
1.2461 
1.5812 

0.4736 
0.8917 
1.2290 
Table 5. Nondimensional natural frequencies in twolayer (0^{0}/90^{0}) laminated composite spherical shell (R_{1}= R_{2}=R)
for varying modes of vibration (R/a=1 and a/h=10).
n 
Source 
m=1 
m =2 
m =3 
m =4 
m =5 
m =6 
1 
Present 
1.4818 
2.6042 
4.3235 
6.2130 
8.0999 
9.8893 

3DElasticity [13] 
1.3997 
2.4387 
4.0531 
5.8455 
7.6895 
8.7973 
2 
Present 
2.5333 
3.2067 
4.6702 
6.4332 
8.2533 
10.006 

3DElasticity [13] 
2.4420 
3.0452 
4.4168 
5.7938 
8.0320 
9.6841 
3 
Present 
4.1621 
4.5650 
5.6853 
7.2038 
8.8718 
10.534 

3DElasticity [13] 
4.0841 
4.4327 
5.4741 
6.9051 
8.5234 
10.226 
4 
Present 
5.9659 
6.2301 
7.0901 
8.3601 
9.8434 
11.383 

3DElasticity [13] 
5.7102 
6.1128 
6.9091 
8.0983 
9.5244 
11.083 
5 
Present 
7.7957 
7.9834 
8.6702 
9.7402 
11.051 
12.465 

3DElasticity [13] 
7.4002 
7.8904 
8.5227 
9.5188 
10.766 
12.177 
6 
Present 
9.5939 
9.7358 
10.306 
11.226 
12.393 
13.692 

3DElasticity [13] 
9.0324 
9.6948 
10.214 
11.065 
12.164 
13.440 
Table 6. Nondimensional natural frequencies in twolayer (0^{0}/90^{0}) laminated composite spherical shell (R_{1}= R_{2}=R)
for different R/a and a/h ratios.
R/a 
Theory 
a/h 

5 
10 
20 
50 
100 

5 
Present 
7.6370 
9.3431 
10.932 
16.7371 
29.0279 

PSDT [45] 
7.6781 
9.3424 
10.923 
16.7059 
29.0271 

ESDT [45] 
7.7826 
9.3759 
10.931 
16.7068 
29.0272 
10 
Present 
7.5733 
9.0750 
9.8931 
11.8618 
16.8058 

PSDT [45] 
7.6122 
9.0738 
9.8893 
11.8560 
16.8218 

FSDT [45] 
7.6482 
9.0991 
9.8978 
11.8575 
16.8222 
50 
Present 
7.5527 
8.9870 
9.5339 
9.7865 
10.0960 

PSDT [45] 
7.5908 
8.9856 
9.5323 
9.7943 
10.1312 

ESDT [45] 
7.6974 
9.0208 
9.5420 
9.7959 
10.1316 
100 
Present 
7.5520 
8.9842 
9.5225 
9.7144 
9.8121 

PSDT [45] 
7.5902 
8.9828 
9.5209 
9.7227 
9.8487 

ESDT [45] 
7.6967 
9.0180 
9.5307 
9.7243 
9.8491 
Plate 
Present 
7.5753 
9.0123 
9.5498 
9.7220 
9.7476 

PSDT [45] 
7.5899 
9.9819 
9.5171 
9.6988 
9.7527 

ESDT [45] 
7.6965 
9.0171 
9.5269 
9.7004 
9.7531 
Table 7. Nondimensional natural frequencies in threelayer (0^{0}/90^{0}/0^{0}) laminated composite spherical shell (R_{1}= R_{2}=R)
for different R/a and a/h ratios.
R/a 
Theory 
a/h 

5 
10 
20 
50 
100 

5 
Present 
8.3515 
12.0792 
15.1567 
20.4682 
31.4974 

PSDT [45] 
8.3200 
12.0613 
15.0499 
20.2525 
31.2192 

ESDT [45] 
8.3425 
12.0412 
15.0365 
20.2601 
31.2189 
10 
Present 
8.2908 
11.8770 
14.4306 
16.6907 
20.6521 

PSDT [45] 
8.2593 
11.8633 
14.3366 
16.5276 
20.4844 

ESDT [45] 
8.2820 
11.8428 
14.3225 
16.5247 
20.4837 
50 
Present 
8.2711 
11.8111 
14.1887 
15.2760 
15.6296 

PSDT [45] 
8.2396 
11.7988 
14.0991 
15.1334 
15.5166 

ESDT [45] 
8.2625 
11.7781 
14.0847 
15.1302 
15.5158 
100 
Present 
8.2705 
11.8090 
14.1811 
15.2296 
15.4460 

PSDT [45] 
8.2390 
11.7968 
14.0916 
15.0876 
15.3352 

ESDT [45] 
8.2619 
11.7760 
14.0772 
15.0845 
15.3343 
Plate 
Present 
8.2878 
11.8281 
14.1997 
15.2359 
15.4063 

PSDT [45] 
8.2388 
11.7961 
14.0891 
15.0724 
15.2742 

ESDT [45] 
8.2617 
11.7754 
14.0747 
15.0692 
15.2734 
Table 8. Nondimensional natural frequencies in threelayer (0^{0}/core/0^{0}) sandwich laminated composite spherical shell (R_{1}= R_{2}=R)
for different R/a ratios.
R/a 
Model 
C 

1 
2 
5 
10 
15 

5 
Present 
5.0011 
5.9644 
8.0349 
10.2930 
11.868 

PSDT [45] 
5.0209 
5.9690 
8.0090 
10.023 
10.249 

ESDT [45] 
5.0205 
5.9683 
8.0076 
10.0216 
10.246 
10 
Present 
5.0480 
5.9841 
8.0227 
10.041 
10.269 

PSDT [45] 
4.8082 
5.7718 
7.8223 
10.035 
11.563 

ESDT [45] 
4.8274 
5.6883 
7.6248 
9.9028 
11.332 
20 
Present 
4.8280 
5.6882 
7.6236 
9.9008 
11.329 

PSDT [45] 
4.8556 
5.7042 
7.6392 
9.9209 
11.353 

ESDT [45] 
4.7585 
5.7225 
7.7681 
9.9693 
11.485 
50 
Present 
4.7771 
5.6150 
7.5237 
9.8689 
11.645 

PSDT [45] 
4.7783 
5.6156 
7.5241 
9.8692 
11.645 

ESDT [45] 
4.8061 
5.6318 
7.5399 
9.8895 
11.669 
100 
Present 
4.7635 
5.5953 
7.4971 
9.8623 
11.741 

PSDT [45] 
4.7630 
5.5944 
7.4956 
9.8600 
11.738 

ESDT [45] 
4.7642 
5.5951 
7.4959 
9.8602 
11.738 
Plate 
Present 
4.7425 
5.7066 
7.7507 
9.9482 
11.460 

PSDT [45] 
4.7615 
5.5923 
7.4931 
9.8610 
11.754 

ESDT [45] 
4.7610 
5.5915 
7.4916 
9.8587 
11.751 
Fig. 2. Variations of natural frequencies with respect to
R/a ratio in laminated composite cylindrical shell.
Fig. 3. Variations of natural frequencies with respect to
R/a ratio in laminated composite spherical shell
Fig. 4. Variations of natural frequencies with respect to
a/h ratio in laminated composite spherical shell.
Fig. 5. Variations of natural frequencies with respect to
modes of vibration in orthotropic spherical shell.
Fig. 6. Variations of natural frequencies with respect to
modes of vibration in two layered (0^{0}/90^{0}) laminated
composite spherical shell.
In the present study, a new fifthorder shear and normal deformation theory is developed and applied for the free vibration analysis of laminated composite and sandwich shells. The present theory includes the effects of both transverse shear and normal deformations. A polynomial type transverse shear strain shape function is used in the displacement field to account for these effects. The fundamental frequency analysis is performed for different types of shell problems, to prove the efficacy and validity of the present theory. Based on the numerical results and discussion, the following conclusions are drawn.
Based on the literature review, illustrated examples, numerical results, and discussion, the authors recommend the use of the present theory for many other problems of composite shells.
Appendix
The elements of stiffness matrix [K] in Eq. (16) are: [A.1]









The elements of mass matrix [M] in Eq. (16) are: [A.2]








where [A.3] is,















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