Free vibration analysis of circumferentially ring-stiffened fiber-metal laminated cylindrical shells: A semi-analytical and finite element study

Abstract

This study examines the free vibration behavior of isotropic and fiber-metal laminated cylindrical shells, with and without circumferential ring stiffeners, under clamped-clamped and clamped-free boundary conditions. A semi-analytical formulation based on Flügge thin-shell theory and the Rayleigh-Ritz method is developed and implemented using an in-house Python code to compute natural frequencies. Four FML configurations-aluminum-carbon/epoxy, aluminum-Kevlar-49/epoxy, aluminum-S-glass/epoxy, and aluminum-E-glass/epoxy-are considered to assess the influence of material hybridization. The formulation is validated through finite element modal analysis in ANSYS Workbench, with results showing close agreement and a maximum error below 7%. Carbon-based FML shells show the highest frequency enhancement due to their superior stiffness, while Kevlar and glass-based FMLs provide moderate improvement with favorable stiffness-to-weight characteristics. The inclusion of circumferential ring stiffeners further increases the fundamental natural frequencies, particularly in lower modes. The main contribution of this work is the development of a unified semi-analytical framework that accounts for asymmetric FML configurations, extension-bending coupling, and discrete stiffeners using a Dirac delta formulation within the Flügge-Rayleigh-Ritz approach.

Keywords

cylindrical shells free vibration fiber-metal laminated circumferential ring stiffeners flügge shell theory modal analysis and boundary conditions

Introduction

Cylindrical shell structures find extensive use in aerospace, marine, and mechanical engineering applications due to their high load-carrying efficiency, stiff structure geometry, and favorable weight-to-strength ratio. Such shell structures are frequently exposed to dynamic excitation sources, such as aerodynamic loading, acoustic pressures, machinery rotation, and fluid structure interactions. With these loading conditions, any resulting vibrations and resonances might cause fatigue damage and failure of the structures. It is therefore critical to predict accurately the natural frequencies and mode shapes of the shell structures.

Traditionally, analysis of vibrations of cylindrical shells was based on classical theories developed by Donnell, Love, Sanders, and Flügge. The theory of Flügge accounted for membrane-bending coupling caused by curvature and thus provides an improved model of moderately thin shells. The classical benchmarks for the problem were established by Leissa,¹ whereas the energy approach for shells was advanced by Soedel.²

The increased requirements for light-weighted structures led to the introduction of FRP composites as substitutes for metallic shells. Therefore, numerous studies have been conducted on the vibrations of composite cylindrical shells. Shi et al.³ have shown the effects of stacking sequence using high-order shell theory. Liu et al.⁴ pointed out the importance of material hybridization in improving vibration characteristics of multi-phase composite materials. Kim et al.⁵ have presented the significance of anisotropy and thickness change with the help of the Haar wavelet approach. Han and Li⁶ focused on the impact of boundary conditions, whereas Chen et al.⁷ have shown the applicability of the Ritz method in analyzing the vibrations of composite shells.

Fiber-metal laminates (FMLs), consisting of stacked metal and composite layers, possess increased fatigue, damage tolerance, and stiffness. The vibration behavior of stiffened FML shells was investigated experimentally and numerically by Nazari et al.⁸ Hemmatnezhad et al.⁹ verified the accuracy of analytical, numerical, and experimental methods in analyzing stiffened composite shells. However, most of the available literature involves symmetric laminates or equivalent material models and does not consider the influence of coupling in asymmetric laminates.

Geometric stiffening is a powerful technique to improve performance. Stiffening rings increase the bending stiffness and prevent the occurrence of global modes. Qing et al.¹⁰ indicated the considerable effect of stiffener spacing on the vibration characteristics. Frequency variation due to stiffener geometry was observed by Gan et al.,¹¹ whereas the effects of shell-stiffener interaction were discussed by Pan et al.¹² The importance of stiffener placement was emphasized by Jafari and Bagheri,¹³ and the effect of stiffeners in increasing natural frequencies was shown by Zhao and Li.¹⁴

Recent efforts have been directed toward increasing the accuracy of analysis and computation. Cammalleri and Castellano¹⁵ suggested a matrix approach to analyze shells with various boundary conditions. Finite elements for the analysis of aluminum shells were provided by Yasniy et al.¹⁶ Lopatin and Morozov¹⁷ highlighted the effect of boundary conditions in laminated shells. Tang et al.¹⁸ proposed the reverberation-ray matrix technique, and the high-precision spectral technique was introduced by Guo et al.¹⁹ Effective numerical techniques for laminated shells were presented by Tornabene et al.²⁰ Theoretical models were provided by Qatu²¹ and Reddy,²² whereas Noda and Yasuda²³ stressed accurate FE modeling.

As reported by Shahgholian-Ghahfarokhi et al.,²⁴ studies have been carried out recently regarding vibration analysis of composite sandwich cylindrical shells having grid core using experimental and numerical approaches. Studies done by Li et al.,²⁵ introduced an analytical method for stiffened cylindrical shells with arbitrary boundary conditions, providing accurate prediction of frequency parameters.^26–37

Despite all of these advancements, there exists a gap in the literature regarding the availability of a semi-analytical approach for investigating asymmetrical FMLs, extension-bending coupled effect, stiffeners and arbitrary boundary conditions simultaneously. The other area of interest is that the vibration behavior of cylindrical shells made of aluminum-Kevlar-49, aluminum-S-glass, and aluminum-E-glass FMLs has not yet been investigated comprehensively.

Moreover, most of the research conducted so far deals with isotropic thin shells or symmetric composite laminates but rarely do they consider the asymmetric fiber-metal laminates having extension-bending coupling property. Most of the analytical models utilize smeared stiffener approaches or ignore the effect of discrete circumferential stiffeners along with the anisotropic properties of laminates under different types of boundary conditions. The vibration characteristics of aluminum-S glass and aluminum-E glass FML cylindrical shells have rarely been studied before.

A semi-analytical Flugge-Rayleigh-Ritz method is proposed in this paper to analyze the vibration characteristics of isotropic and asymmetric FML cylindrical shells. FMLs with aluminum-carbon, aluminum-Kevlar-49, aluminum-S glass, and aluminum-E glass composites were considered in this study. Circumferential ring stiffeners are incorporated into the model using the energy method, and the boundary conditions of clamped-clamped and clamped-free were also considered. This method was compared with finite element mode analysis carried out in ANSYS Workbench.

The originality of this research is in the approach in which the asymmetric fiber metal laminate (FML) cylindrical shells accounting for the coupling of extensional and bending effects, the presence of discrete circumferential stiffeners in terms of the Dirac delta function, and different types of boundary conditions have been analyzed within one semi-analytical formulation. The other aspect which distinguishes this study from others is the application of S-glass and E-glass FMLs.

Materials and shell configurations

Material systems

Five material combinations are considered in order to analyze the effect of hybridization of materials, modulus, and density on free vibration response of cylindrical shells. These are (i) isotropic Aluminum alloy, (ii) aluminum-carbon/epoxy FML, (iii) aluminum-Kevlar-49/epoxy FML, (iv) aluminum-S-glass/epoxy FML, and (v) aluminum-E-glass/epoxy FML. Such a choice would allow direct comparison of metallic and hybrid composite shells with similar geometries.

Material combination of aluminum alloy 6061-T6 is used for the isotropic and metallic layer in all FMLs. Aluminum alloy is assumed to be a homogeneous, isotropic, and linearly elastic material.

The laminae made up of carbon/epoxy material are used for stiffeners due to their high longitudinal modulus, which considerably increases stiffness in axial and circumferential directions. On the other hand, the Kevlar-49/epoxy material can be termed as a lightweight material as compared to carbon fiber due to its increased toughness but lower stiffness.

In order to compare the materials, the materials S-glass/epoxy and E-glass/epoxy have also been included in the analysis. S-glass shows higher stiffness as compared to regular E-glass while providing medium stiffness at lower costs. These materials allow us to study the stiffness/weight ratio of different materials for FML cylindrical shell structures.

All composite laminates have been considered to be orthotropic and linearly elastic materials. Perfect adhesion between metallic and composite layers has been assumed. Properties of all materials used in semianalytical as well as finite element analysis are presented in Table 1.

Table 1.

Material properties used for isotropic and fiber-metal laminated cylindrical shells.

Materials	Aluminum	Carbon-epoxy	Kevlar-49 epoxy	S2-glass	E-Glass
Longitudinal elastic modulus, E₁ (GPa)	68.5	171	78	58	37.65
Transverse elastic modulus, E₂ (GPa)		8.97	5.2	15.5	8.86
Thickness elastic modulus, E₃ (GPa)		8.97	5.2	15.5	8.86
In-plane Poisson’s ratio (ν₁₂, ν₁₃)	0.3	0.32	0.35	0.25	0.28
Thickness Poisson’s ratio (ν₂₃)	0.3	0.45	0.43	0.42	0.34
In-plane shear modulus, G₁₂, G₁₃(GPa)	26	5.17	2.2	6.9	3.22
Thickness shear modulus, G₂₃(GPa)	26	3.64	1.7	5.7	3.30
Density ρ (Kg/m³)	2700	1580	1410	1980	1840

Shell geometry and thin-shell validity

All cylindrical shell geometries share the same geometric dimensions to examine only the effect of material and circumferential reinforcement. The shell geometry is characterized by an axial length L = 350 and a mean radius R = 100 mm. The thickness is h = 3.0 mm for the isotropic aluminum shell and h = 4.0 mm for all FML shells.

The geometric aspect ratios are L/R = 3.5 and h/R ≈ 0.03-0.04, thus fulfilling the classical thin-shell condition (h/R <0.05). Thus, the Flügge thin-shell theory is suitable for the present shell geometries. The Flügge theory is applicable for thin shells in which transverse shear deformations and thickness extension are small. For thick shells (h/R >0.05), higher-order shear deformation theories might be required. However, since all the shell geometries under investigation satisfy the thin-shell condition, such effects are not taken into consideration.

Fiber-metal laminate configuration

The FML of cylindrical shells consists of alternate layers of aluminum plates and unidirectional composites reinforced by carbon, Kevlar-49, S-glass, or E-glass fibres. In all cases, the stacking pattern of the layers is [Al/0°/90°/Al/0°/90°/Al], which is an asymmetric seven-layer pattern. The thickness of each aluminium layer is 0.8 mm, whereas that of each composite layer is 0.4 mm, thus giving a total laminate thickness of 4 mm.

An asymmetric laminate stacking sequence is chosen for developing extension-bending coupling. Such a type of coupling is commonly found in real structures. Extension-bending coupling is considered by the inclusion of the coupling matrix B into the Classical Lamination Theory formulation. Matrices A, B, and D are generated and then used in the semi-analytical and FEA formulations.

Circumferential ring stiffeners

Circumferential ring stiffeners are included to examine the effect of geometric stiffening on the vibration behavior of cylindrical shells. Ring stiffeners are usually used in cylindrical shells in order to increase the bending stiffness of the structure and reduce the global modes.

Six ring stiffeners are equally spaced at an interval of 50 mm along the length of the shell. These stiffeners have a square section with sides measuring 5 × 5 mm. They are made up of aluminum alloy 6061-T6. These stiffeners are placed away from the edges in order to avoid edge effects.

Stiffeners are represented using beam elements that are joined rigidly to the cylindrical shell in the finite element model. Stiffeners contribute stiffness and inertia terms to the governing equations of motion for cylindrical shells through a discrete energy technique in the semi-analytical model.

Structural configurations considered

Based on the defined materials and geometry, 10 cylindrical shell configurations are analyzed: (i) unstiffened and stiffened isotropic aluminum shell, (ii) unstiffened and stiffened aluminum-carbon/epoxy FML shells, (iii) unstiffened and stiffened aluminum-Kevlar-49/epoxy FML shells, (iv) unstiffened and stiffened aluminum-S-glass/epoxy FML shells, and (v) unstiffened and stiffened aluminum-E-glass/epoxy FML shells. All configurations share identical geometric parameters and boundary conditions, enabling direct comparison of material and stiffening effects on vibration characteristics.

Analytical formulation

Assumptions and shell kinematics

Free vibration of cylindrical shells is modeled using Flugge’s theory of thin shell, accounting for coupling between membrane action and bending caused by curvature and is suitable for thin shells. The model follows along the lines of classical formulations presented by Leissa¹ and Soedel.² Displacements are taken from Love’s first approximation, considering only axial displacement and neglecting shear deformation and rotary inertia effects.

In the case of laminated shells, anisotropic and coupled behavior due to extension and bending are modeled using Classical Lamination Theory (CLT). The stiffness is accounted for following the works of Qatu²¹ and Reddy.²²

Since all cases meet the condition of h/R <0. 05, transverse shear deformations can be ignored. Therefore, Flügge’s approach gives a good compromise between computational efficiency and accuracy. More advanced models might be necessary for relatively thick shells. The displacement functions on the middle surface are given by u(x,θ,t), v(x,θ,t), and w(x,θ,t).

Governing equations of flügge thin-shell theory

The strain-displacement equations are defined by

ε_{x} = \frac{\partial u}{\partial x}, ε_{θ} = \frac{1}{R} (\frac{\partial v}{\partial θ} + w), γ_{x θ} = \frac{1}{R} \frac{\partial u}{\partial θ} + \frac{\partial v}{\partial x}

(1)

The equations for curvature are given as follow

κ_{x} = - \frac{\partial^{2} w}{\partial x^{2}}, κ_{θ} = - \frac{1}{R^{2}} (\frac{\partial^{2} w}{\partial θ^{2}} + w), κ_{x θ} = - \frac{2}{R} \frac{\partial^{2} w}{\partial x \partial θ}

(2)

These equations are compatible with the classical shell theory developed by Leissa¹ and Qatu.²¹

Displacement assumptions and ritz expansion

The displacement can be expressed through Fourier series in the circumferential direction and Ritz series in the axial direction as follows

u (x, θ, t) = \sum_{m = 1}^{N} A_{m} \frac{d ϕ_{m} (x)}{dx} \cos (n θ) e^{i ω t}

(3)

v (x, θ, t) = \sum_{m = 1}^{N} B_{m} ϕ_{m} (x) \sin (n θ) e^{i ω t}

(4)

w (x, θ, t) = \sum_{m = 1}^{N} C_{m} ϕ_{m} (x) \cos (n θ) e^{i ω t}

(5)

where n is the circumferential wave number, ω is the natural circular frequency (rad/sec), and

ϕ

_m(x) are admissible functions. The number of terms in the Ritz series, denoted by N, is selected based on convergence requirements. In the present analysis, the circumferential wave number is represented by the term n and is identical to the mode number used in graphical illustrations.

Admissible functions and boundary conditions in ritz formulations

The admissible functions satisfy geometric boundary conditions.

• Clamped-Clamped (C-C) Boundary Condition

For both ends clamped:

ϕ_{m} (0) = ϕ_{m}^{'} (0) = ϕ_{m} (L) = ϕ_{m}^{'} (L) = 0

(6)

With the governing eigenvalue equation

\frac{d^{4} ϕ_{m}}{d x^{4}} = λ_{m}^{4} ϕ_{m}

(7)

• Clamped-Free (C-F) Boundary Condition

For cantilevered shells:

ϕ_{m} (0) = ϕ_{m}^{'} (0) = 0, ϕ_{m}^{″} (L) = ϕ_{m}^{‴} (L) = 0

(8)

The exact eigenfunctions are

ϕ_{m} (x) = \cosh (β_{m} x) - \cos (β_{m} x) - σ_{m} [\sinh (β_{m} x) - \sin (β_{m} x)]

(9)

where β_m satisfies

\cos (β_{m} L) \cosh (β_{m} L) + 1 = 0

(10)

And

σ_{m} = \frac{\sinh (β_{m} L) - \sin (β_{m} L)}{\cosh (β_{m} L) - \cos (β_{m} L)}

(11)

Energy formulation

Hamilton’s principle is written as

δ \int_{t_{1}}^{t_{2}} (T - U) d t = 0

(12)

The strain and kinetic energies are:

U = \frac{1}{2} \int_{0}^{L} \int_{0}^{2 π} (ε^{T} A ε + κ^{T} D κ) Rd θ dx

(13)

T = \frac{1}{2} \int_{0}^{L} \int_{0}^{2 π} ρ h ({\dot{u}}^{2} + {\dot{v}}^{2} + {\dot{w}}^{2}) Rd θ dx

(14)

Composite and FML constitutive relations

The areal density is:

μ = \sum_{k = 1}^{N_{L}} ρ_{k} h_{k}

(15)

The laminate stiffness matrices are:

A_{ij} = \sum_{k = 1}^{N_{L}} \int_{z_{k - 1}}^{z_{k}} {\bar{Q}}_{ij}^{(k)} dz

(16)

B_{ij} = \sum_{k = 1}^{N_{L}} \int_{z_{k - 1}}^{z_{k}} {\bar{Q}}_{ij}^{(k)} zdz

(17)

D_{ij} = \sum_{k = 1}^{N_{L}} \int_{z_{k - 1}}^{z_{k}} {\bar{Q}}_{ij}^{(k)} z^{2} dz

(18)

where z represents the thickness coordinate measured from the mid-plane, and

{\bar{Q}}_{ij}^{(k)}

represents the transformed reduced stiffness terms for the k-th layer with i, j = 1, 2, 6. In the case of asymmetric laminates, B ≠ 0, thus, resulting in coupling between extension and bending, according to Classical Lamination Theory by Qatu²¹ and Reddy.²²

Circumferential ring stiffener modelling

The ring stiffener is considered as an independent element. The strain energy of ring stiffeners is given by:

U_{s} = \frac{1}{2} \int_{0}^{2 π} (E_{s} A_{s} ε_{θ}^{2} + E_{s} I_{eff} κ_{θ}^{2}) Rd θ δ (x - x_{s})

(19)

and the kinetic energy due to the ring stiffener is

T_{s} = \frac{1}{2} \int_{0}^{2 π} ρ_{s} A_{s} {\dot{w}}^{2} Rd θ δ (x - x_{s})

(20)

where δ(⋅) represents the Dirac delta function used for indicating the discrete location of the ring stiffener. This approach is more realistic than smeared models and is consistent with the work of Qing et al.,¹⁰ Gan et al.,¹¹ Pan et al.,¹² and Li et al.²⁵

The total strain energy and total kinetic energy are found by summing up the individual contributions made by each of the stiffeners.

U_{total} = U_{shell} + \sum_{s = 1}^{N_{r}} U_{s,} T_{total} = T_{shell} + \sum_{s = 1}^{N_{r}} T_{s}

(21)

A discrete stiffener approximation through the use of the Dirac delta function provides a good approximation of stiffness and mass distribution. When compared to smeared stiffeners, where stiffness is assumed to be evenly distributed along the shell wall, the present analysis properly describes the presence of the discrete stiffeners. This technique follows similar analyses used in previous studies on stiffened cylindrical shells,^10,11,12,25 and will likely improve the overall accuracy of the analysis.

Rayleigh-Ritz matrix approach

The governing differential equation can be written as

({K - ω}^{2} M) q = 0

(22)

where K represents the stiffness matrix and M represents the mass matrix. The problem is solved using an in-house Python code that makes use of Gauss-Legendre quadrature.

Eigenvalue solution and numerical implementation

The eigenvalue problem is described as

K ϕ {= ω}^{2} M ϕ

(23)

The frequencies are obtained using the eigenvalues

f = \frac{ω}{2 π}

(24)

Finite element modelling

Finite element procedure

FEM analysis has been carried out using ANSYS Workbench 2025 R1 software to study the free vibration characteristics of the isotropic and FML cylindrical shells, including those with or without ring stiffeners. The geometry, material property, stacking sequence, stiffening pattern, and other boundary conditions mentioned in Section 2 have been considered in the model.

The vibration characteristics have been obtained through linear modal analysis for determining natural frequencies and their associated mode shapes. Linear material and small deformation assumptions have been assumed here, and hence geometric nonlinearity and damping have been omitted from the analysis.

Coordinate system and degrees of freedom

A cylindrical coordinate system (x, θ, z) has been used, wherein x, θ, and z are along the axial, circumferential, and radial axes, respectively. A shell node consists of six degrees of freedom (three translations and three rotations), facilitating the modeling of membrane, bending, and coupling effects accurately.

Element selection

SHELL281 elements have been used to mesh the shell. This type of element is suitable for thin and laminated composite shells and can be used to model anisotropic materials. Ring stiffeners are modeled using BEAM188 elements. The stiffeners are attached to the shell through common nodes, thus maintaining the same displacements and, hence, correctly modeling the shell-stiffener interaction.

Material modeling and laminate definition

The aluminum 6061-T6 alloy is considered to be a linear elastic and isotropic material, while the composite laminae are considered to be orthotropic materials, i. e., carbon, Kevlar-49, S-glass, and E-glass. FML shell elements can be considered layered shell sections whose stacking sequence is [Al/0°/90°/Al/0°/90°/Al], where every layer is defined by its thickness, material, and ply angle.

It is also assumed that perfect bonding between metallic and composite layers exists. This assumption is commonly used in laminated composite vibration analysis problems. Nevertheless, from a practical standpoint, interlayer bonding failures may occur because of manufacturing defects or external loadings or any environmental factors. For example, the laminate stiffness will be reduced in cases when delamination or debonding occurs due to cyclic loads. This fact will result in a decrease in the natural frequencies and change in the mode shapes. However, the proposed approach does not include these interfacial damage effects.

Mesh generation

The use of a structured quadrilateral mesh is made in generating the shell geometry, with local mesh refinements in the regions near stiffeners. The selected mesh spacing is about 3 mm, based on convergence studies, leading to a total number of elements ranging from 25,000 to 45,000 depending on the shell configuration.

Boundary conditions

There are two types of boundary conditions:

(i) Clamped-Clamped (C-C): All degrees of freedom are constrained at both ends.

(ii) Clamped-Free (C-F): One end is fully constrained and the other is free.

Both these boundary conditions have been used uniformly throughout the various shell structures.

Modal solution and validation

These models were then analyzed to compute the natural frequencies and mode shapes that were used for the validation of the developed semi-analytical solution.

The results obtained from the finite element analysis concerning the mode shapes coincide with the response expected for the classic cylindrical shell. The low-order modes show dominance in terms of global bending and breathing modes, while high-order modes present variations around the circumference.

Mesh independence study

A mesh convergence study is performed for an unstiffened isotropic aluminum shell under C-C boundary conditions. The fundamental frequency is evaluated for different mesh densities (Table 2).

Table 2.

Mesh convergence for isotropic aluminum shell (C-C condition).

Element size	No. of elements	Frequency (Hz)
6.0	12,500	1220.5
4.0	21,800	1221.4
3.0	32,600	1222.4
2.2	41,200	1223.1

The variation in frequency becomes negligible beyond a mesh size of 3 mm, confirming convergence. This mesh is used for all subsequent simulations to ensure accuracy and computational efficiency.

Numerical results and discussion

This section discusses the natural vibration characteristics of isotropic and fiber-metal laminate cylindrical shells, including rings of circumferential ring stiffeners for C-C and C-F support types using the Flügge-Rayleigh-Ritz formulation. The natural vibration responses predicted by the formulated solution have been validated through finite element analysis performed in ANSYS Workbench software.

The results obtained are tabulated comprehensively in order to cover all combinations of materials systems considered. The discretization of the ring stiffeners used in the present study is similar to previous formulations documented by Qing et al.,¹⁰ Gan et al.,¹¹ and Pan et al.¹²

Unstiffened cylindrical shells under clamped-clamped boundary condition

Vibration characteristics of unstiffened cylindrical shells subjected to clamped-clamped boundary conditions are compiled in Table 3.

Table 3.

Natural frequencies of unstiffened isotropic and FML cylindrical shells with C-C boundary condition.

Mode	Al			Carbon FML			Kevlar FML			S-Glass			E-Glass
Mode	C-C Python code	C-C ANSYS	% Error	C-C Python code	C-C ANSYS	% Error	C-C Python code	C-C ANSYS	% Error	C-C Python code	C-C ANSYS	% Error	C-C Python code	C-C ANSYS	% Error
1	1247.57	1222.40	2.06	1380.66	1361.8	1.38	1325.26	1282.8	3.31	1339.9	1255.5	6.72	1205.76	1221.7	−1.30
2	1285.80	1311.00	−1.92	1606.74	1650.5	−2.65	1502.92	1571.3	−4.35	1559.77	1572.2	−0.79	1457.05	1518.4	−4.04
3	1674.42	1724.30	−2.89	1722.03	1748.7	−1.53	1660.47	1644.2	0.99	1611.15	1591.	1.27	1524.60	1561.2	−2.34
4	1850.28	1817.30	1.81	2360.65	2401.4	−1.69	2276.82	2267.7	0.40	2212.23	2231.7	−0.87	2117.27	2176.4	−2.72
5	2062.67	2061.20	0.07	2387.97	2428.8	−1.68	2301.14	2306.2	−0.22	2286.71	2297.9	−0.49	2246.10	2226.5	0.88
6	2265.04	2266.00	−0.04	2564.77	2565.9	−0.04	2413.07	2403.4	0.40	2343.36	2335.6	0.33	2292.68	2291.9	0.03
7	2423.37	2406.90	0.68	2718.25	2675.	1.62	2575.09	2575.3	−0.01	2598.38	2603.	−0.18	2517.55	2513.2	0.17
8	2556.51	2559.90	−0.13	2972.60	2975.9	−0.11	2720.05	2795.7	−2.71	2759.07	2742.5	0.60	2697.89	2683.3	0.54
9	2903.03	2863.10	1.39	3220.12	3218.9	0.04	3078.60	3073.1	0.18	3087.95	3083.	0.16	2994.63	2979.8	0.50
10	2935.76	2925.20	0.36	3294.75	3351.8	−1.70	3197.09	3174.7	0.71	3171.08	3159.6	0.36	3057.76	3070.8	−0.42

Finite-element analyses reveal good agreement with semi-analytical results in all modes, confirming the validity and stability of the present method. Natural frequency values for finite-length-matrix shells are found to be greater compared to isotropic aluminum shells because of the improved stiffness of FML shells. For the four materials examined here, carbon FML shells possess the greatest natural frequencies, whereas Kevlar, S-glass, and E-glass FML shells follow accordingly, due to the increasing elasticity of reinforcement fibers.

Unstiffened cylindrical shells under clamped-free boundary condition

The natural vibration properties of unstiffened cylinders with clamped-free end conditions are summarized in Table 4.

Table 4.

Natural frequencies of unstiffened isotropic and FML cylindrical shells with C-F boundary condition.

Mode	Al			Carbon FML			Kevlar FML			S-Glass			E-Glass
Mode	C-F Python code	C-F ANSYS	% Error	C-F Python code	C-F ANSYS	% Error	C-F Python code	C-F ANSYS	% Error	C-F Python code	C-F ANSYS	% Error	C-F Python code	C-F ANSYS	% Error
1	496.865	499.83	−0.59	595.08	580.58	2.50	538.49	530.74	1.46	531.75	506.64	4.96	512.19	490.89	4.34
2	574.21	600.42	−4.36	891.87	851	4.80	827.15	799.67	3.44	788.27	763.84	3.20	762.73	752.06	1.42
3	1076.23	1060.30	1.50	1159.21	1115.40	3.93	1054.12	1038.2	1.53	1037.70	1016.1	2.13	1017.48	978.55	3.98
4	1107.67	1103.20	0.41	1545.85	1495.1	3.39	1427.78	1400.3	1.96	1382.67	1364.8	1.31	1359.61	1327.2	2.44
5	1341.73	1326.80	1.13	1564.10	1539.1	1.62	1479.59	1446.2	2.31	1429.13	1390.5	2.78	1402.06	1368.7	2.44
6	1374.67	1380.00	−0.39	1860.94	1831.2	1.62	1798.82	1722.4	4.44	1718.39	1670.1	2.89	1694.09	1637.8	3.44
7	1680.08	1693.60	−0.80	1957.63	1933.8	1.23	1884.20	1817.6	3.66	1817.95	1796.5	1.19	1779.27	1734.6	2.58
8	1877.63	1879.40	−0.09	2053.36	1965.6	4.46	1967.87	1907.5	3.16	1975.12	1943.	1.65	1925.25	1875.8	2.64
9	1935.97	1953.80	−0.91	2689.76	2601.2	3.40	2316.87	2291.3	1.12	2253.41	2220.9	1.46	2246.37	2180.1	3.04
10	2207.11	2203.50	0.61	2736.71	2635.7	3.83	2545.06	2448.7	3.93	2489.72	2405.3	3.51	2433.82	2343.8	3.84

The natural frequency decreases in comparison to the case of fully clamped end conditions because of less restraint at the boundaries and increased flexibility. However, the comparative performance of the different materials is still maintained. The accuracy of the semi-analytical approach in predicting the natural vibration response in comparison with the finite element method is high for all vibration modes, indicating the efficiency of the proposed approach in simulating cantilever structures.

Ring-stiffened cylindrical shells under clamped-clamped boundary condition

The effect of circumferential ring stiffeners on the vibration characteristics of a clamped-clamped cylindrical shell is tabulated below in Table 5.

Table 5.

Natural frequencies of circumferentially ring-stiffened isotropic and FML cylindrical shells with C-C boundary condition.

Mode	Al			Carbon FML			Kevlar FML			S-Glass			E-Glass
Mode	C-C Python code	C-C ANSYS	% Error	C-C Python code	C-C ANSYS	% Error	C-C Python code	C-C ANSYS	% Error	C-C Python code	C-C ANSYS	% Error	C-C Python code	C-C ANSYS	% Error
1	1332.79	1337.5	−0.35	1529.57	1543.6	−0.9	1424.19	1471.8	−3.23	1525.28	1446.8	5.42	1367.16	1415.9	−3.44
2	1540.38	1541.1	−0.05	1668.16	1591.2	4.83	1543.60	1516.4	1.79	1530.73	1524.6	0.40	1479.96	1473.1	0.47
3	1916.30	1910.7	0.29	2299.32	2273.9	1.11	2194.87	2173.	1.01	2159.26	2133.	1.23	2166.40	2099.3	3.20
4	2260.05	2228.5	1.42	2461.87	2462.7	−0.03	2252.86	2347.7	−4.04	2355.77	2348.6	0.31	2222.93	2280.5	−2.52
5	2325.16	2310.8	0.62	2480.04	2513.3	−1.32	2407.13	2416.9	−0.40	2471.70	2459.5	0.50	2366.55	2372.2	−0.24
6	2701.42	2615.9	3.27	2702.43	2767.5	−2.35	2619.98	2635.8	−0.60	2621.68	2610.9	0.41	2508.51	2554.2	−1.79
7	2840.95	2856.1	−0.53	3144.91	3064.6	2.62	2927.86	2928.2	−0.01	3013.06	2957.	1.90	2909.22	2857.4	1.81
8	3000.06	3016.3	−0.54	3430.57	3419.7	0.31	3216.82	3250.3	−1.03	3233.06	3217.5	0.48	3194.44	3158.9	1.13
9	2876.54	3036.1	−5.26	3661.48	3575.7	2.39	3402.45	3404.9	−0.07	3401.05	3408.3	−0.21	3315.08	3319.5	−0.13
10	3120.65	3056.2	2.11	3699.61	3623	2.11	3440.75	3458.6	−0.52	3452.36	3482.2	−0.86	3370.91	3375.8	−0.14

Incorporation of ring stiffeners increases the frequencies for all material combinations, with a greater increase evident in lower vibrational modes. This supports the idea that ring stiffeners have considerable value in enhancing bending stiffness. For the different material systems, the frequencies for carbon based FML shells are maximum, followed by Kevlar, S-glass, and E-glass. The results match those obtained using finite elements analysis and therefore validate the stiffener model.

Ring-stiffened cylindrical shells under clamped-free boundary condition

The vibration characteristics of ring-stiffened cylindrical shells with clamped-free boundary conditions are shown in Table 6.

Table 6.

Natural frequencies of circumferentially ring-stiffened isotropic and FML cylindrical shells with C-F boundary condition.

Mode	Al			Carbon FML			Kevlar FML			S-Glass			E-Glass
Mode	C-F Python code	C-F ANSYS	% Error	C-F Python code	C-F ANSYS	% Error	C-F Python code	C-F ANSYS	% Error	C-F Python code	C-F ANSYS	% Error	C-F Python code	C-F ANSYS	% Error
1	518.4	516.4	0.39	593.45	623.14	−4.76	535.92	581.09	−7.77	511.60	559.71	−8.60	522	546.52	−4.49
2	895.4	925.5	−3.25	1087.94	1059.6	2.67	925.42	985.94	−6.14	979.31	970.85	0.87	984.04	934.52	5.30
3	950.7	938.4	1.31	1104.59	1103	0.14	1040.43	1058.3	−1.69	1012.38	1025.3	−1.26	1018.55	1013.6	0.49
4	1314.8	1325.9	−0.84	1645.22	1622.2	1.41	1550.85	1532.6	1.19	1487.05	1499.8	−0.85	1461.26	1464.1	−0.19
5	1562.1	1584.2	−1.39	1858.65	1856.9	0.09	1736.50	1749.7	−0.75	1733.12	1737.5	−0.25	1697.89	1677.3	1.23
6	1666.8	1666.8	0.00	1923.43	1861.5	3.32	1748.20	1798.8	−2.81	1719.18	1818.8	−5.48	1749.9	1777.4	−1.55
7	1873.2	1872.1	0.06	2004.16	1958.2	2.34	1951.32	1864.9	4.63	1839.90	1843.2	−0.18	1780.49	1791.1	−0.59
8	2004	1982	1.11	2260.50	2275.7	−0.66	2126.64	2170.1	−2.00	2107.23	2128.4	−0.99	2101.67	2093.6	0.39
9	2281.8	2303.7	−0.95	2717.64	2761.3	−1.58	2588.38	2612.5	−0.92	2602.49	2596.8	0.22	2517.87	2518.1	−0.01
10	2357	2334.3	0.97	2924.79	2887.5	1.29	2723.98	2737.1	−0.48	2689.70	2704.4	−0.54	2697.9	2641.8	2.12

Despite the fact that the clamped-free boundary condition shows less stiffness, it can be seen that the presence of the stiffeners leads to an enhanced dynamic behavior. This is especially true for the low-order modes when the mode shapes involve mainly overall deformations. The numerical values from the semi-analytical model show good correlation with those from the finite element analysis.

Effect of material hybridization, geometric stiffening and boundary conditions

In order to determine the percentage change in the fundamental natural frequency resulting from the aforementioned parameters, analysis is carried out, as shown in Table 7. The first mode is selected due to the significant role played by the mode in dynamic performance.

Table 7.

Percentage change in the fundamental natural frequency for different cylindrical shell configurations with C-C and C-F boundary conditions.

Configuration	C-C boundary condition	C-F boundary condition
Carbon FML vs isotropic aluminum (unstiffened)	11.40%	16.15%
Kevlar 49 FML vs isotropic aluminum (unstiffened)	4.94%	6.18%
S-glass FML vs isotropic aluminum(unstiffened)	2.71%	1.37%
E-glass FML vs isotropic aluminum(unstiffened)	−0.06%	−1.8%
Circumferential ring stiffened aluminum vs unstiffened aluminum	9.42%	3.32%
Circumferential ring stiffened carbon FML vs unstiffened carbon FML	13.35%	7.35%
Circumferential ring stiffened kevalr 49 FML vs unstiffened kevlar 49 FML	14.73%	9.50%
Circumferential ring stiffened S-glass FML vs unstiffened S-glass FML	15.24%	10.47%
Circumferential ring stiffened E-glass FML vs unstiffened E-glass FML	15.90%	11.33%
Circumferential ring stiffened carbon FML vs unstiffened aluminum	26.30%	24.70%

Effect of material hybridization

The FML structure boosts the natural frequency compared to isotropic aluminum shells, but the level of this boosting depends heavily on the stiffness-to-mass ratio of the reinforcement material. The FML shell reinforced by carbon fibers shows the largest boost in frequency because of the very high longitudinal modulus of this material. It increases the values of the stiffness matrix elements A and D significantly.

On the other hand, the FML shell reinforced with E-glass fibers shows a much weaker effect. Although E-glass fibers decrease the density, their elastic modulus is lower compared to the previous case. Therefore, the stiffness contribution decreases considerably, while the mass reduction is not enough to compensate for this effect. This leads to a lower boost of frequency.

Thus, we can conclude that the stiffness effect dominates the density one in cylindrical shells vibration. This result corresponds to the well-known observations in the field of laminated composites vibration analysis.^21,22

Influence of geometric stiffening

The installation of circumferential ring stiffeners results in higher natural frequencies due to increased effective bending stiffness. Such a contribution occurs particularly for low-order vibration modes where there is an increased contribution from global deformation, and the effect of the stiffener is very significant.

For FML shells, the interaction between stiffeners and laminate stiffness occurs, leading to enhanced stiffness performance. As a result, both deformation amplitudes decrease, and natural frequencies increase. Such results correspond well with the stiffness-dominant vibration properties of cylindrical shells.

Combined effect of hybridization and stiffening

Boundary conditions have a considerable effect on vibration characteristics. The natural frequencies of clamped-clamped shells are larger because of the stronger constraint, while those of clamped-free shells are smaller owing to the weaker stiffness (Figures 1–6).

Figure 1.

Geometric configuration of the FML cylindrical shell.

Figure 2.

Illustrates a representative finite element model of a circumferentially ring-stiffened fiber-metal laminated cylindrical shell.

Figure 3.

Variation of natural frequency (Hz) with mode number for different material systems under C-C boundary condition (unstiffened cylindrical shell).

Figure 4.

Variation of natural frequency (Hz) with mode number for different material systems under C-F boundary condition (unstiffened cylindrical shell).

Figure 5.

Variation of natural frequency (Hz) with mode number for different material systems under C-C boundary condition (ring-stiffened cylindrical shell).

Figure 6.

Variation of natural frequency (Hz) with mode number for different material systems under C-F boundary condition (ring-stiffened cylindrical shell).

The effect of anisotropic materials in combination with boundary conditions is crucial in determining the vibration characteristics. In the case of clamped-clamped boundary conditions, the effect of structural constraints dominates, thereby reducing the effect of anisotropy. On the contrary, in the clamped-free case, the effect of structural flexibility increases the sensitivity to anisotropic stiffness and extension-bending couplings. These results are consistent with previous findings in the study of vibrations of shells.^6,17

Hybrid materials in combination with stiffened geometry give rise to the maximum frequency increase, especially in the case of carbon FML shells.

Effect of extension-bending coupling (B-matrix)

Since the laminate structure is asymmetric, there is a non-zero value of the B-matrix, thus causing membrane and bending modes to interaction. Such coupling results in higher stiffness and high natural frequencies.

Extension-bending coupling effect is more evident when clamped-free boundary conditions are used because it increases the structural flexibility and laminate anisotropy effect. On the other hand, symmetric laminates lack such coupling and have lower stiffness interaction effects as well as low natural frequencies.

Despite the fact that no B-matrix parameter variation study was made, such a coupling effect is already covered in the asymmetric laminate setup. Natural frequency increase compared to that for symmetric laminates from related literature indicates the extent of the influence.

Convergence study of Rayleigh-Ritz solution

Convergence analysis was performed through a change in the value of N (number of axial admissible functions). These findings have been summarized in Table 8.

Table 8.

Convergence of fundamental natural frequency for Carbon-FML cylindrical shell (C-C condition).

Number of ritz terms(N)	Frequency(Hz)
4	1342.8
6	1362.5
8	1373.9
10	1379.4
12	1380.6

It can be seen that very quick convergence occurs as N increases. The variations are negligible when N >10. This indicates that the solution obtained through the proposed methodology is numerically stable.

While convergence analysis is done on the fundamental mode, it is expected that the same will occur in the case of higher modes as well, due to the added contribution of Ritz terms in axial deformations. For the current analysis, N = 10 appears to be adequate.

Validation and error analysis

The reliability of the suggested semi-analytical model has been demonstrated by comparing its predictions with the finite element solutions through an analysis of mesh independency and errors.

The percentage error stays within permissible limits, not exceeding a deviation of about 7% on average, which proves the adequacy of the developed algorithm. In the case of clamped-clamped boundary conditions, low errors are detected, confirming high prediction stability.

In relation to the clamped-free condition, somewhat higher deviations are revealed, especially for the lower modes. Such discrepancies can be explained by high structural flexibility and dependence on boundary conditions. The lower modes reveal larger errors because of the dominant nature of global deflection, while higher modes reveal better coincidence due to local deflection predominance.

Effect of stiffener number and spacing

A parametric study was performed in order to examine the effect of the number and distance between the stiffeners on an FML cylindrical shell with a C-C support. The results obtained are presented in Table 9 below.

Table 9.

Effect of stiffener number on fundamental natural frequency (Carbon FML, C-C condition).

Number of stiffeners	Spacing(mm)	Frequency (Hz)
0	0	1380.66
4	70	1468.20
6	50	1529.57
8	38	1572.30

As the number of stiffeners increases (or equivalently, when their distance is decreased), the natural frequencies become larger because the bending rigidity becomes larger and the effective length is smaller. Nevertheless, the rate of improvement becomes lower beyond a certain number of stiffeners, suggesting the existence of an optimum case.

The Dirac delta function approach in discretization is quite suitable for modeling the stiffened panels since it is capable of capturing concentrated properties. Contrary to smeared models which consider the uniform distribution of the stiffeners, discrete modeling yields consistent results similar to those obtained using finite element models.^8,9,10

Design-wise, it is important to note that both the number and spacing of stiffeners have significant impact. The increase in stiffeners improves the vibration characteristics; However, an optimum configuration is needed.

Engineering design implications

The findings presented herein are useful for the engineering design of cylindrical shells used in the aerospace, marine, and mechanical industries. Materials and stiffeners may be designed based on vibration requirements, weight restrictions, and budget limitations.

Cylindrical shells made from carbon-based FMLs are suitable for applications requiring high stiffness and vibration resistance. Cylindrical shells made from Kevlar-49 and glass-fiber FMLs have relatively higher natural frequencies while maintaining low density levels, making them ideal for lightweight structures. E-glass FMLs shells are specifically recommended for economical applications where modest performance is adequate.

The use of circumferential stiffeners has proven to be highly beneficial in raising natural frequencies, particularly for lower-order vibrations. However, the effect of adding more stiffeners gradually declines; thus, there is a need to optimize their arrangement. It is important to note that this study only considers linear vibration response in idealized conditions. Future studies may include nonlinearity, thermal loads, and imperfect interfaces, which might influence vibration responses.

Conclusion

The current research presents a semi-analytical as well as a numerical analysis of the free vibration of isotropic and FML cylinders with or without circumferential ring stiffeners, considering two sets of boundary conditions, C-C and C-F. A unified model based on Flügge’s thin shell theory and Rayleigh-Ritz approach has been formulated and analyzed using in-house coded Python software. The laminate constitutive law was modeled based on Classical Lamination Theory, while the stiffener effect was included in the model by means of the discrete energy concept. The results obtained from semi-analytical methods have been compared with the finite element modal analysis performed in ANSYS Workbench.

(1) As seen in the results, the presented semi-analytical model shows reliable predictions of natural frequencies, showing good agreement with FEA solutions and maintaining a maximum error that lies in the permissible range.

(2) FML shells show higher natural frequencies compared to the aluminum shells, with carbon FMLs providing the highest increase due to high stiffness levels.

(3) FML shells reinforced with Kevlar-49, S-glass, and E-glass provide moderate frequency increase because of an appropriate compromise among stiffness, weight, and price.

(4) Ring-type circumferential stiffeners lead to increased fundamental frequencies mainly due to improved global stiffness.

(5) The joint use of FML materials and stiffeners results in the highest frequencies, demonstrating the significant interplay between the shell material and reinforcement geometry.

(6) Different types of boundary conditions have a great impact on vibration characteristics. Thus, C-C shells exhibit higher frequencies, whereas C-F shells are sensitive to deformations and coupling effects.

(7) In addition, extension-bending coupling caused by the asymmetrical layers has an effect on vibration response, especially when flexible boundary conditions are applied.

(8) The convergence and mesh independence tests confirm the stability and precision of the numerical approach.

In the present study, only linear free vibration analysis for small amplitudes is considered. Issues such as geometric nonlinearities, large amplitudes, and material nonlinearities are ignored. Moreover, thermal loads, residual stress, and hygrothermal loads are excluded from the analysis. The case where there are no interfacial defects between the layers, such as debonding, is considered in an idealized situation.

Further research could include the development of the current model in the context of the analysis of nonlinear vibrations by introducing issues such as geometric nonlinearities and large amplitudes. Considering thermal loads and hygrothermal loads would make the model more realistic. In addition, other modeling techniques such as the effect of interfacial defects on the model and optimization of stiffeners’ configuration could be introduced.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Jakeer Hussain Shaik

Appendix

References

Leissa

. Vibration of shells (NASA SP-288). Scientific and technical information office. National Aeronautics and Space Administration, 1973.

Soedel

. Vibrations of shells and plates. 3rd ed. CRC Press, 2004.

Shi

Wang

, et al. Free vibration analysis of moderately thick composite cylindrical shells with arbitrary boundary conditions. Thin-Walled Struct 2022; 170: 108636.

Liu

Duan

Zheng

, et al. Free vibrations of a new three-phase composite cylindrical shell. Aerospace 2023; 10(12): 1007. https://doi.org/10.3390/aerospace10121007

Kim

J-S

Kim

K-S

Kim

K-S

, et al. Free vibration of laminated shells with variable thickness using the haar wavelet method. Appl Math Model 2022; 100: 1–18.

Han

. Free vibration of laminated cylindrical shells with arbitrary boundary conditions. Thin-Walled Struct 2015; 95: 229–237.

Chen

Mei

Wang

. Free vibration analysis of composite cylindrical shells using Ritz method. Compos Struct 2014; 111: 315–325.

Nazari

Malekzadeh

Naderi

. Experimental and numerical vibration of hybrid stiffened fiber-metal laminate cylindrical shells. Journal of Solid Mechanics 2019; 11(1): 105–119.

Hemmatnezhad

Rahimi

Tajik

, et al. Experimental, numerical and analytical investigation of free vibrational behavior of GFRP-Stiffened composite cylindrical shells. Compos Struct 2015; 120: 509–518. https://doi.org/10.1016/j.compstruct.2014.10.011

10.

Qing

Feng

Liu

, et al. Semi-analytical solution for free vibration of stiffened cylindrical shells. J Mech Mater Struct 2006; 1(1): 129–145. https://doi.org/10.2140/jomms.2006.1.129

11.

Gan

Zhang

. Free vibration of ring-stiffened cylindrical shells using a wave propagation approach. J Sound Vib 2009; 326(3-5): 633–646. https://doi.org/10.1016/j.jsv.2009.05.001

12.

Pan

. Free vibration of ring-stiffened thin cylindrical shells with arbitrary boundary conditions. J Sound Vib 2008; 314(3-5): 636–648.

13.

Jafari

Bagheri

. Free vibration of non-uniformly ring stiffened cylindrical shells using analytical, experimental and numerical methods. Thin-Walled Struct 2006; 44(1): 82–90. https://doi.org/10.1016/j.tws.2005.08.008

14.

Zhao

. Free vibration analysis of stiffened composite cylindrical shells using wave propagation method. J Sound Vib 2011; 330(11): 2640–2658.

15.

Cammalleri

Castellano

. A dynamic matrix for the study of free vibrations of thin circular cylindrical shells under different boundary conditions. Design 2023; 7(6): 122. https://doi.org/10.3390/designs7060122

16.

Yasniy

Mykhailyshyn

Pyndus

, et al. Numerical analysis of natural vibrations of cylindrical shells made of aluminum alloy. Mater Sci 2020; 55(4): 502–508. https://doi.org/10.1007/s11003-020-00331-2

17.

Lopatin

Morozov

. Fundamental frequency of the laminated composite cylindrical shell with clamped edges. Int J Mech Sci 2015; 92: 35–43. https://doi.org/10.1016/j.ijmecsci.2014.11.020

18.

Tang

Yao

, et al. Free vibration analysis of circular cylindrical shells with arbitrary boundary conditions by the method of reverberation‐ray matrix. Shock Vib 2016; 2016(1): 3814693. https://doi.org/10.1155/2016/3814693

19.

Guo

Liu

Wang

, et al. A unified strong spectral chebyshev solution for predicting free vibration characteristics of cylindrical shells with stepped thickness and stiffeners. Compos Struct 2024; 330: 117845.

20.

Tornabene

Fantuzzi

Bacciocchi

, et al. The local GDQ method applied to general higher-order theories of doubly-curved laminated composite shells and panels: the free vibration analysis. Compos Struct 2014; 116: 637–660. https://doi.org/10.1016/j.compstruct.2014.05.008

21.

Qatu

. Vibration of laminated shells and plates. Elsevier, 2004.

22.

Reddy

. Mechanics of laminated composite plates and shells: theory and analysis. CRC Press, 2003.

23.

Noda

Yasuda

. Finite element vibration analysis of laminated cylindrical shells. AIAA J 2001; 39(10): 1833–1840.

24.

Shahgholian-Ghahfarokhi

Rahimi

Zarei

, et al. Free vibration analyses of composite sandwich cylindrical shells with grid cores: experimental study and numerical simulation. Mech Base Des Struct Mach 2022; 50(2): 687–706. https://doi.org/10.1080/15397734.2020.1725565

25.

X-Q

Zhang

Yang

X-D

, et al. A unified approach for free vibration analysis of stiffened cylindrical shells with general boundary conditions. Int J Mech Sci 2019; 159: 221–236.

26.

Naeem

Kanwal

Shah

, et al. Vibration characteristics of ring‐stiffened functionally graded circular cylindrical shells. Int Sch Res Not 2012; 2012(1): 232498. https://doi.org/10.5402/2012/232498

27.

Amabili

. Nonlinear vibrations and stability of shells and plates. Cambridge University Press, 2008.

28.

Shen

H-S

Xiang

. Non-linear vibration of functionally graded composite laminated cylindrical shells in hygrothermal environments. Compos Struct 2012; 94(11): 3106–3114.

29.

Hosseini

Arefi

. Vibration analysis of composite cylindrical shells reinforced with stiffeners. Compos Struct 2017; 178: 373–391.

30.

Wang

Dai

. Free vibration of composite cylindrical shells with ring and stringer stiffeners. Thin-Walled Struct 2018; 125: 311–321.

31.

Zhao

Liang

, et al. Free vibration of composite cylindrical shells with orthogonal stiffeners. J Theor Appl Mech. 2022; 60.

32.

Xiang

Zhao

. Free vibration of functionally graded cylindrical shells with ring stiffeners. J Sound Vib 2010; 329(14): 2820–2838.

33.

Khosravi

Jafari

. Free vibration of laminated cylindrical shells with elastic foundations. Compos Struct 2016; 141: 280–292.

34.

Dai

Wang

. Free vibration analysis of stiffened composite shells with arbitrary boundary conditions. Compos Struct 2019; 216: 280–292.

35.

Farshidianfar

Oliazadeh

. Free vibration analysis of circular cylindrical shells: comparison of different shell theories. Int J Mech Appl 2012; 2(5): 74–80. https://doi.org/10.5923/j.mechanics.20120205.04

36.

Jain

Rawat

Kushwah

. Free vibration analysis of circular cylindrical shells. Am J Eng Res 2016; 5(10): 373–380.

37.

Rawat

. Finite element analysis of thin circular cylindrical shells, 2016.