Elsevier

Composite Structures

Volume 120, February 2015, Pages 509-518
Composite Structures

Experimental, numerical and analytical investigation of free vibrational behavior of GFRP-stiffened composite cylindrical shells

https://doi.org/10.1016/j.compstruct.2014.10.011Get rights and content

Abstract

The present research aims to investigate the vibration characteristics of stiffened composite cylindrical shells using experimental, numerical and analytical techniques. The specimens are fabricated from continuous glass fiber (GFRP) using a specially-designed filament winding setup. The theoretical formulation is established based on Sanders’ thin shell theory. In the analytical approach, a smeared method is employed to superimpose the stiffness contribution of the stiffeners with those of shell in order to obtain the equivalent stiffness parameters of the whole panel. Using the Ritz method, the governing eigenvalue equations are obtained and will then be solved for evaluating the natural frequencies of the GFRP-stiffened composite shells. In order to validate the analytical achievements, experimental modal analysis is conducted on a stiffened cylinder. A 3-D finite element model is built for a further validation. This model takes into account the exact geometric configuration of the stiffeners and the shell. Results confirm the accuracy of the analytical method. Furthermore, the influences of changes in the skin thickness and boundary condition are analyzed.

Introduction

Cylindrical shells are one of the most important structural components and can be extensively found in civil, aviation and aerospace industries (e.g. pressure vessels, launch vehicles, re-entry vehicles, aircraft fuselages, spacecrafts, etc.). Their light weight as well as high capacity of load bearing has provided them with such a wide range of engineering applications. Shells are often subjected to dynamic loads which cause vibrations. That is why the vibrational behavior investigation of these structures (e.g. frequencies, mode shapes, and modal forces) is of high importance in structural dynamics. A Great number of investigations have been emerged rapidly during the past decades concerning with the free vibrational analysis of isotropic, composite and functionally graded (FG) cylindrical shells. Excellent reviews conducted on the dynamics of shells, have been collected by Leissa [1] and Qatu [2], [3], [4], [5], [6]. Recently, many computational methods have been developed for studying the vibrational behavior of composite cylindrical shells such as analytical methods [7], [8], [9], [10], [11], [12], differential quadrature method [13], [14], finite element method [15], [16] and discrete singular convolution approach [17]. For further acquiring knowledge on dynamic analysis of composite shells, the reader may be referred to the review work by Qatu et al. [18]. During very recent years, performing various studies on the vibrational behavior of composite cylindrical shells has also been continued. Amabili and Reddy [19] proposed a new higher-order shear deformation nonlinear theory for the doubly-curved shells taking into account the geometric imperfection and also full nonlinear terms in the strain–displacement relations. Then, the large-amplitude forced vibrations of simply supported, laminated circular cylindrical shells are studied. Qu et al. [20] proposed an efficient domain decomposition method for investigating the free, harmonic and transient vibrations of isotropic and composite cylindrical shells subjected to different combinations of classical and non-classical boundary conditions. The theoretical formulation was based on laminated version of Reissner–Naghdi’s shell theory. The displacement functions are assumed to be expanded as Fourier series and orthogonal polynomials. The analytical results were compared with those obtained via finite element program ANSYS and very good agreement was achieved. Based on the first-order shear deformation theory, Jin et al. [21] developed a unified analytical approach using Rayleigh–Ritz method for the vibration analysis of moderately thick composite laminated cylindrical shells subjected to general boundary conditions and arbitrary intermediate ring supports and various lamination schemes. An accurate solution procedure based on the Haar wavelet discretization method (HWDM) was employed for studying the free vibration analysis of laminated composite cylindrical shells with different end supports by Xie et al. [22]; this theoretical formulation was in the framework of Reissner–Naghdi’s shell theory. The governing partial differential equations of motion are first converted into system of ordinary differential equations by the separation of variables and finally solved by means of the HWDM to give the natural frequencies of vibration. The literature review clarifies that the number of publications concerning with experimental modal analysis of thin cylindrical shells are not as notable as those conducted on computational methods. Also, most of the related experimental investigations have been performed on isotropic cylindrical shells [23], [24], [25], [26], [27]. A few studies can be found dealing with experimental modal analysis of composite cylindrical shells. Hosokawa et al. [28] studied numerically and experimentally the free vibrations of angle-ply laminated carbon fiber reinforced plastic (CFRP) cylindrical shells with clamped edges. Wu et al. [29] performed an experimental study as well as finite element analysis on the dynamic response of filament wound pressure vessels filled with liquid. They also provided the primary information which can be used for vibration-based damage detection of composite vessels.

Grid-stiffened composite cylindrical shells are well known structures in many engineering fields, especially aerospace industry. They are cylinders reinforced with different types of stiffening structures either on the inner, outer or both sides of the shell. These stiffeners significantly increase the load resistance of a cylinder without much increase in weight. The selection of stiffener configuration depends on several factors such as the loading condition, cost, and other factors. The promising future of stiffened cylinders with reinforcing grids or ribs, has led to a wide range of research work [30], [31], [32], [33], [34], [35], [36], [37], [38].

Several papers were concerned with the vibrational behavior of isotropic and composite cylindrical shells stiffened with longitudinal and circumferential stiffeners [39], [40], [41], [42], [43], [44]. Other studies are also conducted on the experimental modal analysis of ring and stringer stiffened cylindrical shells [45], [46]. Based on the Carrera unified formulation (CUF) [47], [48], [49], Carrera et al. employed the finite element method for obtaining the natural frequencies of thin-walled cylinders reinforced with longitudinal and transversal stiffeners and under arbitrary boundary conditions [50]. However, the number of publications performed on the buckling and vibrational behavior of grid-stiffened composite cylinders with helical stiffeners is scarce. Kidane et al. [51], [52] derived the buckling loads of a generally cross and horizontal grid-stiffened composite cylindrical shell by developing a smeared method for determination of the equivalent stiffness parameters of a grid-stiffened composite cylindrical shell. The stiffness contribution of the stiffeners was superimposed with the stiffness contribution of the shell to obtain the equivalent stiffness parameters of the whole panel. Then, energy method was implemented to obtain the buckling load for a particular stiffener configuration. Buckling tests were also performed on a stiffened composite cylinder and compared with analytical results and satisfactory agreement was achieved. Yazdani and Rahimi performed experimental investigations on the buckling behavior of composite cylindrical shells with cross stiffeners [53]. They also studied the effects of helical ribs’ number and changes in grid types on the buckling load of these structures [54], [55]. Recently, Rahimi and his co-authors studied the effect of stiffener cross-section profile on the buckling strength of composite stiffened cylindrical shells by implementing the finite element method [56]. More recently, Shi et al. [57] presented the initial buckling and postbuckling responses of axially loaded grid-stiffened composite cylindrical shells with reinforced rectangular or circular cutouts using finite element analysis. Afterwards they extended their previous study and obtained the critical buckling loads of grid-stiffened composite conical cylindrical shells using the minimum potential energy principle [58].

To the best of authors’ knowledge, there is no published research in the literature conducted on the free vibrations of grid-stiffened cylindrical shells with helical ribs unless the very recent ones by [59], [60]. Hemmatnezhad et al. [59] implemented an exact analytical approach (as in [12]) for investigating the vibrational behavior of grid-stiffened composite cylindrical shells considering the flexural behavior of the ribs. A smeared method was employed to superimpose the stiffness contribution of the stiffeners with those of shell in order to obtain the equivalent stiffness parameters of the whole panel. The stiffeners were modeled as a beam and considered to support shear loads and bending moments further to the axial loads which seems to be also an extension of the model used by [51], [52]. Therefore, the corresponding stiffness terms are taken into consideration while obtaining the stiffness matrices due to the stiffeners. Theoretical formulations were based on first order shear deformation shell theory which included the effects of transverse shear deformation and rotary inertia. They also compared their results with those of finite element analysis via ABAQUS software and obtained a satisfactory agreement. Using the boundary layer theory, Li and Qiao [60] presented the nonlinear free vibration and parametric resonance analysis of geodesically-stiffened laminated composite cylindrical shell embedded in an elastic medium and subjected to static or periodic axial forces. The governing equations of motion are developed using Donnel shell theory with von Kàrmàn nonlinearity. An improved smeared stiffener approach is proposed to take the skin-stiffener interaction into account by evaluating the bending and coupling stiffness due to the skin and stiffeners in the skin-stiffener region about a shift in the neutral axis of the stiffeners. Finally the equations of motion are solved by implementing the singular perturbation technique [61], [62] in order to determine the linear and nonlinear frequencies of vibration. However, the need for experimental modal analysis of these structures is still felt.

The main goal of the present paper is to carry out a comparative study based on analytical, experimental and numerical techniques capable of predicting the vibration characteristics of grid-stiffened glass fiber reinforced plastic (GFRP) laminated composite cylindrical shells. The specimens are fabricated from continuous glass fiber using a specially-designed filament winding setup. For the analytical procedure, the theoretical formulation is established based on Sanders’ thin shell theory. A smeared method is employed to superimpose the stiffness contribution of the stiffeners with those of shell in order to obtain the equivalent stiffness parameters of the whole panel. Using energy functional, an appropriate set of displacement functions and applying the Ritz method, the governing eigenvalue equations are derived and the natural frequencies of vibration of GFRP-stiffened composite shells are obtained. A 3-D finite element model is also built using ABAQUS software which takes into consideration the exact geometric configuration of the stiffeners and the shell. In order to validate the accuracy of the analytical and numerical approaches, experimental modal analysis is carried out in order to experimentally estimate the natural frequencies of vibration. Furthermore, the influences of changes in the skin thickness and end supports on the vibration frequencies of GFRP-stiffened composite cylindrical shells are studied.

Section snippets

Equivalent stiffness parameters

Consider a thin GFRP composite cylindrical shell of mean radius R, length L and thickness t, the shell is reinforced with a lozenge-type stiffener structure as shown in Fig. 1. The middle surface of the skin (shell structure) is chosen as the reference surface where an orthogonal coordinate system (x, y, z) is fixed. x and y are in the axial and circumferential directions while z points out to the outward normal to the middle surface. The deformation components of the shell with reference to this

Finite element analysis

3-D models are built for the grid-stiffened GFRP cylindrical shells with the lozenge-type stiffeners using ABAQUS CAE 6.10 finite element analysis software (See Fig. 3). The lozenge-type stiffeners consist of six helical ribs oriented at 30° and −30° angles with regard to the longitudinal axis of the shell/stiffener structure. The stiffeners are assembled into the shell so that the interfacing areas are tied together. Therefore, the ribs and shell become a unique structure. The quadratic planar

Experimental modal analysis (EMA)

The specimens were composed of E-glass fiber and room temperature-curing epoxy resin using a specially-designed filament winding setup. The schematic view of the fabricated GFRP-stiffened and -unstiffened specimens used for the present experimental modal analysis is shown in Fig. 4. Knowing the material properties of fiber and resin and weights of them, the material properties can be obtained by implementing the rule of mixture for composite materials [66]. The nominal material properties for

Results and discussion

The FRF measurements for three different grid points of a stiffened shell are plotted in Fig. 6. Each peak indicates at least one vibration mode. As would be observed from FRF curves, the peaks are not clear enough for frequency values greater than 800 Hz. Also, all the peaks in this region do not really indicate the vibration modes but can be of noise signals. This fact is due to the light weight of the fabricated specimen and any possible influence of additive mass caused by the accelerometer.

Conclusions

Free vibration characteristics of GFRP-stiffened composite cylindrical shells were investigated analytically, numerically and experimentally probably for the first time. The GFRP-stiffened specimens are fabricated using a specially-designed filament winding machine. Results obtained from the three types of analyses meet a good agreement. The differences in some cases are mainly due to the complexity of the structure and errors may occur while fabricating the specimens. Further to these, using

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