Elsevier

Composite Structures

Volume 147, 1 July 2016, Pages 168-184
Composite Structures

Non-linear dynamic instability analysis of laminated composite cylindrical shells subjected to periodic axial loads

https://doi.org/10.1016/j.compstruct.2016.02.064Get rights and content

Abstract

The dynamic instability of thin laminated composite cylindrical shells subjected to harmonic axial loading is investigated in the present work based on nonlinear analysis. The equations of motion are developed using Donnell’s shallow-shell theory and with von Karman-type of nonlinearity. The nonlinear large deflection shallow-shell equation of motions are solved by using Galerkin’s technique that leads to a system of nonlinear Mathieu–Hill equations. Both stable and unstable solutions amplitude of the steady-state vibrations are obtained by applying the Bolotin’s method. The nonlinear dynamic stability characteristics of both symmetric and antisymmetric cross-ply laminates with different lamination schemes are examined. A detailed parametric study is conducted to examine and compare the effects of the magnitude of both tensile and compressive axial loads, aspect ratios of the shell including length-to-radius and thickness-to-radius ratios, and different circumferential wave numbers as well on the parametric resonance particularly the steady-state vibrations amplitude. The present results show good agreement with that available in the literature.

Introduction

Composite structures have been progressed from almost an engineering curiosity to widely used structures in aerospace, automotive, and civil engineering as well as in many other applications in everyday life. Advantages of fiber-reinforced composite materials including their outstanding strength and stiffness particularly the specific stiffness which is the stiffness-to-density make them more attractive for use in weight-sensitive structures such as aircraft or spacecraft structures [1].

When the lightweight structural components are subjected to dynamic loading particularly periodic loads, when the frequency of in-plane dynamic load and the frequency of vibration satisfy certain specific condition, parametric resonance will occur in the structure, which makes the plate or shell structure to enter into a state of dynamic instability [2]. This instability is of concern because it can occur at load magnitudes that are much less than the static buckling load, so a component designed to withstand static buckling may fail in a periodic loading environment. Further, the dynamic instability occurs over a range of forcing frequencies rather than at a single value [2], [3].

The interest to study the dynamic stability behavior of engineering structures dates back to the text by Bolotin [4] which addresses numerous problems on the stability of structures under pulsating loads. According to the general theory of dynamic stability of elastic systems by using Bolotin’s method a set of differential equations of the Mathieu–Hill type are derived, and by seeking periodic solutions using Fourier series expansion the boundaries of unstable regions are determined. An extensive bibliography of the earlier works on parametric response of structures was presented by Evan-Iwanowsky [5].

A detailed research survey on the dynamic stability behavior of plates and shells in which the literature from 1987 to 2005 has been reviewed can be found in the review paper by Sahu and Datta [6].

The dynamic instability regions of laminated anisotropic cylindrical shells were studied by Argento and Scott [3], [7] using a perturbation technique. The shell’s ends were clamped and subjected to axial periodic loading. In the numerical part [7] they discussed the effect of circumferential wave number and magnitude of external axial load on instability regions. Argento [8] then extended this work to compare the instability regions of the shell subjected to pure axial, pure torsional, and combined axial and torsional loadings. Extensive studies of dynamic stability of laminated composite cylindrical shells have been carried out by Ng. et al. [9], [10], [11], [12], [13], [14], using Love’s classical thin shell theory for antisymmetric cross-ply laminate to investigate the effects of different lamination schemes and magnitude of the axial periodic loading [9], and length-to-radius and thickness-to radius ratios [10]. A comparison of different thin shell theories namely, Donnell’s, Love’s and Flugge’s shell theories in predicting dynamically unstable regions of cross-ply laminated cylindrical shells has been provided [11]. Cylindrical panels with transverse shear effects have been studied using Donnell’s shell theory and then extended using first-order shear deformation theory [12]. Donnell’s equation has been used to study thin rotating isotropic cylindrical shells subjected to periodic axial loading [13] and, dynamic instability of laminated cylindrical shells has been studied via the mesh-free kp-Ritz method [14]. Fazilati and Ovesy used finite strip method to study the parametric instability of laminated composite plates and shells [15], subjected to non-uniform in-plane loads [16], moderately thick cylindrical panels with internal cutouts [17] and longitudinally stiffened curved panels with cutout [18] as well.

All these mentioned works are based on linear analysis and so lead to dynamic instability regions. Stability analysis based on classical linear theories provided only an outline of the parameter regimes where nonlinear effects are of importance. According to Popov [19] without adequate non-linear analysis the results in some cases can be inaccurate. “According to linear theory, one expects the vibration amplitudes in the regions of dynamic instability to increase unboundedly with time indeed very rapidly so as to increase exponentially. However, this conclusion contradicts experimental results which reveal that vibrations with steady-state amplitudes exist in the instability regions. As the amplitude increases, the character of the vibrations changes; the speed of the amplitude growth gradually decreases until vibrations of constant (or almost constant) amplitude are finally established” [4].

Some non-linear problems of laminated shells have been addressed in literature including initial post-buckling behavior [20], and free vibration and dynamic analysis of cylindrical and conical shells [21], [22], [23], [24], [25].

A comprehensive literature review covering the period 2003–2013 on non-linear vibrations of shells has been done by Alijani and Amabili [26].

Few works have been conducted considering the non-linear shell theories for dynamic stability problems. Cheng-ti and Lie-dong [27] used Hamilton principle to derive the equation of motion and solved this equation with variational methods to study the effect of large deflection which leads to nonlinear dynamic instability for three typical laminated composite cylindrical shells, viz, Graphite epoxy, E-glass epoxy and ARAAL shells. Their studies were limited to overall trend of variation of the amplitude with these three composites.

To the present authors’ knowledge a comprehensive study which considers the effects of stacking sequence and aspect ratios has not been carried out on the non-linear dynamic instability of thin laminated shells. In the present paper, the Donnell’s shallow-shell theory with von Karman-type of nonlinearity is considered for thin, laminated composite cylindrical shell subjected to harmonic axial loading. Galerkin’s technique is then employed to solve the nonlinear large deflection shallow-shell equations of motion and a system of nonlinear Mathieu–Hill equations are derived. The steady-state amplitudes of both stable and unstable solutions are determined by applying the Bolotin’s method. The parametric studies are performed to investigate and compare the effects of different lamination schemes of symmetric and antisymmetric cross-ply laminated shells, the magnitude of axial loads both tensile and compressive loads, different aspect ratios of the shell including length-to-radius and thickness-to-radius ratios and different circumferential wave numbers as well on the parametric resonance particularly of the steady-state vibrations. The present results show good agreement when compared with that available in the literature and hence can be used as bench mark results for future studies.

Section snippets

Formulation

A thin simply supported laminated composite cylindrical shell, having length L and radius R with respect to the curvilinear coordinates (X,θ,Z) which are assigned in the mid-surface of the shell is considered as shown in Fig. 1.

Here, u, v and w are the displacement components of the shell with reference to this coordinate system in the X,θ,Z, directions, respectively.

The cylindrical shell is subjected to a periodically pulsating load in the axial direction with the axial loading per unit length

Solution for laminated orthotropic shells

Considering the simply supported boundary condition for the studied laminated orthotropic cylindrical shell, the Navier’s double Fourier series with the time-dependent coefficient qmn(t) is chosen to describe the transverse displacement function w0(x,θ,t):w0=m=1n=1qmn(t)sinmπLxcosnθwhere m, n represent the number of axial half waves in corresponding standing wave pattern and the circumferential wave number, respectively.

Fxx is the average axial force at the edge, thus the stress function

Amplitude of vibrations at the principal parametric resonance

As mentioned above Eq. (32) is a nonlinear Mathieu equation where the nonlinear term γqmn3(t) represents the effect of large deflection. According to Liapunov Principle, dynamically unstable region is determined by the linear parts of the Eq. (32) [4] which will be discussed in the next section. Here the focus is set on the parametric resonance of the system. The basic solutions of Mathieu equation include two periodic solutions: i.e. periodic solutions of T and 2T with T=2π/P. The solutions

Dynamic instability regions

The resonance curve is not influenced by nonlinearity of Eq. (29) and as mentioned in the previous section the dynamic instability regions are determined by linear part of Mathieu–Hill equation, and so Eq. (29) can be rewritten as follow:Mmnq¨mn(t)+Kmn-Qmncosptqmn(t)+ηmnqmn3(t)=0whereKmn=Kmn-FsQmnandQmn=FdQmn

The principal region of dynamic instability which corresponds to solution of period 2T is determined by substituting Eq. (36) into Eq. (43) and equating the determinant of the

Numerical results and discussion

Nonlinear dynamic stability characteristics of cross-ply laminated composite cylindrical shells subjected to combined static and periodic axial loads are studied here. The material properties used in the present analysis are chosen in accordance with Ng et al. [9] as E1/E2=40, G12/E2=0.5 and υ12=0.25.

For isotropic cylindrical shells the critical static buckling load in terms of engineering constants is given by Timoshenko and Gere as [29]Nbuc=Eh2R[3(1-ν2)]

The mechanism of dynamic buckling is

Conclusions

In the present work, the non-linear dynamic stability of both symmetric and antisymmetric cross-ply laminated composite cylindrical shells under combined static and periodic axial loading has been studied. Donnell’s shallow-shell equations of motion with von Karman-type of nonlinearity were solved by employing Galerkin’s technique. By applying Bolotin’s method to the governing system of nonlinear Mathieu–Hill equations the amplitudes of both stable and unstable solutions were obtained for

Acknowledgements

The authors thank the Faculty of Engineering and Computer Science of Concordia University, and Natural Sciences and Engineering Research Council of Canada (NSERC) for their financial support for the research work conducted. The NSERC support was provided through the Discovery Grant # 172848-2012 awarded to the second author of the present paper.

References (30)

Cited by (24)

  • Instability analysis of fluid-filled angle-ply laminated circular cylindrical shells subjected to harmonic axial loading

    2023, European Journal of Mechanics, A/Solids
    Citation Excerpt :

    Adopting higher-order shear deformation theory, Pradyumna and Bandyopadhyay (2011) developed a finite element method for the dynamic instability analysis of laminated composite hyperbolic paraboloid bounded by straight lines and conoid shells. Darabi and Ganesan (2018, 2016) carried out a complete nonlinear dynamic instability analysis of laminated composite cylindrical shells (Darabi and Ganesan, 2016) and thin plates (Darabi and Ganesan, 2018) subjected to harmonic in-plane loading. Heydarpour and Malekzadeh (2018) developed a mathematical model to study the dynamic instability behaviour of carbon nanotubes reinforced functionally graded composite cylindrical shells subjected to combined static and periodic axial loads.

View all citing articles on Scopus
View full text