STRUCTURAL RESPONSE OF LAMINATED COMPOSITE SHELLS SUBJECTED TO BLAST LOADING: COMPARISON OF EXPERIMENTAL AND THEORETICAL METHODS

The results from a theoretical and experimental investigation of the dynamic response of cylindrically curved laminated composite shells subjected to normal blast loading are presented. The dynamic equations of motion for cylindrical laminated shells are derived using the assumptions of Love's theory of thin elastic shells. Kinematically admissible displacement functions are chosen to represent the motion of the clamped cylindrical shell and the governing equations are obtained in the time domain using the Galerkin method. The time-dependent equations of the cylindrically curved laminated shell are then solved by the Runge-Kutta-Verner method. Finite element modelling and analysis for the blast-loaded cylindrical shell are also presented. Experimental results for cylindrically curved laminated composite shells with clamped edges and subjected to blast loading are presented. The blast pressure and strain measurements are performed on the shell panels. The strain response frequencies of the clamped cylindrical shells subjected to blast load are obtained using the fast Fourier transformation technique. In addition, the effects of material properties on the dynamic behaviour are examined. The strain-time history curves show agreement between the experimental and analysis results in the longitudinal direction of the cylindrical panels. However, there is a discrepancy between the experimental and analysis results in the circumferential direction of the cylindrical panels. A good prediction is obtained for the response frequency of the cylindrical shell panels.     

Acoustic wave scattering from a composed shell reinforced by a single rib: characteristics of peripheral waves

The steady-state axisymmetrical problem of a plane acoustic wave scattering from a composed shell is considered. The shell has a cylindrical part and two hemispherical endcaps. The rib is a ring of rectangular cross-section that divides the shell into two equal parts. The motion of the shell and the rib is described by the equations of elasticity theory, and the liquid is described by the Helmholtz equation. The solution is obtained numerically by a coupled finite element/boundary element model. Two peripheral waves are generated in the shell: the membrane S0 wave and the bending type water-borne A wave. The form function, acoustic spectrogram and dispersion curves of the phase velocities are presented, and the effect of the rib on the peripheral waves is discussed.     

Acoustic signature of a submarine hull under harmonic excitation

The structural and acoustic responses of a submarine under harmonic force excitation are presented. The submarine hull is modelled as a cylindrical shell with internal bulkheads and ring stiffeners. The cylindrical shell is closed by truncated conical shells, which in turn are closed at each end using circular plates. The entire structure is submerged in a heavy fluid medium. The structural responses of the submerged vessel are calculated by solving the cylindrical shell equations of motion using a wave approach and the conical shell equations with a power series solution. The far-field radiated sound pressure is then calculated by means of the Helmholtz integral. The contribution of the conical end closures on the radiated sound pressure for the lowest circumferential mode numbers is clearly observed. Results from the analytical model are compared with computational results from a fully coupled finite element/boundary element model.     

Characteristics of the vibrational power flow propagation in a submerged periodic ring-stiffened cylindrical shell

By using space-harmonic analysis method, the characteristics of the vibrational power flow propagation in an infinite periodic ring-stiffened cylindrical shell immersed in water are studied. The harmonic motion of the shell and the sound pressure field in the fluid are described by Flügge shell equations and Helmholtz equation, respectively, and four kinds of the rings’ forces and moments are considered. Along the shell axial direction, the propagation of the power flow carried by different internal forces (moments) of the shell wall can be obtained, thus the total power flow in the shell wall and the ratios of the component power flow carried by different shell internal forces (moments) to the total power flow are also studied. It is found that characteristics of the vibrational power flow propagation vary with different circumferential modes order and different frequencies. Moreover, the presence of the stiffeners and structural damping will greatly influence the results.     

Diffraction of a convergent sound wave by an elastic cylindrical shell with a longitudinal fixation

The problem of the diffraction of a zero-order convergent cylindrical wave by a cylindrical shell with a longitudinal fixation along one of its generatrices is considered. The problem is solved on the basis of using the so-called helical waves, which are aperiodic eigensolutions to the equations of the shell motion. The diffraction field is represented in the form of a convergent series in cylindrical harmonics. The method of the solution allows for a generalization to several cases of longitudinal fixation with conditions of different forms. The calculation of the scattering amplitude of the diffraction field is carried out for various frequencies and shell parameters.     

Some approximations in the linear dynamic equations of thin cylinders

Theoretical analysis is performed on the linear dynamic equations of thin cylindrical shells to find the error committed by making the Donnell assumption and the neglect of in-plane inertia. At first, the effect of these approximations is studied on a shell with classical simply supported boundary condition. The same approximations are then investigated for other boundary conditions from a consistent approximate solution of the eigenvalue problem. The Donnell assumption is valed at frequencies high compared with the ring frequencies, for finite length thin shells. The error in the eigenfrequencies from omitting tangential inertia is appreciable for modes with large circumferential and axial wave lengths, independent of shell thickness and boundary conditions.     

DYNAMIC BUCKLING OF A CYLINDRICAL SHELL WITH VARIABLE THICKNESS SUBJECT TO A TIME-DEPENDENT EXTERNAL PRESSURE VARYING AS A POWER FUNCTION OF TIME

In this study, the dynamic buckling of an elastic cylindrical shell with variable thickness, subject to a uniform external pressure which is a power function of time, has been considered. Initially, the fundamental relations and Donnell-type dynamic buckling equation of an elastic cylindrical shell with variable thickness have been obtained. Then, employing Galerkin's method, these equations have been reduced to a time-dependent differential equation with variable coefficients. Finally, applying a special Ritz-type method, the critical static and dynamic loads, the corresponding wave numbers and dynamic factor have been found analytically. Using those results, the effects of the variation of the thickness with a linear, a parabolic or an exponential function in the axial direction and the effect of the variation of the power of time in the external pressure expression are studied using pertinent computations. It is observed that these effects change appreciably the critical parameters of the problem. The present method has been verified, comparing the results of the present work and those of previous works in the literature, for a shell with constant thickness subject to a uniform external pressure varying linearly with time.     

Space harmonic analysis of sound radiation from a submerged periodic ring-stiffened cylindrical shell

An analytical method is developed to study radiated sound power characteristics from an infinite submerged periodically stiffened cylindrical shell excited by a radial cosine harmonic line force. The harmonic motion of the shell and the pressure field in the fluid are described by Flügge shell equations and Helmholtz equation, respectively. By using periodic theory of space harmonic analysis, the response of the periodic structure to harmonic excitations has been obtained by expanding it in terms of a series of space harmonics. Radiated sound power on the shell wall along the axial direction and the influence of different parameters on the results are studied, respectively. A conclusion is drawn that the stiffeners have a great influence at low and high frequencies while have a slight influence at intermediate frequencies for low circumferential mode orders. The work will give some guidelines for noise reduction of this kind of shell.     

Free localized vibrations of a semi-infinite cylindrical shell

Free vibrations of a semi-infinite cylindrical shell, localized near the edge of the shell are investigated. The dynamic equations in the Kirchhoff-Love theory of shells are subjected to asymptotic analysis. Three types of localized vibrations, associated with bending, extensional, and super-low-frequency semi-membrane motions, are determined. A link between localized vibrations and Rayleigh-type bending and extensional waves, propagating along the edge, is established. Different boundary conditions on the edge are considered. It is shown that for bending and super-low-frequency vibrations the natural frequencies are real while for extensional vibrations they have asymptotically small imaginary parts. The latter corresponds to the radiation to infinity caused by coupling between extensional and bending modes.     

ELASTIC WAVE SCATTERING AND DYNAMIC STRESS CONCENTRATIONS IN CYLINDRICAL SHELLS WITH A CIRCULAR CUTOUT

In this paper, based on the theory of elastic wave motion for open cylindrical shell, wave scattering and dynamic stress concentrations in open cylindrical shells with a hole are studied by making use of small parameter perturbation methods and boundary-integral equation techniques. The boundary-integral equations and iterative imminent series of scattered waves around the cavity of the cylindrical shell are derived. By employing this method, the approximately analytical solutions of scattered waves on the edge of cutout are gained. The computational formula for getting the dynamic stress concentration factors on the contour of cavity is developed. As an example, the numerical results of these dynamic stress concentration factors are graphically presented and discussed. The analytical methods put forward in the present work have practical significances for solving the problem of elastic wave scattering and dynamic stress concentrations in cylindrical shells with a circular cutout.     

Effects of strain and higher order inertia gradients on wave propagation in single-walled carbon nanotubes

Dispersion relation of single-walled carbon nanotubes (SWCNTs) is investigated. The governing equations of motion of SWCNTs are derived on the basis of the gradient shell model, which involves one strain gradient and one higher order inertia parameters in addition to two Lamé constants. The present shell model can predict wave dispersion in good agreement with those of molecular dynamic (MD) simulations available in the literature. The effects of two small scale parameters on the angular frequency and phase velocity in the longitudinal, torsional and radial directions are studied in detail. The numerical results show that the angular frequency and phase velocity increase with the increase of strain gradient parameter, whereas decrease with inertia gradient parameter increases. In addition, analytical expressions of the cut-off frequencies and asymptotic phase velocities are given. It is found that the number of cut-off frequencies is dependent on the circumferential wave number, and two asymptotic phase velocities exist for nonzero value of strain gradient parameter, while only one exists when the strain gradient parameter is excluded.     

TRANSIENT DYNAMIC RESPONSE ANALYSIS OF ORTHOTROPIC CIRCULAR CYLINDRICAL SHELL UNDER EXTERNAL HYDROSTATIC PRESSURE

Following Flügge's exact derivation for the buckling of cylindrical shells, the equations of motion for transient dynamic loading of orthotropic circular cylindrical shells under external hydrostatic pressure have been formulated. The normal mode theory is used to provide transient dynamic response for the equations of motion. The effect of shell's parameters, external hydrostatic pressure and material properties on the shell response has been studied in detail. A part of tables and figures are given in this paper.     

Circumferential thermoelastic waves in orthotropic cylindrical curved plates without energy dissipation

In this article, the propagation of guided thermoelastic waves in the circumferential direction of orthotropic cylindrical curved plates subjected to stress-free, isothermal boundary conditions is investigated in the context of the Green-Naghdi (GN) generalized thermoelastic theory (without energy dissipation). The coupled wave equations and heat conduction equation are solved by the Legendre orthogonal polynomial series expansion approach. The convergency of the method is discussed through a numerical example. The dispersion curves of thermal modes and elastic modes are illustrated simultaneously. Dispersion curves of the corresponding pure elastic cylindrical plate are also shown to analyze the influence of the thermoelasticity on elastic modes. The displacement, temperature and stress distributions are shown to discuss the differences between the elastic modes and thermal modes. A thermoelastic cylindrical plate with a different ratio of radius to thickness is considered to discuss the influence of the ratio on the characteristics of circumferential thermoelastic waves.     

Free vibration study of layered cylindrical shells by collocation with splines

The free vibration of circular cylindrical thin shells, made up of uniform layers of isotropic or specially orthotropic materials, is studied using point collocation method and employing spline function approximations. The equations of motion for the shell are derived by extending Love's first approximation theory. Assuming the solution in a separable form a system of coupled differential equations, in the longitudinal, circumferential and transverse displacement functions, is obtained. These functions are approximated by Bickley-type splines of suitable orders. The process of point collocation with suitable boundary conditions results in a generalized eigenvalue problem from which the values of a frequency parameter and the corresponding mode shapes of vibration, for specified values of the other parameters, are obtained. Two types of boundary conditions and four types of layers are considered. The effect of neglecting the coupling between the flexural and extensional displacements is analysed. The influences of the relative layer thickness, a length parameter and a total thickness parameter on the frequencies are studied. Both axisymmetric and asymmetric vibrations are investigated. The effect of the circumferential node number on the vibrational behaviour of the shell is also analysed.     

Free vibrations of cylindrical shells with elastic-support boundary conditions

This paper concerns the free vibrations of cylindrical shells with elastic boundary conditions. Based on the Flügge classical thin shell theory, the equations of motion for the cylindrical shells are solved by the method of wave propagations. The wave numbers are obtained by directly solving an eighth order equation. The elastic-support boundary conditions can be arbitrarily specified in terms of 8 independent sets of distributed springs. All the classical homogeneous boundary conditions can be considered as the special cases when the stiffness for each set of springs is equal to either infinity or zero. The present solutions are validated by the results previously given by other researchers and/or obtained using finite element models. The effects on the frequency parameters of elastic restraints are investigated for shells of different geometrical characteristics.     

Internal resonance of axially moving laminated circular cylindrical shells

The nonlinear vibrations of a thin, elastic, laminated composite circular cylindrical shell, moving in axial direction and having an internal resonance, are investigated in this study. Nonlinearities due to large-amplitude shell motion are considered by using Donnell’s nonlinear shallow-shell theory, with consideration of the effect of viscous structure damping. Differently from conventional Donnell’s nonlinear shallow-shell equations, an improved nonlinear model without employing Airy stress function is developed to study the nonlinear dynamics of thin shells. The system is discretized by Galerkin’s method while a model involving four degrees of freedom, allowing for the traveling wave response of the shell, is adopted. The method of harmonic balance is applied to study the nonlinear dynamic responses of the multi-degrees-of-freedom system. When the structure is excited close to a resonant frequency, very intricate frequency–response curves are obtained, which show strong modal interactions and one-to-one-to-one-to-one internal resonance phenomenon. The effects of different parameters on the complex dynamic response are investigated in this study. The stability of steady-state solutions is also analyzed in detail.     

Free vibrational characteristics of isotropic coupled cylindrical-conical shells

This paper presents the free vibrational characteristics of isotropic coupled conical-cylindrical shells. The equations of motion for the cylindrical and conical shells are solved using two different methods. A wave solution is used to describe the displacements of the cylindrical shell, while the displacements of the conical sections are solved using a power series solution. Both Donnell-Mushtari and Flügge equations of motion are used and the limitations associated with each thin shell theory are discussed. Natural frequencies are presented for different boundary conditions. The effect of the boundary conditions and the influence of the semi-vertex cone angle are described. The results from the theoretical model presented here are compared with those obtained by previous researchers and from a finite element model.     

Numerical stability analysis for the explicit high-order finite difference analysis of rotationally symmetric shells

A numerical stability analysis has been formulated to accompany the already developed explicit high-order finite difference analysis of rotationally symmetric shells subjected to time-dependent impulsive loadings. This already developed analysis utilizes a constant nodal point spacing for the spatial finite difference mesh, with the governing field differential equations formulated in terms of the transverse, meridional, and circumferential displacements as the fundamental variables. The remaining quantities which enter into the natural boundary conditions at each edge of the shell are incorporated into the complete system of equations by defining those quantities at each boundary in terms of the displacements. Surface loadings and inertia forces in each of the three displacement directions of the shell have been considered in the governing equations. Ordinary finite difference representations are used for the time derivatives. All loadings and dependent variables in the circumferential direction of the shell are expressed in Fourier series expansions. The complete system of equations is solved implicitly for the first time increment, while explicit relations are used to determine the three primary displacements within the boundary edges of the shell for the second and succeeding time increments. Separate implicit solutions at each boundary are then used to determine the remaining unspecified primary variables on and outside the boundaries. Subsequently, the remaining primary variables within the boundary edges of the shell and all secondary variables are determined explicitly. Numerical stability (or instability) of numerical solutions for given choices of spatial and time increments is determined by evaluation of the eigenvalues of the explicit coefficient matrix and comparing the maximum eigenvalue with the requirements of a stability criterion developed before by the author. Solutions for typical shells and loadings together with results of stability analyses have been included, and comparisons of the stability requirements and solutions with the requirements and solutions based upon ordinary spatial finite difference representations are included.     

An analytical quasi-exact method for calculating eigenvibrations of thin circular cylindrical shells

A method for calculating the eigenfrequencies and corresponding deformation modes of a thin circular cylindrical shell is presented, based on analytical solutions of Flügge's shell theory equations. The partial differential equations are transformed into algebraic equations which can be solved with high accuracy. Consequently, the results can be considered as quasi-exact. Results of calculations are presented for a shell stiffened at both circular edges. Such boundary conditions are typical of the core barrel of a pressurized water reactor, for instance. Most of the calculated deformation modes show strong gradients of the displacements close to the boundaries.     

DYNAMICS BEHAVIOR OF AXISYMMETRIC AND BEAM-LIKE ANISOTROPIC CYLINDRICAL SHELLS CONVEYING FLUID

The flow-induced vibration characteristics of anisotropic laminated cylindrical shells partially or completely filled with liquid or subjected to a flowing fluid are studied in this work for two cases of circumferential wave number, the axisymmetric, where n=0 and the beam-like, where n=1. The shear deformation effects are taken into account in this theory; therefore, the equations of motion are determined with displacements and transverse shear as independent variables. The present method is a combination of finite element analysis and refined shell theory in which the displacement functions are derived from the exact solution of refined shell equations based on orthogonal curvilinear co-ordinates. Mass and stiffness matrices are determined by precise analytical integration. A finite element is defined for the liquid in cases of potential flow that yields three forces (inertial, centrifugal and Coriolis) of moving fluid. The mass, stiffness and damping matrices due to the fluid effect are obtained by an analytical integration of the fluid pressure over the liquid element. The available solution based on Sanders' theory can also be obtained from the present theory in the limiting case of infinite stiffness in transverse shear. The natural frequencies of isotropic and anisotropic cylindrical shells that are empty, partially or completely filled with liquid as well as subjected to a flowing fluid, are given. When these results are compared with corresponding results obtained using existing theories, very good agreement is obtained.