The buckling of cross-ply laminated non-homogeneous orthotropic composite conical thin shells under a dynamic external pressure

Summary.  The subject of this investigation is to study the buckling of cross-ply laminated orthotropic truncated circular conical thin shells with variable Young's moduli and densities in the thickness direction, subjected to a uniform external pressure which is a power function of time. After obtaining the dynamic stability and compatibility equations we reduce both of them to a time dependent ordinary differential equation with variable coefficient by using Galerkin's method. The critical dynamic and static loading, the corresponding wave numbers, the dynamic factors, critical time and critical impulse are found analytically by applying the Ritz type variational method. The dynamic behavior of cross-ply laminated truncated conical shells is investigated with: (a) lamina that present variations in the Young's moduli and densities, (b) different numbers and ordering of layers, (c) variable semi-vertex angles, and (d) external pressures which vary with different powers of time. It is concluded that all these factors contribute to appreciable effects on the critical parameters of the problem in question. Received June 13, 2002; revised December 10, 2002 Published online: May 8, 2003 This research project was initiated with the support of DAAD when Dr. Sofiyev was in Germany for a research project. The project was successfully accomplished with generous support of TUBITAK (The Scientific and Technical Research Council of Turkey).     

Vibrations of angle-ply laminated circular cylindrical shells subjected to different sets of edge boundary conditions

This paper studies the free vibrations of finite, closed, circular cylindrical shells, made of one or more monoclinic layers. The study is based on the Love-type version of a unified shear-deformable shell theory. This theory enables the trial and testing of different through-thickness transverse shear-strain distributions and, among them, strain distributions that do not involve the undesirable implications of the transverse-shear correction factors. For flexural vibrations, the analytical solution of the corresponding axisymmetric solution is obtained, as a particular case, when it is assumed that the free-vibration pattern is independent of the circumferential co-ordinate parameter. If the appropriate material simplifications are employed, the present analysis yields, as a further particular case, the corresponding free-vibration solution that has already been presented elsewhere for cross-ply laminated cylindrical shells.     

On the transient response of cross-ply laminated circular cylindrical shells

Transient response of simply-supported circular cylindrical shells is investigated using a higher-order shear deformation theory (HSDT). The theory is a modification of the Sanders' shell theory and accounts for parabolic distribution of the transverse shear strains through thickness of the shell and tangential stress-free boundary conditions on the bounding surfaces of the shell. The results obtained using the classical shell theory (CST) and the first-order shear deformation theory (FSDT) are compared with those obtained using the higher-order theory. The state-space approach is used to develop the analytical solutions to the equations of motion of the three theories.     

基于波动法的静水压力下环肋圆柱壳耦合振动特性研究

研究了静水压力下环肋圆柱壳的振动特性。在Flügge理论的基础上,考虑流体的影响,通过变换轴向波数,先后采用波动法和牛顿迭代法得到了不同边界条件下环肋圆柱壳的固有频率值。经过与已有文献数据进行对比,验证了研究的有效性和正确性。通过算例,分析了静水压力、肋条截面尺寸和数目、边界条件等因素对水下环肋圆柱壳固有频率的影响。     

Optimization of clamped plastic shallow shells subjected to initial impulsive loading

An optimization technique is suggested for shallow spherical shells made of a ductile material and subjected to initial impact loading. The shell under consideration is pierced with a central hole and clamped at the outer edge. The optimal design of the shell of piece-wise constant thickness is established under the condition that the maximal residual deflection attains its minimal value for given total weight. The material of the shell is assumed to obey the Tresca yield condition and associated flow law. By the use of the method of mode form motions the problem is transformed into a particular problem of non-linear programming and solved numerically.     

The buckling of FGM truncated conical shells subjected to combined axial tension and hydrostatic pressure

The purpose of this paper is to investigate the elastic buckling of FGM truncated thin conical shells under combined axial tension and hydrostatic pressure. Here axial tensions are separately applied to small and large bases of the truncated conical shell, respectively. It is assumed that the cone is a mixture of metal and ceramic, and that its properties changes as the power and exponential functions of the shell thickness. After giving the fundamental relations, the stability and compatibility equations of an FGM truncated conical shell, subject to combined axial tension and hydrostatic pressure, have been derived. Applying Galerkin’s method general formulas have been obtained for the critical combined and separate loads of FGM conical shells. The appropriate formulas for homogenous and FGM cylindrical shells are found as a special case. Effects of changing shell characteristics, material composition and volume fraction of constituent materials on the critical combined and separate loads of FGM shells with simply supported edges are also investigated. The results obtained for homogeneous cases are compared with their counterparts in the literature.     

On the buckling and vibration of antisymmetric angle-ply laminated circular cylindrical shells

The buckling and free vibration problem of a thin composite antisymmetric angle-ply laminated circular cylindrical shell is studied. The Donnell-type equations of motion, in terms of the shell middle surface displacement components, are used and solved approximately by means of Galerkin's method. Numerical results are presented and the expectation that the bending-stretching coupling phenomenon rapidly dies out as the number of layers increases is confirmed.     

Damage prediction of clamped circular plates subjected to contact underwater explosion

Contact explosion damage prediction of circular steel plates is of interest to the naval architects and warship designers. During contact explosion, a fraction of the explosive energy goes for plastic deformation and fracture of the plate. Experiments were conducted on air backed circular high strength low alloy (HSLA) steel plates to establish this fraction. There was a large amount of plastic deformation preceding fracture in the contact exploded plates. The deformation contour was found to be spherical showing maximum absorption of energy for the depth of bulge attained. The radius of penetration of the test plate was obtained by applying the terminal strain to fracture and the non-uniform strain distribution in the target plate. A case history is presented where the current predictions are compared with the existing empirical methodology and experimental data.     

On the solution of generalized non-linear complex-symmetric eigenvalue problems

This paper brings an attempt toward the systematic solution of the generalized non-linear, complex-symmetric eigenproblem ( K ₀−iω C ₁−ω² M ₁−iω³ C ₂−ω⁴ M ₂−···) ϕ = 0 , with real, symmetric matrices K ₀, C _j, M _j ε R^n×n, which are associated with the dynamic governing equations of a structure submitted to viscous damping, as laid out in the frame of an advanced mode superposition technique. The problem can be restated as ( K _(ω)−ω M _(ω)) ϕ = 0 , where K _(ω)= K and M _(ω)= M are complex-symmetric matrices given as power series of the complex eigenfrequencies ω, such that, if (ω, ϕ ) is a solution eigenpair, ϕ ^T M _(ω) ϕ =1 and ϕ ^T K _(ω) ϕ =ω. The traditional Rayleigh quotient iteration and the more recent Jacobi–Davidson method are outlined for complex-symmetric linear problems and shown to be mathematically equivalent, both with asymptotically cubic convergence. The Jacobi–Davidson method is more robust and adequate for the solution of a set of eigenpairs. The non-linear eigenproblem subject of this paper can be dealt with in the exact frame of the linear analysis, thus also presenting cubic convergence. Two examples help us to visualize some of the basic concepts developed. Three more examples illustrate the applicability of the proposed algorithm to solve non-linear problems, in the general case of underdamping, but also for overdamping combined with multiple and close eigenvalues. Copyright © 2007 John Wiley & Sons, Ltd.     

Springback of circular clamped armor steel plates subjected to spherical air-blast loading

The development of protective elements (plates) that are impulsively loaded requires comprehensive experiments in which the transient dynamic and residual deflections are measured. Whereas the residual plastic deflections are easily measured upon completion of the experiment, transient deflection measurements involve more elaborate setups. Relying on the residual deflection only can be misleading when one tries to assess the peak transient deflection which poses a real hazard to the protected subject. In this study, the peak transient and the residual deflections are compared for a clamped circular armor steel plate subjected to large close-range spherical air-blast loading. The difference between the maximal transient and the residual plastic deflection is addressed here as the elastic springback.     

Nonlinear dynamic buckling of orthotropic cylindrical shells subjected to rapidly applied loads

Dynamic buckling of an orthotropic cylindrical shell which is subjected to rapidly applied compression is considered. A nonlinear differential equation of Donnell–Karman type is derived with the initial imperfection taken into account. An energy method is used to obtain the equation of motion which is then solved numerically by means of a Runge-Kutta method. These numerical results show that the critical load is increased over the corresponding static case. An analytical solution is also obtained for the problem of hydrostatic pressure.     

Dynamic response of cross-ply laminated circular cylindrical shells with various boundary conditions

Summary Closed-form solutions of the dynamic response of cross-ply laminated circular cylindrical shells are developed for arbitrary boundary conditions and under arbitrary loadings. The equations of motion of the classical, first-order and third-order theories are converted into a single-order system of equations by using state variables. To solve for the dynamic response, the biorthogonality conditions of principle modes of the original and adjoint eigenfunctions are used to decouple the state space equation. The study reveals that the disagreement between shear deformation theories in much less than the disagreement between them and the classical theory.     

Symmetricability of pressure stiffness matrices for shells with loaded free edges

For the case of free edges which are loaded, follower forces remaining normal to the middle surface of a shell throughout the deformation history do not have a load potential. In finite element analysis, this results in an unsymmetric pressure stiffness matrix. Depending on the structure of the available computer program, implementation of an equation solver permitting solution of unsymmetric simultaneous systems of algebraic equations may be a tedious task. This explains the significance of the topic of symmetricability of pressure stiffness matrices, turning out to be of special importance in the case of static buckling under the assumption of a linear prebuckling path. At first, incremental equations for tracing the nonlinear load–displacement path are derived. Thereafter, the buckling condition is deduced. Then, it is demonstrated that symmetrization of the pressure stiffness matrix is admissible if the so-obtained ‘buckling pressure’ differs ‘very little’ from the ‘buckling pressure’ resulting from an alternative symmetric ‘buckling matrix’, as is shown to be the case for a simply supported cylindrical shell with a free upper edge, subjected to hydrostatic external pressure. The alternative symmetric ‘buckling matrix’ is a consequence of deleting the virtual work term, causing the unsymmetry of the pressure stiffness matrix, in the expression for the external virtual work. A mechanical interpretation of this virtual work term is given. It is shown to be equal to the difference of virtual work of the original pressure load and of a ‘substitute pressure-field’, of the form of a Fourier series of the former. This explains why, normally, the buckling coefficient resulting from the ‘substitute pressure-field’ represents a good approximation to the buckling coefficient stemming from the original pressure load.     

Frequency analysis of rotating truncated circular orthotropic conical shells with different boundary conditions

In this paper, a study is made of the influences of orthotropic properties and boundary conditions on the free vibrations of a rotating, truncated, circular orthotropic conical shell. The study includes the effects of the Coriolis and centrifugal accelerations and the initial hoop tension. It is based on the Love first-approximation theory and Galerkin procedure. Results are obtained for the frequency characteristics of different orthotropic parameters, rotating velocities, cone angles and boundary conditions. Influences of orthotropic properties and boundary conditions on the relationships between frequency parameter and rotation velocity are discussed for different cone angles. To validate the present analysis, comparisons are made with those available in the open literature and very good agreements are obtained.     

The effect of non-homogeneity on the non-linear buckling behavior of laminated orthotropic conical shells

In this work, the buckling behavior of the cross-ply laminated non-homogeneous orthotropic truncated conical shells in the large deformation under the uniform axial load is studied. Firstly, the basic relations of the cross-ply laminated non-homogeneous orthotropic truncated conical shells are derived using the large deformation theory. Then modified Donnell type non-linear stability and compatibility equations are obtained and solved. A computer program called Maple 14 has been used in the numerical solution. Finally, the influences of the degree of non-homogeneity, the number and ordering of layers and the variations of the conical shell characteristics on the non-linear axial buckling load are investigated. The comparison with available results is satisfactorily good.     

Dynamic stability of stiffened laminated composite plates and shells subjected to in-plane pulsating forces

The dynamic stability of laminated composite stiffened or non-stiffened plates and shells due to periodic in-plane forces at boundaries is investigated in this paper. A three-dimensional (3-D) degenerated shell element and a 3-D degenerated curved beam element are used to model plates/shells and stiffeners, respectively. The characteristic equations to find the natural frequencies, buckling loads and their corresponding mode shapes are obtained from the finite element equation of motion. Then, the method of Hill's infinite determinants or the method of multiple scales is applied to analyse the dynamic instability regions. Numerical results are presented to demonstrate the effects of various parameters, such as skew angle, lamination scheme, stiffened scheme, in-plane force type and curvature of cylindrical shell, on the dynamic stability of stiffened and non-stiffened plates and shells subjected to in-plane pulsating forces at boundaries.     

Viscoelastic damped response of cross-ply laminated shallow spherical shells subjected to various impulsive loads

In this paper, the viscoelastic damped response of cross-ply laminated shallow spherical shells is investigated numerically in a transformed Laplace space. In the proposed approach, the governing differential equations of cross-ply laminated shallow spherical shell are derived using the dynamic version of the principle of virtual displacements. Following this, the Laplace transform is employed in the transient analysis of viscoelastic laminated shell problem. Also, damping can be incorporated with ease in the transformed domain. The transformed time-independent equations in spatial coordinate are solved numerically by Gauss elimination. Numerical inverse transformation of the results into the real domain are operated by the modified Durbin transform method. Verification of the presented method is carried out by comparing the results with those obtained by the Newmark method and ANSYS finite element software. Furthermore, the developed solution approach is applied to problems with several impulsive loads. The novelty of the present study lies in the fact that a combination of the Navier method and Laplace transform is employed in the analysis of cross-ply laminated shallow spherical viscoelastic shells. The numerical sample results have proved that the presented method constitutes a highly accurate and efficient solution, which can be easily applied to the laminated viscoelastic shell problems.     

Dynamic stability analysis of laminated composite cylindrical shells subjected to conservative periodic axial loads

The dynamic stability of thin, laminated cylindrical shells under combined static and periodic axial forces is studied here using Love's theory for thin shells. A system of Mathieu–Hill equations is obtained by a normal-mode expansion of the equations of motion, the stability of which is examined by Bolotin's method. The dynamic instability regions are investigated for different lamination schemes. The effects of the length-to-radius and thickness-to-radius ratios of the cylinder on the instability regions are also examined.     

On buckling of orthotropic compressed plates with circular holes

Buckling and post-buckling behaviour of square plates having central circular holes is analysed with the aid of the finite element method. Four different degrees of orthotropy pertinent to real materials are examined. Both uniaxial and biaxial compression are considered for various types of boundary conditions. Also, buckling of polar orthotropic annular plates is analysed for different boundary conditions and material properties. In the case of small holes a perturbation method is applied to both kinds of plates.     

Dynamic response of pre-stressed fibre metal laminate (FML) circular cylindrical shells subjected to lateral pressure pulse loads

Dynamic response of fiber metal laminate cylindrical shells subjected to initial combined axial load and internal pressure were studied in this paper. First order shear deformation theory (FSDT) was utilized in the shell’s equilibrium equations and strain–displacement relations were based on Love’s first approximation theory. Equilibrium equations for buckling, free and forced vibration problems of the shell were solved using Galerkin method. The influences of FML parameters such as material properties lay up, Metal Volume Fraction (MVF), fiber orientation and initial stresses on dynamic response were investigated. The results were indicated that the FML lay up, has a significant effect on natural frequencies as well as transient dynamic response with respect to various values of MVF as well as pre-stress.