Buckling characteristics and layup optimization of long laminated composite cylindrical shells subjected to combined loads using lamination parameters

The buckling characteristics and layup optimization of long laminated composite cylindrical shells subjected to combined loads of axial compression and torsion are examined on the basis of Flügge’s theory. In the buckling analysis of long laminated composite cylindrical shells, 12 lamination parameters are introduced and used as design variables for layup optimization. Applying a variational approach, the feasible region in the design space of the 12 lamination parameters is numerically obtained. The buckling characteristics are discussed in the design space of the 12 lamination parameters. In the layup optimization, the optimum lamination parameters for maximizing the buckling loads and the laminate configurations for realizing the optimum lamination parameters are determined by mathematical programming methods. It is found that in case of combined loads of axial compression and torsion, the optimum laminate configurations are unsymmetric.     

Large deformation analysis of moderately thick laminated plates on nonlinear elastic foundations by DQM

In this paper, the nonlinear behavior of symmetric and antisymmetric cross ply, thin to moderately thick, elastic rectangular laminated plates resting on nonlinear elastic foundations are studied using differential quadrature method (DQM). The first-order shear deformation theory (FSDT) in conjunction with the Green’s strain and von Karman hypothesis are assumed for modeling the nonlinear behavior. Elastic foundation is modeled as shear deformable with cubic nonlinearity. The differential quadrature (DQ) discretized form of the governing equations with the various types of boundary conditions are derived. The Newton–Raphson iterative scheme is employed to solve the resulting system of nonlinear algebraic equations. Comparisons are made and the convergence studies are performed to show the accuracy of the results even with a few number of grid points. The effects of thickness-to-length ratio, aspect ratio, number of plies, fiber orientation and staking sequence on the nonlinear behavior of cross ply laminated plates with different boundary conditions resting on elastic foundations are studied.     

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.     

Influences of elastic foundations and shear deformations on the buckling behavior of functionally graded material truncated conical shells under axial compression

This article presents an investigation on the buckling of functionally graded (FG) truncated conical shells under an axial load resting on elastic foundations within the shear deformation theory (SDT). The governing equations are solved using the Galerkin method, and the closed-form solution of the axial buckling load for FG conical shells on elastic foundations within the SDT is obtained. Various numerical examples are presented and discussed to verify the accuracy of the closed-form solution in predicting dimensionless buckling loads for FG conical shells on the Winkler–Pasternak elastic foundations within the SDT.     

Analysis of laminated conical shell structures using higher order models

In this paper is presented a numerical method for the structural analysis of laminated conical shell panels using a quadrilateral isoparametric finite element based on the higher order shear deformation theory. The displacement expressions used for the longitudinal and circumferential components of the displacement field are given by power series of the transversal coordinate and the condition of zero stresses in the top and bottom surfaces of the shell is imposed. The shape functions used for the transversal displacement are C¹ conforming and the finite element is a conical/cylindrical panel with 8 nodes and 40 degrees of freedom. The model presented performs static analysis with arbitrary boundary conditions and loads, as well eigenvalue problems (free vibration and buckling). Illustrative examples are presented and discussed.     

Non-linear buckling and postbuckling of shear deformable anisotropic laminated cylindrical shell subjected to varying external pressure loads

Non-linear buckling and postbuckling of a moderately thick anisotropic laminated cylindrical shell of finite length subjected to lateral pressure, hydrostatic pressure and external liquid pressure has been presented in the paper. The material of each layer of the shell is assumed to be linearly elastic, anisotropic and fiber-reinforced. The governing equations are based on a higher order shear deformation shell theory with von Kármán–Donnell-type of kinematic non-linearity and including the extension/twist, extension/flexural and flexural/twist couplings. The non-linear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A singular perturbation technique is employed to determine the buckling pressure and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of perfect and imperfect, moderately thick, anisotropic laminated cylindrical shells with different values of shell parameters and stacking sequence. The results confirm that there exists a circumferential stress along with an associate shear stress when the shell is subjected to external pressure.     

Effects of boundary conditions on buckling and postbuckling responses of composite laminate with various shaped cutouts

The objective of the present work is to study the effects of flexural boundary conditions on buckling and postbuckling behavior of axially compressed quasi-isotropic laminate, (+45/−45/0/90)_2s with various shaped cutouts (i.e., circular, square, diamond, elliptical–vertical and elliptical–horizontal) of various sizes using the finite element method. The FEM formulation is based on first order shear deformation theory and von Karman’s assumptions are used to incorporate geometric nonlinearity. The 3-D Tsai-Hill criterion is used to predict the failure of a lamina while the onset of delamination is predicted by the interlaminar failure criterion. It is observed that the laminates clamped and simply supported on all edges have the highest and lowest buckling and postbuckling strength, respectively, irrespective of cutouts shape and size. It is also noted that a fully clamped laminate with large size elliptical–vertical cutout can take higher compressive buckling load than the laminate without cutout for same boundary condition.     

Analysis of critical buckling load of laminated composites plate with different boundary conditions using FEM and analytical methods

In this study, the critical buckling load of fiber reinforced composite plate was calculated by analytical and finite element methods. The critical buckling loads and composite deformations were obtained on the basis of plate dimensions ratio (L_x/L_y). The deformation behaviour of the plate is shown for modes i = 1,2 and different values of orientation angles [0/θ]₂. As a result of the analysis, for the composite plates with symmetric and antisymmetric layup the critical buckling loads were obtained for different support conditions. It was observed that difference between results increased depending on θ angle, although critical buckling load values obtained by using numerical and finite element methods were nearly same in [0/θ]₂ orientation angles.     

Stochastic post buckling analysis of laminated composite cylindrical shell panel subjected to hygrothermomechanical loading

In this paper, the effect of random system properties on the post buckling load of geometrically nonlinear laminated composite cylindrical shell panel subjected to hygrothermomechanical loading is investigated. System parameters are assumed as independent random variables. The higher order shear deformation theory and von-Karman nonlinear kinematics are used for basic formulation. The elastic and hygrothermal properties of the composite material are considered to be dependent on temperature and moisture concentration using micromechanical approach. A direct iterative based C⁰ nonlinear finite element method in conjunction with first-order perturbation technique proposed by present author for the plate is extended for shell panel subjected to hygrothermomechanical loading to compute the second-order statistics (mean and variances) of laminated composite cylindrical shell panel. The effect of random system properties, plate geometry, stacking sequences, support conditions, fiber volume fractions and temperature and moisture distributions on hygrothermomechanical post-buckling load of the laminated cylindrical shell panel are presented. The performance of outlined stochastic approach has been validated by comparing the present results with those available in the literature and independent Monte Carlo simulation.     

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 linear stability of eccentrically stiffened functionally graded annular spherical shell on elastic foundations

The study deals with the formulation of governing equations of eccentrically stiffened functionally graded materials annular spherical shells resting on elastic foundations and based upon the classical shell theory and the smeared stiffeners technique taking into account geometrical nonlinearity in Von Karman-Donnell sense. The annular spherical shells are reinforced by eccentrically longitudinal and transversal stiffeners made of full metal or full ceramic depending on situation of stiffeners at metal-rich side or ceramic-rich side of the shell respectively. Approximate solutions are assumed to satisfy the simply supported boundary condition and Galerkin method is applied to obtain closed-form relations of bifurcation type of buckling loads. Numerical results are given to evaluate effects of inhomogeneous, dimensional parameters, outside stiffeners and elastic foundations to the buckling of structures.     

Elliptically delaminated buckling near the surface of piezoelectric laminated shells under electric and thermal loads

This paper reports the result of an investigation into the local buckling for the delimitation of an elliptic shape near the surface of piezoelectric laminated shells, where local delaminated sub-shells may be monolayer and multiplayer, and the geometrical axes of elliptically delaminated sub-shells are inconsistent with the geometrical main axes of piezoelectric laminated base-shells. From example calculations, the effects of the geometrical and physical parameters, stacking models of piezoelectric laminated base-shells, elliptically laminated sub-shells, and coupled electric and thermal loads on the local buckling for elliptically delaminated sub-shells near the surface of piezoelectric laminated base-shells are described and discussed.     

On the stability of FGM shells subjected to combined loads with different edge conditions and resting on elastic foundations

In this study, the stability analysis of functionally graded material (FGM) cylindrical, truncated and complete conical shells subjected to combined loads and resting on elastic foundations for two boundary conditions is investigated. The functionally graded material properties are assumed to vary continuously through the thickness of the conical shell. At first, the basic relations, the stability and compatibility equations of the FGM truncated conical shell on the Pasternak-type elastic foundation are obtained. By applying the Galerkin method to the foregoing equations, the critical combined loads of clamped–clamped and sliding–sliding FGM shells on the Pasternak-type elastic foundation are obtained. Finally, carrying out some computations, effects of the elastic foundation, boundary conditions, the variation of shell characteristics and material composition profiles on the values of critical combined loads have been studied.     

Closed-form analysis of the buckling loads of symmetrically laminated orthotropic plates considering elastic edge restraints

The initial buckling loads of symmetrically laminated rectangular orthotropic plates under uniaxial compression are determined in a closed-form analytical manner. The considered laminates are simply supported at all edges and furthermore subjected to elastic rotational restraints at the unloaded edges. The analysis approach which is based on the assumptions of Classical Laminate Plate Theory employs a representation for the transverse displacements consisting of a system of trigonometric functions in both inplane coordinate directions. Due to its simplicity, a closed-form solution for the critical buckling load is possible which is useful for preliminary design purposes. The presented approach is compared to closed-form analytical and numerical data available in the literature and is found to be in favorable agreement.     

Vibrations and buckling of composite orthotropic cylindrical shells with nonuniform axial loads

The vibrational response of orthotropic composite cylindrical shells, subjected to circumferentially nonuniform axial loads, is investigated based on Flügge-type field equations. The use of a complex finite Fourier transform provides a simple method for handling any arbitrary nonuniform load but introduces modal coupling between the transformed equations. For simply supported boundaries (conditions SS3) the determination of the critical buckling load reduces to finding the eigenvalues of a finite matrix. Two different nonuniform loads are considered, having forms proportional to (1+2cos θ) and (θ*−θ), where is the Heaviside function, θ is the circumferential coordinate and aθ* is the width of an axial strip of the shell of radius a. Computed results indicate the sensitivity of the critical buckling loads and free vibrational frequency to the type of nonuniform load and the material lay-ups of the cylinders.     

Nonlinear bending and postbuckling of FGM cylindrical panels subjected to combined loadings and resting on elastic foundations in thermal environments

A nonlinear analysis is presented for FGM cylindrical panels resting on elastic foundations subjected to the combined actions of uniform lateral pressure and compressive edge loads in thermal environments. The two cases of postbuckling of initially pressurized FGM cylindrical panels and of nonlinear bending of initially compressed cylindrical panels are considered. Heat conduction and temperature-dependent material properties are both taken into account. Material properties of functionally graded materials (FGMs) are assumed to be graded in the thickness direction based on Mori-Tanaka micromechanics model. The formulations are based on a higher order shear deformation theory and von Kármán strain displacement relationships. The panel-foundation interaction and thermal effects are also included. The governing equations are solved by a singular perturbation technique along with a two-step perturbation approach. The numerical illustrations concern the postbuckling behavior and the nonlinear bending response of FGM cylindrical panels with two constituent materials resting on Pasternak elastic foundations. The effects of volume fraction index, temperature variation, foundation stiffness as well as initial stress on the postbuckling behavior and the nonlinear bending response of FGM cylindrical panels are discussed in detail.     

Similitude invariants and scaling laws for buckling experiments on anti-symmetrically laminated plates subjected to biaxial loading

Establishing the similitude between the model and prototype rigorously is a necessary step in designing an experiment efficiently. So far, to the best of the authors’ knowledge, no one has ever derived the similitude invariant for anti-symmetric cross- and angle-ply laminated plates subjected to biaxial loading before. This research paper is the first to establish the similitude invariant of anti-symmetric cross- and angle-ply laminated plates by applying the similitude transformation to the governing differential equations of buckling directly. Then the scaling laws for buckling loads of laminated plates subjected to biaxial loads are derived. But in reality, either due to the complexity of the scaling laws or to economize on costly experiments, it may not be feasible to construct the model conforming to the scaling laws completely, therefore partial similitude is investigated theoretically and approximate scaling laws are recommended. The buckling loads of the prototype predicted from the scaling laws are then compared with the available theoretical values. The complete similitude cases show exact agreement between results predicted from the scaling laws and the available analytical solutions. For partial similitude cases, the models distorted in stacking sequences, number of plies, and material properties are studied and the approximate scaling laws which yield good agreement are recommended.     

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.     

Nonlinear thermo-mechanical response of eccentrically stiffened Sigmoid FGM circular cylindrical shells subjected to compressive and uniform radial loads using the Reddy's third-order shear deformation shell theory

Based on Reddy's third-order shear deformation shell theory, this paper presents an analytical approach to investigate the nonlinear thermo-mechanical response of imperfect Sigmoid FGM circular cylindrical shells surrounded on elastic foundations and reinforced by outside metal stiffeners. The eccentrically stiffened S-FGM shells are subjected to axial compressive load and uniform radial load in thermal environment. Using the stress function, Bubnov–Galerkin method, the paper proposes the formula for forces and moments taking into account the thermal stress in both the shells and stiffeners. The obtained results are validated by comparing with other results reported in the literature.     

A semi-analytical finite element model for the analysis of laminated 3D axisymmetric shells: Bending, free vibration and buckling

A general semi-analytical finite element model is developed for bending, free vibration and buckling analysis of shells of revolution made of laminated orthotropic elastic material. The 3D elasticity theory is used and the equations of motion are obtained by expanding the displacement field and load in the Fourier series in terms of the circumferential coordinate, θ. The coefficients of the expansion are functions of (r, z), and they are approximated using the finite element method. This leads to a semi-analytical finite element in the (r, z) plane. The element is validated by comparing the present results with the analytical and numerical solutions available in the literature.