Torsional buckling of cross-ply laminated orthotropic composite cylindrical shells subject to dynamic loading

This study considers torsional buckling of cross-ply laminated orthotropic composite cylindrical thin shells under loads, which is a power function of time. The modified Donnell type dynamic stability and compatibility equations are obtained first. These equations are subsequently reduced to a time dependent differential equation with variable coefficients by using Galerkin's method. The critical parameters are found analytically by applying the Ritz type variational method. According to theoretical solutions, numerical analyses are done.     

Nonlinear dynamic buckling of symmetrically laminated cylindrically orthotropic shallow spherical shells

Summary In this paper, a model of cusped catastrophe at nonlinear dynamic buckling of a symmetrically laminated cylindrically orthotropic shallow spherical shell is presented. The shell is subjected to an axisymmetrical load. Effects of transverse shear are taken into account. Effects of the shear modulus, geometry and parameters of the material on the nonlinear dynamic buckling are discussed. Received 2 April 1997; accepted for publication 27 November 1997     

An analysis of the post-buckling of laminated plates of symmetric cross-ply

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Creep buckling of cross-ply symmetric laminated cylindrical panels

A creep buckling analysis of cross-ply symmetric laminated cylindrical panels is given in this paper.By means of theoretical analysis,a method to determine the critical load of creep buckling of the panels with simply supported boundary conditons is obtained.     

Influence of rotatory inertia on dynamic stability of the viscoelastic symmetric cross-ply laminated plates

The dynamic stability problem of the symmetrically laminated cross-ply plates made of the viscoelastic Voigt–Kelvin material, compressed by time-dependent stochastic membrane forces, is investigated. The effect of rotatory inertia is included in the present formulation. It is assumed that all elastic moduli have the same retardation times. By using the direct Liapunov method, bounds of the almost sure stability of cross-ply plates as a function of viscous damping coefficient, retardation time, variances of the stochastic forces, ratio of the principal lamina stiffnesses, number of layers, plate aspect ratio, cross-ply ratio and intensity of the deterministic components of axial loading are obtained. Numerical calculations are performed for the Gaussian process with a zero mean, as well as an harmonic process with random phase.     

Non-linear axisymmetric static analysis of orthotropic thin annular plates

This investigation deals with the geometrically non-linear axisymmetric static analysis of elastic orthotropic thin annular plates subjected to uniformly distributed and ring loads. Non-linear differential equations in terms of the transverse displacement w and stress function ψ have been employed. Both w and ψ are expanded in finite power series and the orthogonal point collocation method has been used to obtain discretised algebraic equations from the governing differential equations. Results have been presented for annular plates with and without a rigid plug in the hole; simply supported, clamped and free outer edge conditions; three orthotropic parameters β; two annular ratios and two loading conditions. Results have also been presented illustrating the effect of an elastic rotational constraint at the edge and the effect of a prescribed inplane displacement of the edge.     

Dynamic analysis of laminated cross-ply composite non-circular thick cylindrical shells using higher-order theory

Here, the dynamic analysis of laminated cross-ply composite non-circular thick cylindrical shells subjected to thermal/mechanical load is carried out based on higher-order theory. The formulation accounts for the variation of the in-plane and transverse displacements through the thickness, abrupt discontinuity in slope of the in-plane displacements at the interfaces, and includes in-plane, rotary inertia terms, and also the inertia contributions due to the coupling between the different order displacement terms. The strain–displacement relations are accurately accounted for in the formulation. The shell responses are obtained employing finite element approach in conjunction with direct time integration technique. A detailed parametric study is carried out to bring out the effects of length and thickness ratios, eccentricity parameters and number of layers on the thermal/mechanical response characteristics of non-circular shells.     

On the vibration of laminated nonhomogeneous orthotropic shells

In this paper, an analytical procedure is given to study the free vibration of the laminated homogeneous and non-homogeneous orthotropic conical shells with freely supported edges. The basic relations, the modified Donnell type motion and compatibility equations have been derived for laminated orthotropic truncated conical shells with variable Young’s moduli and densities in the thickness direction of the layers. By applying the Galerkin method, to the basic equations, the expressions for the dimensionless frequency parameter of the laminated homogeneous and non-homogeneous orthotropic truncated conical shells are obtained. The appropriate formulas for the single-layer and laminated complete conical and cylindrical shells made of homogeneous and non-homogeneous, orthotropic and isotropic materials are found as a special case. Finally, the influences of the non-homogeneity, the number and ordering of layers and the variations of the conical shell characteristics on the dimensionless frequency parameter are investigated. The results obtained for homogeneous cases are compared with their counterparts in the literature.     

Thermoelastoplastic axisymmetric stress-strain state of laminated orthotropic shells

A method is proposed for the numerical analysis of the thermoelastoplastic stress-strain state of laminated shells of revolution, made of isotropic and orthotropic materials, under axisymmetric loading. The method is based on the Kirchhoff-Love hypotheses for a layer stack, the theory of deformation along paths of small curvature for isotropic materials, and Hills theory of flow with isotropic hardening for orthotropic materials. The problem is solved by the method of successive approximations. A numerical example is given.__________Translated from Prikladnaya Mekhanika, Vol. 40, No. 12, pp. 84–91, December 2004.     

Nonlinear bending of simply supported symmetric laminated cross-ply rectangular plates

Based on the von Kármán-type theory of plates,nonlinear bending problems of simplysupported symmetric laminated cross-ply rectangular plates under the combined action ofpressure and inplane load are investigated in this paper.The solution which satisfies thegoverning equations and boundary conditions is obtained by using the double Fourier seriesmethod.     

Thermoelastic equilibrium of laminated,orthotropic shells with temperature-dependent characteristics

Kiev Highway Institute. Translated from Prikladnaya Mekhanika, Vol. 26, No. 9, pp. 36–43, September, 1990.     

Non-linear studies of orthotropic shallow spherical shells on elastic foundation

Governing non-linear integro-differential equations for cylindrically orthotropic shallow spherical shells resting on linear Winkler-Pasternak elastic foundations, undergoing moderately large deformations are presented. Three problems, namely, non-linear static deflection response, non-linear dynamic deflection response and dynamic snap-through buckling of orthotropic shells have been investigated. The influences of material orthotropy, foundation parameters and shell-material damping on the deflection response are determined for the clamped and the simply- supported immovable edge conditions accurately. Orthotropy, foundation interaction and material damping play significant roles in improving the load carrying capacity of the shell structures.     

Nonlinear thermo-elastic buckling characteristics of cross-ply laminated joined conical–cylindrical shells

Here, the nonlinear thermo-elastic buckling/post-buckling characteristics of laminated circular conical–cylindrical/conical–cylindrical–conical joined shells subjected to uniform temperature rise are studied employing semi-analytical finite element approach. The nonlinear governing equations, considering geometric nonlinearity based on von Karman’s assumption for moderately large deformation, are solved using Newton–Raphson iteration procedure coupled with displacement control method to trace the pre-buckling/post-buckling equilibrium path. The presence of asymmetric perturbation in the form of small magnitude load spatially proportional to the linear buckling mode shape is assumed to initiate the bifurcation of the shell deformation. The study is carried out to highlight the influences of semi-cone angle, material properties and number of circumferential waves on the nonlinear thermo-elastic response of the different joined shell systems.     

Vibrations of structurally orthotropic laminated shells under thermal power loading

On the basis of the linearized version of equations obtained in a geometrically nonlinear statement and describing the nonaxisymmetric strain of nonshallow sandwich structure orthotropic shells under thermal power loading, the Rayleigh–Ritz method with polynomial approximation of displacements and shear strains is used to solve the problem of small free vibrations of axisymmetrically thermally preloaded freely supported three-layer conical shell. The causes of dynamical fracture of the shell under study are revealed.     

Theoretical and experimental studies on nonlinear oscillations of symmetric cross-ply composite laminated plates

The nonlinear oscillations and resonant responses of the symmetric cross-ply composite laminated plates are investigated theoretically and experimentally. The governing equations of motion for the composite laminated plate are derived by using the von Karman type equation, Reddy’s third-order shear deformation plate theory, and Galerkin method with the geometric nonlinearity. The four-dimensional averaged equation is obtained by using the method of multiple scales. The frequency-response functions are analyzed under the consideration of strongly coupled of two modes. The influences of the resonance case on the softening and hardening type of nonlinearity are analyzed with different parameters for the composite laminated plates. The numerical results indicate that there exist the hardening and softening types of the composite laminated plate in the specific resonant case. The variation of the response amplitudes is studied for the composite laminated plate under combined the transverse and in-plane excitations. A sweep frequency experiment is performed to obtain the hardening and softening nonlinearities of a composite laminated plate. The experimental results coincide with the numerical results qualitatively. The influences of the excitation amplitudes on the softening and hardening types of nonlinearity are also analyzed for the composite laminated plate. The amplitude spectrums of the test plate also demonstrate that the change of the nonlinear dynamic responses may be caused by the subharmonic resonance.     

Non-linear modal equations for thin elastic shells

In this paper we derive non-linear modal equations for thin elastic shells of arbitrary geometry. Geometric non-linearities are accounted for by utilizing the strain-displacement relations of the Sanders-Koiter non-linear shell theory. Arbitrary initial imperfections are accounted for and the shell thickness is free to vary within the limits of thin shell theory. The derivation gives the coefficients of the modal equations as integral expressions over the surface of the shell. The resulting equations are well-suited for practical applications. Weighting factors are introduced to allow for reduction of our results to the Love shell theory and to the Donnell approximation. The equations are specialized for a finite simply supported circular cylinder and numerical results are compared to those previously published in the literature.     

Non-linear static and dynamic response of spherical shells

In the present investigation an analytical technique using Chebyshev series expansion has been presented and used to study the non-linear transient behavior of shallow spherical shells with and without damping. The two coupled non-linear differential equation governing the shallow shell behavior are initially linearised using Taylor's series expansion. Results have been presented for three types of transient loadings, namely, step function, N-shaped pulse and sine wave pulse. It is shown that accurate results can be obtained using a five terms Chebyshev series expansion which is unlikely in the conventional methods.     

An exact symplectic solution for the dynamic analysis of shear deformable antisymmetric angle-ply laminated plates

From the mixed variational principle, by the selection of the state variables and its dual variables, the Hamiltonian canonical equation for the dynamic analysis of shear deformable antisymmetric angle-ply laminated plates is derived, leading to the mathematical frame of symplectic geometry and algorithms, and the exact solution for the arbitrary boundary conditions is also derived by the adjoint orthonormalized symplectic expansion method. Numerical results are presented with the emphasis on the effects of length/thickness ratio, arbitrary boundary conditions, degrees of anisotropy, number of layers, ply-angles and the corrected coefficients of transverse shear.