International Journal of Structural Stability and Dynamics

Using Fourier series as shape functions practically eliminates the ill-conditioning problem associated with high-order polynomials in structural analyses as confirmed by the newly constructed condition number diagram. This paper extends the Fourier p-elements to study the dynamic stability of frame structures. Both conservative and follower forces are considered. The results compare very well to the exact method of dynamic stiffness. It is found that the Fourier p-elements outperform the dynamic stiffness method in terms of versatility in applications and numerical stability at the very low and high ends of the frequency spectrum. New results of follower tension are given. Follower tension buckling under uniformly distributed follower tension is originally reported.

In this paper, static and dynamic stabilities of a cantilever laminated composite beam, having a linear translation spring as elastic support whose position is changeable from the free end to midspan of the beam, subjected to periodic vertical end loading, are examined. The beam is assumed to be an Euler beam and the finite element model used can accommodate symmetric and antisymmetric lay-ups. Solutions referred to as combination resonance are investigated for the dynamic stability analysis. The effects of length-to-thickness ratio, the variation of cross-section in one direction, orientation angle, static and dynamic load parameters, stiffness and position of the elastic support on stability are examined.

This paper deals with the out-of-plane free vibrations of circular strips with linearly varying breadth. In deriving the differential equations for such strips, the effects of the rotatory and torsional inertias and shear deformation are considered. The differential equations are numerically solved to calculate the natural frequencies and mode shapes. In the numerical examples, three end constraints, i.e. clamped–clamped, clamped-hinged and hinged–hinged ends, are considered. The five lowest frequency parameters and mode shapes are presented. The effects of the rotatory and torsional inertias, and shear parameter on the natural frequencies are evaluated. Parametric studies are carried out for the influence of following parameters of the strip on the natural frequencies: subtended angle, section ratio, thickness ratio, and slenderness ratio. Also presented are the experimental validations of the seven lowest predicted natural frequencies. The natural frequencies obtained by this study agree well with those by the finite element method for both the flexural and torsional modes.

Accurate free vibration frequencies and mode shapes are presented for complete sets of moderately thick, cantilevered skew plates of triangular, trapezoidal and parallelogram shape. These accurate results are obtained by using the Ritz method applied to the Mindlin plate theory. Two sets of functions are employed simultaneously for each of the three dependent variables: transverse displacement (w) and bending rotations (ϕ_x and ϕ_y). One set is the widely used algebraic polynomials. The other is the set of corner functions which provide the proper stress singularities in the reentrant clamped-free corner, and accelerates the convergence of the solutions. The extensive frequencies presented are exact to the four digits shown. Corresponding mode shapes are also shown, by means of nodal patterns, most of which are novel in the published literature.

In this paper, the elastic lateral-torsional buckling of tapered cantilever strip beams acted by a tip load is investigated. Analytical solutions of the buckling problem are derived for cantilevers with a linearly varying depth. It is shown that the corresponding governing differential equation is a particular case of the general confluent equation, whose solution is given in terms of confluent hypergeometric functions. The buckling load corresponds to the lowest positive root of a 4×4 characteristic determinant arising from the introduction of the general solution of the differential equation into the boundary conditions. Finally, one discusses the optimization of the cantilever geometrical shape (for constant volume), as far as its resistance to lateral-torsional buckling is concerned.

An analytic method is presented to study the secondary instability (bifurcation) and mode jumping of thermomechanically deep buckled composite laminates. Unlike most ad hoc approaches, the governing partial differential equations (PDEs) and constitutive relations are rigorously derived from an asymptotically correct, geometrically nonlinear theory. A novel and simpler solution approach is developed to solve the two coupled fourth-order PDEs, namely, the compatibility equation and the dynamic governing equation. The generalized Galerkin method is used to solve boundary value problems corresponding to antisymmetric angle-ply and cross-ply composite plates. The variety of possible modal interactions is expressed in an explicit and concise form by transforming the coupled nonlinear governing equations into a system of nonlinear ordinary differential equations (ODEs). The comparison between the present method and the finite element analysis (FEA) shows a pretty good match in their numerical results in the primary postbuckling region. While the FEA may lose its convergence when solution comes close the secondary bifurcation point, the analytic approach has the capability of exploring deeply into the post-secondary buckling realm and capture the mode jumping phenomenon for various combinations of plate configurations and in-plane boundary conditions. Qualitatively different propagations of buckling pattern are observed before and after mode jumping. Free vibration along the stable primary postbuckling and the jumped equilibrium paths are also studied.

A variational formulation of the Ritz method is used to establish an eigenvalue problem for the local buckling behavior of composite plates elastically restrained (R) along their four edges (the RRRR plates) and subjected to biaxial compression, and the explicit solution in terms of the rotational restraint stiffness (k) is presented. Based on the different boundary and loading conditions, the explicit local buckling solution for the rotationally restrained plates is simplified to several special cases (e.g. the SSSS, SSCC, CCSS, CCCC, SSRR, RRSS, CCRR, and RRCC plates) under biaxial compression (and further reduced to uniaxial compression) with a combination of simply-supported (S), clamped (C), and/or restrained (R) edge conditions. The deformation shape function is presented by using the unique harmonic functions in both the axes to account for the effect of elastic rotational restraint stiffness (k) along the four edges of the orthotropic plate. A parametric study is conducted to evaluate the influences of the loading ratio (α), the rotational restraint stiffness (k), the aspect ratio (γ), and the flexural-orthotropy parameters (α_OR and β_OR) on the local buckling stress resultants of various rotationally restrained plates, and design plots with respect to these parameters are provided. The present explicit local buckling solution of the elastically restrained composite plates and the associated design plots can be employed to facilitate design analysis of composite structures (e.g. stiffened panels, thin-walled structures, and honeycomb cores).

The present work deals with the stability analysis of shear deformable functionally graded rectangular plates subjected to thermo-mechanical loads. The material properties of the functionally graded plates are assumed to vary continuously through the thickness, according to a simple power law distribution of the volume fraction of the constituents. An analytical approach based on fast converging Chebyshev polynomials is presented. The formulation is based on first-order shear deformation plate theory and von-Karman nonlinearity. The temperature-dependent thermal and mechanical properties of FGM plate are considered. The variations in the buckling load/temperature due to various parameters are studied. The postbuckling response of the clamped FGM plate, subjected to in-plane edge compressive mechanical loading (uniaxial, biaxial, shear), thermal (uniform in-plane temperature, transverse temperature gradient) and thermo-mechanical loading are presented. It is observed that volume fraction index greatly affects the buckling load/temperature of the plate.

Using Green's strains based on Sanders' approximation, the postbuckling behavior of cross-ply laminated composite conical and joined conical-cylindrical shells subjected to thermo-mechanical loads are studied employing the semi-analytical finite element method. The formulation is based on the first-order shear deformation theory and the field consistency principle. The Newton–Raphson iteration technique coupled with the displacement control method is used for the solution of nonlinear governing equations to trace the pre- and post-buckling equilibrium paths. A small magnitude load spatially proportional to the linear buckling mode shape is employed as asymmetric perturbation to initiate the bifurcation of the shell deformation. The influence of semi-cone angle and number of circumferential waves on prebuckling/postbuckling characteristics of cross-ply laminated conical and joined conical-cylindrical shells are investigated.