Analytical studies for buckling and vibration of weld-bonded beam shells of rectangular cross-section

An approach to analytical solution is presented for vibration and buckling of thin-walled tubular beam shells typical of automotive structures, which are fabricated by joining sheet metal stampings along the two longitudinal edges with periodic spot welds, adhesive bonding, or combination of spot welds and bonding, known as weld bonding. Solutions are obtained for such beam shells of rectangular cross-section with two opposite ends simply supported. The beam shell is modeled as an assembly of the constituent walls and Levy-type formulation is used to obtain a series solution for the transverse displacement of each of the walls. The challenge of expressing the discrete point support conditions at the spot welds by a continuous function is addressed using the flexibility function approach used in literature. The flexibility function, used earlier to represent the flexibility distribution along weld-bonded edges of rectangular plates with periodic spot welds, is used here. The characteristic equations are obtained by satisfying the displacement, slope, shear, and moment equilibrium at the mating edges of the walls including the two weld-bonded edges and the compatibility conditions at the spot-weld locations. This approach to analytical solution, described here for thin-walled beam shells of rectangular cross-section, can be suitably adopted for more general cross-sections and joints along non-symmetric edges. A parametric study is undertaken to show the effect of aspect ratio of the beam shell, adhesive joint parameters, and the number of spot welds on the elastic buckling loads and the natural frequencies. Such parametric studies can be of use to designers in arriving at an optimal joint configuration of weld-bonded beam shells from buckling and vibration considerations.     

The secondary buckling and post-secondary-buckling behaviours of rectangular plates

Secondary buckling and post-secondary-buckling behaviours are theoretically studied for simply-supported rectangular plates, whose primary buckling modes of deflection contain more than half-waves in the load acting direction. Modal coupling effects more complex than one two-term-coupling effects are incorporated into the secondary buckling and post-secondary-buckling analyses. Then, unstable or stable symmetric secondary branching points are found on the post-primary-buckling paths and “snap through” motions involving an abrupt change in wave-form are shown to be possible. Wave-form variations along post-secondary-buckling paths are also disclosed by means of a numerical analysis of equilibrium paths.     

Elastic buckling of plates with eccentrically positioned rectangular perforations

The conjugate load/displacement method is used to analyze square plates with perforations, and results are presented for geometries with rectangular holes offset from one or both axes of symmetry. Results are given for uniaxial and shear loading, and simply supported and fully fixed boundaries. Design guidance for the location of perforations and the effect of longitudinal or lateral slots is explained. The potential to increase the elastic critical load for an unperforated plate by the introduction of a small perforation is reported.     

Pseudospectral analysis of buckling and free vibration of rectangular Mindlin plates

A study of buckling and free vibration of rectangular Mindlin plates is presented. The analysis is based on the pseudospectral method, which uses basis functions that satisfy the boundary conditions. The equations of motion are collocated to yield a set of algebraic equations that are solved for the critical buckling load and for the natural frequencies in the presence of the in-plane loads. Numerical examples of rectangular plates with SS-C-SS-C boundary conditions are provided for various aspect ratios and thickness ratios, which show good agreement with those of the classical plate theory when the thickness ratio is very small. This paper was recommended for publication in revised form by Associate Editor Eung-Soo Shin Jinhee Lee received B.S. and M.S. degrees from Seoul National University and KAIST in 1982 and 1984, respectively. He received his Ph.D. degree from the University of Michigan, Ann Arbor in 1992 and joined the Dept. of Mechanical and Design Engineering of Hongik University in Choongnam, Korea. His research interests include inverse problems, pseudospectral method, vibration and dynamic systems.     

Elastoplastic buckling of rectangular plates in biaxial compression/tension

This paper examines the elastoplastic buckling of a rectangular plate, with various boundary conditions, under uniform compression combined with uniform tension (or compression) in the perpendicular direction. The analysis is based on the standard linear buckling equations and material behaviour is modelled by the small strain J₂ flow and deformation theories of plasticity. A detailed parametric study has been made for Al 7075 T6 over a range of plate geometries (a/b=0.25–4,a/h≈20–100) and with three sets of boundary conditions (four simply supported boundaries and the symmetric combinations of clamped/simply supported sides). For sufficiently thin plates we recover with both theories the classical elastic results. However, for thicker plates there is a remarkable difference in the buckling loads predicted by these two theories. Apart from the expected observation that deformation theory gives lower critical stresses than those obtained from the flow theory, we report on the existence of an optimal loading path for the deformation theory model. Buckling loads attained along the optimal path—specified by particular compression/tension ratios—are the highest possible over the entire space of loading histories. By contrast, no similar optimum has been found with the flow theory. This discrepancy in the buckling behaviour, obtained from the two competing plastic theories, provides a possibly new illustration of the plastic buckling paradox.     

Vibration analysis of rectangular plates with edge V-notches

A method is presented for accurately determining the natural frequencies of plates having V-notches along their edges. It is based on the Ritz method and utilizes two sets of admissible functions simultaneously, which are (1) algebraic polynomials from a mathematically complete set of functions, and (2) corner functions duplicating the boundary conditions along the edges of the notch, and describing the stress singularities at its sharp vertex exactly. The method is demonstrated for free, square plates with a single V-notch. The effects of corner functions on the convergence of solutions are shown through comprehensive convergence studies. The corner functions accelerate convergence of results significantly. Accurate numerical results for free vibration frequencies and nodal patterns are tabulated for V-notched square plates having notch angle α=5° or 30° at different locations and with various notch depths. These are the first known frequency and nodal pattern results available in the published literature for rectangular plates with V-notches.     

Thermal buckling analysis of cross-ply laminated rectangular plates under nonuniform temperature distribution: A differential quadrature approach

Differential quadrature method (DQM) is implemented for analyzing the thermal buckling behavior of the symmetric cross-ply laminated rectangular thin plates subjected to uniform and/or non-uniform temperature fields. The approach includes two steps: (1) solving the problem of in-plane thermo-elasticity to obtain the in-plane force resultants and (2) solving the buckling problem under the force distribution obtained in the previous step. Solution procedures are numerically performed by discretizing the governing differential equations and boundary conditions using DQM method. Applying the developed DQ formulation, the buckling loads are obtained for several sample plates. The numerical results compared well with those available in the literature as well as those obtained by ABAQUS. Parametric studies are conducted to investigate the influence of some important parameters including the plate aspect ratio, cross-ply ratio, and stiffness ratio on the critical temperature and mode shape of buckling.     

Thermal buckling of functionally graded rectangular plates subjected to partial heating

We present the thermal buckling analysis of functionally graded rectangular plates subjected to partial heating in a plane and uniform temperature rise through its thickness. The plate is simply supported for out-of-plane deformation and perfectly clamped for in-plane deformation. It is assumed that the functionally graded material properties such as the coefficient of linear thermal expansion and Young's modulus are changed individually in the thickness direction of the plate with the power law, while Poisson's ratio is assumed to be constant. Analytical developments consist of two stages. First, the nonuniform in-plane resultant forces are determined by solving a plane thermoelastic problem. Then the critical buckling temperatures of the plates with the predetermined resultant forces are calculated as the generalized eigenvalue problem which is constructed by using the Galerkin method. Finally, the effects of material inhomogeneity, aspect ratio, and heated region on the critical buckling temperatures are examined.     

Vibration characteristics and power transmission of coupled rectangular plates with elastic coupling edge and boundary restraints

Coupled-plate structures are widely used in the practical engineering such as aeronautical,civil and naval engineering etc.Limited works can be found on the vibration of the coupled-plate structure due...     

FEM buckling analysis of quasi-isotropic symmetrically laminated rectangular composite plates with a square/rectangular cutout

A numerical study was conducted using the finite element method to determine the effects of square and rectangular cutouts on the buckling behavior of a 16-ply quasi-isotropic graphite/epoxy symmetrically laminated rectangular composite plate. The square/ rectangular cutouts were subjected to uniaxial compression loading. This study addresses the effects of the size of the square/rectangular cutout, orientation of the square/rectangular cutout, plate aspect ratio (a/b), and plate length/thickness ratio (a/t) on the buckling behavior of the symmetrically laminated rectangular composite plate under uniaxial compression loading. Buckling loads were computed for seven different quasi-isotropic laminate configurations [0°/+45°/?45°/90°]_2s, [15°/+60°/?30°/?75°]_2s, [30°/+75°/?15°/?60°]_2s, [45°/+90°/0°/?45°]_2s, [60°/?75°/+15°/?30°]_2s, [75°/?60°/+30°/?15°]_2s, [90°/?45°/+45/°0°]_2s. Results showed that the magnitudes of the buckling loads decrease with increasing cutout positioned angle as well as c/b and d/b ratios for plates with a rectangular cutout. The symmetrically laminated quasi-isotropic [0°/+45°/?45°/90°]_2s composite plate is stronger than all other symmetrically analyzed laminated quasi-isotropic composite plates. The magnitudes of the buckling loads of a rectangular composite plate with square/rectangular cutout decrease with increasing plate aspect ratio (a/b) and plate length/thickness (a/t) ratio.     

Vibration of rectangular Mindlin plates resting on non-homogenous elastic foundations

This paper is concerned with the vibration behaviour of rectangular Mindlin plates resting on non-homogenous elastic foundations. A rectangular plate is assumed to rest on a non-homogenous elastic foundation that consists of multi-segment Winkler-type elastic foundations. Two parallel edges of the plate are assumed to be simply supported and the two remaining edges may have any combinations of free, simply supported or clamped conditions. The plate is first divided into subdomains along the interfaces of the multi-segment foundations. The Levy solution approach associated with the state space technique is employed to derive the analytical solutions for each subdomain. The domain decomposition method is used to cater for the continuity and equilibrium conditions at the interfaces of the subdomains. First-known exact solutions for vibration of rectangular Mindlin plates on a non-homogenous elastic foundation are obtained. The vibration of square Mindlin plates partially resting on an elastic foundation is studied in detail. The influence of the foundation stiffness parameter, the foundation length ratio and the plate thickness ratio on the frequency parameters of square Mindlin plates is discussed. The exact vibration solutions presented in this paper may be used as benchmarks for researchers to check their numerical methods for such a plate vibration problem. The results are also important for engineers to design plates supported by multi-segment elastic foundations.     

Vibration and buckling of generally laminated composite plates with arbitrary edge conditions

A method is developed for the analysis of free vibration and buckling of generally laminated composite plates having arbitrary edge conditions, such as clamped, simply supported or free. The procedure is an extension of the Ritz method utilizing a strain energy functional containing both bending and stretching effects and accommodating arbitrary ply stacking sequences. Displacement functions are taken in the form of polynomials, and an algorithm for satisfying the geometric boundary conditions is presented. Numerical results are compared with those of other researchers in order to establish the correctness and effectiveness of the method. Some additional new results are also presented.     

Vibration analysis of rectangular Mindlin plates with internal line supports using static Timoshenko beam functions

In this paper, the vibration characteristics of rectangular Mindlin plates with internal line supports in one or two directions are studied by using the Rayleigh–Ritz method. The static Timoshenko beam functions are employed as admissible functions, which are composed of a set of transverse deflection functions and a set of rotation-angle functions due to bending. The static Timoshenko beam functions are derived from a point-supported strip of unit width taken from the particular plate under consideration in a direction parallel to the edge of the plate and acted upon by a series of static sinusoidal loads distributed along the length of the strip. It can be seen that the suggested approach is very simple mathematically, and each of the beam functions is only a sine or cosine function plus a polynomial function of not more than the third order. A unified program can be easily prepared, because the changes in boundary conditions, number and locations of internal line supports and thickness ratio of the plate will result in a corresponding change of only the coefficients of the polynomials. Both high accuracy and low computational cost have been verified by convergence and comparison studies. In addition, it can be seen that the admissible functions presented in this paper can also properly describe the discontinuity conditions of the shear forces at the line supports. Therefore, accurate results can be expected for the analysis of dynamic response and internal force distribution of the plate.     

Thermal buckling and dynamic characteristics of composite plates under pressure load

Journal of Mechanical Science and Technology - The effect of geometrical nonlinearities due to pressure load on the thermal buckling and dynamic characteristics of composite plates are investigated...     

Accurate buckling loads of thin rectangular plates under parabolic edge compressions by the differential quadrature method

The buckling of thin rectangular plates with nonlinearly distributed loadings along two opposite plate edges is analyzed by using the differential quadrature (DQ) method. The problem is considerably more complicated since it requires that first the plane elasticity problem be solved to obtain the distribution of in-plane stresses, and then the buckling problem be solved. Thus, very few analytical solutions (the only one available in the literature is for rectangular plates with all edges simply supported) have been available in the literature thus far. Detailed formulations and solution procedures are given herein. Nine combinations of boundary conditions and various aspect ratios are considered. Comparisons are made with a few existing analytical and/or finite element data. It has been found that a fast convergent rate can be achieved by the DQ method with non-uniform grids and very accurate results are obtained for the first time. It has also been found that the DQ results, verified by the finite element method with NASTRAN, are not quite close to the newly reported analytical solution. A possible reason is given to explain the difference.     

Three-dimensional vibration, buckling and bending analyses of thick rectangular plates based on discrete singular convolution method

The main objective of this study is to give a numerical solution of three-dimensional analysis of thick rectangular plates. The analysis uses discrete singular convolution (DSC) method. Free vibration, bending and buckling of rectangular plates have been studied in this paper. Regularized Shannon's delta (RSD) kernel is selected as singular convolution to illustrate the present algorithm. In the proposed approach, the derivatives in both the governing equations and the boundary conditions are discretized by the method of DSC. The obtanied results are compared with those of other numerical methods. It is found that the convergence of the DSC approach is very good and the results agree well with those obtained by other researchers.     

Bending and buckling of thick symmetric rectangular laminates using the moving least-squares differential quadrature method

In this paper, the moving least-squares differential quadrature (MLSDQ) method is applied to the bending and buckling analyses of moderately thick symmetric laminates based on the first-order shear deformation theory (FSDT). The transverse deflection and two rotations of the laminate are independently assumed with the moving least-squares (MLS) approximation. The weighting coefficients used in the MLSDQ approximation are obtained through a fast computation of the MLS shape functions and their partial derivatives. Numerical examples are illustrated to study the accuracy, stability and convergence of the MLSDQ method. The typical displacements, stresses and critical buckling loads of various laminated plates are presented and compared with the analytical values. Effects of support size, order of the complete basis functions and node irregularity on the numerical accuracy are investigated.     

Unsteady flow between rectangular plates and between circular plates

An analysis is presented of the unsteady flow of a viscous incompressible fluid between two parallel infinitely long rectangular plates and between two parallel circular plates. The lower plate is fixed and the upper plate moves towards the lower plate. Full Navier-Stokes equations are used to obtain the pressure distribution as a function of the film thickness h(t) and the velocity $\text{h}\text{?}\text{(t)}$ of the upper plate. The inertia is taken into account and, for a given load on the upper plate, the sinkage relation between h and t is determined for various values of the Reynolds number. The departure from the classical inertialess solution is exhibited for various values of the two dimensionless parameters involved, one characterizing the load and the other gravity.     

The elastic buckling of circular plates

How to compute upper and lower bound buckling loads based on potential energy and complementary potential energy methods is well known. The present paper is an extension of the method presented in Ref. (1), to deal with statically indeterminate problems of infinitely many degrees. The method presented here utilizes the Rayleigh quotient and an improved upper bound termed the Timoshenko quotient to obtain a lower bound.     

Analytical solutions for buckling of rectangular plates under non-uniform biaxial compression or uniaxial compression with in-plane lateral restraint

The objective of this work is to examine the effect of a non-uniform distribution of the applied edge loads on their net critical value with reference to the title problem. This forms a sequel to an earlier work on uniaxial compression of plates without any in-plane restraint at lateral edges to suppress the Poisson effect. A rigorous superposition approach is employed for plane stress analysis of the loaded plate, and the resulting non-uniform in-plane stress field is fully accounted for in the subsequent stability analysis which is based on Galerkin's approach with a complete set of admissible functions. Pertinent results are presented to highlight the influence of various non-uniform distributions as well as an in-plane restraint at the lateral edges.