Y. Xiang

Transverse vibration of thick rectangular plates. I: Comprehensive sets of boundary conditions

Abstract Recently, comprehensive research work has been carried out on the free vibration analysis of thick rectangular plates. Due to its practical importance, a vast array of research results will be presented in a series of four papers. These papers will present comprehensive sets of accurate vibration frequencies for thick rectangular plates subjected to: 21 boundary conditions involving all possible combinations of clamped, simply supported and free edges (part I); oblique internal line supports (part II); internal ring supports (part III) and in-plane isotropic pressure (part IV). In this study, sets of mathematically complete two-dimensional polynomials are assumed in the displacement and rotational functions to approximate the appropriate mode shapes. The energy function derived using Mindlins plate theory is minimized using the Rayleigh-Ritz procedure which leads to the governing eigenvalue equations. In this part (part I), sets of reasonably accurate vibration frequencies are presented for a wide range of aspect ratios a b and relative thickness ratios t b for each boundary condition. The present results, where possible, are compared with other existing established values.

Exact vibration solution for initially stressed Mindlin plates on Pasternak foundations

Closed-form solutions for the vibration problem of initially stressed thick rectangular plates as described by Mindlin theory are presented. The plates are simply supported and resting on Pasternak foundations. The subset problem of buckling of Mindlin plates on Pasternak foundations is automatically solved by setting the frequency parameter to be equal to zero. The solution also applies to Winkler foundations where the shear modulus of the Pasternak model is taken to be zero. The closed-form solutions should be useful for checking the accuracy of numerical solutions.

Buckling of rectangular mindlin plates with internal line supports

This paper considers the elastic buckling of rectangular Mindlin plates under normal inplane forces. The plates may have any number of internal line (straight or curved) supports that span between any two edges. Based on the energy functional derived from the incremental total potential energy approach, the newly developed pb-2 Rayleigh-Ritz method is applied for solution. The special feature of this method is the pb-2 Ritz function which consists of the product of (1) a two-dimensional polynomial function and (2) a basic function formed from taking the product of the equations of the boundaries and internal line supports; with each equation raised to appropriate powers as shown herein. The method is convenient for analysts since no discretization is required and very accurate results can be obtained. Buckling load intensity factors for rectangular plates with various boundary conditions and various thickness and aspect ratios are tabulated for designers. These may also serve as benchmark values for the testing of other numerical methods.

Buckling solutions for Mindlin plates of various shapes

In the open literature, there are many formulae, tables and design charts for use in estimating the buckling loads of thin plates. However, buckling solutions for thick plates, apart from those belonging to rectangular and axisymmetric plate shapes, are rather scarce. Prompted by the lack of results, this paper presents generic buckling solutions for isotropic inplane loaded Mindlin plates of regular polygonal, elliptical, semicircular and annular plates. These numerical solutions should help designers to save time rather than having to analyse the stability of moderately thick plates.

Exact buckling solutions for composite laminates: proper free edge conditions under in-plane loadings

SummaryThis paper considers the elastic buckling of symmetric cross-ply laminated rectangular plates with two parallel edges simply supported, one edge free and the remaining edge free, simply supported or clamped. The first-order shear deformation plate theory is used in the analysis. An error apparently made by previous researchers for boundary conditions at free edges subjected to in-plane loads is corrected. Closed-form buckling factors are obtained using a generalised Levy-type solution method to solve the differential equations which govern the buckling behaviour of the laminates. Comparisons are made with previously published results, and the differences between buckling factors obtained with the appropriate and inappropriate free edge conditions are examined. The variation of buckling factors with plate aspect ratio, thickness ratio and the number of layers is investigated. Sets of first-known buckling solutions for cross-ply laminates are reported in design charts and tables.

Formulation of Mindlin-Engesser model for stiffened plate vibration

In this paper, a Mindlin-Engesser model is developed for the vibration analysis of moderately thick plates with arbitrarily oriented stiffeners. The theoretical derivation incorporates the Mindlin theory to account for the effects of transverse shear deformation and rotary inertia of plates, and the Engesser theory to account for the shear deformation of stiffeners with the inclusion of torsion effect. In the method of solution, the resulting energy functionals are minimized using the Ritz procedure with a set of admissible two-dimensional functions expressed in the form of simple polynomials. The key kinematic feature of these shape functions is that they are boundary oriented and no boundary losses are introduced as in discretization methods. With the aim of demonstrating the applicability and versatility of the method, numerical examples including plates of various shapes with arbitrarily oriented stiffeners are presented. Several findings and conclusions regarding the method have been highlighted and discussed.

Flexural vibration of shear deformable circular and annular plates on ring supports

This paper presents an analysis of free flexural vibration of circular and annular Mindlin plates lying on multiple internal concentric ring supports. The Rayleigh-Ritz method with an admissible displacement function expressed in terms of a set of simple polynomials is used to obtain the governing eigenvalue equation. The natural frequencies of circular and annular Mindlin plates with one or two ring supports having inner and outer peripheries with different combinations of free, simply supported and clamped conditions are determined. A comprehensive set of vibration results is presented and, where possible, these have been verified with existing solutions published in the open literature. No previous theoretical results for thick annular plates supported with internal rings are known to exist. Thus the study should be useful and valuable to engineers and researchers.

Free vibration of isosceles triangular mindlin plates

The free flexural vibration of thick isosceles triangular plates based on the Mindlin shear deformation theory is investigated. The pb-2 Rayleigh-Ritz method, proposed earlier by the authors, is employed for solution. New sets of vibration frequency parameters for isosceles triangular Mindlin plates with various apex angles, thickness-to-width ratios and different combinations of free, simply supported and clamped edges are presented. These results are valuable to designers and may serve as the benchmark values for future numerical techniques and software packages in thick plate analyses. Since no results are available to date for such thick triangular plates, comparisons of results can only be made in the special case of small thickness-to-width ratios with existing thin plate results. In all cases, they are found to be in very close agreement. The influences of shear deformation and rotatory inertia on the frequency parameters are examined.

A global approach for vibration of thick trapezoidal plates

A global pb-2 Rayleigh-Ritz method is used to study the free vibration behaviour of trapezoidal Mindlin plates with different combinations of boundary conditions. The Mindlin theory is used to account for the effects of shear deformation and rotary inertia. The governing energy functional for the plate is derived and minimized to arrive at the eigenvalue equation. In the process, the shape functions are assumed in a set of two-dimensional mathematically complete polynomials which satisfy the geometric boundary conditions of the plate. This procedure has been implemented numerically to compute the vibration frequencies for trapezoidal Mindlin plates. Convergence and comparison studies have been carried out to verify the accuracy of this method. Sets of first known results have been presented for trapezoidal Mindlin plates of varying thickness, aspect ratio and boundary constraint. These new results enhance the database of vibration solutions.

Single-equation yield surfaces for monosymmetric and asymmetric sections

Single-equation yield surfaces are presented for channel, tee, single-and double-angle sections under combined axial force and biaxial moments. The equations give very good approximations to the accurate yield surfaces while maintaining smoothness, convexity and continuity requirements in three-dimensional space. The equations derived may be implemented in an elasto-plastic analysis program for modelling the nonlinear global behaviour of steel frame structures comprising one or more of these types of section.