Differential quadrature method (DQM) for studying initial imperfection effects and pre- and post-buckling vibration of plates

The effects of initial geometric imperfection and pre- and post-buckling deformations on vibration of isotropic rectangular plates under uniaxial compressive in-plane load have been studied. The differential equations of plate motions, using the Mindlin theory and Von-Karman stress-strain relations for large deformations, were extracted. The solution of nonlinear differential equations was assumed as the summation of dynamic and static solutions. Due to a large static plate deflection as compared with its vibration amplitude, the differential equations were solved in two steps. First, the static equations were solved using the differential quadrature method and the arc-length strategy. Next, considering small vibration amplitude about the deformed shape and eliminating nonlinear terms, the natural frequencies were extracted using the differential quadrature method. The results for different initial geometric imperfection and different boundary conditions reflect the impact of the mentioned factors on the natural frequencies of plates

Buckling and Vibration of a Stepped Plate

This study analyses the elastic stability and free vibration of a simply supported stepped plate under combined loading conditions defined by the parameter α. Mathematical identification of these phenomena has been made using Levy\u27s method as implemented in the conditions of equivalent fictitious load. The buckling coefficient k and the frequency parameter λ of the stepped plate were verified according to literature sources. Influential parameters of stability and the free vibration of the stepped plate under combined load were identified. It has been concluded that the buckling coefficient primarily depends on the relative thickness Δt, while the frequency parameter λ was significantly affected by the position of discontinuity Δb. Pure bending (α=2) induces several buckling modes for the same plate geometry with respect to uniform compression (α=0), thus creating a considerable technological stability reserve, particularly at higher discontinuities. Formulation of the frequency parameter enables us to choose the optimum geometry with minimal susceptibility to the appearance of free vibration in the plate

Free transverse vibrations of orthotropic thin rectangular plates with arbitrary elastic edge supports

A so-called Spectro-Geometric Method (SGM) is presented for the free transverse vibration analysis of orthotropic thin rectangular plates with arbitrary elastic supports along each of its edges, a class of problems which are rarely attempted in the literature. Regardless of boundary conditions, the displacement function is invariably and simply expressed, in spectral form, as a trigonometric series expansion with an accelerated polynomial rate of convergence as compared with the conventional Fourier series. All the unknown expansion coefficients are treated as the generalized coordinates, and determined using the Rayleigh-Ritz technique. This work allows a capability of modeling a wide spectrum of orthotropic thin rectangular plate under a variety of boundary conditions, and changing the boundary conditions as easily as modifying the material properties or dimensions of the plates. The accuracy and reliability of the SGM prediction are demonstrated though numerical examples. The SGM prediction can be readily and directly extended to other more complicated boundary conditions involving non-uniform restraints, point supports, partial supports and their combinations

Static Response and Buckling Loads of Multilayered Composite Beams Using the Refined Zigzag Theory and Higher-Order Haar Wavelet Method

The paper presents a review of Haar wavelet methods and an application of the higher-order Haar wavelet method to study the behavior of multilayered composite beams under static and buckling loads. The Refined Zigzag Theory (RZT) is used to formulate the corresponding governing differential equations (equilibrium/stability equations and boundary conditions). To solve these equations numerically, the recently developed Higher-Order Haar Wavelet Method (HOHWM) is used. The results found are compared with those obtained by the widely used Haar Wavelet Method (HWM) and the Generalized Differential Quadrature Method (GDQM). The relative numerical performances of these numerical methods are assessed and validated by comparing them with exact analytical solutions. Furthermore, a detailed convergence study is conducted to analyze the convergence characteristics (absolute errors and the order of convergence) of the method presented. It is concluded that the HOHWM, when applied to RZT beam equilibrium equations in static and linear buckling problems, is capable of predicting, with a good accuracy, the unknown kinematic variables and their derivatives. The HOHWM is also found to be computationally competitive with the other numerical methods considered

THERMAL BUCKLING ANALYSIS OF FUNCTIONALLY GRADED CIRCULAR PLATE RESTING ON THE PASTERNAK ELASTIC FOUNDATION VIA THE DIFFERENTIAL TRANSFORM METHOD

In this paper, we propose a thermal buckling analysis of a functionally graded (FG) circular plate exhibiting polar orthotropic characteristics and resting on the Pasternak elastic foundation. The plate is assumed to be exposed to two kinds of thermal loads, namely, uniform temperature rise and linear temperature rise through thickness. The FG properties are assumed to vary continuously in the direction of thickness according to the simple power law model in terms of the volume fraction of two constituents. The governing equilibrium equations in buckling are based on the Von-Karman nonlinearity. To obtain the critical buckling temperature, we exploit a semi-numerical technique called differential transform method (DTM). This method provides fast accurate results and has a short computational calculation compared with the Taylor expansion method. Furthermore, some numerical examples are provided to consider the influence of various parameters such as volume fraction index, thickness-to-radius ratio, elastic foundation stiffness, modulus ratio of orthotropic materials and influence of boundary conditions. In order to predict the critical buckling temperature, it is observed that the critical temperature can be easily adjusted by appropriate variation of elastic foundation parameters and gradient index of FG material. Finally, the numerical results are compared with those available in the literature to confirm the accuracy and reliability of the DTM to determine the critical buckling temperature