Free vibration analysis of variable thickness thin and moderately thick plates with elastically restrained edges by DQM

A differential quadrature (DQ) procedure is developed for free vibration analysis of variable thickness moderately thick plates with edges elastically restrained against translation and rotation. The governing equations are based on the first order shear deformation theory and the rotary inertia effects are considered. Comparisons with known thin plate and uniform thickness Mindlin plate solutions are carried out to verify the applicability and accuracy of the analysis. Plates with linear or nonlinear varying thickness in one or two directions can be considered. It is demonstrated that using this DQ procedure, classical boundary conditions such as simply supported, clamped and free edges for variable thickness, thin as well as moderately thick plates can be simulated without any numerical difficulties. The effects of different combinations of constraints at edges, aspect ratio, thickness-to-length ratio, and stiffness parameter values on accuracy and convergence behaviors of the plates are presented. Some new results are presented for bi-linearly variable thickness plates with elastically restrained edges.     

Vibration of general triangular composite plates with elastically restrained edges

Triangular fibre reinforced composite plates are important structural elements in modern engineering structures. In this paper a computationally efficient and accurate numerical model is presented for the study of free vibration behaviour of anisotropic triangular plates with edges elastically restrained against rotation and translation. The approach developed is based on the Rayleigh–Ritz method and the use of non-orthogonal right triangular co-ordinates. The deflection of the plate is approximated by a set of beam characteristic orthogonal polynomials generated using the Gram–Schmidt procedure. Several examples are solved and some results which correspond to particular cases are compared with existing values in the literature. New results are also presented for single layer composite plates with different fibre orientations and combinations of boundary conditions. For some plates, mode shapes of free vibration are also shown. Selected new transverse vibration mode shapes are presented to illustrate the effects of boundary constraints, aspect ratio and fibre orientation.The method can be applied to a wide range of elastic restraint conditions, any aspect ratio and for higher modes. The effect of the fibre orientation on the natural frequencies for plates with these restraint conditions are also considered.     

Buckling of circular plates with an internal ring support and elastically restrained edges

Presented herein are the exact buckling solutions of circular plates with a concentric internal ring support and elastically restrained edges. This study shows the existence of buckling mode switching with respect to the radius of the internal ring support. Generally the plate buckles in an axisymmetric mode but when the ring support radius becomes small, the plate may buckle in an asymmetric mode. The cross-over ring support radius varies from 0.081 to 0.152 times the plate radius, depending on the rotational stiffness of the elastic restraint at the edges.     

Buckling strength of steel plating with elastically restrained edges

The twin aims of the present study are to investigate the buckling strength characteristics of steel plating elastically restrained at their edges and also to develop simple design formulations for buckling strength as a function of the torsional rigidity of support members that provide the rotational restraints along either one set of edges or all (four) edges. The characteristic equation for the buckling strength of steel plating that is elastically restrained along either long or short edges while the other edges are simply supported was derived by an analytical method. Using the computed results obtained by directly solving the buckling characteristic equation, closed-form expressions of the buckling strength of the plating with one set of edges elastically restrained while the other set of edges is simply supported are derived empirically by curve fitting. Based on the insights developed in the present study, approximate equations for the buckling strength for plating with all edges elastically restrained are proposed as a function of a relevant combination of the three simpler edge condition cases (i.e., long edges elastically restrained/short edges simply supported, long edges simply supported/short edges elastically restrained, and all edges simply supported). The effect of distortion of support members before the plating buckles is also approximately accounted for. The validity of the proposed closed-form buckling strength design formulations is studied by a comparison with theoretical and numerical solutions.     

A meshfree model without shear-locking for free vibration analysis of first-order shear deformable plates

Difficulty in imposing essential boundary conditions in the standard element-free Galerkin method (EFG) is due to the lack of Kronecker’s delta function property of shape functions generated by moving least square approximation (MLS). In this paper, we further apply a meshfree model based on the moving Kriging interpolation method (MK) to free vibration analysis of first-order shear deformable plates. The deflection and two rotation field variables of plate are approximated by the MK method, which is employed to construct the shape functions having the delta function property. With this approach, the drawback in enforcement of the boundary conditions caused by the MLS is now avoided. The present formulation is based on the first-order shear deformation plate theory (FSDT) associated with an effective elimination of the shear-locking phenomenon completely, and hence the approach is applicable to both moderately thick and thin plates. Numerical examples considering various aspect ratios and different boundaries are examined and solutions on natural frequencies obtained by the present method are then compared with existing reference solutions, and very good agreements are observed.     

Free vibration of quadrilateral laminated plates with carbon nanotube reinforced composite layers

The free vibration behavior of quadrilateral laminated thin-to-moderately thick plates with carbon nanotube reinforced composite (CNTRC) layers is studied. The governing equations are based on the first-order shear deformation theory (FSDT). The solution procedure is based on transforming the governing differential equations from an arbitrary straight-sided physical domain to a regular computational one, and discretization of the spatial derivatives by employing the differential quadrature method (DQM) as an efficient and accurate numerical tool. Four different profiles of single walled carbon nanotubes (SWCNTs) distribution through the thickness of layers are considered, which are uniformly distributed (UD) and three others are functionally graded (FG) distributions. The fast rate of convergence of the presented approach is numerically demonstrated and to show its high accuracy, wherever possible comparison studies with the available results in the open literature are performed. Then, the effects of volume fraction of carbon nanotubes (CNTs), geometrical shape parameters, thickness-to-length and aspect ratios, different kinds of CNTs distribution along the layers thickness and different boundary conditions on the natural frequencies of laminated plates are studied.     

Nonlinear vibration of thin plates with initial stress by spline finite strip method

The spline finite strip method and the incremental time–space finite element procedure are used to analyse large amplitude vibration of plates with initial stresses. Two improvements for the procedure are presented. The free vibration and the internal resonance of plates with initial stress as well as the forced vibration of plates with damping and initial stress are computed. The results compared favourably with those available in other publications.     

Nonlinear free vibration of skew nanoplates with surface and small scale effects

The nonlinear free flexural vibration of skew nanoplates is studied by considering the influences of free surface energy and size effect (small scale) simultaneously. The formulations are derived based on classical plate theory (CPT) in conjunction with nonlocal and surface elasticity theories using Hamilton's principle. Green's strain tensor together with von Kármán assumptions is employed to model the geometrical nonlinearity. The free surfaces are modeled as two-dimensional membranes adhering to the underlying bulk material without slipping. The solution algorithm is based on the transformation of the governing differential equation from the physical domain to a rectangular computational one, and discretization of the spatial derivatives by employing the differential quadrature method (DQM) as an efficient and accurate numerical tool. The effect of small scale parameter and surface effect together with the geometrical parameters and boundary conditions on the nonlinear frequency parameters of the skew nanoplates are studied.     

Free vibration analysis of circular thin plates with stepped thickness by the DSC element method

Novel formulation is presented by using the discrete singular convolution (DSC) for free vibration analysis of circular thin plates with uniform and stepped thickness. Different from the commonly used ones in literature, regularity conditions are not needed at the circular plate center point to avoid singularity. DSC circular and annular thin plate elements are established. For the DSC circular plate element with radius of R₁, the stiffness equation is first formulated in region [−R₁, R₁] with even number of nodes and then reduced to region [0, R₁] by using either symmetric or anti-symmetric conditions. The proposed DSC circular and annular plate elements are used for obtaining frequencies of uniform/stepped circular thin plates or annular thin plates with different boundary conditions. Comparison of the present DSC results to existing analytic and numerical solutions verifies the proposed formulations. The present research extends the DSC method to free vibration of circular thin plates with stepped thicknesses.     

Finite strip method for the free vibration and buckling analysis of plates with abrupt changes in thickness and complex support conditions

The free vibration problem of a stepped plate supported on non-homogeneous Winkler elastic foundation with elastically mounted masses is formulated based on Hamilton's principle. The stepped plate is modelled by finite strip method. To overcome the problem of excessive continuity of common beam vibration functions at the location of abrupt change of plate thickness, a set of C¹ continuous functions have been chosen as the longitudinal interpolation functions in the finite strip analysis. The C¹ continuous functions are obtained by augmenting the relevant beam vibration modes with piecewise cubic polynomials. As these displacement functions are built up from beam vibration modes with appropriate corrections, they possess both the advantages of fast convergence of harmonic functions as well as the appropriate order of continuity. The method is further extended to the buckling analysis of rectangular stepped plates. Numerical results also show that the method is versatile, efficient and accurate.