Classical Thin Plate Theory Research Articles

Hamiltonian systems of propagation of elastic waves and localized vibrations in the strip plate

In this paper, based on Lagrange–Germanian theory of elastic thin plates, applying the method in Hamiltonian state space, the elastic waves and vibrations when the boundary of the two lateral sides of the strip plate are free of traction are investigated, and the process of analysis and solution are proposed. The existence of all kinds of vibration modes and wave propagation modes is also analyzed. By using eigenfunction expansion method, the dispersion relations of waveguide modes in the strip plate are derived, and the comparisons with the dispersion relations directly obtained by the classical theory of thin plates are also presented. At last, the results are analyzed and discussed.

Dynamic response of a flexible plate on saturated soil layer

An analytical approach is developed to study the dynamic response of a flexible plate on single-layered saturated soil. The analysis is based on Biot's two-phased theory of poroelasticity and also on the classical thin-plate theory. First, the governing differential equations for saturated soil are solved by the use of Hankel transform. The general solutions of the skeleton displacements, stresses, and pore pressures, derived in the transformed domain, are subsequently incorporated into the imposed boundary conditions, which leads to a set of dual integral equations describing the corresponding mixed boundary value problem. These governing integral equations are finally reduced to the Fredholm integral equations of the second kind and solved by standard numerical procedures. The accuracy of the present solution is validated via comparisons with existing solutions for an ideal elastic half-space. Furthermore, some numerical results are presented to show the influences of the layer depth, the plate flexibility, and the soil porosity on the dynamic compliances.

Bending Solutions for Inhomogeneous and Laminated Elastic Plates

A procedure has been developed in previous papers for constructing exact solutions of the equations of linear elasticity in a thick plate of inhomogeneous isotropic linearly elastic material in which the elastic moduli depend in any specified manner on a coordinate normal to the plane of the plate. The essential idea is that any solution of the classical thin plate or classical laminate theory equations (which are two-dimensional theories) generates, by straightforward substitutions, a solution of the three-dimensional elasticity equations for the homogeneous material. Recently this theory has been formulated in terms of functions of a complex variable. It was shown that the displacement and stress fields in the inhomogeneous material could be expressed in terms of four complex potentials that are analytic functions of the complex variable ζ = x + iy in the mid-plane of the plate. However, the analysis performed so far applies only to the case of a plate with traction-free upper and lower faces. The present paper extends these solutions to the case where the plate is bent by a pressure distribution applied to a face.

Complex Variable Solutions for Inhomogeneous and Laminated Elastic Plates

A procedure has been developed in previous papers for constructing exact solutions of the equations of linear elasticity in a plate (not necessarily thin) of inhomogeneous isotropic linearly elastic material in which the elastic moduli depend in any specified manner on a coordinate normal to the plane of the plate. The essential idea is that any solution of the classical thin plate or classical laminate theory equations (which are two-dimensional theories) generates, by straightforward substitutions, a solution of the three-dimensional elasticity equations for the inhomogeneous material. In this paper we formulate this theory in terms of complex potentials. It is shown that the displacement and stress in the inhomogeneous material can be expressed in terms of four complex potentials that are analytic functions of ζ = x + iy. Expressions for stress and moment resultants are derived in terms of these complex potentials. As examples we solve the problems of biaxial stress in an infinite plate and of an in-plane point force in an infinite plate and at the boundary of a semiinfinite plate. We also formulate the general boundary-value problem for a half-plane and develop a procedure to determine the complex potentials for an inhomogeneous plate in terms of the complex potentials for the corresponding plane-strain problem.

Vibration of shear deformable plates with variable thickness — first-order and higher-order analyses

This work presents accurate numerical calculations of the natural frequencies for elastic rectangular plates of variable thickness with various combinations of boundary conditions. The thickness variation in one or two directions of the plate is taken in polynomial form. The first-order shear deformation plate theory of Mindlin and the higher-order shear deformation plate theory of Reddy have been applied to the plate analysis. The governing equations and the boundary conditions are derived using the dynamic version of the principle of minimum of the Lagrangian function. The solution is obtained by the extended Kantorovich method. This approach is combined with the exact element method for the vibration analysis of members with variable flexural rigidity, which provides for the derivation of the exact dynamic stiffness matrix of varying cross-sections strips. The large number of numerical examples demonstrates the applicability and versatility of the present method. The results obtained by both shear deformation theories are compared with those obtained by the classical thin plate theory and with published results.

Buckling of Circular Mindlin Plates with an Internal Ring Support and Elastically Restrained Edge

This paper is concerned with the elastic buckling problem of circular Mindlin plates with a concentric internal ring support and elastically restrained edge. In solving this problem analytically, the circular plate is first divided into an annular segment and a core circular segment at the location of the internal ring support. Based on the Mindlin plate theory, the governing differential equations for the annular and circular segments are then solved exactly and the solutions brought together by using the interfacial conditions. New exact critical buckling loads of circular Mindlin plate with an internal ring support and elastically restrained edge are presented for the first time. The optimal radius of the internal ring support for maximum buckling load is also found. An approximate relationship between the buckling loads of such circular plates based on the classical thin plate theory and the Mindlin plate theory is also explored.

Natural frequency analysis of a sandwich panel with soft core based on a refined shear deformation model

The natural frequency of a thick rectangular sandwich panel composed of orthotropic facesheets and a soft core was studied based on a refined shear deformation model. The shear deformation of the sandwich panel was described by a polynomial function. The effect of transverse shear modulus of the facesheets and core on flexural vibration of the panel was investigated. Comparison was made among classical thin plate theory, linear shear (low order) deformation theory and the refined shear (high order) deformation model. Results from finite element analysis were also provided to verify the theoretical predictions. It was shown that the refined shear deformation model provided a better prediction on the natural frequency of vibration of a sandwich panel than thin plate model or low order deformation model.

Bending Solutions of Sectorial Thick Plates Based on Reissner Plate Theory

ABSTRACT This paper is concerned with the bending of sectorial and annular sectorial thick plates with the radial edges simply supported. The Reissner plate theory has been adopted to cater for the effect of transverse shear deformation. In solving the aforementioned plate problems, we derive the relationships between the bending solutions of the Reissner plate theory and the classical thin plate theory. The relationships enable one to deduce the bending solutions of the Reissner plate theory using the corresponding classical thin plate results, which can be determined readily or are already available in the literature. Several sectorial plate problems are solved, and the Reissner plate results are checked against existing numerical results as well as results obtained using other higher-order plate theories.

Vibration Analysis of Rectangular Plates with One or More Guided Edges via Bicubic B-Spline Method

A simple and accurate method is proposed for the vibration analysis of rectangular plates with one or more guided edges, in which bicubic B-spline interpolation in combination with a new type of basis cubic B-spline functions is used to approximate the plate deflection. This type of basis cubic B-spline functions can satisfy simply supported, clamped, free, and guided edge conditions with easy numerical manipulation. The frequency characteristic equation is formulated based on classical thin plate theory by performing Hamilton's principle. The present solutions are verified with the analytical ones. Fast convergence, high accuracy and computational efficiency have been demonstrated from the comparisons. Frequency parameters for 13 cases of rectangular plates with at least one guided edge, which are possible by approximate or numerical methods only, are presented. These results are new in literature.

Stability of variable thickness shear deformable plates—first order and high order analyses

This work presents analysis of the buckling loads for thick elastic rectangular plates with variable thickness and various combinations of boundary conditions. Both the first order and high order shear deformation plate theories have been applied to the plate's analysis. The effects of higher order non-linear strain terms (curvature terms) are considered as well. The governing equations and the boundary conditions are derived using the principle of minimum of potential energy. The solution is obtained by the extended Kantorovich method. This approach is combined with the exact element method for the stability analysis of compressed members with variable cross-section, which provides for the derivation of the exact stiffness matrix of tapered strips including the effect of in-plane forces. The results from the two shear deformation theories are compared with those obtained by the classical thin plate's theory and with published results.

Bending of a sector-shaped annular plate with continuous thickness variation along the radial direction

This paper is concerned with the analysis of the problem in the title based on Kirchhoff's (classical thin) plate theory. A class of sector-shaped annular plates whose straight radial edges are simply-supported is considered. The boundary conditions at the plate circular edges may be quite general, and between these two edges the plate has varying radial thickness. Closed-form solutions for the elastic bending problem of sector-shaped annular plates have been derived based on the small parameter method and Levy-type approach. The accuracy of the present model is demonstrated via problems for which the solutions and numerical results are available, and results are also presented for a variety of problems for which solutions are not available in the literature.

Modeling and Design of Microjet Actuators

A computational model is developed to aid the design of microelectromechanical systems (MEMS) for use in active turbulence control. The focus here is on micro-actuators and, in particular, a design employed by syntheticjet devices. This consists of a diaphragm within a cavity that, by its piezoinduced motion, creates an ejection of fluid through an orifice in the cavity’s lid. The diaphragm is modeled using classical thin-plate theory, with the stiffness of the attached piezodevice incorporated. For numerical economy, the fluid motion within the cavity is not modeled; instead, the pressure is calculated with the perfect gas law. However, in the orifice, where viscous forces are more dominant, one-dimensional Navier‐Stokes equations are solved. The actuator system is modeled in its entirety. All that is required to calculate the outlet jet velocity is the input voltage applied to the piezodevice. The numerical model is validated against experimental data for synthetic-jet devices and used to predict their optimal dimensions. An alternative mode of forcing the diaphragm is proposed that does not suffer from the drawbacks inherent in synthetic-jet operation at MEMS scale. This mode generates a jump in cavity pressure, creating a pufflike jet disturbance. This concept is explored with the aim of uncovering practical issues and simple design guidelines.

High-order free vibration of sandwich panels with a flexible core

Free vibration analysis of sandwich panels with a flexible core based on the high-order sandwich panel theory approach is presented. The mathematical formulation uses the Hamilton principle and includes derivation of the governing equations along with the appropriate boundary conditions. The formulation embodies a rigorous approach for the free vibration analysis of sandwich plates with a general construction, having high-order effects owing to the non-linear patterns of the in-plane and the vertical displacements of the core through its height. As such, it improves on the available classical and high-order theories. The formulation uses the classical thin plate theory for the face sheets and a three-dimensional elasticity theory or equivalent one for the core. The analyses are valid for any type of loading scheme, localized as well as distributed, and distinguish between loads applied at the upper or the lower face. It can also deal with any type of boundary conditions that may be different at the upper and the lower face sheets at the same edge. The effects of the rotary inertia of the various constituents of the sandwich construction are included. Two types of computational models are considered. The first model uses the vertical shear stresses in the core in addition to the displacements of the upper and the lower face sheets as its unknowns. The second model assumes a polynomial description of the displacement fields in the core that is based on the displacement fields of the first model. In this case the unknowns are the coefficients of these polynomials in addition to the displacements of the various face sheets. The two computational models have been validated numerically through a very good comparison with the well known classical and high-order plate theories. The numerical study consists of free vibration eigenmodes of two typical simply-supported panels, including higher modes that cannot be detected by other high-order computational models, and a parametric study that compares the results of the various computational models and the first-order shear deformable results.

Geometrically nonlinear axisymmetric response of thin circular plate under piezoelectric actuation

The objective of this study is to present an approximate analytical one-term Galerkin solution for the problem of nonlinear deflection, thermal buckling and natural frequencies of a three-layer thin circular plate made of an isotropic elastic core with piezoelectric layers bonded to its faces. The analysis is restricted to axisymmetric moderately large deflection of the plate subjected to a thermal load, radial edge load or edge displacement and actuated by applying an electric potential across a piezoelectric layer. The accuracy of this solution is assessed by comparison with the numerical series solution using collocation. The direct piezoelectric effect is neglected in the governing equations. The piezoelectric layers are assumed to be very thin and having Young's modulus and density much smaller than those of the elastic core. Hence, their stiffness and inertia are neglected. The rotational and inplane inertia and the shear deformation are neglected in the Von Karman type classical thin plate theory used in the analysis. Comparison with the collocation solution establishes that the Galerkin solution yields quite accurate results for the problems considered.

High order behavior of sandwich plates with free edges––edge effects

The edge effects of a sandwich plate with a “soft” core and free edges, i.e. the plate is supported only at the lower face-sheet, and the upper face-sheet and the core are free of stresses at their edges, using the high order approach (HSAPT), are presented. The two-dimensional analysis consists of a mathematical formulation that uses the classical thin plate theory for the face-sheets and a three-dimensional elasticity theory for the core. The governing equations and the required boundary conditions are derived explicitly through variational principals, yielding a system of eight partial differential equations. The non-homogeneous differential equations system is numerically solved using a modification of the extended Kantorovich method (MEKM). The model presented enables a two-dimensional solution of the stress and displacement fields when subjected to a general scheme of loads. It is applicable to any type of boundary conditions that can be applied separately on each face-sheet and on the core. A numerical study is presented, and it examines the behavior and the two-dimensional stress field of a sandwich plate with free edges, at the upper face-sheet and core, subjected to thermal and uniformly distributed loads, for various boundary conditions at the lower face-sheet. For completeness, the MEKM solution of the two-dimensional high order model is verified through comparison with a three-dimensional Finite Element model revealing good correlation. Furthermore, the problems involved in the construction of an appropriate three-dimensional FE model of a full scale sandwich plate that require large computer resources are discussed. The numerical study yields that the peeling (normal) stresses, which reach their maximum values at the edges of the sandwich plate, using a one-dimensional analysis, varies also in the transverse direction from a maximum value in the middle of the edge, descending towards the corners. Moreover, the nature of variation along the boundaries strongly depends on the type of loading and the transverse boundary conditions. The substantial variation of the stress field in the transverse direction clearly shows the necessity of a two-dimensional analysis and the inefficiencies of the one-dimensional model.

Forced Vertical Vibration of Circular Plate in Multilayered Poroelastic Medium

This paper considers the vertical vibrations of an elastic circular plate in a multilayered poroelastic half space. The plate is subjected to axisymmetric timeharmonic vertical loading and its response is governed by the classical thin-plate theory. The contact surface between the plate and the multilayered half space is assumed to be smooth and either fully permeable or impermeable. The half space under consideration consists of a number of layers with different thicknesses and material properties and is governed by Biot’s poroelastodynamic theory. The vertical displacement of the plate is represented by an admissible function containing a set of generalized coordinates. Contact stress and pore pressure jump are established in terms of generalized coordinates through the solution of flexibility equations based on the influence functions corresponding to vertical and pore pressure loading. Solutions for generalized coordinates are obtained by establishing the equation of motion of the plate through the application of Lagrange’s equations of motion. Selected numerical results are presented to portray the influence of various parameters on dynamic interaction between an elastic plate and a multilayered poroelastic half space.

An exact solution for the bending of thin rectangular plates with uniform, linear, and quadratic thickness variations

In this paper, the analysis of the titled problem is based on classical thin-plate theory, and its numerical solution is carried out by using the small parameter method and Lévy-type approach. The thin rectangular plate considered herein is simply supported on two opposite edges. The boundary conditions at the other two edges may be quite general, and between these two edges the plate may have varying thickness. Closed-form solutions have been developed for the static response of isotropic rectangular plates with non-uniform thickness variation and subjected to arbitrary loading. The accuracy of the present model is demonstrated via problems for which the exact solutions and numerical results are available, and results are also compared with those obtained by using the finite-difference method.

Beam Element Based on a Higher-Order Shear Deformation Theory

The finite element models of the Kirchhoff (classical thin) plate theory and Mindlin plate theory are now standard. The Mindlin plate theory includes the effect of transverse shear deformation. This theory is called the first-order shear deformation theory. The first author developed a beam element based on a displacement formulation of Mindlin plate theory. The objective of this paper is to present a formulation for plate element based on the refined plate theory of Shimpi. The refined plate theory of Shimpi uses higher-order polynomials in the expansion of the displacement components through the thickness of plate. Furthermore, the transverse displacement ω is given by ω = ωb + ωs, where ωb and ωs are its components due to bending and shear, respectively. Several numerical experiments are performed and show that the present element gives excellent results for both thin and thick plates.

217 面内圧縮荷重を受ける直交異方性不均質長方形板の座屈解析

A buckling analysis is carried out for inhomogeneous rectangular plates under in-plane compression. It is assumed that inhomogeneous material property of Young's modulus of elasticity is changed in the thickness direction with the power law, while Poisson's ratio is assumed to be constant. Based on the Kirchhoff (or classical thin) plate theory, the fundamental equations system can be derived by introducing the technique of the newly defined position of the reference plane. The critical buckling loads of the simply supported rectangular plate are presented using the derived fundamental relations. Effects of the inhomogeneous Young's modulus of elasticity, material orthotopy, aspect ratio and thickness of the plate are discussed.

Calculation of Moments on Top Slab in Single-Cell Box Girders

A new approach is presented for the calculation of moments on top slab in single-cell box girders. Using the classic thin plate theory, trigonometric series solutions for bending moments of the top slab under local distributed load and concentrated load with elastically clamped boundary conditions are derived. A numerical example is given to demonstrate the application of the proposed method.