Thin Plate Problems Research Articles

A frequency domain formulation of the singular boundary method for dynamic analysis of thin elastic plate

This paper presents a frequency domain formulation of the singular boundary method (SBM) for the analysis of thin elastic plates subjected to dynamic loads. In the SBM, the fundamental solutions for the dynamic thin plate problem in frequency domain are taken as basis functions. To isolate the singularities, we derive the origin intensity factors (OIFs) for the kernels of the SBM which allows the source points to coincide with the collocation points. For the transient analysis, the exponential window method (EWM) is applied in conjunction with the SBM to obtain the time-domain result. To demonstrate the efficiency and accuracy of the present method, several examples under harmonic and transient loads are carried out in comparison with the exact solutions and finite element solutions.

Evaluation of thin square bending plate using IGA Galerkin

This paper presents an evaluation of thin bending plate problems using IGA Galerkin. It focuses on square plates under uniform load with different boundary conditions so that B-Spline is sufficient to generate basis functions. To demonstrate the performance of IGA Galerkin formulation, two numerical tests are presented.

Isogeometric Analysis for Thin Square Bending Plate using Collocation Method

This paper discusses the evaluation of rectangular thin plate problem with uniform load using Isogeometric analysis (IGA) collocation method. The implementation details of the proposed method for problems with different boundary conditions are shown. The numerical results confirm that the proposed method represents an efficient formulation and shows good behavior for thin plate structures.

A high accurate simulation of thin plate problems by using the method of approximate particular solutions with high order polynomial basis

In this paper, a closed-form particular solution of polyharmonic splines has been obtained for high order partial differential operators. Instead of using complex derivation, the new particular solution is derived simply by adding or subtracting several available particular solutions. The proposed particular solution is further coupled with polynomial basis for numerically solving thin plate problems. The relationship between number of nodes and order of polynomials are fully studied. Numerical examples with irregular domains are presented to demonstrate the effectiveness of the proposed algorithm.

A virtual element method for the vibration problem of Kirchhoff plates

The aim of this paper is to develop a virtual element method (VEM) for the vibration problem of thin plates on polygonal meshes. We consider a variational formulation relying only on the transverse displacement of the plate and propose anH2(Ω) conforming discretization by means of the VEM which is simple in terms of degrees of freedom and coding aspects. Under standard assumptions on the computational domain, we establish that the resulting scheme provides a correct approximation of the spectrum and prove optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. Finally, we report several numerical experiments illustrating the behaviour of the proposed scheme and confirming our theoretical results on different families of meshes. Additional examples of cases not covered by our theory are also presented.

Flexural stability analysis of stiffened plates using the finite element method

A four-noded stiffened plate element has been developed which has all the advantages and efficiency of an isoparametric element to model arbitrary shaped plates, but without the disadvantage of the shear-locking problem inherent in the isoparametric element. Another unique feature is that the arbitrary placement of the stiffener inside the plate element is without any restriction of its orientation. The boundary conditions have been incorporated in a general manner so as to accommodate the curved as well as the straight-edged boundaries. The element has been used for stability analysis of arbitrary shaped stiffened plates. Novelty: In this work, a plate bending element is proposed, which can model any arbitrary shape as efficiently as an isoparametric element. As it does not include the shear deformation, thin plate problems can be considered without any numerical difficulties as observed in isoparametric elements. This element is generalized to accommodate any arbitrary shapes of the plate geometry. The mesh divisions for plates with irregular boundaries using the finite element method are sometimes difficult. However, this element eliminates such complexities as the mesh divisions are done in the mapped square plate. The stiffener is modeled so that it can be of any shape, dispositions and be arbitrarily placed on the plate.

The complex variable reproducing kernel particle method for bending problems of thin plates on elastic foundations

In this paper, the complex variable reproducing kernel particle method (CVRKPM) for solving the bending problems of isotropic thin plates on elastic foundations is presented. In CVRKPM, one-dimensional basis function is used to obtain the shape function of a two-dimensional problem. CVRKPM is used to form the approximation function of the deflection of the thin plates resting on elastic foundation, the Galerkin weak form of thin plates on elastic foundation is employed to obtain the discretized system equations, the penalty method is used to apply the essential boundary conditions, and Winkler and Pasternak foundation models are used to consider the interface pressure between the plate and the foundation. Then the corresponding formulae of CVRKPM for thin plates on elastic foundations are presented in detail. Several numerical examples are given to discuss the efficiency and accuracy of CVRKPM in this paper, and the corresponding advantages of the present method are shown.

Isogeometric Galerkin in rectangular plate bending problem based on UI approach

Abstract This paper presents the development of Isogeometric Analysis for plate bending problems based on unified and integrated (UI) approach, which is a modification of Reissner-Mindlin plate theory for solving thick to thin plate problems. In Reissner-Mindlin, the total displacement and two rotations are independent of each other, while in this UI approach the total displacement is split into bending displacement and shear displacement which causes the rotations, curvatures and shear deformations can be defined as first, second and third derivatives of bending displacement, respectively. The virtual work of Galerkin Method is used to define bending stiffness and shear stiffness of the element. Several convergence tests were conducted to observe the performance of unified and integrated approach in rectangular plate of different types of boundaries conditions. The result of thick and thin plate showed good results despite of low number of element with fourth degree of polynomial or increasing degree of polynomial with only one element.

Solutions of vibration problems for thin infinite plates subjected to harmonic loads

New closed form solutions for harmonic vibrations of infinite Kirchhoff plates subjected to a constant harmonic ring load, a constant harmonic circular load and an alternating harmonic circular load are derived. Two different approaches are used to define the closed form solutions. The first approach uses the integration of the harmonic point force and the addition theorem for Bessel functions, while the second approach applies the Hankel transform to solve the inhomogeneous partial differential equation of the Kirchhoff plate theory. The new closed form particular solutions can especially be used in Trefftz like methods and extend their field of application.

Kantorovich-Euler Lagrange-Galerkin’s method for bending analysis of thin plates

In this work, the Kantorovich method is applied to solve the bending problem of thin rectangular plates with three simply supported edges and one fixed edge subject to uniformly distributed load over the entire plate surface. In the method, the plate bending problem is presented using variational calculus. The total potential energy functional is found in terms of a displacement function constructed using the Kantorovich procedure, as the product of an unknown function of x (f(x)) and a coordinate basis function in the y direction that satisfies the displacement end conditions at y = 0, y = b. The Euler-Lagrange differential equation is determined for this functional. The Galerkin method is then used to obtain the unknown function f(x). Bending moment curvature relations are used to find the bending moments and their extreme values. The results obtained agree remarkably well with literature. The effectiveness of the method is demonstrated by the marginal relative error obtained for one term displacement solutions. http://dx.doi.org/10.4314/njt.v36i2.5

Quasi-Green’s function approach to fundamental frequency analysis of elastically supported thin circular and annular plates with elastic constraints

Free vibration analysis of homogeneous and isotropic thin circular and annular plates with discrete elements such as elastic ring supports is considered. The general form of quasi- -Green’s function for thin circular and annular plates is obtained. The nonlinear characteristic equations are defined for thin circular and annular plates with different boundary conditions and different combinations of the core and support radius. The continuity conditions at the ring supports are omitted based on the properties of Green’s function. The fundamental frequency of axisymmetric vibration has been calculated using the Newton-Raphson method and calculation software. The obtained results are compared with selected results presented in literature. The exact frequencies of vibration presented in a non-dimensional form can serve as benchmark values for researchers to validate their numerical methods when applied for uniform thin circular and annular plate problems.

An anisotropic Mumford–Shah model

A variational model is introduced for the segmentation problem of thin structures, like tubes or thin plates, in an image. The energy is based on the Mumford–Shah model with a surfacic term perturbed by a Finsler metric. The formulation in the special space of functions with bounded variations is given and, in order to get an energy more adapted for numerics, a result of Γ-convergence is proved.

A Fortran implementation of isogeometric analysis for thin plate problems with the penalty method

Purpose As one of the earliest high-level programming languages, Fortran with easy accessibility and computational efficiency is widely used in the engineering field. The purpose of this paper is to present a Fortran implementation of isogeometric analysis (IGA) for thin plate problems. Design/methodology/approach IGA based on non-uniform rational B-splines (NURBS) offers exact geometries and is more accurate than finite element analysis (FEA). Unlike the basis functions in FEA, NURBS basis functions are non-interpolated. Hence, the penalty method is used to enforce boundary conditions. Findings Several thin plate examples based on the Kirchhoff-Love theory were illustrated to demonstrate the accuracy of the implementation in contrast with analytical solutions, and the efficiency was validated in comparison with another open method. Originality/value A Fortran implementation of NURBS-based IGA was developed to solve Kirchhoff-Love plate problems. It easily obtained high-continuity basis functions, which are necessary for Kirchhoff formulation. In comparison with theoretical solutions, the numerical examples demonstrated higher accuracy and faster convergence of the Fortran implementation. The Fortran implementation can well solve the time-consuming problem, and it was validated by the time-consumption comparison with the Matlab implementation. Due to the non-interpolation of NURBS, the penalty method was used to impose boundary conditions. A suggestion of the selection of penalty coefficients was given.

Closed-form solutions for free vibration of rectangular FGM thin plates resting on elastic foundation

This article presents closed-form solutions for the frequency analysis of rectangular functionally graded material (FGM) thin plates subjected to initially in-plane loads and with an elastic foundation. Based on classical thin plate theory, the governing differential equations are derived using Hamilton’s principle. A neutral surface is used to eliminate stretching–bending coupling in FGM plates on the basis of the assumption of constant Poisson’s ratio. The resulting governing equation of FGM thin plates has the same form as homogeneous thin plates. The separation-of-variables method is adopted to obtain solutions for the free vibration problems of rectangular FGM thin plates with separable boundary conditions, including, for example, clamped plates. The obtained normal modes and frequencies are in elegant closed forms, and present formulations and solutions are validated by comparing present results with those in the literature and finite element method results obtained by the authors. A parameter study reveals the effects of the power law index n and aspect ratio a/b on frequencies.

New analytic solutions for static problems of rectangular thin plates point-supported at three corners

This paper deals with the bending of rectangular thin plates point-supported at three corners using an analytic symplectic superposition method. The problems are of fundamental importance in both civil and mechanical engineering, but there were no accurate analytic solutions reported in the literature. This is attributed to the difficulty in seeking the solutions that satisfy the governing fourth-order partial differential equation with the free boundary conditions at all the edges as well as the support conditions at the corners. In the following, the Hamiltonian system-based equation for plate bending is formulated, and two types of fundamental problems are analytically solved by the symplectic method. The analytic solutions of the plates point-supported at three corners are then obtained by superposition, where the constants are obtained by a set of linear equations. The solution procedure presented in this paper offers a rigorous way to yield analytic solutions of similar problems. Some numerical results, validated by the finite element method, are shown to provide useful benchmarks for comparison and validation of other solution methods.

Natural Frequencies of Rectangular Plate With- and Without-Rotary Inertia

A nine-node isoparametric plate element, in conjunction with first-order shear deformation theory, was used for free vibration analysis of rectangular plates. Both thick and thin plate problems were solved for various aspect ratios and boundary conditions. In this work, the primary focus is on the effect of rotary inertia on the natural frequencies of rectangular plates. It is found that rotary inertia significantly affects thick plates, while it can be ignored for thin plates. The numerical convergence is very rapid and based on a comparison with data from the literature; it is proposed that the present formulation can yield highly accurate results. Finally, some numerical solutions are provided here, which may serve as benchmarks for future research on similar problems.

A Review of Skin Buckling Theory in Composite Members

This paper provides a comprehensive review of various methods used for skin buckling analysis in composite components. The skin buckling phenomenon is one of the governing criteria in composite design. It is a kind of contact buckling in which partial sections of skin buckle away from the filler material. In general, the problem can be modelled as a thin plate (skin) in unilateral contact with elastic medium (filler material). The theoretical analysis of contact buckling is complicated due to the nonlinearity arising from changing contact regions. To simplify the calculations, the filler material was usually modelled as a tensionless elastic foundation. The skin buckling coefficient varies in terms of the relative foundation stiffness factors. Because the Eigen-value method is not applicable to nonlinear systems, the finite element (FE) method was usually employed for post-buckling analysis, while initial buckling performance was investigated through analytical or semi-analytical methods such as rigid foundation model, infinite plate model and finite plate model. The compressive buckling and shear buckling problems for thin plates resting on tensionless foundations have been solved successfully. However, there are still urgent needs for future research on the topic. For example, the load carrying capacity of the buckling plates needs to be formulated for practical application. Complicated problems with complex loadings and/or corrugated skins need further investigation as well.

Free vibration analysis of rectangular plates with central cutout

AbstractA nine-node isoparametric plate element in conjunction with first-order shear deformation theory is used for free vibration analysis of rectangular plates with central cutouts. Both thick and thin plate problems are solved for various aspect ratios and boundary conditions. In this article, primary focus is given to the effect of rotary inertia on natural frequencies of perforated rectangular plates. It is found that rotary inertia has significant effect on thick plates, while for thin plates the rotary inertia term can be ignored. It is seen that the numerical convergence is very rapid and based on comparison with experimental and analytical data from literature, it is proposed that the present formulation is capable of yielding highly accurate results. Finally, some new numerical solutions are provided here, which may serve as benchmark for future research on similar problems.

Green's function for infinite thin plate with elliptic hole under bending heat source and interaction between elliptic hole and crack under uniform bending heat flux

ABSTRACTGreen's function is derived for the bending problem of an infinite thin plate with an elliptic hole under a bending heat source. Then the interaction problem between an elliptic hole and a crack in a thin plate under uniform bending heat flux is analyzed. First, the complex variable method is developed for the thermoelastic problem of bending. Then an exact solution in explicit form is derived for the Green's function by using the complex variable method. Distributions of temperature moment, heat flux moments, bending moments along the hole edge are shown in figures. For solving the interaction problem, a solution for an infinite thin plate with an adiabatic elliptic hole under uniform bending heat flux, and two Green's functions of the plate under a bending heat source couple and a bending dislocation are given. The interaction problem then reduces into singular integral equations using the Green's functions and the principle of superposition. After the equations are solved numerically, the moment intensity factors at crack tips are presented in the figures.

Solving the transverse bending problem of thin elastic orthotropic plates with form factor interpolation method

The article deals with the transverse bending problem of thin elastic orthotropic plates with combined boundary conditions. The form factor interpolation method is used for deformation calculations. Analytical expressions of maximum deflection surfaces are obtained. It is proposed to use these expressions as the reference solutions. The example of the solution to the problem of transverse bending plate in the form of a parallelogram is given.