Plate Bending Problems Research Articles

求解弹性地基上自由矩形中厚板弯曲问题的辛-叠加方法

Based on the symplectic superposition method proposed in recent years, the bending problems of free-edge rectangular thick plates resting on elastic foundations were analytically solved. The original problem was split into 3 subproblems corresponding to the bending problems of rectangular thick plates with 2 opposite edges slidingly clamped and resting on elastic foundations, which were solved with the symplectic geometry method. The analytic solution of the original problem was then obtained through superposition. Compared to the conventional analytic approaches such as the semi-inverse method, the symplectic superposition method has the advantages of both rationality of the symplectic method and regularity of the superposition method. The solution procedure starts from the basic equations of elasticity, and a rigorous derivation yields the analytic solutions, thus extending the scope of problems to be solved. The present method can serve as an effective analytic approach to complex boundary value problems of high-order partial differential equations in elasticity, as represented by the rectangular plate problems.

A Posteriori Estimates for Conforming Kirchhoff Plate Elements

We derive a residual a posteriori estimator for the Kirchhoff plate bending problem. We consider the problem with a combination of clamped, simply supported, and free boundary conditions subject to...

Поведение напряжений в угловой точке ортотропной пластины при изгибе

Поведение напряжений в угловой точке ортотропной пластины при изгибе

Combined Semianalytical and Numerical Static Plate Analysis. Part 1: Formulation of the Problem and Approximation Models

The distinctive paper is devoted to solution of multipoint (particularly, two-point) boundary problem of plate analysis (Kirchhoff model) based on combined application of finite element method (FEM) and discrete-continual finite element method (DCFEM). As is known the Kirchhoff-Love theory of plates is a two-dimensional mathematical model that is normally used to determine the stresses and deformations in thin plates subjected to forces and moments. The given domain, occupied by considering structure, is embordered by extended one. The field of application of DCFEM comprises fragments of structure (subdomains) with regular (constant or piecewise constant) physical and geometrical parameters in some dimension (“basic” dimension). DCFEM presupposes finite element mesh approximation for non-basic dimension of extended domain while in the basic dimension problem remains continual. FEM is used for approximation of all other subdomains (it is convenient to solve plate bending problems in terms of displacements). Discrete (within FEM) and discrete-continual (within DCFEM) approximation models for subdomains are under consideration.

Combined Semianalytical and Numerical Static Plate Analysis. Part 2: Resultant Multipoint Boundary Problem

The distinctive paper is devoted to solution of multipoint boundary problem of plate analysis (Kirchhoff model) based on combined application of finite element method (FEM) and discrete-continual finite element method (DCFEM). As is known the Kirchhoff-Love theory of plates is a two-dimensional mathematical model that is normally used to determine the stresses and deformations in thin plates subjected to forces and moments. The given domain, occupied by considering structure, is embordered by extended one. The field of application of DCFEM comprises fragments of structure (subdomains) with regular (constant or piecewise constant) physical and geometrical parameters in some dimension (“basic” dimension). DCFEM presupposes finite element mesh approximation for non-basic dimension of extended domain while in the basic dimension problem remains continual. FEM is used for approximation of all other subdomains (it is convenient to solve plate bending problems in terms of displacements). Coupled multilevel approximation model for extended domain and resultant multipoint boundary problem are constructed. Brief information about software systems and verification samples are presented as well.

A Decomposition Result for Kirchhoff Plate Bending Problems and a New Discretization Approach

A new approach is introduced for deriving a mixed variational formulation for Kirchhoff plate bending problems with mixed boundary conditions involving clamped, simply supported, and free boundary ...

The Morley-Type Virtual Element for Plate Bending Problems

We propose a simple nonconforming virtual element for plate bending problems, which has few local degrees of freedom and provides the optimal convergence in $$H^2$$ -norm. Moreover, we prove the optimal error estimates in $$H^1$$ - and $$L^2$$ -norm. The nonconforming virtual element is constructed for any order of accuracy, but not $$C^0$$ -continuous. It is worth mentioning that, for the lowest-order case on triangular meshes the simplified nonconforming virtual element coincides with the well-known Morley element, so it can be taken as the extension of the Morley element to polygonal meshes. Finally, we verify the optimal convergence in $$H^2$$ -norm for the nonconforming virtual element by some numerical tests.

Separate control meshes for displacements and rotations for a shear locking free isogeometric Reissner‐Mindlin plate

AbstractIn the analysis of plate and shell bending problems using an isogeometric Reissner‐Mindlin approach, transversal shear locking effects may occur especially for thin structures. One possibility to overcome locking effects is to increase the polynomial order of the NURBS basis functions. However, there are certain examples where this method shows some deficiencies, like oscillations. For low polynomial degrees, there exist only a few effective concepts for the elimination of locking effects. One is the enhanced assumed strain (EAS) method which is used in finite element formulations and which is sensitive to distorted element geometries. Beirão da Veiga et al. [1] introduced a new approach for a Reissner‐Mindlin plate formulation where the displacements and rotations of the mesh are approximated using different control meshes. The physical space of the structure always remains the same. Hence, the method is in accordance to the isoparametric paradigm. However, the shape functions for the approximation of the displacements and the rotations may have different polynomial degrees and number of control points. In this way, the compatibility requirement for pure bending is fulfilled and shear locking is avoided. The method is tested for an isogeometric Reissner‐Mindlin plate formulation, which is based on a degenerated shell formulation [2]. Basic examples are chosen and the results are compared to the unaltered isogeometric Reissner‐Mindlin plate and the finite element Method using MITC elements. The results show that the method has similar accuracy and efficiency as the MITC element and is also applicable for skew element geometries. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The Kantorovich method applied to bending, buckling, vibration, and 3D stress analyses of plates: A literature review

ABSTRACTThis study presents a thorough review of the applications of the Kantorovich method to several plate problems. The main objective of this review is to compile an up-to-date list of studies that employ the Kantorovich method, which is a semi-analytical numerical method, to bending, buckling, vibration, and three-dimensional elasticity problems of plates. The reviews highlight the derivations of the governing equations, which are written in form of ordinary differential equations, and the solution methods used to solve those equations. This review should be helpful for researchers and engineers to quickly gain an overview of the application of the method to thin-walled structures.

On the characterisation of polar fibrous composites when fibres resist bending

This study aims to initiate research for the invention of methods appropriate for characterisation of fibre-reinforced materials that exhibit polar material behaviour due to fibre bending resistance. It thus focuses interest in the small strain regime, where there are examples of particular deformations for which non-polar linear elasticity fails to distinguish clearly the nature of a fibrous composite or even to account for the presence of fibres. Particular attention is accordingly given to the solution of the polar material version of the pure bending problem of transverse isotropic or special orthotropic plates with embedded fibres resistant in bending. It is seen that pure bending deformation enables polar fibre-reinforced materials to generate constant couple stress-field which, in turn, endorses uniqueness of the solution of the corresponding boundary value problem. In this context, by appropriately extending the validity of Clapeyron's theorem within the regime of polar linear elasticity for fibre-reinforced materials, it is shown that the solution of well-posed linear elasticity boundary value problems that generate a constant couple-stress field is unique. The well-known uniqueness of solution of conventional, non-polar linear elasticity boundary value problems is, in fact, a particular case in which the generated constant value of the couple-stress field is zero.

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.

An Improved Interpolating Complex Variable Meshless Method for Bending Problem of Kirchhoff Plates

An improved interpolating complex variable moving least squares (IICVMLS) method is proposed for numerical simulations of structures, in which a complete basis function and singular weight function are used to form a new basis function through the orthogonalization process. In this method, a new shape function which has the property of Kronecker [Formula: see text] function is derived to build the interpolating function. Based on the IICVMLS method, an improved interpolating complex variable element free Galerkin (IICVEFG) method is obtained for bending problem of Kirchhoff plates. In the IICVEFG method, the essential boundary conditions can be satisfied directly, and thus the final discrete matrix equation is more concise than that in the non-interpolating complex variable element free Galerkin methods. Hence, the proposed meshless method is more accurate and efficient than conventional complex variable meshless methods. Numerical examples of bending problem of Kirchhoff plates are presented to validate the advantages of the IICVEFG method.

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.

The improved complex variable element-free Galerkin method for the analysis of Kirchhoff plates

Based on the improved complex variable moving least-squares (ICVMLS) approximation, the improved complex variable element-free Galerkin (ICVEFG) method for the bending problem of Kirchhoff plate is presented. Compared with the moving least-squares (MLS) approximation, in the ICVMLS approximation, the approximation function of two-dimensional problems can be obtained with one-dimensional basis function, then the computational efficiency of the shape functions is higher. Compared with the meshless methods based on the MLS approximation, under the same node distributions, the ones using the ICVMLS approximation can obtain the solutions with higher computational accuracy; and under the similar computational accuracy, the ones using the ICVMLS approximation have higher computational efficiency. The ICVMLS approximation is used to form the approximation function of the deflection of a Kirchhoff plate, the Galerkin weak form of the bending problem of Kirchhoff plates is adopted to obtain the discretized system equations, and the penalty method is employed to enforce the essential boundary conditions, then the corresponding formulae of the ICVEFG method for the bending problem of Kirchhoff plates are presented. By computing and analyzing four typical examples, it is shown that the ICVEFG method of Kirchhoff plates in this paper is efficient. And the computational precision of the numerical solutions is analyzed to select the basis function, weight function, scaling factor, node distribution and penalty factor in the ICVEFG method. Numerical examples show that the method in this paper has better convergence and higher accuracy. When the quadratic polynomial basis function and the cubic or quartic spline weight function are used, and d m a x = 4.2−4.4 , the ICVEFG method of Kirchhoff plates in this paper can obtain the solutions with high computational accuracy. And more nodes are distributed in the problem domain, higher computational accuracy of the solution will obtained, which shows that the method in this paper has better convergence.

A New Strain Based Sector Finite Element for Plate Bending Problems

The purpose of this paper is to present the formulation of a new sector finite element based on the strain approach for the numerical analysis of circular thin plate bending. The element is named SBSPK and has four nodes and three degrees of freedom per node (3 d.o.f./node). From several numerical examples, it is shown that convergence can be achieved with the use of only a small number of finite elements. The results obtained are compared with analytical and available numerical solutions.

Local radial basis function collocation method for bending analyses of quasicrystal plates

• Meshless formulations are developed for static and dynamic bending of QC plates. • The phonon-phason coupling effects are studied numerically. • The influence of elastic foundation and variable plate thickness is investigated. • The response of QC plates to impact loading is simulated. The local radial basis function collocation method (LRBFCM) is proposed for plate bending analysis in orthorhombic quasicrystals (QCs) under static and transient dynamic loads. Three common types of the plate bending problems are considered: (1) QC plates resting on Winkler foundation (2) QC plates with variable thickness and (3) QC plates under a transient dynamic load. According to the Reissner–Mindlin plate bending theory, there is allowed to simulate the behavior of the two excitations in QC plates, phonon and phason, by 2D strong formulations for the system of governing equations. The governing equations, which describe the phason displacements, are based on Agiasofitou and Lazar elastodynamic model. Numerical results demonstrate the effect of the elastic foundation, as well as plate thickness on the phonon and phason characteristics in this paper. For the transient dynamic analysis, the influence of the phason friction coefficients on the responses of QC plate to transient dynamic loads is also studied.

Mathematical Simulation of the Problem of the Pre-Critical Sandwich Plate Bending in Geometrically Nonlinear One Dimensional Formulation

In this paper we consider the geometrically nonlinear problem of determining the equilibrium position of a sandwich plate consisting of two external carrier layers and located between transversely soft core, connected with carrier layer by means of adhesive joint. We investigate the generalized statement of the problem. For its numerical implementation we offer a two-layer iterative process and investigate the convergence of the method. Numerical experiments are carried out for the model problem.

Method of particular solutions using polynomial basis functions for the simulation of plate bending vibration problems

The traditional polynomial expansion method is deemed to be not suitable for solving two- and three-dimensional problems. The system matrix is usually singular and highly ill-conditioned due to large powers of polynomial basis functions. And the inverse of the coefficient matrix is not guaranteed for the evaluation of derivatives of polynomial basis functions with respect to the differential operator of governing equations. To avoid these troublesome issues, this paper presents an improved polynomial expansion method for the simulation of plate bending vibration problems. At first, the particular solutions using polynomial basis functions are derived analytically. Then these polynomial particular solutions are employed as basis functions instead of the original polynomial basis functions in the method of particular solutions for the approximated solutions. To alleviate the conditioning of the resultant matrix, we employ the multiple-scale method. Numerical experiments compared with analytical solutions, solutions by the Kansa’s method, and reference solutions in references confirm the efficiency and accuracy of the proposed method in the solution of Winkler and thin plate bending problems including irregular shapes.

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

ADVANCES IN SHAPE-FREE FINITE ELEMENT METHODS: A REVIEW

As the most important tool for simulation and computation, the finite element method has been widely applied in engineering and scientific problems. However, this powerful method still suffers from some inherent deficiencies, such as the sensitivity problem to mesh distortion and so on, which have not been completely solved yet. This paper systematacially introduces some research developments on new finite element methods, i.e., the shape-free finite element methods, including the hybrid stress-function elements for plane and 2D fracture problems, the hybrid displacement-function elements for Reissner-Mindlin plate bending problem, and new unsymmetric finite element for plane and 3D elasticity. Based on the existing hybrid stress and unsymmetric finite element methods, some advanced techniques, including the analytical trial function method, new natural coordinate method, generalized conforming technique, etc., were adopted in the aforementioned new approaches. The resulting new finite element models possess high precision and good robustness. In particular, they can keep their level of performance even in extremely distorted meshes. Furthermore, some historical challenges in the finite element method, such as the limitation defined by the MacNeal's theorem, the edge effect problem of the Resissner-Mindlin plate, were also successfully solved. At the end of this paper, some features of the new methods and further development are discussed.