Bending of orthotropic rectangular thin plates with two opposite edges clamped

2019 ◽  
Vol 234 (6) ◽  
pp. 1220-1230 ◽  
Author(s):  
Yangye He ◽  
Chen An ◽  
Jian Su
Keyword(s):  
Boundary Conditions ◽  
Integral Transform ◽  
High Order ◽  
Thin Plates ◽  
Constant Thickness ◽  
Order Partial Differential Equation ◽  
Algebraic Equations ◽  
Simply Supported ◽  
Linear Algebraic Equations ◽  
Analytical Integration

This work presents integral transform solutions of the bending problem of orthotropic rectangular thin plates with constant thickness, subject to five sets of boundary conditions: (a) fully clamped; (b) three edges clamped and one edge simply supported; (c) three edges clamped and one edge free; (d) two opposite edges clamped, one edge simply supported, and one edge free; and (e) two opposite edges clamped and two edges free. By adopting eigenfunctions of Euler–Bernoulli beams with corresponding boundary conditions for each direction of the plate, the governing fourth-order partial differential equation is integral transformed into a system of linear algebraic equations. Boundary conditions at the free edges are treated exactly by carrying out integral transform in the boundary formulations, which are incorporated in the transformed governing equations by integration by parts. The numerical difficulties with the high-order beam functions are overcome by using modified exponential forms, thus limiting the eigenfunctions to the range between −2 and 2. Analytical integration forms are used for the integrals of the coefficients of the transformed equations, further avoiding numerical difficulties with large high-order eigenvalues. The accuracy and convergence of the solutions are shown through numerical examples in comparison with available solutions in the literature and with finite element solutions obtained by using Abaqus program.

New Analytical Solutions of Buckling Problem of Rotationally-Restrained Rectangular Thin Plates

2019 ◽  
Vol 11 (10) ◽  
pp. 1950101 ◽  
Author(s):  
Salamat Ullah ◽  
Jinghui Zhang ◽  
Yang Zhong
Keyword(s):  
Boundary Conditions ◽  
Integral Transform ◽  
Integral Transformation ◽  
Solution Procedure ◽  
Thin Plates ◽  
Algebraic Equations ◽  
Transform Method ◽  
Linear Algebraic Equations ◽  
Title Problem ◽  
Finite Integral

A double finite sine integral transform method is employed to analyze the buckling problem of rectangular thin plate with rotationally-restrained boundary condition. The method provides more reasonable and theoretical procedure than conventional inverse/semi-inverse methods through eliminating the need to preselect the deflection function. Unlike the methods based on Fourier series, the finite integral transform directly solves the governing equation, which automatically involves the boundary conditions. In the solution procedure, after performing integral transformation the title problem is converted to solve a fully regular infinite system of linear algebraic equations with the unknowns determined by satisfying associated boundary conditions. Then, through some mathematical manipulation the analytical buckling solution is elegantly achieved in a straightforward procedure. Various edge flexibilities are investigated through selecting the rotational fixity factor, including simply supported and clamped edges as limiting situations. Finally, comprehensive analytical results obtained in this paper illuminate the validity of the proposed method by comparing with the existing literature as well as the finite element method using (ABAQUS) software.

Influence Functions in the Numerical Analysis of Bending of Thin Elastic Plates

1984 ◽  
Vol 8 (2) ◽  
pp. 103-114 ◽  
Author(s):  
Mohammed F.N. Mohsen ◽  
Ali A. Al-Gadhib ◽  
Mohammed H. Baluch
Keyword(s):  
Boundary Conditions ◽  
Influence Function ◽  
Infinite Plate ◽  
Thin Plates ◽  
Influence Functions ◽  
Algebraic Equations ◽  
Elastic Plates ◽  
The Real ◽  
Linear Algebraic Equations ◽  
The Moment

A numerical method for the linear analysis of thin plates of arbitrary plan form and subjected to arbitrary loading and boundary conditions is presented in this paper. This method is an extension of the Wu-Altiero method [1] where use has been made of the force influence function for an infinite plate, whereas the work contained in this paper is based on the use of the moment influence function of an infinite plate. The technique basically involves embedding the real plate into a fictitious infinite plate for which the moment influence function is known. N points are prescribed at the plate boundary at which the boundary conditions for the original problem are collocated by means of 2N fictitious moments placed around contours outside the domain of the real plate. A system of 2N linear algebraic equations in the unknown moments is obtained. The solution of the system yields the unknown moments. These may in turn be used to compute deflection, moments or shear at any point in the thin plate. Finally, the method is extended to include influence functions of both concentrated forces and concentrated moments. This is obtained by applying concentrated moments and forces simultaneously on the contours located outside the domain of the plate.

On the Interaction at Anti-Flat Deformation of Stress Concentrators of the Type of Cracks and Stringers with Regard to the Layer Manufactured from Miscellaneous Materials

2019 ◽  
Vol 828 ◽  
pp. 81-88
Author(s):  
Nune Grigoryan ◽  
Mher Mkrtchyan
Keyword(s):  
Integral Transform ◽  
Composite Layer ◽  
Singular Integral Equations ◽  
Original System ◽  
Fourier Integral ◽  
Quadrature Formulas ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Special Cases ◽  
Tangential Forces

In this paper, we consider the problem of determining the basic characteristics of the stress state of a composite in the form of a piecewise homogeneous elastic layer reinforced along its extreme edges by stringers of finite lengths and containing a collinear system of an arbitrary number of cracks at the junction line of heterogeneous materials. It is assumed that stringers along their longitudinal edges are loaded with tangential forces, and along their vertical edges - with horizontal concentrated forces. In addition, the cracks are laden with distributed tangential forces of different intensities. The case is also considered when the lower edge of the composite layer is free from the stringer and rigidly clamped. It is believed that under the action of these loads, the composite layer in the direction of one of the coordinate axes is in conditions of anti-flat deformation (longitudinal shift). Using the Fourier integral transform, the solution of the problem is reduced to solving a system of singular integral equations (SIE) of three equations. The solution of this system is obtained by a well-known numerical-analytical method for solving the SIE using Gauss quadrature formulas by the use of the Chebyshev nodes. As a result, the solution of the original system of SIE is reduced to the solution of the system of systems of linear algebraic equations (SLAE). Various special cases are considered, when the defining SIE and the SLAE of the task are greatly simplified, which will make it possible to carry out a detailed numerical analysis and identify patterns of change in the characteristics of the tasks.

LEAST-SQUARES MESHFREE COLLOCATION METHOD

2012 ◽  
Vol 09 (02) ◽  
pp. 1240031 ◽  
Author(s):  
BO-NAN JIANG
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Least Squares ◽  
Collocation Method ◽  
Present Method ◽  
Collocation Methods ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Linear Algebraic Equations ◽  
Dual Variables

A least-squares meshfree collocation method is presented. The method is based on the first-order differential equations in order to result in a better conditioned linear algebraic equations, and to obtain the primary variables (displacements) and the dual variables (stresses) simultaneously with the same accuracy. The moving least-squares approximation is employed to construct the shape functions. The sum of squared residuals of both differential equations and boundary conditions at nodal points is minimized. The present method does not require any background mesh and additional evaluation points, and thus is a truly meshfree method. Unlike other collocation methods, the present method does not involve derivative boundary conditions, therefore no stabilization terms are needed, and the resulting stiffness matrix is symmetric positive definite. Numerical examples show that the proposed method possesses an optimal rate of convergence for both primary and dual variables, if the nodes are uniformly distributed. However, the present method is sensitive to the choice of the influence length. Numerical examples include one-dimensional diffusion and convection-diffusion problems, two-dimensional Poisson equation and linear elasticity problems.

Isogeometric analysis of boundary integral equations: High-order collocation methods for the singular and hyper-singular equations

2016 ◽  
Vol 26 (08) ◽  
pp. 1447-1480 ◽  
Author(s):  
Matthias Taus ◽  
Gregory J. Rodin ◽  
Thomas J. R. Hughes
Keyword(s):  
Integral Equations ◽  
Isogeometric Analysis ◽  
Computer Aided Design ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
High Order ◽  
Collocation Methods ◽  
Algebraic Equations ◽  
Integration Schemes ◽  
Linear Algebraic Equations

Isogeometric analysis is applied to boundary integral equations corresponding to boundary-value problems governed by Laplace’s equation. It is shown that the smoothness of geometric parametrizations central to computer-aided design can be exploited for regularizing integral operators to obtain high-order collocation methods involving superior approximation and numerical integration schemes. The regularization is applicable to both singular and hyper-singular integral equations, and as a result one can formulate the governing integral equations so that the corresponding linear algebraic equations are well-conditioned. It is demonstrated that the proposed approach allows one to compute accurate approximate solutions which optimally converge to the exact ones.

The Post-Buckling Behaviour of Simply-Supported Square Plates

1969 ◽  
Vol 20 (3) ◽  
pp. 203-222 ◽  
Author(s):  
A. C. Walker
Keyword(s):  
Ultimate Load ◽  
Test Results ◽  
Algebraic Equations ◽  
Load Prediction ◽  
Simply Supported ◽  
Expansion Technique ◽  
Linear Algebraic Equations ◽  
Out Of Plane ◽  
Post Buckling ◽  
Plane Direction

SummaryThe post-buckling behaviour of flat square plates loaded along two opposite straight edges is analysed using the von Kármán equations, trigonometric series and Galerkin’s method. The unloaded edges are considered to be either free to distort in the plane of the plate or maintained straight but allowed to move bodily; all edges are simply-supported in the out-of-plane direction. The non-linear algebraic equations are solved approximately using a McLaurin’s series expansion technique which facilitates the development of explicit expressions for ultimate load, end shortening and stiffness. The effects of initial geometric imperfections are also studied and it is shown that the results for ultimate load prediction and end shortening are in good agreement with test results.

Flexural problems of circular ring plates and sectorial plates. I

1960 ◽  
Vol 56 (1) ◽  
pp. 75-95 ◽  
Author(s):  
W. A. Bassali ◽  
M. A. Gorgui
Keyword(s):  
Circular Ring ◽  
Concentric Circle ◽  
Thin Plates ◽  
Constant Thickness ◽  
Concentrated Load ◽  
Simply Supported ◽  
General Line ◽  
Special Cases ◽  
Edge Conditions ◽  
Elastically Restrained

ABSTRACTIn this paper explicit expressions in closed forms are first obtained for the complex potentials and deflexion at any point of a circular annular plate under various edge conditions when the plate is acted upon by general line loadings distributed along the circumference of a concentric circle. These solutions are then used to discuss the bending of a circular plate with a central hole under a concentrated load or a concentrated couple acting at any point of the plate. Solutions for singularly loaded sectorial plates bounded by two arcs of concentric circles and two radii are also derived when the plate is simply supported along the straight edges. The boundary conditions along the circular edges include the cases of a free boundary as well as the elastically restrained boundary which covers the usual rigidly clamped and simply supported boundaries as special cases. The usual restrictions relating to the small deflexion theory of thin plates of constant thickness are assumed. Limiting forms of the resulting solutions are investigated.

scholarly journals Hybrid algorithm Newton method for solving systems of nonlinear equations with block Jacobi matrix

2020 ◽  
pp. 208-217
Author(s):  
O.M. Khimich ◽  
◽  
V.A. Sydoruk ◽  
A.N. Nesterenko ◽  
◽  
...  
Keyword(s):  
Nonlinear Equations ◽  
Newton Method ◽  
Jacobi Matrix ◽  
Sparse Matrices ◽  
High Order ◽  
New Method ◽  
Systems Of Nonlinear Equations ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
High Convergence Rate

Systems of nonlinear equations often arise when modeling processes of different nature. These can be both independent problems describing physical processes and also problems arising at the intermediate stage of solving more complex mathematical problems. Usually, these are high-order tasks with the big count of un-knows, that better take into account the local features of the process or the things that are modeled. In addition, more accurate discrete models allow for more accurate solutions. Usually, the matrices of such problems have a sparse structure. Often the structure of sparse matrices is one of next: band, profile, block-diagonal with bordering, etc. In many cases, the matrices of the discrete problems are symmetric and positively defined or half-defined. The solution of systems of nonlinear equations is performed mainly by iterative methods based on the Newton method, which has a high convergence rate (quadratic) near the solution, provided that the initial approximation lies in the area of gravity of the solution. In this case, the method requires, at each iteration, to calculates the Jacobi matrix and to further solving systems of linear algebraic equations. As a consequence, the complexity of one iteration is. Using the parallel computations in the step of the solving of systems of linear algebraic equations greatly accelerates the process of finding the solution of systems of nonlinear equations. In the paper, a new method for solving systems of nonlinear high-order equations with the Jacobi block matrix is proposed. The basis of the new method is to combine the classical algorithm of the Newton method with an efficient small-tile algorithm for solving systems of linear equations with sparse matrices. The times of solving the systems of nonlinear equations of different orders on the nodes of the SKIT supercomputer are given.

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