Solution of Some Problems in Bending of Thin Clamped Plates by Means of the Method of Muskhelishvili

1957 ◽  
Vol 24 (2) ◽  
pp. 295-298
Author(s):  
L. I. Deverall
Keyword(s):  
Conformal Mapping ◽  
Mapping Function ◽  
Elliptic Functions ◽  
Transverse Load ◽  
Rectangular Plates ◽  
Convergence Properties ◽  
Clamped Plate ◽  
Central Maximum ◽  
Method Of Solution ◽  
Square Plate

Abstract In this paper, the complex variable method of N. I. Muskhelishvili is applied to the problem of bending for small deflections of a thin, isotropic, homogeneous, clamped plate with transverse load. The functional equation involved in Muskhelishvili’s method is solved by means of series expansions. The necessary conformal mapping functions are found from the Schwarz-Christoffel formula or expansion of elliptic functions. For uniformly loaded square and rectangular plates, the central (maximum) deflection obtained by the method of Muskhelishvili is compared to the corresponding deflections obtained by other methods. Deflections for the square plate are given for three and five terms, respectively, of the series expansion of the conformal mapping function in order to estimate convergence properties of the method of solution. The solution for a uniformly loaded clamped plate of equilateral triangular planform is also discussed. Central deflection for this case is given.

Solution of the wedge entry problem by numerical conformal mapping

1972 ◽  
Vol 56 (1) ◽  
pp. 173-192 ◽  
Author(s):  
O. F. Hughes
Keyword(s):  
Conformal Mapping ◽  
Functional Dependence ◽  
Mapping Function ◽  
Computer Method ◽  
Analytic Method ◽  
Inviscid Fluid ◽  
Numerical Conformal Mapping ◽  
Method Of Solution ◽  
Entire Flow

An accurate quasi-analytic method of solution is presented for the classical hydrodynamics problem of the constant-velocity entry of a prismatic wedge into a weightless incompressible inviscid fluid. The method uses the Wagner function W, which reduces the problem to the determination of a mapping function Λ = [Lscr ](W) for the hodograph. [Lscr ](W) is constructed by using the hodograph for an unsymmetric diamond together with a modifying or ‘preparatory’ trans-formation. A computer method of conformal mapping is developed and is used to obtain this latter transformation. Results are presented for the case of a 90° wedge and show that the solution is both more accurate than previous solutions, having an error of less than 1 %, and more complete, as it portrays the entire flow field and furnishes information about the functional dependence among the variables.

Vibration of Rectangular Plates by the Ritz Method

1950 ◽  
Vol 17 (4) ◽  
pp. 448-453 ◽  
Author(s):  
Dana Young
Keyword(s):  
Ritz Method ◽  
Normal Modes ◽  
Approximate Solutions ◽  
The Other ◽  
Rectangular Plates ◽  
Elastic Plates ◽  
Modes Of Vibration ◽  
Ritz’S Method ◽  
Square Plate ◽  
Set Up

Abstract Ritz’s method is one of several possible procedures for obtaining approximate solutions for the frequencies and modes of vibration of thin elastic plates. The accuracy of the results and the practicability of the computations depend to a great extent upon the set of functions that is chosen to represent the plate deflection. In this investigation, use is made of the functions which define the normal modes of vibration of a uniform beam. Tables of values of these functions have been computed as well as values of different integrals of the functions and their derivatives. With the aid of these data, the necessary equations can be set up and solved with reasonable effort. Solutions are obtained for three specific plate problems, namely, (a) square plate clamped at all four edges, (b) square plate clamped along two adjacent edges and free along the other two edges, and (c) square plate clamped along one edge and free along the other three edges.

Some Thin-Plate Problems by the Sine Transform

1951 ◽  
Vol 18 (2) ◽  
pp. 152-156
Author(s):  
L. I. Deverall ◽  
C. J. Thorne
Keyword(s):  
Thin Plate ◽  
The Other ◽  
Rectangular Plates ◽  
Specific Load ◽  
General Solutions ◽  
Arbitrary Load ◽  
Sine Transform ◽  
Edge Conditions ◽  
Method Of Solution ◽  
Load Function

Abstract General expressions for the deflection of thin rectangular plates are obtained for cases in which two opposite edges have arbitrary but given deflections and moments. The sine transform is used as a part of the method of solution, since solutions can be found for an arbitrary load for each set of edge conditions at the other two edges. Even for the classical cases, the use of the sine transform makes the process of solving the problem much easier. The six general solutions given are those which arise from all possible combinations of physically important edge conditions at the other two edges. Solutions for a specific load function can be found by integration or by the use of a table of sine transforms. Tables useful in application of the method to specific problems are included.

Large Elastic-Plastic Deformation Analysis of Rectangular Plates

2002 ◽  
Author(s):  
Reza Naghdabadi ◽  
Mohsen Shahi
Keyword(s):  
Lateral Displacement ◽  
Deformation Analysis ◽  
Plastic Behavior ◽  
Rectangular Plates ◽  
Computationally Efficient ◽  
Plastic Deformations ◽  
Elastic Plastic ◽  
Various Boundary Conditions ◽  
Method Of Solution ◽  
Suitable Combination

The purpose of this paper is to find a fast and simple solution for the large deformation of rectangular plates considering elastic-plastic behavior. This analysis contains material and geometric nonlinearities. For geometric nonlinearity the concept of load analogy is used. In this method the effect of nonlinear terms of lateral displacement is considered as suitable combination of additional fictitious lateral load, edge moment and in-plane forces acting on the plate. Variable Material Property (V.M.P.) method has been used for analysis of material nonlinearity. In this method, the basic relations maintain the form of stress-strain elastic formula, while material properties are modified to take into account the path-dependency involved in elastic-plastic deformations. Therefore, the solution of a von-Karman plate enduring large elastic-plastic deformations is reduced to that of an equivalent elastic plate undergoing small deformations. The method of solution employed in this study is computationally efficient and can easily be used for various boundary conditions and loadings.

Constructal design applied to geometrical evaluation of rectangular plates with inclined stiffeners subjected to uniform transverse load

2019 ◽  
Author(s):  
Vinícius Torres Pinto ◽  
◽  
Marcelo Langhinrichs Cunha ◽  
Grégori da Silva Troina ◽  
Kauê Louro Martins ◽  
...  
Keyword(s):  
Transverse Load ◽  
Rectangular Plates ◽  
Constructal Design

scholarly journals The Schwarz-Christoffel Conformal Mapping for “Polygons” with Infinitely Many Sides

2008 ◽  
Vol 2008 ◽  
pp. 1-20 ◽  
Author(s):  
Gonzalo Riera ◽  
Hernán Carrasco ◽  
Rubén Preiss
Keyword(s):  
Conformal Mapping ◽  
Mapping Function ◽  
Mathematical Software ◽  
Explicit Formulas ◽  
Line Segments ◽  
Hyperelliptic Surfaces ◽  
Open Questions ◽  
Upper Half Plane ◽  
Koch Snowflake ◽  
Boundary Curves

The classical Schwarz-Christoffel formula gives conformal mappings of the upper half-plane onto domains whose boundaries consist of a finite number of line segments. In this paper, we explore extensions to boundary curves which in one sense or another are made up of infinitely many line segments, with specific attention to the “infinite staircase” and to the Koch snowflake, for both of which we develop explicit formulas for the mapping function and explain how one can use standard mathematical software to generate corresponding graphics. We also discuss a number of open questions suggested by these considerations, some of which are related to differentials on hyperelliptic surfaces of infinite genus.

scholarly journals Bending of a ship's skin panel loaded along the axis of symmetry

2021 ◽  
pp. 14-19
Author(s):  
Т.П. Кныш ◽  
М.В. Сухотерин ◽  
С.О. Барышников
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Double Fourier Series ◽  
Transverse Load ◽  
Bending Moments ◽  
Approximate Methods ◽  
Loaded Part ◽  
Axis Of Symmetry ◽  
Square Plate ◽  
3D Shapes

Задача изгиба прямоугольной панели обшивки от действия распределенной по оси симметрии поперечной нагрузки не имеет точного решения в конечном виде в виду сложности краевых условий и вида нагрузки. Использование другими авторами различных приближенных методов оставляет открытым вопрос о точности полученных результатов. Целью исследования является получение точного решения с помощью гиперболо-тригонометрических рядов по двум координатам. Для этого используется метод бесконечной суперпозиции указанных рядов, которые в отдельности удовлетворят лишь части граничных условий. Порождаемые ими невязки взаимно компенсируются в ходе итерационного процесса и стремятся к нулю. Частное решения представлено двойным рядом Фурье. Точное решение достигается увеличением количества членов в рядах и числа итераций. При достижении заданной точности процесс прекращается. Получены численные результаты для прогибов и изгибающих моментов для квадратной пластины при различной длине загруженной части оси пластины. Представлены 3D-формы изогнутой поверхности пластины и эпюры изгибающих моментов. The problem of bending a rectangular skin panel from the action of a transverse load distributed along the axis of symmetry does not have an exact solution in the final form due to the complexity of the boundary conditions and the type of load. The use of various approximate methods by other authors leaves open the question of the accuracy of the results obtained. The aim of the study is to obtain an exact solution using hyperbolo-trigonometric series in two coordinates. To do this, we use the method of infinite superposition of these series, which individually satisfy only part of the boundary conditions. The residuals generated by them are mutually compensated during the iterative process and tend to zero. The quotient of the solution is represented by a double Fourier series. The exact solution is achieved by increasing the number of terms in the series and the number of iterations. When the specified accuracy is reached, the process stops. Numerical results are obtained for deflections and bending moments for a square plate with different lengths of the loaded part of the plate axis. 3D shapes of the curved surface of the plate and diagrams of bending moments are presented.

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