Clamped Semicircular Plate Under Uniform Bending Load

1955 ◽  
Vol 22 (1) ◽  
pp. 129
Author(s):  
S. Woinowsky-Krieger
Keyword(s):  
Exact Solution ◽  
Approximate Methods ◽  
Bending Load ◽  
Distribution Of Stresses

Abstract The semicircular plate subjected to bending usually is considered as a particular case of a sectorial plate and one introduces polar co-ordinates to discuss the deflection of the plate and the corresponding distribution of stresses. If the semicircular plate is clamped along the boundary the application of approximate methods becomes necessary to this end. It is worthy of note that a rather simple exact solution can be given in this latter case by making use of bipolar instead of polar co-ordinates.

Solution of heat and mass transfer problems with non-linear coefficients

2019 ◽  
Vol 5 (4) ◽  
pp. 10-20
Author(s):  
Boris G. Aksenov ◽  
Yuri E. Karyakin ◽  
Svetlana V. Karyakina
Keyword(s):  
Mass Transfer ◽  
Exact Solution ◽  
Numerical Methods ◽  
Heat And Mass Transfer ◽  
Unknown Function ◽  
Comparison Theorems ◽  
Upper And Lower Bounds ◽  
Maximum Deviation ◽  
Approximate Methods ◽  
Nonmonotonic Dependence

Equations, which have nonlinear nonmonotonic dependence of one of the coefficients on an unknown function, can describe processes of heat and mass transfer. As a rule, existing approximate methods do not provide solutions with acceptable accuracy. Numerical methods do not involve obtaining an analytical expression for the unknown function and require studying the convergence of the algorithm used. The value of absolute error is uncertain. The authors propose an approximate method for solving such problems based on Westphal comparison theorems. The comparison theorems allow finding upper and lower bounds of the unknown exact solution. A special procedure developed for the stepwise improvement of these bounds provide solutions with a given accuracy. There are only a few problems for equations with nonlinear nonmonotonic coefficients for which the exact solution has been obtained. One of such problems, presented in this article, shows the efficiency of the proposed method. The results prove that the proposed method for obtaining bounds of the solution of a nonlinear nonmonotonic equation of parabolic type can be considered as a new method of the approximate analytical solution having guaranteed accuracy. In addition, the proposed here method allows calculating the maximum deviation from the unknown exact solution of the results of other approximate and numerical methods.

scholarly journals An optical determination of the variation of stress in a thin rectangular plate subjected to shear

1912 ◽  
Vol 86 (587) ◽  
pp. 291-319 ◽  
Keyword(s):  
Exact Solution ◽  
Shear Stresses ◽  
Approximate Methods ◽  
Riveted Joints ◽  
Simplifying Assumptions ◽  
Optical Determination ◽  
Angular Displacements ◽  
Constructional Work ◽  
Bounding Surfaces

The determination of the distribution of shear stress in a body subjected to the action of an arbitrary set of forces applied to its bounding surfaces is often of great importance in constructional work. In most cases mathematical difficulties do not allow of an exact solution, while the simplifying assumptions which are often made to reduce these difficulties lead to formulæ which are probably not correct even to a first approximation. It is important, therefore, to have experimental determinations to compare with the values calculated by approximate methods to determine to what extent the assumptions are correct, and also, if possible, to afford data for better approximations in cases which do not allow of mathematically exact solution. The shear stresses in riveted joints, the stiffened webs of plate girders and such like structures, afford examples of practical problems of extreme complexity from the mathematical standpoint, which are dealt with in practice by rough approximations, using large factors of safety fixed by experience of the behaviour of similar structures, in order to allow for contingencies due to ignorance of the actual stresses and accidental overloading. The experimental determination of the state of stress in a body as inferred measurements of the strains produced at its surface affords a means, which has been used by various investigators, to determine the condition of stress in a body subjected to the action of forces applied at its boundary. In shear problems, with which the present paper deals, this usually involves the measurement of small angular displacements, but these strains are so minute for the materials of construction at ordinary working stresses that it is usually found necessary to make experiments on other materials like indiarubber, plasticine and the like, which afford strains sufficiently large for accurate measurement, while their elastic properties are such that they may be considered to behave like more intractable materials with fair accuracy.

Comparison of exact and approximate methods for analysing vibrational relaxation regions

1961 ◽  
Vol 10 (1) ◽  
pp. 33-47 ◽  
Author(s):  
P. A. Blythe
Keyword(s):  
Exact Solution ◽  
Shock Waves ◽  
Numerical Integration ◽  
Vibrational Relaxation ◽  
Relaxation Frequency ◽  
Rayleigh Line ◽  
Relaxation Equation ◽  
Approximate Methods ◽  
Exact And Approximate Methods

The validity of various solutions for the vibrational relaxation region in shock-waves, and of the assumptions on which they are based, has been assesed by comparison with an exact solution obtained by numerical integration of the relaxation equation, and also by use of the Rayleigh-line equations. Estimates of errors in the values of the relaxation frequency, determined by means of these solutions, are given.

Transverse Vibrations of Beams, Exact Versus Approximate Solutions

1981 ◽  
Vol 48 (4) ◽  
pp. 923-928 ◽  
Author(s):  
J. R. Hutchinson
Keyword(s):  
Exact Solution ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Circular Cross Section ◽  
Approximate Solutions ◽  
Approximate Methods ◽  
Transverse Vibrations ◽  
Elasticity Equations ◽  
Beam Approximation ◽  
Free Beam

An exact solution for the natural frequencies of vibration of a finite length free-free beam with a circular cross section is found and compared to approximate solutions. This exact solution is a series solution of the general linear elasticity equations which converges to correct natural frequencies. Correctness of the frequencies is established by comparison to previous experiments. Comparison of the exact to approximate solutions is made with the Pochhammer-Chree approximation, the Timoshenko beam approximation and the Pickett approximation. The comparisons clearly show the range of applicability of the approximate methods as well as their accuracy. The correct shear coefficient for use in the Timoshenko beam approximation is investigated and conclusions which differ with, yet at the same time complement, those of previous researchers are reached.

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.

scholarly journals Analytical Analysis of Distribution of Bending Stresses in Layers of Plywood with Numerical Verification

2019 ◽  
Vol 70 (1) ◽  
pp. 77-88 ◽  
Author(s):  
Andrzej Makowski
Keyword(s):  
Bending Strength ◽  
Plate Theory ◽  
Load Capacity ◽  
Bending Test ◽  
Accurate Estimation ◽  
Numerical Verification ◽  
Classical Plate Theory ◽  
Bending Load ◽  
Bending Stresses ◽  
Distribution Of Stresses

The study presents methods for accurate estimation of bending stresses in the 3-point flexural bending test of plywood, i.e. a wood-based laminate with an alternate crosswise ply configuration. The characteristic bending strength (MOR) and mean modulus of elasticity (MOE) of standard beech plywood was determined using European Standard bending tests EN 310. Correlations were determined between empirically determined bending moduli of the plywood and material moduli of the veneer layer. Calculations were conducted based on the classical plate theory for thin panels comprising the theory of elasticity including the Kirchhoff-Love hypothesis. Rigidity of individual layer was established theoretically in the axial configuration of transformed rigidity matrix values. Numerical laminate models were developed and simulation tests were conducted. Results of experimental and analytical studies were verified using the Finite Element Method (FEM). Analyses were performed in two plywood cross-band arrangement variants. An analysis of the distribution of stresses in individual layers of plywood used an analytical and numerical method assuming the plywood specimen to be a rhombic-anisotropic material. It was found that the bending load capacity of plywood depends on the configuration of individual layers (veneers). Values of stresses originating from bending do not only depend on the distance of the considered plywood layer from the middle layer but also on stiffness in the direction of operating stresses. Bending strength varies in individual directions of the plywood panel. Therefore, the distribution of stresses in individual layers differs from that resulting from the stress distribution for homogeneous isotropic materials. Results are presented in the form of tables, bitmaps, graphs and photographs. The tests were conducted based on the BFU-BU-18 standard beech plywood thickness of 18 mm.

scholarly journals Analytic formulas for the natural frequencies of hinged structures with taking into account the dead weight

2018 ◽  
Vol 230 ◽  
pp. 02016 ◽  
Author(s):  
Yurii Krutii ◽  
Mykola Suriyaninov ◽  
Victor Vandynskyi
Keyword(s):  
Exact Solution ◽  
Solid Mechanics ◽  
Physical Phenomenon ◽  
Analytical Form ◽  
Variable Coefficients ◽  
Vibration Frequencies ◽  
Bending Vibrations ◽  
Approximate Methods ◽  
Dead Weight ◽  
The Dead

Free bending vibrations of hinged vertical uniform rod with taking into account the dead weight are investigated. The research is based on the exact solution of the partial differential vibration equation with variable coefficients. This approach allows to get more reliable picture of rod’s vibration because only the exact solution carries information of a qualitative nature and forms the most complete picture of the physical phenomenon under consideration. The frequencies equation of problem was written in dimensionless form and the way of its root finding is shown. It has been shown that the problem of determination the nature frequencies of structures is reduced to finding corresponding dimensionless vibration coefficients from equation. The formulas for the first three vibration frequencies of structures were obtained in analytical form. An analytic relationship between the frequencies with and without taking into account the dead weight of the structures was established. The nature of the dependence of frequencies on the value of the longitudinal load was revealed. The presence of conclusive analytical formulas for determining the vibration frequencies of hinged vertical structures with taking into account the dead weight is a real alternative for using the approximate methods for solving this class of problems of solid mechanics.

Equilibrium Temperatures in a Boundary-Layer Flow Over a Flat Plate of Absorbing-Emitting Gas

1968 ◽  
Vol 90 (2) ◽  
pp. 257-266 ◽  
Author(s):  
Yehuda Taitel ◽  
J. P. Hartnett
Keyword(s):  
Boundary Layer ◽  
Exact Solution ◽  
Flat Plate ◽  
Interaction Mechanism ◽  
Approximate Solutions ◽  
Equilibrium Temperature ◽  
Constant Heat Flux ◽  
Adiabatic Wall ◽  
Approximate Methods ◽  
Layer Flow

The effect of radiation on the equilibrium temperature for a flow of emitting-absorbing gas over a flat plate is studied. Three methods of solution are formulated: An approximate solution for a thin boundary layer, a similarity solution for the limiting case when the boundary layer is optically thick, and an exact solution. Emphasis is put on the study of the recovery or adiabatic wall case, where conduction to the wall is balanced by the net radiation away from the wall. Results are reported for the limiting cases of a black plate and completely reflective plate and for a unit Prandtl number. The exact solution reflects very favorably on the use of the approximate methods and points out clearly the conditions for which the approximate solutions are applicable. Results are also reported for the equilibrium wall temperature for the case of constant heat flux and for the recovery factor in the case of blowing and suction; both for optically thin boundary layers. Special attention is put on the interaction mechanism and the role of the emitting-absorbing coefficient on this process. It is shown that, for small absorption coefficient, high wall emissivity, and Mach number, the results approach the case where the gas is transparent.

Incompatible Graded Finite Elements for Analysis of Nonhomogeneous Materials

2018 ◽  
Vol 86 (2) ◽  
Author(s):  
Asmita Rokaya ◽  
Jeongho Kim
Keyword(s):  
Exact Solution ◽  
Finite Element ◽  
Finite Elements ◽  
Computation Time ◽  
Plane Elasticity ◽  
Functionally Graded ◽  
Two Dimensional ◽  
Graded Materials ◽  
Bending Load ◽  
Nonhomogeneous Materials

A six-node incompatible graded finite element is developed and studied. Such element is recommended for use since it is more accurate than four-node compatible element and more efficient than eight-node compatible element in two-dimensional plane elasticity. This paper presents comparison between six-node incompatible (QM6) and four-node compatible (Q4) graded elements. Numerical solution is obtained from abaqus using UMAT capability of the software and exact solution is provided as reference for comparison. A graded plate with exponential and linear gradation subjected to traction and bending load is considered. Additionally, three-node triangular (T3) and six-node triangular (T6) graded elements are compared to QM6 element. Incompatible graded element is shown to give better performance in terms of accuracy and computation time over other element formulations for functionally graded materials (FGMs).

Exact analysis of simply supported rhombic plates under uniform pressure

1974 ◽  
Vol 76 (1) ◽  
pp. 381-388 ◽  
Author(s):  
K. Rajaiah ◽  
Akella Kameswara Rao
Keyword(s):  
Exact Solution ◽  
Complex Variables ◽  
Transverse Load ◽  
Uniform Pressure ◽  
Approximate Methods ◽  
Simply Supported ◽  
Series Form ◽  
Exact Analysis ◽  
Present Solution

AbstractThe simply supported rhombic plate under transverse load has received extensive attention from elasticians, applied mathematicians and engineers. All known solutions are based on approximate procedures. Now, an exact solution in a fast converging explicit series form is derived for this problem, by applying Stevenson's tentative approach with complex variables. Numerical values for the central deflexion and moments are obtained for various corner angles. The present solution provides a basis for assessing the accuracy of approximate methods for analysing problems of skew plates or domains.

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