Buckling Under Locally Hydrostatic Pressure

1946 ◽  
Vol 13 (3) ◽  
pp. A198-A200
Author(s):  
G. H. Handelman
Keyword(s):  
Differential Equation ◽  
Hydrostatic Pressure ◽  
Boundary Conditions ◽  
Pressure Distribution ◽  
Buckling Loads ◽  
The Third ◽  
Basic Assumptions ◽  
Basic Differential Equation ◽  
Discrete Values ◽  
Definition Of

Abstract The instability of a simple pin-supported beam when subject to a locally hydrostatic pressure distribution, the intensity of which is a function of distance along the beam only, is discussed in this paper. In particular, the pressure is assumed to follow a polynomial law given by Equation [1]. The problem, basic assumptions, and precise definition of locally hydrostatic pressure are stated in the first section of the paper. The second section contains a discussion of the equations of equilibrium from which the basic differential equation is derived. The solutions of this differential equation satisfy the boundary conditions only for certain discrete values of the parameters involved, and these values in turn define the buckling loads. In the third section the buckling loads are tabulated for several cases in comparison with the buckling loads for the same beam subject only to an end thrust. The appendix contains a mathematical discussion of the solution of the basic differential equation and a derivation of the formula for the buckling loads.

The Bending, Buckling, and Flexural Vibration of Simply Supported Polygonal Plates by Point-Matching

1961 ◽  
Vol 28 (2) ◽  
pp. 288-291 ◽  
Author(s):  
H. D. Conway
Keyword(s):  
Hydrostatic Pressure ◽  
Boundary Conditions ◽  
Flexural Vibration ◽  
Frequency Equation ◽  
Numerical Results ◽  
Special Technique ◽  
Point Matching ◽  
Buckling Loads ◽  
Simply Supported ◽  
Flexural Vibrations

The bending by uniform lateral loading, buckling by two-dimensional hydrostatic pressure, and the flexural vibrations of simply supported polygonal plates are investigated. The method of meeting the boundary conditions at discrete points, together with the Marcus membrane analog [1], is found to be very advantageous. Numerical examples include the calculation of the deflections and moments, and buckling loads of triangular square, and hexagonal plates. A special technique is then given, whereby the boundary conditions are exactly satisfied along one edge, and an example of the buckling of an isosceles, right-angled triangle plate is analyzed. Finally, the frequency equation for the flexural vibrations of simply supported polygonal plates is shown to be the same as that for buckling under hydrostatic pressure, and numerical results can be written by analogy. All numerical results agree well with the exact solutions, where the latter are known.

Analytical solution for the buckling of rectangular plates under uni-axial compression with variable thickness and elasticity modulus in the y-direction

2009 ◽  
Vol 224 (1) ◽  
pp. 33-41 ◽  
Author(s):  
M Saeidifar ◽  
S N Sadeghi ◽  
M R Saviz
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Power Series ◽  
Boundary Conditions ◽  
Variable Thickness ◽  
Elasticity Modulus ◽  
Plate Motion ◽  
Transverse Displacement ◽  
Buckling Loads ◽  
Simply Supported

The present study introduces a highly accurate numerical calculation of buckling loads for an elastic rectangular plate with variable thickness, elasticity modulus, and density in one direction. The plate has two opposite edges ( x = 0 and a) simply supported and other edges ( y = 0 and b) with various boundary conditions including simply supported, clamped, free, and beam (elastically supported). In-plane normal stresses on two opposite simply supported edges ( x = 0 and a) are not limited to any predefined mathematical equation. By assuming the transverse displacement to vary as sin( mπ x/ a), the governing partial differential equation of plate motion will reduce to an ordinary differential equation in terms of y with variable coefficients, for which an analytical solution is obtained in the form of power series (Frobenius method). Applying the boundary conditions on ( y = 0 and b) yields the problem of finding eigenvalues of a fourth-order characteristic determinant. By retaining sufficient terms in power series, accurate buckling loads for different boundary conditions will be calculated. Finally, the numerical examples have been presented and, in some cases, compared to the relevant numerical results.

Measurement of small values of hydrostatic pressure difference / Pomiar małych wartości różnicy ciśnień hydrostatycznych

2012 ◽  
Vol 57 (1) ◽  
pp. 157-167
Author(s):  
Krzysztof Broda ◽  
Wiktor Filipek
Keyword(s):  
Hydrostatic Pressure ◽  
Pressure Distribution ◽  
Atmospheric Pressure ◽  
Free Water ◽  
Reynolds Numbers ◽  
Measurement Results ◽  
Order Of Magnitude ◽  
Hydraulic Gradients ◽  
Accuracy Of Measurement ◽  
Definition Of

In order to describe the fluid flow through the porous centre, made of identical spheres, it is necessary to know the pressure, but in fact - the pressure distribution. For the flows in the range that was traditionally called laminar flow (i. e. for Reynolds numbers (Bear, 1988; Duckworth, 1983; Troskolański, 1957) from the range 0,01 to 3) it is virtually impossible with the use of the tools directly available on the market. Therefore, many scientists who explore this problem have concentrated only on the research of the velocity distribution of the medium that penetrates the intended centre (Bear, 1988) or pressure distribution at high hydraulic gradients (Trzaska & Broda, 1991, 2000; Trzaska et al., 2005). It may result from the inaccessibility to the measurement methods that provide measurement of very low hydrostatic pressures, such as pressure resulting from the weight of liquid located in the gravitational field (Duckworth, 1983; Troskolański, 1957). The pressure value c. 10 Pa (Troskolański, 1957) can be generated even by 1 mm height difference between the two levels of the free water surface, which in fact constitutes the definition of gauging tools of today measuring the level of the hydrostatic pressure. Authors proposed a method of hydrostatic pressure measurement and devised a gauging tool. Then a series of tests was conducted aiming at establishing what is the influence of various factors, such as temperature, atmospheric pressure, velocity of measurement completion, etc. on the accuracy and method of measurements. A method for considerable reduction of hysteresis that occurs during measurement was also devised. The method of measurement of small hydrostatic difference measurements allows for the accuracy of measurement of up to 0.5 Pa. Measurement results can be improved successfully by one order of magnitude, which for sure would entail necessary temperature stabilization of the tool. It will be more difficult though to compensate the influence of atmospheric pressure on the measurement process.

scholarly journals Existence Theory for Nonlinear Third-Order Ordinary Differential Equations with Nonlocal Multi-Point and Multi-Strip Boundary Conditions

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 281 ◽  
Author(s):  
Ahmed Alsaedi ◽  
Mona Alsulami ◽  
Hari Srivastava ◽  
Bashir Ahmad ◽  
Sotiris Ntouyas
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Existence Theory ◽  
Ulam Stability ◽  
Point Boundary ◽  
Arbitrary Domain ◽  
Third Order ◽  
Order Ordinary Differential Equation ◽  
The Third ◽  
Nonlocal Nonlinear

We investigate the solvability and Ulam stability for a nonlocal nonlinear third-order integro-multi-point boundary value problem on an arbitrary domain. The nonlinearity in the third-order ordinary differential equation involves the unknown function together with its first- and second-order derivatives. Our main results rely on the modern tools of functional analysis and are well illustrated with the aid of examples. An analogue problem involving non-separated integro-multi-point boundary conditions is also discussed.

scholarly journals Analysis on free vibration and critical buckling load of a FGM porous rectangular plate

2021 ◽  
Vol 39 (2) ◽  
pp. 317-325
Author(s):  
Zhaochun Teng ◽  
Pengfei Xi
Keyword(s):  
Differential Equation ◽  
Elastic Modulus ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Rectangular Plate ◽  
Natural Frequencies ◽  
Gradient Index ◽  
Rectangular Plates ◽  
Buckling Loads ◽  
Governing Differential Equation

The properties of functionally gradient materials (FGM) are closely related to porosity, which has effect on FGM's elastic modulus, Poisson's ratio, density, etc. Based on the classical theory of thin plates and Hamilton principle, the mathematical model of free vibration and buckling of FGM porous rectangular plates with compression on four sides is established. Then the dimensionless form of the governing differential equation is also obtained. The dimensionless governing differential equation and its boundary conditions are transformed by differential transformation method (DTM). After iterative convergence, the dimensionless natural frequencies and critical buckling loads of the FGM porous rectangular plate are obtained. The problem is reduced to the free vibration of FGM rectangular plate with zero porosity and compared with its exact solution. It is found that DTM gives high accuracy result. The validity of the method is verified in solving the free vibration and buckling problems of the porous FGM rectangular plates with compression on four sides. The results show that the elastic modulus of FGM porous rectangular plate decreases with the increase of gradient index and porosity. Furthermore, the effects of gradient index and porosity on dimensionless natural frequencies and critical buckling loads are further analyzed under different boundary conditions with constant aspect ratio, and the effects of aspect ratio and load on dimensionless natural frequencies under different boundary conditions.

scholarly journals Analytical solution to the problem of heat transfer using boundary conditions of the third kind

Vestnik IGEU ◽  
2019 ◽  
pp. 67-74
Author(s):  
A.V. Eremin ◽  
K.V. Gubareva
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Heat Flux ◽  
Boundary Conditions ◽  
Flux Density ◽  
Heat Flux Density ◽  
Temperature Function ◽  
The Third ◽  
Approximate Analytical Solutions ◽  
Wide Range

Non-stationary heat transmission within solid bodies is described using parabolic and hyperbolic equations. Currently, numerical methods for studying the processes of heat and mass transfer in the flows of liquids and gases have disseminated. Modern programs allow the automatic construction of computational grids, solutions to the systems of equations and offer a wide range of tools for analysis. Approximate analytical solutions have significant advantages compared to numerical ones. In particular, the solutions obtained in an analytical form allow performing parametric analysis of the system under study, configuration and programming of measurement devices, etc. Based on the joint use of additional desired function and additional boundary conditions in the integral method of heat balance, a method of mathematical modeling for the heat transfer process in a plate under symmetric boundary conditions of the third kind has been developed. Using the heat flux density as a new desired function, the method for solving heat conduction problems with boundary conditions of the third kind has been proposed. Finding a solution to the partial differential equation with respect to the temperature function presents integrating an ordinary differential equation with respect to the heat flux density on the surface of the studied zone. It has been shown that isotherms appear on the surface of the plate with a certain initial velocity which depends on the heat transfer intensity. The calculation results have been compared to the exact solution. The presented method can be used in determining the heat flux density of buildings and heating devices, finding heat losses during convective heat transfer and designing heat transfer equipment. The results can be applied to increase the validity and reliability of the calculation of actual losses and balance of thermal energy. The method reliability, validity and a high degree of approximation with about 3% inaccuracy have been demonstrated. The accuracy of the solution depends on the number of approximations performed and is determined by the degree of the approximating polynomial.

ABOUT THE PROBLEM OF CLASSIFICATION OF THE CONCEPTS OF INFORMATICS STUDIED AT SECONDARY SCHOOL

2018 ◽  
pp. 4-7
Author(s):  
S. I. Zenko
Keyword(s):  
Secondary School ◽  
Foreign Language ◽  
Computer Science ◽  
Parts Of Speech ◽  
The Third ◽  
The Cross ◽  
Definition Of

The article raises the problem of classification of the concepts of computer science and informatics studied at secondary school. The efficiency of creation of techniques of training of pupils in these concepts depends on its solution. The author proposes to consider classifications of the concepts of school informatics from four positions: on the cross-subject basis, the content lines of the educational subject "Informatics", the logical and structural interrelations and interactions of the studied concepts, the etymology of foreign-language and translated words in the definition of the concepts of informatics. As a result of the first classification general and special concepts are allocated; the second classification — inter-content and intra-content concepts; the third classification — stable (steady), expanding, key and auxiliary concepts; the fourth classification — concepts-nouns, conceptsverbs, concepts-adjectives and concepts — combinations of parts of speech.

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