Bending of Thin Ring-Sector Plates

1951 ◽  
Vol 18 (4) ◽  
pp. 359-363
Author(s):  
L. I. Deverall ◽  
C. J. Thorne
Keyword(s):  
Continuous Function ◽  
Bending Moment ◽  
Circular Ring ◽  
Circular Plates ◽  
Thin Ring ◽  
Special Cases ◽  
Edge Conditions

Abstract General expressions for the deflection of plates whose planform is a sector of a circular ring are given for cases in which the straight edges have arbitrary but given deflection and bending moment. The solutions are given for all combinations of physically important edge conditions on the two circular edges. Sectors of circular plates are included as special cases. Solutions are given for a general load which is a continuous function of r, and a sectionally continuous function of θ, where r and θ are the usual polar co-ordinates with the pole at the center of the ring. Several specific examples are given.

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.

An Approximate Analysis of Circular Plates With Variable Thickness

1982 ◽  
Vol 104 (3) ◽  
pp. 533-535
Author(s):  
A. K. Naghdi
Keyword(s):  
Variable Thickness ◽  
Simple Technique ◽  
Bending Moment ◽  
The Other ◽  
Uniform Pressure ◽  
Approximate Analysis ◽  
Circular Plates ◽  
Numerical Examples ◽  
Classic Theory ◽  
Rotationally Symmetric

Based on classic theory of beams and certain modifications, a simple technique is derived in order to obtain an approximate value of the maximum bending moment in a rotationally symmetric circular plate with a variable thickness. It is assumed that one of the two concentric boundaries of the plate is clamped, and the other is free. Numerical examples for both cases of constant and variable thickness plates subject to uniform pressure or rim line loading are presented.

Stationary distributions for dams with additive input and content-dependent release rate

1977 ◽  
Vol 9 (03) ◽  
pp. 645-663 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Continuous Function ◽  
Probability Measure ◽  
Stationary Distribution ◽  
Release Rate ◽  
Zero Set ◽  
Stationary Distributions ◽  
Special Cases ◽  
Necessary And Sufficient ◽  
Limiting Case ◽  
Finite Dam

Conditions are derived under which a probability measure on the Borel subsets of [0, ∞) is a stationary distribution for the content {Xt } of an infinite dam whose cumulative input {At } is a pure-jump Lévy process and whose release rate is a non-decreasing continuous function r(·) of the content. The conditions are used to find stationary distributions in a number of special cases, in particular when and when r(x) = x α and {A t } is stable with index β ∊ (0, 1). In general if EAt , < ∞ and r(0 +) > 0 it is shown that the condition sup r(x)>EA 1 is necessary and sufficient for a stationary distribution to exist, a stationary distribution being found explicitly when the conditions are satisfied. If sup r(x)>EA 1 it is shown that there is at most one stationary distribution and that if there is one then it is the limiting distribution of {Xt } as t → ∞. For {At } stable with index β and r(x) = x α , α + β = 1, we show also that complementing results of Brockwell and Chung for the zero-set of {Xt } in the cases α + β < 1 and α + β > 1. We conclude with a brief treatment of the finite dam, regarded as a limiting case of infinite dams with suitably chosen release functions.

scholarly journals Extension of Castigliano’s method for isotropic beams

2020 ◽  
Vol 231 (11) ◽  
pp. 4621-4640
Author(s):  
Juergen Schoeftner
Keyword(s):  
Shear Force ◽  
Strain Energy ◽  
Axial Stress ◽  
Bending Moment ◽  
Axial Direction ◽  
Present Contribution ◽  
Special Cases ◽  
Dummy Load ◽  
Complementary Strain ◽  
Castigliano’S Theorem

Abstract In the present contribution Castigliano’s theorem is extended to find more accurate results for the deflection curves of beam-type structures. The notion extension in the context of the second Castigliano’s theorem means that all stress components are included for the computation of the complementary strain energy, and not only the dominant axial stress and the shear stress. The derivation shows that the partial derivative of the complementary strain energy with respect to a scalar dummy parameter is equal to the displacement field multiplied by the normalized traction vector caused by the dummy load distribution. Knowing the Airy stress function of an isotropic beam as a function of the bending moment, the normal force, the shear force and the axial and vertical load distributions, higher-order formulae for the deflection curves and the cross section rotation are obtained. The analytical results for statically determinate and indeterminate beams for various load cases are validated by analytical and finite element results. Furthermore, the results of the extended Castigliano theory (ECT) are compared to Bernoulli–Euler and Timoshenko results, which are special cases of ECT, if only the energies caused by the bending moment and the shear force are considered. It is shown that lower-order terms for the vertical deflection exist that yield more accurate results than the Timoshenko theory. Additionally, it is shown that a distributed load is responsible for shrinking or elongation in the axial direction.

KNIFE-EDGE CONDITIONS AND THE MACRODYNAMICS OF SMALL OPEN ECONOMIES

2002 ◽  
Vol 6 (2) ◽  
pp. 307-335 ◽  
Author(s):  
Stephen J. Turnovsky
Keyword(s):  
Growth Model ◽  
Open Economies ◽  
Small Open Economies ◽  
Knife Edge ◽  
Generic Structure ◽  
Equilibrium Dynamics ◽  
Trade Offs ◽  
Special Cases ◽  
Edge Conditions ◽  
Endogenous Growth Model

Macrodynamic models of small open economies are inevitably characterized by “knife-edge conditions,” meaning that certain parameters are constrained for a viable equilibrium to exist. This paper examines the macrodynamic structure of such an economy and considers the role played by various standard knife-edge conditions. The dynamic model presented is sufficiently general so as to provide a unifying framework within which alternative models can be embedded. We identify three important models as special cases of this generic structure: (i) The traditional stationary Ramsey model, (ii) the endogenous growth model, and (iii) the nonscale growth model. We consider three margins along which knife-edge conditions are imposed. These include (i) preference parameters, (ii) production and employment characteristics, and (iii) openness of international financial markets. These restrictions are shown to play key roles in determining the equilibrium dynamics, and how the economy responds to various shocks. The existence of trade-offs between these knife-edge conditions is discussed.

Experimental Study on the Characteristics of Bolted Pipe Flange Connection Under Bending Moment and Internal Pressure

2020 ◽  
Author(s):  
Xing Zheng ◽  
Toshiyuki Sawa ◽  
Mei Feng ◽  
Honggui Ding
Keyword(s):  
Experimental Study ◽  
Internal Pressure ◽  
Bending Moment ◽  
Wind Load ◽  
Force Distribution ◽  
Flange Connection ◽  
Special Cases ◽  
Minimum Residual ◽  
Spiral Wound

Abstract After a bolted gasketed pipe flange connection is assembled, the pipe flange connection is usually subjected to some additional loads such as bending moment, own weight, wind load and so on. These additional loads will lead to changing the axial bolt force distribution of the pipe flange connection and the distribution will become more and more scattered. As a result, the minimum residual axial bolt force will be much smaller and the minimum contact gasket stress will decrease, so a leakage is easy to occur in the connection. In special cases such as earthquakes, the bolted pipe flange connection is usually subjected to a high bending moment. Then sometimes leakage accidents occur. In order to promote the safety of the connections and to avoid them being broken under the earthquakes, in the present paper, the equivalent pressure and the assembly efficiency in the pipe flange connection of class 150 4″ are measured experimentally. The leak rates of the connection using spiral-wound gasket when a bending moment was applied or not applied were measured to elicit the equivalent pressure. Moreover, some tightening procedures such as JIS B 2251, ASME PCC-1 Legacy and GB/T 38343 were applied to tighten the pipe flange connection. The axial bolt force distribution, the assembly efficiency based on the target axial bolt force and the assembly efficiency based on tightness parameter of the connection when bending moment was applied or not applied were measured, and the results are compared. As a result, the equivalent pressure under a given bending moment is obtained, and a difference of the equivalent pressure between our results and Kellogg’s results is demonstrated. In addition, the new assembly efficiency based on the tightness parameter is also measured under a given bending moment as well as internal pressure. Using the equivalent pressure and the assembly efficiency obtained in the present paper, a new design will be possible for pipe flange connections under bending moment.

AXISYMMETRIC VIBRATION OF POLAR ORTHOTROPIC CIRCULAR PLATES OF QUADRATICALLY VARYING THICKNESS RESTING ON ELASTIC FOUNDATION

2005 ◽  
Vol 05 (03) ◽  
pp. 387-408 ◽  
Author(s):  
N. BHARDWAJ ◽  
A. P. GUPTA
Keyword(s):  
Elastic Foundation ◽  
Ritz Method ◽  
Three Dimensional ◽  
Normal Modes ◽  
Mode Shapes ◽  
Axisymmetric Vibration ◽  
Circular Plates ◽  
Varying Thickness ◽  
Edge Conditions ◽  
Polar Orthotropic

This paper is concerned with the axisymmetric vibration problem of polar orthotropic circular plates of quadratically varying thickness and resting on an elastic foundation. The problem is solved by using the Rayleigh–Ritz method with boundary characteristic orthonormal polynomials for approximating the deflection function. Numerical results are computed for frequencies, nodal radii and mode shapes. Three-dimensional graphs are also plotted for the first four normal modes of axisymmetric vibration of plates with free, simply-supported and clamped edge conditions for various values of taper, orthotropy and foundation parameters.

Nonlinear Axisymmetric Static Analysis of Shallow Spherical Shells on Winkler-Pasternak Foundation

1987 ◽  
Vol 109 (1) ◽  
pp. 28-34 ◽  
Author(s):  
R. K. Jain ◽  
Y. Nath
Keyword(s):  
Static Analysis ◽  
Nonlinear Static Analysis ◽  
Normal Displacement ◽  
Pasternak Foundation ◽  
Chebyshev Series ◽  
Circular Plates ◽  
Spherical Shells ◽  
Linear Elastic ◽  
Annular Plates ◽  
Edge Conditions

In the present investigation nonlinear static analysis of thin axisymmetric circular plates, annular plates and shallow spherical shells resting on linear elastic Winkler-Pasternak foundation under uniformly distributed normal loads, has been carried out. Donnell-type governing differential equations expressed in terms of normal displacement and stress function have been employed and solved using Chebyshev series. A convergence study for Chebyshev series has been conducted. The influence of foundation stiffness parameters (K and G) on the response of circular plates, annulus and spherical shells has been studied for both the clamped and simply supported immovable edge conditions. A few typical snap-through results for shells are also included.

Parametric Instability of a Rotating Circular Ring With Moving, Time-Varying Springs

2005 ◽  
Vol 128 (2) ◽  
pp. 231-243 ◽  
Author(s):  
Sripathi Vangipuram Canchi ◽  
Robert G. Parker
Keyword(s):  
Parametric Instability ◽  
Circular Ring ◽  
Time Varying ◽  
Ring Gear ◽  
Geometric Description ◽  
Bending Vibrations ◽  
System Parameters ◽  
Ring Rotation ◽  
Special Cases ◽  
Instability Regions

Parametric instabilities of in-plane bending vibrations of a rotating ring coupled to multiple, discrete, rotating, time-varying stiffness spring-sets of general geometric description are investigated in this work. Instability boundaries are identified analytically using perturbation analysis and given as closed-form expressions in the system parameters. Ring rotation and time-varying stiffness significantly affect instability regions. Different configurations with a rotating and nonrotating ring, and rotating spring-sets are examined. Simple relations governing the occurrence and suppression of instabilities are discussed for special cases with symmetric circumferential spacing of spring-sets. These results are applied to identify possible conditions of ring gear instability in example planetary gears.

scholarly journals Generalized proportional fractional integral Hermite–Hadamard’s inequalities

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Tariq A. Aljaaidi ◽  
Deepak B. Pachpatte ◽  
Thabet Abdeljawad ◽  
Mohammed S. Abdo ◽  
Mohammed A. Almalahi ◽  
...  
Keyword(s):  
Continuous Function ◽  
Differential Equations ◽  
Fractional Calculus ◽  
Approximation Theory ◽  
Fractional Differential Equations ◽  
Fractional Integral ◽  
Fractional Operators ◽  
Uniqueness Of Solutions ◽  
Special Cases ◽  
Fractional Differential

AbstractThe theory of fractional integral inequalities plays an intrinsic role in approximation theory also it has been a key in establishing the uniqueness of solutions for some fractional differential equations. Fractional calculus has been found to be the best for modeling physical and engineering processes. More precisely, the proportional fractional operators are one of the recent important notions of fractional calculus. Our aim in this research paper is developing some novel ways of fractional integral Hermite–Hadamard inequalities in the frame of a proportional fractional integral with respect to another strictly increasing continuous function. The considered fractional integral is applied to establish some new fractional integral Hermite–Hadamard-type inequalities. Moreover, we present some special cases throughout discussing this work.

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