On the Existence of a Solution for a Solid Circular Plate Bilaterally Supported Along Two Antipodal Boundary Arcs and Loaded by a Central Transverse Concentrated Force

2000 ◽  
Vol 68 (5) ◽  
pp. 809-812 ◽  
Author(s):  
G. Monegato ◽  
A. Strozzi
Keyword(s):  
Circular Plate ◽  
Plate Theory ◽  
Reaction Force ◽  
Mechanical Analysis ◽  
Square Root ◽  
Existence Of A Solution ◽  
Concentrated Force ◽  
Boundary Constraints ◽  
Title Problem ◽  
The One

A purely flexural mechanical analysis is presented for a thin, solid, circular plate, deflected by a central transverse concentrated force, and bilaterally supported along two antipodal periphery arcs, the remaining part of the boundary being free. This problem is modeled in terms of a singular integral equation of the Prandtl type, which possesses a unique solution expressed in terms of a reaction force containing a factor exhibiting square root endpoint singularities. This solution is then shown not to respect the requested boundary constraints. It is therefore concluded that, within the framework of the purely flexural plate theory, the title problem cannot admit the weighted L2 solution here examined. It cannot, however, be excluded that a solution to the title problem exists, which possesses stronger endpoint singularities than those examined in this paper, or is of a more general form than the one considered here.

Circular solid plate supported along an edge arc and deflected by a central transverse force

2001 ◽  
Vol 215 (4) ◽  
pp. 389-404 ◽  
Author(s):  
A Strozzi ◽  
P Vaccari
Keyword(s):  
Integral Equation ◽  
Unique Solution ◽  
Fredholm Integral Equation ◽  
Equilibrium Condition ◽  
Plate Theory ◽  
Reaction Force ◽  
Square Root ◽  
Plate Edge ◽  
Concentrated Force ◽  
Rotational Equilibrium

A purely flexural structural analysis is carried out for a thin solid circular plate, deflected by a static central transverse concentrated force, and simply supported along an edge arc, the remaining part of its periphery being free. This problem is modelled in terms of a Fredholm integral equation of the first kind, where the kernel is expressed analytically, and where the unknown function is the reaction force along the support. The initial equation is then modified into a new Fredholm integral equation of the first kind, which implicitly respects the condition imposed on the plate edge deflections by the rigidity of the support, but which has still to be coupled with the translational and rotational equilibrium conditions. By showing that a certain operator is a contraction mapping, it is demonstrated that this new integral equation coupled only with the translational equilibrium condition possesses a unique solution expressed in terms of a smooth function with square root singularities at the support ends. It is also shown that this unique solution, when expressed via Chebyshev polynomials, does not fulfil the rotational equilibrium condition, apart from the limit case when the plate is axisymmetrically supported. It is concluded that, in the framework of the purely flexural plate theory, the title problem does not possess any smooth solution with square root singularities at the ends. An approximate solution is nevertheless computed with the collocation method, by accepting limited undulations of the plate periphery.

Mechanical Analysis of a Thin Solid Circular Plate Deflected by Transverse Periphery Forces and by a Central Load

1995 ◽  
Vol 209 (2) ◽  
pp. 77-86 ◽  
Author(s):  
E Dragoni ◽  
A Strozzi
Keyword(s):  
Circular Plate ◽  
Series Solution ◽  
Analytical Form ◽  
Mechanical Analysis ◽  
Basic Model ◽  
Sinusoidal Load ◽  
Load Effects ◽  
Title Problem ◽  
Plate Loading ◽  
Central Load

A thin, solid, circular plate deflected by a number of transverse, concentrated, periphery forces, not necessarily angularly equispaced or equally oriented, and sustaining a central, transverse, equilibrating load, is mechanically analysed via a purely flexural model. This problem is interpreted as a angularly dephased combination of a relatively simple model. This fundamental scheme consists of a plate loaded by a single periphery force, by a central load of equal intensity and opposite direction, and by a sinusoidal periphery line loading whose wavelength equals the plate border and whose intensity renders the plate loading self-equilibrated. When a sequence of basic schemes, possibly referring to loads of different intensity, is combined, respecting the condition that the resultant of the periphery loads is equilibrated by a central force alone, the sinusoidal load effects vanish and the title problem is recovered. A series solution in terms of plate deflections is obtained for the basic model, whose coefficients are analytically evaluated via a computer algebra package. The series sum is expressed in finite terms involving the dilogarithmic function, valid over the whole plate region. In particular, the series is summed in analytical form for the whole plate periphery, along which the boundary deflections are expressed in closed form.

A Simple Calculation Method of the Load Factor and a Bolt Preload Determination Satisfying Allowable Leak Late for Bolted Pipe Flange Connections With Gaskets Subjected to Internal Pressure

2018 ◽  
Author(s):  
Toshiyuki Sawa ◽  
Koji Sato ◽  
Toshio Mabuchi
Keyword(s):  
Internal Pressure ◽  
Circular Plate ◽  
Calculation Method ◽  
Plate Theory ◽  
Reaction Force ◽  
Simple Calculation ◽  
Couple Problem ◽  
Load Factor ◽  
Real Rate ◽  
Bolt Preload

In designing bolted pipe flange connections with gaskets under internal pressure, it is important to predict an actual residual contact gasket stress in the connections. For estimating the reduced gasket stress, it is needed how to know the load factor of the connections with gaskets. In the previous paper (2017PVP), for predicting the load factor of the connections with gaskets, a new model was proposed using a circular plate theory. However, the rigidity of the flange hub was assumed and it is necessary to improve the model for calculation. In the present paper, a simple and more accurate calculation method is proposed using a circular plate theory taking into account the reaction force distribution at the gasket interfaces and the effect of flange hub. In addition, the effect of the flange hub is analyzed as a couple problem between a cylindrical shell (hub) and a circular plate. The obtained results of the load factor in the connections are in a fairly good agreement with those obtained from FEM. In the numerical calculations, the values of the load factor for JIS 10K flange connections and ASME flange connections with compressed sheet and spiral wound gaskets (from 2” to 24”) are shown. Using the obtained load factor, the residual contact gasket stress and an amount of gas leakage are predicted. For verification of the simple calculation method for obtaining the load factor and FEM results, experiments to measure the load factor and the amount of the leakage were conducted for 24” connection. The calculated results are compared with the experimental method. In addition, an issue how to determine the bolt preload for satisfying a give allowable real rate is demonstrated.

Exact deflection expressions for a thin solid circular plate loaded by periphery couples

2001 ◽  
Vol 215 (3) ◽  
pp. 341-351 ◽  
Author(s):  
A Nobili ◽  
A Strozzi ◽  
P Vaccari
Keyword(s):  
Closed Form ◽  
Circular Plate ◽  
Series Solution ◽  
Analytical Form ◽  
Mechanical Analysis ◽  
Boundary Line ◽  
Plate Edge ◽  
Line Load ◽  
Title Problem ◽  
Radial Axis

A mechanical analysis is carried out for a thin, solid, circular plate, deflected by a series of periphery-concentrated couples with a radial or circumferential axis. Although such couples need not be of equal intensity or angularly equispaced, they must constitute a self-equilibrated system of couples. This problem is decomposed into a combination of two basic models, the first of which considers a single periphery couple with a radial axis, and the second addresses an edge couple with a circumferential axis. In both models the concentrated border couple is equilibrated by a sinusoidal boundary line load of proper intensity, whose wavelength equals the plate edge. When such basic configurations are combined, respecting the condition that the system of concentrated couples be self-equilibrated, the effects of the sinusoidal loads cancel out, and the title problem is recovered. A classical series solution in terms of purely flexural plate deflections is achieved for the two basic models, where the series coefficients are computed with the aid of an algebraic manipulator. For both models, the series is summed in analytical form over the whole plate region. Closed-form deflection formulae can thus be easily derived from the two basic models for any combination of self-equilibrated edge couples, where some selected relevant situations are developed in detail.

scholarly journals Influence of 1D and 2D Carbon Fillers and Their Functionalisation on Crystallisation and Thermomechanical Properties of Injection Moulded Nylon 6,6 Nanocomposites

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Fabiola Navarro-Pardo ◽  
Ana L. Martínez-Hernández ◽  
Victor M. Castaño ◽  
José L. Rivera-Armenta ◽  
Francisco J. Medellín-Rodríguez ◽  
...  
Keyword(s):  
Mechanical Analysis ◽  
Thermomechanical Properties ◽  
Reduced Form ◽  
Scanning Calorimetry ◽  
Graphene Nanocomposites ◽  
Moulding Process ◽  
Nylon 6,6 ◽  
Injection Moulding Process ◽  
The One ◽  
Carbon Fillers

Carbon nanotubes (CNTs) and graphene were used as reinforcing fillers in nylon 6,6 in order to obtain nanocomposites by using an injection moulding process. The two differently structured nanofillers were used in their pristine or reduced form, after oxidation treatment and after amino functionalisation. Three low nanofiller contents were employed. Crystallisation behaviour and perfection of nylon 6,6 crystals were determined by differential scanning calorimetry and wide angle X-ray diffraction, respectively. Crystallinity was slightly enhanced in most samples as the content of the nanofillers was increased. The dimensionality of the materials was found to provide different interfaces and therefore different features in the nylon 6,6 crystal growth resulting in improved crystal perfection. Dynamical, mechanical analysis showed the maximum increases provided by the two nanostructures correspond to the addition of 0.1 wt.% amino functionalised CNTs, enhancing in 30% the storage modulus and the incorporation of 0.5 wt.% of graphene oxide caused an increase of 44% in this property. The latter also provided better thermal stability when compared to pure nylon 6,6 under inert conditions. The superior properties of graphene nanocomposites were attributed to the larger surface area of the two-dimensional graphene compared to the one-dimensional CNTs.

scholarly journals Qualitative analysis of an autocatalytic chemical reaction model with decay

2014 ◽  
Vol 144 (2) ◽  
pp. 427-446 ◽  
Author(s):  
Jun Zhou ◽  
Junping Shi
Keyword(s):  
Steady State ◽  
Chemical Reaction ◽  
Reaction Diffusion ◽  
Steady State Solution ◽  
Reaction Model ◽  
Existence Of A Solution ◽  
Positive Steady State ◽  
Chemical Reaction Model ◽  
The One ◽  
Necessary And Sufficient

In this paper, we revisit a reaction—diffusion autocatalytic chemical reaction model with decay. For higher-order reactions, we prove that the system possesses at least two positive steady-state solutions; hence, it has bistable dynamics similar to the system without decay. For the linear reaction, we determine the necessary and sufficient condition to ensure the existence of a solution. Moreover, in the one-dimensional case, we prove that the positive steady-state solution is unique. Our results demonstrate the drastic difference in dynamics caused by the order of chemical reactions.

scholarly journals Core overshooting under the light of fluid dynamics

2019 ◽  
Vol 82 ◽  
pp. 153-165
Author(s):  
M. Rieutord
Keyword(s):  
Fluid Dynamics ◽  
Viscous Dissipation ◽  
Mixing Layer ◽  
Mechanical Analysis ◽  
Scale Height ◽  
Early Type ◽  
Pressure Scale ◽  
Early Type Stars ◽  
The One

We discuss the possible contraints that are brought about by a fluid mechanical analysis of the overshooting phenomenon at the interface of convective cores and radiative envelopes of early-type stars. We investigate an improvement of Roxburgh’s criterion by taking into account the viscous dissipation but show that this criterion remains not stringent enough to be predictive. We then discuss the thickness of the overshooting layer and show that all estimates, including the one of Zahn (1991), lead to a very thin mixing layer typically less than a percent of the pressure scale height.

Extended Stokes series: laminar flow through a loosely coiled pipe

1978 ◽  
Vol 86 (1) ◽  
pp. 129-145 ◽  
Author(s):  
Milton Van Dyke
Keyword(s):  
Laminar Flow ◽  
Flow Speed ◽  
Square Root ◽  
Mean Flow ◽  
Dean Number ◽  
Curvature Ratio ◽  
Wide Range ◽  
Square Root Singularity ◽  
Flow Through ◽  
The One

Dean's series for steady fully developed laminar flow through a toroidal pipe of small curvature ratio has been extended by computer to 24 terms. Analysis suggests that convergence is limited by a square-root singularity on the negative axis of the square of the Dean number. An Euler transformation and extraction of the leading and secondary singularities at infinity render the series accurate for all Dean numbers. For curvature ratios no greater than$\frac{1}{250} $, experimental measurements of the laminar friction factor agree with the theory over a wide range of Dean numbers. In particular, they confirm our conclusion that the friction in a loosely coiled pipe grows asymptotically as the one-quarter power of the Dean number based on mean flow speed. This contradicts a number of incomplete boundary-layer analyses in the literature, which predict a square-root variation.

On the Design of a Long-Stroke Beam-Based Compliant Mechanism Providing Quasi-Constant Force

2019 ◽  
Author(s):  
Pietro Bilancia ◽  
Alessandro Geraci ◽  
Giovanni Berselli
Keyword(s):  
Compliant Mechanism ◽  
Design Method ◽  
Reaction Force ◽  
Constant Force ◽  
Physical Prototype ◽  
Flexible Behavior ◽  
Improved Design ◽  
The One ◽  
Long Stroke ◽  
Integrated Software

Abstract In this paper the design of a linear long-stroke quasi-constant force compliant mechanism (CM) is presented and discussed. Starting from a flexure-based slider-crank mechanism, providing the required constant force within a rather limited deflection range, the paper reports about the shape optimization carried out with the specific aim of extending the available CM operative range. The proposed device is suitable in several precision manipulation systems, which require to maintain a constant-force at their contact interface with the manipulated object. Force regulation is generally achieved by means of complex control algorithms and related sensory apparatus, resulting in a flexible behavior but also in high costs. A valid alternative may be the use of a purposely designed CM, namely a purely mechanical system whose shape and dimensions are optimized so as to provide a force-deflection behavior characterized by zero stiffness. In the first design step, the Pseudo-Rigid Body (PRB) method is exploited to synthesize the sub-optimal compliant configuration, i.e. the one characterized by lumped compliance. Secondly, an improved design alternative is evaluated resorting to an integrated software framework, comprising Matlab and ANSYS APDL, and capable of performing non-linear structural optimizations. The new embodiment makes use of a variable thickness beam, whose shape and dimensions have been optimized so as to provide a constant reaction force in an extended range. Finally, a physical prototype of the beam-based configuration is produced and tested, experimentally validating the proposed design method.

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