Torsion of Noncylindrical Shafts of Circular Cross Section

1950 ◽  
Vol 17 (3) ◽  
pp. 275-282
Author(s):  
H. J. Reissner ◽  
G. J. Wennagel
Keyword(s):  
Cross Sections ◽  
Inverse Method ◽  
Direct Method ◽  
Circular Cross Section ◽  
Theory Of Elasticity ◽  
Elementary Functions ◽  
Single Term ◽  
Circumferential Displacement ◽  
Basic Differential Equation ◽  
Technical Interest

Abstract The theory of torsion of noncylindrical bodies of revolution, initiated by J. H. Michell and A. Föppl, is stated by a basic differential equation of the circumferential displacement and by a boundary condition of the shear stress along the generator surface. The solution of these two equations by the “direct” method of first assuming the boundary shape has not lent itself to closed solutions in terms of elementary functions, so that only approximation, infinite series, and experimental methods have been applied. A semi-inverse method analogous to Saint Venant’s semi-inverse method for cylindrical bodies has the disadvantage of the restriction to special boundary shapes but the advantage of exact solutions by means of elementary functions. By this method, bodies of conical, ellipsoidal, and hyperbolic boundary shapes have been obtained in a simple analysis. One class of integrals leading to other boundary shapes seems not to have been analyzed up to now, namely, the integrals in the form of a product of two functions of, respectively, axial (z) and radial (r) co-ordinates. A first suggestion of this possibility was given in Love’s treatise on the mathematical theory of elasticity. In the present paper, the classes of boundary shapes, displacements, and stress distributions are investigated analytically and numerically. The extent of the numerical investigation contains only the results of single-term integrals for full and hollow cross sections of technical interest. The detailed analysis of the boundary shapes, following from series integrals, presents essential mathematical obstacles. Overcoming these difficulties might lead to a multitude of solutions of interesting boundary shapes, and stress and strain distribution.

scholarly journals Asymptotic Behavior of Stable Structures Made of Beams

2021 ◽  
Author(s):  
Georges Griso ◽  
Larysa Khilkova ◽  
Julia Orlik ◽  
Olena Sivak
Keyword(s):  
Asymptotic Behavior ◽  
Cross Section ◽  
Cross Sections ◽  
Linear Elasticity ◽  
Circular Cross Section ◽  
Linear Elasticity Theory ◽  
Linear Elastic ◽  
The Cross ◽  
Small Rotation ◽  
Stable Structures

AbstractIn this paper, we study the asymptotic behavior of an $\varepsilon $ ε -periodic 3D stable structure made of beams of circular cross-section of radius $r$ r when the periodicity parameter $\varepsilon $ ε and the ratio ${r/\varepsilon }$ r / ε simultaneously tend to 0. The analysis is performed within the frame of linear elasticity theory and it is based on the known decomposition of the beam displacements into a beam centerline displacement, a small rotation of the cross-sections and a warping (the deformation of the cross-sections). This decomposition allows to obtain Korn type inequalities. We introduce two unfolding operators, one for the homogenization of the set of beam centerlines and another for the dimension reduction of the beams. The limit homogenized problem is still a linear elastic, second order PDE.

scholarly journals Effect of liquid cooling on PCR performance with the parametric study of cross-section shapes of microchannels

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Yousef Alihosseini ◽  
Mohammad Reza Azaddel ◽  
Sahel Moslemi ◽  
Mehdi Mohammadi ◽  
Ali Pormohammad ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Cross Section ◽  
Cross Sections ◽  
Heat Sink ◽  
Evaluation Criteria ◽  
Circular Cross Section ◽  
Microchannel Heat Sink ◽  
Liquid Cooling ◽  
Maximum Heat ◽  
Maximum Heat Transfer

AbstractIn recent years, PCR-based methods as a rapid and high accurate technique in the industry and medical fields have been expanded rapidly. Where we are faced with the COVID-19 pandemic, the necessity of a rapid diagnosis has felt more than ever. In the current interdisciplinary study, we have proposed, developed, and characterized a state-of-the-art liquid cooling design to accelerate the PCR procedure. A numerical simulation approach is utilized to evaluate 15 different cross-sections of the microchannel heat sink and select the best shape to achieve this goal. Also, crucial heat sink parameters are characterized, e.g., heat transfer coefficient, pressure drop, performance evaluation criteria, and fluid flow. The achieved result showed that the circular cross-section is the most efficient shape for the microchannel heat sink, which has a maximum heat transfer enhancement of 25% compared to the square shape at the Reynolds number of 1150. In the next phase of the study, the circular cross-section microchannel is located below the PCR device to evaluate the cooling rate of the PCR. Also, the results demonstrate that it takes 16.5 s to cool saliva samples in the PCR well, which saves up to 157.5 s for the whole amplification procedure compared to the conventional air fans. Another advantage of using the microchannel heat sink is that it takes up a little space compared to other common cooling methods.

Numerical Investigation of Cross-Section on Aluminum A6063 Thin-Walled Structures under Low-Velocity Impact Loading

2014 ◽  
Vol 1019 ◽  
pp. 96-102
Author(s):  
Ali Taherkhani ◽  
Ali Alavi Nia
Keyword(s):  
Energy Absorption ◽  
Cross Section ◽  
Cross Sections ◽  
Peak Load ◽  
Circular Cross Section ◽  
Absorption Capacity ◽  
Energy Absorption Capacity ◽  
Thin Walled Structures ◽  
Thin Walled ◽  
Crush Strength

In this study, the energy absorption capacity and crush strength of cylindrical thin-walled structures is investigated using nonlinear Finite Elements code LS-DYNA. For the thin-walled structure, Aluminum A6063 is used and its behaviour is modeled using power-law equation. In order to better investigate the performance of tubes, the simulation was also carried out on structures with other types of cross-sections such as triangle, square, rectangle, and hexagonal, and their results, namely, energy absorption, crush strength, peak load, and the displacement at the end of tubes was compared to each other. It was seen that the circular cross-section has the highest energy absorption capacity and crush strength, while they are the lowest for the triangular cross-section. It was concluded that increasing the number of sides increases the energy absorption capacity and the crush strength. On the other hand, by comparing the results between the square and rectangular cross-sections, it can be found out that eliminating the symmetry of the cross-section decreases the energy absorption capacity and the crush strength. The crush behaviour of the structure was also studied by changing the mass and the velocity of the striker, simultaneously while its total kinetic energy is kept constant. It was seen that the energy absorption of the structure is more sensitive to the striker velocity than its mass.

Effect of Gold Nanoparticles Size on Detection of Methylphosphonic Acid Hydrolysis Product of Nerve Agent

2021 ◽  
Vol 25 (7) ◽  
pp. 1-7
Author(s):  
Fellyzra Elvya Pojol ◽  
Buong Woei Chieng ◽  
Keat Khim Ong ◽  
Rashid Jahwarhar Izuan Abd ◽  
Mohd Junaedy Osman ◽  
...  
Keyword(s):  
Gold Nanoparticles ◽  
Inverse Method ◽  
Direct Method ◽  
Hydrolysis Product ◽  
Spherical Shape ◽  
Methylphosphonic Acid ◽  
Visible Spectroscopy ◽  
Transmission Electron ◽  
Citrate Reduction ◽  
Chloride Trihydrate

Citrate reduction of gold (III) chloride trihydrate (HAuCl4) is commonly used method to synthesise citrate-capped gold nanoparticles (cit-AuNPs). In this study, the sequence of reagents addition was modified (“inverse” method) to synthesise smaller size of cit-AuNPs than the standard Turkevich method (“direct” method). Ultraviolet-visible spectroscopy (UV-vis) and field emission transmission electron microscopy (FETEM) confirmed the formation of cit-AuNPs. The cit-AuNPs synthesized using “inverse” method are smaller in size (14.0 ± 3.03 nm) with uniform spherical shape compared to “direct” method (23.5 ± 7.52 nm). Smaller particles size of cit-AuNPs provide higher efficiency and sensitivity for detection of methylphosphonic acid (MPA) via colorimetric incorporated with image processing with a linear range from 2.5 to 12.5 mM and a low detection limit of 6.28 mM at shorter detection period (24 to 30 s).

scholarly journals GEOMETRIC MODELING OF A SHAPE OF PARALLELOGRAM PLATES IN A PROBLEM OF FREE VIBRATIONS USING CONFORMAL RADII

2021 ◽  
pp. 143-159
Author(s):  
A.A. Chernyaev ◽  
Keyword(s):  
Cross Sections ◽  
Geometric Modeling ◽  
Interpolation Method ◽  
Geometric Characteristic ◽  
Free Vibrations ◽  
Theory Of Elasticity ◽  
Constant Thickness ◽  
Basic Frequency ◽  
Plate Vibrations ◽  
The Subject

The paper considers a method of geometric modeling applied when solving basic twodimensional problems of the theory of elasticity and structural mechanics, in particular the applied problems of engineering. The subject of this study is vibrations of thin elastic parallelogram plates of constant thickness. To determine a basic frequency of vibrations, the interpolation method based on the geometric characteristic of the shape of plates (membrane, cross sections of a rod) is proposed. This characteristic represents a ratio of interior and exterior conformal radii of the plate. As is known from the theory of conformal mappings, conformal radii are those obtained by mapping of a plate onto the interior and exterior of a unit disk. The paper presents basic terms, tables, and formulas related to the considered geometric method with a comparative analysis of the curve diagrams obtained using various interpolation formulas. The original computer program is also developed. The main advantage of the proposed method of determining the basic frequency of plate vibrations is a graphic representation of results that allows one to accurately determine the required solution on the graph among the other solutions corresponding to the considered case of parallelogram plates. Although there are many known approximate approaches, which are used to solve the considered problems, only geometric modeling technique based on the conformal radii ratio gives such an opportunity.

scholarly journals Particle separation of a modified feed nozzle hydrocyclone under different operating parameters

2021 ◽  
Vol 11 (5) ◽  
pp. 159-170
Author(s):  
Zsolt Hegyes ◽  
Máté Petrik ◽  
L. Gábor Szepesi
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Rectangular Cross Section ◽  
Circular Cross Section ◽  
Particle Separation ◽  
Baffle Plate ◽  
Important Data ◽  
Preliminary Research ◽  
All Speeds ◽  
Plate Solution

During the operation of the hydrocyclone the cut size diameter is the most important data. This is connected to feed rate, which is closely related to the feed cross section. Preliminary research has revealed that square cross-section is more effective than circular cross-section. The research compared 2 types of feed cross sections at 5 different feed rates. One is a standard rectangular cross-section and the other is a square cross-section that narrows with a baffle plate. Preliminary calculations for cut size diameter have shown that better particle separation at all speeds can be achieved with the baffle plate solution. In both types, the increased velocity created decreased cut size diameter. During the simulation, the baffle plate did not cause any abnormalities in the internal pressure and velocity distributions. The simulation revealed that the particles did not behave as previously calculated.

scholarly journals Self-Inductance of the Circular Coils of the Rectangular Cross-Section with the Radial and Azimuthal Current Densities

Physics ◽  
2020 ◽  
Vol 2 (3) ◽  
pp. 352-367
Author(s):  
Slobodan Babic ◽  
Cevdet Akyel
Keyword(s):  
Cross Sections ◽  
Special Functions ◽  
Rectangular Cross Section ◽  
The Self ◽  
Thin Wall ◽  
Elementary Functions ◽  
Special Cases ◽  
Current Densities ◽  
New Formula ◽  
Azimuthal Current

In this paper, we give new formulas for calculating the self-inductance for circular coils of the rectangular cross-sections with the radial and the azimuthal current densities. These formulas are given by the single integration of the elementary functions which are integrable on the interval of the integration. From these new expressions, we can obtain the special cases for the self-inductance of the thin-disk pancake and the thin-wall solenoids that confirm the validity of this approach. For the asymptotic cases, the new formula for the self-inductance of the thin-wall solenoid is obtained for the first time in the literature. In this paper, we do not use special functions such as the elliptical integrals of the first, second and third kind, nor Struve and Bessel functions because that is very tedious work. The results of this work are compared with already different known methods and all results are in excellent agreement. We consider this approach novel because of its simplicity in the self-inductance calculation of the previously-mentioned configurations.

Non-Boussinesq gravity currents and surface waves generated by lock release in a circular-section channel: theoretical and experimental investigation

2019 ◽  
Vol 869 ◽  
pp. 610-633 ◽  
Author(s):  
L. Chiapponi ◽  
M. Ungarish ◽  
D. Petrolo ◽  
V. Di Federico ◽  
S. Longo
Keyword(s):  
Free Surface ◽  
Surface Waves ◽  
Cross Section ◽  
Cross Sections ◽  
Density Ratio ◽  
Circular Cross Section ◽  
Gravity Currents ◽  
Numerical Code ◽  
Long Distance ◽  
Front Speed

We present a combined theoretical and experimental study of lock-release inertial gravity currents (GCs) propagating in a horizontal channel of circular cross-section with open-top surface in the non-Boussinesq regime. A two-layer shallow-water (SW) model is developed for a generic shape of the cross-section with open top, and then implemented in a finite difference numerical code for the solution in a circular-cross-section channel of the type used in the experiments. The model predicts propagation with (almost) constant speed for a fairly long distance, accompanied by a depression of the ambient free open-top surface behind the front of the current. Sixteen experiments were conducted with a density ratio $r=0.587{-}0.939$ in full-depth and part-depth release conditions, measuring the front speed and the free-surface time series at four cross-sections. The channel was a circular tube 409 cm long, with a radius of 9.5 cm; the lengths of the locks were 52 and 103.5 cm. Density contrast was obtained by adding sodium chloride and dipotassium phosphate to fresh water. The theoretical values of the front speed and of the depression overestimate the experimental values, but they predict correctly their trend for varying parameters and provide reliable insights into the underlying mechanisms. In particular, we demonstrate that the circular cross-section increases the speed of propagation as compared to the standard rectangular cross-section case (for the same initial height and density ratio). The discrepancies between the SW predictions and the present experiments are of the same order of magnitude as those of previously published results for simpler systems (Boussinesq, rectangular). In addition to the depression, which is a wave bound to, and following the front of, the GC, the system also displays two kinds of free-surface waves, namely the initial bump (its amplitude is of the same order as the depression) and some short-length and low-amplitude waves in the tail of the bump. These free waves propagate with a celerity well predicted by the ‘fast’ eigenvalues of the mathematical model. Comparison is provided with the celerity of a solitary wave. It is expected that discrepancies between theory and experiments can be partly attributed to the presence of these waves. The reported insights and SW prediction method can be applied to a variety of cross-sections of practical interest (triangles, trapezoids, etc.).

The Elastic Plane With a Circular Insert, Loaded by a Tangentially Directed Force

1962 ◽  
Vol 29 (2) ◽  
pp. 362-368 ◽  
Author(s):  
M. Hete´nyi ◽  
J. Dundurs
Keyword(s):  
Closed Form ◽  
Elastic Properties ◽  
Classical Theory ◽  
Theory Of Elasticity ◽  
Elastic Plane ◽  
Elementary Functions ◽  
Explicit Formulas ◽  
Concentrated Force

The problem treated is that of a plate of unlimited extent containing a circular insert and subjected to a concentrated force in the plane of the plate and in a direction tangential to the circle. The elastic properties of the insert are different from those of the plate, and a perfect bond is assumed between the two materials. The solution is exact within the classical theory of elasticity, and is in a closed form in terms of elementary functions. Explicit formulas are given for the components of stress in Cartesian co-ordinates, and also in polar co-ordinates at the circumference of the insert.

scholarly journals The torsion of a semi-infinite elastic solid by an elliptical stamp

1968 ◽  
Vol 9 (1) ◽  
pp. 36-45
Author(s):  
Mumtaz K. Kassir
Keyword(s):  
Field Equation ◽  
Classical Theory ◽  
Cross Sections ◽  
Plane Surface ◽  
Boundary Surface ◽  
Elastic Solid ◽  
Theory Of Elasticity ◽  
Circular Area

The problem of determining, within the limits of the classical theory of elasticity, the displacements and stresses in the interior of a semi-infinite solid (z ≧ 0) when a part of the boundary surface (z = 0) is forced to rotate through a given angle ω about an axis which is normal to the undeformed plane surface of the solid, has been discussed by several authors [7, 8, 9, 1, 11, and others]. All of this work is concerned with rotating a circular area of the boundary surface and the field equation to be solved is, essentially, J. H. Mitchell's equation for the torsion of bars of varying circular cross-sections.

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