scholarly journals The torsion and flexure of shafting with keyways or cracks

1932 ◽  
Vol 138 (836) ◽  
pp. 607-634 ◽  
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Torsional Rigidity ◽  
Circular Cross Section ◽  
Transverse Load ◽  
Torsion Problem ◽  
Circular Section ◽  
Physical Conditions ◽  
The Cross ◽  
Straight Lines

The object of the paper is to investigate the properties of shafts of circular cross-section into which keyways or slits have been cut, first when subjected to torsion, and second when bent by a transverse load at one end. The torsion problem for similar cases has been treated by several writers. Filon has worked out an approximation to the case of a circular section with one or two keyways ; in his method the boundary of the cross-section was a nearly circular ellipse and the boundaries of the keyways were confocal hyperbolas. In particular he considered the case when the hyperbola degenerated into straight lines starting from the foci. The solution for a circular section with one keyway in the form of an orthogonal circle has been obtained by Gronwall. In each case the solution has been obtained by the use of a conformal trans­formation and this method is again used in this paper, the transformations used being ρ = k sn 2 t . ρ = k 1/2 sn t , ρ = k 1/2 sn 1/2 t where ρ = x + iy , t = ξ + i η. No work appears to have been done on the flexure problem which is here worked out for several cases of shafts with slits. 2. Summary of the Problems Treated . We first consider the torsional properties of shafts with one and with two indentations. In particular cases numerical results have been obtained for the stresses at particular points and for the torsional rigidity. The results for one indentation and for two indentations of the same width and approximately the same depth have been compared. We next consider the solution of the torsion problem for one, two or four equal slits of any depth from the surface towards the axis. The values of the stresses have not been worked out in these cases since the stress is infinite at the bottom of the slits. This in stress occurs because the physical conditions are not satisfied at the bottom of the slits, but as had been pointed out by Filon this does not affect the validity of the values of the torsional rigidity. We compare the effect on the torsional rigidity of the shaft of one, two and four slits of the same depth in particular cases. We also compare the results for one slit with those obtained by Filon by another method, and find very good agreement which is illustrated by a graph. The reduction in torsional rigidity due to a semicircular keyway is compared with that due to a slit of approximately the same depth. Finally the distortion of the cross-sections at right angles to the planes is investigated, and in this, several interesting and perhaps unexpected features appear. The relative shift of the two sides of the slits is calculated in several cases.

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.

Analysis of Twisting Stiffness for a Multifiber Composite Layer

1974 ◽  
Vol 41 (3) ◽  
pp. 658-662 ◽  
Author(s):  
C. W. Bert ◽  
S. Chang
Keyword(s):  
Cross Section ◽  
Least Squares Method ◽  
Rectangular Cross Section ◽  
Torsional Rigidity ◽  
Composite Layer ◽  
Circular Cross Section ◽  
Relaxation Method ◽  
Fiber Composites ◽  
Numerical Results ◽  
Torsion Problem

The twisting stiffness of a rectangular cross section consisting of a single row of solid circular cross-section fibers embedded in a matrix is analyzed. The problem is formulated as a Dirichlet torsion problem of a multielement region and solved by the boundary-point least-squares method. Numerical results for a single-fiber square cross section compare favorably with previous relaxation-method results. New numerical results for three and five-fiber composites suggest that the torsional rigidity of a multifiber composite can be approximated from the torsional rigidities of single and three-fiber models.

scholarly journals Flexure with shear and associated torsion in prisms of uni-axial and asymmetric cross-sections

1938 ◽  
Vol 237 (776) ◽  
pp. 161-229 ◽  
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Axial Symmetry ◽  
Harmonic Functions ◽  
Transverse Load ◽  
Constant Cross Section ◽  
Principal Axes ◽  
The Cross ◽  
Axes Of Symmetry ◽  
Perpendicular Axis

It is now well over eighty years ago since Barre de Saint-Venant reduced the problem of the beam of constant cross-section under the action of a single transverse load to the search for plane harmonic functions satisfying a certain condition round the boundary of the cross-section. The solutions due to Saint-Venant, which include the rectangular, elliptic and circular cross-sections, are all cases in which the cross-sections have two axes of symmetry at right angles, meeting of necessity in the centroid of the cross-section, and along these axes the single transverse load is resolved. These axes are principal axes, and his solution depends upon this fact. Some less useful solutions exist for the load along one axis of certain beams of such bi-axial symmetry of cross-section, the solutions not yet being known for the load along the perpendicular axis.

Resonant Gas Oscillations in a Linear Area Variation Cavity: Rectangular Versus Circular Cross Section

2015 ◽  
Vol 138 (1) ◽  
Author(s):  
Sonu K. Thomas ◽  
T. M. Muruganandam
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Second Harmonic ◽  
Circular Cross Section ◽  
High Amplitude ◽  
Rectangular Section ◽  
Closed Cavity ◽  
Circular Section ◽  
Resonance Frequencies ◽  
Area Variation

Resonant gas oscillations in a linear area variation closed cavity are investigated, for two duct cross sections: rectangular and circular. The resonance frequencies were similar for both the ducts. Increased drive amplitude produced higher distortions in the waveform. It was found that both resonators exhibited commensurate behavior. This is opposed to noncommensurate behavior observed in nonuniform circular cross section resonators. The rectangular section duct had higher energy than circular section duct, in second harmonic for the same drive amplitude. The results reveal that in order to achieve shockless high amplitude pressure oscillations in a duct, both nonuniform area variation and circular cross section are required.

Modeling of Strain and Stress States for Straight Rods with Particular Cross Sections Subjected to Torsion

2013 ◽  
Vol 837 ◽  
pp. 699-704
Author(s):  
Marilena Glovnea ◽  
Cornel Suciu
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Circular Cross Section ◽  
Segment Length ◽  
Circular Arc ◽  
Tangential Stresses ◽  
Applied Torque ◽  
The Cross ◽  
Stress States ◽  
Model Strain

In the case of shaft-hub joints with cylindrical pins found in both macro and micro-devices, a longitudinal gutter with an almost half-circular cross section is practiced along the length of the shaft segment. The center of the circular arc is placed on the circular edge of the cross section. The present paper aims to model strain and stress states within such a shaft, when the material elastic properties are known along with shaft segment length and applied torque. Using the MathCad environment, 3D and constant stress level plots were obtained for the distribution of tangential stresses over the cross section. After application of torque, the transverse cross sections shift and become anti-symmetric as illustrated by the obtained 3D and constant strain plots.

scholarly journals Saint-Venant’s torsion of thin-walled nonhomogeneous open elliptical cross section

2021 ◽  
Vol 11 (5) ◽  
pp. 151-158
Author(s):  
István Ecsedi ◽  
Ákos József Lengyel ◽  
Attila Baksa ◽  
Dávid Gönczi
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Torsional Rigidity ◽  
Elliptical Cross Section ◽  
Curvilinear Coordinates ◽  
Torsion Problem ◽  
Cross Sectional ◽  
Elliptical Cross ◽  
Thin Walled ◽  
The One

This paper deals with the Saint-Venant’s torsion of thin-walled isotropic nonhomogeneous open elliptical cross section whose shear modulus depends on the one of the curvilinear coordinates which define the cross-sectional area of the beam. The approximate solution of torsion problem is obtained by variational method. The usual simplification assumptions are used to solve the uniform torsion problem of bars with thin-walled elliptical cross-sections. An example illustrates the application of the derived formulae of shearing stress and torsional rigidity.

Material tailoring in functionally graded rods under torsion

2014 ◽  
Vol 228 (18) ◽  
pp. 3283-3295 ◽  
Author(s):  
F Liaghat ◽  
MR Hematiyan ◽  
A Khosravifard
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Torsional Rigidity ◽  
Optimization Methods ◽  
Arbitrary Cross Section ◽  
Functionally Graded ◽  
Material Tailoring ◽  
Graded Material ◽  
The Cross ◽  
Torsional Analysis

Material tailoring in functionally graded isotropic hollow rods of arbitrary cross section under torsion is studied. The purposes of material tailoring pursued in this paper are divided into two categories. In the first category, we find the variation of the volume fractions of constituents of a functionally graded member under torsion to obtain an appropriate distribution of shear stress over the cross section. In the second category, the torsional rigidity of a rod with a pre-defined mass is maximized by appropriate determination of the variation of constituents of the functionally graded material. Hollow rods are studied in this paper since they have higher torsional rigidity compared to solid members with the same mass. Meshless numerical methods are used for torsional analysis of the cross sections. Moreover, numerical optimization methods are used for material tailoring of the rods. Several examples with different cross sections are presented to investigate the usefulness of the proposed technique on achieving the mentioned purposes.

Experimental Studies Of Multilayer Glass Structures On Compression, Compression With Bending And Simple Bending

2019 ◽  
pp. 22-30
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Applied Load ◽  
Experimental Studies ◽  
Strength Properties ◽  
Research Topic ◽  
Normative Values ◽  
Laminated Glass ◽  
The Cross ◽  
Experimental Values

The work of multilayer glass structures for central and eccentric compression and bending are considered. The substantiation of the chosen research topic is made. The description and features of laminated glass for the structures investigated, their characteristics are presented. The analysis of the results obtained when testing for compression, compression with bending, simple bending of models of columns, beams, samples of laminated glass was made. Overview of the types and nature of destruction of the models are presented, diagrams of material operation are constructed, average values of the resistance of the cross-sections of samples are obtained, the table of destructive loads is generated. The need for development of a set of rules and guidelines for the design of glass structures, including laminated glass, for bearing elements, as well as standards for testing, rules for assessing the strength, stiffness, crack resistance and methods for determining the strength of control samples is emphasized. It is established that the strength properties of glass depend on the type of applied load and vary widely, and significantly lower than the corresponding normative values of the strength of heat-strengthened glass. The effect of the connecting polymeric material and manufacturing technology of laminated glass on the strength of the structure is also shown. The experimental values of the elastic modulus are different in different directions of the cross section and in the direction perpendicular to the glass layers are two times less than along the glass layers.

scholarly journals Total cross sections of eγ → $$ eX\overline{X} $$ processes with X = μ, γ, e via multiloop methods

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Roman N. Lee ◽  
Alexey A. Lyubyakin ◽  
Vyacheslav A. Stotsky
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Elliptic Integral ◽  
High Energy ◽  
Calculation Methods ◽  
Complete Elliptic Integral ◽  
Total Cross Sections ◽  
Multiloop Calculation ◽  
The Cross ◽  
Analytical Expressions

Abstract Using modern multiloop calculation methods, we derive the analytical expressions for the total cross sections of the processes e−γ →$$ {e}^{-}X\overline{X} $$ e − X X ¯ with X = μ, γ or e at arbitrary energies. For the first two processes our results are expressed via classical polylogarithms. The cross section of e−γ → e−e−e+ is represented as a one-fold integral of complete elliptic integral K and logarithms. Using our results, we calculate the threshold and high-energy asymptotics and compare them with available results.

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.

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