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.

Deflection of Beams of Varying Cross Section

1937 ◽  
Vol 4 (2) ◽  
pp. A49-A52
Author(s):  
Miklós Hetényi
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Original Problem ◽  
New Method ◽  
Constant Cross Section ◽  
Deflection Curve ◽  
Force System ◽  
Uniform Cross Section ◽  
The Cross

Abstract This paper calls attention to a new method of dealing with deflections of beams, the cross sections of which vary by steps. It is shown that the effect of this variation on the shape of the deflection curve can be represented by a properly chosen force system acting on a beam of uniform cross section. There is no approximation involved in this substitution, whereby the original problem is reduced to one of computing deflections of beams of constant cross section.

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.

The Centre of Shear of Aerofoil Sections

1953 ◽  
Vol 57 (508) ◽  
pp. 235-237 ◽  
Author(s):  
John A. Jacobs
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Lateral Surface ◽  
The Other ◽  
Principal Axes ◽  
Unstrained State ◽  
Uniform Cross Section ◽  
Single Force ◽  
The Cross ◽  
Fixed End

Consider a cantilever beam of uniform cross section whose generators are parallel to the z-axis and whose lateral surface is free from surface tractions. The line of centroids of the cross sections in the unstrained state is taken as the z-axis, and the x- and y-axes are the principal axes of the cross section at the centroid of the fixed end z = 0.The other end of the beam (z = l) is subject to forces which reduce to a single force with components (Wx, Wv, 0), transverse to the z-axis, acting through the load point L of this end section (see Fig. 1). The co-ordinates of L are taken as (p, q, l).

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 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.

ISOSPIN DECOMPOSITION OF pp → NNππ CROSS SECTIONS — DO WE SEE A SIGN OF Δ(1600) EXCITATION IN THE nnπ+π+ CHANNEL?

2009 ◽  
Vol 24 (02n03) ◽  
pp. 450-453
Author(s):  
◽  
T. SKORODKO ◽  
M. BASHKANOV ◽  
D. BOGOSLOWSKY ◽  
H. CALÉN ◽  
...  
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Differential Cross ◽  
Pion Production ◽  
Significant Contribution ◽  
Differential Cross Sections ◽  
Pp Collisions ◽  
Theoretical Predictions ◽  
Order Of Magnitude ◽  
The Cross

The two-pion production in pp-collisions has been investigated in exclusive measurements from threshold up to Tp = 1.36 GeV . Total and differential cross sections have been obtained for the channels pnπ+π0, ppπ+π-, ppπ0π0 and also nnπ+π+. For intermediate incident energies Tp > 1 GeV , i.e. in the region, which is beyond the Roper excitation but at the onset of ΔΔ excitation the total ppπ0π0 cross section falls behind theoretical predictions by as much as an order of magnitude near 1.2 GeV, whereas the nnπ+π+ cross section is a factor of five larger than predicted. A model-unconstrained isospin decompostion of the cross section points to a significant contribution of an isospin 3/2 resonance other than the Δ(1232). As a possible candidate the Δ(1600) is discussed.

scholarly journals The Cross Section for the J = 0 → 2 Rotational Excitation of Hydrogen by Slow Electrons

1969 ◽  
Vol 22 (6) ◽  
pp. 715 ◽  
Author(s):  
RW Crompton ◽  
DK Gibson ◽  
AI McIntosh
Keyword(s):  
Momentum Transfer ◽  
Cross Section ◽  
Cross Sections ◽  
Vibrational Excitation ◽  
The Cross ◽  
Excitation Processes ◽  
Slow Electrons ◽  
And Diffusion ◽  
Momentum Transfer Cross Section ◽  
Section Analysis

The results of electron drift and diffusion measurements in parahydrogen have been analysed to determine the cross sections for momentum transfer and for rotational and vibrational excitation. The limited number of possible excitation processes in parahydrogen and the wide separation of the thresholds for these processes make it possible to determine uniquely the J = 0 → 2 rotational cross section from threshold to 0.3 eV. In addition, the momentum transfer cross section has been determined for energies less than 2 eV and it is shown that, near threshold, a vibrational cross section compatible with the data must lie within relatively narrow limits. The problems of uniqueness and accuracy inherent in the swarm method of cross section analysis are discussed. The present results are compared with other recent theoretical and experimental determinations; the agreement with the most recent calculations of Henry and Lane is excellent.

scholarly journals Supplemental Material: Plate boundary trench retreat and dextral shear drive intracontinental fault-slip histories: Neogene dextral faulting across the Gabbs Valley and Gillis Ranges, Central Walker Lane, Nevada

2020 ◽  
Author(s):  
J. Lee ◽  
et al.
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Plate Boundary ◽  
Fault Slip ◽  
Cross Sectional ◽  
Walker Lane ◽  
Dextral Shear ◽  
The Cross

<div>Figure 6. Interpretative cross sections illustrating the cross-sectional geometry of several paleovalleys. See Figure 3 for location of all cross sections and Figure 8 for location of cross section CCʹ. Cross sections AAʹ and BBʹ are plotted at the same scale, and cross section CCʹ is plotted at a smaller scale. Figure 6 is intended to be viewed at a width of 45.1 cm.</div>

scholarly journals Analysis of Chemical Composition Homogeneity in the Cross-section of the Rods Produced from Alloys of 6xxx Group

2020 ◽  
Vol 66 (3) ◽  
pp. 139-148
Author(s):  
Maja Vončina ◽  
Peter Cvahte ◽  
Ana Kračun ◽  
Tilen Balaško ◽  
Jožef Medved
Keyword(s):  
Corrosion Resistance ◽  
Chemical Composition ◽  
Cross Section ◽  
Cross Sections ◽  
Intermetallic Phases ◽  
Scanning Calorimetry ◽  
Good Corrosion Resistance ◽  
Energy Dispersion Spectrometer ◽  
High Casting Speed ◽  
The Cross

AbstractThe alloys from Al–Mg–Si system provide an excellent combination of mechanical properties, heat treatment at extrusion temperature, good weldability, good corrosion resistance and formability. Owing to the high casting speed of rods or slabs, the solidification is rather non-equilibrium, resulting in defects in the material, such as crystalline segregations, the formation of low-melting eutectics, the unfavourable shape of intermetallic phases and the non-homogeneously distributed alloying elements in the cross-section of the rods or slabs and in the entire microstructure. The inhomogeneity of the chemical composition and the solid solution negatively affects the strength, the formability in the warm and the corrosion resistance, and can lead to the formation of undesired phases due to segregation in the material. In this experimental investigation, the cross-sections of the rods from two different alloys of the 6xxx group were investigated. From the cross-sections of the rods, samples for differential scanning calorimetry (DSC) at three different positions (edge, D/4 and middle) were taken to determine the influence of inhomogeneity on the course of DSC curve. Metallographic sample preparation was used for microstructure analysis, whereas the actual chemical composition was analysed using a scanning electron microscope (SEM) and an energy dispersion spectrometer (EDS).

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