Saint-Venant Decay Rates for the Rectangular Cross Section Rod

2004 ◽  
Vol 71 (3) ◽  
pp. 429-433 ◽  
Author(s):  
N. G. Stephen ◽  
P. J. Wang
Keyword(s):  
Finite Element ◽  
Aspect Ratio ◽  
Transfer Matrix ◽  
Cross Section ◽  
Cross Sections ◽  
Rectangular Cross Section ◽  
Decay Rates ◽  
Elastic Rod ◽  
Cross Sectional ◽  
Element Transfer

A finite element-transfer matrix procedure developed for determination of Saint-Venant decay rates of self-equilibrated loading at one end of a semi-infinite prismatic elastic rod of general cross section, which are the eigenvalues of a single repeating cell transfer matrix, is applied to the case of a rectangular cross section. First, a characteristic length of the rod is modelled within a finite element code; a superelement stiffness matrix relating force and displacement components at the master nodes at the ends of the length is then constructed, and its manipulation provides the transfer matrix, from which the eigenvalues and eigenvectors are determined. Over the range from plane stress to plane strain, which are the extremes of aspect ratio, there are always eigenmodes which decay slower than the generalized Papkovitch-Fadle modes, the latter being largely insensitive to aspect ratio. For compact cross sections, close to square, the slowest decay is for a mode having a distribution of axial displacement reminiscent of that associated with warping during torsion; for less compact cross sections, slowest decay is for a mode characterized by cross-sectional bending, caused by self-equilibrated twisting moment.

The shape of a Möbius band

1993 ◽  
Vol 440 (1908) ◽  
pp. 149-162 ◽  
Keyword(s):  
Aspect Ratio ◽  
Asymptotic Solution ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Elastic Rods ◽  
Large Deflections ◽  
Mechanical Equilibrium ◽  
Elastic Rod ◽  
Flexible Material ◽  
Möbius Band

The shape of a Möbius band made of a flexible material, such as paper, is determined. The band is represented as a bent, twisted elastic rod with a rectangular cross-section. Its mechanical equilibrium is governed by the Kirchhoff–Love equations for the large deflections of elastic rods. These are solved numerically for various values of the aspect ratio of the cross-section, and an asymptotic solution is found for large values of this ratio. The resulting shape is shown to agree well with that of a band made from a strip of plastic.

scholarly journals Cross-Sectional Geometry and the Intermetallics Structure in a High Pressure Die Cast Mg-Al Alloy

2010 ◽  
Vol 638-642 ◽  
pp. 1579-1584 ◽  
Author(s):  
A.V. Nagasekhar ◽  
Carlos H. Cáceres ◽  
Mark Easton
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Rectangular Cross Section ◽  
Circular Cross Section ◽  
Al Alloy ◽  
Cross Sectional ◽  
The Core ◽  
Die Cast ◽  
Increased Strength ◽  
Scanning Electron

Specimens of rectangular and circular cross section of a Mg-9Al binary alloy have been tensile tested and the cross section of undeformed specimens examined using scanning electron microscopy. The rectangular cross sections showed three scales in the cellular intermetallics network: coarse at the core, fine at the surface and very fine at the corners, whereas the circular ones showed only two, coarse at the core and fine at the surface. The specimens of rectangular cross section exhibited higher yield strength in comparison to the circular ones. Possible reasons for the observed increased strength of the rectangular sections are discussed.

scholarly journals The steady propagation of a semi-infinite bubble into a tube of elliptical or rectangular cross-section

2002 ◽  
Vol 470 ◽  
pp. 91-114 ◽  
Author(s):  
ANDREW L. HAZEL ◽  
MATTHIAS HEIL
Keyword(s):  
Aspect Ratio ◽  
Cross Section ◽  
Capillary Number ◽  
Stokes Equations ◽  
Rectangular Cross Section ◽  
Scaling Law ◽  
Propagation Speed ◽  
Cross Sectional ◽  
Steady Propagation ◽  
Rectangular Tubes

This paper investigates the propagation of an air finger into a fluid-filled, axially uniform tube of elliptical or rectangular cross-section with transverse length scale a and aspect ratio α. Gravity is assumed to act parallel to the tube's axis. The problem is studied numerically by a finite-element-based direct solution of the free-surface Stokes equations.In rectangular tubes, our results for the pressure drop across the bubble tip, Δp, are in good agreement with the asymptotic predictions of Wong et al. (1995b) at low values of the capillary number, Ca (ratio of viscous to surface-tension forces). At larger Ca, Wong et al.'s (1995b) predictions are found to underestimate Δp. In both elliptical and rectangular tubes, the ratio Δp(α)/Δp(α = 1) is approximately independent of Ca and thus equal to the ratio of the static meniscus curvatures.In non-axisymmetric tubes, the air-liquid interface develops a noticeable asymmetry near the bubble tip at all values of the capillary number. The tip asymmetry decays with increasing distance from the bubble tip, but the decay rate becomes very small as Ca increases. For example, in a rectangular tube with α = 1.5, when Ca = 10, the maximum and minimum finger radii still differ by more than 10% at a distance 100a behind the finger tip. At large Ca the air finger ultimately becomes axisymmetric with radius r∞. In this regime, we find that r∞ in elliptical and rectangular tubes is related to r∞ in circular and square tubes, respectively, by a simple, empirical scaling law. The scaling has the physical interpretation that for rectangular and elliptical tubes of a given cross-sectional area, the propagation speed of an air finger, which is driven by the injection of air at a constant volumetric rate, is independent of the tube's aspect ratio.For smaller Ca (Ca < Ca), the air finger is always non-axisymmetric and the persisting draining flows in the thin film regions far behind the bubble tip ultimately lead to dry regions on the tube wall. Ca increases with increasing α and for α > αˆ dry spots will develop on the tube walls at all values of Ca.

scholarly journals Shape Optimization of Bone-Bonding Subperiosteal Devices with Finite Element Analysis

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
Takeshi Ogasawara ◽  
Masayoshi Uezono ◽  
Kazuo Takakuda ◽  
Masanori Kikuchi ◽  
Shoichi Suzuki ◽  
...  
Keyword(s):  
Finite Element ◽  
Cross Section ◽  
Bonding Strength ◽  
Clinical Performance ◽  
Rectangular Cross Section ◽  
Optimum Shape ◽  
Bone Surface ◽  
Element Analysis ◽  
Cross Sectional ◽  
Rapid Acquisition

Subperiosteal bone-bonding devices have been proposed for less invasive treatments in orthodontics. The device is osseointegrated onto a bone surface without fixation screws and is expected to rapidly attain a bone-bonding strength that successfully meets clinical performance. Hence, the device’s optimum shape for rapid and strong bone bonding was examined in this study by finite element analyses. First, a stress analysis was performed for a circular rod device with an orthodontic force parallel to the bone surface, and the estimate of the bone-bonding strength based on the bone fracture criterion was verified with the results of an animal experiment. In total, four cross-sectional rod geometries were investigated: circular (Cr), elliptical (El), semicircular (Sc), and rectangular (Rc). By changing the height of the newly formed bone to mimic the progression of new bone formation, the estimation of the bone-bonding strength was repeated for each geometry. The rod with the Rc cross section exhibited the best performance, followed by those with the Sc, El, and Cr cross sections, from the aspects of the rapid acquisition of strength and the strength itself. Thus, the rectangular cross section is the best for rod-like subperiosteal devices for rapid bone bonding.

scholarly journals Effect of Non-Uniform Torsion on Elastostatics of a Frame of Hollow Rectangular Cross-Section

2018 ◽  
Vol 68 (2) ◽  
pp. 35-52
Author(s):  
Justín Murín ◽  
Mehdi Aminbaghai ◽  
Vladimír Goga ◽  
Vladimír Kutiš ◽  
Juraj Paulech ◽  
...  
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Cross Section ◽  
Cross Sections ◽  
Rectangular Cross Section ◽  
Twist Angle ◽  
Additional Degree ◽  
Linear Elastic ◽  
Solid Finite Elements ◽  
Stress And Deformation

AbstractIn this paper, results of numerical simulations and measurements are presented concerning the non-uniform torsion and bending of an angled members of hollow cross-section. In numerical simulation, our linear-elastic 3D Timoshenko warping beam finite element is used, which allows consideration of non-uniform torsion. The finite element is suitable for analysis of spatial structures consisting of beams with constant open and closed cross-sections. The effect of the secondary torsional moment and of the shear forces on the deformation is included in the local finite beam element stiffness matrix. The warping part of the first derivative of the twist angle due to bimoment is considered as an additional degree of freedom at the nodes of the finite elements. Standard beam, shell and solid finite elements are also used in the comparative stress and deformation simulations. Results of the numerical experiments are discussed, compared, and evaluated. Measurements are performed for confirmation of the calculated results.

Natural Frequencies of Pre-Twisted Airfoil Blades

2017 ◽  
Author(s):  
Neeraj Kavan Chakshu ◽  
Sunil K. Sinha
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Free Boundary ◽  
Cross Section ◽  
Cross Sections ◽  
Natural Frequencies ◽  
Rectangular Cross Section ◽  
Element Analysis ◽  
Free Boundary Conditions ◽  
Airfoil Blades

In this paper, the natural frequencies of pre-twisted cantilever blades of various angles of twist having different airfoil cross sections in the NACA 6 series have been determined. The main objectives of this paper are to replicate the results previously published for the similar types of blades but with the assumption of a uniform rectangular cross-section and to compare it with the results obtained for blades with more refined airfoil cross-sections. Cantilevered type clamped-free boundary conditions have been used in this paper for all blades. The comparison of the natural frequencies among different airfoils of the same NACA series has also been described in the paper in order to find out if any parameter of the airfoil such as camber, maximum thickness etc have any significant role in changing the frequencies of the beam. Commonly used commercial codes for finite element analysis have been used to determine these results.

scholarly journals Theory of “dissolution” and “condensation” of the physical geometric characteristics of an arbitrary cross-section under the action of torsion with bending

2019 ◽  
Vol 15 (4) ◽  
pp. 261-270
Author(s):  
Vladimir I. Kolchunov ◽  
Aleksej I. Demyanov ◽  
Nikolay V. Naumov
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Strain State ◽  
Rectangular Cross Section ◽  
Arbitrary Cross Section ◽  
Large Circle ◽  
Stress Strain ◽  
Cross Sectional ◽  
Tangential Stresses ◽  
Stress Strain State

Aim of research - to continue the development of methods for determining the stress-strain state of rods during torsion using materials resistance methods. Methods. A new approach for determining tangential torsional stresses for arbitrary cross sectional rods, based on simplified assumptions of material resistance, is proposed. The main feature of this approach is the approximation of rectangular or any complex cross section of reinforced concrete structures by describing a large circle around the cross section and splitting it into small squares with circles inscribed into them. Results. Three theorems have been formulated, the first of which relates the accumulation of tangential stresses (increments) from the edges of a rectangle to the middle of a rectangular section with the formula for determining tangent stresses for round sections. The second theorem allows to establish a connection between the tangential stresses calculated for each of the small squares-circles and the tangent stresses of the large circle through their increments. The third theorem makes it possible to find tangential stresses for each of the small square circles. The proposed approach allows to remove the need to use special tables for the calculation and not only in the elastic stage. It also makes it possible to separate the stress-strain state in the whole set of round cross-sections from the additional field caused by the deplanation of the rectangular cross-section. In addition, the proposed approach makes it possible to take into account the concentration of angular deformations in the incoming angles and other places with changing geometric parameters.

scholarly journals Engineering methods for calculation of elements from combined glued timber

2020 ◽  
Vol 11 (2) ◽  
pp. 79-90
Author(s):  
S. I. Bilyk ◽  
D. V. Mykhaіlovskyi
Keyword(s):  
Finite Element ◽  
Cross Section ◽  
Software Package ◽  
Strength Class ◽  
Rectangular Cross Section ◽  
Analytical Calculation ◽  
The Finite Element Method ◽  
Cross Sectional ◽  
Numerical Studies ◽  
Reduced Modulus

Extensive world experience in the implementation of building structures made of timber, in particular glued timber, for various purposes confirms the feasibility of their use. This is facilitated by the fact that glued timber effectively accumulates the positive properties of timber as a structural material level the shortcomings of solid timber. One of the types of constructions of glued timber are constructions of combined glued timber. Taking into account the structure and features of elements of combined glued timber of rectangular cross section, for a detailed analysis of the stress-strain state, a method is proposed, which consists in applying to standard formulas for calculating the reduced cross-sectional characteristics: reduced area, reduced moment of inertia, reduced moment of resistance. To calculation, the elements of combined glued timber of rectangular cross section according to the second limit state (serviceability), it is proposed to use the reduced modulus of elasticity of the section to the boards of the outer layers. To analyze the proposed method, a number of numerical studies of beams of combined and glued timber of the same strength class using analytical calculation methods and using the finite element method in the software package LIRA-CAD, using three-dimensional and flat finite elements. Numerical studies show that the results of calculations of beams of combined glued and glued timber of the same strength class differ within 20% in the direction of increasing the values of deflections and normal stresses in the elements of combined glued timber. Finite element calculations in the software package LIRA-CAD beams of combined and glued timber of the same strength class modeled volumetric and flat showed almost complete coincidence of results with a discrepancy of up to 2%, which suggests the need to significantly simplify the modeling, set elements from glued timber with flat finite elements. It is confirmed that the analytical calculation of beams of combined glued timber is recommended to be carried out according to the proposed method. The proposed technique allows to take into account the thickness and mechanical characteristics for the strength class of each board of which the glued cross section of the element, which significantly expands the range of use of combined glued timber. The high level of coincidence (within 5%) of the proposed analytical method with determination of the given cross-sectional characteristics with the results obtained by the finite element method for different cross-sections and spans of beams is confirmed, which allows to assert the expediency of its application in engineering calculations. In addition, the modeling of structures made of combined glued timber is possible with rod elements with the provision of the reduced modulus of elasticity according to the proposed method, which greatly simplifies the calculation of complex rod systems.

Automated 3D measurement of fiber cross section morphology in handsheets

2012 ◽  
Vol 27 (2) ◽  
pp. 264-269 ◽  
Author(s):  
Christian Lorbach ◽  
Ulrich Hirn ◽  
Johannes Kritzinger ◽  
Wolfgang Bauer
Keyword(s):  
Image Analysis ◽  
Cross Section ◽  
Cross Sections ◽  
Image Plane ◽  
3D Measurement ◽  
Cross Sectional ◽  
Fiber Cross Section ◽  
Fiber Wall ◽  
Sectional Shape ◽  
Cross Section Geometry

Abstract We present a method for 3D measurement of fiber cross sectional morphology from handsheets. An automated procedure is used to acquire 3D datasets of fiber cross sectional images using an automated microtome and light microscopy. The fiber cross section geometry is extracted using digital image analysis. Simple sample preparation and highly automated image acquisition and image analysis are providing an efficient tool to analyze large samples. It is demonstrated that if fibers are tilted towards the image plane the images of fiber cross sections are always larger than the true fiber cross section geometry. In our analysis the tilting angles of the fibers to the image plane are measured. The resulting fiber cross sectional images are distorted to compensate the error due to fiber tilt, restoring the true fiber cross sectional shape. We use an approximated correction, the paper provides error estimates of the approximation. Measurement results for fiber wall thickness, fiber coarseness and fiber collapse are presented for one hardwood and one softwood pulp.

The Exact One-Dimensional Theory for End-Loaded Fully Anisotropic Beams of Narrow Rectangular Cross Section

2001 ◽  
Vol 68 (6) ◽  
pp. 865-868 ◽  
Author(s):  
P. Ladeve`ze ◽  
J. G. Simmonds
Keyword(s):  
Cross Section ◽  
Plane Stress ◽  
Rectangular Cross Section ◽  
Dimensional Theory ◽  
Explicit Formulas ◽  
Cross Sectional ◽  
Rectangular Shape ◽  
One Dimensional ◽  
Anisotropic Elastic ◽  
Derivatives Of

The exact theory of linearly elastic beams developed by Ladeve`ze and Ladeve`ze and Simmonds is illustrated using the equations of plane stress for a fully anisotropic elastic body of rectangular shape. Explicit formulas are given for the cross-sectional material operators that appear in the special Saint-Venant solutions of Ladeve`ze and Simmonds and in the overall beamlike stress-strain relations between forces and a moment (the generalized stress) and derivatives of certain one-dimensional displacements and a rotation (the generalized displacement). A new definition is proposed for built-in boundary conditions in which the generalized displacement vanishes rather than pointwise displacements or geometric averages.

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