On the theory of rods. I. Derivations from the three-dimensional equations

1974 ◽  
Vol 337 (1611) ◽  
pp. 451-483 ◽  
Keyword(s):  
Cross Section ◽  
Constitutive Relations ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Beam Theory ◽  
Explicit Calculation ◽  
Approximation Procedure ◽  
Elastic Rods ◽  
Variable Cross ◽  
Free Energy Function

Starting with the three-dimensional theory of classical continuum mechanics, some aspects of both the nonlinear and the linear theories of elastic rods are discussed. Detailed attention is given to the derivation of constitutive equations for the linear isothermal theory of elastic rods of an isotropic material and of variable cross-section, deduced by an approximation procedure from the three-dimensional equations. Explicit linear constitutive relations are obtained for straight isotropic circular rods of non-uniform cross-section; the explicit calculation is carried out (in terms of an approximate specific Gibbs free energy function) in four distinct parts, since the complete system of equations involved separate into those appropriate for extensional, torsional and two flexural modes of deformation. A system of displacement differential equations is derived for flexure of a beam of variable circular cross-section; they reduce to those of the Timoshenko beam theory when the radius of the cross-section is constant.

Axially Symmetric Waves in Elastic Rods

1960 ◽  
Vol 27 (1) ◽  
pp. 145-151 ◽  
Author(s):  
R. D. Mindlin ◽  
H. D. McNiven
Keyword(s):  
Cross Section ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Elastic Rods ◽  
Axially Symmetric ◽  
Elastic Rod ◽  
One Dimensional ◽  
Axial Shear ◽  
Wave Numbers ◽  
Complex Wave

A system of approximate, one-dimensional equations is derived for axially symmetric motions of an elastic rod of circular cross section. The equations take into account the coupling between longitudinal, axial shear, and radial modes. The spectrum of frequencies for real, imaginary, and complex wave numbers in an infinite rod is explored in detail and compared with the analogous solution of the three-dimensional equations.

SAF file: It's really three dimensional file

2018 ◽  
Vol 14 (1) ◽  
pp. 1
Author(s):  
Prof. Dr. Jamal Aziz Mehdi
Keyword(s):  
Cross Section ◽  
Root Canal ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Circular Motion ◽  
Root Canal Treatment ◽  
Metal Core ◽  
Rotating Blade

The biological objectives of root canal treatment have not changed over the recentdecades, but the methods to attain these goals have been greatly modified. Theintroduction of NiTi rotary files represents a major leap in the development ofendodontic instruments, with a wide variety of sophisticated instruments presentlyavailable (1, 2).Whatever their modification or improvement, all of these instruments have onething in common: they consist of a metal core with some type of rotating blade thatmachines the canal with a circular motion using flutes to carry the dentin chips anddebris coronally. Consequently, all rotary NiTi files will machine the root canal to acylindrical bore with a circular cross-section if the clinician applies them in a strictboring manner

Boundary element method in numerical study of three-dimensional Stokes flows in channels of arbitrary shape

2012 ◽  
Vol 9 (1) ◽  
pp. 94-97
Author(s):  
Yu.A. Itkulova
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Cross Section ◽  
Numerical Study ◽  
Three Dimensional ◽  
Cylindrical Tube ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Periodic Flows ◽  
Element Method

In the present work creeping three-dimensional flows of a viscous liquid in a cylindrical tube and a channel of variable cross-section are studied. A qualitative triangulation of the surface of a cylindrical tube, a smoothed and experimental channel of a variable cross section is constructed. The problem is solved numerically using boundary element method in several modifications for a periodic and non-periodic flows. The obtained numerical results are compared with the analytical solution for the Poiseuille flow.

On the Flexural-Torsional Behavior of a Straight Elastic Beam Subject to Terminal Moments

1993 ◽  
Vol 60 (2) ◽  
pp. 498-505 ◽  
Author(s):  
Z. Tan ◽  
J. A. Witz
Keyword(s):  
Coordinate System ◽  
Cross Section ◽  
Equilibrium Equation ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Elastic Beam ◽  
Cartesian Coordinate System ◽  
Kinematic Constraints ◽  
Torsional Behavior ◽  
Cartesian Coordinate

This paper discusses the large-displacement flexural-torsional behavior of a straight elastic beam with uniform circular cross-section subject to arbitrary terminal bending and twisting moments. The beam is assumed to be free from any kinematic constraints at both ends. The equilibrium equation is solved analytically with the full expression for curvature to obtain the deformed configuration in a three-dimensional Cartesian coordinate system. The results show the influence of the terminal moments on the beam’s deflected configuration.

Bending Vibrations of Variable Section Beams

1956 ◽  
Vol 23 (1) ◽  
pp. 103-108
Author(s):  
E. T. Cranch ◽  
Alfred A. Adler
Keyword(s):  
Cross Section ◽  
Rectangular Cross Section ◽  
Circular Cross Section ◽  
Beam Theory ◽  
Cantilever Beams ◽  
Constant Depth ◽  
Bending Vibrations ◽  
Variable Section

Abstract Using simple beam theory, solutions are given for the vibration of beams having rectangular cross section with (a) linear depth and any power width variation, (b) quadratic depth and any power width variation, (c) cubic depth and any power width variation, and (d) constant depth and exponential width variation. Beams of elliptical and circular cross section are also investigated. Several cases of cantilever beams are given in detail. The vibration of compound beams is investigated. Several cases of free double wedges with various width variations are discussed.

Geometry Effects in Eulerian/Granular Simulation of a Turbulent FCC Riser with a (kg-?g)-KTGF Model

2010 ◽  
Vol 8 (1) ◽  
Author(s):  
Hamid Reza Nazif ◽  
Hassan Basirat Tabrizi ◽  
Farhad A Farhadpour
Keyword(s):  
Cross Section ◽  
Large Scale ◽  
Rectangular Cross Section ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Good Prediction ◽  
Model Predictions ◽  
Secondary Circulations ◽  
Sharp Corners ◽  
Granular Simulation

Three-dimensional, transient turbulent particulate flow in an FCC riser is modeled using an Eulerian/Granular approach. The turbulence in the gas phase is described by a modified realizable (kg-?g) closure model and the kinetic theory of granular flow (KTGF) is employed for the particulate phase. Separate simulations are conducted for a rectangular and a cylindrical riser with similar dimensions. The model predictions are validated against experimental data of Sommerfeld et al (2002) and also compared with the previously reported LES-KTGF simulations of Hansen et al (2003) for the rectangular riser. The (kg-?g)-KTGF model does not perform as well as the LES-KTGF model for the riser with a rectangular cross section. This is because, unlike the more elaborate LES-KTGF model, the simpler (kg-?g)-KTGF model cannot capture the large scale secondary circulations induced by anisotropic turbulence at the corners of the rectangular riser. In the cylindrical geometry, however, the (kg-?g)-KTGF model gives good prediction of the data and is a viable alternative to the more complex LES-KTGF model. This is not surprising as the circulations in the riser with a circular cross section are due to the curvature of the walls and not due to the presence of sharp corners.

scholarly journals Computational Study of Zigzag Spacer Design with Elliptical Cross-Section Filaments

2020 ◽  
Vol 307 ◽  
pp. 01047
Author(s):  
Gohar Shoukat ◽  
Farhan Ellahi ◽  
Muhammad Sajid ◽  
Emad Uddin
Keyword(s):  
Mass Transfer ◽  
Cross Section ◽  
Cross Sections ◽  
Computational Study ◽  
Flow Direction ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Two Dimensional ◽  
Permeate Flux ◽  
Elliptical Cross

The large energy consumption of membrane desalination process has encouraged researchers to explore different spacer designs using Computational Fluid Dynamics (CFD) for maximizing permeate per unit of energy consumed. In previous studies of zigzag spacer designs, the filaments are modeled as circular cross sections in a two-dimensional geometry under the assumption that the flow is oriented normal to the filaments. In this work, we consider the 45° orientation of the flow towards the three-dimensional zigzag spacer unit, which projects the circular cross section of the filament as elliptical in a simplified two-dimensional domain. OpenFOAM was used to simulate the mass transfer enhancement in a reverse-osmosis desalination unit employing spiral wound membranes lined with zigzag spacer filaments. Properties that impact the concentration polarization and hence permeate flux were analyzed in the domain with elliptical filaments as well as a domain with circular filaments to draw suitable comparisons. The range of variation in characteristic parameters across the domain between the two different configurations is determined. It was concluded that ignoring the elliptical projection of circular filaments to the flow direction, can introduce significant margin of error in the estimation of mass transfer coefficient.

Sign in / Sign up

Export Citation Format

Share Document