One-dimensional model of fourth order for rods with loading on lateral boundary: The case of rectangular cross section

2016 ◽  
Vol 22 (12) ◽  
pp. 2269-2287 ◽  
Author(s):  
Erick Pruchnicki
Keyword(s):  
Cross Section ◽  
Fourth Order ◽  
Displacement Field ◽  
Rectangular Cross Section ◽  
Transverse Dimension ◽  
Lateral Boundary ◽  
Dimensional Model ◽  
Equilibrium Equations ◽  
Lagrange Equations ◽  
One Dimensional

We propose deducing from three-dimensional elasticity a one dimensional model of a beam when the lateral boundary is not free of traction. Thus the simplification induced by the order of magnitude of transverse shearing and transverse normal stress must be removed. For the sake of simplicity we consider a beam with rectangular cross section. The displacement field in rods can be approximated by using a Taylor–Young expansion in transverse dimension of the rod and we truncate the potential energy at the fourth order. By considering exact equilibrium equations, the highest-order displacement field can be removed and the Euler–Lagrange equations are simplified.

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.

Simulation on Composite Backscattering from Rough Surface with Target above it

2012 ◽  
Vol 490-495 ◽  
pp. 603-607
Author(s):  
Wei Tian ◽  
Xin Cheng Ren
Keyword(s):  
Rough Surface ◽  
Root Mean Square ◽  
Cross Section ◽  
Correlation Length ◽  
Rectangular Cross Section ◽  
Mean Square ◽  
Method Of Moment ◽  
Backscattering Coefficient ◽  
One Dimensional ◽  
Surface Fluctuation

One-dimensional Gaussion rough surface is simulated and employed by Monte Carlo Method, the composite backscattering from one-dimensional Gaussion rough surface with rectangular cross-section column above it is studied using Method of Moment. The curves of composite backscattering coefficient with scattering angle and frequency of incident wave are simulated by numerical calculation, the influence of the root mean square and the correlation length of rough surface fluctuation, the height between the center of the rectangular cross-section column and the rough surface, the length and the width of the rectangular cross-section column is discussed. The characteristic of the composite back-scatting from one-dimensional Gaussion rough surface with a rectangular cross-section column above it is obtained. The results show that the influences of the root mean square and the correlation length of rough surface fluctuation, the height between the center of the rectangular cross-section column and the rough surface, the width of the rectangular cross-section column on the composite backscattering coefficients are obvious while the influences of the length of the rectangular cross-section column on the complex backscattering coefficient is less.

scholarly journals A one-dimensional tearing mode equation for pedestal stability studies in tokamaks

2018 ◽  
Vol 84 (3) ◽  
Author(s):  
J. W. Connor ◽  
R. J. Hastie ◽  
C. Marchetto ◽  
C. M. Roach
Keyword(s):  
Cross Section ◽  
Approximate Equation ◽  
Stability Studies ◽  
Dimensional Model ◽  
Tearing Mode ◽  
One Dimensional ◽  
Tearing Modes ◽  
The Stability ◽  
Poloidal Flux

Starting from expressions in Connor et al. (Phys. Fluids, vol. 31, 1988, p. 577), we derive a one-dimensional tearing equation similar to the approximate equation obtained by Hegna & Callen (Phys. Plasmas, vol. 1, 1994, p. 2308) and Nishimura et al. (Phys. Plasmas, vol. 5, 1998, p. 4292), but for more realistic toroidal equilibria. The intention is to use this approximation to explore the role of steep profiles, bootstrap currents and strong shaping in the vicinity of a separatrix, on the stability of tearing modes which are resonant in the H-mode pedestal region of finite aspect ratio, shaped cross-section tokamaks, e.g. the Joint European Torus (JET). We discuss how this one-dimensional model for tearing modes, which assumes a single poloidal harmonic for the perturbed poloidal flux, compares with a model that includes poloidal coupling Fitzpatrick et al. (Nucl. Fusion, vol. 33, 1993, p. 1533).

scholarly journals Well-Posed Lateral Boundary Conditions for Spectral Semi-Implicit Semi-Lagrangian Schemes: Tests in a One-Dimensional Model

2009 ◽  
Vol 137 (1) ◽  
pp. 315-330 ◽  
Author(s):  
F. Voitus ◽  
P. Termonia ◽  
P. Bénard
Keyword(s):  
Boundary Conditions ◽  
Weather Prediction ◽  
Implicit Time ◽  
Lateral Boundary ◽  
Dimensional Model ◽  
One Dimensional ◽  
Lateral Boundary Conditions ◽  
Boundary Treatment ◽  
Truncation Scheme ◽  
Well Posed

Abstract The aim of this paper is to investigate the feasibility of well-posed lateral boundary conditions in a Fourier spectral semi-implicit semi-Lagrangian one-dimensional model. Two aspects are analyzed: (i) the complication of designing well-posed boundary conditions for a spectral semi-implicit scheme and (ii) the implications of such a lateral boundary treatment for the semi-Lagrangian trajectory computations at the lateral boundaries. Straightforwardly imposing boundary conditions in the gridpoint-explicit part of the semi-implicit time-marching scheme leads to numerical instabilities for time steps that are relevant in today’s numerical weather prediction applications. It is shown that an iterative scheme is capable of curing these instabilities. This new iterative boundary treatment has been tested in the framework of the one-dimensional shallow-water equations leading to a significant improvement in terms of stability. As far as the semi-Lagrangian part of the time scheme is concerned, the use of a trajectory truncation scheme has been found to be stable in experimental tests, even for large values of the advective Courant number. It is also demonstrated that a well-posed buffer zone can be successfully applied in this spectral context. A promising (but not easily implemented) alternative to these three above-referenced schemes has been tested and is also presented here.

Non-one-dimensional combustion of specimens with rectangular cross section

1983 ◽  
Vol 19 (4) ◽  
pp. 377-380 ◽  
Author(s):  
V. A. Vol'pert ◽  
A. V. Dvoryankin ◽  
A. G. Strunina
Keyword(s):  
Cross Section ◽  
Rectangular Cross Section ◽  
One Dimensional

scholarly journals Cross-Sectional Analysis of the Resistance of Rc Members Subjected to Bending With/without Axial Force

2021 ◽  
Author(s):  
Marek Lechman
Keyword(s):  
Cross Section ◽  
Axial Force ◽  
Rectangular Cross Section ◽  
Bending Moment ◽  
Equilibrium Equations ◽  
Cross Sectional ◽  
Cross Sectional Analysis ◽  
Nonlinear Stress ◽  
The Cross ◽  
Sectional Analysis

The paper presents section models for analysis of the resistance of RC members subjected to bending moment with or without axial force. To determine the section resistance the nonlinear stress-strain relationship for concrete in compression is assumed, taking into account the concrete softening. It adequately describes the behavior of RC members up to failure. For the reinforcing steel linear elastic-ideal plastic model is applied. For the ring cross-section subjected to bending with axial force the normalized resistances are derived in the analytical form by integrating the cross-sectional equilibrium equations. They are presented in the form of interaction diagrams and compared with the results obtained by testing conducted on RC columns under eccentric compression. Furthermore, the ultimate normalized bending moment has been derived for the rectangular cross-section subjected to bending without axial force. It was applied in the cross-sectional analysis of steel and concrete composite beams, named BH beams, consisting of the RC rectangular core placed inside a reversed TT welded profile. The comparisons made indicated good agreements between the proposed section models and experimental results.

Calculation of Self Inductance and Mutual Inductance of Coils with Rectangular Cross Section

2014 ◽  
Vol 9 (1) ◽  
pp. 10-14
Author(s):  
Aleksandr Ivanov
Keyword(s):  
Numerical Calculation ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Mutual Inductance ◽  
One Dimensional ◽  
Dimensional Integral

An inductance and mutual inductance of coils with rectangular cross section is given in a form of one-dimensional integral. The formula is appropriate for numerical calculation of the inductances with given accuracy

One-Dimensional Theories of Wave Propagation and Vibrations in Elastic Bars of Rectangular Cross Section

1966 ◽  
Vol 33 (3) ◽  
pp. 489-495 ◽  
Author(s):  
M. A. Medick
Keyword(s):  
Wave Propagation ◽  
Cross Section ◽  
Isotropic Material ◽  
Rectangular Cross Section ◽  
One Dimensional ◽  
Anisotropic Elastic

A method for constructing rational, one-dimensional theories of various orders of approximation, descriptive of wave propagation and vibrations in anisotropic elastic bars of rectangular cross section, is presented. As illustrations, several approximate theories are derived which are applicable to extensional motion in rectangular bars of isotropic material.

Numerical Three-Phase Model of the Two-Dimensional Steamflood Process

1970 ◽  
Vol 10 (04) ◽  
pp. 405-417 ◽  
Author(s):  
N.D. Shutler
Keyword(s):  
Fluid Flow ◽  
Cross Section ◽  
Oil Recovery ◽  
Vertical Cross Section ◽  
Recovery Efficiency ◽  
Two Dimensional ◽  
Dimensional Model ◽  
One Dimensional ◽  
Oil Water ◽  
Three Phases

Abstract This paper describes a numerical mathematical model that is a significant extension of a previously published one-dimensional model of the steamflood published one-dimensional model of the steamflood process. process. The model describes the simultaneous flow of the three phases - oil, water and gas - in two dimensions. Interphase mass transfer between water and gas phases is allowed, but the oil is assumed nonvolatile and the hydrocarbon gas insoluble in the liquid phases. The model allows two-dimensional heat convection within the reservoir and two-dimensional heat conduction in a vertical cross-section spanning the oil sand and adjacent strata. Example calculations are presented which, on comparison with experimental results, tend to validate the model. Steam overriding due to gravity effects is shown to significantly reduce oil recovery efficiency in a thick system while jailing to do so in a thinner system. A study of the effect of capillary pressure indicates that failure to scale capillary forces in laboratory models of thick sands may lead to optimistic recovery predictions, while properly scaled capillary forces may be sufficiently low as to play no important role in oil recovery. Calculations made with and without vertical permeability show that failure to account for vertical fluid flow can lead to predictions of pessimistic oil recovery efficiency. pessimistic oil recovery efficiency Introduction Mathematical tools of varying complexity have been used in studying the steamflood process. A "simplified" class of mathematical models has served primarily as aids in engineering design. A more comprehensive class of models has improved understanding of the nature of the process. The model described in this report is of the latter class, but it is more comprehensive than any previously published model. published model. All previously available calculations of the steamflood process are confined to one space dimension in their treatments of fluid flow. Thus all previous models necessary ignore all effects of gravity reservoir heterogeneity, and nonuniform initial fluid-phase distributions on fluid flow in a second dimension. This model, an extension of a previously published model accounts for heat and previously published model accounts for heat and fluid transfer in two space dimensions and, hence, can evaluate these effects on simultaneous horizontal and vertical flow. While the model can describe the areal performance of a steamflood (in which case the heat transfer is described in three dimensions), this aspect will not be considered in this paper. Rather, this paper will describe the model in its application to a vertical cross-section through the reservoir and will consider some preliminary investigations to demonstrate the importance of being able to simultaneously account for horizontal and vertical fluid flow. Mathematical details are given in appendices. MATHEMATICAL DESCRIPTION OF STEAMFLOODING Darcy's law provides expressions for the velocities of the three phases (oil, water and gas), which, when combined with oil, water and gas mass balances give the partial differential equations governing Now of the three phases within a reservoir sand: OIL PHASE ..(1) WATER PHASE ..(2) SPEJ P. 405

scholarly journals Effect of local stiffeners and warping constraints on the buckling of symmetric open thin-walled beams with high warping stiffness

Meccanica ◽  
2021 ◽  
Author(s):  
G. Piana ◽  
E. Lofrano ◽  
A. Carpinteri ◽  
G. Ruta
Keyword(s):  
Cross Section ◽  
Bending Stiffness ◽  
Dimensional Model ◽  
One Dimensional ◽  
Practical Applications ◽  
Thin Walled ◽  
Compressive Buckling ◽  
The Way

AbstractLocal stiffeners affect the behaviour of thin-walled beams (TWBs). An in-house code based on a one-dimensional model proved effective in several instances of compressive buckling of TWBs but gave counterintuitive results for locally stiffened TWBs. To clarify the matter, we investigated TWBs with multi-symmetric double I cross-section, widely used in practical applications where high bending stiffness is required. Several samples were manufactured and stiffened on purpose, closing them over a small portion of the axis at different places. The samples were tested with end constraints accounting for various warping conditions. The experimental and numerical outputs from a commercial FEM code gave a key to overcome the unexpected results by the in-house code, paving the way for further studies.

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