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.

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.

On a One-Dimensional Theory of Finite Torsion and Flexure of Anisotropic Elastic Plates

1981 ◽  
Vol 48 (3) ◽  
pp. 601-605 ◽  
Author(s):  
E. Reissner
Keyword(s):  
Linear Theory ◽  
Beam Theory ◽  
Bending Moment ◽  
Elastic Plates ◽  
Equilibrium Equations ◽  
Dimensional Theory ◽  
Cross Sectional ◽  
One Dimensional ◽  
Slender Beams ◽  
Anisotropic Elastic

Equations for small finite displacements of shear-deformable plates are used to derive a one-dimensional theory of finite deformations of straight slender beams with one cross-sectional axis of symmetry. The equations of this beam theory are compared with the corresponding case of Kirchhoff’s equations, and with a generalization of Kirchhoff’s equations which accounts for the deformational effects of cross-sectional forces. Results of principal interest are: 1. The equilibrium equations are seven rather than six, in such a way as to account for cross-sectional warping. 2. In addition to the usual six force and moment components of beam theory, there are two further stress measures, (i) a differential plate bending moment, as in the corresponding linear theory, and (ii) a differential sheet bending moment which does not occur in linear theory. The general results are illustrated by the two specific problems of finite torsion of orthotropic beams, and of the buckling of an axially loaded cantilever, as a problem of bending-twisting instability caused by material anisotropy.

Anisotropic Elastic Tubes of Arbitrary Cross Section Under Arbitrary End Loads: Separation of Beamlike and Decaying Solutions

2004 ◽  
Vol 72 (4) ◽  
pp. 500-510
Author(s):  
P. Ladevèze ◽  
J. G. Simmonds
Keyword(s):  
Cross Section ◽  
Arbitrary Cross Section ◽  
Constant Thickness ◽  
Cross Sectional ◽  
Edge Zone ◽  
One Dimensional ◽  
Elastic Tubes ◽  
Anisotropic Elastic ◽  
Relative Errors ◽  
Complex Valued

First approximation analytical solutions are constructed for finite and semi-infinite, fully anisotropic elastic tubes of constant thickness h and arbitrary cross section, subject to purely kinetic or purely kinematic boundary conditions. Final results contain relative errors of O(h∕R), where R is some equivalent cross sectional radius. Solutions are decomposed into the sum of an exact beamlike or Saint-Venant solution, treated in Ladevèze et al. (Int. J. Solids Struct., 41, pp. 1925–1944, 2004) and extended in an appendix; a rapidly decaying edge-zone solution; and a slowly decaying semi-membrane-inextensional-bending (MB) solution. Explicit conditions on the boundary data are given that guarantee decaying solutions. The MB solutions are expressed as an infinite series of complex-valued exponential functions times real-valued one-dimensional eigenfunctions which satisfy a fourth-order differential equation in the circumferential coordinate and depend on the pointwise cross sectional curvature only.

Long waves in a straight channel with non-uniform cross-section

2015 ◽  
Vol 770 ◽  
pp. 156-188 ◽  
Author(s):  
Patricio Winckler ◽  
Philip L.-F. Liu
Keyword(s):  
Boundary Layer ◽  
Wave Propagation ◽  
Cross Section ◽  
Three Dimensional ◽  
Channel Cross Section ◽  
Cross Sectional ◽  
New Model ◽  
One Dimensional ◽  
Uniform Channel ◽  
The Cross

A cross-sectionally averaged one-dimensional long-wave model is developed. Three-dimensional equations of motion for inviscid and incompressible fluid are first integrated over a channel cross-section. To express the resulting one-dimensional equations in terms of the cross-sectional-averaged longitudinal velocity and spanwise-averaged free-surface elevation, the characteristic depth and width of the channel cross-section are assumed to be smaller than the typical wavelength, resulting in Boussinesq-type equations. Viscous effects are also considered. The new model is, therefore, adequate for describing weakly nonlinear and weakly dispersive wave propagation along a non-uniform channel with arbitrary cross-section. More specifically, the new model has the following new properties: (i) the arbitrary channel cross-section can be asymmetric with respect to the direction of wave propagation, (ii) the channel cross-section can change appreciably within a wavelength, (iii) the effects of viscosity inside the bottom boundary layer can be considered, and (iv) the three-dimensional flow features can be recovered from the perturbation solutions. Analytical and numerical examples for uniform channels, channels where the cross-sectional geometry changes slowly and channels where the depth and width variation is appreciable within the wavelength scale are discussed to illustrate the validity and capability of the present model. With the consideration of viscous boundary layer effects, the present theory agrees reasonably well with experimental results presented by Chang et al. (J. Fluid Mech., vol. 95, 1979, pp. 401–414) for converging/diverging channels and those of Liu et al. (Coast. Engng, vol. 53, 2006, pp. 181–190) for a uniform channel with a sloping beach. The numerical results for a solitary wave propagating in a channel where the width variation is appreciable within a wavelength are discussed.

The Uniform Flexure of an Incomplete Tore

1949 ◽  
Vol 2 (4) ◽  
pp. 469
Author(s):  
W Freiberger ◽  
RCT Smith
Keyword(s):  
General Solution ◽  
Cross Section ◽  
Harmonic Functions ◽  
Rectangular Cross Section ◽  
Three Dimensional ◽  
Elasticity Problems ◽  
Three Dimensional Elasticity ◽  
Derivatives Of ◽  
Special Case

In this paper we discuss the flexure of an incomplete tore in the plane of its circular centre-line. We reduce the problem to the determination of two harmonic functions, subject to boundary conditions on the surface of the tore which involve the first two derivatives of the functions. We point out the relation of this solution to the general solution of three-dimensional elasticity problems. The special case of a narrow rectangular cross-section is solved exactly in Appendix II.

Numerical Investigation on Diffusion Slot Hole With Various Cross-Sectional End Shapes

2017 ◽  
Vol 139 (9) ◽  
Author(s):  
Bai-Tao An ◽  
Jian-Jun Liu
Keyword(s):  
Cross Section ◽  
Film Cooling ◽  
Separation Bubble ◽  
Rectangular Cross Section ◽  
Pressure Ratio ◽  
Smooth Transition ◽  
Shape Variation ◽  
Cross Sectional ◽  
Cooling Effectiveness ◽  
Shaped Hole

The diffusion hole constructed on a slot-type cross section has the potential to obtain high film cooling performance. However, the end shape of the cross section can greatly affect film cooling characteristics. This study examined eight cases of diffusion slot holes with various cross-sectional end shapes. The comparison of the eight diffusion slot holes and a typical fan-shaped hole was performed with a flat plate model using a three-dimensional (3D) steady computational fluid dynamics (CFD) method. The rectangular cross section had an aspect ratio of about 3.4. The end shape variation can be described based on sidewall contraction location, size, and form. The simulations were performed under an engine-representative condition of mainstream inlet Mach number 0.3 and turbulence intensity 5.2%. The simulated results showed that a strip separation bubble caused by inlet “jetting effect” occurs near the downstream wall of the diffusion slot hole and interacts with the diffusion flow. The different end shape of the rectangular cross section leads to different sidewall static pressure and exit velocity profiles, thereby produces three cooling effectiveness patterns, single-peak, bipeak, and tripeak patterns. The tripeak pattern produces higher cooling effectiveness and relatively uniform film coverage. The structure with moderate contraction and smooth transition on two sides of the downstream wall favors creation of a tripeak pattern. Compared with the fan-shaped hole, the discharge coefficient of diffusion slot hole is slightly small in low pressure ratio range, the pressure loss ratio has little difference.

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.

Film Cooling Investigation of a Slot-Based Diffusion Hole

2016 ◽  
Author(s):  
Bai-Tao An ◽  
Jian-Jun Liu ◽  
Si-Jing Zhou ◽  
Xiao-Dong Zhang ◽  
Chao Zhang
Keyword(s):  
Cross Section ◽  
Film Cooling ◽  
Rectangular Cross Section ◽  
Density Ratio ◽  
Vortex Structures ◽  
Cross Sectional ◽  
Cooling Effectiveness ◽  
Film Cooling Effectiveness ◽  
Blowing Ratio ◽  
Shaped Hole

This paper presents a new configuration of discrete film hole, i.e., the slot-based diffusion hole. Retaining the similar diffusion features to a traditional diffusion hole, the slot-based diffusion hole transforms the cross section of circle for the traditional diffusion hole to a flattened rectangle with respect to the equivalent cross-sectional area. Consequently, the exit width of the new hole is effectively enlarged. To verify the film cooling effectiveness, a low speed flat plate experimental facility incorporated with Pressure Sensitive Paint (PSP) measurement technique was employed to obtain the adiabatic film cooling effectiveness. The experiments were performed with hole pitch to diameter ratio p/D=6 and density ratio DR=1.38. The blowing ratio was varied from M=0.5 to M=2.5. A fan-shaped hole and two slot-based diffusion holes were tested and compared. Three-dimensional numerical simulation was employed to analyze the flow field in detail. The experimental results showed that the area averaged effectiveness of two slot-based diffusion holes is significantly higher than that of the fan-shaped hole when the blowing ratio exceeds 1.0. The slot-based diffusion hole demonstrates the great advantage over the fan-shaped hole at hole exit and maintains this to far downstream. The numerical results showed that the ends shape of the flattened rectangular cross section has large influences on film distribution patterns and downstream vortex structures. The semi-circle and straight line ends shapes lead to a bi-peak and a single-peak effectiveness pattern, respectively. The optimal ends shape can regulate the vortex structures and improve the film cooling effectiveness further.

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.

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.

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