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Thin-Walled Hollow Shafts

Thin-Walled Hollow Shafts

In analyzing a thin-walled hollow shaft subjected to torsional loading, a segment with width dx is isolated for examination. Despite its equilibrium state, this segment faces torsional shearing forces at its ends. These forces are quantitatively described by the product of the longitudinal shearing stress on the segment's minor surface and the area of this surface, leading to the concept of shear flow. This shear flow is consistent throughout the structure, indicating a uniform distribution of...

Gauss's Law: Cylindrical Symmetry

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A charge distribution has cylindrical symmetry if the charge density depends only upon the distance from the axis of the cylinder and does not vary along the axis or with the direction about the axis. In other words, if a system varies if it is rotated around the axis or shifted along the axis, it does not have cylindrical symmetry. In real systems, we do not have infinite cylinders; however, if the cylindrical object is considerably longer than the radius from it that we are interested in,...

Deformation in a Circular Shaft

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One of the distinctive characteristics of circular shafts is their ability to maintain their cross-sectional integrity under torsion. In other words, each cross-section continues to exist as a flat, unaltered entity, simply rotating like a solid, rigid slab. To understand the distribution of shearing stress within such a shaft, consider a cylindrical section inside this circular shaft. This section has a length of L and a radius of R, with one end fixed. The radius of the cylindrical section is...

Torsion of Noncircular Members

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Circular shafts undergoing torsional stress maintain their cross-sectional integrity due to their axisymmetric nature. This symmetry ensures an even distribution of stress, allowing the shaft to withstand torsion without distorting. In contrast, square bars, lacking this axial symmetry, experience significant distortion across their cross-sections when subjected to torsion, with the exception of along their diagonals and at lines connecting midpoints. A detailed examination of a cubic element...

Relation between Poisson's ratio, Modulus of Elasticity and Modulus of Rigidity

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Deformation occurs in axial and transverse directions when an axial load is applied to a slender bar. This deformation impacts the cubic element within the bar, transforming it into either a rectangular parallelepiped or a rhombus, contingent on its orientation. This transformation process induces shearing strain. Axial loading elicits both shearing and normal strains. Applying an axial load instigates equal normal and shearing stresses on elements oriented at a 45° angle to the load axis.

Deformations in a Transverse Cross Section

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When a material is subjected to uniaxial stress, it elongates or contracts in the direction of the applied force, and also undergoes changes in the perpendicular directions. This behavior is crucial for understanding how materials behave under stress and is governed by mechanical properties such as Poisson's ratio v, which measures the ratio of transverse strain to axial strain.
As the material stretches, it expands or contracts in orthogonal directions to the load. This phenomenon varies...

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Finite Element Modeling for the Simulation of the Quasi-Static Compression of Corrugated Tapered Tubes

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Published on: January 6, 2023

Differential theory for anisotropic cylindrical objects with an arbitrary cross section.

Philippe Boyer¹

¹Département d’Optique P.M. Duffieux, Institut FEMTO-ST, CNRS UMR 6174, Université de Franche-Comté, Besançon, France. philippe.boyer@univ‑fcomte.fr

Journal of the Optical Society of America. A, Optics, Image Science, and Vision

|April 19, 2013

Summary

This summary is machine-generated.

This study extends differential theory for anisotropic cylinders with arbitrary cross-sections. The new method accurately calculates scattering matrices and validates results against circular cylinder diffraction.

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Area of Science:

Electromagnetism and Optics
Computational Physics
Materials Science

Background:

The differential theory is a key tool for analyzing electromagnetic wave scattering.
Anisotropic cylindrical structures present unique challenges due to their complex material properties and geometry.
Existing methods may struggle with arbitrary cross-sections and the inclusion of the origin within the structure.

Purpose of the Study:

To extend the differential theory for analyzing electromagnetic scattering from anisotropic cylinders with arbitrary cross-sections.
To develop a robust computational framework applicable to both cases: origin outside and origin inside the cylinder.
To validate the extended theory by comparing numerical results with known cases, such as diffraction by a circular cylinder.

Main Methods:

Extension of the differential theory to accommodate anisotropic cylindrical structures.
Development of a scattering matrix propagation algorithm for cylinders not containing the origin.
Application of Fourier-Bessel expansions to describe external fields.
Utilization of a semi-analytical theory for cylinders containing the origin.
Adaptation of the scattering matrix propagation algorithm for the case where the origin is inside the cylinder.

Main Results:

The extended differential theory successfully handles anisotropic cylinders with arbitrary cross-sections.
Numerical results demonstrate good agreement with established solutions for circular cylinder diffraction.
The theory is effectively applied to analyze diffraction by an elliptical cylinder, showcasing its versatility.

Conclusions:

The developed extended differential theory provides a powerful and accurate method for analyzing electromagnetic scattering from complex anisotropic cylindrical structures.
The approach is versatile, handling different geometric configurations and material properties.
This work offers a valuable tool for researchers and engineers in electromagnetics and related fields.