The Torsion of Elastic Cylinders with Regular Curvilinear Cross Sections

Bassali, W. A.

doi:10.1002/sapm1959381232

Cited by 4 publications

(5 citation statements)

References 9 publications

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“…7 in the following page. Refer to Bassali [3] for the details of the solution to the torsion problem.…”

Section: The Conformal Mapping Methodsmentioning

confidence: 99%

“…The first fraction in (2.5.14) is of the denominator to the numerator gives each parenthesis a power of -3. Expanding each parenthesis with the use of Binomial theorem (2.5.16), (2.5.18), and (2.5.20) for the case n = 2 is 0 for the case n = 2, is given by can use (2.5.6) to find 0 I for the case n = 2, which is[3] for the computation of the general result with any n = 2, 5.22) and (2.5.23) gives the torsional rigidity for any value n as follows:In this section, we use conformal mapping and Fourier series to solve a torsion problem for special curves where 2Note we change the contours n Γ as n C to avoid confusion of the gamma function ( ) cardioid as shown inFig. 8below.…”

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confidence: 99%

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Applications to boundary value problems

2018

CRM Monograph Series

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In this thesis, we solved the Saint-Venant's torsion problem for beams with different cross sections bounded by simple closed curves using various methods. In addition, we solved the flexure problem of beams with certain curvilinear cross sections. The first method was derived by Bassali and Obaid. We focused on cross sections bounded by hyperbolas, circular groves, lemniscate of Booth, and sectorial cross sections. The second method used Tchebycheff polynomials to solve the torsion problem corresponding to the circle and ellipse. The third method used conformal mapping to derive the solution of different cross sections bounded by curvilinear edges. The flexure problem has been reduced to six boundary value problems; three are Dirichlet and three are Neumann problems. We derived the flexure functions

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“…7 in the following page. Refer to Bassali [3] for the details of the solution to the torsion problem.…”

Section: The Conformal Mapping Methodsmentioning

confidence: 99%

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confidence: 99%

Applications to boundary value problems

2018

CRM Monograph Series

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“…Similarly, we also get as expected. Refer to Bassali [3] for the details of the solution to the torsion problem.…”

Section: Cross Sectionsmentioning

confidence: 99%

11 Applications to Boundary Value Problems

1983

Mathematics in Science and Engineering

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In this thesis, we solved the Saint-Venant's torsion problem for beams with different cross sections bounded by simple closed curves using various methods. In addition, we solved the flexure problem of beams with certain curvilinear cross sections.The first method was derived by Bassali and Obaid. We focused on cross sections bounded by hyperbolas, circular groves, lemniscate of Booth, and sectorial cross sections. The second method used Tchebycheff polynomials to solve the torsion problem corresponding to the circle and ellipse. The third method used conformal mapping to derive the solution of different cross sections bounded by curvilinear edges.The flexure problem has been reduced to six boundary value problems; three are Dirichlet and three are Neumann problems. We derived the flexure functions corresponding to a certain boundary. v ACKNOWLEDGEMENTS

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“…With the usual convention = 0 ( 8 > v) we see that the lower end of the range in ( 5 ) may be taken as v = 1 instead of v = nm. (3 To prove the two identities (24) and (37) in the paper we consider the function…”

Section: Apporidix Imentioning

confidence: 99%

“…The classical torsion problem has long been a favorite of the physicist and of the applied matheniatician. Ileal and complex variable methods have been applied to the problem for various forms of boundary including polygons, angles, sectors, liniacons, leniniscates and others; references are to be found in standard textbooks and comprehensive reviews mentioned in previous papers by one of thc authors [ 2 ] , [3], [a] and in a recent paper by the authors [5].…”

Section: Introductionmentioning

confidence: 99%

On the Torsion of Elastic Cylindrical Bars

Bassali

Obaid

1981

Z Angew Math Mech

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Certain trigonometric formulae and combinatorial identities are used to derive exact and closed expressions for the stress functions, torsional rigidities and peripheral shearing stresses of certain isotropic cylinders under torsion. The cross sections are bounded by the closed curves \documentclass{article}\pagestyle{empty}\begin{document}$ r = a\left| {\sin \frac{\theta }{n}} \right|^n \left({ - \pi < \theta \mathbin{\lower.3ex\hbox{$\buildrel<\over {\smash{\scriptstyle=}\vphantom{_x}}$}} \pi } \right) $\end{document} where n is a positive integer (n > 1). Complex variable methods are applied to obtain the complex torsion functions, torsional rigidities and shearing stresses when the sections are bounded by the contours \documentclass{article}\pagestyle{empty}\begin{document}$ r = a\cos ^n \frac{\theta }{n}\left({ - \frac{{n\pi }}{2} < \theta \mathbin{\lower.3ex\hbox{$\buildrel<\over {\smash{\scriptstyle=}\vphantom{_x}}$}} \frac{{n\pi }}{2}} \right) $\end{document}, where n is a positive number (0 < n ≦ 2). Numerical results are presented in the form of tables and graphs illustrating the variation of torsional rigidities and the distribution of tangential shearing stresses along the edges of the cross sections. The cardioid and lemniscate of Bernoulli are included as special cases.

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The Torsion of Elastic Cylinders with Regular Curvilinear Cross Sections

Cited by 4 publications

References 9 publications

Applications to boundary value problems

Applications to boundary value problems

11 Applications to Boundary Value Problems

On the Torsion of Elastic Cylindrical Bars

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