Torsion of the Truncated Hollow Orthotropic Elastic Body of Rotation
ISTVÁN ECSEDI, ATTILA BAKSA, MARWEN HABBACHI
Institute of Applied Mechanics,
University of Miskolc,
H-3515 Miskolc-Egyetemváros, Miskolc,
HUNGARY
Abstract: - This paper deals with the torsion of the body of rotation. The meridian section of the body is
bounded by two ellipses and two straight lines which are perpendicular to the axis of rotation of the body. The
material of the body is elastic and cylindrical orthotropic. To solve the torsion problem, the theory of the
torsion of shafts of varying circular cross-sections is used, which was developed by Mitchell and Töppl. An
analytical solution is given for the shearing stresses and circumferential displacement. A numerical example
illustrates the application of the presented analytical solution. The results of this paper can be used as a
benchmark solution to verify the accuracy of the results computed by finite element simulations and finite
different methods.
Key-Words: - Torsion of body of rotation, orthotropic, elastic, variable cross-section
Received: March 12, 2023. Revised: August 12, 2023. Accepted: September 13, 2023. Published: October 13, 2023.
1 Introduction
The torsional deformation of a body of rotation is a
very important topic in the mechanics of structures,
[1], [2], [3]. The book, [4], and the book, [5],
presents many works on the torsion of elastic shafts
of varying circular cross section. For a body of
rotations whose boundary surfaces are coordinate
surfaces of an orthogonal curvilinear coordinate
system the closed-form solutions are derived from
the torsional boundary value problem, [6]. There are
several works on the problem of the torsion
deformations of elastic bodies of rotation, [7], [8],
[9], [10]. It is not the aim of this paper to provide a
detailed list of the literature on this topic. In this
paper, the torsion of the hollow truncated body of
rotation is considered. The meridian section of the
orthotropic elastic body is bounded by two ellipses
and two straight lines that are perpendicular to the
axis of rotation of the body. The formulation of the
torsional boundary value problem is given in the
cylindrical coordinate system (Figure 1). The
meridian section of the truncated hollow body of
rotation is shown in Figure 2. The body of rotation
occupies the space domain in the three-
dimensional space
(1)
where
(2)
(3)
(4)
(5)
In equations (2), (3), (4) and (5) , are the
shear flexibility coefficients of the material of the
body of rotation, respectively.
Fig. 1: Cylindrical coordinates .
The boundary curve of the body is (see Figure
2)
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István Ecsedi, Attila Baksa, Marwen Habbachi
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(6)
(7)
(8)
(9)
(10)
Fig. 2: Meridian section of the truncated hollow
body of rotation
2 Formulation of the Problem
Michell-Föppl’s theory of the torsion of shafts of
varying circular cross-sections is based on the
following displacement field, [1], [2], [3], [4], [7].
(11)
The unit vectors of the cylindrical coordinate system
are denoted by , and (Figure 1). The non-
zero infinitesimal strains are the shearing strains
and
(12)
, (13)
where
(14)
For orthotropic homogeneous linearly elastic
material according to Hooke’s law, we have for the
shearing stresses and
(15)
Since there are no body forces, the condition of the
mechanical equilibrium is described by the
following stress equilibrium equation (Figure 2)
(16)
Let the general solution of equilibrium equation (9)
in terms of first-order stress function
can be represented as
(17)
A combination of equations (12), (13), (15) and (16)
gives
(18)
The solution of this partial differential equation
which satisfies the boundary condition
(19)
(20)
is as follows
(21)
Application of formula (17) gives for the shearing
stresses
(22)
(23)
The resultant of shearing stress
(24)
Determination of the function is based
on the following equation
(25)
The solution of the system of partial differential
equations (25) for
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(26)
The expression of the circumferential displacement
is
(27)
3 Numerical Example
The following data is used in the numerical example
The contour lines of the stress function
and the function of are shown
in Figure 3 and Figure 4. The contour lines of the
shearing stress are resultant and are presented in
Figure 5.
A simple computation gives the following values of
stress resultant
(28)
, (29)
, (30)
, (31)
4 Conclusions
An analytical solution is presented for the problem
of torsion of the truncated body of rotation. The
curved boundary surfaces of the considered body
are ellipsoids of rotation. The material of the body
of rotation is linearly elastic homogenous and
orthotropic. It is assumed that the deformations are
small and the formulation of the linearized theory of
elasticity can be used. A numerical example
illustrates the presented theory of the torsion of the
orthotropic body of rotation.
Fig. 3: The contour lines of stress function .
Fig. 4: The contour lines of the stress function
.
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Fig. 5: The contour lines of the shear stress
resultant.
References:
[1] Saada, A.S.: Elasticity. Theory and its
Application. London, Pergamon Press. (1974).
[2] Sokolnikoff, I.S.: Mathematical Theory of
Elasticity. McGraw-Hill, New York, (1956).
[3] Timoshenko, S.P. Goodier J.N.: Theory of
Elasticity. New York, McGraw Hill, 3rd
Edition, (1970).
[4] Arutynyan, N.H. Abramyan, B.L.: Torsion of
Elastic Bodies. in Russian, Moscow, 1963.
[5] Lekhniskii, S.G.: Torsion of Anisotropic and
Non-Homogenous Beams. in Russian, Moscow
(1971).
[6] Ecsedi, I.: A special case of the problem of
hollow-core solids of revolution. Acta
Technica Academiae Scientiarum Hungarica,
Vol. 98, No. 3-4, pp. 273-293, 1985.
[7] Szoljanik Kassa, K.V.: Torsion of Shafts of
Varying Circular Section, in Russian, Moscow:
Gosztelizdat 1949.
[8] Föppl, A.: Die Torsion von runden Staben mit
veranderlichen Durchmesser. Sitzungsb. Bayer
Akad. Wiss. Munchen, 35. pp. 249-305,
(1905).
[9] Banshchikova, I. A study of creep of
orthotropic rods under torsion using the
method of characteristic parameters. J Appl
Mech Tech Phy 64, 146–158 2023. doi:
10.1134/S0021894423010169
[10] Tsai CL, Wang CH, Hwang SF, Chen WT,
Cheng CY. The torsional rigidity of an
orthotropic bar of unidirectional composite
laminate part I: Theoretical solution. Journal of
Composite Materials. 2023;57(10):1729-1742.
doi: 10.1177/00219983231161819
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- István Ecsedi and Attila Baksa carried out the
investigation and the formal analysis. István
Ecsedi has implemented the algorithm for all the
examples.
- Attila Baksa and Marwen Habbachi were
responsible for the validation and visualization of
the results.
All authors have been writing the paper with
original draft, review, and editing.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
The author(s) received no financial support for the
research, authorship, and/or publication of this
article. Furthermore, on behalf of all authors, the
corresponding author states that there is no conflict
of interest.
Conflict of Interest
The authors have no conflict of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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