 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
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| Total articles in the database: | | 12854 |
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| In English (Mech. Solids): | | 4810 |
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| N. A. Bazarenko, "Operator method for solving the plane elasticity problem for a strip with periodic cuts," Mech. Solids. 42 (4), 630-639 (2007) |
| Year |
2007 |
Volume |
42 |
Number |
4 |
Pages |
630-639 |
| Title |
Operator method for solving the plane elasticity problem for a strip with periodic cuts |
| Author(s) |
N. A. Bazarenko (Rostov State Academy of Agricultural Machinery Industry, pl. Strany Sovetov 1, Rostov-on-Don, 344023, Russia) |
| Abstract |
In the present paper, we use the conformal mapping z/c=ζ−2asinζ (a, c − const, ζ=u+iv) of the strip {|v|≤v0, |u|<∞} onto the domain D, which is a strip with symmetric periodic cuts. For the domain D, in the orthogonal system of isometric coordinates u, v, we solve the plane elasticity problem. We seek the biharmonic function in the form F=Cψ0+Sψ0*+x(Cψ1−Sψ2)+y(Cψ2+Sψ1), where C(v) and S(v) are the operator functions described in [1] and ψ0(u),…,ψ2(u) are the desired functions. The boundary conditions for the function F posed for v=±v0 are equivalent to two operator equations for ψ1(u) and ψ2(u) and to two ordinary differential equations of first order for ψ0(u) and ψ0*(u) [2]. By finding the functions ψj(u) in the form of trigonometric series with indeterminate coefficients and by solving the operator equations, we obtain infinite systems of linear equations for the unknown coefficients. We present an efficient method for solving these systems, which is based on studying stable recursive relations. In the present paper, we give an example of analysis of a specific strip (a=1/4, v0=1) loaded on the boundary v=v0 by a normal load of intensity p. We find the particular solutions corresponding to the extension of the strip by the longitudinal force X∞ and to the transverse and pure bending of the strip due to the transverse force Y∞ and the constant moment M∞, respectively. We also present the graphs of normal and tangential stresses in the transverse cross-section x=0 and study the stress concentration effect near the cut bottom. |
| References |
| 1. | N. A. Bazarenko,
"An Operator Method for Solving
the Plane Problem of the Theory of Elasticity,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 3, 73-83 (2000)
[Mech. Solids (Engl. Transl.)]. |
| 2. | N. A. Bazarenko,
"Operator Method for Solving the Plane Elasticity Problem
in a Domain Bounded by Second-Order Curves,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 5, 50-61 (2003)
[Mech. Solids (Engl. Transl.)]. |
| 3. | L. V. Kantorovich and V. I. Krylov,
Approximate Methods of Higher Analysis
(Fizmatgiz, Moscow-Leningrad, 1962; Interscience Publishers, New York, 1958). |
| 4. | M. Abramowitz and I. A. Stegun,
Handbook of Mathematical Functions,
with Formulas, Graphs, and Mathematical Tables
(Dover, New York, 1972; Nauka, Moscow, 1979). |
| 5. | H. Neuber,
Kerbspannungslehre
(Springer, Berlin, 1958)
[in German]. |
|
| Received |
24 May 2004 |
| Link to Fulltext |
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