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Актуальные проблемы авиации и космонавтики - 2016. Том 1
УДК 539.3
THE CALCULATION OF THE EXACT VALUE OF THE COEFFICIENT OF THE REDUCTION LENGTH FOR GENERALIZED FORMULA EULER'S
I. R. Kravchuk, E. S. Romanova, N. V. Vasilyev, A. N. Tokoyakov Scientific Supervisor - R. A. Sabirov
Reshetnev Siberian State Aerospace University 31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660037, Russian Federation E-mail: [email protected]
A review of the literature on the generalization of the Euler critical force calculation by applying the coefficient length for any boundary conditions. A special case of calculating the exact value of the coefficient of reduction length.
Keywords: rod - beam, stability, strength of materials.
ВЫЧИСЛЕНИЕ ТОЧНОГО ЗНАЧЕНИЯ КОЭФФИЦИЕНТА ПРИВЕДЕНИЯ ДЛИНЫ ДЛЯ ОБОБЩЕННОЙ ФОРМУЛЫ ЭЙЛЕРА
И. Р. Кравчук, Е. С. Романова, Н. В. Васильев, А. Н. Токояков Научный руководитель - Р. А. Сабиров
Сибирский государственный аэрокосмический университет имени академика М. Ф. Решетнева
Российская Федерация, 660037, г. Красноярск, просп. им. газ. «Красноярский рабочий», 31
E-mail: [email protected]
Выполнен обзор литературы по обобщению формулы Эйлера вычисления критической силы с помощью применения коэффициента приведения длины для любых граничных условий. Рассмотрен частный случай вычисления точного значения коэффициента приведения длины.
Ключевые слова: стержень, устойчивость, сопротивление материалов.
A generalized formula Euler for calculating the critical force buckling rectilinear rod
l
lpr
A generalized formula Euler for calculating the critical force buckling rectilinear rod obtained in the course study of materials resistance based second order differential equation obtained by the calculation scheme.
^^ (2)
Here: Pkr - the critical force; E - Young's modulus; J - The moment of inertia of the cross section of the rod; lpr - The reduced length of the rod; v = v(x) - Deflection function.
Generalization of the formula (1) is administering the reduced equal length for different occasions securing rod
lpr = Цl, (3)
where ц - coefficient of the reduction length, and l - the length of the rod.
Секция «Механика конструкций ракетно-космической техники»
Comparing the allowable buckling to form the stability of the rod with pinchers and simply supported ends, prescribed ^. For example, in [1, p. 173], the length of the rod is divided into four equal parts and the shape of the two inner parts same as the shape of the rod with hinge-fixed supports. Similarly, "... I rack to the case, in which both ends are clamped and can not rotate, the middle portion of the rod length l /2 is in the same conditions as the rod at the ends of the supported-hinge" [2, p. 627]. Therefore, for this case it is adopted ^ = 0,5. That is, the authors is determined the coefficient ^ approximately by comparing the buckling.
To quote from [3, p. 16]: ". . . The solution strength of materials courses objectives are based on the second-order equation of the type (2). Here we get from (2) to the fourth order equation, as this will give the solution more generic and allow to extend it to other boundary conditions. Whereas, by differentiating (1) twice; get
EJ
d 4v
dx
4 = Pkr
d 2v
J„2 •
(4)
Note that in [3] we obtain the equation (4) by formal differentiation without explaining the physical meaning of this procedure, and apply a generalization of the formula for solving the problems of sustainability.
In [3, p. 24] we find that the concept of "reduced length" was introduced by Felix Yasinsky [4] that for arbitrary boundary conditions can be treated as equivalent to the length of the bar having at the ends of the hinge support.
In [5, p. 339] is also taken into account the reception comparison buckling to take account of boundary conditions. In [6] new methods of solving problems of strength of materials, in solving problems of the theory of stability of rods, used sustainability equation (2). Performed approximation of differential equation finite difference method, and the problem reduces to the problem of eigenvalues; used numerical calculations of the stability of the rod; and considered only rods with hinge bearings at the ends. Rigidly clamped and the free edge of the author can not be taken into account, since it uses the equation (2).
The dependence of the critical force of conditions anchorages rod in [7, pp. 442], [8, p. 450], [9, p. 380], using the concept of reduced length extends to bars having a support not only at the ends of the rod, and inside his flight. We indicate textbook [10, 433 pp.], And wherein the output is used in the calculation of the equation (4); in the calculation of the exact values obtained coefficient ^ .
Get the exact value of the coefficient ^, in a simpler way than in [10, p. 433], only for the particular problem. Consider clamped rod on both sides. We form the function of a possible buckling in the form
v( x)=111 - cos2T
(5)
in the interval 0 < x < l (see Figure), substituting that into (4) we have the critical power
P =
4EJ n
2
l
2
(6)
Deflection
Compare (6) and (1) determine thatlpr = l2 /4 . Using (3), that is |212 = l2 /4, we get the exact value of the length of the coefficient | = 0,5 for this task. It should be noted that the substitution of (5) in equation (2) does not give solutions.
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References
1. Timoshenko S. P. Strength of materials. V.2. More complex questions of theory and problems. M.-L. : OGIZ Gostekhizdat, 1946. 456 p.
2. Belyaev N. M. Strength of materials. State Publishing House technical and theoretical literature. M. : 1956. 856 p.
3. Volmir A. S. Stability of elastic systems. M. : Fizmatgiz, 1963. 880 p.
4. Yasinski F. S. On the resistance to buckling. 1894 Selected works on the stability of compressed rods. M. : Gostekhizdat, 1952.
5. Timoshenko S. P., Gera J. The mechanics of materials. M. : Mir, 1976. 670 p.
6. Varvak P. M. New methods for solving problems of strength of materials. M. : Vishcha school, 1977. 160 p.
7. Feodosyev V. I. Strength of materials. M. : Nauka, 1986. 560 p.
8. Pisarenko G. S., Yakovlev A. P., Matveev V. V. Handbook of resistance of materials. Kiev :. Nauk. Dumka, 1988. 736 p.
9. Gorshkov A. G., Troshin V. N., Shalashilin V. I. Strength of materials. M. : Fizmatlit, 2002. 544 p.
10. Birger I. A., Mavlyutov R. R. Strength of materials : Textbook. allowance. M. : Nauka, 1986.
560 p.
© Kravchuk I. R., Romanova E. S., Vasilyev N. V., Tokoyakov A. N., 2016
