Geodesic
The modeling of loss in stability of a heavy rod placed on a rigid basis
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 8 (2016) no. 1, pp. 34-42 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Stability of the ponderable elastic column lying on an absolutely rigid basis is investigated. The ends of the column are closed up and do not turn at a core bend. The column contracts by longitudinal force. Previously a weightless column with freight in the middle was considered. Research is conducted at rather small values of deflections of the column. Curvature of a curved axis of the column is accepted equal by the second derivative of function of deflections. The problem is solved in the linearized statement. For definition of deflections of a curved axis of the column the linear differential equation of the fourth order is used. The nontrivial private solution of the differential equation under the boundary conditions corresponding to fixing conditions is only with an accuracy of indefinite coefficient. The isoperimetric condition expressing the column length invariance at a bend is applied to determination of this coefficient. The critical force with which the column loses stability is found. Cases of a full and partial bend of the column are considered. For determination of critical force, besides the condition of the existence of the nontrivial solution of the differential equation of a bend, the law of conservation of mechanical energy is applied. It is established that the critical force and configuration of a curved axis of the column depends not only on length of the column and geometry of cross section, but also on column material density.
Keywords: modeling of loss of stability; stability of a horizontal rod. 
@article{VYURM_2016_8_1_a4,
     author = {Yu. I. Dorogov},
     title = {The modeling of loss in stability of a heavy rod placed on a rigid basis},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matematika, mehanika, fizika},
     pages = {34--42},
     year = {2016},
     volume = {8},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VYURM_2016_8_1_a4/}
}
TY  - JOUR
AU  - Yu. I. Dorogov
TI  - The modeling of loss in stability of a heavy rod placed on a rigid basis
JO  - Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika
PY  - 2016
SP  - 34
EP  - 42
VL  - 8
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/VYURM_2016_8_1_a4/
LA  - ru
ID  - VYURM_2016_8_1_a4
ER  - 
%0 Journal Article
%A Yu. I. Dorogov
%T The modeling of loss in stability of a heavy rod placed on a rigid basis
%J Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika
%D 2016
%P 34-42
%V 8
%N 1
%U http://geodesic.mathdoc.fr/item/VYURM_2016_8_1_a4/
%G ru
%F VYURM_2016_8_1_a4
Yu. I. Dorogov. The modeling of loss in stability of a heavy rod placed on a rigid basis. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 8 (2016) no. 1, pp. 34-42. http://geodesic.mathdoc.fr/item/VYURM_2016_8_1_a4/

[1] Vol'mir A. S., The stability of deformable systems, Nauka Publ., M., 1967, 984 pp. (in Russ.)

[2] Panovko Ya. G., Gubanova I. I., Stability and oscillations of elastic systems, Nauka Publ., M., 1987, 352 pp. (in Russ.) | MR

[3] Rabotnov Yu. N., Deformable Solid Body Mechanics, Nauka Publ., M., 1988, 712 pp. (in Russ.)

[4] Dorogov Yu. I., “The stability of the shape of an unattached elastic rod with stiff flanges on its ends”, Journal of Applied Mathematics and Mechanics, 77:3 (2013), 338–345 | DOI | MR | Zbl

[5] Dorogov Yu. I., Stroitel'naya mekhanika i raschyet sooruzheniy, 2013, no. 3, 16–21 (in Russ.)

[6] Dorogov Yu. I., “Buckling of the Column with the Bent Butts”, Composite Mechanics and Design, 18:2 (2012), 255–266 (in Russ.)

[7] Dorogov Yu. I., “Buckling of the Elastic Column with Failing Support”, Composite Mechanics and Design, 16:1 (2010), 84–96 (in Russ.)

[8] Dorogov Yu. I., “The Longitudinal Bend of a Column with Failing Attachments”, Composite Mechanics and Design, 16:4 (2010), 575–586 (in Russ.)