Geodesic
Stability of bars with variable rigidity
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 6 (2010), pp. 65-69

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A heterogeneous in length bar with a variable cross-section is considered. The axis of a bar, which joins the centers of gravity of all the cross-sections, is a straight line. The bar is compressed by a longitudinal force applied to the center of gravity of the boundary cross-section. The article describes the case of stability loss of the straight-line form of equilibrium of a bar, when both, linear and curved forms are possible. Approximate analytical formulas for critical compressive force in four cases of boundary conditions for periodically heterogeneous bar are obtained. In case of a bar with a stepped variation of its cross-section and which consists of only one period (the limiting case) the comparison of results, computed using obtained formulas, with exact solutions of stability equation known before is made. 
@article{VMUMM_2010_6_a14,
     author = {V. I. Gorbachev and O. B. Moskalenko},
     title = {Stability of bars with variable rigidity},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {65--69},
     publisher = {mathdoc},
     number = {6},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_2010_6_a14/}
}
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V. I. Gorbachev; O. B. Moskalenko. Stability of bars with variable rigidity. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 6 (2010), pp. 65-69. http://geodesic.mathdoc.fr/item/VMUMM_2010_6_a14/