Geodesic
The integral formula in problems of the stability of inhomogeneous rods
Čebyševskij sbornik, Tome 22 (2021) no. 3, pp. 345-352

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A heterogeneous in length bar with a variable cross-section is considered. The bar is compressed by a variable longitudinal force which is distributed along its axis. The article describes the case of stability loss of the straight form of equilibrium of a bar, when both, linear and curved forms are possible. The critical combination of rigidity and longitudinal force is the result of an integral representation for the solution of the given stability equation with variable coefficients by the aid of the solution of similar equation but with constant coefficients. The integral representation includes the Green function of the given equation which can be obtained by the method of perturbations. The example of compiling of the equation for critical loading is reduced.
Keywords: Elasticity, stability, inhomogeneous rod, average method, the integral formula. 
@article{CHEB_2021_22_3_a19,
     author = {V. I. Gorbachev and A. A. Ruban},
     title = {The integral formula in problems of the stability of inhomogeneous rods},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {345--352},
     publisher = {mathdoc},
     volume = {22},
     number = {3},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2021_22_3_a19/}
}
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V. I. Gorbachev; A. A. Ruban. The integral formula in problems of the stability of inhomogeneous rods. Čebyševskij sbornik, Tome 22 (2021) no. 3, pp. 345-352. http://geodesic.mathdoc.fr/item/CHEB_2021_22_3_a19/