Variational formulations of the integral equation of stability of elastic bars

• Main Author: Kupavtsev Vladimir Vladimirovich
• Format: Article
• Language: English
• Published: Moscow State University of Civil Engineering (MGSU) 2012-10-01
• Series: Vestnik MGSU
• Subjects: kernel symmetrization; kernel; minimax problem; elastic bar; critical loading; variational formulation; stability; integral equation
• Online Access: http://vestnikmgsu.ru/files/archive/issues/2012/9/ru/20.pdf

The author considers the variational formulations of the problem of stability of non-uniformly compressed rectilinear elastic bars that demonstrate their variable longitudinal bending rigidity in the event of different classical conditions of fixation of bar ends. Identification of the critical bar loading value is presented as a minimax problem with respect to the loading parameter and to the transversal displacement of the bar axis accompanied by the loss of stability. The author demonstrates that the critical value of the loading parameter may be formulated as a solution to the dual minimax problem. Further, the minimax formulation is transformed into the problem of identification of eigenvalues in the bilinear symmetric and continuous form, which is equivalent to the identification of eigenvalues of a strictly positive, linear and completely continuous operator. The operator kernel is presented in the form of symmetrization of the non-symmetric kernel derived in an explicit form. Within the framework of the problem considered by the author, the bar ends are fixed as follows: (1) both ends are rigidly fixed, (2) one end is rigidly fixed, while the other one is pinned, (3) one end is rigidly fixed, while the other one is attached to the support displaceable in the transverse direction, (4) one end is rigidly fixed, while the other one is free, (5) one end is pinned, while the other one is attached to the support displaceable in the transverse direction, (6) both ends are pinned.