Application of Khas’minskii’s limit theorem to the buckling problem of a column with random initial deflections
Authors:
B. P. Videc and J. Lyell Sanders Jr.
Journal:
Quart. Appl. Math. 33 (1976), 422-428
DOI:
https://doi.org/10.1090/qam/99658
MathSciNet review:
QAM99658
Full-text PDF Free Access
Abstract |
References |
Additional Information
Abstract: An approximate asymptotic expression is obtained for the buckling load of an imperfect column resting on a nonlinear elastic foundation. The result holds for a large range of imperfection shapes, which are assumed to be stationary random functions of position. The asymptotic analysis is based on application of Khas’minskii’s limit theorem to equations for the slowly varying part of the deflection of the column. Previous results obtained for Gaussian imperfection shapes are shown to be valid also for the larger class of random imperfections considered here.
W. B. Fraser and B. Budiansky, The buckling of a column with random initial deflections, J. Appl. Mech. 36, 232–240 (1969)
J. C. Amazigo, B. Budiansky and G. F. Carrier, Asymptotic analysis of the buckling of imperfect columns on nonlinear elastic foundations, Int. J. Solids Struct. 6, 1341–1356 (1971)
J. C. Amazigo, Buckling of stochastically imperfect columns on nonlinear elastic foundations, Quart. Appl. Math. 29, 403–410 (1971)
R. Z. Khas’minskii, On stochastic processes defined by differential equations with a small parameter, Theory Prob. Applic. 211–228 (1966)
R. Z. Khas’minskii, A limit theorem for the solution of differential equations with random right-hand sides, Theory Prob. Applic., 390–406 (1966)
N. Krylov and N. Bogoliubov, Introduction to non-linear mechanics, Princeton University Press, Princeton, New Jersey, 1947
- N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic methods in the theory of non-linear oscillations, Translated from the second revised Russian edition, International Monographs on Advanced Mathematics and Physics, Hindustan Publishing Corp., Delhi, Gordon and Breach Science Publishers, New York, 1961. MR 0141845
- Bernard P. Videc, NONLINEAR STOCHASTIC BOUNDARY VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS, ProQuest LLC, Ann Arbor, MI, 1974. Thesis (Ph.D.)–Harvard University. MR 2940449
- G. C. Papanicolaou, Stochastic equations and their applications, Amer. Math. Monthly 80 (1973), 526–545. MR 317407, DOI https://doi.org/10.2307/2319609
W. B. Fraser and B. Budiansky, The buckling of a column with random initial deflections, J. Appl. Mech. 36, 232–240 (1969)
J. C. Amazigo, B. Budiansky and G. F. Carrier, Asymptotic analysis of the buckling of imperfect columns on nonlinear elastic foundations, Int. J. Solids Struct. 6, 1341–1356 (1971)
J. C. Amazigo, Buckling of stochastically imperfect columns on nonlinear elastic foundations, Quart. Appl. Math. 29, 403–410 (1971)
R. Z. Khas’minskii, On stochastic processes defined by differential equations with a small parameter, Theory Prob. Applic. 211–228 (1966)
R. Z. Khas’minskii, A limit theorem for the solution of differential equations with random right-hand sides, Theory Prob. Applic., 390–406 (1966)
N. Krylov and N. Bogoliubov, Introduction to non-linear mechanics, Princeton University Press, Princeton, New Jersey, 1947
N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic methods in the theory of non-linear oscillations, Hindustan Publishing Co., Delhi, India, 1961
B. P. Videc, Nonlinear stochastic boundary value problems for ordinary differential equations, Ph.D. Thesis, Harvard University, Cambridge, Mass., 1974
G. C. Papanicolaou, Stochastic equations and their applications, Amer. Math. Monthly 80, 542 (1973)
Additional Information
Article copyright:
© Copyright 1976
American Mathematical Society
