Random Vibration of Flexible, Uncertain Beam Element

A stochastic finite element method is developed to analyze the random responses of geometrically nonlinear beams and frames with combined uncertain material and geometric properties under simultaneous spatial and temporal random excitations. The method is based on an equivalent linearization scheme, combined with the mean‐centered second‐order perturbation technique and the modal expansion approach. The procedure can be straightforwardly extended to incorporate other existing finite elements. Examples include large‐amplitude free vibration and dynamic random response of three simply supported beams and a portal frame with combined uncertain material and geometric properties. For the case of beams and frame with no structural uncertainties, the results obtained are in good agreement with alternative solutions. For the case of beams and frame with structural uncertainties, alternative representative results are also obtained using Monte Carlo simulation to compare and validate the present solution and method.

[1]  W. D. Mark,et al.  Random Vibration in Mechanical Systems , 1963 .

[2]  Humberto Contreras,et al.  The stochastic finite-element method , 1980 .

[3]  R. Herbert.  Random Vibrations of a Nonlinear Elastic Beam , 1964 .

[4]  E. Vanmarcke.  Probabilistic Modeling of Soil Profiles , 1977 .

[5]  Wing Kam Liu,et al.  Probabilistic finite elements for nonlinear structural dynamics , 1986 .

[6]  C. W. S. To.  THE RESPONSE OF NONLINEAR STRUCTURES TO RANDOM EXCITATION , 1984 .

[7]  Masanobu Shinozuka,et al.  Random Eigenvalue Problems in Structural Analysis , 1971 .

[8]  Masanobu Shinozuka,et al.  Stochastic Finite Element Methods in Dynamics , 1986 .

[9]  M. Shinozuka,et al.  Random fields and stochastic finite elements , 1986 .

[10]  T. Y. Yang,et al.  Buckled plate vibrations and large amplitude vibrations using high-order triangular elements , 1983 .

[11]  J. E. Goldberg,et al.  On the Bernoulli-Euler Beam Theory with Random Excitation , 1960 .

[12]  Wilfred D. Iwan,et al.  On the existence and uniqueness of solutions generated by equivalent linearization , 1978 .

[13]  Pol D. Spanos,et al.  Galerkin-Based Response Surface Approach for Reliability Analysis , 1990 .

[14]  Chuh Mei,et al.  Nonlinear multimode response of clamped rectangular plates to acoustic loading , 1986 .

[15]  Ted Belytschko,et al.  Probabilistic finite elements for transient analysis in nonlinear continua , 1985 .

[16]  P-T. D. Spanos,et al.  Formulation of Stochastic Linearization for Symmetric or Asymmetric M.D.O.F. Nonlinear Systems , 1980 .

[17]  R. Fox,et al.  Rates of change of eigenvalues and eigenvectors. , 1968 .

[18]  C. W. S. To.  Random Vibration of Nonlinear Systems , 1987 .

[19]  R. Ghanem,et al.  Stochastic Finite Element Expansion for Random Media , 1989 .

[20]  Weiqiu Zhu,et al.  Random Vibration: A Survey of Recent Developments , 1983 .

[21]  Mircea Grigoriu,et al.  STOCHASTIC FINITE ELEMENT ANALYSIS OF SIMPLE BEAMS , 1983 .

[22]  J B Roberts,et al.  Techniques for nonlinear random vibration problems , 1984 .

[23]  Jagmohan Humar,et al.  Dynamic of structures , 1989 .

[24]  A. Kiureghian,et al.  Methods of stochastic structural dynamics , 1986 .

[25]  S. Utku,et al.  Stochastic linearization of multi‐degree‐of‐freedom non‐linear systems , 1976 .

[26]  R. Herbert.  On the stresses in a nonlinear beam subject to random excitation , 1965 .

[27]  J. D. Collins,et al.  The eigenvalue problem for structural systems with statistical properties. , 1969 .

[28]  P. Seide.  Nonlinear Stresses and Deflections of Beams Subjected to Random Time Dependent Uniform Pressure , 1976 .

[29]  Masanobu Shinozuka,et al.  Response Variability of Stochastic Finite Element Systems , 1988 .

[30]  J. B. Roberts,et al.  Literature Review : Nonlinear Random Vibration in Mechanical Systems , 1988 .

[31]  A. B. Mason,et al.  Equivalent linearization for systems subjected to non-stationary random excitation , 1980 .

[32]  Wing Kam Liu,et al.  Random field finite elements , 1986 .