A Decoupled Uncertainty Propagation to Buckling Loads and Shapes from Spatially Random Mechanical Properties

A method is developed to quantify the variability in the buckling loads and mode shapes of columns, plates, and frames with uncertain mechanical properties, i.e., flexural rigidity, foundation flexibility, and modulus of elasticity. The mechanical properties are considered to be Gaussian or lognormal random fields. The method is developed within the finite element framework, with the vector of flexural rigidities or foundation flexibilities associated with the different elements described by a multivariate distribution corresponding to the underlying random field. The method involves Taylor series-based first-order uncertainty propagation to find the means and variances of the buckling loads and the mean vector and covariance matrix of the buckling mode shapes. Jacobi’s formula for the derivative of a determinant is used to find the variance of the buckling loads prior to finding the covariance matrix of the mode shapes. This decouples the uncertainty quantification of the buckling loads from the mode shapes, making the proposed method simpler than existing techniques. The method is illustrated using a simply supported column, plate, and frame and is validated against results from Monte Carlo simulations. It is shown that the method is capable of accurately quantifying the uncertainty in the first several buckling modes. For a column lying on a flexible foundation, the method accurately detects the switching of the buckling mode shape from half-sine to full-sine wave mode. Further, at the value of foundation flexibility where mode switching occurs, corresponding to a repeated eigenvalue and nonunique mode shape, the method gives a family of solutions for the mean and covariance matrix of the mode shape.