Eigenvalue Uncertainty in Stressed Structures

This paper develops a method for calculating the statistics of the natural frequencies and mode shapes of vibration for a structure acted upon by an external static loading which results in the structure being stressed for the eigenvalue analysis. The particular complexity of this problem centers around the fact that in order to formulate a geometric stiffness matrix for the structure it is necessary, in general, to solve a static response problem to obtain each structural member's axial forces. This geometric stiffness matrix is then added to the elastic stiffness matrix and the standard eigenvalue problem is solved to obtain natural frequencies and mode shapes. The first layer is associated with the uncertainty in the values of the axial forces used in the formulation of the geometric stiffness matrix and the second layer is associated with the solution of the eigenvalue problem using the random mass and elastic stiffness matrices.