Evaluating the Structural Stability of Existing Reticulated Shells Using the Minimum Eigenvalue of the Tangent Stiffness Matrix

Abstract

The design of a reticulated shell is usually dominated by the checks of the structural stability under a few load cases, and evaluating the structural stability is crucial for health monitoring of existing reticulated shells. Corresponding to each given dominant load, the minimum eigenvalue of the structural tangent stiffness matrix is employed as a stability indicator of the reticulated shell. For an existing metal reticulated shell damaged by corrosion, a novel method is then proposed to estimate its magnitude by applying static loads to a small number of nodes and obtaining the displacement responses of a limited number of degrees of freedom. When the member cross-sectional areas deteriorate, analytical relationships between the variation in the stability indicator and the resulting changes in the structural elastic and geometric stiffness matrices are established. The change in the geometric stiffness matrix is theoretically determined to generally have a negligible effect. The perturbation analysis of the structural tangent stiffness matrix further indicates that the stability indicator can be estimated using a few eigenvalues and eigenvectors of the structural elastic stiffness matrix. A strategy based on static loadings is then proposed to determine the specified eigenvalues and eigenvectors of the elastic stiffness matrix of an existing reticulated shell by analyzing the relationship between the elastic stiffness matrix and the induced displacements. Optimizations of the magnitude and location of static loads and the selection of displacement measuring points are introduced. Two illustrative examples of existing structures, including a reticulated shallow shell and a vaulted space truss, are established by introducing randomly generated deviations of the member cross-sectional areas into their design models. The proposed method is employed to evaluate their structural stabilities corresponding to given dominant loads. The accuracy and validity of the method are verified by comparing the results obtained with their exact solutions determined based on numerical models.