Determining failure modes devoid of redundant elements for pin-bar structures using null space of equilibrium matrix

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• Search for minimal cut sets of pin-bar structures based on equilibrium matrix. • Two lemmas characterizing the cut sets are proposed and proved. • All failure modes devoid of redundant elements can be determined. • The calculation efficiency of structural reliability is significantly improved. The reliability of a pin-bar structure is dominated only by the failure modes devoid of redundant elements, that is, the structure will become geometrically variant only when all the failed elements in the failure mode are removed. A novel method is developed to search for all the combinations of such elements in a pin-bar structure, i.e., the minimal cut sets of system reliability. Based on the matrix consisting of the null space basis vectors of the structural equilibrium matrix (abbr. null space basis matrix), two lemmas indicating the characteristics of the minimal cut sets are proposed and proved. By investigating the row vectors of the null space basis matrix and their linear dependence, whether single or multiple elements can constitute a minimal cut set can be evaluated. A strategy to search for all the minimal cut sets is proposed based on the maximal linearly independent subsets of the row vectors of the null space basis matrix. A procedure is also provided to calculate the system reliability of pin-bar structures only using the obtained minimal cut sets. Two numerical examples are employed to verify the validity and computational efficiency of the proposed method.