The exact stability stiffness matrix for the analysis of multi-cracked frame structures

• Frames in presence of an arbitrary number of concentrated cracks are modelled.
• The exact buckling loads and relevant modes for damaged frames are evaluated.
• The exact closed-form expression of the stability stiffness matrix is proposed.
• The Wittrick and Williams algorithm is applied to provide exact solutions.
• No additional variables are introduced at the cracked sections.

In this paper the exact stability stiffness matrix of an Euler–Bernoulli column in presence of an arbitrary number of concentrated cracks is derived. The procedure for the evaluation of the stability stiffness matrix is based on the exact closed form solution of the buckling modes of the multi-cracked column, derived by the present authors in a previous paper. The knowledge of the exact stability stiffness matrix of the multi-cracked beam allows the direct evaluation of the exact global stability stiffness matrix of damaged frame structures. Furthermore, the exact evaluation of the buckling loads and the corresponding buckling modes, consistent with the distributed parameter model, are obtained through the application of the well-known Wittrick–Williams algorithm. The great advantage of the proposed approach is that the degrees of freedom of the overall frame structure are exactly the same of the equivalent undamaged structure irrespective of the number of the concentrated damages. A numerical application, aiming at validating the exact solution of a framed structure, in presence of concentrated cracks, is reported. The influence of multiple cracks on the critical load and the corresponding buckling mode of the frame under study is assessed.