An Analytical Study on the Buckling Behavior of Cracked Column Based on the Physical Property of the Dirac delta Function

The Dirac delta function provides a simple and effective tool for representing point loads and singularities in structural problems, often leading to closed-form solutions. In this study, buckling of simple double-ended columns with one and two cracks has been analyzed analytically. Although in recent years this issue has attracted the attention of many researchers, the methods presented to solve the problem usually have a significant computational load. Therefore, in this study, a new approach has been used to solve the problem using the property of the Dirac's delta function. This approach simplifies the problem-solving process and significantly reduces the computational cost. Based on this, the crack is modeled with a bilateral behaviour via Dirac delta function. This model takes into account the crack closure effect on buckling behaviour of column by introducing a suitable switching criterion, which allows each crack to be open or closed depending on the sign of the axial strain at the crack centre. The proposed method was used to finding the buckling load, determining the effects of crack stiffness for both one and two-crack scenarios, and accounting for the effect(s) of crack opening and closing on the buckling load. For validation purposes, the finite element software SAP2000 was utilized.