On the application of a simple computational model for slender transversely cracked beams in buckling problems

This paper discusses the implementation of a simplified computational model that is widely used for the computation of transverse displacements in transversely cracked slender beams into the Euler\u27s elastic flexural buckling theory. Two alternatives are studied instead of solving the corresponding differential equations to obtain exact analytical expressions for the buckling load ▫$P_{cr}$▫ due to the complexity of this approach. The first approach implements wisely selected polynomials to describe the behavior of the structure, which allows the derivation of approximate expressions for the critical buckling load. Although the relevance of the results strongly depends on the proper prime selection of the polynomial, it is shown that the later upgrading of the polynomials can lead to even more reliable results. The second approach is a purely numerical approach and presents the geometrical stiffness matrix for a beam finite element with a transverse crack. To support the discussed approaches, numerical examples covering several structures with different boundary conditions are briefly presented. The results obtained with the presented approaches are further compared with the values from enormous 2D finite elements models, where a detailed description of the crack was achieved with the discrete approach. It is evident that the drastic difference in the computational effort is not reflected in the significant differences in the results between the models