This paper presents considerations on the failure of axially compressed thin-walled composite structures with open sections (using an top-hat section structure). The analyzed columns were made of composite material — carbon fiber reinforced polymer (CFRP) using the autoclave technique. The study was conducted in a comprehensive approach using both experimental methods and advanced numerical simulations based on the finite element method (in ABAQUS® program). Investigations were performed up to the total failure of the structure, while considering the phenomena directly contributing to the failure (damage initiation, damage evolution, delamination). During the conducted experimental studies, both the postbuckling equilibrium paths obtained from the universal testing machine and the acoustic emission signals were registered. Regarding the simultaneously conducted numerical simulations, two independent advanced damage models (progressive failure analysis — PFA and cohesive zone model — CZM) were used, allowing for a thorough analysis of the complex mechanism of the composite material failure phenomenon. Both numerical and experimental results show high agreement. This paper provides a more comprehensive understanding how failure occurs such that the design of the analyzed composite sections. The most important aspect of this paper is the potential to enable to understand the failure of the top-hat stiffeners during post buckling such that made it possible to utilize the stable, undamaged region of the post-buckling region. In addition, the paper conducts an advanced study based on the simultaneous use of two independent damage models, allowing in particular for a thorough analysis of the damage with delamination phenomenon detection, on structures with a complex shape of the end cross-sections of the structure.