The study investigated short, thin-walled Z-shaped carbon–epoxy laminate columns. Z-columns were compressed while considering the eccentric force realized from the center of gravity of the column section. The study involved performing a nonlinear analysis of the structures with implemented geometric imperfections reflecting the first buckling modes. The nonlinear analysis was performed by using the Tsai–Wu criterion to determine the effort of the composite material. The computations were run until the critical parameter was reached in the Tsai–Wu criterion, allowing for a description of the failure initiation mechanism in the composite material. The first signs of damage to the composite material were determined by using the acoustic emission method. Based on the results, postcritical equilibrium paths of the numerical models were determined. The equilibrium paths were then compared with the experimental characteristics of real structures. The numerical results and experimental findings show a satisfactory agreement. The results confirmed that the numerical models were adequate for estimating the performance of composite structures in the postcritical range, depending on the amplitude of compressive load eccentricity. The research topic undertaken is important because the thin-walled structure design relates to actual loads which, in most cases, differ from the idealized theoretical load conditions.