The presence of an open hole or low velocity impact damage in composite stiffened panels introduces high local stresses that may initiate material failures and lead to catastrophic failure of the structure or component before Euler buckling occurs. In the case of compressive loading, near‐surface instability or fiber microbuckling in the 0° plies may occur where high stress gradients exist. In this study, the critical load is estimated by using the maximum stress failure criterion, giving a conservative estimate for the failure load. An alternative method, the Soutis–Fleck cohesive zone fracture model, is also employed, where the fiber microbuckling growing from the edge of the hole is mathematically represented by a through‐the‐thickness crack and fracture mechanics concepts are applied. In both approaches, the unnotched compressive strength is required, which is estimated by using a fiber instability model based on a 3D stability theory of deformable bodies. Theoretical predictions are in good agreement with experimental data, giving confidence to the present analytical approach.