Abstract Superconducting magnetic levitation force measurements on large field cooled cylindrical MgB₂ bulks with different diameters and thicknesses are reported. For these experiments, a special set-up permitting one to measure forces up to 500 N was used. In contradiction to previous measurements, the obtained force hysteresis cycles could not be reproduced with the analytical mean field model proposed by Bernstein et al (2017 Supercond. Sci. Technol. 30 065007). The failure of the model has been attributed to surface barriers effects which were not taken into account in the model. This last one was accordingly modified in order that the measured force cycles could be reproduced. Contrary to most other models describing surface barriers effects, the modified model suggests that above a threshold field anti-vortices and not vortices enter the superconductor. This behaviour is related to the storage by the superconductor of the mechanical work done by the operator. In addition, it has turned out that the threshold field is a decreasing function of the critical surface current density of the samples. As a consequence, the surface barriers effects occur only if this quantity and the critical current density are large enough. Otherwise, the internal magnetic field of the superconductor could be computed and was seen to be a decreasing function of the thickness of the superconductors.