•Different strategies for calibration of a cyclic model for steel are investigated.•Monotonic, quasi-static cyclic and pseudo-dynamic tests on a column were performed.•The influence of the type of calibration test on the prediction is assessed.•A multi-objective optimisation approach accounting for several tests is proposed. Accurate response predictions of steel structures subjected to earthquake loading involve the use of models able to simulate properly the cyclic behaviour of the regions where nonlinear phenomena take place. In case of full-strength joints, they are represented by the members connected, which may present softening response due to local buckling. Even though a number of phenomenological models have been developed in the last decades, their calibration seems to have received less attention. Usually, calibration is based on matching the experimental and numerical cyclic responses under loading protocols proposed by standards. Since these were not developed to this aim, the predictive capability of so calibrated models deserves investigation. In this work, a calibration procedure based on the minimisation of response misfit is presented and critically discussed, with reference to an experimental programme carried out at the University of Salerno. Different assumptions about the function to minimise, the ultimate rotation and number of objectives are compared and analysed. The main outcome of this investigation is that a calibration based on cyclic response only is not robust, since its accuracy under different loading conditions may deteriorate. The introduction of the monotonic test in a multi-objective framework may be effective, and its accuracy is confirmed by the results of pseudo-dynamic tests.