Abstract A family of hierarchical shell elements based on assumed strain formulation has been developed to enhance the performance of displacement-based hierarchical shell elements typically up to polynomial order six. For thin shells, hierarchical assumed strain elements with five degrees-of-freedom (two rotations and three mid-plane translations) and six degrees of freedom (three translations on the top and bottom surfaces of the shell) per hierarchical mode have been formulated. Hierarchical in-plane modes are adaptively selected on the basis of their ability to reduce solution errors. For thick shells both in-plane and through-the-thickness mode selection is controlled by the process of adaptivity. Numerical experiments have been carried out to test the usefulness of assumed strain formulation in the framework of the p-method combined with selective polynomial escalation. The sensitivity to element distortion and influence of super-parametric formulation on the rate of convergence have been numerically assessed.