Stiffening by means of an elastic core is demonstrated to increase significantly the resistance of a cylindrical shell to buckling from an external impulsive pressure. Both theory and experiments show that the elastic core constrains the shell to absorb more energy by membrane plastic flow. Inclusion in the theory of a continuously changing tangent modulus of the shell material leads to a critical core modulus which depends on the hoop strain and explains the increased wave numbers observed in the experiments with stiff cores.