Different types of loss of stability of elastic structures are usually illustrated by simple models. This paper presents a family of structures which can demonstrate four cases of the double cusp catastrophe by the variation of a parameter. The path defined by this parameter is calculated in the diagram of the classes of the double cusp catastrophe. In one of the four cases the primary equilibrium path of the perfect structure intersects a secondary equilibrium surface at the critical point. In the other cases there are two secondary equilibrium paths, one of which belongs to critical state in all points. In all four cases the equilibrium paths of the imperfect structure are determined as well as the critical load in terms of the value of the simplest imperfections.