Abstract This paper is concerned with the problem of designing ellipsoidal domes so that they are in a membrane state of stress. The investigation is based on a method proposed by Vekua, in which an adjustment of the gravity of the shell is used to ensure a membrane state. The theory is specialized to ellipsoidal shells and some numerical results of the gravity distribution (i.e. thickness of the shell), and the corresponding stresses and linear elastic displacements are presented. However, it should be noted that the solutions obtained in this way are not unique.