## 1 Introduction  
The FGM structures presented in this work were obtained by mixing two different component materials, in this case, a metal and a ceramic, in such a way that a continuous variation over the thickness direction of the material properties was verified, in contrast with the laminated composite material structures. Metal-ceramic FGM plates and shells are widely used in engineering in aircraft, space vehicles, and other engineering applications.  
Depending on the load level, structures can experience material nonlinearity known as elastoplasticity, which should be avoided because of large deformations resulting from elastoplastic behaviour in isotropic and FGM structures.  
In recent years, many research studies on FGM structures were done. Here, we focused our attention on axisymmetric-shell-related studies: [PERSON] et al. [1], [PERSON] et al. [2], [PERSON] et al. [3] and [PERSON] et al. [4] studied circular plates in linear and geometrically nonlinear behaviour using different theories and to provide different analyses: static bending, dynamic and buckling.  
Regarding the structural optimization involving FGM plates and shells of revolution, there is a lack of publications specifically devoted to elastoplastic behaviour.  
Elastoplasticity analysis has been less investigated in FGM structures. [PERSON] and [PERSON] [5] presented a model for FGM circular plates based on a physical neutral surface and higher-order shear deformation theory. [PERSON] et al. [6] presented a formulation for linear and nonlinear static bending and elastoplasticity analysis of functionally graded axisymmetric plate/shell-type structures under mechanical loading. [PERSON] et al. [7] presented structural and sensitivity analyses, allowing for the material distribution and sizing optimization of functionally graded material structures. The finite element model used in this work was the same as the one previously used by the authors in references [6; 7]. Other studies that dealt with the optimization of FGM of different types of structures and objective functions are as follows: [PERSON] et al. [8], [PERSON] and [PERSON]. [9], [PERSON] and [PERSON] [10], [PERSON] and [PERSON] [11], [PERSON] et al. [12], [PERSON] et al. [13] and [PERSON] et al. [14]. In what concerns the elastoplastic behaviour of axisymmetric shells, the work of [PERSON] and [PERSON] [15] can be mentioned and was used here for comparison and validation purposes.  
In this paper, we present the elastoplastic analysis and subsequent thickness optimization results of functionally graded axisymmetric structures subjected to axisymmetric loading. To avoid the elastoplastic behaviour of the structures, it was necessary to find the minimum thicknesses (which correspond to the minimum mass of the structure) that provide an elastic behaviour for a certain load level. This kind of optimization problem is not encountered in the literature, and therefore, it was the main objective of this work.  
The solutions were obtained using a finite element model based on [PERSON] et al. [16]. The analyses of some illustrative optimization applications were performed, and the results are presented and discussed.