## 1 Introduction  
The stress-strain analysis of cantilever beams is frequently present in engineering practice. Therefore, a large number of researchers have been working in these fields. Initial research included analysis of the elastic deflection of a beam with an assumption of small deformations and displacements [1]. Further research, which included large elastic deformations of cantilever beams loaded by concentrated force at the free end, involved the resolving of differential equations applying elliptic integrals [2]. The complexity of this solution directed the following research towards the development of different semi-analytical methods for finding the solution for the problem of large elastic deformations of cantilever beams [3]. Additionally, the elastic deflection of beams with linearly variable depth of the cross-section and semi-rigid supports was analyzed [4]. Different numerical methods for solving the fourth order ordinary differential equation of beam deflection were developed, and they are systematized in the reference [5]. Lately, elastic deflection of non-prismatic cantilever beams [6], as well as thin-walled beams with a curvilinear open cross-section, have been analytically and numerically studied [7]. Besides the elastic behavior, the materials possess plastic behavior where irreversible strains occur. The broadening of research on the elastoplastic deformation of cantilever beams is thus required. Research of the elastoplastic deflection of cantilever beams has been focused on the analysis of the beams with constant rectangular cross-section and elastic-perfectly plastic material without hardening in the plastic domain [8; 9; 10; 11]. Numerical-analytical methods for solving the elastoplastic deflection of prismatic cantilever beams subjected to axial force and bending moment at the free end were proposed, and they included elastic-perfectly plastic materials [12] and elastoplastic materials with strain hardening [13]. Research also covered the influence of the inclined load on the large deflection of elastoplastic cantilever beams [14]. The elastoplastic deflection of cantilever beams with rectangular cross-section loaded by shear force at the free end were analyzed for different non-linear material models [15; 16; 17]. Finite deflection of a slender cantilever beam with a predefined load application locus was analyzed using numerical methods for beams with constant rectangular cross-section and elastic-perfectly plastic material [18; 19]. Simply supported beams with linearly variable depth of the cross-section were investigated in [20]. Analyses of elastoplastic deflection of cantilever beams mainly treated prismatic rectangular beams. In the case of circular cross-sections, the plastic moment-curvature relationship was developed for elastic-perfectly plastic material [21; 22; 23]. An approximate solution for plastic the moment-curvature relationship which replaces a circular cross-section with the equivalent rectangular one was proposed in [24]. In recent years, piezoelectric materials have drawn attention and been implemented in structural health monitoring systems. Therefore, an explicit exact analytical solution for the shape deformation control of smart laminated cantilever piezo composite hybrid plates and beams was proposed [25]. The solution covers a variety of complex loading systems. Although this solution is universal, covering both isotropic and orthotropic structures, it was developed for elastic materials and can hardly be implemented for the elastoplastic strain hardening deflection of cantilever beams with linearly variable cross-section, which is the subject of this paper. The exact solution of the elastoplastic deflection of cantilever beams subjected to shear force at the free end was not developed due to the complexity of the mathematical model.