( 3 a$_{33}$ − ε$_{0}$ ) ϕ ′′ zz + d$_{31}$ a$_{33}$ w ′′ = 0 (14) Q el z = ∫ A E$_{z}$ ( x, z ) dA M el z = A zE$_{z}$ ( x, z ) dA w = w ′ = 0 , (15) ( c$_{11}$I − d 2 $_{31}$I a$_{33}$ − f 2 1133 a$_{33}$ A ) w ′′ − d$_{31}$ a$_{33}$ M el z − f$_{1133}$ a$_{33}$ Q el z = 0 , ( c$_{11}$I − d 2 $_{31}$I a$_{33}$ − f 2 1133 a$_{33}$ A ) w ′′′ − d$_{31}$ a$_{33}$ ∂ ∂x M el z − (16) f$_{1133}$ a$_{33}$ ∂ ∂x Q el z = P$_{∗}$, ϕ ′ z = 0 , (17) f 2 1133 a$_{33}$ > ( c$_{11}$ − d 2 31 a$_{33}$ ) h 2 12 , Y = 1 − d 2 31 c$_{11}$a$_{33}$ − 12 f 2 1133 c$_{11}$a$_{33}$h $_{2}$, where Q el z and M el z are the electric transverse shear force and the electric bending moment, respectively: with the associated boundary conditions at x = 0 at x = l and at z = ± h/ where P$_{∗}$ is a force applied to the right edge of the cantilever. The equations (13), (14) and the boundary conditions (15) - (17) couple the mechanical displacement w and the electric potential ϕ . From the equation (13) it is seen that piezoelectric and flexoelectric effects increase the bending rigidity of the beam ( a$_{33}$ < 0) and for the beams with large cross-section area and consequently large its moment of inertia, the term Af 2 $_{1133}$/a$_{33}$ plays insignificant role. However, for small cross-sectional dimensions, when where h is the thickness of the beam, the flexoelectric effect becomes noticeable. The normalized Young modulus is 7