The homogenization of composite beams is based on the fact that inelastic defects of the material produce additional fields of strains, that are equivalent to eigenstrains in an identical bull elastic background beam with effective virgin stiffness, [1]. The generalized Hooke's law shows the two-field approach, however, with an a' priori volume coupling present through the nonlinear constitutive relations, the rate form is more appropriate: epsilon=E(-1)sigma+<(epsilon)over bar>, gamma=G(-1)tau+<(gamma)over bar>. In the present beam theories stress components sigma = sigma(xx) and tau = tau(zz) = tau(zz) are considered, epsilon and gamma denote the total amount of strain and shear angle, E and G are Young's modulus and shear modulus, respectively, and <(epsilon)over bar> and <(gamma)over bar> denote the distribution of imposed and non-compatible inelastic strain and inelastic shear angle. Since beam theories deal with stress and strain resultants, it is important to define proper resultants of these eigenstrains, i.e., inelastic curvatures <(kappa)over bar> and inelastic averaged shear angles <(gamma)over bar>(m). The motion of an elastic-plastic homogeneous shear-deformable beam excited by a lateral load q(x, t) is described by the following set of partial differential equations: B Psi,(zz)-S(Psi+w,(x))=B<(kappa)over bar>,(z)-S<(gamma)over bar>(m), mu w-S(Psi,(x)+w,(xx))=q-S<(gamma)over bar>(m,x), where w is the lateral deflection, Psi denotes the cross-sectional rotation, B and S are the bending and the shear stiffness, respectively, and mu denotes the mass per unit of length.
