Effects of end-plate on the lateral buckling of doubly symmetrical I-section cantilever beam with free end restrained laterally are analyzed using the software COMSOL and linear shell finite elements. The torsional stiffness of the end-plate prevents the free warping of flanges and decreases the warping effective length of the cantilever beam. A parametric study is conducted on 3231 cantilever beams under uniform bending to propose an approximative formula to determine the warping effective length factor which depends on the ratio between the torsional constant of the end-plate and the warping stiffness of the beam. The small standard deviation and high coefficient of determination show a very good correlation between analytical formulas and numerical results. Numerical applications are applied to analyze some cantilever beams subjected to uniform bending to demonstrate the reliability of the proposed formula and the effects of the end-plate on the enhancement of the global stability of cantilever beams with free end restrained laterally. Keywords: cantilever lateral buckling; end-plate; edge stiffener; flexural-torsional buckling; warping effective length; warping effective length factor.

Lightweight structures with soft inclusion material, such as hollow core slabs, foam sandwich wall, pervious pavement ... are widely used in construction engineering for sustainable goals. Voids and soft inclusion can be modeled as a very soft material, while the main material is modeled with its original rigidity, which is so much higher than inclusion's one. In consequence, highly contrast bi-phase structure attracts the interests of scientists and engineers. One important demand is how to build a homogeneous equivalent model to replace the multi-phase structure which requires much resources and time to perform structure analysis. Various homogenization schemes have succeeded in establishing a homogeneous substitution model for composite materials which fulfill the scale separation condition (characteristic length of heterogeneity is very small in comparison to structure dimensions). Herein, elastic stiffness matrix of a homogeneous model which replaces a bi-phase material is computed by a higher-order homogenization scheme. A non-homogeneous boundary condition (a polynomial inspired from Taylor series expansion) is used in computation. Homogeneous substitution model constructed from this computation process, can give engineers a fast and effective tool to predict the behavior of bi-phase structure. Instead of a classical Cauchy continuum, second gradient model is selected as a potential candidate for substituting the composite material behavior because of the separation scale (volume ratio of inclusion to matrix phase reaches unit). Keywords: generalized continuum; second-gradient medium; higher-order homogenization; non-homogeneous boundary conditions; representative volume element.