Let X be a real normed space, then C( X) is functionally convex (brie y, F-convex), if T(C)  R is convex for all bounded linear transformations T 2 B(X; R); and K( X) is functionally closed (brie y, F-closed), if T(K)  R is closed for all bounded linear transformations T 2 B(X; R). We improve the Krein-Milman theorem on  nite dimensional spaces. We partially prove the Chebyshev 60 years old open problem. Finally, we introduce the notion of functionally convex functions. The function f on X is functionally convex (brie y, F-convex) if epi f is a F-convex subset of X  R. We show that every function f: (a; b) 􀀀 ! R which has no vertical asymptote is F-convex.

The bamboo structure can be generally viewed as a Functionally Graded composite material constituted by long and aligned cellulose fibres embedded in a lignin matrix. Analysing the transversal section of a bamboo culm, one can observe that the fibre distribution is variable through its thickness. The non-uniform distribution of fibres prevents the direct application of equations used to model the behaviour of composite materials, as the rule of mixtures equations for strength and modulus of elasticity. These equations assume, besides the perfect bonding between fibre and matrix, uniform distribution of the fibres in the matrix. In bamboo, the fibre distribution follows an organized pattern with a higher concentration of fibres on the outer surface of the culm. Establishing how this variation occurs, the basic equations from the composite materials approach can be modified in order to model the mechanical behaviour of bamboo. This paper presents the meso-structure analyses of bamboo culms through Digital Image Analysis. The variation of the volume fraction of the cellulose fibres across the transversal section of the bamboo is established. The developed methodology is successfully applied to study the volume fraction variation of fibres in two different samples of bamboo species Phyllostachys heterocycla pubenscens, commonly know as "Moso".      

In this article, thermal stability of beams made of functionally graded material (FGM) is considered. The derivations of equations are based on the one-dimensional theory of elasticity. The material properties vary continuously through the thickness direction. Tanigawa's model for the variation of Poisson's ratio, the modulus of shear stress, and the coefficient of thermal expansion is considered. The equilibrium and stability equations for the functionally graded beam under thermal loading are derived using the variational and force summation methods. A beam containing six different types of boundary conditions is considered and closed form solutions for the critical normalized thermal buckling loads related to the uniform temperature rise and axial temperature difference are obtained. The results are reduced to the buckling formula of beams made of pure isotropic materials….