The stress state of a rectangular cross section beam made of a material with porous structure at pure bending

The article is devoted to the problem of rational distribution of porosity along the height of the cross section of a loaded structural element at pure bending. We consider pure bending of the beam which is made of a material (steel) with a porous structure of variable porosity along the height of cross section. The research is limited to the case of an elastic deformation under the assumption that the maximum stress in the beam does not exceed the yield stress of the material. It is known that Young's modulus is a function of material porosity.
The porosity is assumed to be variable along the height of the cross section. Therefore, Young's modulus is also a function of the height coordinate of the beam cross section.
The problem of bending is reduced to the problem of bending of a beam which is made of a heterogeneous material with a variable elastic characteristic along the height of cross section and variable yield stress.
According to Hooke's law, at the tension-compression state of bent beam, the law of stress variation along the height of cross section is determined, which makes it possible to calculate the value of the bending moment.
The aim of the study is to select the distribution law of both Young's modulus and, hence, porosity along the height of beam cross section in order to obtain the greatest possible bending moment under restrictions on the stress and porosity.
A numerical solution of the problem corroborates that the bending moment is 63% higher at rationally chosen porosity than that at average porosity. The results of calculations indicate an optimization of the porosity distribution over the cross section.