Accurate solution of some I-beam optimization problems

Abstract

The problem of proportioning an assembled steel I-beam made of three sheets is wellknown. It is included in nearly every course of study where the analysis of steel components is involved. In the presented case we consider finding the depth of a beam with a predefined modulus of resistance, based on the condition that the cross sectional area should be minimum. Such a depth of the beam is generally called optimum. At the same time, the minimum depth of the beam, which can be found from the stiffness conditions, can be considered in addition to its optimum depth. Such a problem is generally solved by empirical approaches with no strict formulation nor a proper mathematical analysis. The paper poses the problem as that of parametric mathematical programming with inequality constraints which appear as the condition of strength in bending, the condition of strength in shear, and the stiffness condition. Several variations which differ in the functional relationships between the depth of the beam¿s web and its thickness have been analyzed.