# Closed form solution of optimal design of rectangular reinforced concrete sections

M.H.F.M. Barros (Department of Civil Engineering, Faculty of Sciences and Technology, University of Coimbra, Portugal)
A.F.M. Barros (Department of Mechanical Engineering, Instituto Superior Técnico, Lisbon, Portugal)
C.A. Ferreira (Department of Civil Engineering, Faculty of Sciences and Technology, University of Coimbra, Portugal)

ISSN: 0264-4401

Article publication date: 1 October 2004

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## Abstract

The aim of this paper is to obtain the analytical solution for the optimal design of reinforced concrete sections under ultimate design. The equilibrium equations of the section under bending moment and axial force in rupture are derived. The ultimate conditions are considered either in the steel or in the concrete according to the concrete design codes. The definition of the strains and stresses in the materials is based on the use of Heaviside functions. With this definition the equilibrium equations are described by unique equations. The optimization can then be developed with any design variables in the geometric definition, as area of the reinforcement and location. The optimization is developed with yielding of tensile steel and crushing of concrete. Although this is the current situation in reinforced concrete design, future developments of the model can include other steel and concrete conditions. Cost optimization and variable materials strength ratio are possible applications of the model. The interest of the model is the use of closed form unique equilibrium equations in the optimization of reinforced concrete sections. Numerical examples of the optimization of a rectangular section with minimum reinforcing steel area and economic bending moment are presented. The originality of the paper is the use of Heaviside functions in the definition of the ultimate strains in the reinforced concrete section. Unique equations for the objective function and restrictions are derived. The paper is useful for the design of reinforced concrete. The equations derived can be implemented into computer programs.

## Citation

Barros, M.H.F.M., Barros, A.F.M. and Ferreira, C.A. (2004), "Closed form solution of optimal design of rectangular reinforced concrete sections", Engineering Computations, Vol. 21 No. 7, pp. 761-776. https://doi.org/10.1108/02644400410565298

## Publisher

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Emerald Group Publishing Limited