Eccentricity-Based Optimization Procedure for Strength Design of RC Sections under Compression and In-Plane Bending Moment D. López-Martín 1 ; J. F. Carbonell-Márquez 2 ; L. M. Gil-Martín 3 ; and E. Hernández-Montes 4 Abstract: The strength design of reinforced concrete (RC) rectangular sections for combined compression and in-plane bending with two levels of reinforcement is indeterminate: three unknowns are to be solved, but with only two equilibrium equations; an additional condition is necessary to solve the problem. The additional condition leads to the finding of a minimum reinforcement-concrete ratio. This paper proposes a new approach based on the equivalent eccentricity of the applied compressive load. Different domains are reported, each of which is associated with given values of eccentricity and axial load. Analytical expressions for the domain boundaries are established, and a simple procedure is described to outline the conditions corresponding to the optimal reinforcement. The main advantage of this procedure is its simplicity, which allows manual computations. Some examples employing reinforcement sizing diagrams illustrate the validity of this approach. DOI: 10.1061/(ASCE)ST.1943-541X.0000794. © 2013 American Society of Civil Engineers. Author keywords: Reinforced concrete; Optimal reinforcement; Strength design; Equivalent eccentricity; Structural optimization. Introduction One of the most commonly studied topics in schools of engineering is the ultimate strength proportioning of a reinforced concrete (RC) rectangular cross section subjected to a combination of axial compressive load and bending moment. The widespread use of concrete and reinforcing steel in buildings constructed in the twentieth century meant that this problem has been dealt with in many books, as well as being included in every concrete design code. The problem is difficult to resolve because numerous variables govern the equations, and it is usually necessary to iterate in order to find its solution. Therefore, the designer has to rely on intuitive experience to fix some of these variables to obtain the most appro- priate reinforcement. When experience is not enough, a wide range of existing literature also provides many simplified or trial-and- error procedures based on tables or abacuses, which help in finding a design solution. Recent studies provide many different approaches to getting the optimal solution for the reinforcement design. Some researchers have tried to find the optimum based on the cost of every compo- nent of the section (i.e., concrete and steel). Barros et al. (2005) investigated the cost optimization of rectangular RC sections using the nonlinear MC90 equation. Barros et al. (2012) studied the minimal cost problem of a rectangular section in simple bend- ing where the objective function is the cost of raw materials and the variables are the section depth and the steel reinforcement areas. Lee and Ahn (2003) and Camp et al. (2003) also employed genetic algorithms to perform a discrete optimization of the flexural design of RC frames, both of which included material and construction costs. Other approaches assume that the rectangular dimensions of the cross section are given and the optimal solution for the reinforce- ment in ultimate strength design needs to be found. Thereby, Hernández-Montes et al. (2004, 2005) presented a new design ap- proach called Reinforcement Sizing Diagrams (RSD), which shows the infinite number of solutions for top and bottom reinforcement that provide the required ultimate strength for sections subject to combined axial load and moment. Because RSD represents an in- finite number of solutions, the optimal (or minimum) reinforcement may be identified. Also, Aschheim et al. (2007) employed this RSD technique to define optimal domains with respect to axial-bending load coordinates according to provisions of Eurocode 2 (EC2) (CEN 2001). Ultimately, the observation of the characteristics of optimal solutions led Hernández-Montes et al. (2008) to the de- velopment of the Theorem of Optimal Section Reinforcement (TOSR). This work provides the additional conditions to be im- posed in the equilibrium equations to achieve an optimal design of reinforcement. Although Hernández-Montes et al. (2008) described and proved the additional conditions to be implemented, each of which has a special suitability depending on the applied loads. As a corollary to the mentioned theorem, Hernández-Montes et al. (2008) proposed to check every condition in the problem in question and select the one that provides the optimal solution. In this paper, a procedure similar to the one that Aschheim et al. (2007) exposed is given according to EC2 specifications that ad- dress the problem from the point of view of many traditional con- crete textbooks: depending on the equivalent eccentricity of the applied compressive load, this approach will provide an additional 1 Associate Professor, Dept. of Structural Mechanics, Univ. of Granada (UGR), Campus Universitario de Fuentenueva, 18072 Granada, Spain. 2 Ph.D. Candidate, Dept. of Structural Mechanics, Univ. of Granada (UGR), Campus Universitario de Fuentenueva, 18072 Granada, Spain (corresponding author). E-mail: jfcarbonell@ugr.es 3 Associate Professor, Dept. of Structural Mechanics, Univ. of Granada (UGR), Campus Universitario de Fuentenueva, 18072 Granada, Spain. 4 Full Professor, Dept. of Structural Mechanics, Univ. of Granada (UGR), Campus Universitario de Fuentenueva, 18072 Granada, Spain. Note. This manuscript was submitted on July 3, 2012; approved on December 17, 2012; published online on December 19, 2012. Discussion period open until February 23, 2014; separate discussions must be sub- mitted for individual papers. This paper is part of the Journal of Struc- tural Engineering, © ASCE, ISSN 0733-9445/04013029(9)/$25.00. © ASCE 04013029-1 J. Struct. Eng. J. Struct. Eng. Downloaded from ascelibrary.org by UGR/E.U ARQUITECTURA TECNICA on 10/23/13. Copyright ASCE. For personal use only; all rights reserved.