OPTIMUM REINFORCEMENT FOR DUCTILE RESPONSE OF LRC BEAMS USING A SIMPLE GA R. C. Yu and G. Ruiz ETSI de C. y P., Universidad de Castilla-La Mancha, Avda. Camilo Jos´ e Cela s/n, 13071 Ciudad Real, Spain Abstract. The objective of this work is to apply an optimization technique based on the use of a Genetic Algorithm to evaluate the minimum reinforcement in lightly reinforced concrete (LRC) beams for given material properties, geometry and loading conditions. It adopts the effective slip length model, which reproduces numerically the mechanical response of a given reinforced beam, as the tool that generates fitness values needed by the GAs procedure. The speed and accuracy of the model compensate with the large amount of calculations required by GAs. The methodology is demonstrated through the searching of minimum ratio for a reinforced prismatic beam loaded at three points at various conditions. Resumen. Esta comunicaci´ on aplica una t´ ecnica de optimizaci´ on basada en un algoritmo gen´ etico (GA) para hallar la armadura m´ ınima de una viga de hormig´ on armado dadas las propiedades de los materiales, la geometr´ ıa y las condiciones de apoyo. Utilizamos el modelo de la longitud efectiva de anclaje, el cual reproduce num´ ericamente la respuesta mec´ anica de un aviga armada dada, como la herramienta que genera los valores de ajuste necesarios para los procesos internos del GA. La velocidad y precisi´ on del modelo compensa la gran cantidad de c´ alculos que el GA tiene que realizar. Validamos la metodolog´ ıa realizando varios ejemplos de b´ usqueda de la armadura m´ ınima para una viga prism´ atica que resiste flexi´ on en tres puntos. 1 INTRODUCTION In a reinforced concrete design, in order to prevent instan- taneous catastrophic failure without warning, the capacity of a section or a structure as a whole to undergo a reason- able amount of plastic deformation without significant loss of strength during its collapse is preferred. This means min- imum reinforcement requirements are necessary to provide a ductile response, ensure adequate beforehand knowledge of an impending failure at overloads, and prevent excessive crack widths at service load (in the case of strongly rein- forced members). Various researchers have dedicated themselves to the issue of minimum reinforcement from different viewpoints [1–5]. In all cases, empirical formulas were proposed for practicing engineers. Nevertheless, when new conditions do not fall within the application range of those formulas, the traditional trial-and-error would be the only way to proceed. With these concerns in mind, we would like adopt Genetic Algorithms (GAs), which have been proved effective when “guessing” is needed for decision making, to establish a systematic way, or a black box for the searching of the minimum reinforcement. The basic underlying principle of GAs is that of the Dar- winian evolutionary principle of natural selection, wherein the fittest members of a species survive and are favored to produce offspring. First introduced by Holland [6], GAs have since been extensively used to solve optimization problems where conventional methods are either inapplicable or inef- ficient, ranging from optimal design of stacking sequence of laminate composites [7–11], shape and structural optimiza- tion, [12–14], to parameter identification [15, 16], and so forth. But they nevertheless tend to be computationally ex- pensive, since a set of samples with certain degree of vari- ety is needed to start and evolve the searching process. If the fitness value has to be fed by even a moderate finite ele- ment method calculation, the amount of calculations needed would render the usage unrealistic. There are two avenues to sidestep this difficulty. One is to use a fast method for computing the fitness value of each member; the other is to avoid the necessity of computing the fitness value for all the members. The method of Pichler, Lackner and Mang [16] belongs to the second category. They incorporated GAs with the trained Artificial Neural Network (ANN) to provide an estimate of optimal solutions, which finally are going to be assessed by an additional FE analysis. The methodology so indicated seem to pave a way for GAs to be used in other large scale finite element analysis. Nevertheless, to start with, we adopt here the first methodology, i.e., using a fast numerical method to feed the objective function for the GAs. Actually, a simple GA (no overlapping of searching samples) is chosen to perform the optimization process. GAs have been previously adopted by Coello and his co- workers [17, 18] for minimum reinforcement but in a sense of reducing the costs of concrete, steel, etc.; Rafiq [19] ap- plied GAs to optimal design and detailing of reinforced con- crete biaxial columns in identifying the optimal reinforce- ment bar sizes and bar detailing arrangements, but neither of them adopted ductility as the design objective nor is the concept of fracture mechanics referred to. 535 ANALES DE MECÁNICA DE LA FRACTURA Vol. 22 (2005)