Structural Optimization 7, 9]-102 Q Springer-Verlag 1994 Minimum cost design of reinforced concrete beams using continuum-type optimality criteria A. Adamu and B.L. Karihaloo School of Civil & Mining Engineering, The University of Sydney, NSW 2006, Australia G.I.N. Rozvany FB 10, Essen University, Postfach 10 37 64, D-45117 Essen, Germany Abstract This paper outlines a general procedure for obtain- ing, on the basis of continut~m-type optimality criteria (COC), economic designs for reinforced concrete beams under various de- sign constraints. The costs to be minimized include those of con- crete, reinforcing steel and formwork. The constraints consist of limits on the maximum deflection, and on the bending and shear strengths. However, the formulation can easily cater for other types of constraints such as those on axial strength. Conditions of cost minimality are derived using calculus of variation on an augmented Lagrangian. An iterative procedure based on opti- mality criteria is applied to a test example involving a reinforced concrete propped cantilever beam whose cross-section varies con- tinuously. Numerical examples are presented in which the design variables are both the width and the depth or the depth alone, and the optimal costs are compared. The solution of the test ex- ample with depth alone as the design variable is confirmed by an alternative approach using discretized continuum-type optimality criteria (DCOC). 1 Introduction Efficient design of the members of all reinforced concrete structures is essential to make effective use of the materi- als and to achieve a minimum cost of construction involving costs of concrete, steel and formwork. As pointed out by Kar- ihaloo (1993), labour cost may be included in each ingredient. The design variables are chosen in such a way that the cal- culated behaviour satisfies all structural and non-structural requirements according to Limit State Design (LSD). The essential features of Limit State Design Codes for reinforced concrete structures issued by international (e.g. CEB) or na- tional (e.g. Australian Standards Association) organizations are very similar. Thus the developments reported in this pa- per, although based on the CEB Code (CEB 1990), can be used in other countries without major alterations. The de- sign loads can, however, vary significantly from country to country, because of variation in climatic and seismological conditions. The optimal design process is generally iterative in na- ture, each iteration involving (i) the analysis of the structure under prescribed design loads for a known current design and (ii) redesign in which the design variables are modified to re- duce the value of the objective function without violating any constraints. On the basis of the techniques used in minimiz- ing the objective function, structural optimization methods may be grouped into two major categories. • Direct minimization, using mathematical programming methods (Kanagasundaram and Karihaloo 1990; Kari- haloo 1991; Karihaloo and Kanagasundaram 1986, 1988). • Indirect minimization using methods based on optimality criteria (Berke 1970; Venkayya, Khot and Berke 1973; Khot et al. 1978; Fleury 1979; Grierson and Moharrami 1993; Khot 1981; Rozvany et al. 1989, 1990; Zhou and Rozvany 1992/93; Rozvany and Zhou 1993). The former generally involves repeated computation of the objective and constraint functions and their gradients with respect t6 the design variables until a local minimum of the objective function is obtained. In the latter meth- ods, necessary (and/or sufficient) conditions of optimality are derived mathematically and then a solution satisfying these conditions is found either explicitly (analytical solu- tions) or iteratively (numerical solutions). Methods based on optimality criteria (OC) have been classified (Rozvany 1989; Rozvany et al. 1989, 1990; Rozvany and Zhou 1993) into dis- crete (DOC) and continuum-type (COG) methods depend- ing on whether the formulation involves only design variables (DOC), or design variables and real and virtual stress resul- tants (COC). For practical design problems, a diseretized ver- sion of continuum-type optimality criteria methods (DCOC) has been developed (Zhou and Rozvany 1992/93). The purpose of this paper is to demonstrate the use of COC on the model problem of a singly reinforced beam of rectangular section with a view to using later DCOC for large practical reinforced concrete structures. The design constraints such as the maximum deflection in the span, flex- ural and shear constraints (shear with regard to web crush- ing) are included according to design codes (CEB 1990; SAA 1988; Warner et al. 1988). After formulating the objective function to suit the solution technique based on COC, the necessary conditions for optimality are derived, and an iter- ative procedure for the solution of the resulting optimization problem is outlined. As a test example, a propped cantilever beam subjected to a uniformly distributed load is considered. Results are obtained when the design variables are both the width and depth or the depth alone, and the optimal costs are compared. For an independent verification of the solution to one of the test example, the DCOC is also used. A sum-