New approach to optimization of reinforced concrete beams
Introduction
The search for an effective design of concrete structures is not a new subject. Recently, this problem has received much attention, particularly due to wide spread of concrete materials in structural engineering. In most applications, however, the aim has been at finding an optimum weight of a structure for given design conditions. An almost standard task of designing structures for their maximum strength/weight ratio has been addressed by a number of researches [3], [8], [9], to cite a few.
To explore other areas of the optimization space we focused on one of the key quantities decisive for the efficient structural design. Quite naturally, we selected the total price of a structure to guard the optimization process.
In our previous work [6] we suggested that an optimization problem, which reduces the cost of a structure, can be tackle very efficiently using the genetic algorithm based optimizers [4], [7]. Successively, we attempted to reduce the final cost of a structure by first modifying its shape and then including the effect of bending. In other words we were concerned with the objective function consisting of twelve design variables only. In this particular case even a simple genetic algorithm was able to deliver appreciable results for a reasonable number of iterations.
Including the effect of shear reinforcement, however, caused this algorithm to fail as it became prohibitively expensive. The reason is linked to a high complexity of the optimization problem manifested by 21 mutually independent variables entering the optimization procedure. The present paper overcomes this obstacle by implementing a version of the augmented simulated annealing method [5].
Section 2 outlines formulation and evaluation of the objective function. Review of the augmented simulated annealing method is provided in Section 3. Section 4 then describes several optimization strategies we pursued. Results are derived for a simple continuous steel reinforced concrete beam.
Section snippets
Objective function
As intimated in the introductory part we search for a configuration characterized by a minimum price, which yet complies with all selected allowable strength and serviceability limits.
To introduce the subject we first present formulation of the desired objective function in the formsubjected to following constraintsIn Eq. (1) Vc is the volume of concrete and Ws is the weight of steel; Pc and Ps are the price of concrete per unit volume
Optimization techniques
Before proceeding with formulation of the actual optimization procedure we first introduce the data structure for individual design variables. Next we briefly review essentials of genetic algorithms and their implementation in conjunction with the augmented simulated annealing method.
Results and discussion
As an example we selected a continuous beam subjected to a uniformly distributed load according to Fig. 12. Due to symmetry, only one half of the beam was analyzed. Distribution of internal forces (bending moment and shear force) were found using the finite element method, recall Fig. 3. The required amount of steel follows from Eq. (9). The price Pc=1350.0 Kč/m3 for concrete and Ps=50.0 Kč/kg for steel was assumed (Kč––czech crowns).
In [6] this problem was already solved for two specific
Acknowledgements
Financial support was provided by the GAČR 103/97/P040 grant and by the research project J04/98:210000003.
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