Keywords: economic optimization, heuristics, concrete structures, structural design.

The design of economic concrete structures is up to date much conditioned by the experience of structural engineers. Most design office procedures are based on the adoption of cross-section dimensions and material grades based on sanctioned common practice. Once the structure is defined, it follows the analysis of stress resultants and the computation of the reinforcement that satisfy the limit states prescribed by concrete codes. Should the initial design be insufficient, the structure is redefined on a trial and error basis. The methods of structural optimization are an alternative to designs based on experience. They may be classified into exact methods and heuristic methods. The second group is the heuristic methods, whose recent development is linked to the evolution of artificial intelligence procedures. Examples of search algorithms are genetic algorithms, simulated annealing, threshold accepting, tabu search, ant colonies, etc. As regards RC structures, early applications in 1997 include the work of Coello et al. [1], who applied genetic algorithms to RC beams. Recently, there have been a number of RC applications, which optimize RC beams and building frames by genetic algorithms.

The structures studied in this work are frame bridges which are usually built of RC in road construction. The method followed has consisted first in the development of a computer module that checks the feasibility of given solutions. Dimensions, materials and steel reinforcement have been taken as variables. This module computes the cost of a given solution and checks all the relevant limit states. Simulated annealing, threshold accepting and tabu search are then used to search the solution space. The problem of structural concrete optimization that is put forward is an economic optimization. It deals with the minimization of the cost function, which is the sum of unit prizes multiplied by the measurements of construction units. Solutions have to comply with all the structural constraints, which are all the limit states that the structure has to satisfy.

Three heuristic methods have been used to explore the solution space. The first search method used in this work is the simulated annealing (SA henceforth), that was originally proposed by Kirkpatrick et al. [2]. The SA algorithm is based on the analogy of crystal formation from masses melted at high temperature and let cool slowly. The algorithm starts with a feasible random solution. This solution is changed by a small random move of the values of the variables. Greater cost solutions are accepted when a 0 to 1 random number is smaller than the expression , where is the cost increment and is the current temperature. The current solution is then checked against structural constraints. The second search method used in this work is the threshold accepting (TA henceforth), that was proposed by Dueck and Scheuer [3] in 1990 as an alternative to the SA algorithm. The algorithm also starts with a feasible solution randomly generated. Greater cost solutions are accepted when the cost increment is smaller than the current threshold. The current solution is then checked against structural constraints. The third search method used in the present paper is the tabu search (TS henceforth), that was proposed by Glover and Laguna [4] as a general purpose metaheuristic for solving difficult combinatorial problems. Tabu search is a metaheuristic that guides a local heuristic search procedure beyond local optimality by the use of its adaptive memory. Typically, TS stores attributes of recent moves in a list and it makes them tabu for a series of next moves. The heuristic local search has similarities to simulated annealing, for which random moves of the values of a number of the variables are performed and new solutions are accepted even when they increase the cost but the annealing accepting criteria is met.

The example studied relates to RC frame bridges used in road construction This problem has 50 design variables. Variables include the depth of the walls and slabs; 3 different grades of concrete for the 3 types of elements; and 44 types of reinforcement bars and bar lengths following a standard setup. All variables are discrete in this analysis. Structural restrictions considered followed standard provisions for Spanish design of this type of structure that include checks of the limit states of flexure and shear for the stress envelopes due to the traffic loads and the earth fill. Although fatigue of concrete and steel is rarely checked in road structures, it was considered since this ultimate limit state cannot be neglected. The SA-TA-TS algorithms were applied to a bridge box road frame of 13.00 m of horizontal free span, 6.17 m of vertical free span and 1.50 m of earth cover. The comparison of the 3 methods shows that TA-SA-TS give similar results, although running times for TA-TS are almost double than the SA times. The cost of best TA solution is 4616 euros/m. The depth of the top slab is 0.95 m of C25 (25 MPa of characteristic strength), which represents a slender span/depth ratio of 13.68. As regards deflections and fatigue limit states, it is shown that neglecting both limit states leads to 3.6% more economical solutions, but obviously unsafe. It is finally concluded that the comparison of the SA-TA-TS algorithms applied to the design of bridge frames shows similar results in terms of cost. Additionally, the study of RC bridge frames shows the potential use of heuristic algorithms for the advanced design of real concrete structures.

1

C.A. Coello, A.D. Christiansen and F. Santos, "A simple genetic algorithm for the design of reinforced concrete beams", Engineering with Computers, 13, 185-196, 1997. doi:10.1007/BF01200046

2

S. Kirkpatrick, C.D. Gelatt and M.P. Vecchi, "Optimization by simulated annealing", Science, 220(4598), 671-680, 1983. doi:10.1126/science.220.4598.671

3

G. Dueck and T. Scheuer, "Threshold accepting: A general purpose optimization algorithm superior to simulated annealing", Journal of Computation Physics, 90, 161-175, 1990. doi:10.1016/0021-9991(90)90201-B

4

F. Glover and M. Laguna, "Tabu Search", Kluwer Academic Publishers, Boston, 1997.

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