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The present paper describes the application of an evolutionary algorithm to the optimum design of the reinforcement of timber beams using FRP laminates and sheets. The objective function is the material cost of the strengthening and is subjected to ten constraints derived from the ultimate limit states for flexural and shear behaviour as well as the serviceability limit states. A genetic algorithm is used and the optimization problem is transformed into an unconstrained one by means of an adaptive penalty function. The design variables are the CFRP and GFRP mechanical properties and dimensions and they are encoded in a binary chromosome: type of composite material (CFRP or GFRP), reinforcement mechanical properties and geometric configuration. The search space for the minimum cost consists of 65 billion possible solutions. The crossover operator switches randomly between a fenotype crossover and flat crossover. An adaptive mutation scheme has been as well as an elitism criterion. The algorithm has been used for obtaining optimum designs in several specific load and geometry cases of glued laminated timber beams. The objective is finding whether there are specific reinforcement configurations more feasible for a certain loading situations: short or long beams and lower or higher loading increments. Five cases have been analysed. In the first three cases the length of the beams has constant values of 2, 2.5 and 3 m, whereas the value of loading was variable. In the latter case, the value of the load was fixed and the length of the beam was variable. The analysis of the results shows that the GFRP reinforcement is more efficient than CFRP for designs governed by shear failure, whereas CFRP is more effective in the case of flexural failure and deflection controlled strengthening of timber beams.
adaptive operators, FRP strengthening, genetic algorithm, structural optimization, timber structures
[1] Triantafillou, T.C., Shear reinforcement of wood using FRP materials. Journal of Materials in Civil Engineering, 9(2), pp. 22–25, 1997.https://doi.org/10.1061/(asce)0899-1561(1997)9:2(65)
[2] Bru, D., Baeza, F.J., Varona, F.B., García-Barba, J. & Ivorra, S., Static and dynamic properties of retrofitted timber beams using glass fibre reinforced polymers. Materials and Structures, 49(1), pp. 181–191, 2014.https://doi.org/10.1617/s11527-014-0487-0
[3] Holland, J.H., Adaptation in natural and artificial systems, MIT Press: Cambridge, Mass., 1975.
[4] Perera, R. & Varona, F.B., Flexural and shear design of FRP plated RC structures using a genetic algorithm. ASCE Journal of Structural Engineering, 135(11), pp. 1418–1429, 2009.https://doi.org/10.1061/(asce)0733-9445(2009)135:11(1418)
[5] Bru, D., Baeza, F.J., Varona, F.B. & Ivorra, S., Numerical and experimental evaluation of FRP reinforcement on the mechanical behaviour of timber beams. Proceedings of the 16th European Conference on Composite Materials, pp. 1–8, 2014.
[6] Yang, Y., Liu, J. & Xiong, G., Flexural behaviour of wood beams strengthened with HFRP. Construction and Building Materials, 43¸ pp. 118–124, 2013. https://doi.org/10.1016/j.conbuildmat.2013.01.029
[7] Gen, M. & Cheng, R., A survey of penalty techniques in genetic algorithms. Proceedings of the 1996 International Conference on Evolutionary Computation, pp. 804–809, 1996.
[8] Radcliffe, N.J., Equivalence class analysis of genetic algorithms. Complex Systems, 5, pp. 183–205, 1991.
[9] Juvandes, L.F.P. & Barbosa, R.M.T., Bond analysis of timber structures strengthened with FRP systems. Strain: An International Journal for Experimental Mechanics, 48, pp. 124–135, 2012.https://doi.org/10.1111/j.1475-1305.2011.00804.x