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The optimal design of an arch girder of variable curvature and stiffness by means of control theory. (English) Zbl 1427.74086

Summary: The problem of optimal design of the statically indeterminate arch girder which constitutes the primary structural system of the arch bridge is presented. The task is to determine the optimal shape of the axis of the arch girder, as well as the optimal distribution of the cross section height, ensuring the minimum arch volume as well as fulfillment of the standard requirements. This optimisation task, with numerous control functions and constraints, is formulated as a control theory problem with maintaining the formal structure of the minimum principle and then transformed to the multipoint boundary value problem and solved by means of numerical methods. The numerical results are obtained with optimal control methods, using the Dircol software. Since the changes in the shape and cross-section of the arch affect the distribution of the dead and moving loads transferred on the girder from the bridge deck, the optimisation procedure is combined with the finite element method analysis, which together with the complexity of the multidecision arch optimisation problem accounts for the novelty of the proposed approach. The numerical analysis reveals that the optimal girder shape is the frame-arched structure, with considerable lengths of straight sections and only short arch elements, in the areas of the application of concentrated forces and moments. The presented method can be successfully extended to optimisation of structures with different static schemes and load categories taken into account.

MSC:

74K10 Rods (beams, columns, shafts, arches, rings, etc.)
35Q74 PDEs in connection with mechanics of deformable solids
49Q12 Sensitivity analysis for optimization problems on manifolds

Software:

DIRCOL
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Szefer, G.; Mikulski, L., Optimal design of elastic arches with I cross-section, Engineering Transactions, 32, 4, 467-480 (1984) · Zbl 0559.73087
[2] Szefer, G.; Mikulski, L., Optimal design of elastic arches with rectangular cross section, SM archives, 10, 2, 171-185 (1985) · Zbl 0592.73120
[3] Marano, G. C.; Trentadue, F.; Petrone, F., Optimal arch shape solution under static vertical loads, Acta Mechanica, 225, 3, 679-686 (2014) · Zbl 1331.74139 · doi:10.1007/s00707-013-0985-0
[4] Trentadue, F.; Marano, G. C.; Vanzi, I.; Briseghella, B., Optimal arches shape for single-point-supported deck bridges, Acta Mechanica, 229, 5, 2291-2297 (2018) · Zbl 1390.74121 · doi:10.1007/s00707-017-2084-0
[5] Farshad, M., On optimal form of arches, Journal of The Franklin Institute, 302, 2, 187-194 (1976) · doi:10.1016/0016-0032(76)90022-3
[6] Hu, C.; Wan, Y.; ShangGuan, X., A new practice in the design of arch axis, Proceedings of the 6th International Conference on Arch Bridges, Fuzhou University
[7] Osserman, R., How the Gateway Arch got its shape, Nexus Network Journal, 12, 2, 167-189 (2010) · Zbl 1202.00088 · doi:10.1007/s00004-010-0030-8
[8] Houšt, V.; Eliáš, J.; Miča, L., Shape optimization of concrete buried arches, Engineering Structures, 48, 716-726 (2013) · doi:10.1016/j.engstruct.2012.11.037
[9] Habbal, A., Direct approach to the minimization of the maximal stress over an arch structure, Journal of Optimization Theory and Applications, 97, 3, 551-578 (1998) · Zbl 0914.73038 · doi:10.1023/A:1022685908429
[10] Kumarci, K.; Dehkordy, P. K.; Mahmodi, I., Optimum shape in brick masonry arches under dynamic loads by cellular automata, Journal of Civil Engineering, 37, 1, 73-90 (2009)
[11] Park, J.; Chun, Y.-H.; Lee, J., Optimal design of an arch bridge with high performance steel for bridges using genetic algorithm, International Journal of Steel Structures, 16, 2, 559-572 (2016) · doi:10.1007/s13296-016-6024-y
[12] Pouraminian, M.; Ghaemian, M., Shape optimisation of concrete open spandrel arch bridges, Gradjevinar, 67, 12, 1177-1185 (2015)
[13] Geering, H. P., Optimal Control with Engineering Applications (2007), Berlin, Germany: Springer, Berlin, Germany · Zbl 1121.49001
[14] Pontryagin, L. S.; Boltyanskii, V. G.; Gamkrelidze, R. V., The Mathematical Theory of Optimal Processes (1962), New York: Wiley-Interscience, New York · Zbl 0102.32001
[15] Laskowski, H., Optimal design of structural elements as a control theory problem, Technical Transactions. Civil Engineering, 114, 6, 119-134 (2017)
[16] Kropiowska, D., Selected problems of the optimal design of bar systems within the formal structure of the minimum principle (2016), Cracow: Cracow University of Technology Publishing House, Cracow
[17] Mikulski, L., Control structure in optimization problems of bar systems, International Journal of Applied Mathematics and Computer Science, 14, 4, 515-529 (2004) · Zbl 1137.93356
[18] Laskowski, H.; Mikulski, L., Control Theory in Composite Structure Optimizing, Measurement Automation Monitoring, 55, 6, 346-351 (2009)
[19] EN 1991-2 Eurocode 1:Actions on Bridges-Part 2: Traffic loads on bridges
[20] Rakowski, G.; Solecki, R., Curved bars. Statical analysis (1965), Arkady, Warszawa
[21] Gawecki, A., Mechanics of materials and bar structures (1998), Poznan: Poznan University of Technology Publishing House, Poznan · Zbl 0956.74026
[22] von Stryk, O., Numerical hybrid optimal control and related topics [Dissertation, thesis] (2000), Technische Universität München
[23] von Stryk, O., Users guide for Dircol (Version 2.1): a direct collocation method for the numerical solution of optimal control problems, Simulation and Systems Optimization Group (2002), Technische Universität Darmstadt
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