Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Bulletin of the Polish Academy of Sciences Technical Sciences

The Journal of Polish Academy of Sciences

6 Issues per year

IMPACT FACTOR 2016: 1.156
5-year IMPACT FACTOR: 1.238

CiteScore 2016: 1.50

SCImago Journal Rank (SJR) 2016: 0.457
Source Normalized Impact per Paper (SNIP) 2016: 1.239

Open Access
See all formats and pricing
More options …
Volume 62, Issue 1


Graph based discrete optimization in structural dynamics

B. Blachowski
  • Institute of Fundamental Technological Research of the Polish Academy of Sciences, 5b Pawinskiego St., 02-106 Warsaw, Poland
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ W. Gutkowski
  • Corresponding author
  • Institute of Mechanized Construction and Rock Mining, 6/8 Racjonalizacji St., 02-673 Warsaw, Poland
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2014-03-25 | DOI: https://doi.org/10.2478/bpasts-2014-0011


In this study, a relatively simple method of discrete structural optimization with dynamic loads is presented. It is based on a tree graph, representing discrete values of the structural weight. In practical design, the number of such values may be very large. This is because they are equal to the combination numbers, arising from numbers of structural members and prefabricated elements. The starting point of the method is the weight obtained from continuous optimization, which is assumed to be the lower bound of all possible discrete weights. Applying the graph, it is possible to find a set of weights close to the continuous solution. The smallest of these values, fulfilling constraints, is assumed to be the discrete minimum weight solution. Constraints can be imposed on stresses, displacements and accelerations. The short outline of the method is presented in Sec. 2. The idea of discrete structural optimization by means of graphs. The knowledge needed to apply the method is limited to the FEM and graph representation.

The paper is illustrated with two examples. The first one deals with a transmission tower subjected to stochastic wind loading. The second one with a composite floor subjected to deterministic dynamic forces, coming from the synchronized crowd activities, like dance or aerobic.

Keywords : discrete structural optimization; combinatorial optimization; structural dynamics; stochastic loading; problem oriented optimization; graphs


  • [1] M.J. Turner, “Design of minimum-mass structures with specified natural frequencies”, AIAA J. 5 (3), 406-412 (1967).CrossrefGoogle Scholar

  • [2] K.C. Tang, M.Q. Brewster, E.J. Haug, B.R. McCart, and T.D. Streeter, “Optimal design of structures with constraints on natural frequency”, AIAA J. 8 (6), 1012-1019 (1970).Google Scholar

  • [3] Z. Mroz, “Optimum design of elastic structures subjected to dynamic, harmonically-varying loads”, Ang. Math. Mech. 50, 303-309 (1970).Google Scholar

  • [4] B.L. Pierson, “A survey of optimal structural design under dynamic constraints”, Int. J. Numerical Methods in Engineering 4 (4), 491-499 (1972).Google Scholar

  • [5] O.G. McGee and K.F. Phan, “On the convergence quality of minimum-weight design of large space frames under multiple dynamic constraints”, Structural Optimization 4, 156-164 (1992).Google Scholar

  • [6] R. Grandhi, “Structural optimization with frequency constraints - a review”, AIAA J. 31 (1870), 2296-2303 (1993).Google Scholar

  • [7] J.S. Arora, “Methods for discrete variable structural optimization”, in Recent Advances in Optimal Structural Design, ed. S.A Burns, pp. 1-40, ASCE Publication, Reston, 2002.Google Scholar

  • [8] L.C. Lee, B. Castro, and P.W. Partridge, “Minimum weight design of framed structures using a genetic algorithm considering dynamic analysis”, Latin American J. Solids and Structures 3, 107-123 (2006).Google Scholar

  • [9] W. Gutkowski, “Structural optimization with discrete design variables”, Eur. J. Mech., Solids A 16, 107-126 (1997).Google Scholar

  • [10] B. Blachowski and W. Gutkowski, “Discrete structural optimization by removing redundant material”, Engineering Optimization 40 (7), 685-694 (2008).Google Scholar

  • [11] B.S. Kang, W.S. Choi, and G.J. Park, “Structural optimization under equivalent static loads transformed from dynamic loads based on displacement”, Computers and Structures 79, 145-154 (2001).Google Scholar

  • [12] W.S. Choi, K.B. Park, and G.J. Park, “Calculation of equivalent static load and its application”, Trans. SMIRT Washington DC 16, paper #1111 (2001).Google Scholar

  • [13] M. Papadrakakis, N.D. Lagaros, and V. Plevris, “Multiobjective optimization of skeletal structures under static and seismic loading conditions”, Engineering Optimization 34 (6), 645-669 (2002).Google Scholar

  • [14] A. Norkus and R. Karkauskas, “Truss optimization under complex constraints and random loading”, J. Civil Engineering and Management 10 (3), 217-226 (2004).Google Scholar

  • [15] H.A. Jensen and M. Beer, “Discrete-continuous variable structural optimization of systems under stochastic loading”, Structural Safety 32 (5), 293-304 (2010).Google Scholar

  • [16] H.Y. Guo and Z.L. Li, “Structural topology optimization of high-voltage transmission, tower with discrete variables”, Structural and Multidisciplinary Optimization 43 (6), 851-861 (2011).Google Scholar

  • [17] W. Gutkowski, J. Bauer, and Z. Iwanow, “Support number and allocation for optimum structure”, Discrete Structural Optimization, Proc. IUTAM Symp. 1, 168-177 (1993).Google Scholar

  • [18] Z. Iwanow, “An algorithm for finding an ordered sequence of values of a discrete linear function”, Control and Cybernetics 6, 238-249 (1990).Google Scholar

  • [19] B. Blachowski and W. Gutkowski, “A hybrid continuousdiscrete approach to large discrete structural optimization problems”, Structural and Multidisciplinary Optimization 41 (6), 965-977 (2010).Google Scholar

  • [20] B. Blachowski and W. Gutkowski, “Revised assumptions for monitoring and control of 3D lattice structures”, 11th Pan- American Congress Applied Mechanics PACAM XI 1, CDROM (2010).Google Scholar

  • [21] J. Gu, Z.D. Ma, and G.M. Hulbert, “A new load-dependent Ritz vector method for structural dynamics analyses: quasistatic Ritz vectors”, Finite Elements in Analysis and Design 36 (3), 261-278 (2000).Google Scholar

  • [22] B. Blachowski, “Model based predictive control of guyed mast vibration”, J. Theoretical and Applied Mechanics 45 (2), 405-423 (2007).Google Scholar

  • [23] W. Borutzky, Bond Graph Methodology: Development and Analysis of Multidisciplinary Dynamic System Models, SCS Publishing House, Erlangen, 2004.Google Scholar

  • [24] A. Buchacz, “Modifications of cascade structures in computer aided design of mechanical continuous vibration bar systems represented by graphs and structural numbers”, J. Materials Processing Technology 157-158, 45-54 (2004).Google Scholar

  • [25] J.A. Bondy and U.S.R. Murty, Graph Theory, Springer, Berlin, 2008.Google Scholar

  • [26] E. Wilson, Three-Dimensional Static and Dynamic Analysis of Structures, 3rd edition, Computers and Structures Inc., Berkeley, 2002.Google Scholar

  • [27] H.A. Buchholdt, Structural Dynamics for Engineers, Thomas Telford, London, 1997.Google Scholar

  • [28] L. Carassale and G. Piccardo, “Double modal transformation and wind engineering applications”, J. Engineering Mechanics 127 (5), 432-439 (2001).Google Scholar

  • [29] E.N. Strommen, Theory of Bridge Aerodynamics, Springer, Berlin, 2006.Google Scholar

  • [30] M. Petyt, Introduction to Finite Vibration Analysis, 2nd edition, Cambridge University Press, Cambridge, 2010.Google Scholar

  • [31] W. Chen and T. Atsuta, Theory of Beam-columns, Space Behavior and Design, vol. 2, J. Ross Publishing, London, 2008.Google Scholar

  • [32] A.L. Smith, S.J. Hicks, and P.J. Devine, “Design of floors for vibration: a new approach”, The Steel Construction Institute, New York, 2009.Google Scholar

  • [33] M. Szczepaniak and T. Burczynski, “Swarm optimization of stiffeners locations in 2D structures” , Bull. Pol. Ac.: Tech. 60 (2), 241-246 (2012). Google Scholar

About the article

Published Online: 2014-03-25

Published in Print: 2014-03-01

Citation Information: Bulletin of the Polish Academy of Sciences: Technical Sciences, Volume 62, Issue 1, Pages 91–102, ISSN (Print) 0239-7528, DOI: https://doi.org/10.2478/bpasts-2014-0011.

Export Citation

This content is open access.

Comments (0)

Please log in or register to comment.
Log in