Backtrack Method with Applications to Dso

  • J. Farkas
  • K. Jármai
Part of the International Centre for Mechanical Sciences book series (CISM, volume 373)


The backtrack discrete mathematical programming method is described giving a detailed flow chart. If a continuous mathematical method is used and discrete series of values are given for variables, the discrete optima can be determined by a complementary discretization which is also explained. Optimum design problems of stiffened and cellular plates, tubular trusses, welded box beams and welded steel silos are treated. In these applications the discrete variables appear in various forms. In the cost function the material and fabrication (welding) costs are formulated. It is shown that the optimum number of ribs in stiffened or cellular plates depends on the fabrication cost factor. In the optimization of trusses it is verified that the use of the Euler buckling formula gives unsafe solutions and the optimum geometry depends on the profile shape of compression members. In the multiobjective optimization of welded box beams the deflection is formulated as the third objective function in addition to the cost and weight functions. The systematic incorporation of the cost analysis in the optimization procedure is shown in the case of a welded steel silo. The detailed strength and cost calculation is carried out for the main structural parts of a silo for several discrete values of the height/diameter ratio to find the optimum one.


Multiobjective Optimization Plate Element Local Buckling Fabrication Cost Stiffened Plate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 4.1
    Walker,R.J.: An enumerative technique for a class of combinatorical problems, in: Proc. of Symposia in Appl. Math. Amer.Math.Soc.Providence, R.I. 10(1960), 91–94.Google Scholar
  2. 4.2
    Golomb,S.W. and L.D.Baumert: Backtrack programming, J. Assoc. Computing Machinery, 12 (1965), 516–524.MathSciNetCrossRefMATHGoogle Scholar
  3. 4.3
    Bitner,J.R. and E.M.Reingold: Backtrack programming techniques, Communications of ACM, 18 (1975), 651–656.CrossRefMATHGoogle Scholar
  4. 4.4
    Lewis,A.D.M.: Backtrack programming in’welded girder deign, in: Proc. 5th Annual SHARE-ACM-IEEE Design Automation Workshop, Washington, 1968, 28/1–28/9.Google Scholar
  5. 4.5
    Annamalai,N.: Cost optimization of welded plate girders. Dissertation, Purdue Univ. Indianapolis, Ind. 1970.Google Scholar
  6. 4.
    Farkas,J. and L.Szabó: Optimum design of beams and frames of welded I-sections by means of backtrack programming. Acta Techn. Hung. 91(1980),121–135Google Scholar
  7. 4.7
    Knuth,D.E.: Estimating the efficiency of backtrack programs. Mathematics of Computation 29 (1975), 121–136.MathSciNetCrossRefMATHGoogle Scholar
  8. 4.8
    Farkas,J.: Optimum Design of Metal Structures. Akadémiai Kiadó, Budapest, Ellis Horwood, Chichester, 1984.Google Scholar
  9. 4.9
    Järmai,K.: Optimal design of welded frames by complex programming method. Publ.Techn.Univ.Heavy Ind. Ser.C. Machinery, 37 (1982), 79–95.Google Scholar
  10. 4.10
    Farkas,J.: Cost comparisons of plates stiffened on one side and cellular plates. Welding in the World 30 (1992), 132–137.Google Scholar
  11. 4.
    Pahl,G.and K.H. Beelich: Kostenwachstumgesetze nach Aenlichkeitsbeziehungen fir Schweissverbindungen, in: VDI-Bericht Nr. 457.1982, Diisseldorf, 129–141.Google Scholar
  12. 4.12
    Farkas,J.: Discussion to “Simplified analysis for cellular structures” by Evans,H.R. and Shanmugam,N.E.: J,Struct. Eng. ASCE 111 (1985), 2268–2271.Google Scholar
  13. 4.13
    Timoshenko,S. and S. Woinowsky-Krieger: Theory of plates and shells. 2“ ed. McGraw Hill, New York-Toronto-London, 1959.Google Scholar
  14. 4.14.
    Usami,Ts. and Y.Fukumoto: Local and overall buckling of welded box columns. J.Struct.Div. Proc. ASCE (1982), 525–541.Google Scholar
  15. 4.15
    Zhou,J.L. and A.Tits: User’s guide for FSQP Version 3.0: a Fortran code for solving optimization problems. Systems Research Center, University of Maryland, Techn. Report SRC-TR-90–60 rlf, College Park. 1992.Google Scholar
  16. 4.16
    Farkas,J.: Minimum cost design of tubular trusses considering buckling and fatigue constraints, in: Tubular Structures. 3`’ Int. Symposium, 1989, Lappeenranta, E. Niemi, P. Mäkeläinen (eds), Elsevier, London, 1990, 451–459.Google Scholar
  17. 4.17
    Farkas,J.: Techno-economic considerations in the optimum design of welded structures. Welding in the World 29 (1991), 295–300.Google Scholar
  18. 4.18
    Farkas,J. and K.Järmai: Minimum cost design of laterally loaded welded rectangular cellular plates. In Structural Optimization ‘83 World Congress, Rio de Janeiro. Proc. Vol. 1. 1993, 205–212.Google Scholar
  19. 4.19
    Ott,H.H. and V. Hubka: Vorausberechnung der Herstellkosten von Schweisskonstruktionen. (Fabrication cost calculation of welded structures), in: Proc.Int.Conference on Engineering Design ICED, 1985. Hamburg. Ed. Heurista, Zürich, 1985, 478–487.Google Scholar
  20. 4.20
    COSTCOMP, Programm zur Berechnung der Schweisskosten. Deutscher Verlag fir Schweisstechnik, Düsseldorf, 1990.Google Scholar
  21. 4.21.
    Bodt,H.J.M.: The global approach to welding costs. The Netherlands Institute of Welding, The Hague, 1990.Google Scholar
  22. 4.22
    American Petroleum Institute: API Bulletin on Design of flat plate structures. Bul. 2V, 1st, ed. 1987.Google Scholar
  23. 4.23
    Eurocode 3. (EC3) Design of steel structures. Part 1.1. Brussels, CEN - European Committee for Standardization. 1992.Google Scholar
  24. 4.24
    Khot,N.S. and L.Berke: Structural optimization using optimality criteria methods, in: New directions in optimum structural design. Eds. Atrek,E., Gallagher,R.H. et al. Wiley and Sons, Chichester, New York, etc.1984,.47–74.Google Scholar
  25. 4.25
    Vanderplaats,G.N. and F.Moses: Automated design of trusses for optimum geometry. Journal of Structural Division Proc. ASCE 98 (1972), 671–690.Google Scholar
  26. 4.26
    Saka,M.P.: Shape optimization of trusses. Journal of Structural Division Proc. ASCE 106 (1980), 1155–1174.Google Scholar
  27. 4.27
    Amir,H.M. and T.Hasegawa: Shape optimization of skeleton structures using mixed-discrete variables. Structural Optimization 8 (1994), 125–130.CrossRefGoogle Scholar
  28. 4.28
    Farkas,J.: Optimum design of circular hollow section beam-columns, in: Proceedings of the Second International Offshore and Polar Engineering Conference, San Francisco, 1992. ISOPE, Golden, Colorado, USA. 494–499.Google Scholar
  29. 4.29
    Saka,M.P.: Optimum design of pin jointed steel structures with practical applications. Journal of Structural Division Proc. ASCE 116 (1990), 2599–2620CrossRefGoogle Scholar
  30. 4.30
    Farkas,J.and K.Jârmai: Savings in weight by using CHS or SHS instead of angles in compressed struts and trusses, in: Tubular Structures VI. Proceedings of the 6th International Symposium, Melbourne, 1994. Eds. Grundy,P.,Holgate,A.,Wong,B. Balkema, Rotterdam–Brookfield. 417–422.Google Scholar
  31. 4.31
    Dutta,D. and K-G.Wiürker: Handbuch Hohlprofile in Stahlkonstruktionen. Köln, TÜV Rheinland GmbH, 1988.Google Scholar
  32. 4.32
    Wardenier,J., Kurobane,Y. et al.: Design guide for circular hollow section joints under predominantly static loading. Köln, TÜV Rheinland, 1991.Google Scholar
  33. 4.33
    Rondal,J., Würker, K-G. et al.: Structural stability of hollow sections. Köln, TÜV Rheinland. 1992.Google Scholar
  34. 4.34
    Saka,M.P.. Optimum geometry design of roof trusses by optimality criteria method. Computers and Structures 38 (1991), 83–92.CrossRefMATHGoogle Scholar
  35. 4.
    Koumousis,V.K.. Lay-out and sizing design of civil engineering structures in accordance with the Eurocodes, in: Topology Design of Structures. Eds. Bendsoe,M.P. and C.A. Mota Soares. Dordrecht-Boston-London: Kluwer:1992, 103116.Google Scholar
  36. 4.36
    Packer,J.A., J. Wardenier et al.. Design guide for rectangular hollow section joints under predominantly static loading. Köln: TÜV Rheinland, 1992.Google Scholar
  37. 4.37
    Hasegawa,A.,H. Abo et al.. Optimum cross-sectional shapes of steel compression members with local buckling, in: Proc. JSCE Structural Engineering/ Earthquake Engineering 2(1985),121–129.Google Scholar
  38. 4.38
    Eschenauer,H., Koski,J. and Osyczka,A., Multicriteria Design Optimization. Springer, Berlin, etc. 1990.CrossRefMATHGoogle Scholar
  39. 4.39
    Koski, J.: Bicriterion optimum design method for elastic trusses. Acta Polytechnica Scandinavica, Mechanical Engineering Series No.86. Helsinki, 1984.Google Scholar
  40. 4.40
    Osyczka,A.: Multicriterion Optimization in Engineering. Ellis Horwood, Chichester, 1984.Google Scholar
  41. 4.41
    Farkas,J.: Fabrication aspects in the optimum design of welded structures. Structural Optimization 4 (1991), 51–58.CrossRefGoogle Scholar
  42. 4.42
    Jârmai,K.: Single-and multicriteria optimization as a tool of decision support system. Computers in Industry 11 (1989), 249–266.CrossRefGoogle Scholar
  43. 4.43
    Jârmai,K.: Application of decision support system on sandwich beams verified by experiments. Computers in industry 11 (1989), 267–274.CrossRefGoogle Scholar
  44. 4.44
    Jârmai,K.: Decision support system on IBM PC for design of economic steel structures, applied to crane girders. Thin-walled Structures 10 (1990), 143–159.CrossRefGoogle Scholar
  45. 4.45
    Farkas,J. and K.Jârmai: Multiobjective optimal design of welded box beams. Microcomputers in Civil Engng 10 (1995), 249–255.CrossRefGoogle Scholar
  46. 4.46
    Martens,P. ed.: Silo-Handbuch. Berlin, Ernst and Sohn, 1988.Google Scholar
  47. 4.47
    Gaylord,E.H.jr. and Gaylord,Ch.N.: Design of steel bins for storage of bulk solids. Prentice Hall,Inc. Englewood Cliffs, New Jersey, 1984.Google Scholar
  48. 4.48
    Trahair,N.S.,Abel,A. et al.: Structural design of steel bins for bulk solids. Australian Institute of Steel Construction, Sydney, 1983.Google Scholar
  49. 4.49
    Teng,J.G. and J.M.Rotter: Recent research on the behaviour and design of steel silo hoppers and transition junctions. Journal of Constructional Steel Research, 23 (1992), 313–343CrossRefGoogle Scholar
  50. 4.50
    Farkas,J.: Discussion to “Elastic behaviour of isolated column-supported ringbeams” by Rotter, J.M.- J. Constructional Steel Research 4(1984),235–252. J.C.S.R 5 (1985), 239–242.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 1997

Authors and Affiliations

  • J. Farkas
    • 1
  • K. Jármai
    • 1
  1. 1.University of MiskolcMiskolcHungary

Personalised recommendations