Automatic evaluation of plastic collapse conditions for planar frames with vertical irregularities


The plastic load and failure modes of vertically irregular planar frames are studied by means of an original software code developed in the agent-based programming environment NetLogo with a user-friendly interface. The proposed method lies in the limit analysis framework and is based on the generation of elementary collapse mechanisms and on their linear combination aimed at minimizing the collapse load factor. The considered irregularities consist in the absence of an arbitrary column in a regular grid of the frame and require considering additional elementary mechanisms, here presented for the first time, with respect to those associated to the corresponding regular frame. A further novelty of the method is the adoption, in the linear combination of elementary mechanisms, of negative coefficients, which, as better shown in the applicative section, is fundamental to grasp the actual collapse mechanism in irregular frames. The minimization procedure is efficiently performed by means of genetic algorithms, which allow computing both the collapse load factor and the correspondent failure mode with great accuracy and in a very short computing time. Many applications have been performed considering seismic load scenarios. Finally, by means of a parametric study, some general considerations on the weakest configurations of this typology of vertically irregular frames are provided.

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  1. 1.

    Mokhtar-zadeh A, Kaveh A (1999) Optimal plastic analysis and design of frames; graph-theoretical methods. Comput Struct 73:485–496

    Article  MATH  Google Scholar 

  2. 2.

    Kaveh A (1995) Structural mechanics: graph and matrix methods. In: Research studies press, 2nd edn, John Wiley, Exeter

    Google Scholar 

  3. 3.

    Baker J, Horne MR, Heyman J (1956) The steel skeleton plastic behavior and design, vol 2. Cambridge at the University Press, Cambridge

    Google Scholar 

  4. 4.

    Lyamin AV, Sloan SW (2002) Lower bound limit analysis using non-linear programming. Int J Numer Methods Eng 55(5):573–611

    Article  MATH  Google Scholar 

  5. 5.

    Pisano AA, Fuschi P, De Domenico D (2013) Peak loads and failure modes of steel-reinforced concrete beams: predictions by limit analysis. Eng Struct 56:477–488

    Article  Google Scholar 

  6. 6.

    Larsen KP, Poulsen PN, Nielsen LO (2012) Limit analysis of 3D reinforced concrete beam elements. J Eng Mech 138(3):286–296

    Article  Google Scholar 

  7. 7.

    Roca P, López-Almansa F, Miquel J, Hanganu A (2007) Limit analysis of reinforced masonry vaults. Eng Struct 29(3):431–439

    Article  Google Scholar 

  8. 8.

    Sloan SW (1988) Lower bound limit analysis using finite elements and linear programming. Int J Numer Anal Methods Geomech 12(1):61–77

    Article  MATH  Google Scholar 

  9. 9.

    Lyamin AV, Sloan SW (2002) Upper bound limit analysis using linear finite elements and non-linear programming. Int J Numer Anal Methods Geomech 26(2):181–216

    Article  MATH  Google Scholar 

  10. 10.

    Neal BG, Symonds PS (1952) The rapid calculation of plastic collapse loads for a framed structure. In: Proceedings of the Institution of Civil Engineers London, vol 1:2, pp 58–71

  11. 11.

    Neal BG, Symonds PS (1951) The calculations of collapse loads for framed structures. J Inst Civ Eng 35:21–40

    Article  Google Scholar 

  12. 12.

    Charnes A, Greenberg HJ (1951) Plastic collapse and linear programming. Bull Am Math Soc 57(6):480 (abstract)

  13. 13.

    Livesley RK (1977) Linear programming in structural analysis and design. In: Gallagher RH et al (eds) Optimum structural design. Wiley, New York (Chapter 6)

    Google Scholar 

  14. 14.

    Cohn MZ, Maier G eds (1979) Engineering plasticity by mathematical programming. Pergamon Press Ltd, New York

  15. 15.

    Watwood VB (1979) Mechanism generation for limit analysis of frames. J Struct Div ASCE 109:1–15

    Google Scholar 

  16. 16.

    Gorman MR (1981) Automated generation for limit analysis of frames. Proc ASCE ST7:1350–1354

    Google Scholar 

  17. 17.

    Deeks AJ (1996) Automatic computation of plastic collapse loads for frames. Comput Struct 60:91–102

    Article  MATH  Google Scholar 

  18. 18.

    Krabbenhoft K, Damkilde L (2003) A general non-linear optimization algorithm for lower bound limit analysis. Int J Numer Methods Eng 56 (2):165–184

    Article  MATH  Google Scholar 

  19. 19.

    Zhang Y-G, Lu M-W (1995) An algorithm for plastic limit analysis. Comput Methods Appl Mech Eng 126 (3–4):333–341

    Article  Google Scholar 

  20. 20.

    Gaudrat VF (1991) A Newton type algorithm for plastic limit analysis. Comput Methods Appl Mech Eng 88 (2):207–224

    Article  MATH  Google Scholar 

  21. 21.

    Wang H, Ohmori H (2013) Elasto-plastic analysis based truss optimization using Genetic Algorithm. Eng Struct 50:1–12

    Article  Google Scholar 

  22. 22.

    Kaveh A, Jahanshahi M (2008) Plastic limit analysis of frames using ant colony systems. Comput Struct 86:(11–12), pp. 1152–1163

    Google Scholar 

  23. 23.

    Chintanapakdee C, Chopra AK (2004) Seismic response of vertically irregular frames: response history and modal pushover analyses. J Struct Eng 130(8):1177–1185

    Article  Google Scholar 

  24. 24.

    Mahdi T, Gharaie VS (2011) Plan irregular RC frames: comparison of pushover with nonlinear dynamic analysis. Asian J Civ Eng 12(6):679–690

    Google Scholar 

  25. 25.

    Greco A, Cannizzaro F, Pluchino A (2017) Seismic collapse prediction of frame structures by means of genetic algorithms. Eng Struct 143:152–168. (ISSN: 0141–0296)

    Article  Google Scholar 

  26. 26.

    Goldberg DE (1989) Genetic algorithms in search, optimization and machine learning. Addison-Wesley, Boston

    Google Scholar 

  27. 27.

    Holland JH (1992) Adaptation in natural and artificial systems. MIT Press, Cambridge

    Google Scholar 

  28. 28.

    Gen M, Cheng R (2000) Genetic algorithms and engineering optimization, vol 7. John Wiley, Hoboken

    Google Scholar 

  29. 29.

    Man K-F, Tang K-S, Sam K (1996) Genetic algorithms: concepts and applications in engineering design. IEEE Trans Indus Electronics 43(5):519–534

    Article  Google Scholar 

  30. 30.

    Kaveh A, Khanlari K (2003) Collapse load factor for rigid-plastic analysis of frames using a genetic algorithm. In: Topping BHV (ed) Proceedings of the seventh international conference on the application of artificial intelligence to civil and structural engineering, Paper 33. Civil-Comp Press, Stirlingshire, UK.

  31. 31.

    Kaveh A, Khanlari K (2004) Collapse load factor of planar frames using a modified genetic algorithm. Commun Numer Methods Eng 20:911–925

    Article  MATH  Google Scholar 

  32. 32.

    Kaveh A, Jahanshahi M (2004) Plastic analysis of planar frames using kinematic method and genetic algorithm. Asian J Civ Eng (Build Hous) 5(3–4):145–160

    Google Scholar 

  33. 33.

    Kohama Y, Takada T, Kozawa N, Miyamura A (1997) Collapse analysis of rigid frame by genetic algorithm. In: Proceedings of OPT197. Wessex Institute of Technology, Rome, pp 456–461

  34. 34.

    Kaveh A, Jahanshahi M (2006) Plastic design of frames using heuristic algorithms. In: Topping BHV, Montero G, Montenegro R (eds) Proceedings of the eighth international conference on computational structures technology, Paper No 108. Civil-Comp Press, Stirlingshire

    Google Scholar 

  35. 35.

    Jahanshahi M, Pouraghajan M, Pouraghajan M (2013) Enhanced ACS algorithms for plastic analysis of planar frames. Comput Methods Civ Eng 4:65–82

    Google Scholar 

  36. 36.

    Kaveh A, Jahanshahi M, Khanzadi M (2008) Plastic analysis of frames using genetic algorithm and ant colony algorithm. Asian J Civ Eng 3:9227–9246

    Google Scholar 

  37. 37.

    Jahanshahi M, Maleki E, Ghiami A (2016) On the efficiency of artificial neural networks for plastic analysis of planar frames in comparison with genetic algorithms and ant colony systems. Neural Comput Appl.

    Google Scholar 

  38. 38.

    Kaveh A, Ghafari MH (2015) Plastic analysis of planar frames using CBO and ECBO algorithms. Int J Optim Civ Eng 5(4):479–492

    Google Scholar 

  39. 39.

    Rajeev S, Krishnamoorthy CS (1992) Discrete optimization of structures using genetic algorithms. J Struct Eng 118(5):1233–1250

    Article  Google Scholar 

  40. 40.

    Hofmeyer H, Davila Delgado JM (2015) Co-evolutionary and genetic algorithm based building spatial and structural design. AI EDAM 29:351–370

    Google Scholar 

  41. 41.

    Rafiq MY (2000) A design support tool for optimum building concept generation using a structured genetic algorithm. Int J Comput Integr Des Constr 2(2):92–102

    Google Scholar 

  42. 42.

    Turrin M, Von Buelow P, Stouffs R (2011) Design explorations of performance driven geometry in architectural design using parametric modelling and genetic algorithms. Adv Eng Inform 25(4):656–675

    Article  Google Scholar 

  43. 43.

    Mares C, Surace C (1996) An application of genetic algorithms to identify damage in elastic structures. J Sound Vib 195(2):195–215

    Article  Google Scholar 

  44. 44.

    Chou J-H, Ghaboussi J (2001) Genetic algorithm in structural damage detection. Comput Struct 79(14):1335–1353

    Article  Google Scholar 

  45. 45.

    Chiozzi A, Milani G, Tralli A (2017) Fast kinematic limit analysis of FRP-reinforced masonry vaults. I: general genetic algorithm-NURBS-based formulation. J Eng Mech 143(9):04017071-1–04017071-13

    Article  Google Scholar 

  46. 46.

    Wilensky U (1999) NetLogo. Center for connected learning and computer-based modeling. Northwestern University, Evanston. Accessed 1 Dec 2017

  47. 47.

    Mazzolani F, Piluso V (1997) Plastic design of seismic resistant steel frames. Earthq Eng Struct Dyn 26:167–191

    Article  Google Scholar 

  48. 48.

    CSI analysis reference manual for SAP2000, 2009  Etabs Safe and CsiBridge, Computers and Structures Inc., Berkeley (ISO No. GEN062708M1 Rev.4)

  49. 49.

    Caliò I, Greco A, D׳Urso D (2014) Free vibrations of spatial Timoshenko arches. J Sound Vib 333(19):4543–4561

    Article  Google Scholar 

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Greco, A., Cannizzaro, F. & Pluchino, A. Automatic evaluation of plastic collapse conditions for planar frames with vertical irregularities. Engineering with Computers 35, 57–73 (2019).

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  • Limit analysis
  • Elementary mechanisms method
  • Genetic algorithms
  • NetLogo
  • Irregular frames