Structural and Multidisciplinary Optimization

, Volume 47, Issue 2, pp 157–174 | Cite as

Reinforcement layout design for concrete structures based on continuum damage and truss topology optimization

  • Oded Amir
  • Ole Sigmund
Research Paper


This article presents a new procedure for the layout design of reinforcement in concrete structures. Concrete is represented by a gradient-enhanced continuum damage model with strain-softening and reinforcement is modeled as elastic bars that are embedded into the concrete domain. Adjoint sensitivity analysis is derived in complete consistency with respect to path-dependency and the nonlocal model. Classical truss topology optimization based on the ground structure approach is applied to determine the optimal topology and cross-sections of the reinforcement bars. This approach facilitates a fully digital work flow that can be highly effective, especially for the design of complex structures. Several test cases involving two- and three-dimensional concrete structures illustrate the capabilities of the proposed procedure.


Reinforced concrete Topology optimization Continuum damage Truss optimization 



The work of the first author was funded by the Danish Council for Independent Research | Technology and Production Sciences. The work of the second author was funded by Villum Fonden via the NextTop project. These supports are gratefully acknowledged. We wish to thank Claus B.W. Pedersen for several fruitful discussions and for his important comments. We also thank Krister Svanberg for the FORTRAN MMA code.


  1. Batoz JL, Dhatt G (1979) Incremental displacement algorithms for nonlinear problems. Int J Numer Methods Eng 14:1262–1267MathSciNetzbMATHCrossRefGoogle Scholar
  2. Bažant ZP, Belytschko TB, Chang T (1984) Continuum theory for strain-softening. ASCE J Eng Mech 110(12):1666–1692CrossRefGoogle Scholar
  3. Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202CrossRefGoogle Scholar
  4. Bendsøe MP, Sigmund O (2003) Topology optimization—theory, methods and applications. Springer, BerlinGoogle Scholar
  5. Bogomolny M, Amir O (2012) Conceptual design of reinforced concrete structures using topology optimization with elasto-plastic material modeling. Int J Numer Methods Eng 90(13):1578–1597. doi: 10.1002/nme.4253 zbMATHCrossRefGoogle Scholar
  6. Bruggi M (2009) Generating strut-and-tie patterns for reinforced concrete structures using topology optimization. Comput Struct 87(23–24):1483–1495CrossRefGoogle Scholar
  7. Červenka J, Papanikolaou VK (2008) Three dimensional combined fracture-plastic material model for concrete. Int J Plast 24(12):2192–2220zbMATHCrossRefGoogle Scholar
  8. Chang T, Taniguchi H, Chen W (1987) Nonlinear finite element analysis of reinforced concrete panels. ASCE J Struct Eng 113:122–140CrossRefGoogle Scholar
  9. Dombernowsky P, Søndergaard A (2009) Three-dimensional topology optimisation in architectural and structural design of concrete structures. In: Proceedings of the International Association for Shell and Spatial Structures (IASS) symposium, Valencia, SpainGoogle Scholar
  10. Drucker DC, Prager W (1952) Soil mechanics and plastic analysis or limit design. Q Appl Math 10(2):157–165MathSciNetzbMATHGoogle Scholar
  11. Feenstra PH, de Borst R (1996) A composite plasticity model for concrete. Int J Solids Struct 33:707–730zbMATHCrossRefGoogle Scholar
  12. fib Task Group 44 (2008) Practitioners’ guide to finite element modelling of reinforced concrete structures. International Federation for Structural Concrete (fib), Lausanne, SwitzerlandGoogle Scholar
  13. Kato J, Ramm E (2010) Optimization of fiber geometry for fiber reinforced composites considering damage. Finite Elem Anal Des 46(5):401–415CrossRefGoogle Scholar
  14. Kato J, Lipka A, Ramm E (2009) Multiphase material optimization for fiber reinforced composites with strain softening. Struct Multidisc Optim 39(1):63–81CrossRefGoogle Scholar
  15. Kwak HG, Noh SH (2006) Determination of strut-and-tie models using evolutionary structural optimization. Eng Struct 28(10):1440–1449CrossRefGoogle Scholar
  16. Lemaître J, Desmorat R (2005) Engineering damage mechanics. Springer, BerlinGoogle Scholar
  17. Liang Q, Xie Y, Steven G (2000) Topology optimization of strut-and-tie models in reinforced concrete structures using an evolutionary procedure. ACI Struct J 97(2):322–330Google Scholar
  18. Lubliner J, Oliver J, Oller S, Oñate E (1989) A plastic-damage model for concrete. Int J Solids Struct 25:299–326CrossRefGoogle Scholar
  19. Marti P (1985) Truss models in detailing. Concr Int 7:66–73Google Scholar
  20. Mazars J, Pijaudier-Cabot G (1989) Continuum damage theory—application to concrete. ASCE J Eng Mech 115(2):345–365CrossRefGoogle Scholar
  21. Michaleris P, Tortorelli DA, Vidal CA (1994) Tangent operators and design sensitivity formulations for transient non-linear coupled problems with applications to elastoplasticity. Int J Numer Methods Eng 37:2471–2499zbMATHCrossRefGoogle Scholar
  22. Moen CD, Guest JK (2010) Reinforced concrete analysis and design with truss topology optimization. In: Proceedings of the 3rd fib international congress, Washington DC, USAGoogle Scholar
  23. Peerlings RHJ, de Borst R, Brekelmans WAM, de Vree JHP (1996) Gradient enhanced damage for quasi-brittle materials. Int J Numer Methods Eng 39(19):3391–3403zbMATHCrossRefGoogle Scholar
  24. Phillips D, Zienkiewicz O (1976) Finite element nonlinear analysis of concrete structures. ICE Proc 61(1):59–88CrossRefGoogle Scholar
  25. Schlaich J, Schafer K, Jennewein M (1987) Toward a consistent design of structural concrete. PCI J 32(3):74–150Google Scholar
  26. Sigmund O, Bendsøe MP (2004) Topology optimization: from airplanes to nano-optics. In: Stubkjær K, Kortenbach T (eds) Bridging from technology to society. Technical University of Denmark, Lyngby, DenmarkGoogle Scholar
  27. Sigmund O, Torquato S (1997) Design of materials with extreme thermal expansion using a three-phase topology optimization method. J Mech Phys Solids 45(6):1037–1067MathSciNetCrossRefGoogle Scholar
  28. Stromberg LL, Beghini A, Baker WF, Paulino GH (2011) Application of layout and topology optimization using pattern gradation for the conceptual design of buildings. Struct Multidisc Optim 43:165–180CrossRefGoogle Scholar
  29. Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24:359–373MathSciNetzbMATHCrossRefGoogle Scholar
  30. Victoria M, Querin OM, Martí P (2011) Generation of strut-and-tie models by topology optimization using different material properties in tension and compression. Struct Multidisc Optim 44:247–258CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringTechnical University of DenmarkLyngbyDenmark

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