Abstract
Optimization of frame structures composed of beams, columns and joints is considered. The problem is to find the optimal combination of standard cross sections from a provided catalog. The approach taken utilizes the Discrete Material Optimization (DMO) method to parameterize the problem and optimize using a gradient based method. It has roots in continuum topology optimization and thus strong parallels are drawn hereto in terms of methodology. The MATLAB implementation can take mass, compliance and stress criteria into account. In addition continuous joint stiffness design variables will indicate whether the joint should be rigid or pinned. Issues related to the non-convexity of the design spaces and the numerous local minima are discussed. The numerical results with benchmark models of varying complexity successfully validate the method as a design tool.
Similar content being viewed by others
References
Balling RJ (1991) Optimal steel frame design by simulated annealing. J Struct Eng 117(6):1780–1795. https://doi.org/10.1061/(ASCE)0733-9445(1991)117:6(1780)
Bazoune A, Khulief Y, Stephen N (2003) Shape functions of three-dimensional timoshenko beam element. J Sound Vib 259(2):473–480. https://doi.org/10.1006/jsvi.2002.5122
Bendsøe M, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer, Berlin
Bendsøe MP (1989) Optimal shape design as a material distribution problem. Structural Optimization 1 (4):193–202. https://doi.org/10.1007/BF01650949
Bruggi M (2008) On an alternative approach to stress constraints relaxation in topology optimization. Struct Multidiscip Optim 36(2):125–141. https://doi.org/10.1007/s00158-007-0203-6
Bruyneel M (2011) SFP-A new parameterization based on shape functions for optimal material selection: application to conventional composite plies. Struct Multidiscip Optim 43(1):17–27. https://doi.org/10.1007/s00158-010-0548-0
Bruyneel M, Duysinx P (2005) Note on topology optimization of continuum structures including self-weight. Struct Multidiscip Optim 29(4):245–256. https://doi.org/10.1007/s00158-004-0484-y
CEN (2007a) EN 1993: Design of steel structures - Part 1-8: Design of joints
CEN (2007b) EN 1993: Design of steel structures - Part 1-9: Fatigue
Cheng G, Guo X (1997) ε-relaxed approach in structural topology optimization. Structural Optimization 13(4):258–266. https://doi.org/10.1007/BF01197454
Cook RD, Malkus DS, Plesha ME, Witt RJ (2002) Concepts and applications of finite element analysis, 4th edn. John Wiley & Sons, Inc, USA
Dorn WS, Gomory RE, Greenberg HJ (1964) Automatic design of optimal structures. J de Mecanique 3:25–52
Fredricson H, Johansen T, Klarbring A, Petersson J (2003) Topology optimization of frame structures with flexible joints. Struct Multidiscip Optim 25(3):199–214. https://doi.org/10.1007/s00158-003-0281-z
Gibiansky L, Sigmund O (2000) Multiphase composites with extremal bulk modulus. J Mech Phys Solids 48(3):461–498. https://doi.org/10.1016/S0022-5096(99)00043-5
Huang M, Arora J (1997) Optimal design of steel structures using standard sections. Structural optimization 14(1):24–35. https://doi.org/10.1007/BF01197555
Huber G, Tschemmernegg F (1998) Modelling of beam-to-column joints. J Constr Steel Res 45(2):199–216. https://doi.org/10.1016/S0143-974X(97)00072-2
Hvejsel CF, Lund E (2011) Material interpolation schemes for unified topology and multi-material optimization. Struct Multidiscip Optim 43(6):811–825. https://doi.org/10.1007/s00158-011-0625-z
Jármai K, Farkas J, Kurobane Y (2006) Optimum seismic design of a multi-storey steel frame. Eng Struct 28(7):1038–1048. https://doi.org/10.1016/j.engstruct.2005.11.011
Jenkins WM (1992) Plane Frame Optimum Design Enviornment Based on Genetic Algorithm. J Struct Eng 118(November):3103–3112. https://doi.org/10.1061/(ASCE)0733-9445(1992)118:11(3103)
Kirsch U (1990) On singular topologies in optimum structural design. Structural Optimization 2(3):133–142. https://doi.org/10.1007/BF01836562
París J, Navarrina F, Colominas I, Casteleiro M (2009) Topology optimization of continuum structures with local and global stress constraints. Struct Multidiscip Optim 39(4):419–437. https://doi.org/10.1007/s00158-008-0336-2
Sekulovic M, Salatic R, Nefovska M (2002) Dynamic analysis of steel frames with flexible connections. Comput Struct 80(11):935–955. https://doi.org/10.1016/S0045-7949(02)00058-5
Sigmund O, Torquato S (1997) Design of materials with extreme thermal expansion using a Three-Phase topology. J Mech Phys Solids 45(6):1037–1067. https://doi.org/10.1016/S0022-5096(96)00114-7
Stegmann J, Lund E (2005) Discrete material optimization of general composite shell structures. Int J Numer Methods Eng 62(14):2009–2027. https://doi.org/10.1002/nme.1259
Stolpe M, Svanberg K (2001) An alternative interpolation scheme for minimum compliance topology optimization. Struct Multidiscip Optim 22(2):116–124. https://doi.org/10.1007/s001580100129
Torii AJ, Lopez RH, Miguel LFF (2015) Modeling of global and local stability in optimization of truss-like structures using frame elements. Struct Multidiscip Optim 51(6):1187–1198. https://doi.org/10.1007/s00158-014-1203-y
Tortorelli D, Michaleris P (1994) Design sensitivity analysis: overview and review. Inverse Prob Eng 1(1):71–105. https://doi.org/10.1080/174159794088027573
Yu-pin H, Yong-cun Z, Shu-tian L, Hou YP (2013) Optimization of standard cross-section type selection in steel frame structures based on gradient methods. 30(1):454–462. https://doi.org/10.6052/j.issn.1000-4750.2011.06.0388
Zhang Y, Hou Y, Liu S (2013) A new method of discrete optimization for cross-section selection of truss structures. Eng Optim 46(8):1052–1073. https://doi.org/10.1080/0305215X.2013.827671
Acknowledgements
The authors wish to acknowledge Christian Frier Hvejsel, PhD for his contributions to the work presented in this paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Krogh, C., Jungersen, M.H., Lund, E. et al. Gradient-based selection of cross sections: a novel approach for optimal frame structure design. Struct Multidisc Optim 56, 959–972 (2017). https://doi.org/10.1007/s00158-017-1794-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-017-1794-1