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Optimized reinforcement distribution in reinforced concrete structures under plane stress state

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The strut-and-tie model is widely used for analysis and design of reinforced concrete structures. To apply this model, it is necessary to define strut-and-tie systems that represent the flow of stresses generated in the analyzed structure. In many situations, this strut-and-tie model is defined through an evolutionary structural optimization (ESO) considering linear isotropic material. The results obtained from this model are not always satisfactory Query ID="Q1" Text="Please confirm the inserted city name and country name in affiliations 1 and 2 are correct and amend if necessary." because of the nonlinear behavior of the concrete, mainly because of the great difference in the behavior under tension and compression. Because of this, an evolutionary algorithm is developed in this article to define the optimized reinforcement distribution in reinforced concrete structures under plane stress state, considering the nonlinearity of the materials. This algorithm adopts the same principle as the ESO algorithm; however, it does not eliminate the mesh element that discretizes the analyzed domain, but it eliminates the reinforcement of the elements that do not meet an efficiency criterion. A three-node triangular finite element is used for nonlinear analysis of the reinforced concrete structure under plane stress state. The structure domain is discretized by this element, which can be of four types: with reinforcement in two fixed orthogonal directions, only in one direction, or without reinforcement. During the evolutionary process of the algorithm using a rejection criterion based on the level of reinforcement strain, the elements with reinforcement are changed to elements without reinforcement or with reinforcement in only one direction. Three practical applications were evaluated to verify the efficiency of the algorithm proposed in this article. Two of them had their results compared with results provided in the literature, verifying the efficiency of the proposed algorithm.

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  • Almeida VS, SimonettI HL, Neto LO (2013) Comparative analysis of strut-and-tie models using Smooth evolutionary structural optimization. Eng Struct 56:1665–1675

    Article  Google Scholar 

  • Andreassen E, Clausen A, Schevenels M, Lazarov BS, Sigmund O (2011) Efficient topology optimization in matlab using 88 lines of code. Struct Multidisc Optim 43(1):1–6

    Article  Google Scholar 

  • Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202

    Article  Google Scholar 

  • CEB/FIP MODEL CODE (2010) Fip Bulletin 55: Model Code 2010, First complete draft—vol 1. In: Internacional Federation for Structural Concrete.

  • Chen H, Yi WJ, Hwang HJ (2018) Cracking strut-and-tie model for shear strength evaluation of reinforced concrete deep beams. Eng Struct 163:396–408

    Article  Google Scholar 

  • Dashlejeh AA, Arabzadeh A (2019) Experimental and analytical study on reinforced concrete deep beams. Int J Struct Eng.

    Article  Google Scholar 

  • Farghaly AS, Benmokrane B (2013) Shear behavior of FRP-reinforced concrete deep beams without web reinforcement. J Composites Constr 17(6):0401315

    Article  Google Scholar 

  • Huang Z, Burgess IW, Plank RJ (1999) Non-linear analysis of reinforced concrete slabs subjected to fire. ACI Struct J 96(1):127–135

    Google Scholar 

  • Huang Z, Burgess IW, Plank RJ (2003) Modelling membrane action of concrete slabs in composite buildings in fire. Part I: Theoretical development. J Struct Eng 8(129):1093–1102

    Article  Google Scholar 

  • Huang Z, Burgess IW, Plank RJ (2003b) Modelling membrane action of concrete slabs in composite buildings in fire. Part II: validations. J Struct Eng ASCE 129(8):1103–1112

    Article  Google Scholar 

  • Lanes RM, Greco M (2013) Application of a topological evolutionary optimization method developed through python script. Sci Eng J 22:1–11

    Google Scholar 

  • Lanes RM, Greco M, Guerra MBBF (2019) Strut-and-tie models for linear and nonlinear behavior of concrete based on topological evolutionary structure optimization. Ibracon Struct Mater J 12:87–100

    Google Scholar 

  • Liang QQ, Steven GP (2002) A performance-based optimization method for topology design of continuum structures with mean compliance constraints. Comput Methods Appl Mech Eng 191:1471–1489

    Article  Google Scholar 

  • Liang QQ, Uy B, Steven GP (2002) Performance-based optimization for strut-and-tie modeling of structural concrete. J Struct Eng 128(6):815–823

    Article  Google Scholar 

  • Liang QQ, Xie YM, Steven GP (2000) Topology optimization of strut-and-tie models in reinforced concrete structures using an evolutionary procedure. ACI Struct J 97(2):322–330

    Google Scholar 

  • Pantoja JC, Pecin TG, Vaz LE, Martha LF (2010) Reliability analysis of concrete structures applied to strut-and-tie model. Advances and trends in structural engineering. Mechanics and Computation. Taylor & Francis Group, London, pp 775–778

    Google Scholar 

  • Querin OM, Steven GP, Xie YM (2000) Evolutionary structural optimisation using an additive algorithm. Finite Elem Anal Des 34(3–4):291–308

    Article  Google Scholar 

  • Rots JG, Kusters GMA, Blaauwendraad J (1984) The need for fracture mechanics options in finite element models for concrete structures. In: Damjanic et al (eds) Proceedings, Int. Conf. On Computer Aided Analysis and Design of Concrete Structures. Pineridge Press, Part 1, pp 19–32

  • Schlaich J, Schäfer K, Jennewein M (1987) Toward a consistent design of structural concrete. PCI J 32(3):75–150

    Article  Google Scholar 

  • Schäfer K, Schläich J (1991) Design and detailing of structural concrete using strut-and-tie models. Struct Eng 69(6):1991

    Google Scholar 

  • Sigmund O (2001) A 99 line topology optimization code written in Matlab. Struct Multidisc Optim 21(2):120–127

    Article  Google Scholar 

  • Souza RA (2004) Concreto Estrutural: análise e dimensionamento de elementos com descontinuidades. Doctoral thesis. USP - Polytechnic School of the University of São Paulo. Engineering and Foundations Department. São Paulo (in Portuguese)

  • Tan KH, Tong K, Tang CY (2001) Direct strut-and-tie model for prestressed deep beams. J Struct Eng 127(9):1076–1084

    Article  Google Scholar 

  • Tjhin TN, Kuchma DA (2002) Computer-based tools for design by strut-and tie method: advances and challenges. ACI Struct J 99(5):586594

    Google Scholar 

  • Wang GL, Meng SP (2008) Modified strut-and-tie model for prestressed concrete deep beams. Eng Struct 30(12):3489–3496

    Article  Google Scholar 

  • Xia Y, Langelaar M, Hendriks MAN (2020) A critical evaluation of topology optimization results for strut-and-tie modeling of reinforced concrete. Comput Aided Civil Infrastruct Eng 35(8):850–869

    Article  Google Scholar 

  • Xie YM, Steven GP (1993) A simple evolutionary procedure for structural optimization. Comput Struct 49:885–896

    Article  Google Scholar 

  • Xie YM, Steven GP (1996) Evolutionary structural optimization for dynamic problems. Comput Struct 58:1067–1073

    Article  Google Scholar 

  • Zhang N, Tan KH (2007) Direct strut-and-tie model for single and continuous deep beams. Eng Struct 29(11):2987–3001

    Article  Google Scholar 

  • Zhang N, Tan KH (2010) Effects of support settlement on continuous deep beams and STM modeling. Eng Struct 32(2):361–372

    Article  MathSciNet  Google Scholar 

  • Zhang H, Liu X, Yi W (2018) Experimental investigation on stress redistribution and load-transfer paths of shear walls with openings. J Struct Eng.

    Article  Google Scholar 

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The authors would like to thank the Federal University of Ouro Preto (UFOP/PROPEC), FAPEMIG ,and CNPq for collaboration.

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Correspondence to Amilton Rodrigues da Silva.

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da Silva, A.R., Ladeira, A.H. Optimized reinforcement distribution in reinforced concrete structures under plane stress state. Struct Multidisc Optim 65, 205 (2022).

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