Figure 1
Concrete responses for uniaxial loading (a) in tension (b) in compression [1[1] ABAQUS. Abaqus analysis user’s manual, Version 6.10, Dassault Systèmes, 2010.]
Figure 2
Surface rupture in the plane of stresses [13[13] LEE, J.; FENVES, G. L. Plastic-damage model for cyclic loading of concrete structures, Journal of Engineering Mechanics, Vol. 124, No. 8, p. 892-900, 1998.]
Figure 3
Corresponding values for the parameter [13[13] LEE, J.; FENVES, G. L. Plastic-damage model for cyclic loading of concrete structures, Journal of Engineering Mechanics, Vol. 124, No. 8, p. 892-900, 1998.]
Figure 4
Strut-and-tie models according to Silva and Giongo (adapted from [24[24] SILVA, R. C.; GIONGO, J.S. Modelo de Bielas e Tirantes Aplicados a Estruturas de Concreto Armado. São Carlos: EESC-USP, 2000.])
Figure 5
Strut-and-tie Model of a simply supported beam ([5[5] AMERICAN CONCRETE INSTITUTE. ACI 318/05 - Building Code Requirements for Structural Concrete and Commentary, APPENDIX A: Strut-And-Tie Models”. Detroit, 2005.] and [18[18] MACGREGOR, J.G. Reinforced concrete mechanicas and desing. New Jersey. Prentice Hall, 1997. ])
Figure 6
Removal of the element from mesh by the method of optimization [19[19] FRANÇA, M. B. B.; GRECO, M.; LANES, R. M.; ALMEIDA, V. S.; Topological optimization procedure considering nonlinear material behavior for reinforced concrete designs. Computers and Concrete. Volume 17, Issue 1, pp.141-156, 2016. ]
Figure 7
Simply supported deep beam with a hole [21[21] SCHLAICH, J; SCHAFER, K; JENNEWEIN, M. Toward a consistent design of structural concrete. PCI-Journal, vol. 32, nr.3, p. 74-150, May/June, 1987.]
Figure 8
Optimum topologies by ESO and stress distribution obtained for the element, according to FEM (a) considering the linear and (b) non-linear behavior
Figure 9
The results are presented in the literature (a) in Schlaich et al. [21[21] SCHLAICH, J; SCHAFER, K; JENNEWEIN, M. Toward a consistent design of structural concrete. PCI-Journal, vol. 32, nr.3, p. 74-150, May/June, 1987.] by the process of the load path (b) and (c) in Liang et al. [15[15] LIANG, Q. Q.; XIE Y. M.; STEVEN, G.P. Topology optimization of strut-and-tie models in reinforced concrete structures using an evolutionary procedure. ACI Struct. J. 97(2): 322-330, 2000.] by the method ESO (d) and (e) in Almeida et al. [3[3] ALMEIDA, V. S.; SIMONETTI, H. L.; NETO, L. O. Análise de modelos de bielas e tirantes para estruturas de concreto armado via uma técnica numérica. Revista Ibracon de Estruturas e Materiais, Volume 6, 139-157, 2013.] by the SESO method of topological optimization
Figure 10
Strut-and-tie model with nodes and member numbers for Example 1
Figure 11
Geometry of strut for Example 1
Figure 12
Reinforcement sketch for Example 1
Figure 13
Bridge column [15[15] LIANG, Q. Q.; XIE Y. M.; STEVEN, G.P. Topology optimization of strut-and-tie models in reinforced concrete structures using an evolutionary procedure. ACI Struct. J. 97(2): 322-330, 2000.]
Figure 14
Optimum topologies by ESO and stress distribution required for the element, according to FEM (a) for linear and (b) non-linear behavior
Figure 15
Strut-and-tie model with nodes and member numbers for Example 2
Figure 16
Reinforcement sketch for Example 2
Figure 17
Corbel in a column [15[15] LIANG, Q. Q.; XIE Y. M.; STEVEN, G.P. Topology optimization of strut-and-tie models in reinforced concrete structures using an evolutionary procedure. ACI Struct. J. 97(2): 322-330, 2000.]
Figure 18
Solution considering the linear behavior of the material and (b) Solution considering the non-linear behavior of the material
Figure 19
Strut-and-tie model (a) considering the linear behavior and (b) considering the nonlinear behavior. Dotted lines representing struts and continuous lines representing ties
Figure 20
Strut-and-tie model with nodes and member numbers for Example 3
Figure 21
Reinforcement sketch for Example 3
Table 1
Identification of nodes of the Example 1
Table 2
Design of the compression bars of the Example 1
Table 3
Characteristics of the footings
Table 4
Identification of nodes of the Example 2
Table 5
Design axial forces in the bars of the Example 2
Table 6
Identification of nodes of the Example 3
Table 7
Design axial forces in the struts of the Example 3
Table 8
Design axial traction forces in the ties of the Example 3
