Global Journal of Researches in Engineering, E: Civil & Structural, Volume 23 Issue 2

Topology Optimization: Applications of VFLSM and SESO in the Generation of Three- Dimensional Strut-and-Tie Models Hélio Luiz Simonetti α , Valério S. Almeida σ , Virgil Del Duca Almeida ρ , Luttgardes de Oliveira Neto Ѡ , Marlan D. S. Cutrim ¥ & Vitor Manuel A. Leitão § Abstract- This article presents the analysis of Strut-and-Tie Model (STM) in reinforced concrete 3D structures based on the study of topological optimization, so that the problem is formulated with the Smooth-ESO (SESO) discrete method, whose removal heuristic is bidirectional with discrete optimization procedure, and the Velocity Field Level Set Method (VFLSM), which is an inheritance of the classical continuum Level Set Method (LSM), but advances the design limits with a velocity field constructed from the rate of the design variables and base functions. The proposed approach is to couple both methods in conjunction with the Method of Moving Asymptotes (MMA), used to control the various design constraints that are the minimization of compliance and the Von Mises stress that has demonstrated more rational STM results. Additionally, it has been formulated a methodology for the automatic generation of optimal of 3D STM by using sensitivity analysis obtaining via derivatives of the Von Mises stress fields, finding the force paths prevailing compression in the directions of the strut and the tensile in the directions of the ties for the reinforcement insertion. All the codes are implemented with Matlab software and several comparison examples: Deep beam with opening, a pile cap, a bridge pier, and a single corbel, are presented to validate the present formulations and the results are compared with the literature. I. I ntroduction n the field of structural engineering, most concrete linear elements are designed by the classical theory of Bernoulli hypothesis. For a real physical analysis about behavior of these bending elements it is common to use the Strut-and-Tie Model (STM) that is a generalization of the classical analogy of the truss beam model. This analogy is shown by Ritter and Morsch at the beginning of the twentieth century, associated with the Reinforced Concrete (RC) beam in an equivalent truss structure (regions B, Fig. 1). The bar elements represent the fields of tensile and the compressed struts emerged inside the structural element as bending effects. The analogy has been improved and is still used by the technical standards in the design of reinforced concrete beams in flexural and shear force and laying down various criteria for determining safe limits in its procedures. However, the application of this hypothesis for any structural element can lead to over or under sizing of certain parts of the structure. The Bernoulli hypothesis is valid for parts of the frame that there is no interference from other regions, such as sections near the columns, changing in geometry or other areas where the influence of strain due to shear efforts is not negligible. In this line, there are regions where the assumptions of Bernoulli do not adequately represent the bending structural behavior and the stress distribution. Structural elements such as beams, walls and pile caps and special areas such as beam-column connection, openings in beams and geometric discontinuities are examples. These regions, denominated “discontinuity regions D”, are limited to distances of the dimension order of structural adjacent elements (Saint Venant’s principle), that the shear stresses are applicable and the distribution of strains in the cross section is not linear. From the 80's, a Professor at the University of Stuttgart and other collaborators presented several researches that evaluated these regions more adequately, as [1], [2], [3], and other researchers as [4], [5] and [6]. The pioneering work by [1] describes the STM more generally, covering the equivalent truss models and including these regions and special structural elements. The analogy used in the STM uses the same idea of the classical theory in order to define bars representing the flow of stress trying to create the shortest and more logical path loads. Several experimental evaluations have been studied to validate the STM applied to the RC design, as [7], [8], [9] and [10]. I Keywords: reinforced concrete, topology optimization, strut-and-tie model, SESO, VFLSM. © 2023 Global Journals Global Journal of Researches in Engineering ( ) E Volume XxXIII Issue II Version I 23 Year 2023 Author α : Ph.D, Department of Mathematics, Instituto Federal de Educação, Ciência e Tecnologia de Minas Gerais IFMG. e-mail: helio.simonetti@ifmg.edu.br Author σ : Ph.D, Department of Geotechnical and Structural Engineering, School of Engineering, University of São Paulo (EPUSP). e-mail: valerio.almeida@usp.br Author ρ : MSc, Department of Automation and Control Engineering, Instituto Federal de Educação, Ciência e Tecnologia de Minas Gerais – IFMG. e-mail: virgil.almeida@ifmg.edu.br Author Ѡ : Doctor, Department of Civil and Environmental Engineering - School of Engineering –UNESP. e-mail: lutt@feb.unesp.br Author ¥ : MSc, Department of Structure Engineering and Geotechnics of the Polytechnic School of the University São Paulo –USP. e-mail: marlancutrin@usp.br Author § : Prof. Doctor, CERIS, Instituto Superior Técnico, Universidade de Lisboa. e-mail: vitor.leitao@tecnico.ulisboa.pt