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

The STM is recognized as a rational approach to the design of discontinuity regions and is incorporated in several current codes, such as ASCE- ACI 445 on Shear and Torsion [11], [12], [13] and [14]. These code provisions still require improvement due to uncertainties in the selection of optimal struts-and-ties, especially in the case of complex geometry or general load application conditions. Because of its simple model and needs the experience of the designer to select and distribute the elements of the model in order to represent the stresses path in a better way, it becomes evident the use of more reliable and automatic tools for defining its geometric and structural configuration. Fig. 1: D and B regions To overcome these difficulties and improve the efficiency in building the optimal STM in RC structures, the theory of Topology Optimization (TO) has been used for two decades as an alternative and systematic approach consolidating itself as a fruitful path of design related research, once facilitates the shaping of materials under certain conditions. Many methods have been proposed for the solution of TO applied to STM, highlighting the use of the classical SIMP: [15], [16], [17], or ESO (Evolutionary Structural Optimization): [18], [19], [20], Liang et al. [21,22,23], Chen et al. [24], Zhong et al. [25], or variants, like BESO, Shobeiri et al. [26], RESO (Refined ESO), Leu et al. [27] or SESO proposed by the present authors, Almeida et al. [28]. SESO is based on the philosophy that if an element is not really necessary for the structure, its contribution to the structural stiffness is gradually diminished until it does not have any influence in the structure; that is, its removal is done smoothly, not radically as in the ESO method, that have been showed more efficient and robust and less sensitive to the discretization than ESO and faster than BESO, causing a decrease of the checkerboard formation. In the last decade, the Level Set Method (LSM) has been highlighted in the field of TO, different from the conventional element wise density-based methods. LSM has clearer and smoother results and are flexible for complex topological changing, citing the pioneer’s works of [29], [30] and [31]. The method describes the topological path by an implicit shape evolutive sequence by using a higher dimensional function to the design space for achieving the minimum energy under design constraints. Several other schemes have been included in the standard LSM to improve performance and achieve better results for general applications, like [32], [33]. Wang and Kang [34,35] proposed the Velocity Field Level Set Method (VFLSM) which has been proved to be more efficient to deal with multiple constraints and design variables than LSM, but few works have been applied to STM by using VFLSM. OT in solving problems in the field of 3D STM is not much explored for general D-regions, discouraged by the instabilities (checkerboard problem) inherent to SIMP, ESO/BESO or the complex formulation and high processing time of LSM/VFLSM. Thus, for stabilizing and accelerating the TO solution, several mathematical optimization methods have been proposed, such as Optimality Criteria, by Huang et al. [36] with BESO, Augmented Lagrangian [37] or [38] with Level-Set, Lagrangian multiplier by [39] and [40] with LSM, or the Method of Moving Asymptotes (MMA), by [41] with SIMP. In the present work, aiming at the solution of 3D STM in general reinforced concrete problems, the SESO methods whose advantages are easy implementation and decrease of the checkerboard effect and the VFLSM, which deals well with shape and topological optimizations, are formulated together with the MMA optimization method to accelerate and stabilize 3D STM. It is also noteworthy new approach of sensitivity analysis is incorporated in these formulations for the automatic generation of struts-and-ties based on partial derivatives Global Journal of Researches in Engineering © 2023 Global Journals ( ) E Volume XxXIII Issue II Version I 24 Year 2023 Topology Optimization: Applications of VFLSM and SESO in the Generation of Three-Dimensional Strut-and-Tie Models