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

Taking the local calculation of the derivative of the von Mises stress of the element with respect to the components of the stress vector described respectively as: ( ) 11 = 1 2 (2 11 − 22 − 33 ) ( ) 22 = 1 2 (2 22 − 11 − 33 ) ( ) 33 = 1 2 (2 33 − 11 − 22 ) ( ) 12 = 3 12 2 ( ) 23 = 3 23 2 ( ) 31 = 3 31 2 Considering Eq.(13) and making ( ( )) 11 > 0 then the elements are preponderantly tensioned (blue color - ties) while ( ( )) 33 < 0 are preponderantly compressed (green color - strut). The flowchart presented in Fig. 4 shows the original methodology presented in this section with the approach of using element sensitivity for automatic generation of STMs via stress derivatives, when a target volume is reached, the stopping criterion is reached. A set of techniques has not yet been presented in scientific articles on 3D models, so the results obtained in item 4 are compared with those proposed by [16], [26] and [45]. Highlights that the VFLSM method required a neighborhood filter to define the tensile (blue) and compression (green) regions. This filter is due to intermediate values that occur in continuous TO methods such as the intermediate densities that occur in the SIMP methodology. IV. N umerical E xamples The following examples of structures engineering focus on TO base on minimizing compliance for STMs. The geometry and boundary conditions for numerical applications are represented for each case. All numerical examples were processed on a Core i7-2370, 8th Gen notebook, 2.8 GHz CPU with 20.0 GB (RAM). a) Example 1 – Deep Beam with Opening The example presents a simply supported deep beam with an opening at the bottom left corner. The beam has its span three times its height and it is defined in [46], where the simple bending structural behavior is no longer considered. A vertical downward force F=3000 kN is applied eccentrically at the top edge as shown in Fig. 5. The structure is discretized with a total of 65,420 hexahedral elements (SESO) and 65,420 tetrahedral elements (LSM) (Fig. 5 shows the design domain and its boundary conditions). In this configuration, the force in off-center position and the opening positioned near the left low end create a situation that changes the internal stress flow in the structure, between the load and the supports. The tie elements, resulting from tensile stresses, are positioned at the extremities of the strut elements, resulting from compressive stresses, geometrically defining the final model. (13) Global Journal of Researches in Engineering © 2023 Global Journals ( ) E Volume XxXIII Issue II Version I 28 Year 2023 Topology Optimization: Applications of VFLSM and SESO in the Generation of Three-Dimensional Strut-and-Tie Models