## 2 Methods and Materials  
### Representative Volume Element and Structure Parametrization  
In this paper, the hollow core concrete slabs reinforced with steel bars are considered. The hollow cores are stadium shaped. The geometry of the structures replicates the commercial products of construction companies (manufacturers of prefabricated structures), which produces the precast HC slabs, for instance, SPK from Konbet [23] or HC from Pekabex [24]. The typical widths are 1200 mm and common heights of the slabs range from 150 to 500 mm, respectively. Because the scope of the paper is to show the efficiency of the proposed problem reduction via an elaborate numerical approach, minor attention has been given to material modeling and linear elasticity was assumed. The typical values of the material constants have been used, namely, for concrete and steel, a [PERSON]'s modulus of 30 Gpa and 210 Gpa and [PERSON]'s ratio of 0.2 and 0.3 have been used, respectively. Due to the homogenization method used, namely the Garbowski and Gajewski algorithm [9, 10] it is necessary to isolate the representative volume element (RVE). The periodic geometry of HC slabs defines the RVE geometry which was used in a natural way. The width of the RVE slab corresponds to the vertical distances between the concrete ribs. Moreover, it was assumed that the RVE has a square base. The example of the RVE geometry is presented in Figure 2a. In the numerical homogenization technique used, the finite element (FE) stiffness matrix of the RVE must be computed; here, the Abaqus FEA was used for computing the stiffness matrix. The steel bars were modelled by using two-node, 3-dimenssional truss elements (T3D2 according to Abaqus FEA [25]), while the concrete volume was modelled by using solid general purpose linear brick elements with reduced integration (C3 DRR according to Abaqus FEA [25]). These choices of element types are recommended and align with the best industry practices. Truss elements are particularly suitable for structures where the primary mode of load transfer is axial, such as in beams, columns, and trusses. Solid brick elements are well-suited for modeling concrete structures as they can capture both the tensile and compressive behaviors, as well as shear deformation and confinement effects. Moreover, the reduced integration helps to mitigate shear locking issues. Figure 1: The flowchart graphically representing the research study. In the RVE, the reinforcement was taken into consideration by using a special numerical technique called embedded region [7; 25]. In this technique, the interaction between steel and concrete is accounted for indirectly by using truss FE elements for modeling steel bars. Here, the stiffness of the steel truss FE is added to the concrete solid FE. In such an approach, an ideal bond between the materials is assumed. An alternative technique, not considered here, is the extrusion technique, in which steel is modeled by solid elements; the interaction is accounted by employing the contact techniques. The first approach is beneficial, because it does not require dense mesh in the area of the reinforcement and higher convergence is ensured. In the study, the RVE is built based on the set of six parameters, which are simultaneously the ANN input parameters, see Figure 2b, i.e., \(h\): slab height, \(a\): width of a periodic part, \(a_{1}\): position of the steel reinforcement, \(r\): hole radius, \(A_{s1}\): cross-sectional area of steel reinforcement and \(l\): vertical dimension of the hole. The parameters having the greatest impact on the mechanical behavior of the HC slabs were selected (excluding material properties). They will have a direct impact on the laminate stiffnesses obtained from the numerical homogenization procedure, which form the output of the ANN. The ANN output is represented by the laminate mechanical properties matrix, explained in Section 2.3. In Table 1, the lower and upper boundaries of the ANN input parameters assumed were demonstrated.