## 4 Case-Study Building
### Modal-DBD Design Steps
#### 4.2.7 Refined Numerical Modeling of Log-Walls and Derivation of the Backbone Parameters  
In order to define the backbone parameters of interest for design purposes, a detailed analysis of each main log-wall of Figure 4 was first carried out with the support of refined Finite Element (FE) numerical models. To this aim, each P1-to-P4 was subdivided into three parts in the building elevation, and namely detected as GF (Ground Floor), FF (First Floor) and SF (Second Floor) respectively. Each log-wall portion was thus modeled using the ABAQUS/Explicit software [30], and separately assessed under in-plane lateral loads (i.e., Figure 2). First, to ensure the consistency and accuracy of FE models reproducing each log-wall portion, preliminary analyses that are not included in this paper, for the sake of clarity, were carried out by modeling a full-size wall having the same geometrical and mechanical features of the test specimens presented in [19]. The numerically obtained load-displacement curve was compared with [19], for a validation of FE assumptions. Accordingly, the same modeling strategy was extended to P1-to-P4 shear log-walls in Figure 4.  
Basically, the whole 3D modeling approach was based on previous research studies, see [20; 25] and the detail view in Figure 6. Each FE model was realized via a set of timber components, as obtained from 8-node, linear brick, solid elements with reduced integration (C3D8R) of the ABAQUS element library. Orthogonal log-walls (i.e., outriggers) able to mechanically interact with the shear log-walls were accounted in a fixed total length of 1 m [20]. Simplified rectangular cross-sections (thus deprived of top/bottom notches and protrusions) were used to describe each solid log [20], while the corner joints were accurately reproduced in their nominal geometry. The nominal gap was set in \(t_{gap}=1\) mm [20; 31]. Rigid ideal restraints were taken into account in place of the base angular brackets [20]. C24-resistance class spruce was then numerically described based on the mechanical properties in Table 1, via an orthotropic, equivalent elasto-plastic material. The resistance parameters in the directions of interest were accounted in the form of Hill plastic law, as also reported in [25; 32]. More in detail, the compressive resistance perpendicular to the grain (Y-local axis of Figure 6) was set in \(f_{c,90}\) = 3.29 MPa. The other Hill stress ratios were then calculated along the remaining directions (X and Z directions, and XY, XZ and YZ planes for the shear components) were defined so as to reproduce the mean compressive resistance parallel to the grain (\(f_{0,c}\) = 30 MPa, X-axis), the mean compressive resistance perpendicular to the grain (\(f_{90,c}\) = 3.29 MPa, Z-axis) and the mean shear strength \(f_{v}\) = 3.96 MPa (XY, XZ, YZ) for timber. Based on Table 1, the above mean values were calculated in accordance with [33], that is:  
\[f_{m}=\frac{f_{k}}{1-1.645 COV[f_{k}]}; \tag{16}\]  
with \(COV[f_{m}]=10\%\) for compression perpendicular to the grain and \(\mathrm{COV[f_{m}]}=0.8\mathrm{g}\) (25%) for compression parallel to the grain. Regarding the contact interactions that were included as a key boundary non-linearity of the reference FE assembly, the \"general contact\" surface-to-surface formulation was used, as also reported in [20; 25]. In the case of overlapping logs, the main input parameter was represented by the static friction coefficient, that was set to \(\mu\) = 0.5 for the parametric simulations [19; 20]. The FE description of door/window openings, finally, was based on the drawings of Figure 4 and on past numerical studies [19; 20; 31]. For each log-wall, both an in-plane lateral displacement and a constant vertical pre-compressive load were simultaneously introduced, as also in accordance with [19; 20]. Such pressure amplitude was separately calculated, for each wall, so as to reproduce a uniform pre-compressive magnitude. The horizontal in-plane lateral displacement was imposed to the lateral surface of the top beam.