## 3 Modal-DBD Method
### Modal-DBD Method Applied to Blockhaus Systems
#### 3.4.2 Preliminary Assessment of the Modal-DBD Assumptions and Requisites for Blockhaus Log-Walls  
As emphasized in Section 3.1, the basic assumption of the Modal-DBD approach for a given case-study Blockhaus building, is that the in-plane stiffness of each shear-wall could be assumed as proportional to the wall length [11]. At the preliminary stage of the Modal-DBD extension to Blockhaus systems, accordingly, it was hence verified the validity of this mandatory requisite. Such a consideration can be essentially based on the refined FE numerical models of single carpentry joints that have been reported in [25], by accounting for the compression resistance limits of timber logs in use (i.e., for all the contact regions of corner joints in which crushing can occur). The mentioned FE approach proved in fact to offer an excellent correlation between joint stiffness estimated from literature experiments or by the corresponding FE approaches. As far as the in-plane lateral stiffness of a single log-wall is taken into account, it is then convenient to refer to the schematic representation in Figure 3, where the mechanical behavior is schematized for a single log-wall under in-plane lateral loads and vertical compressive pressures. Figure 2: Typical examples of (**a**) corner joint features and (**b**) in-plane lateral response for timber Blockhaus shear-walls under monotonic loads (figure reproduced from [21] with permission from Elsevier®, Copyright License Agreement n. 4681430084070, October 2019). Let us assume that the typical \(L\times H\) log-wall is thus obtained by the superposition of \(n\) timber members that are in contact along their top-bottom surfaces, and a sill log joined at the foundation level. For calculation purposes, the foundation restraints (typically consisting in regularly spaced steel angular brackets) can be assumed as indefinitely rigid, for the base log of each shear-wall. The main logs are then pre-compressed by a uniformly distributed load \(p\) and subjected to an in-plane lateral load \(F\). Under the so-imposed design loads, the expected top lateral displacement is given by the sum of sequential relative displacements of \(n\) overlapping logs, as well as by the possible local deformation (i.e., initial crushing of timber logs, in the region of corner notches, etc.) that can manifest in further flexibility of the system. The analytical model of Figure 3, in particular, assumes that [25]:  
* The corner joints are represented in the form of a series of linear elastic springs, with equivalent stiffness \(K_{el}\). The latter (when experiments are not available) can be estimated by means of detailed FE models of corner joints;
* The presence of \(n\) tolerance gaps is taken into account in the region of corner joints (with \(t_{gap}\) their average amplitude in the range from 0.5 to 1 mm), as an additional contribution for the total displacement estimation;
* Static friction effects are accounted in the form of \(f\)-Coulomb forces;
* Finally, the carpentry joints are considered at the logs ends only (i.e., possible internal joints and restraints are fully disregarded). The force equilibrium condition for the system in Figure 3 requires that:  
\[F=f+2K_{el}d_{i} \tag{1}\]  
where the friction contribution is given by:  
\[f=\mu pL \tag{2}\]  
with \(\mu\) the static friction coefficient. From Equation (1), it follows that the in-plane lateral force \(F\) can be also rewritten as:  
\[F=K_{tot}d_{top}=f+2K_{el}d_{i}=\mu pL+2K_{el}d_{i} \tag{3}\]  
Figure 3: Mechanical model of a typical log-house shear-wall under combined in-plane compression and lateral loads (front view). Reproduced from [25] under the terms and condition of CC BY-NC 4.0 license.