*6.2. Base Restraints*

The actual boundary condition of timber members (and in particular the base restraint of the lateral members for the PO setup in Figure 3) represents, in the same way of mechanical interactions, a relevant influencing parameter for the examined joints. In this paper, three different boundary conditions are thus taken into account for the typical TTC specimen, including:


The BC#1 model, in this sense, coincides with the reference modelling strategy described in Section 3 and validated in Section 4. Variations for the BC#2 and BC#3 models are then represented by the restraint detailing only, with identical material properties and contact formulations.

In Figure 18, selected FE results are proposed for a given joint under different boundary conditions. According to Section 6.2, no relevant variations were observed for the elastic stage of the collected force-slip curves of Figure 18a, as well as for most of the examined joint configurations. However, depending on the arrangement of STSs in use, variations in the vertical reaction force were numerically predicted with up to a +30% increase of the BC#1 value, for the BC#2 and BC#3 conditions.

**Figure 17.** Analysis of boundary condition effects on the PO performance of TTC joints.

**Figure 18.** Analysis of boundary effects on the PO numerical response of TTC joints with inclined STSs. In evidence, the (**a**) vertical and (**b**) horizontal base reaction force, as a function of the measured slip for a selected TTC joint (ABAQUS/Explicit).

In addition, major variations were observed especially in terms of reaction forces in the horizontal direction for the BC#2 restraint, due to the use of unreliable boundary conditions for the standard PO setup (see Figure 18b).

#### *6.3. Friction Coe*ffi*cient*

At a final stage of this concise sensitivity study, the effects of friction phenomena are explored, with a particular attention for the timber-to-timber interface (Figure 6a). To this aim, the reference FE models in Section 3 are still taken into account, while the static friction coefficient for the mechanical contacts in use are progressively modified. According to [5], it is in fact known that friction at the timber interfaces should be considered for screws subjected to shear-tension stresses only (i.e., TTC joints with positive inclination α for the STSs, based on the convention of this paper). On the other side, any friction mechanism should be disregarded for STSs under shear-compression stresses (STSs with negative α). In this case, the central and lateral timber members of the PO setup are in fact expected to separate from each other, and thus enabling the development of possible attritive interactions.

Following Figure 7 and Equation (2), the force contributions are thus separately analyzed in this paper, for the examined TTC joints. For comparative purposes, more in detail, the input value for µ*timber* is progressively modified in the range from 0 and 0.8. *μ*

α

As expected, major variations of µ*timber* for TTC joints with an imposed shear-compression stress regime (α < 0) were found to have negligible effects on the collected force-slip contributions, given that: *μ* α

$$F = F\_{screw} \tag{6}$$

with:

$$F\_{\text{timber}} \approx \mathbf{0} \tag{7}$$

Figure 19 presents an example of the so-measured force-slip curves, with a focus on the S#1 joint with α = −15◦ . As shown in Figure 19a, minimum variations can be observed in the collected curves, even in presence of marked modifications for µ*timber*. Regarding the force contribution *Ftimber* sustained by the timber members, see Figure 19b, this is estimated as a limited part of the total *F*, thus agreeing with Equations (6) and (7). ௧ ≈ 0 α − *μ*

**Figure 19.** Analysis of static friction effects on the PO numerical response of TTC joints with inclined STSs. In evidence, the (**a**) vertical and (**b**) horizontal base reaction force, as a function of the measured slip (ABAQUS/Explicit).

*μ μ μ* In order to further investigate such an effect for different screw arrangements, finally, the FE parametric study (with µ*timber* = 0) was extended to several TTC joints under shar-compressive stresses. In Figure 20, comparative numerical results are proposed for the S#1 specimens as a function of µ*timber*. The parametric numerical results were post-processed from the collected force-slip curves according to Figure 8. As far as the relevant mechanical parameters are taken into account for them, their trend with µ*timber* can be investigated.

Δ Figure 20, more in detail, shows the percentage variation ∆ given by Equation (5), in terms of:


For clarity of presentation, the FE models with µ*timber* = 0 are set as a reference condition for ∆ calculations.

α α

*μ*

*μ* α

*μ*

*μ* Δ

*μ*

α

*μ* ≈

*μ*

α

*μ*

*μ*

α

**Figure 20.** Percentage variation (Equation (4)) of performance indicators for TTC joints with inclined STSs, as a function of the timber-to-timber friction coefficient (S#1 joints under shear-compression): (**a**) maximum force; (**b**) load-bearing contribution of the STSs and (**c**) serviceability stiffness (ABAQUS/Explicit).

α Regarding the total ultimate resistance *Fmax* of S#1 joints in Figure 20a) for example, it is possible to see that *Fmax* progressively increases as far as µ*timber* increases, for a given α. A relatively regular

*μ*

Δ *μ*

−

α

trend can be observed for the collected FE dots, as also suggested by the linear fitting curves. At the same time, however, it is possible to see that the increase of is indirectly proportional to α, thus maximum benefits deriving from additional frictional phenomena can be expected for STSs with limited inclination α only (α = 15◦ , in this study).

Such an outcome is strictly related to the occurrence, at failure, of local damage mechanisms in timber that can be further magnified especially for high α values (see also Section 4). As far as the typical static friction coefficients of interest for TTC systems are taken into account (i.e., µ*timber* ≈ 0.25–0.5), moreover, it is interesting to notice that the predicted *Fmax* values show a mean + 20–30% variation for STSs joints under shear-compression loads. This result from Figure 20a is thus a further confirmation of the relatively high sensitivity of ultimate resistance predictions for the examined TTC joints, under a standard PO setup.

When the shear force contribution that is sustained by the STSs only is taken into account, see Figure 20c, the same order of percentage variation is observed for various α values. In the figure, in particular, ∆*Fscrew* = 100% coincides with *Fmax* for the whole TTC specimen when µ*timber* = 0. Otherwise, the progressive increase of frictional effects with µ*timber* lead to a mostly linear increase of the total resistance *Fmax* in Figure 20a. As a result of such a kind of phenomenon, the load-bearing contribution of the STSs (in percentage terms) progressively decreases with µ*timber*, with variations that can be expected up to −20% compared to frictionless TTC joints (Figure 20b).

Finally, when the serviceability stiffness *Kser* is taken into account in Figure 20c, an opposite trend can be noticed for the collected FE results, as a function of and µ*timber*. This is in line with the general expectations and past literature efforts on the topic, where the serviceability stiffness of a given TTC joint reasonably increases when increasing the inclination α of the STSs. As a further remark for the FE results in Figure 20c, it can be noticed a relatively scattered variation of *Kser* estimates with µ*timber*, as far as α increases (i.e., Figure 14).

#### **7. Conclusions**

In this paper, the structural performance of timber-to-timber composite (TTC) joints with inclined self-tapping screws (STSs) was numerically investigated. The finite element (FE) numerical modelling assumptions were validated towards past experimental results of literature, by taking into account different arrangement and features for the STSs joints, including serviceability stiffness and ultimate resistance comparisons with analytical methods of literature. Through the FE parametric investigation, as shown, a key role was assigned to timber material properties but especially interface damage contacts in the region of fasteners. Major advantage was taken from the use of a surface-based cohesive zone modelling (CZM) damage interaction, so as to capture possible local effects and damage mechanisms in the examined TTC joint components.

For the examined small-scale TTC specimens under a standard push-out (PO) setup, in particular, an average scatter of −25% or +10% was generally observed for the load-bearing estimations in shear-compression and shear-tension respectively. Major deviations of FE models from the literature tests were mainly observed for the TTC specimens characterized by the presence of STSs with high inclination α (±40◦ or ±45◦ , in the current study), hence suggesting possible numerical issues due to mostly local effects, as well as possible uncertainties on the material properties and on the idealized description of the reference PO test setup.

On the other side, the collected FE estimations were always found to offer enhanced predictions for various STSs arrangements, compared to analytical models of literature. The ultimate resistance values, in particular, were generally strongly underestimated by analytical calculations, for various inclinations of STSs.

Based on extended parametric FE calculations, the actual sensitivity of PO numerical predictions to a series of relevant input parameters (and in particular the CZM damage parameters, the actual boundary condition of timber members and the effect of friction phenomena) was further emphasized. In doing so, major advantage was taken from the analysis of resultant forces that are expected to be

sustained (through the whole PO monotonic loading stage) by the steel screws or by timber components. The reciprocal mechanical interaction of the involved load-bearing members was thus explored.

**Author Contributions:** This paper results from a joint collaboration of all the involved authors. C.B.: conceptualization, analysis, validation, draft preparation, review; M.S. and M.F.: investigation, draft preparation, review. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
