*5.3. Experimental and Analytical Assessment of Serviceability Sti*ff*ness Predictions (Kser)*

The elastic stiffness *Kser* is then estimated for the examined TTC joints, based on Equation (3) and the collected numerical force-slip curves. In Figure 14, the FE stiffness values are reported for the S#1-to-S#3 type (average) or S#4 type of specimens, as a function of α. Comparisons are proposed towards the past experimental data from [6], as well as the enhanced analytical formulation proposed in [6].

In general, a rather close correlation can be observed for the stiffness trend of most of the S#1, S#2 and S#3 configurations in Figure 14. However, major scatter of the FE predictions to the experiments can be still observed especially for high α values, both for shear-tension and shear-compression loading conditions. In the case of X-shaped joints, even a more pronounced sensitivity can be observed in terms of stiffness estimations as a function of α, compared to the S#1-to-S#3 joints and the respective experimental data from [6]. In terms of analytical assessment, finally, the same numerical predictions in Figure 14 are comprised, for the majority of joint configurations, between the single/double stiffness predictions derived from [6].

As far as the percentage scatter is calculated from Equation (5), it is interesting to notice in Figure 15 that such a variation is less regular than in the case of maximum force predictions, when the inclination α modifies. For the experimental data in Figure 15a, the scatter trend is mostly regular for all the S#*n* joints, thus suggesting a certain stability of material properties and mechanical assumptions for the FE models in use. Major sensitivity can be perceived in Figure 15b,c, as far as the single stiffness or double stiffness analytical model from [6] is taken into account.

α

α

α

**Figure 14.** Comparison of numerical (ABAQUS/Explicit) and analytical [6] serviceability stiffness estimates for TTC joints with inclined STSs: (**a**) S#1-to-S#3 (average) or (**b**) S#4 joints.

**Figure 15.** Percentage scatter of stiffness values for TTC joints with inclined STSs (Equation (5)), as obtained from the FE numerical analyses (ABAQUS/Explicit) and by literature [6]: (**a**) experimental data, (**b**) single stiffness analytical model, (**c**) double stiffness analytical model.

#### **6. Parametric FE Investigation**

Besides the rather close correlation in Section 5 for the FE predictions and the experimental and analytical results of literature, the sensitivity of the modelling technique to some influencing properties was further assessed. As a reference configuration, the typical FE model herein considered is thus characterized by input properties according to Section 3.

δ

δ

δ

## *6.1. Mechanical Interactions and CZM Damage Parameters*

A first insight is dedicated to the effects of mechanical interactions, with a special care of the STSs in use. It was shown in Section 3 the key role of the soft layer with the CZM contact interaction, as well as of reliable material property definitions. In this sense, the "upper limit condition" for the parametric study is assumed as a "tie" rigid constraint that is used to replace the CZM interaction for the soft layer (Figure 6). In other words, any kind of possible damage propagation in the region of screws (with the exception of possible material degradation in the timber and steel components) is fully disregarded. The "lower limit condition", at the same time, is set to coincide with the CZM formulation in Section 3 (with δ<sup>u</sup> = 4 mm, Table 2). Among these two conditions, further FE analyses are carried out with the CZM input parameters of Section 3 (Table 2), but progressively increasing the reference failure displacement δ<sup>u</sup> in the range from 4mm to a maximum of 10 × 4 = 40 mm. From a practical point of view, such a variation in δ<sup>u</sup> represents a residual capacity of the soft-layer to provide a certain mechanical interaction between each STS and the surrounding timber. Such an input value was in fact magnified so as to reproduce an ideal bonding condition with a weak mechanical degradation for the soft-layer interface, even under large slip amplitudes (Figure 6e). Selected numerical results are proposed in Figure 16 for two different screw arrangements (S#1 joints), in terms of measured vertical (*F*) and horizontal (*H*) base reaction forces as a function of the measured slip *s*. δ δ δ

α − α **Figure 16.** Analysis of mechanical interaction and CZM effects on the PO numerical response of TTC joints with inclined STSs (S#1). Vertical (*F*) and horizontal (*H*) reaction forces as a function of slip, for (**a**,**b**) α= −15◦ and (**c**,**d**) α= 45◦ (ABAQUS/Explicit).

α − Major variations between the comparative plots are represented by the screw inclination, with α = −15◦ and +45◦ .

The use of CZM interfaces, as also expected, proved to have a key effect on the collected mechanical responses for the selected STS configurations. This was observed especially towards the "upper limit

δ

δ

δ δ

δ

condition" characterized by the use of rigid "tie" constraint. In the latter case, the FE outcomes were in fact typically associated to unreliable local and global effects for the examined TTC joints, with a consequent marked increase of both the calculated serviceability stiffness *Kser* and ultimate resistance *Fmax*. Such a combination of phenomena, finally, was also found associated to a remarkable modification of the measured reaction forces (see for example Figure 16b–d).

Regarding the CZM interaction and failure, on the other side, major issues were represented by the accurate calibration of input parameters for damage initiation and evolution. While the nominal resistance values of timber (Table 1) can be reasonably taken into account for the CZM damage initiation, the characterization of its damage law would in fact necessarily require dedicated studies at the component level, and possibly the support of small-scale experiments.

Under the assumption of Table 2, the modification in δ<sup>u</sup> was commonly associated to a rather constant elastic response for the examined TTC joints, but to a marked decrease of residual resistance and stiffness for most of the tested configurations. Such an effect can be notice in Figure 16. As far as the critical displacement δ<sup>u</sup> for the CZM interaction increases, a reduced slope can be observed for the descending arm of the collected force-slip curves. Compared to the available experimental data from [6], a reliable fitting of degreasing arms for the comparative force-slip curves was observed in the range of δ<sup>u</sup> = 6–7mm. This fitting value δu, however, results from a numerical calibration in which the nominal mechanical properties of timber are taken into account (Tables 1 and 2). Accordingly, further refined, multi-objective and multi-parameter calculations should be carried out in this direction. Moreover, given that the CZM failure data were found to do not affect the initial stage of the collected force-slip curves (and thus the calculated serviceability stiffness and ultimate resistance for the examined joints), the reference value δ<sup>u</sup> = 4mm could be taken into account for preliminary conservative calculations on timber members with similar mechanical properties/class.
