*4.2. Service Type and Slip Modulus*

In Figure 10, the average % variation of slip modulus *Kser* is shown as a function of the moisture content, the bonding direction to the grain and the type of adhesive.

**Figure 10.** Average experimental variation of slip modulus *Kser* for specimens with (**a**) parallel or (**b**) perpendicular rod-to-grain arrangement.

The service class, as shown, generally resulted in severe modification of mechanical parameters for the tested specimens, both for epoxy or polyurethane bonded rods, with more pronounced effects for loading parallel to the grain (Figure 10a). In the perpendicular direction, a rather stable variation of average stiffness results can be observed in Figure 10b for both the adhesive types.

#### *4.3. Service Class and Load-Bearing Capacity*

The above quantitative comparisons were further supported by the qualitative analysis of experimental outcomes. In general terms, the analysis of BiR performances under different service conditions can be summarized as follows.

For service class 1:


For service class 2:


Finally, for service class 3:


In this regard, the analysis of test results can take advantage of existing empirical formulations that have been proposed for glued-in-rods with parallel or perpendicular rod-to-grain orientation. Among the literature efforts for the analytical analysis and design of BiR connections, the failure load of parallel rod-to-grain arrangement can be for example estimated as [15]:

$$F\_{\rm ax,0} = \pi \, L \left( f\_v d\_{equ} + k(d+e)e \right) \tag{6}$$

The empirical equation has been proposed by Feligioni et al. [15] to predict the pull-out strength at failure (in N), where *L* is the joint length (mm); *f<sup>v</sup>* is the shear strength of wood (MPa); *d* is the rod diameter (mm); *dequ* the smaller between the hole diameter *d<sup>h</sup>* and the rod diameter *d* multiplied by 1.25 (mm); *e* is the joint thickness (mm); *k* is a parameter proposed in 0.086 or 1.213 (based on experimental fitting), for adhesives with brittle or ductile behaviour respectively.

For comparative studies with the current experimental results, the above parameters are set:

$$d\_{equ} = \min(14; 1.25 \times 10) = 12.5 \,\text{mm} \tag{7}$$

$$f\_{v,k} = 1.2 \times 10^{-3} d\_{eq}^{-0.2} \rho^{1.5} = 10.65 \text{ MPa} \tag{8}$$

with *ρ* = 600 kg/m<sup>3</sup> the average density of wood specimens (±25); *e* = 2 mm, *L* = 60 mm.

From Equation (6), the comparative analysis is carried out towards the average (mean) experimental failure loads earlier discussed, for various configurations of specimens. The findings are summarized in Figure 11, as a function of the analytical vs. experimental failure load, the adhesive type and the moisture/service class. As far as the service class 1 is taken into account, it is possible to see that the analytical to experimental ratio is in the order of the unit. This suggests a certain correlation of literature model with the current experiments. Besides, the analysis of higher moisture content tends to progressively overestimate the analytical failure force for the examined specimens, as shown in Figure 11 for both the adhesive types. Furthermore, the high moisture content reveals also pronounced effects due to the input *k* coefficients calibrated from [15].

**Figure 11.** Average experimental failure load versus the ratio of analytical prediction, as obtained for specimens with (**a**) epoxy or (**b**) polyurethane adhesive and parallel rod-to-grain arrangement.

For specimens with perpendicular rod-to-glue arrangement, the analytical model proposed by Yeboah et al. is considered [30]. The model, in particular, assumes that the load-bearing capacity is given by:

$$F\_{\rm ax,90,mean} = f\_{v,90,mean} \pi \, d\_h \mathcal{L} \tag{9}$$

with the limit applicability condition of *L* < 15*d<sup>h</sup>* .

The empirical model of Equation (9) agrees with the experimental trends earlier discussed in Figure 8. As far as the moisture content increases and the adhesive degradation of mechanical properties increases, Equation (9) itself severely overestimates the expected axial force at failure for the tested joints. In Figure 12, the empirical derivation of material strength is shown for epoxy or polyurethane specimens, as obtained from the mean experimental results.

**Figure 12.** Inverse experimental derivation of *fv,90,mean* strength, based on Equation (9), for epoxy or polyurethane specimens with perpendicular rod-to-glue arrangement.

#### **5. LEFM-Based Analytical Model**

#### *5.1. State-of-Art*

Theoretical approaches, based on the stress distribution in the joint, have been used to describe the laws governing the mechanical behaviour of connections by glued-in rods. One of the pioneering works was by Volkersen [31], who developed an elastic analysis of the shear distribution in a single lap joint. However, the substrates were assumed to respond to

the load only in tension and the adhesive to respond only in shear. A further development was made by Goland and Reissner [32], who included the influence of a bending moment in the connection in their calculation model. Later, Hart-Smith [33] introduced elastic-plastic stress distribution of anisotropic materials in the analysis. Depending on the ductility of the bond-line, these traditional strength analyses will be more or less accurate.

More recently, the behaviour of glued-in rods was investigated within the framework of fracture mechanics. In this approach, a pre-existing crack in the joint is assumed to lead to a stress singularity so that the traditional maximum stress criterion can be no longer applied. For instance, in accordance with LEFM, Serrano [11] proposed an evaluation model of the load bearing capacity for a single glued-in rod, assuming that failure of the joint could occur when the energy release rate is equal to the fracture energy. In the same year, Gustafsson [17] took into consideration the damage amount preceding the failure of a joint through an approach based on nonlinear fracture mechanics (NLFM), and essentially based on the mode II fracture energy. Thus, many empirical or theoretical design calculations could be found in literature to estimate either the shear strength (especially in studies based on elastic stress analysis) or the fracture energy in mode II. It should be noted that most existing studies were either based on experiments or numerical investigations, but rarely combined both approaches [21].

#### *5.2. Model Definition*

In order to check this assumption, the study of the fracture behaviour of BiR connection is proposed within the framework of equivalent LEFM, which is well known to be useful to characterize the quasibrittle failure of load-bearing components [21].

The reference mechanical model is schematized in Figure 13, with evidence of the required geometrical and mechanical parameters in the detailed view.

**Figure 13.** Mechanical model for the analysis of force transmission and deformation behaviour in BiR connections.

Considering that the joint acts as a fibre in the matrix, where the fibre represents a steel glued-in bar and the matrix consists of the wood log as in Figure 13, the following model of shear stresses distribution along the joint is proposed in this study:

$$\tau(\mathbf{x}) = \frac{A\_s}{b\_b} \omega \left( (\sigma\_{\text{s,max}} - q) \frac{\cosh(\omega \mathbf{x})}{\sinh(\omega \mathbf{L})} + q \frac{\cosh(\omega (L-\mathbf{x}))}{\sinh(\omega \mathbf{L})} \right) \tag{10}$$

where *A*<sup>0</sup> is cross-area of the wooden part; *b<sup>k</sup>* is the mean width of the adhesive layer; *σ*0,*max* represents the maximum normal stress in the BiR; *x* is the length coordinate; *L* denotes the length of the adhesive joint (anchorage length); ω is a correction factor that can be estimated from:

*ω*

$$r^2 = \frac{1}{p} \tag{11}$$

and:

$$p = \left(\frac{E\_s A\_s E\_{w} A\_w}{E\_w A\_w + E\_s A\_s}\right) \frac{h\_b}{G\_b h\_b} \quad \left(\text{in mm}^2\right) \tag{12}$$

$$q = \left(\frac{E\_{\rm s}}{E\_{\rm w}A\_{\rm w} + E\_{\rm s}A\_{\rm s}}\right) F\_{\rm ax} \quad \text{(in MPa)}\tag{13}$$

By integrating Equation (10) to get a strain energy release rate, the J-integral method could be implemented directly. Further, the problem analysis and the definition of an accurate behaviour model for BiR connections should be necessarily based on local stress distribution which has been obtained in the framework of DIC system (i.e., Figure 7).

According to the known maximum shear strength of wood, it is possible to predict the bearing capacity of the BiR connection, i.e., maximum pull-out force, as:

$$F\_{\rm ax} = \frac{\pi(\mathbf{x}) \cdot \frac{b\_b}{\omega} \cdot \frac{\sinh(\omega \cdot \mathbf{L})}{\cosh(\omega \cdot \mathbf{x})}}{1 + \left(\frac{E\_s \cdot A\_s}{E\_w \cdot A\_w + E\_s \cdot A\_s}\right) \cdot \left(\frac{\cosh(\omega \cdot (\mathbf{L} - \mathbf{x})) - \cosh(\omega \cdot \mathbf{x})}{\cosh(\omega \cdot \mathbf{x})}\right)} \tag{14}$$

From the developed analytical model, the variation of the shear force, stress and strain in BiR connections can be thus predicted along the bonding length *L*. Selected examples are proposed in Figure 14.

**Figure 14.** Analytical prediction of shear (**a**) force, (**b**) stress and (**c**) strain in BiR connections with epoxy or polyurethane adhesives (examples for 9% moisture and parallel rod-to-glue arrangement).

#### *5.3. Assessment of Analytical Predictions*

The proposed analytical model is further assessed by taking advantage of the available test results and of nominal material properties earlier discussed. Parametric calculations were carried out on the grouped specimens (average estimates) in terms of shear force and stress, by changing the adhesive type, arrangement and environment condition.

In this regard, the analytical model proved to offer reliable estimates for both the parameters of force and stress agreeing with the general trends of Figure 14. Comparative examples are proposed in Figure 15, in terms of force or stress, as obtained for grouped specimens as a function of their service class. It should be noted that the error ratio R is found both to overestimate or underestimate the expected parameter, for all the types of BiR specimens. Most importantly, however, is that the collected *R* values confirm the rather small scatter for all the analytical predictions, thus confirming the validity and accuracy of the approach.

**Figure 15.** Average experimental failure load versus the ratio of analytical prediction, as obtained for specimens with (**a**) epoxy or (**b**) polyurethane adhesive and parallel rod-to-grain arrangement.

#### **6. Conclusions**

In this paper, the mechanical performance of bonded-in rod (BiR) connections for structural timber applications has been explored experimentally and analytically.

The experimental investigations revealed, significant differences in the observed mechanical behaviour of BiR connections, by changing the adhesives type, the bonding arrangement and the wet climate exposure. The pull-out test set-up and the use of a digital image correlation (DIC) system, in particular, helped to obtain results in support of the definition of generalized design tools for this type of connections.

The experimental study, in most of the cases, exhibited a failure mechanism of the connections in the wood, in the vicinity of the wood–adhesive interface. As such, a study of the stress field along this interface was performed with the use of a newly developed linear elastic fracture mechanics (LEFM) formulation. In addition to the shear stress, expected for this kind of connection, the stress field analysis revealed the existence of normal stress (to the interface), which was relevant at the onset of the failure.

The observed rheological behaviour of adhesive types in use further indicates that (in terms of reliability) special attention should be paid to joints exposed to extreme climatic conditions. The current study provided useful information about the short-term behaviour of bonded-in-rods. However, the long-term behavioural analysis of BiR connections requires further investigations, in order to check the mechanical performance of this repair process according to service classes defined in the European timber design codes. Most importantly, additional requirements in standards should be included, or certification from the adhesive manufacturer should be sought, to ensure the safe use of this type of joints in practical applications. In this regard, further research efforts will be dedicated to the in-depth analysis of mechanical parameters and their sensitivity to severe environment conditions, so as to include additional configurations and parameters of technical interest.

**Author Contributions:** This research paper results from a joint collaboration of the involved authors. More precisely J.B. and M.K.B. carried out the experiments; C.B. and V.R. contributed to the postprocessing of results; V.R. supervised all the research activities. All authors have read and agreed to the published version of the manuscript.

**Funding:** The EU-COST Action CA18120 is gratefully acknowledged for providing financial support to the first and third authors (J.B. visitor at Aarhus University, Denmark and C.B. visitor at University of Zagreb, Croatia), in the framework of CERTBOND Short Term Scientific Missions—STSM grants.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data supporting this research study will be made available upon request.

**Acknowledgments:** This publication is based upon work from EU-COST Action CA18120 (CERTBO ND—https://certbond.eu/ (accessed on 16 February 2021)), supported by COST (European Cooperation in Science and Technology—https://www.cost.eu/ (accessed on 16 February 2021)). MDPI is also acknowledged for waived APCs (C.B.).

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

#### **Appendix A**

Experimental force–displacement curves for the investigated BiR specimens, grouped by adhesive type, rod-to-glue arrangement and service class.

**Figure A1.** Experimental force–displacement results for specimens with (**a**) epoxy or (**b**) polyurethane glue, under 18% moisture and parallel rod-to-grain arrangement.

**Figure A2.** Experimental force–displacement results for specimens with (**a**) epoxy or (**b**) polyurethane glue, under 18% moisture and bonding and perpendicular rod-to-grain arrangement.

**Figure A3.** Experimental force–displacement results for specimens with (**a**) epoxy or (**b**) polyurethane glue, under 27% moisture and bonding and parallel rod-to-grain arrangement.

**Figure A4.** Experimental force–displacement results for specimens with (**a**) epoxy or (**b**) polyurethane glue, under 27% moisture and bonding and perpendicular rod-to-grain arrangement.

#### **References**

