*3.2. Reinforced Models*

In the models of series **B10** and **BP10**, reinforced with steel connectors, no cracks on walls typical for unreinforced models were observed for the whole range of loading. Displacements of interconnected wall panels were unnoticeable in the initial phase of loading. At a given moment, a rapid increase in displacements was clearly visible to the naked eye. However, it was still possible to continue the loading of the models until the moment of failure. failure was rapid and caused shearing of the joint and a distinct vertical displacement (by ca. 17 mm) of the wall web—see Figure 7b. The wall settled on the wooden protection. Models at the point of failure are shown in Figure 7a. The failure of the models of series **B10** and **BP10** was caused by the yielding and bending of steel flat profiles in the vicinity of the contact surface (Figure 7c,d). Spalling of masonry units beneath each connector was observed at the wall edge (see arrows in Figure 7c,d). The measured length of spalling areas was ca. 15 mm. However, no shear fracture of the connector was observed in the mortar laid in bed joints due to the holes in the flat profile. Mortar penetrating through the holes was not subjected to shearing. It acted as a dowel and prevented displacement. For **B10** models, an increase in displacements was observed at lower values of the loading force when compared to **BP10** models.

**Figure 7.** Failure of reinforced models: (**a**) damaged model (B10\_1), (**b**) damaged model with dimensioned displacement between bed joints (B10\_2), (**c**) typical bending of punched flat profile near the contact surface (B10\_1), (**d**) typical bending of punched flat profile near the contact surface (BP10\_3).

Bent connectors were removed from damaged models, inspected and their permanent deformation was evaluated—see Figure 8. The shape of connectors was originally flat in areas of anchoring in joints. However, permanent deformation occurred in the central area where connectors crossed the wall joints. A permanent displacement uu perpendicular to thflate connector axis was observed in the section marked eu. Additionally, the representative total extension of each connector δ<sup>u</sup> was calculated. A permanent displacement eu in models B10 ranged from 20 mm to 27 mm, and the mean was 23 mm (23t) at the mean displacement uu between 8 mm and 17 mm and a mean of 11 mm (11t). A permanent displacement eu in models BP10 ranged from 20 mm to 29 mm, and the mean was 23 mm (23t). The vertical displacement uu was between 8 mm and 17 mm, and the mean was 12 mm (12t). Deformation seemed to be identical despite the shape of the connectors. The only reported difference was the position of the deformed area regarding the mid-length of the connector. Displacements observed for some connectors were of the order of ±20 mm with respect to the mid-length of the connector. As no regularity caused by, e.g., their position in joints was found, the above was assumed to be the effect of precisely made joints. The measured geometry of the connectors in each model and the mean values are presented in Table 8.

**Figure 8.** Deformed connectors removed from damaged test models after tests: (**a**) punched flat profiles in the wall B10\_2, (**b**) punched widened flat profiles in the wall BP10\_2.


**Table 8.** Measured geometry of deformed connectors.


**Table 8.** *Cont.*

As the in case of unreinforced joints, phases of reinforced joints can be presented in diagrams illustrating the relationship between the load N and relative (mutual) displacement u of joined walls—see Figure 9.

**Figure 9.** Relationship between the total force and mean displacement of joints.

Until the crack on the contact area that appeared under the maximum load Ncr = Nu = 7.3–12.3 kN for models**B10** and 12.5–16.5 kN for models**BP10**, an increment in displacement was nearly proportional and that phase was defined as the elastic phase. A clear increase in displacements and a drop in force to Nd = 3.4–5.0 kN in models **B10** and 8.9–10.5 kN in models **BP10** was observed after cracking in the failure phase. When the force Nd was reached in the failure phase, the joint demonstrated the capacity to take load, and a small hardening was noticed. The failure of the models caused by excessive displacements was observed under the maximum load Ncr = Nu = 2.3–9.2 kN for models **B10** and 10.6–14.8 kN for models **BP10**. Thus, a drop in the residual force of the maximum force was ca. 35% for models **B10** and only 15% for models **BP10**. Connectors **B10** produced lower values of the force in individual phases. Loading at the time of cracking was lower by 76%, and the maximum loading was lower by as much as 82%. Moreover, the residual force was lower by 63% when compared to the force determined for unreinforced models. Displacements in the reinforced models at the greatest force were lower only by 18% compared to the unreinforced joint. Displacements in reinforced joints greater than 100% were found under the residual force at the end of the failure phase. When compared to the unreinforced models, the cracking force acting on the models with connectors **BP10** with a widened central part was lower by 62% than in the model with the traditional joint. The maximum cracking force acting on the reinforced models was lower by 71% than in the case of the unreinforced models. Furthermore, the residual force was greater by more than 63%. Displacements in the reinforced models at the greatest force were lower by 15% than in the unreinforced joint. Moreover, displacements slightly greater by 4% than in unreinforced models were observed under the residual force at the end of

the failure phase. A twofold widening of the connector in the models **BP10** resulted in ca. 60% increase in forces Nu and over 100% increase in forces Nd and Nr when compared to results obtained for models **B10**. Displacements in the models with wider connectors were as expected—almost identical in the elastic phase and lower by 30%–50% in the failure phase. The observed phases were the basis of a multi-linear diagram illustrating the *N*–*u* relationship for joints in AAC walls—see Figure 10. The elastic phase was defined within the loading range 0–Ncr = Nu, and the failure phase within the range Nu–Nd–Nr.

**Figure 10.** Approximation of work of reinforced joint between masonry walls (connector (*1*)).

Values of forces and corresponding displacements are presented in Tables 9 and 10. Joint stiffnesses were determined in each phase according to Equations (1)–(3) and they are presented in Table 11. A linear approximation of results is shown in Figure 10.


**Table 9.** Test results for reinforced joints (forces).

**Table 10.** Test results for reinforced joints (displacements).


**Table 11.** Test results for reinforced joints (stiffness).


Validation of the Model Representing Reinforced Joints in Walls

Like for unreinforced joints, the obtained results were generalised. The following assumptions were made:

	- i. the elastic phase observed in the load range0–Ncr = Nu;
	- ii. the failure phase observed in the load range Nu–Nd–Nr.

The behaviour of the joint was described with simplified solutions found in the literature [20,21]. According to the cited papers, connectors were working as bars fixed on both sides, and the value of the force causing the displacement *u* could be expressed as follows:

$$V = \frac{12E\_s I}{e^3}u\_\prime \tag{13}$$

the corresponding bending moment in the connector is equal to:

$$M = \frac{6E\_s I}{c^2} u\_\prime \tag{14}$$

where *EI* is the flexural stiffness of the connector, *u* is the relative displacement of the connector ends and *e* is the representative length of the connector (distance between points of contraflexure).

Stress values of extreme fibres in the connector fixed in bed joints increased proportionally to the displacement u. For some displacements uel, stress at extreme fibres reached the yield point, and the bending moment and the shearing force were expressed via the following equations:

$$\mathcal{M}\_{\rm cl} = f\_{\rm y} \mathcal{W}\_{\rm cl} = \frac{6E\_{\rm s}I}{\mathcal{e}\_{\rm cl}^2} \mu\_{\rm cl\prime} V\_{\rm cl} = \frac{2\mathcal{M}\_{\rm cl}}{\mathcal{e}\_{\rm cl}}.\tag{15}$$

where*Wel* is the elastic indicator of the transverse bending of the connector section, *fy* is the representative yield point of steel in the connector and *eel* is the connector length in the elastic phase.

An increase in the relative displacements of the ends of connectors was observed with the yielding of the total section of the connector, resulting in the highest bending moment and the greatest shearing force equal to:

$$\mathcal{M}\_{\rm pl} = f\_{\rm y} \mathcal{W}\_{\rm pl} = \frac{6E\_{\rm s}I}{c\_{\rm pl}^2} \mu\_{\rm pl}, \; V\_u = \frac{2M\_{\rm pl}}{c\_{\rm pl}},\tag{16}$$

where *Wpl* is the plastic index of the transverse bending of the connector section, *fy* is the representative yield point of steel in the connector and *eel* is the connector length in the plastic phase.

An increase in relative displacements could cause the spalling of the wall beneath the connector and an increase in the length of connectors. This, in turn, could produce a noticeable drop in the force in the joints. As in previous phases, bending moments in connectors and shearing forces were determined from the following relationship:

$$M\_d = f\_y \\ N\_{pl} = \frac{6E\_s I}{e\_d^2} \\ \mu\_{d\prime} \; V\_d = \frac{2M\_d}{e\_d} \; \tag{17}$$

Tests demonstrated that a further increase in relative displacements could cause an increase in the forces in joints. In that phase, displacements were so considerable that connectors could work in a flexible and also a tendon mode. Consequently, friction force was generated between the joined walls. The bending moment and shearing forces in the joint can be expressed as:

$$M\_{\rm u} = f\_{\rm y} \\ W\_{\rm pl} = \frac{6E\_{\rm s}I}{\mathcal{e}\_{\rm u}^2} u\_{\rm u} \; V\_{\rm u} = \frac{2M\_{\rm u}}{\mathcal{e}\_{\rm u}} \; \tag{18}$$

And the axial force in the joint induced by tendon work was:

$$T = E\_s A \frac{\delta\_u}{c\_{1l}} \, ^\prime \tag{19}$$

where δ*<sup>u</sup>* is the extension of the connector, determined from the following equation:

$$
\delta\_{\mathsf{u}} = \sqrt{\mathsf{e}\_{\mathsf{u}}^{2} + \mathsf{u}\_{\mathsf{u}}^{2}} - \mathsf{e}\_{\mathsf{u}}.\tag{20}
$$

The horizontal and vertical components of force, being the effects of the tendon work (at α ≈ 0), were equal to:

$$\begin{aligned} T\_{\mu} &= T \cos \alpha \approx T\_{\prime} \\ T\_{v} &= T \sin \alpha \approx 0. \end{aligned} \tag{21}$$

Taking into account the tendon work of connectors, the load capacity of reinforced joints in walls can be expressed as:

$$V\_{\rm u} = \frac{2f\_{\rm y}\mathcal{W}\_{\rm pl}}{e\_{\rm u}}\mu\_{\rm c} + \alpha n\_{\rm c}E\_{\rm s}A\frac{\delta\_{\rm u}}{e\_{\rm u}}\mu\_{\rm \prime} \tag{22}$$

where μ is the friction coefficient α is the empirical coefficient, *eu* is the average length of the connector (distance between points of contraflexure acc. to Table 8) and *nc* = 5 is the number of connectors.

The corresponding displacement is expressed by the following relationship:

$$
\mu\_u = \frac{f\_y \mathcal{W}\_{pl} e\_u^2}{6E\_s I} \beta\_\prime \tag{23}
$$

where μ is the friction coefficient and β is the empirical coefficient.

Forces in the failure phase can be determined similarly.

$$V\_d = \frac{2f\_y \mathcal{W}\_{pl}}{\mathcal{e}\_{\mathcal{U}}} \mathfrak{n}\_c + \alpha\_1 \mathfrak{n}\_c E\_s A \frac{\delta\_{\mathcal{U}}}{\mathcal{e}\_{\mathcal{U}}} \mathfrak{n}\_r \tag{24}$$

$$
\mu\_d = \frac{f\_y \mathcal{W}\_{\text{pl}} e\_u^2}{6E\_s I} \beta\_{1\prime} \tag{25}
$$

$$V\_I = \frac{2f\_y \mathcal{W}\_{pl}}{\mathcal{e}\_{\mathcal{U}}} n\_\mathcal{E} + \alpha\_2 n\_\mathcal{E} E\_\mathcal{S} A \frac{\delta\_\mathcal{U}}{\mathcal{e}\_{\mathcal{U}}} \mu\_\prime \tag{26}$$

$$
\mu\_r = \frac{f\_y \mathcal{W}\_{pl} \mathfrak{e}\_u^2}{6E\_s I} \beta\_2. \tag{27}
$$

Above equations included not only the mechanical parameters of the connectors (*E, fy*) but also the measured length of connectors *eu*—the distance between points of contraflexure. However, this approach is not unconditional. The length of connectors measured in the tests was ca. 23t. The authors in [20] determined experimentally that the length of connectors from flat profiles was (1.6–2.5)*t*, provided that masonry units below the connector were not crushed as observed in the models made of AAC. Like for unreinforced models, the values of empirical coefficients were calculated using results from material tests and tests on individual elements. Boundary values of mean coefficients α, α1, α2, β, β1, β<sup>1</sup> were determined at the significance level α = 0.8. As the sample size was small, the relationship expressed by the relationship in Equation (11) was used. Lower and upper values from the confidence interval of mean coefficients are presented in Table 12.


**Table 12.** Validation of empirical coefficients of the model with reinforced wall joints.

Following the procedure conducted for unreinforced joints, two values defining lower and upper limits of confidence intervals matched each of the six coefficients (Table 12). Thus, there were <sup>6</sup> 2 different combinations (without any repetitions) for coefficients. Similarly, as for unreinforced joints, the minimum value of the mean percentage error (MPE) [19] was applied as a selection criterion separately for forces and displacements. Optimal values of those coefficients were calculated from 15 combinations. For the values of coefficients in the shaded cells in Table 12, the minimum MPE for forces and displacements in connectors B10 was equal to 22%. For connectors BP10, the MPE for forces and displacements was 11%. Using results from the model and standard tests, empirical relationships describing the work of joints in particular phases are presented in Table 13, and calculated values and empirically obtained values are compared in Table 14 and Figure 10.


**Table 13.** Relationships expressing the work of reinforced joints in walls.

**Table 14.** Compared tests results and own calculations for the standard model.


For standard connectors **B10** without widening, calculated forces determining coordinates of particular phases were lower than those obtained during tests. The difference for the maximum force was equal to 31%, and for the aggregate interlocking force −23%. The value of the force Nr in the failure phase was lower by 33% than the empirical value. Similar results were obtained for connectors **BP10**. Determined force values were lower than experimental ones. The maximum force Nu was lower by 16%, and forces Vd and Vr in the failure phase were lower by 10% and 18%, respectively, when compared to forces determined experimentally. Calculated displacements of joints with connectors **B10** varied significantly. The calculated displacement at failure was lower by 48% than the experimentally determined values. Moreover, displacements in the failure phase corresponding to the force Vd were greater by over 55% than experimental values, and calculated displacements were greater only by 10%. For connectors **BP10**, displacements at the maximum force were underestimated at a level of

over 65%, and overestimated by only 3% under the force Vd. Differences in calculated and measured displacements at failure were equal to just 7%.

The obtained results, particularly for forces, can be used to estimate, with safe margins, the forces in joints and to verify SLS conditions where no guidelines can be applied. As for unreinforced joints, the greatest differences were observed for displacements. The recommended relationships can cause a significant underestimation of displacement at failure, even at the level of ca. 50%.

## **4. Conclusions**

Tests described in this paper are a part of a piece of complex research work conducted at the Silesian University of Technology. This paper presents results from testing three types of wall joints: a traditional mortar bonding (URM), joints with punched steel flat profiles (**B10**) and with connectors of genuine shape (**BP10**) protected by the patent.

The failure process and crack development on the walls bonded with mortar were mild and included three phases. Distinct wall cracks near the joint were observed prior to failure. Failure and cracking of models with steel elements, apart from lower load capacity, were completely different. No cracks preceding the wall destruction were observed, but there were rapid displacements and a drop in loading. For perforated flat profiles used as steel connectors, significantly lower values were obtained when compared to the models with mortar bonding. Forces at the time of cracking were lower by 62% (**BP10**) and 76% (**B10**), and the difference at the maximum force was 82% (**BP10**) and 71% (**B10**). Reinforced models were less deformed in the elastic phase. Differences at the maximum force were 18% (**B10**) and 15%(**BP10**). Greater differences were observed for displacements prior to the failure. Displacements in the models with reinforced joints **B10** were greater by over 100% than in unreinforced models. Generally, the same displacements were reported for the models with connectors **BP10**. A twofold widening of the connector in models **BP10** resulted in a ca. 60% increase in maximum forces when compared to results obtained for models **B10**. Displacements in the models with wider connectors were as expected and almost identical in the elastic phase and lower by 30%–50% in the failure phase.

Particular phases of joint work were determined and defined, and an empirical approach was proposed to determine the forces and displacement of wall joints using the results from less complicated standard tests. Values of cracking and failure forces were estimated with a safety margin for unreinforced joints. Moreover, they differed by 15% and 9% in comparison to the test results. On the basis of relationships described in the literature [20,21], a technical solution was proposed, which included the determination of forces producing cracks on the contact area and maximum forces in joints between walls reinforced with punched flat profiles. Due to the small number of elements per series, differences in the safe estimation of forces were of the order of 31% for maximum forces in connector **B10**, and 26% in connector **BP10**.

Work should be continued and additional test models should be constructed to define the statistically empirical parameters of models. Then, the results of validation can be expected to provide lower differences in extreme values. Moreover, FEM (Finite Element Method)-based analyses seem to be necessary to determine the real work of joints, particularly to determine their real length (e). The target model should also give consideration to the phase of joint weakening and to the estimation of forces Ncr, Nd and Nu and corresponding displacements with satisfactory accuracy.

**Author Contributions:** Conceptualization, R.J. and I.G.; methodology, R.J. and I.G.; validation, R.J.; formal analysis, I.G.; investigation, I.G.; writing—original draft preparation, R.J.; writing—review and editing, I.G.; visualization, R.J. and I.G.; supervision, R.J. and I.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** The research was financed from the own funds of the Department of Building Structures and Department of Structural Engineering Silesian University of Technology and project: NB-323/RB-2/2017 Experimental tests of joints in masonry walls made of autoclaved aerated concrete, financed by Solbet Company.

**Acknowledgments:** The authors would like to express particular thanks to Solbet and NOVA companies for valuable suggestions and the delivery of masonry units, mortar, and connectors which were used to prepare test models and perform tests.

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