*3.1. Unreinforced Models*

The behaviour of all unreinforced models was similar. No cracking noise and no visible splitting on the lateral surfaces of elements were noticed in the initial phase of loading. Non-dilatation strain in some parts of the wall was observed. That phase lasted until the appearance of the first diagonal cracks in the adjacent vicinity of wall joints—see Figure 4a,b. Load increments caused the distinct development of cracks present at the location of joints and propagation towards the reinforced concrete column which transferred loading (Figure 4c). The greatest force was registered in that phase. Continued loading led to the distinct growth of mutual displacements and the rotation of joined walls. The joint was removed after failure—see Figure 4d. Almost vertical shearing of elements forming the bond was found. No clear damage was reported in the case of other elements.

**Figure 4.** Destruction of models of series P (**a**) a first crack on the reference model P\_2, (**b**) a first crack on the reference model P\_6, (**c**) joint after failure P\_5, (**d**) joint after failure P\_3.

The cracking mechanism for elements is also visible on diagrams illustrating the relationship between the load *N* and relative (mutual) displacement *u* of bonded walls—see Figure 5. Until cracking of the contact surface observed under the load *N*cr = 27.3–54.1 kN, increments in relative displacements *u* were almost directly proportional, and thus the working phase of the joint was called the elastic phase. After cracking in the post-elastic phase, stiffness was reduced. However, joints still had the capacity to take the load.

**Figure 5.** Relationship between the total force and mean displacement for test results and calculations.

This phase was completed at maximum force values within the range Nu = 38.6–59.8 kN. Continued attempts at loading in the failure phase resulted in a clear drop in the values of forces registered by a dynamometer, and an increment in relative displacements. Force was close to zero, and the joint had the capacity to take some load. In this phase, forces were called aggregate interlocking forces with values of Nag = 14.1–31.1 kN. Further increment in joint displacements caused a minor load increase and hardening. The last registered forces, called residual forces, preceded the failure that resulted in the total splitting of bonded elements and their mutual rotation. Their value ranges were Nr = 8.4–42.9 kN. Forces and corresponding displacements are presented in Tables 2 and 3, and the linear approximation of results is shown in Figure 6. Joint stiffness was determined in each phase according to Equations (1)–(3) and they are presented in Table 4:


**Table 2.** Test results for joints between unreinforced walls.


**Table 3.** Test results for joints between unreinforced walls (displacements).

**Figure 6.** Approximation of work of unreinforced joints between masonry walls.


**Table 4.** Test results for joints between unreinforced walls (joint stiffness).

Joint stiffness in the elastic phase,

$$K\_{\rm f} = \frac{N\_{cr}}{u\_{cr}} \, ^{\prime}$$

joint stiffness in the post-elastic phase,

$$K\_p = \frac{N\_u - N\_{cr}}{u\_u - u\_{cr}},\tag{2}$$

joint stiffness in the failure phase,

$$K\_r = \frac{|N\_r - N\_u|}{u\_r - u\_u}.\tag{3}$$

Validation of the Model with Unreinforced Wall Joints

Performed tests were used to generalize the obtained results by proposing the so-called standard model [17]. The following assumptions were made:

	- i. The elastic phase observed in the load range 0–Ncr;
	- ii. The post-elastic phase observed in the load range Ncr–Nu;
	- iii. The failure phase observed in the load range Nu–Nag–Nr.

The following empirical relationships were recommended to determine forces and displacements in particular phases:

Forces and displacements in the elastic phase:

$$N\_{cr} = \alpha\_1 \tau\_{cr,RL} A\_\prime \tag{4}$$

$$
\mu\_{\mathcal{U}} = \mathcal{N}\_{\mathcal{U}} / \mathcal{K}\_{\mathcal{U}} = \mathcal{N}\_{\mathcal{U}} / \alpha \mathcal{K}\_{\mathcal{R}\mathcal{L}}.\tag{5}
$$

Forces and displacements in the post-elastic phase:

$$N\_u = \beta\_1 \tau\_{u,RL} A\_\prime \tag{6}$$

$$
\mu\_u = \mathfrak{u}\_{\mathcal{T}} + (\mathcal{N}\_u - \mathcal{N}\_{\mathcal{C}\tau}) / \mathcal{K}\_p = \mathfrak{u}\_{\mathcal{C}\tau} + (\mathcal{N}\_u - \mathcal{N}\_{\mathcal{C}\tau}) / \mathcal{g}\mathcal{K}\_{t\prime} \tag{7}
$$

where *A* = 0.26 m<sup>2</sup> is the joint area and α, α1, β and β<sup>1</sup> are empirical coefficients.

Shear parameters determined during tests on diagonal compression performed in compliance with ASTM E519-81 were τ*cr,RL* = 0.192 MPa, τ*u,RL* = 0.196 N/mm<sup>2</sup> and stiffness *KRL* = 117.1 MN/m were used as reference values in above equations. At the beginning of the failure phase, residual and aggregate interlocking forces were determined from the following equations:

$$N\_{\rm I} = \gamma \tau\_{\rm u, RL} A\_{\prime} \tag{8}$$

$$N\_{u\chi} = \gamma\_1 \tau\_{u,RL} A\_\prime \tag{9}$$

where γ and γ<sup>1</sup> are empirical coefficients.

Displacements corresponding to the aggregate interlocking force were determined from the following empirical relationship:

$$
u\_{\mathfrak{F}} = \omega \tau\_{\mathfrak{u}, \text{RL}} / \mathcal{K}\_{\text{RL}\text{-}\prime} \tag{10}$$

where ω is the empirical coefficient.

Values of empirical coefficients were calculated using the results from material tests and tests on individual elements. Furthermore, boundary values of mean coefficients α, α1, β, β1, γ, γ<sup>1</sup> and ω were determined at the significance level α = 0.8 to create the reference model [18]. As the sample size was small *n* < 30, the following relationship was used:

$$P\left(\overline{\mathbf{x}} - t\_{1-\alpha/2} \frac{S}{\sqrt{n}} < m < \overline{\mathbf{x}} + t\_{1-\alpha/2} \frac{S}{\sqrt{n}}\right) = 1 - \alpha,\tag{11}$$

where: *<sup>x</sup>* <sup>=</sup> (*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*)/*<sup>n</sup>* is the mean value of the random sample, *<sup>S</sup>* <sup>=</sup> (*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*) 2 /(*n* − 1) is the standard deviation of the sample, *t*1−α/2 are statistics with Student's t-distribution and n-1 degrees of freedom.

Lower and upper values from the confidence interval of mean coefficients are presented in Table 5.


**Table 5.** Validation of empirical coefficient of the model with unreinforced wall joints.

In the failure phase, during which dry shear fracture of separating walls was observed, the joint behaviour was mapped on the basis of standard behaviours specified in PN-EN 1052-3:2004. Those tests included measurements of relative displacements of two masonry units joined with mortar and determination of fracture energy of the joint *GII <sup>f</sup>* <sup>=</sup> 2.37 <sup>×</sup> <sup>10</sup>−<sup>4</sup> MN/m [13], which could be used to describe the behaviour of the brittle material in the failure phase in accordance with the continuum fracture mechanism. The failure phase was described on the basis of observations using two sections with forces varying from *Nu* to *Nag*, and then from *Nag* to *Nr* at corresponding displacements *uu*, *uag* and *ur*. Assuming that fracture energy per joint area *GIIj <sup>f</sup>* (expressed as the area below the diagram shown in Figure 6) was equal to fracture energy *GII <sup>f</sup>* , obtained from standard tests, the displacement corresponding to the residual force *ur* was determined from the relationship:

$$AG\_{f}^{\text{II}} = AG\_{f}^{\text{II}\text{\%}} = \frac{1}{2}(N\_{\text{ll}} - N\_{\text{ad}})(\boldsymbol{u}\_{\text{ad}} - \boldsymbol{u}\_{\text{u}}) + (N\_{\text{ad}} - N\_{\text{r}})(\boldsymbol{u}\_{\text{ad}} - \boldsymbol{u}\_{\text{u}})$$

$$+ \frac{1}{2}(N\_{\text{ad}} - N\_{\text{r}})(\boldsymbol{u}\_{\text{r}} - \boldsymbol{u}\_{\text{ad}}) \tag{12}$$

$$\Rightarrow \boldsymbol{u}\_{\text{r}} = \frac{2C\_{f}^{\text{II}}A - N\_{\text{u}}(\boldsymbol{u}\_{\text{d}\text{g}} - \boldsymbol{u}\_{\text{u}}) + N\_{\text{d}\text{g}}\boldsymbol{u}\_{\text{u}} + N\_{\text{r}}(\boldsymbol{u}\_{\text{d}\text{g}} - 2\boldsymbol{u}\_{\text{u}})}{\left(N\_{\text{d}} - N\_{\text{r}}\right)}.$$

Following that procedure, two values defining the lower and upper limits of yjr confidence intervals matched each of the seven coefficients (Table 5). Maximum and minimum values of displacement expressed by the relationship in (12) depended on the previously used values and could be considered as independent variables. Thus, there were <sup>7</sup> 2 different combinations (without any repetitions) for coefficients. The minimum value of the mean square error calculated separately for forces and displacements was applied as a selection criterion. Optimal values of those coefficients were calculated from 21 combinations. A Mean Percentage of Error (MPE) was calculated [19] *MPE* = <sup>1</sup> *N* <sup>5</sup> *N*=1 *xobs*−*xcal xobs* The minimum MPEs for calculated forces and displacements with respect to the coefficients listed in shaded cells in Table 5 were 16% (for force) and −6% (for displacement).

As a result, empirical relationships based on the results from model and standard tests, describing the work of joints in particular phases, are presented in Table 6, and calculated values and empirically obtained values are compared in Table 7 and Figure 5.


**Table 6.** Relationships describing the work of unreinforced joints between walls.

**Table 7.** Compared test results and our own calculations for the standard model.


Following the assumptions, calculated forces determining coordinates for particular phases of joint work were smaller than those obtained from tests, which was consistent with the assumptions. Considering the force causing cracks, the difference was 15%, and for the failure force, it was –9%. The biggest differences were found in the failure phase. Then, the calculated values Nag and Nr were lower by 36% and 44% than mean empirical values. For relative displacement in the elastic phase, the calculated displacement differed from the average empirical value by only 3%, and by 7% in case of the greatest force. In the failure phase, displacements corresponding to forces Nag and Nr differed by 12% and 17%, respectively. The delivered results were sufficient to predict forces with satisfactory accuracy and thus to verify properly the SLS conditions for joints. Greater differences were found for displacements, which are crucial for verifying SLS conditions. The biggest discrepancy was obtained for the maximum load.
