# Testing Joints between Walls Made of AAC Masonry Units

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- Determination of the cracking and failure mechanisms of joints between AAC walls;
- Comparison of load capacity of wall joints using traditional masonry bonds and steel connectors;
- Optimization of the shape of a steel connector.

## 2. Programme of Our Own Tests

**P**had traditional masonry joints between the web and the flange (Figure 1a). Those elements were regarded as reference models, whose mechanical parameters and behaviour at loading and failure were compared with results from other tests. In two other series, joints between webs and flanges were made with steel connectors (wall geometry acc. to Figure 1b). They were single punched flat profiles in series

**B10**(Figure 1c), and modified flat profiles with a widened central part in series

**BP10**(Figure 1d). This solution was proposed on the basis of our own tests [7] on perforated connectors. The widening of the central part was intended to increase the flexural capacity of the connector and its stiffness. The proposed shape is copyrighted via an application to the Polish Patent Office [8]. Joints made of galvanized perforated steel with a thickness of 1 mm were used in both series.

_{c}= 2.97 N/mm

^{2}, and the modulus of elasticity was E

_{m}= 2040 N/mm

^{2}. The initial shear value determined according to the code PN-EN 1052-3:2004 [11] and presented in [12] was f

_{vo}= 0.31 N/mm

^{2}.

^{2}. Mortar for thin joints was used in the tested elements for the AAC blocks. This mortar is dedicated to the erection of AAC masonry walls Additional tests on steel connectors—see Figure 3—were conducted according to the standard [16]. Three elements were chosen randomly from each series of connectors and placed in the jaws of a testing machine. The basic mechanical parameters of the connectors were determined by controlling the displacement gain. The measurement of strains was non-contact with a video extensometer MEVIX 200. Strain was measured using a base with the length L

_{e}= 53.5 mm in standard connectors and L

_{e}= 75.0 mm in thickened connectors. Figure 3 illustrates (stress σ –strain ε) relationships. The stress–strain relationship of tested connectors was found to have no clear yield point. Therefore, results were approximated with a bi-linear relationship. A theoretical yield point f

_{y}was determined at the intersection of straight lines. The slope of the tangent straight line presented within the range of 0–f

_{y}was assumed as the mean initial modulus of elasticity E

_{s}. Tensile strength was determined at failure of specimens, and the tangent of the straight-line slope within the range of f

_{y}–f

_{t}was determined as the mean secant modulus of elasticity E

_{t}. Test results for connectors and the research programme are compared in Table 1.

## 3. Test results and Analysis

#### 3.1. Unreinforced Models

_{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.

_{u}= 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 N

_{ag}= 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 N

_{r}= 8.4–42.9 kN. Forces and corresponding displacements are presented in Table 2 and Table 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:

#### Validation of the Model with Unreinforced Wall Joints

- a)
- A non-linear relationship N–u determined from tests could be replaced with a multi-linear relationship expressing all observed phases:
- i.
- The elastic phase observed in the load range 0–N
_{cr}; - ii.
- The post-elastic phase observed in the load range N
_{cr}–N_{u}; - iii.
- The failure phase observed in the load range N
_{u}–N_{ag}–N_{r}.

- b)
- It was suggested that all material parameters used in the model should be specified using standard and normalised methods;
- c)
- The model would be subjected to statistical validation on the basis of performed tests.

^{2}is the joint area and α, α

_{1}, β and β

_{1}are empirical coefficients.

_{cr,RL}= 0.192 MPa, τ

_{u,RL}= 0.196 N/mm

^{2}and stiffness K

_{RL}= 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:

_{1}are empirical coefficients.

_{1}, β, β

_{1}, γ, γ

_{1}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:

^{−4}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 N

_{u}to N

_{ag}, and then from N

_{ag}to N

_{r}at corresponding displacements u

_{u}, u

_{ag}and u

_{r}. Assuming that fracture energy per joint area ${G}_{f}^{IIj}$ (expressed as the area below the diagram shown in Figure 6) was equal to fracture energy ${G}_{f}^{II}$, obtained from standard tests, the displacement corresponding to the residual force u

_{r}was determined from the relationship:

_{ag}and N

_{r}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 N

_{ag}and N

_{r}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.

#### 3.2. Reinforced Models

**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.

_{u}perpendicular to thflate connector axis was observed in the section marked e

_{u}. Additionally, the representative total extension of each connector δ

_{u}was calculated. A permanent displacement e

_{u}in models B10 ranged from 20 mm to 27 mm, and the mean was 23 mm (23t) at the mean displacement u

_{u}between 8 mm and 17 mm and a mean of 11 mm (11t). A permanent displacement e

_{u}in models BP10 ranged from 20 mm to 29 mm, and the mean was 23 mm (23t). The vertical displacement u

_{u}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.

_{cr}= N

_{u}= 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 N

_{d}= 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 N

_{d}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 N

_{cr}= N

_{u}= 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 N

_{u}and over 100% increase in forces N

_{d}and N

_{r}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–N

_{cr}= N

_{u}, and the failure phase within the range N

_{u}–N

_{d}–N

_{r}.

#### Validation of the Model Representing Reinforced Joints in Walls

- d)
- A non-linear relationship N–u determined from tests was replaced with a multi-linear relationship expressing all observed phases:
- i.
- the elastic phase observed in the load range 0 – N
_{cr}= N_{u}; - ii.
- the failure phase observed in the load range N
_{u}–N_{d}–N_{r}.

- e)
- It was suggested that all material parameters used in the model should be specified using standard and normalised methods;
- f)
- An elastic and perfectly plastic model of the connector was used;
- g)
- The model would be subjected to statistical validation on the basis of performed tests.

_{el}, stress at extreme fibres reached the yield point, and the bending moment and the shearing force were expressed via the following equations:

_{el}is the elastic indicator of the transverse bending of the connector section, f

_{y}is the representative yield point of steel in the connector and e

_{el}is the connector length in the elastic phase.

_{pl}is the plastic index of the transverse bending of the connector section, f

_{y}is the representative yield point of steel in the connector and e

_{el}is the connector length in the plastic phase.

_{u}is the extension of the connector, determined from the following equation:

_{u}is the average length of the connector (distance between points of contraflexure acc. to Table 8) and n

_{c}= 5 is the number of connectors.

_{y}) but also the measured length of connectors e

_{u}—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, α

_{1}, α

_{2}, β, β

_{1}, β

_{1}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.

**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 V

_{d}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 V

_{d}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 V

_{d}. Differences in calculated and measured displacements at failure were equal to just 7%.

## 4. Conclusions

**B10**) and with connectors of genuine shape (

**BP10**) protected by the patent.

**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.

**B10**, and 26% in connector

**BP10**.

_{cr}, N

_{d}and N

_{u}and corresponding displacements with satisfactory accuracy.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Geometry and details of test models and: (

**a**) traditional masonry bond, P series, (

**b**) walls with steel joints (B10 and BP10 series), (

**c**) joining method with a punched flat bar, (

**d**) joining method with a punched widened flat profile (mm).

**Figure 2.**A scheme and photo of the test stand (longitudinal wall (1a), transverse wall (1b), reinforced concrete column transferring shear load (2), reinforced concrete pillars limiting horizontal deformation (3), horizontal support (4), system of the hydraulic cylinder and the force gauge used to induce shear stress (5), force gauge, vertical reaction (6), horizontal tie (7), steel frame (8)).

**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.

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

**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).

**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.

Name of Series | Type of Joint | A mm ^{2} | I mm ^{4} | f_{y/}f_{t}N/mm ^{2} | E_{s}/E_{t}N/mm ^{2} | Number of Models |
---|---|---|---|---|---|---|

P | Traditional masonry bond | -- | -- | -- | -- | 6 |

B10 | Punched steel flat profile B × t = 22 × 1 mm | 22 | 1.83 | 236/408 | 93467/743 | 3 |

BP10 | Punched widened flat profile B × t = 44 × 1 mm | 44 | 3.67 | 207/345 | 84686/1503 | 3 |

_{y}), tensile strength (f

_{t}), mean initial modulus of elasticity (E

_{s}), mean secant modulus of elasticity (E

_{t}).

Model | Force at the Time of Cracking | Maximum Force | Aggregate Interlocking Force | Residual Force | ||||
---|---|---|---|---|---|---|---|---|

N_{cr,i} | N_{cr,mv} | N_{u,i} | N_{u,mv} | N_{ag} | N_{ag,mv,i} | N_{r,i} | N_{r,mv} | |

kN | kN | kN | kN | kN | kN | kN | kN | |

P_1 | 27.3 | 39.2 | 56.3 | 50.7 | 31.1 | 24.9 | 20.7 | 16.2 |

P_2 | 42.6 | 50.0 | 14.7 | 10.2 | ||||

P_3 | 31.2 | 38.6 | 25.5 | 13.8 | ||||

P_4 | 54.1 | 59.8 | -- | 8.36 | ||||

P_5 | 35.1 | 48.1 | -- | -- | ||||

P_6 | 45.1 | 51.6 | 28.264 | 27.9 |

Model | Displacement at the Time of Cracking | Displacement Right before Failure | Displacement at Aggregate Interlocking Force | Residual Displacement | ||||
---|---|---|---|---|---|---|---|---|

u_{cr,i} | u_{cr,mv} | u_{u,i} | u_{u,mv} | u_{ag,i} | u_{ag,mv} | u_{r,i} | u_{r,mv} | |

mm | mm | mm | mm | mm | mm | mm | mm | |

P_1 | 0.07 | 0.09 | 0.31 | 0.23 | 2.43 | 2.08 | 6.36 | 5.58 |

P_2 | 0.12 | 0.25 | 1.95 | 6.97 | ||||

P_3 | 0.12 | 0.16 | 2.22 | 5.64 | ||||

P_4 | 0.07 | 0.17 | -- | 6.72 | ||||

P_5 | 0.06 | 0.10 | -- | -- | ||||

P_6 | 0.08 | 0.36 | 1.71 | 2.22 |

Model | Elastic Joint Stiffness | Post-Elastic Joint Stiffness | Residual Joint Stiffness | |||
---|---|---|---|---|---|---|

K_{t,i} | K_{t,mv} | K_{p,i} | K_{p,mv} | K_{r,i} | K_{r,mv} | |

MN/m | MN/m | MN/m | MN/m | MN/m | MN/m | |

P_1 | 413 | 496 | 119 | 123 | 5.89 | 7.39 |

P_2 | 341 | 60 | 5.93 | |||

P_3 | 268 | 163 | 4.51 | |||

P_4 | 804 | 52.8 | 7.86 | |||

P_5 | 562 | 322 | -- | |||

P_6 | 590 | 23 | 12.75 |

Model | x_{i} | ||||||
---|---|---|---|---|---|---|---|

${\mathit{\alpha}}_{\mathit{i}}=\frac{{\mathit{K}}_{\mathit{t},\mathit{i}}}{{\mathit{K}}_{\mathit{R}\mathit{L}}}$ | ${\mathit{\beta}}_{\mathit{i}}=\frac{{\mathit{K}}_{\mathit{p},\mathit{i}}}{{\mathit{K}}_{\mathit{t},\mathit{i}}}$ | ${\mathit{\alpha}}_{1,\mathit{i}}=\frac{{\mathit{N}}_{\mathit{c}\mathit{r},\mathit{i}}}{{\mathit{\tau}}_{\mathit{c}\mathit{r},\mathit{R}\mathit{L}}\mathit{A}}$ | ${\mathit{\beta}}_{1,\mathit{i}}=\frac{{\mathit{N}}_{\mathit{u},\mathit{i}}}{{\mathit{\tau}}_{\mathit{u},\mathit{R}\mathit{L}}\mathit{A}}$ | ${\mathit{\gamma}}_{\mathit{i}}=\frac{{\mathit{N}}_{\mathit{r},\mathit{i}}}{{\mathit{\tau}}_{\mathit{u},\mathit{R}\mathit{L}}\mathit{A}}$ | ${\mathit{\gamma}}_{1,\mathit{i}}=\frac{{\mathit{N}}_{\mathit{a}\mathit{g},\mathit{i}}}{{\mathit{\tau}}_{\mathit{u},\mathit{R}\mathit{L}}\mathit{A}}$ | ${\mathit{\omega}}_{\mathit{i}}=\frac{{\mathit{u}}_{\mathit{a}\mathit{g},\mathit{i}}{\mathit{K}}_{\mathit{R}\mathit{L}}}{{\mathit{\tau}}_{\mathit{u}.\mathit{R}\mathit{L}}\mathit{A}}$ | |

P_1 | 3.51 | 0.29 | 0.55 | 1.10 | 0.4 | 0.6 | 5.61 |

P_2 | 2.90 | 0.17 | 0.85 | 0.98 | 0.2 | 0.3 | 4.49 |

P_3 | 2.27 | 0.61 | 0.62 | 0.76 | 0.3 | 0.5 | 5.13 |

P_4 | 6.83 | 0.07 | 1.08 | 1.17 | 0.2 | -- | -- |

P_5 | 4.78 | 0.57 | 0.70 | 0.94 | -- | -- | -- |

P_6 | 5.01 | 0.04 | 0.90 | 1.01 | 0.5 | 0.6 | 3.94 |

n | 6 | 6 | 6 | 6 | 5 | 4 | 4 |

x | 4.22 | 0.29 | 0.79 | 0.99 | 0.32 | 0.49 | 4.80 |

S | 1.7 | 0.2 | 0.2 | 0.1 | 0.2 | 0.1 | 0.7 |

${t}_{1-\alpha /2}$ | 1.48 | 1.48 | 1.48 | 1.48 | 1.53 | 1.64 | 1.64 |

$\overline{x}-{t}_{1-\alpha /2}\frac{S}{\sqrt{n}}$ | 3.22 | 0.14 | 0.67 | 0.91 | 0.21 | 0.37 | 4.20 |

$\overline{x}+{t}_{1-\alpha /2}\frac{S}{\sqrt{n}}$ | 5.22 | 0.44 | 0.91 | 1.08 | 0.43 | 0.60 | 5.39 |

Joint Phase | Force | Stiffness | Displacement |
---|---|---|---|

Elastic phase | ${N}_{cr}=0.67{\tau}_{u,RL}A$ | ${K}_{t}=3.22{K}_{RL}$ | ${u}_{cr}={N}_{cr}/3.22{K}_{RL}$ |

Post-elastic phase | ${N}_{u}=0.91{\tau}_{u,RL}A$ | ${K}_{p}=0.14{K}_{RL}$ | ${u}_{u}=\left({N}_{u}-{N}_{cr}\right)/0.14{K}_{RL}$ |

Failure phase | ${N}_{ag}=0.37{\tau}_{u,RL}A$ | $\left({N}_{u}-{N}_{ag}\right)/\left({u}_{u}-{u}_{ag}\right)$ | ${u}_{ag}=5.39{\tau}_{u,RL}A/{K}_{RL}$ |

${N}_{r}=0.21{\tau}_{u,RL}A$ | ${K}_{r}=\left({N}_{u}-{N}_{r}\right)/\left({u}_{r}-{u}_{u}\right)$ | ${u}_{r}=\frac{2{G}_{f}^{II}A-{N}_{u}\left({u}_{ag}-{u}_{u}\right)+{N}_{ag}{u}_{u}+{N}_{r}\left({u}_{ag}-2{u}_{u}\right)}{\left({N}_{ag}-{N}_{r}\right)}$ |

Test Results for | Calculated Results for | ||||||
---|---|---|---|---|---|---|---|

Forces | Forces | ||||||

N_{cr,mv}kN | N_{u,mv}kN | N_{ag,mv}kN | N_{r,mv}kN | N_{cr,cal}kN | N_{u,cal}kN | N_{ag,cal}kN | N_{r,cal}kN |

39.2 | 50.7 | 24.9 | 16.2 | 33.3 | 46.3 | 19.0 | 10.7 |

Displacements | Displacements | ||||||

u_{cr,mv}mm | u_{u,mv}mm | u_{ag,mv}mm | u_{r,mv}mm | u_{cr,cal}mm | u_{u,cal}mm | u_{u,cal}mm | u_{r,cal}mm |

0.09 | 0.243 | 2.08 | 5.58 | 0.09 | 0.24 | 2.34 | 6.53 |

Model | Layer of Connectors | ||||||
---|---|---|---|---|---|---|---|

Distance between Points of Contraflexure | Relative Displacement of Connector Ends | Connector Extension | |||||

e_{u,i} | e_{u,mv} | u_{u,i} | u_{u,mv} | ${\mathit{\delta}}_{\mathit{u},\mathit{i}}=\sqrt{{\mathit{e}}_{\mathit{u}}^{2}+{\mathit{u}}_{\mathit{u}}^{2}}-{\mathit{e}}_{\mathit{u}}$ | δ_{u,mv} | ||

mm | mm | mm | mm | mm | mm | ||

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

B10_1 | a | 23 | 22 | 9 | 9 | 1.70 | 1.8 |

b | 21 | 8 | 1.47 | ||||

c | 20 | 9 | 1.93 | ||||

d | 24 | 9 | 1.63 | ||||

d | 21 | 10 | 2.26 | ||||

B10_2 | a | 26 | 24 | 11 | 10 | 2.23 | 2.2 |

b | 26 | 10 | 1.86 | ||||

c | 25 | 11 | 2.31 | ||||

d | 22 | 10 | 2.17 | ||||

d | 20 | 10 | 2.36 | ||||

B10_3 | a | 21 | 23 | 11 | 14 | 2.71 | 3.7 |

b | 23 | 13 | 3.42 | ||||

c | 23 | 15 | 4.46 | ||||

d | 23 | 12 | 2.94 | ||||

d | 27 | 17 | 4.91 | ||||

23(23t) | -- | 11(11t) | -- | 2.57(2.57t) | |||

BP10_1 | a | 19 | 24 | 12 | 13 | 3.47 | 3.33 |

b | 27 | 14 | 3.41 | ||||

c | 24 | 13 | 3.29 | ||||

d | 29 | 11 | 2.02 | ||||

d | 23 | 15 | 4.46 | ||||

BP10_2 | a | 23 | 23 | 17 | 15 | 5.60 | 4.22 |

b | 22 | 14 | 4.08 | ||||

c | 27 | 14 | 3.41 | ||||

d | 22 | 13 | 3.55 | ||||

d | 23 | 15 | 4.46 | ||||

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

BP10_3 | a | 22 | 23 | 12 | 10 | 3.06 | 2.01 |

b | 23 | 10 | 2.08 | ||||

c | 19 | 9 | 2.02 | ||||

d | 26 | 8 | 1.20 | ||||

d | 23 | 9 | 1.70 | ||||

23(23t) | -- | 13(13t) | -- | 3.28(2.57t) |

Model | Cracking Force | Force at Failure | Dowel Force | Residual Force | ||||
---|---|---|---|---|---|---|---|---|

N_{cr,i} | N_{cr,mv} | N_{u,i} | N_{u,mv} | N_{d} | N_{d,mv,i} | N_{r,i} | N_{r,mv} | |

kN | kN | kN | kN | kN | kN | kN | kN | |

B10_1 | 12.3 | 9.3 | 12.3 | 9.3 | 5.01 | 4.5 | 6.68 | 6.1 |

B10_2 | 8.41 | 8.41 | 5.02 | 9.20 | ||||

B10_3 | 7.27 | 7.27 | 3.39 | 2.32 | ||||

BP10_1 | 15.9 | 14.9 | 15.9 | 14.9 | 8.86 | 9.6 | 14.8 | 12.7 |

BP10_2 | 16.5 | 16.5 | 10.5 | 12.6 | ||||

BP10_3 | 12.4 | 12.4 | 9.36 | 10.6 |

Model | Displacement at the Time of Cracking | Displacement Right before Failure | Displacement at Dowel Force | Residual Displacement | ||||
---|---|---|---|---|---|---|---|---|

u_{cr,i} | u_{cr,mv} | u_{u,i} | u_{u,mv} | u_{d,i} | u_{d,mv} | u_{r,i} | u_{r,mv} | |

mm | mm | mm | mm | mm | mm | mm | mm | |

B10_1 | 0.07 | 0.19 | 0.07 | 0.19 | 1.83 | 1.94 | 11.50 | 11.45 |

B10_2 | 0.08 | 0.08 | 0.68 | 10.33 | ||||

B10_3 | 0.41 | 0.41 | 3.32 | 12.52 | ||||

BP10_1 | 0.04 | 0.19 | 0.04 | 0.19 | 0.45 | 1.33 | 4.15 | 5.80 |

BP10_2 | 0.05 | 0.05 | 1.61 | 7.04 | ||||

BP10_3 | 0.49 | 0.49 | 1.94 | 6.22 |

Model | Elastic Joint Stiffness | Residual Joint Stiffness | ||
---|---|---|---|---|

K_{t,i} | K_{t,mv} | K_{r,i} | K_{r,mv} | |

MN/m | MN/m | MN/m | MN/m | |

B10_1 | 180 | 100 | 0.496 | 0.327 |

B10_2 | 102 | 0.077 | ||

B10_3 | 17.8 | 0.409 | ||

BP10_1 | 432 | 259 | 0.269 | 0.378 |

BP10_2 | 319 | 0.553 | ||

BP10_3 | 25.6 | 0.312 |

Model | x_{i} | |||||
---|---|---|---|---|---|---|

${\mathit{\alpha}}_{\mathit{i}}=\frac{{\mathit{V}}_{\mathit{u},\mathit{i}}-\frac{2{\mathit{f}}_{\mathit{y}}{\mathit{W}}_{\mathit{p}\mathit{l}}}{{\mathit{e}}_{\mathit{u}}}{\mathit{n}}_{\mathit{c}}}{{\mathit{n}}_{\mathit{c}}{\mathit{E}}_{\mathit{s}}\mathit{A}\frac{{\mathit{\delta}}_{\mathit{u}}}{{\mathit{e}}_{\mathit{u}}}\mathit{\mu}}$ | ${\mathit{\beta}}_{\mathit{i}}=\frac{6{\mathit{E}}_{\mathit{s}}\mathit{I}{\mathit{u}}_{\mathit{u},\mathit{i}}}{{\mathit{f}}_{\mathit{y}}{\mathit{W}}_{\mathit{p}\mathit{l}}{\mathit{e}}_{\mathit{u}}^{2}}$ | ${\mathit{\alpha}}_{1\mathit{i}}=\frac{{\mathit{V}}_{\mathit{d},\mathit{i}}-\frac{2{\mathit{f}}_{\mathit{y}}{\mathit{W}}_{\mathit{p}\mathit{l}}}{{\mathit{e}}_{\mathit{u}}}{\mathit{n}}_{\mathit{c}}}{{\mathit{n}}_{\mathit{c}}{\mathit{E}}_{\mathit{s}}\mathit{A}\frac{{\mathit{\delta}}_{\mathit{u}}}{{\mathit{e}}_{\mathit{u}}}\mathit{\mu}}$ | ${\mathit{\beta}}_{1\mathit{i}}=\frac{6{\mathit{E}}_{\mathit{s}}\mathit{I}{\mathit{u}}_{\mathit{d},\mathit{i}}}{{\mathit{f}}_{\mathit{y}}{\mathit{W}}_{\mathit{p}\mathit{l}}{\mathit{e}}_{\mathit{u}}^{2}}$ | ${\mathit{\alpha}}_{2\mathit{i}}=\frac{{\mathit{V}}_{\mathit{r},\mathit{i}}-\frac{2{\mathit{f}}_{\mathit{y}}{\mathit{W}}_{\mathit{p}\mathit{l}}}{{\mathit{e}}_{\mathit{u}}}{\mathit{n}}_{\mathit{c}}}{{\mathit{n}}_{\mathit{c}}{\mathit{E}}_{\mathit{s}}\mathit{A}\frac{{\mathit{\delta}}_{\mathit{u}}}{{\mathit{e}}_{\mathit{u}}}\mathit{\mu}}$ | ${\mathit{\beta}}_{2\mathit{i}}=\frac{6{\mathit{E}}_{\mathit{s}}\mathit{I}{\mathit{u}}_{\mathit{r},\mathit{i}}}{{\mathit{f}}_{\mathit{y}}{\mathit{W}}_{\mathit{p}\mathit{l}}{\mathit{e}}_{\mathit{u}}^{2}}$ | |

B10_1 | 0.01117 | 0.10 | 0.00421 | 2.73 | 0.00580 | 17.2 |

B10_2 | 0.00744 | 0.12 | 0.00422 | 1.02 | 0.00819 | 15.4 |

B10_3 | 0.00636 | -- | 0.00268 | -- | -- | 18.7 |

n | 3 | 2 | 3 | 2 | 2 | 3 |

x | 0.00832 | 0.11 | 0.003702 | 1.88 | 0.00699 | 17.11 |

S | 0.00253 | 0.0146 | 0.0008880 | 1.21 | 0.00169 | 1.63 |

${t}_{1-\alpha /2}$ | 1.89 | 3.08 | 1.89 | 3.08 | 3.08 | 1.89 |

$\overline{x}-{t}_{1-\alpha /2}\frac{S}{\sqrt{n}}$ | 0.00557 | 0.081 | 0.00274 | −0.75 | 0.0033 | 15.33 |

$\overline{x}+{t}_{1-\alpha /2}\frac{S}{\sqrt{n}}$ | 0.01107 | 0.145 | 0.00467 | 4.50 | 0.0107 | 18.89 |

BP10_1 | 0.00619 | 0.06 | 0.00328 | 0.00573 | ||

BP10_2 | 0.00645 | 0.08 | 0.00397 | 2.43 | 0.00484 | 10.6 |

BP10_3 | 0.00476 | 0.00348 | 2.91 | 0.00401 | 9.4 | |

n | 3 | 2 | 3 | 2 | 3 | 2 |

$\overline{x}$ | 0.00580 | 0.07 | 0.00358 | 2.67 | 0.00486 | 9.98 |

S | 0.000911 | 0.0 | 0.000356 | 0.3 | 0.000859 | 0.877 |

${t}_{1-\alpha /2}$ | 1.89 | 3.08 | 1.89 | 3.08 | 1.89 | 3.08 |

$\overline{x}-{t}_{1-\alpha /2}\frac{S}{\sqrt{n}}$ | 0.0048 | 0.0319 | 0.00319 | 1.93 | 0.0039 | 8.1 |

$\overline{x}+{t}_{1-\alpha /2}\frac{S}{\sqrt{n}}$ | 0.0068 | 0.1014 | 0.0040 | 3.41 | 0.0058 | 11.9 |

Joint Phase | Force | Stiffness | Displacement |
---|---|---|---|

Connector B10 | |||

Elastic phase | ${V}_{u}=\frac{2{f}_{y}{W}_{pl}}{{e}_{u}}{n}_{c}+{0.0056}_{c}{E}_{s}A\frac{{\delta}_{u}}{{e}_{u}}\mu $ | ${K}_{t}={V}_{u}/{u}_{u}$ | ${u}_{u}=0.145\frac{{f}_{y}{W}_{pl}{e}_{u}^{2}}{6{E}_{s}I}$ |

Failure phase | ${V}_{d}=\frac{2{f}_{y}{W}_{pl}}{{e}_{u}}{n}_{c}+0.0027{n}_{c}{E}_{s}A\frac{{\delta}_{u}}{{e}_{u}}\mu $ | ${K}_{r}=\left({V}_{u}-{V}_{r}\right)/\left({u}_{r}-{u}_{u}\right)$ | ${u}_{d}=4.50\frac{{f}_{y}{W}_{pl}{e}_{u}^{2}}{6{E}_{s}I}$ |

${V}_{r}=\frac{2{f}_{y}{W}_{pl}}{{e}_{u}}{n}_{c}+0.0023{n}_{c}{E}_{s}A\frac{{\delta}_{u}}{{e}_{u}}\mu $ | ${u}_{r}=18.9\frac{{f}_{y}{W}_{pl}{e}_{u}^{2}}{6{E}_{s}I}$ | ||

Connector BP10 | |||

Elastic phase | ${V}_{u}=\frac{2{f}_{y}{W}_{pl}}{{e}_{u}}{n}_{c}+0.0048{n}_{c}{E}_{s}A\frac{{\delta}_{u}}{{e}_{u}}\mu $ | ${K}_{t}={V}_{u}/{u}_{u}$ | ${u}_{u}=0.10\frac{{f}_{y}{W}_{pl}{e}_{u}^{2}}{6{E}_{s}I}$ |

Failure phase | ${V}_{d}=\frac{2{f}_{y}{W}_{pl}}{{e}_{u}}{n}_{c}+0.0032{n}_{c}{E}_{s}A\frac{{\delta}_{u}}{{e}_{u}}\mu $ | ${K}_{r}=\left({V}_{u}-{V}_{r}\right)/\left({u}_{r}-{u}_{u}\right)$ | ${u}_{d}=1.93\frac{{f}_{y}{W}_{pl}{e}_{u}^{2}}{6{E}_{s}I}$ |

${V}_{r}=\frac{2{f}_{y}{W}_{pl}}{{e}_{u}}{n}_{c}+0.0039{n}_{c}{E}_{s}A\frac{{\delta}_{u}}{{e}_{u}}\mu $ | ${u}_{r}=8.1\frac{{f}_{y}{W}_{pl}{e}_{u}^{2}}{6{E}_{s}I}$ |

Test results for connector B10 | Calculations for connector B10 | ||||

forces | forces | ||||

N_{cr,mv} = N_{u,mv}kN | N_{d,mv}kN | N_{r,mv}kN | N_{cr,cal} = N_{u,cal}kN | N_{d,cal}kN | N_{r,cal}kN |

9.34 | 4.47 | 6.07 | 6.44 | 3.45 | 4.06 |

Displacements of connector B10 | Calculated displacements of connector B10 | ||||

u_{cr,mv} = u_{u,mv}mm | u_{ag,mv}mm | u_{r,mv}mm | u_{cr,cal} = u_{u,cal}mm | u_{u,cal}mm | u_{r,cal}mm |

0.19 | 1.94 | 11.45 | 0.10 | 3.01 | 12.6 |

Test results for connector BP10 | Calculations for connector BP10 | ||||

force | force | ||||

N_{cr,mv} = N_{u,mv}kN | N_{d,mv}kN | N_{r,mv}kN | N_{cr,cal} = N_{u,cal}kN | N_{d,cal}kN | N_{r,cal}kN |

14.94 | 9.59 | 12.69 | 12.6 | 8.65 | 10.43 |

Displacements of connector BP10 | Calculated displacements of connector BP10 | ||||

u_{cr,mv} = u_{u,mv}mm | u_{ag,mv}mm | u_{r,mv}mm | u_{cr,cal} = u_{u,cal}mm | u_{u,cal}mm | u_{r,cal}mm |

0.19 | 1.33 | 5.80 | 0.07 | 1.29 | 5.40 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jasiński, R.; Galman, I.
Testing Joints between Walls Made of AAC Masonry Units. *Buildings* **2020**, *10*, 69.
https://doi.org/10.3390/buildings10040069

**AMA Style**

Jasiński R, Galman I.
Testing Joints between Walls Made of AAC Masonry Units. *Buildings*. 2020; 10(4):69.
https://doi.org/10.3390/buildings10040069

**Chicago/Turabian Style**

Jasiński, Radosław, and Iwona Galman.
2020. "Testing Joints between Walls Made of AAC Masonry Units" *Buildings* 10, no. 4: 69.
https://doi.org/10.3390/buildings10040069