**5. Analysis of Test Results**

*5.1. Components of Stress State Based on the AE Effect*

Values of stress *P* were determined at each measuring point using the empirical relationships which describe changes in mean hydrostatic stresses as a function of changes in the relative velocity of ultrasonic waves and propagation times of ultrasonic waves, which were determined in stage I and presented in the paper [7]. Mean values of hydrostatic stress *P* expressed as the maps of stress at different stress levels (0.25*σ*3max, 0.50*σ*3max, 0.75*σ*3max) are shown in Figures 14–16.

**Figure 14.** Mean hydrostatic stress values *P* at load *σ*<sup>3</sup> = 0.25*σ*3max: (**a**) model I-1, (**b**) model II-1, (**c**) model III-1.

**Figure 15.** Mean hydrostatic stress values *P* at load *σ*<sup>3</sup> = 0.50*σ*3max: (**a**) model I-1, (**b**) model II-1, (**c**) model III-1.

**Figure 16.** Mean hydrostatic stress values *P* at load *σ*<sup>3</sup> = 0.75*σ*3max: (**a**) model I-1, (**b**) model II-1, (**c**) model III-1.

The distribution of mean hydrostatic stresses *P* in all the test models indicated the predominating compression (*P* > 0) in the masonry units. Mean hydrostatic stresses were clearly decreasing in some areas adjacent to the head joints. Only at some individual points did mean hydrostatic stress represent tension (*P* < 0).

By reference to the paper [11], the qualitative analysis for the obtained results was performed in a comprehensive way using all the test results and then was constrained to a limited number of points. The comprehensive method included *n* = 315 (the model of series I) or 308 (the models of series II or III) measured passing times of ultrasonic wave at each analyzed stress level. The results for the clearly disturbed areas were also taken into account. In the method using a limited number of points stress was estimated only on the basis of the points located in the central area of the masonry units. In that way, the measuring points were considerably reduced to 45 for the model I, and to 44 for the models of series II and III.

At first, the velocity of ultrasonic waves was determined under air-dry conditions according to the following relationship (17). A relative difference in the passing time of the ultrasonic wave was then determined at other stress values. Determination of the acoustoelastic coefficient *δ<sup>P</sup>* from the Equation (15) was the next step. At the end the stress *P* was obtained from the converted relationship (10). Table 6 demonstrates the calculated stresses.

**Table 6.** Calculated mean values of hydrostatic stress in the wall using all measuring points.


Values of coefficients *δ<sup>P</sup>* depended on the density of AAC, however, these differences were relatively small (*δ<sup>P</sup>* <sup>=</sup> −0.0635–−0.0640 mm2/N). Mean values of hydrostatic stress were evidently increasing with an increase in vertical stress values, which showed that compressive stress predominated in the compressed wall. The stress values calculated for the individual models were close to each other only when stress values were relatively low, that is, 0.25*σ*3max and 0.50*σ*3max. Stresses in the model III-1 determined by the AE differed by maximum 12%. At 0.75*σ*3max the stress values did not differ by more than 8%.

The same procedure was repeated in the method based on the limited number of results (from central areas of the masonry units). Analogous to the method, which was based on all the test results, velocities of ultrasonic waves under air-dry conditions were determined at first, and then a relative difference in the passing time of the ultrasonic wave and the stress values *P* were calculated from the converted relationship (10). The coefficient *δ<sup>P</sup>* was the same as the value specified in Table 6. The obtained values of hydrostatic stress *P* are presented in Table 7.

The stress values were much lower at a limited number of measuring points. When stresses were the lowest, that is, equal to 0.25*σ*3max, the stresses determined with the AE method at the minimum number of points were lower by no more than 31% (the model II-1). Mean hydrostatic stresses at higher stresses (0.50*σ*3max and 0.75*σ*3max) were underestimated by a maximum of 18%.


**Table 7.** Results of calculations of normal stress *σ*<sup>3</sup> in the wall using a limited number of measuring points.

By knowing mean hydrostatic stresses and stresses *σ*<sup>3</sup> determined from the Equation (12) and presented in the paper [11], horizontal stresses *σ*<sup>1</sup> could be determined from the relationship

$$P = \frac{1}{3}(\sigma\_1 + \sigma\_2 + \sigma\_3) = \frac{1}{3}(\sigma\_1 + \sigma\_3) \to \sigma\_1 = 3P - \sigma\_3. \tag{20}$$

where *P*—mean hydrostatic stress, *σ*3—normal stress perpendicular to the plane of bed joints, *σ*1—normal stress parallel to the plane of bed joints.

The values of stress *P* and stress *σ*<sup>3</sup> shown in Tables 6 and 7 and presented in the paper [11], were the base to determine stresses *σ*<sup>1</sup> which are summarized in Table 8.



#### *5.2. Numerical FEM Model*

The numerical FEM model was necessary to perform the comprehensive analysis of the determined mean values of hydrostatic stress *P* and normal stress *σ*<sup>1</sup> (determined indirectly on the basis of known values of stress *σ*1). This model was used to determine mean values of hydrostatic stress from the components of the stress state. As it was demonstrated in the paper [11], the defined relationships between stress and strain were similar to the linear relationship. Hence, the linear-elastic FEM micro-model was sufficient for that purpose. The model included nominal geometric dimensions and boundary conditions. The model was 726 mm high, 500 mm wide, and 180 mm thick. It was supported along its bottom edge using the roller supports in each node, except for the middle one with the blocked horizontal movement. Five-node finite elements with 4 degrees of freedom for each node were used for calculations in a plane stress state (2D, PSS). The masonry units were modelled separately, for which the modulus of elasticity was *E*<sup>B</sup> = 2039 N/mm<sup>2</sup> and Poisson's ratio was ν<sup>B</sup> = 0.21 (cf Table 1). Mortar in joints was also modelled separately, and the finite elements took the following parameters *E*<sup>m</sup> = 6351 N/mm2 and ν<sup>m</sup> = 0.18 [52]. Due to linear elasticity of the FEM models, the model was subjected to unit loads *q* = 1 kN/m, and the stress values at higher loads were determined using superpositioning of load states. The numerical FEM models are shown in Figure 17. The calculated data presented as the maps of vertical stresses *σ*<sup>3</sup> and *σ*<sup>1</sup> of unit loads are illustrated in Figure 18.

The stress–strain relationships for all the test elements and the FEM models were compared as shown in Figure 19. These curves indicate that the behavior of the test models was almost linear until the moment of cracking. Strains began to increase much faster than in the linear-elastic FEM model when the stresses were >0.75*σ*3max. Differences in calculated and experimentally determined moduli of elasticity did not exceed 10%.

**Figure 18.** FEM calculations for the test walls: (**a**) stresses *σ*x in the model of series I without a head joint, (**b**) stresses *σ*y in the model of series I without a head joint, (**c**) stresses *σ*x in the model of series II with the head joint at mid-length of the element, (**d**) stresses *σ*y in the model of series II with the head joint at mid-length of the element, (**e**) stresses *σ*x in the model of series III with the head joint in 1/4 length of the masonry unit, (**f**) stresses *σ*y of the model of series III with the head joint in 1/4 length of the masonry unit.

**Figure 19.** Compared relationships between stress and strain (*σ*3–εy) for all tested models and numerical FEM models.

The data for components of the stress states *P*, *σ*1, *σ*<sup>3</sup> obtained on the basis of the FEM calculations are compared in Table 9. The results are compared in Table 10.


**Table 9.** FEM-based calculations for mean stresses *P* and *σ*1.

**Table 10.** Compared mean values *P* and *σ* obtained from the tests and FEM calculations.


The maximum difference in mean stresses *σ*<sup>3</sup> perpendicular to the plane of bed joints, which were determined for all the measuring points, was 3% at the stress levels 0.25*σ*3max–0.50*σ*3max. The biggest difference was observed under compressive stress equal to 0.75*σ*3max. When the number of measurements was limited to central areas of the masonry units, significantly greater differences were noticed, The highest mean overestimation

of the results exceeding 32% was found at the stress level of 0.75*σ*3max. Almost the same results were obtained for the hydrostatic stress *P* in the wall. For all the measuring points, the differences did not exceed 3% at the stress levels of 0.25*σ*3max–0.50*σ*3max. Mean stresses were greater by 31% also under higher compressive stress equal to 0.75*σ*3max.

The greatest variation of the results was found under the stresses *σ*<sup>1</sup> which were parallel to the plane of bed joints. A higher number of measurements caused in this case a higher degree of inconsistency between the results. Under the compressive stresses 0.25*σ*3max–0.50*σ*3max, the stress values were overestimated by ca. 44–50%. An increase in mean values of stresses to the level of 0.75*σ*3max caused that the overestimation of stresses was reduced to ca. 19%. The best results were obtained for the limited number of measuring points. Then, at the stress levels of 0.25*σ*3max and 0.75*σ*3max, the stresses determined with the NDR technique did not significantly vary from the stresses obtained from the FEM calculations. The highest overestimation of the stresses of the order of 28% was found for mean stresses equal to 0.50*σ*3max.

The best agreement with the FEM calculations was reached when the maximum number points were used for vertical stresses *σ*<sup>3</sup> and mean hydrostatic stresses *P*. The measurements limited to central areas of the masonry units resulted in bigger differences in the results when compared to the numerical results. The contradictory tendency was noticed for the stresses *σ*1, under which the biggest differences in the results were obtained when the maximum number of points were used. Limiting the measurements only to the central areas caused a clear drop in the stress values which were empirically determined.

The results were obtained from the methodology of determining the coefficients AE (*β*<sup>113</sup> *η*P), which was conducted on relatively small specimens subjected to the load which eliminated additional stress components and boundary disorders. The stress distribution in real masonry structures (on which the main tests were performed) is significantly disturbed by the presence of head and bed joints, the shape, and interaction with other masonry units. The results were close to the FEM calculations when the measurements were taken in the central area of the masonry units at the least disturbed stress state. A narrower spread of the calculated and test results was the immediate effect. It should be remembered that the plane stress state assumed for the analyses is observed locally in central areas of the masonry units. Additional stresses *σ*<sup>2</sup> = 0 perpendicular to the front plane of the masonry are found in the edge and support areas, which has an impact on mean hydrostatic stresses. The proposed procedure cannot be applied for the whole range of stress values without its prior calibration. At relatively low stresses 0.25*σ*3max and 0.50*σ*3max, the test results were similar to the calculated results. The most significant differences were obtained for the stresses of 0.75*σ*3max, and at this level NDT tests can be performed.

#### *5.3. Model Update*

It is more favorable to perform in practice only the tests restrained to central areas of the masonry units. Such an approach reduces the effect of disturbances created at the element edges due to the presence of bed and head joints. As shown in point 6.2, the NDT technique based on the AE effect and the stresses determined for the central areas of masonry units are expected to provide inconsistency in all the determined stresses. Assuming that the results obtained from the FEM calculations correctly estimate the stress values in the masonry units, the stress values *σ*<sup>3</sup> in the masonry units were at first corrected. For that purpose, mean quotients presented in Table 9 were applied. It was the base to calculate the mean quotient of stresses determined by the NDT and FEM techniques. On this basis, the mean coefficient equal to *α*<sup>3</sup> = 0.75 was determined. Coefficients of stresses *P* and *σ*<sup>1</sup> were determined similarly. These values were *α<sup>P</sup>* = 0.76 and *α*<sup>1</sup> = 1.09. The update empirical values to determine stress in the wall can be expressed as

$$
\omega\_3 = \frac{1}{\alpha\_3} \frac{(c\_p - c\_{p0})}{c\_{p0} \beta\_{113}},
\tag{21}
$$

$$P = \frac{1}{\mathfrak{a}\_P} \frac{\left(\mathfrak{c}\_p - \mathfrak{c}\_{p0}\right)}{\mathfrak{c}\_{p0}\delta\_P},\tag{22}$$

$$
\sigma\_1 = \frac{1}{a\_1} (3P - \sigma\_3). \tag{23}
$$

The results obtained by the NDT technique before and after validation and by the FEM methods are compared in Figures 20–22.

**Figure 20.** Comparison of stress values determined by the NDT and FEM methods for the model I-1: (**a**) mean hydrostatic stress; (**b**) normal stress perpendicular to the plane of bed joints; (**c**) normal stress parallel to the plane of bed joints.

**Figure 21.** Comparison of stress values determined by the NDT and FEM methods for the model II-1: (**a**) mean hydrostatic stress; (**b**) normal stress perpendicular to the plane of bed joints; (**c**) normal stress parallel to the plane of bed joints.

The stresses *σ*<sup>3</sup> and *P* in the updated model were underestimated by no more than 15%. On the other hand, the underestimation of the stresses *P* parallel to the plane of head joints *σ*<sup>1</sup> did not exceed 6%.

**Figure 22.** Comparison of stress values determined by the NDT and FEM methods for the model III-1: (**a**) mean hydrostatic stress; (**b**) normal stress perpendicular to the plane of bed joints; (**c**) normal stress parallel to the plane of bed joints.

The described validation resulted in mean stresses in the wall which were comparable to the data determined by the FEM technique after taking at least *n* > 44 measurements in the central parts of the wall. However, taking the measurements at so many points (at a relatively low variation) can be troublesome in practice. That is why it is necessary to specify the minimum number of measuring points, at which the obtained results are reliable with reference to the defined confidence level [53]. Therefore, the minimum number of measuring points was define assuming that:

1. the general population had the normal distribution N(*μ*, *σ*),


Based on these assumptions, the calculations were performed under the stresses equal to 0.75*σ*3max. Only the stresses *σ*<sup>3</sup> and *P* calculated from the relationships (21) and (23) were considered. It was not necessary to specify the number of samples on the basis of the stress *σ*<sup>1</sup> as it was not an independent variable. The obtained number of samples is shown in Table 11.

**Table 11.** Minimum number of measuring points to determine the stresses *σ*<sup>3</sup> and *P*.


When the tests were focused on determining compressive stress *σ*<sup>3</sup> during the in-situ tests on the wall made of AAC masonry units, the minimum number of measurements was estimated to be *n*<sup>0</sup> = 8. On the other hand, when the aim of the tests is to determine the complex state of stress, the minimum number of measurements should not be lower than *n*<sup>0</sup> = 23. When the results were expressed in 1 m2 of the wall, then the minimum number of measuring points required to determine stresses *σ*<sup>3</sup> should not be lower than *<sup>n</sup>*<sup>0</sup> = 8·(1/0.726·0.5) = 22 measurements/m2, and in case of mean hydrostatic stresses *<sup>n</sup>*<sup>0</sup> = 23·(1/0.726·0.5) = 61 measurements/m2.

The proposed update method was intended to determine mean stresses in the wall, which were crucial for diagnostic tests for structures. Development of the complete model which can be used to define characteristics and design values to verify the estimated structural safety, should include the non-linear FE model and the application of FORM procedures [54,55].

#### **6. Conclusions**

This paper is a continuation of the tests [11,14] concerning the use of the ultrasonic (UPV) techniques, in particular the acoustoelastic (AE) method to detect stresses in a structure by means of the non-destructive technique (NDT). The tests were focused on the commercially produced autoclave aerated concrete (AAC) which is characterized by high homogeneity and repeatability of the parameters. Considering different purposes, the tests were carried out in two stages. In Stage I, the test procedure was specified and the acoustoelastic coefficient *δ<sup>P</sup>* was determined. This coefficient specified the relationships between the mean hydrostatic stresses *P* and the velocity of the longitudinal ultrasonic wave propagation *cp*.

The non-standard cuboidal specimens 180 × 180 × 120 mm were used for the calibration purposes. They were tested at the in-house developed test stand [37] which can be used to exert the biaxial compression. Based on the tests on AAC of different densities, the impact of relative humidity *w* and density ρ was included using the correlations presented in paper [14]. These considerations resulted in formulating the relationship δP(ρ). Verifica-

tion of the discussed procedure was performed in Stage II, in which the complex stress state was to be determined. This stage based on the results from previous test [11] performed on small AAC walls having a nominal density of 600 kg/m3. The models differed in the position of head joints without mortar and were classified into series I, II, and III. The measured velocities of ultrasonic wave propagation were analyzed under various compressive stresses: 0.25*σ*3max, 0.50*σ*3max and 0.75*σ*3max. The performed measurements (*n* = 308–315) were used to define the coefficients AE δ<sup>P</sup> = −0.1632–−0.0281. The data obtained from the AE method were compared with the data calculated for the linear-elastic FEM models of the walls. For the mean values of hydrostatic stress *P,* the stresses were underestimated at the order of 3% at 0.25*σ*3max. Under higher compressive stresses 0.50*σ*3max, the stresses P obtained by the AE method were greater by 2% than the calculated mean values. Under the highest analyzed stresses equal to 0.75*σ*3max, the empirically determined stresses were greater by over 19% than the calculated values. By knowing the stresses *P* and the stresses *σ*<sup>3</sup> perpendicular to the plane of head joins presented in the paper [11], the stresses *σ*<sup>1</sup> could be determined. These results were compared with the values obtained by the FEM calculations under various compressive stresses. Each time the values were overestimated. The stress values *σ*<sup>1</sup> at 0.25*σ*3max were overestimated by 44%. An increase in vertical loads to the values of 0.50*σ*3max and 0.75*σ*3max caused that the stress values determined with the AE method were greater by 50% and 19% compared to the data obtained from the FEM method. These discrepancies were caused by disorders of the stress state in the real structure and they considerably differed from the stress state, under which the coefficient AE (*β*<sup>113</sup> and δP) was determined.

It is not effective to use so many measuring points in practice (as a high number of points and results from the measurements have to be prepared and captured). For that reason, it was suggested that the measuring points were constrained only to the central areas of each masonry units which reduced the number of measurements to *n* = 45 and 44.

A similar comparison as for all the measurements produced considerably higher underestimations of the mean stresses *σ*<sup>3</sup> by 13–32%, and the stresses *P* by 3–19%. These values are not desirable taking into account safety of the structure. Hence, a decision was made to validate the model using the numerical FEM model by defining the coefficients *α*<sup>3</sup> = 0.75, *α*<sup>P</sup> = 0.76, and *α*<sup>1</sup> = 1.09. The stresses in the validated model were underestimated by no more than 15% under the stresses *σ*<sup>3</sup> and *P*. On the other hand, under the stresses *P* parallel to the plane of head joints *σ*<sup>1</sup> the underestimation did not exceed 6%.

In summary:


Also, the minimum number of measurements were defined to ensure reliability of the results at the pre-determined measurement error at the specified level of confidence. If the tests are to measure normal stresses in the plane of bed joints, then the minimum required number of measuring points is 22 measuring points/m2. The tests focused on the analysis of the complex state of stresses require the minimum number of measurements equal to 61 measurements/m2.

Specifying the detailed guidelines for in-situ tests for structures at the present stage of analyses of masonry structures is impossible. It is required to conduct additional tests on slender walls to determine the bending effect (varied stress state in the wall) and to improve the methodology of selecting the measuring points. The selection method of measuring points used to evaluate both the complex and the uniaxial stress state [11] may prove to be inadequate for bending. The double-sided access to the structure can be another problem. Hence, further tests are planned to be performed on the AE coefficient AE (β133) in the AAC wall with one-sided access.

**Author Contributions:** Conceptualization, R.J. and K.S.; Methodology, R.J.; Software, K.S. and P.K.; Validation, K.S. and P.K.; Formal analysis, R.J.; Investigation, R.J., K.S., and P.K.; Data curation, K.S.; Writing—original draft preparation, R.J., K.S., and P.K.; Writing—review and editing, R.J., K.S., and P.K.; Visualization, R.J.; Supervision, R.J.; Project administration, R.J. All authors have read and agreed to the published version of the manuscript.

**Funding:** The research reported in this paper was co-financed by the European Union from the European Social Fund in the framework of the project "Silesian University of Technology as a Center of Modern Education based on research and innovation" POWR.03.05.00-00-Z098/17. Part of the costs related to this research was co-financed by the Laboratory of Civil Engineering Faculty and Department of Building Structures at the Silesian University of Technology. Krzysztof Stebel was partially financed by the grant from SUT-subsidy for maintaining and developing the research potential in 2021.

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

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data available on request due to restrictions eg privacy or ethical The data presented in this study are available on request from the corresponding author. The data are not publicly available due to subsequent analyzes and publications.

**Acknowledgments:** Authors would like to thank students of the Silesian University of Technology— Magdalena Lempa, Piotr Dors, Kamil Mosz, Wojciech Weber—who participated in the preparatory works and basic tests of the research stand. Special thanks go to the experts involved in the project: Wociech Mazur (the ARAMIS software), Jan Pizo ´n (technology of concrete) for their help in preparing the test elements and interpreting the results. The author would like to express particular thanks to Solbet company for valuable suggestions and the delivery of masonry units, mortar which were used to prepare test models and perform tests.

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

#### **References**

