*Statistical Estimation of Stress in Walls*

The practical application of that method requires further tests mainly on location of measuring points and their minimal number. However, assuming only measuring points for central area of each masonry unit are used to determine stress in the masonry, then boundary values of strength could be determined with the probability that the obtained results were not lower than experimentally obtained results. Only values from the range of 0–0.5σ3max were used for the calculations. The selected range seems to be the most reasonable because at the operational stage force values in real walls can correspond to maximum stress of the order of 50% of the calculated compressive strength of the wall fd. Thus, load-carrying capacity of the real wall [43] depends not on absolute values of compressive force generating stress σ3, but on the stability expressed by the reduction factor for load-carrying capacity (Φ1,2 and Φ2m). Boundary values in confidence intervals of the mean value [53] (at *n* > 30 and unknown variance σ) were determined form the general relationship at the statistical significance α = 0.1:

$$P\left(t\_p - \mu\_{1-\alpha/2}\frac{S}{\sqrt{n}} < t\_{p\text{cal}} < t\_p + \mu\_{1-\alpha/2}\frac{S}{\sqrt{n}}\right) = 1 - \alpha\tag{42}$$

where: *t*p—mean time of wave propagation, *S*—standard deviation of propagation velocity for the specimen. *u*1−<sup>α</sup>/2—statistics with the random variable at the normal distribution N(0.1). When *n* < 30, the statistics *t*1−α/2 with the Student's *t*-distribution and *n*-1 degrees of freedom should be applied.

Only the upper value of confidence interval is suitable for practical applications, which in this case can be associated with the quantile of the order of 95%. In other words, the upper limit of the confidence interval for the mean value was assumed because it is commonly used in the construction sector. Stress values were determined with the AE method using calculated values of passing time of the wave. The obtained results were compared with true mean stress values of the masonry wall. Values for upper confidence intervals for passing time *t*pcal and calculated stress values σ3cal using the AE method are presented in Table 7 and compared with stress results obtained from testing the models σ3obs. In that way, we obtain some estimation of the deviation between test and calculated results at the specified confidence level.


**Table 7.** Compared results from tests and upper values of confidence intervals.

Taking into account the statistical estimation of stress, it was underestimated but values were significantly reduced. It can suggest with the probability of not greater than 5% that determination of stress in the walls from central areas of the masonry units with the slightest disturbances will cause underestimation of the mean stress at 0.25σ3max by 18%, and at 0.50σ3max by ca. 12%. That underestimation can be acceptable for masonry structures.
