**4. Prestressing Force Distribution Using Bayesian Approach**

Measured released stresses {Δσc,10; Δσc,20; and Δσc,30} in prestressed concrete sleepers were tested using the Q-Q-plot method as one of the principal testing methods, and approximated using normally distributed data that were randomly generated using the MC simulations. Additionally, over 1000 simulations were performed. The results of Q-Q-plot testing can be seen in Figure 9. The data with the best agreement with the linear regression parameter R<sup>2</sup> = 0.96 were gained from the 30 mm deep saw-cuts, in which the highest stress relief was reached (approximately 72% of total assumed initial normal stress). This corresponds with the calibration factor k30/120 = 1.39. On the other hand, a 10 mm deep saw-cut released only 12% of initial normal stress. For this depth, the calibration factor is k10/120 = 8.40.

**Figure 9.** Q-Q-plot tests for released normal stress (MPa): (**a**) Δσc,10; (**b**) Δσc,20; and (**c**) Δσc,30.

The random vector of prestressing force {P (imm)} = {P (10 mm); P (20 mm); P (30 mm)} based on a measured data set of released normal stress is assumed according to Equation (9). In this paper, only worst fitted data (a saw-cut depth of 10 mm) and best-fitted data (a saw-cut depth of 30 mm) were chosen for graphical interpretation on histograms. PD and CD functions are illustrated in Figures 10 and 11. Consequently, some differences between the functions P (10 mm) and P (30 mm) are obvious from the basic statistical parameters of both of chosen data sets, as listed in Table 5.

(**b**)

**Figure 10.** Δσc,10—(**a**) histogram of generated data and (**b**) PDF and CDF.

**Table 5.** Statistical parameters for P (imm).


**Figure 11.** Δσc,30—(**a**) histogram of generated data and (**b**) PDF and CDF.

The primary predictive data set of calculated prestressing force {Pcalc} can be set as a joint prior distribution function f (μ, σ), according to Equation (4), or prior information. The random vector {P (imm)} derived from normal stress releasing, determined given the depth of a saw-cut, can be used in the Bayesian theorem as a likelihood or conditional information. This function is based on a measured data set which specifies and moves assumed calculation. The resulting {Ppost = P} as a final distribution prestressing force can also be signed as a posterior data distribution f" (μ, σ). These data were derived from Equation (5) using MC simulation and can be graphically interpreted in PD and CD functions of {Ppost}. The resulting shape and parameters of posterior probability distribution f" (μ, σ) depends on the distribution function of the measured data sets, as in Figures 12 and 13. The final statistics of the {P} probability distribution function regarding the depth of a saw-cut are presented in Table 6.

**Figure 12.** Δσc,10—(**a**) CDF of data set and (**b**) PDF of data set.



**Figure 13.** Δσc,30—(**a**) CDF of data set and (**b**) PDF of data set.

#### **5. Discussion**

Our investigation suggests that a relatively small intervention into the prestressed member can cause sufficient local normal stress relief. Intervention in the form of a maximum 30 mm deep saw-cut is insignificant compared to the dimensions of the crosssection of the prestressed member; the global structural integrity is not affected. Moreover, the saw-cut method can be performed without the application of an external load. Absence

of additional external load leads to easier residual prestressing force derivation from the obtained results.

Undoubtedly, prestressing force is the decisive factor in the assessment of existing prestressed concrete structures. However, it is very difficult to obtain its exact value at the time of testing. In addition to the standard analytical evaluation that should be used in all cases, some experimental methods have been verified and applied worldwide [17,18,20,21,26,27]. Nevertheless, these methods provide different results compared to the analytically calculated value. The reason for this is the wide range of factors that affect prestress losses, including the creep and shrinkage of concrete and steel relaxation. It is also possible that the corrosion effect or issues related to inadequate concrete or duct grout quality could influence the analysis.

Generally, Monte Carlo simulation is known as a technique that constructs probability distributions for the possible outcomes of decisions. In the presented study, the MCS was applied to the generation of random variable vectors which were normally distributed. The Bayesian concept can more precisely define the estimated residual value of prestressing force at a certain time Pm (t). Usually, the evaluation can only take analytically derived prestress losses into account using the standard approach, as in Eurocode 2 [25], which is similar to the presented study. However, standard calculation of residual prestressing force value is not often sufficient or adequate to deal with such important parameters for global structure reliability evaluation.

#### **6. Conclusions**

In our paper, there is an obvious coincidence between the prior {Pcalc} and posterior {Ppost} probability distribution functions, with only a 95% confidence level. The reason for such close agreement of both probability functions is that the sleeper specimens were in a very good state after being stored for one year in a covered warehouse without any service or deterioration factors that could affect their structural condition. Moreover, they were kept in a relatively stable environment which inevitably affected the volumetric changes in the concrete. This is why prior and posterior functions are in such good agreement. The kurtosis of posterior functions was higher in both cases, especially for the best-fitted curve for Ppost (30 mm), due to the strong consistency of the experimental results that form likelihood function. Naturally, if the shape, distribution and displacement of the predictive function of prestress losses on the x-axis were distant from the likelihood function, the posterior probability distribution function would be different.

Unquestionably, it is very important to present the results of methods for determining the state of prestressing in statistical form. This makes it possible to define the probability of the obtained state of prestressing, which is a crucial aspect of the evaluation of the load-carrying capacity of the existing prestressed concrete structure. The implementation of the probability approach to determine prestress losses can be an effective tool, since the evaluation process of existing prestressed concrete structures considers model uncertainties and any possible deteriorations.

**Author Contributions:** Conceptualization, M.M. and J.K.; data curation, M.M. and J.K.; validation, J.K. and M.M.; formal analysis, M.M.; measurements, M.M. and J.K.; writing—original draft preparation, J.K. and M.M.; writing—review and editing, J.K.; visualization, J.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research project was supported by the Slovak Grant Agency (VEGA) under contracts No. 1/0048/22 and No. 1/0306/21 and by the Cultural and Educational Grant Agency (KEGA) under contract No. 020ŽU-4/2021.

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

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

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


