**1. Introduction**

When assessing existing prestressed concrete structures, the determination of the residual prestressing force is an essential and inevitable task [1,2]. Prestressing force value obviously decreases over time and its determination should consider all potential prestress losses [3,4]. However, acquiring crucial knowledge about the exact value of prestressing force acting on the structure is quite difficult. Of course, for reliable assessment additional information considering the structure's condition needs to be obtained [5,6]; it is possible to collect this data using common testing methods [7]. This general knowledge includes many material properties, such as information about the real geometry of structural members and reinforcement, damage and deterioration (obtained from visual inspections); data containing the effect of significant overloading of the structure, etc. [8,9].

Generally, the analytical or numerical calculation of prestressing force value is the standard approach. Therefore, required input data consists of the age of the structure, its geometry, reinforcement parameters, and layout. Moreover, necessary material properties can be obtained using a wide range of standard material testing procedures [10–12]. However, all collected data that affect the prestressing force have a natural strongly stochastic

**Citation:** Moravˇcík, M.; Kral'ovanec, J. Determination of Prestress Losses in Existing Pre-Tensioned Structures Using Bayesian Approach. *Materials* **2022**, *15*, 3548. https://doi.org/ 10.3390/ma15103548

Academic Editor: Krzysztof Schabowicz

Received: 22 April 2022 Accepted: 12 May 2022 Published: 16 May 2022

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**Copyright:** © 2022 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 (https:// creativecommons.org/licenses/by/ 4.0/).

character, especially due to effects such as the rheology of concrete (creep and shrinkage) or steel relaxation [13].

In Bayesian philosophy, the analytic calculation (the primary method of prestress determination) can be considered as the prior hypothesis, together with its probability. Nevertheless, other new or additional relevant information can also be desirable and useful, especially regarding the unknown value of a prestressing force that is acting on the structure [14–16]. Likewise, several indirect techniques or structural tests can be used, such as the saw-cut method or structural response method [17–19]. These methods are based on observation of the structural behavior after the application of a known load. In the case of the saw-cut method, normal stress (strain) relief is observed. On the other hand, the structural response method evaluates prestressing based on the measurement of deflection, strain (normal stress) change or width of crack resulting from the external load [20,21]. All new relevant information can be taken into account and combined with the prior probabilistic model using updated techniques. These results are so-called posterior probabilistic models, which may be used to obtain an enhanced assessment of current prestressing force.

Updating the probability distribution of a basic variable is commonly based on the Bayesian approach described briefly below. Two individual events, A and B, are studied. The conditional probability P (A|B) of event A, given event B has occurred with a non-zero probability P (B), is defined as:

$$\text{P (A} \mid \text{B)} = \text{P (A} \cap \text{B)} / \text{P (B)} = \text{[P (A)} \times \text{P (B \mid A)]} / \text{P (B)} \tag{1}$$

where P (A|B) is a conditional probability, i.e., the probability of event A occurring given that B is true. It is also called the posterior probability of A given B. P (B|A) is also a conditional probability, i.e., the probability of event B occurring given that A is true. It can also be interpreted as the likelihood of A, given a fixed B, because P (B|A) = L (A|B) × P (A) and P (B) are the probabilities of observing A and B, respectively, without any given conditions; they are known as the marginal probability or prior probability. A and B must be different events.

The Bayesian theorem can also be applied to the hypothesis verification tool, and graphically interpreted using probability density functions as presented in Figure 1.

**Figure 1.** Bayesian concept.

In fact, the input data are random variables, thus the probability distribution function can be used to represent the data {x1, x2, ... , xn} with distribution parameters of θ. The concept of conjugacy in Bayesian statistics is used. Conjugacy occurs if the posterior distribution is in the same family of probability density functions as the prior belief, but with new parameter values. These values are updated to reflect what has been understood from the obtained data. Bayes' theorem, expressed in terms of probability distributions, appears as:

$$\text{if } (\boldsymbol{\theta} \mid \mathbf{data}) = [\mathbf{f} \mid \mathbf{(\theta)} \times \mathbf{f} \mid \mathbf{(data} \mid \mathbf{\theta})] / [\int \mathbf{f} \mid \mathbf{(data} \mid \mathbf{\theta}) \times \mathbf{f} \mid \mathbf{(\theta)} \times \mathbf{d} \boldsymbol{\theta}] \tag{2}$$

where f (θ|data) is the posterior distribution for the parameter θ; f (data|θ) is the sampling density for the data, which is proportional to the likelihood function, only differing by a constant that makes it a proper density function; f (θ) is the prior distribution for the parameter <sup>θ</sup>; and f (data) is the marginal probability, f (data) = . f (data|θ) <sup>×</sup> f (θ) <sup>×</sup> <sup>d</sup>θ.

A number of closed-form solutions for Equation (2) can be found for special types of probability distribution functions known as the natural conjugate distributions. In cases where no analytical solution is available, first order reliability method (FORM)/second order reliability method (SORM) techniques can be used to assess the posterior distribution. If the normal–normal conjugate family N (μ,σ2) is taken into account, Bayes' theorem leads to the posterior distribution for μ and σ<sup>2</sup> given the observed data to take the form:

$$\mathbf{p}\left(\mu,\sigma^{2}\mid\mathbf{x}\_{1},\mathbf{x}\_{2},\dots,\mathbf{x}\_{n}\right) = \left[\mathbf{p}\left(\mu,\sigma^{2}\right)\times\mathbf{p}\left(\mathbf{x}\_{1},\mathbf{x}\_{2},\dots,\mathbf{x}\_{n}\mid\mu,\sigma^{2}\right)\right] / \text{Normalizing Constant} \tag{3}$$

where p (μ,σ2) is the joint prior distribution function. The likelihood function for (μ,σ2) is proportional to the sampling distribution of the data, L (μ,σ2) ∝ p (x1, x2, ... , xn|μ,σ2), so that the posterior distribution can be re-expressed in proportional form. The symbol ∝ means "proportional to". According to the Joint Committee on Structural Safety (JCSS) Probabilistic Model Code described in [14] and [22], Equation (3) can be expressed for normal distribution in engineering-acceptable form for μ and σ, given as:

$$\mathbf{f}'(\boldsymbol{\mu}, \sigma) = \mathbf{k} \times \sigma^{-\left[\left[\delta(\mathbf{n}') + \mathbf{v}' + 1\right]} \exp\left\{-\left[\left(1/2 \times \sigma^2\right)\right] \times \left[\mathbf{v}' \times \left(\mathbf{s}'\right)^2 + \mathbf{n}' \times \left(\mu - \mathbf{m}'\right)^2\right]\right\} \tag{4}$$

where k is the normalizing constant; δ (n ) = 0 for n = 0; δ (n ) = 1 for n > 0; m is the sample mean; s' is the sample standard deviation; n is the sample size; and v = n − 1 is the number of degrees of freedom. Then, the predictive value of {X} can be found from:

$$\{\mathbf{X}\} = \mathbf{m}^{\prime\prime} + \mathbf{t}\_{\mathbf{v}^{\prime\prime}} \times \mathbf{s}^{\prime\prime} \times \{1 + 1/\mathbf{n}^{\prime\prime}\}^{0.5} \tag{5}$$

where tv has a central Student's t-distribution.

The present study is based on results from the application of the non-destructive saw-cut method performed on pre-tensioned members—e.g., railway sleepers. The method aims to isolate concrete block from acting forces by means of saw-cuts. Residual prestressing force is subsequently calculated from normal stress relief initiated by sawing [17,18]. The ratio of isolation of concrete block is dependent on the parameters of saw-cuts—depth and axial distance [23]. One of the main advantages of this technique is that it has negligible local impact on the prestressed concrete structure. When using the saw-cut method on an unloaded structure, the prestressing force value is easy to calculate because the determination of normal stress resulting from the dead load is obvious. However, if the investigated member is also loaded by an external load, additional normal must be taken into account [20,21].

#### **2. Analytical Calculation of Prestressing Force Value**

The analysis was performed on a prestressed concrete sleeper, which is one of the standard pre-tensioned members produced in the manufacturing process. In particular, our specimen is sleeper type B70 W-49G, as illustrated in Figure 2.

**Figure 2.** Analyzed prestressed concrete sleeper B70 W-49G.

Declared basic parameters entered into the analytical calculation of prestressing force Pm (t) (kN) are listed in Table 1.

**Table 1.** Characteristics of analyzed sleeper.


The prestressed concrete sleeper was designed from concrete class C50/60 [24]. Therefore, the modulus of elasticity, Ecm = 37,000 MPa, was assumed. Calculation of prestressing force Pm (t) in time t = 365 days was performed according to the Eurocode 2—Slovak national implementation STN EN 1992-1-1 [25]. Prestressing of the sleeper was provided by eight wires with a smooth surface and a diameter of 7 mm. The initial prestressing force was derived from Equation (6) considering the value of initial stress in the wire, σp,in = 1380 MPa. The bending moment due to self-weight is MGo = 0.798 kNm.

$$
\sigma\_{\rm pm,max} = \min \left[ 0.80 \times \mathbf{f}\_{\rm pk}; 0.90 \times \mathbf{f}\_{\rm p,0.1k} \right] \tag{6}
$$

Long-term prestress losses have the highest influence on the prestressing force value's decrease over time. They were determined using standard Equations (7) and (8). Prestress losses due to steel relaxation are:

$$
\Delta\sigma\_{\rm p.r.} \text{ (t.t.}\\ \text{)} = -\sigma\_{\rm pi} \times \text{k}\_1 \times \rho\_{1000} \text{ [\%]} \times 10^{-5} \times \text{e}^{\mu} \times \text{k}^2 \times \text{[t/1000]}\\ \text{0}^{0.75 \times (1 - \mu)} = -169.3 \text{ MPa} \tag{7}
$$

$$
\text{where } \text{k}\_1 = 5.39; \text{k}\_2 = 6.7; \rho\_{1000} = 8.0; \mu = 0.79.}
$$

Δσp,r+s+c = −{0.8 × Δσp,r (t,t0) + εcs (t,t0) × Ep + (Ep/Ecm) × ϕ (t,t0) × σc,(Pm0+G0) (t,t0)}/{1 + (Ep/Ecm) × (Ap/Ac) × [1 + (Ac/Ic) × ep 2] <sup>×</sup> [1 + 0.8 <sup>×</sup> <sup>ϕ</sup> (t,t0)} = <sup>−</sup>310.3 MPa (8)

where εcs (t,t0) = 5.01; and ϕ (t,t0) = 1.69.

Corresponding prestressing force, considering the prestress losses after 365 days in one prestressing wire, is Pm (t) = 41.2 kN.

All analytical systems contain a certain degree of variability. When these systems are formed by a combination of random variables, the resulting variability of the system generally cannot be found in a closed-form approach. An alternative approach that allows the estimation of variability in a system, given the variability of its components, is Monte Carlo simulation (MCS). In this study, MCS was used to determine the total variability that defines all Bayesian systems of probability functions. In our case, a total of 1000 simulations were applied.

The current random vector of prestressing force {Pcalc} derived from the analytical calculation has been assumed as a normally distributed function, as in [15], based on a random generation with a known mean value of 40.96 kN and a standard deviation of 10.20 kN. In the first approximation, this level corresponds to the estimated variation coefficient of 25%. Of course, it is appropriate to have all components in Equation (7) or (8) as a random variable. Specifically, they have mean values according to Table 1, and the strength and modulus of elasticity parameters have a common coefficient of variation (CV), CV = 5%, and for the cross-sectional parameters, CV = 3%. However, in this simulation, the consequence of the central limiting theorem was applied. Estimated prestress force is presented using a histogram, probability density function (PDF) and cumulative distribution function (CDF) in Figures 3 and 4. It can be considered as the joint prior probability function, as in Equation (3). Statistical parameters are listed in Table 2.

**Figure 3.** Normally distributed values of Pcalc.

**Figure 4.** *Cont*.

**Figure 4.** (**a**) Histogram and CDF of Pcalc; and (**b**) PDF and CDF of Pcalc.

**Table 2.** Statistical parameters for Pcalc.


#### **3. Experimental Program—Saw-Cut Method**

The application of the saw-cut method on prestressed concrete sleepers consisted of three saw-cuts. Their axial distance was 120 mm, and sawing was performed gradually (depths of 10, 20 and 30 mm). The maximal depth of saw-cuts was chosen with regard to the layout of prestressing wires in the sleepers, as we intended to avoid cutting them and affecting the structural integrity of pre-tensioned members. For the experiment, the upper edge of the specimen with a straight and smooth surface in the mid-span area was chosen. This location provided suitable conditions for the installation of strain gauges and subsequent measurement of strain release after sawing. The measurement is presented in Figure 5. For strain recording, linear foil strain gauges HBM LY41-50/120 made of ferritic steel (temperature matching code "1": 10.8 × <sup>10</sup>−6/K) with a measuring grid length of 50.0 mm and a total length of 63.6 mm were installed. The position of strain gauges can be seen in Figure 6. The prestressed concrete sleepers were supported by two lines at a distance of 0.1 m from the ends. Supports were provided using so-called steel rollers; as a consequence, the specimens behaved as simply supported beams with an effective length of 2.4 m.

All the equations and assumptions mentioned are based on the linear distribution of normal stress which can be assumed in the case of the uncracked prestressed concrete structure. Therefore, in the case of an already pre-cracked structure, such an assumption should not be considered. After sawing, some local nonlinearities could be observed, but the area adjacent to installed strain gauges should not be significantly influenced given an axial distance of 120 mm.

(**a**)

(**b**)

**Figure 5.** The measurement: (**a**) application of saw-cuts and (**b**) view of saw-cuts.

**Figure 6.** Mid-span area: position of saw-cuts (SC) and strain gauges (SG).

Normal stress readings from the measurement are displayed in Figure 7 and listed in Table 3. Evaluation of obtained results was based on the real value of the concrete's modulus of elasticity, which was determined using removed cylindrical samples (37.4 GPa). The real modulus of elasticity value was in compliance with Eurocode 2 [25].



Depending on the depth of a saw-cut, experimentally determined prestressing force value Pexp,i can be calculated according to Equation (9):

$$P\_{\rm exp,i} = \langle \mathbf{k}\_{\rm i} \times \Delta \sigma\_{\rm c,I} - [(\mathbf{M}\_{\rm Ci0} \times \mathbf{z}\_{\rm upp})/\mathbf{I}\_{\rm c}] \rangle / \mathbf{CS} \tag{9}$$

where ki is the "calibration factor of the depth of saw-cut" determined according to a parametric study based on nonlinear numerical simulations. This constant represents the ratio between released normal stress after the application of saw-cuts and initial normal stress in a prestressed concrete member. The deeper a saw-cut is in the shorter axial distance we choose, the more normal stress is released, and the calibration factor has a lower value. More information about the calibration factor and its determination can be found in [23]. Δσc,i is the released normal stress value dependent on the depth of a saw-cut, and CS is a cross-sectional function, as in Equation (10):

$$\rm CS = -(8/A\_c) - \left[ (-4 \times \rm e\_{p,bott} + 4 \times \rm e\_{p,upper} \times \rm z\_{upper})/I\_c \right] \tag{10}$$

where ep,bott is the distance from the cross-section center to the center of the bottom wires; and ep,upp is the distance from the cross-section center to the center of the upper wires. All cross-sectional parameters used in Equation (10) were considered as random variables according to Table 4. A histogram and CDF of the cross-sectional function CS can be seen in Figure 8.

**Table 4.** Cross-sectional parameters.


**Figure 8.** Histogram and CDF of CS parameter.
