1 Introduction

Since the beginning of experiments in plain and reinforced concrete, it has been the goal to gain insights into the internal mechanisms of structural behaviour. For that, various measuring techniques with different qualities have been developed. In recent years, distributed fibre optic sensing (DFOS) experienced a high increase in applications in civil engineering. In particular, the backscatter-based coherent optical frequency domain reflectometry is of great importance. Many advantageous properties compared to conventional methods led to an increasingly frequent use of the technology in numerous areas such as medical instrumentation, aerospace and industrial applications, but also structural health monitoring [1,2,3,4] and measuring technology in various civil engineering applications [5].

The choice of the sensor, the kind of application and the measuring procedure are known to have significant influence on the quality of the generated data [6,7,8,9]—the post-processing of the data takes at least as important a role. Although the application of the sensor technology and the interpretation of measuring results has been subject of multiple publications (e.g., [1,2,3,4,5, 9,10,11,12,13]), the handling and processing of the generated data has not been documented thoroughly. Some publications focus on selected post-processing methods, such as the detection and elimination of strain anomalies or outliers [14,15,16,17,18], the use of the moving average to change the spatial resolution and smoothen data [6], polynomial fitting [19], Savitzky-Golay filtering [20] or the use of splines (only for strain gauges) [21] to fit strain distributions, but a holistic approach that compares several methods and shows their advantages and disadvantages in the context of further calculation has not yet been published. For this reason, the software solution FOS Evaluator has been developed to post-process recorded measurement data with a bundled variety of functions, including data selection, reduction, and smoothing. The process comprises different mathematical operations and a wide range of methods to make the best possible use of the experimental data. Furthermore, the software includes various mathematical functions—e.g., for the calculation of normal and bond stresses, crack detection or crack opening—which can be executed after data selection, reduction, and smoothing to gain deeper insights into the investigated structure.

In this paper, multiple methods for performing adequate post-processing of experimental data generated by DFOS (which are all implemented in FOS Evaluator) are presented and their varying influence on the measurements is systematically analysed. Suggestions for possible post-processing sequences are provided and the application of FOS Evaluator on real test data is demonstrated.

2 Fibre optic sensing in structural concrete

Fibre optic-based sensors, which have been available since the late 1980s, have a wide range of applications and can be distinguished by the intended measurands or the underlying technology. These sensors can measure various physical quantities, including strain, temperature, and pressure. The two most widely used fibre optic sensing technologies are fibre gratings and sensors using the backscattering of the fibre material. Both technologies are based on the recording of frequency shifts within the sensor fibre caused by external effects, with the main difference being the number and distribution of separate measurement points along the optical fibre.

Fibre gratings, such as Fibre Bragg Gratings [22, 23] (FBGs), are used for discrete measurements (similar to strain gauges) with a measuring length of around 10 mm, while systems based on backscattered signals are used for quasi-continuous measurements along the fibre length. By multiplexing the FBGs along the optical fibre, quasi-continuous measurement at several points is also possible [24], yet requiring a prior definition of the measurement points. In many cases, defining measurement points beforehand can be challenging, especially because cracks in concrete structures occur randomly. In such scenarios, in particular for research purposes or structural health monitoring, it is beneficial to have continuously distributed measurement points along the fibre with high spatial resolution.

The other most common types of sensors detect strain variations based on the backscattered parts of the induced light (Brillouin [25], Raman [26], or Rayleigh backscattering [27]). Systems based on Brillouin or Raman backscatter are suitable for long-range applications (measuring length of several kilometres, e.g., in pipelines or geothermal systems [28]), while systems based on Rayleigh backscatter are more appropriate for shorter distances (measuring length of around 100 m) [26, 29, 30]. One advantage of systems using Rayleigh backscatter is their millimetre-scale spatial resolution of measurement points along the sensor [31] compared to the metre-scale resolution of Brillouin or Raman systems [26, 32]. The sensing accuracy of systems using Rayleigh backscatter is about 1 µε and of those that use Raman or Brillouin backscatter about 30 µε [32, 33].

Recently, systems based on Rayleigh backscatter using an optical frequency domain reflectometer (OFDR [34]) have been successfully used in reinforced concrete research to primarily record strain changes under static and cyclic loadings [35, 36]. Furthermore, fibre optic sensors can also be utilized for crack detection [9, 37]. For this purpose, the sensor fibres can be attached directly to the concrete surface [38, 39] or embedded within the matrix [40]. Crack detection with DFOS within the matrix also offers advantages over photogrammetric methods, as these only record the crack on the concrete surface. For structural elements in bending or shear testing, it is not possible to accurately determine the local stress state in the longitudinal and transverse reinforcement solely through external measurements. In such cases, using fibre optic sensors for internal measurements is an appropriate and promising solution. In reinforced concrete testing, it is convenient to apply the fibres to the surface of the reinforcement, which is either steel [17, 33, 39, 41] or non-metallic reinforcement [3, 9, 42]. First, this has been demonstrated in laboratory-scale tests [18, 43], which were followed by successful implementation in real-scale applications [44,45,46].

Measuring the strain evolution of reinforcement and concrete throughout the experiment is essential for the understanding of structural behaviour, for deriving efficient and safe design concepts based on all-encompassing physical models [47,48,49], and for verifying the accuracy of complex nonlinear numerical models. Historically, only the loads and external deformations (e.g., deflections, crack widths) have been recorded in experiments, however, strains of the reinforcement as well as the concrete are also of great importance. Conventional measurement technologies rely on discrete measuring points or smeared values and are therefore limited in their informative value [13]. Strain gauges, for example, measure strain exclusively over a discrete, predefined length, while linear sensors (LVDT) measure discrete deformation which often considers and condenses a larger, predefined influencing area. Both, strain gauges and LVDT, are also difficult to apply inside material tests or structural component tests without having a major impact on essential structural mechanisms like e.g., the bond behaviour. For this reason, fibre optic measurement systems are an advantageous addition to conventional measurement technologies because of their minimal invasiveness. Continuous strain measurements can help to gain more insight into the physical effects and the structural behaviour. The general structure of the measuring technology and its principle are shown in Fig. 1.

Fig. 1
figure 1

Structure and general principle of DFOS

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In the example shown, the measuring fibre (yellow) is applied to the reinforcement of a test specimen under a tension load. The sensing fibre is scanned by a laser and the backscatter is evaluated. The basic principles have already been thoroughly described in other publications (e.g., [17, 32]). With increasing load (F1 and F2), cracks develop in the reinforced concrete specimen (Fig. 1). These cracks can be detected with the help of the measuring fibre as local strain peaks in the reinforcement. By using a pair of measuring fibres—one at the top and one on the bottom of the reinforcement—the curvature of the reinforcement can be assessed to determine dowel action, which is essential for understanding the internal load transfer mechanisms in RC members subjected to shear loading [41, 49, 50]. Depending on the measuring configuration (e.g., gauge length, fibre material, adhesive), varying degrees of accuracy can be achieved [7, 51].

3 Software solution and functionality

3.1 Background

Advantageous properties like high resolution and high measuring frequency helped fibre optic sensors to gain more attention for monitoring structural properties. At the same time, these properties can impede the effectiveness of the evaluation of experimental results. On the one hand, the high sampling rate and the low spacing between the measuring points generate an enormous amount of data. An exemplary measuring of a fibre with a length of 1 m and a gauge length of 0.65 mm (distance between measurement points) with a frequency of 20 Hz over a period of only 20 min generates already over 36 million data points. On the other hand, the small gauge length can lead to spatial variability which generates a distortion of strain measurements [6, 52]. Glitches in the strain readings, so-called Strain-Reading Anomalies [14, 53], generate unplausible strain values that can often reach a multiple of the yield strain of the investigated material. In addition to that, depending on the type of fibre used, the high sensitivity can result in local strain peaks/anomalies, especially due to punctual influences such as cracking. Often, the used sensors already suffer damage during casting. The vulnerability of the sensitive fibre sensor can be reduced by using a more robust fibre [17] or a protective device inside the concrete [40]. However, as a result, a part of the strain that occurs may not be captured. The reason for this can be the reduced bond of the sensor itself to the investigated material or the shear deformations of the coating [7, 54]. Consequently, the raw measured strain often is contaminated and has a limited informative value. These negative side effects—the large amount of data and the partial distortion—make the evaluation of the recorded data more difficult. Against this background, a software solution FOS Evaluator was developed including various functions to counteract these problems and simplify the effective usage of measured strain values via DFOS. The requirements for the program vary depending on the test setup, so that various operations have been implemented.

3.2 Post-processing approach

The application and handling of DFOS is thoroughly discussed in literature, whereas the post-processing of the acquired data is only partly documented for individual methods (cf. Sect. 1). For this reason, the software solution FOS Evaluator has been developed to support and facilitate the application of fibre optic technology in the field of civil engineering by combining different post-processing functions in one place. The proposed workflow that can be achieved through the program FOS Evaluator is presented in Fig. 2.

Fig. 2
figure 2

Workflow of FOS Evaluator for post-processing data generated by DFOS

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The first column in the workflow diagram shows the general operations to process the recorded data, which comprise data selection, reduction, and smoothing. Depending on the desired result, the processed data can be used for various implemented calculation operations, which are listed in the right column of the diagram. Both main functions of FOS Evaluator, data operations and evaluation, are presented in the following sections.

3.2.1 General information

The software FOS Evaluator was developed in the context of tests carried out at the Institute of Structural Concrete (IMB) at the RWTH Aachen University that were instrumented with DFOS. As a measuring device an ODiSI (Optical Distributed Sensor Interrogator) 6108 using the Ryleigh backscattering principle was utilized [55]. To understand the post-processing method in depth, knowledge of the general properties of the measurement output file is necessary. The proprietary ODiSI software gathers the experimentally obtained data and enables an export as a tab-separated text file. This file contains a header with general information such as the name of the test, length of the sensor, software version, and measuring frequency. Below this, the optional gauges are located on the horizontal axis, followed by the position of the measuring points on the x-axis (length axis). The time since the test started is arranged vertically on the y-axis (index, time axis). For each entry in this matrix, a strain measurement is given.

3.2.2 Restoring data

During the measurement with DFOS, local failures and/or disturbances of the measuring fibre can lead to measurement errors, which are stored in the created file as “NaN” values.

Since these values can lead to complications while processing, various functions have been implemented to deal with them during import. The faulty values can be interpolated in different ways, replaced, or deleted. The influence of the different operations is shown in Fig. 3. While replacing the “NaN” values with zeros is the fastest solution, the generated course is no longer continuous. Apparently, the spline interpolation with third order polynomials generates the best result, but compared to the linear interpolation, it is more prone to errors since its applicability is dependent on the number of available data points. For this reason, the further investigations shown in this paper were carried out with the linear interpolation.

Fig. 3
figure 3

Various operations to interpolate missing data points (marked with × on the fibre)

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3.2.3 Data reduction

To accelerate the (post-)processing, it is beneficial to reduce the amount of data that is going to be processed. For this, several functions for data reduction have been integrated into the software. Two distinct parameters of the strain measurement can be targeted during data reduction—the temporal and the spatial dimension. The first is specified by the measurement frequency, the latter is given by the gauge length. Both properties can be adjusted before the measurement takes place and thus, it seems plausible to choose a lower frequency or a larger gauge length if the aim is to reduce the amount of measuring data. In some cases, this would be enough to gather the necessary information. Since most measuring devices are capable of high sample frequencies, whereas the gauge length can often only be adjusted to a few predefined options, it is preferable and sometimes unavoidable to use the maximum capacities of the equipment and filter for needed information afterwards. The presented software is designed for such a task and optimizes the workflow with large data sets.

The most important function handling high-frequent data is resampling, which can be used to change the frequency of data and thereby reduce its amount. This method is of great importance for long-term tests through which a huge amount of data is acquired. In the context of DFOS, the reduction of high frequencies is of particular interest. This can be achieved by combining measurement values over a certain time window and averaging them so that no information gets omitted. For example, by changing the measuring frequency from 20 to 2 Hz, the data can easily be reduced by 90%. This becomes especially relevant if the data is going to be used with external spreadsheet applications like Microsoft Excel. Additionally, the reduction process can speed up the following smoothing or calculation operations. Nevertheless, it has to be kept in mind that an excessive reduction of the frequency and thus, the amount of data, can negatively influence the significance of the measurement results. Especially when investigating high frequency loads in seismic or vibration tests, the data reduction can lead to a loss of information and should be handled with care.

An analogous function was implemented for the length axis of the strain measurement. This means the gauge length can be altered afterwards by combining and averaging a selectable amount of strain values over the length axis.

3.2.4 Smoothing

Fibre optic sensing is characterised by its high measurement sensitivity. The system used in the following investigations combined with a polyimide-coated fibre can sense approx. ± 1 µε. This kind of fibre has a high sensitivity compared to other fibres, e.g., acrylate-coated fibres. The reason for this is the coating of the fibre. With increasing thickness, the coating often experiences shear deformation, which leads to less recorded strain in the sensor core [7]. Furthermore, the sensitivity of the fibre is highly dependent on the application method, which influences the strain transfer [6]. Since concrete is an inhomogeneous material, the measurement quality of the fibre can vary depending on the position inside the specimen. Even small changes in the matrix (e.g., distribution of aggregates, microcracking) can have a great impact on the measured strain values. Due to the spatial variability and the high sensitivity of the components, the captured strain values can fluctuate over time/location and generate Strain-Reading Anomalies [53]. The additional interpolation of missing values during import can result in the original distribution not always being reproduced without errors. This can be a major obstacle for further evaluation and calculation. Thus, the second main aspect of FOS Evaluator focuses on the smoothing of strain measurements to facilitate further post-processing. The presented program contains various approaches for smoothing since the changing requirements depend on the intended use or calculations. The influence of said smoothing methods is shown in Fig. 4 with the respective smoothing parameters varied. For better visibility, the smoothing operations have only been applied to the length axis. For the time axis, the smoothing would lead to similar results. The underlying raw data was obtained from an anchorage test specimen as described in Sect. 5.1. The end anchorage of the reinforcement bars was defined as the length of the support lb = 8 cm. The fibre was applied to the side of one reinforcement bar with a measuring range extending over the support (cf. Fig. 4b). The smoothed data is the strain measured at the maximum test load and the resulting pull out of the reinforcement bar.

Fig. 4
figure 4

Overview of implemented smoothing methods and variation of controllable parameters: a LOWESS, variation of window size as a fraction of total number of data points fr; b detail of the anchorage on which the measurement results shown were recorded; c and d Moving average, variation of window size n and type (“simple” or “weighted”); e and f Splines, variation of evaluation points as a fraction of total number of data points k and variation of smoothing factor sf; g and h Savitzky-Golay filter, variation of window size n and polynomial degree p; i and j Bessel filter, variation of degree n and frequency f

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A robust method to smoothen scatter plots is LOWESS (Locally Weighted Scatterplot Smoothing) [56]. It is a non-parametric approach that does not require a prior assumption of a mathematical function to be approximated and is based on the principle of two-fold local regression. In the first step, a window size corresponding to a subset of the total measured data is defined. Then, step by step, a polynomial fit of degree p (here p = 1) is performed for each point using the Weighted Least Squares (WLS) method, taking into account the number of nearest neighbours (corresponding to the specified window size). The weights are distributed in such a way that the immediate neighbours to the studied point on the considered axis have a higher impact than more distant ones. Each data point is replaced by the approximated value. In the second step, a robust fit is performed on the already smoothed data points. Weights in the regression consider how far the initial data points lay from the already approximated function. Closer points are assigned a greater weight than highly divergent points so that the latter have little effect on the shape of the created curve. This process can be repeated several times to vary the amount of smoothing. The result is a curve that smoothens out outliers effectively (see Fig. 4a).

An additional way to smoothen out unwanted peaks is the moving average. This method has already been successfully implemented by multiple authors [6, 52]. The principle is calculating the moving average of a specified number of strain measurements over one of the two axes (time or length axis). Each individual entry is averaged over the definable window size (i.e., sum of left and right neighbours) and replaced by the obtained value. The program includes two variants of the moving average—the simple and the weighted moving average. In contrast to the simple variant, the weighted moving average assigns every value of the moving window a weight, which is the biggest for the value in the middle of each window. As a result, outliers have a higher impact compared to the simple moving average, so that the basic shape of the strain curve gets altered less (see Fig. 4b).

An interpolation with splines can be used to continuously connect measured data points and thus enable an evaluation of the function at arbitrary points. Instead of a single polynomial function of degree p = n − 1 (n = number of data points) to describe the strain curve, the function consists of polynomials defined in sections, which connect the individual data points (nodes). Continuity conditions guarantee a continuous connection of the individual sections at the nodes. A spline function of degree p must be (p − 1)-times continuously differentiable, so that the differentiability of the overall function is ensured. Common splines are polynomials of fifth (quintic splines) or third degree (cubic splines). Splines are used within the program for various purposes, i.e., various calculations (cf. Sect. 4). At this point, however, they are used to smoothen the data by evaluating the interpolated strain curve only at certain points or by reducing the number of nodes. The successful application of splines for the emulation and smoothing of strain curves has already been shown in [21]. Through this method, a great variability of smoothing can be achieved with individually suited results (see Fig. 4e, f). The variation of the polynomial degree of the spline is not depicted because the exact course of the strain distribution is reproduced either way.

A Savitzky-Golay filter is a smoothing filter based on polynomial regression [57]. Similar to a moving average, a variable subset of measurement points is used to perform a local fit of a polynomial function (degree p). The centre point of the evaluation window is then replaced by the smoothed value. This smoothing method is characterized by its closeness to the original course since high frequencies in the measurement course are often taken into account (see Fig. 4g, h).

Furthermore, two frequency filters are available in the program for data smoothing (Butterworth filter and Bessel filter). These filters are so-called low-pass filters, which filter out higher frequencies from a signal. Frequencies lower than the selectable cut-off frequency f remain unchanged, whereas frequencies above the cut-off frequency are only considered with decreasing intensity. The strength of the drop after the cut-off frequency is indicated by the degree n of the filter. Fluctuations in the course which deviate strongly from the basic course are thus filtered out. Regularly recurring frequencies, such as those caused by noise, can also be effectively filtered out (see Fig. 4i, j).

Smoothing of the strain data is essential for processing and calculating, but at the same time must be individually adjusted. While often useful, it is not always necessary to smoothen the generated data, especially if there is the possibility of falsifying the data. A crucial factor is the cracking of concrete, which generates peaks in the strain distribution analogous to those generated by anomalies of the measuring equipment itself. Factors that influence the quality of the generated data or the necessity of post-processing are the used fibre (e.g., polyimide-coated, acrylate-coated, or even reinforced), the gauge length, and the type of application (on the concrete or the reinforcement). Furthermore, it is important to plan the order of operations. For example, for crack detection, it would be counterproductive to smoothen out every peak, since relevant changes in measured strain could be misleadingly deleted instead of detected.

4 Calculations

The program can be used to calculate various mechanical quantities using the recorded strains (e.g., strain, stress, bond stress, change in length, slip). At this point, only the quantities that can be calculated directly from the recorded strain will be discussed. In addition, the calculations presented will be limited to strain measurements of the reinforcement. The slip, i.e., the relative displacement of the reinforcement and the concrete, cannot be recorded by one measuring fibre alone. For this reason, the evaluation is limited for the time being to a change in length of the material under investigation.

4.1 Stress

To calculate the stress σ(x), a linear-elastic material behaviour is assumed (Eq. 1), so that the stress curves of the reinforcement or the uncracked concrete can be determined by multiplying the strain ε with the young’s modulus E.

$$\upsigma \left(x\right)=E\cdot \varepsilon \left(x\right)$$
(1)

4.2 Bond stress

The bond stress can be derived from the longitudinal stress profile of reinforcement, by exploiting the stress gradient. The calculation itself must be divided into several steps for this purpose. By forming the equilibrium of forces on a differential rebar element, the relationship given in Eq. (2) can be derived between the stress gradient in the reinforcing bar and the bond stress τ(x).

$$\uptau \left(x\right)=\frac{\mathrm{d}{\upsigma }_{s}\left(x\right)}{\mathrm{d}x}\cdot \frac{{d}_{s}}{4}=\frac{\mathrm{d}{\upvarepsilon }_{s}\left(x\right)}{\mathrm{d}x}\cdot {E}_{s}\cdot \frac{{d}_{s}}{4}$$
(2)

Assuming a linear-elastic stress–strain relationship, the bond stress can be expressed as a function of the reinforcement strain (Eq. 2) (e.g., [58]). This relationship is relevant for the further steps of the calculations and especially necessary when describing the strains with mathematical functions. As described previously, the required strain function ε(x) is recorded with DFOS and subsequently converted into the stress function σ(x) (Eq. 1). The derivative of the stress can then be done numerically or analytically by previous approximation with a function. In the present case, splines are used for the computer-aided calculation in order to reproduce the distribution of the stress. The continuity conditions and the differentiability of the splines allow a simple calculation of the derivative. Subsequently, the course can be multiplied by the constant ds/4 to obtain the bond stress τ(x).

4.3 Elongation

Furthermore, changes in length of a material can be calculated by integrating the strains over a given interval from a to b (Eq. 3). This can be used e.g., for assessing crack openings.

$$\Delta l\left(b\right)-\Delta l\left(a\right)=\underset{a}{\overset{b}{\int }}\varepsilon \left(x\right)\mathrm{d}x$$
(3)

Additionally, if the elongation of the surrounding concrete is measured as well, the slip between reinforcement and concrete can be calculated (Eq. 4).

$$s\left(x\right)={\Delta l}_{s}\left(x\right)-{\Delta l}_{c}\left(x\right)$$
(4)

Since the bond stress (Eq. 3) and the slip between reinforcement and concrete (Eq. 4) can be obtained theoretically by using DFOS, by knowing the initial slip (or translative movement of the reinforcement bar) a bond stress-slip relationship can be established.

4.4 Crack detection

In the current version of the program, crack detection is possible by finding local maxima in the measured concrete strain distribution that exceed the maximum tensile strain of the concrete. Alternatively, the cracking of the concrete can be monitored by defining a strain threshold ∆ε. ∆ε can be obtained by subtracting successive strain values for every given time step. If ∆ε exceeds the defined threshold, a crack has occurred. ∆ε can be associated to the released energy. Then, the crack opening can be determined by integrating the incremental strain profile over the influenced area (cf. Eq. 3) [9].

4.5 Export

To complement processing, the edited and/or the newly calculated data can be exported. On the one hand, the data object used within the program can be output, so the size of the file is significantly smaller compared to the import and a considerably faster re-import is possible. On the other hand, there is the option to save the data as a Microsoft Excel file or as comma-separated values to enable further external processing.

5 Examples of application

To present the effectiveness and versatility of the software solution, the next section will present test results that were recorded with the fibre optic sensor system and post-processed with the methods presented above using FOS Evaluator. For an in-depth analysis of its implemented features, the approach was applied to various experimental regimes examining various materials and important aspects of concrete structures. In the following, each of these studies is presented and discussed.

5.1 Anchorages

5.1.1 Test setup and instrumentation

The first campaign focused on the bond behaviour of steel-reinforced concrete. To test the anchorage of ribbed reinforcing bars at the support, three-point bending tests were conducted. For this purpose, three beams were produced, each consisting of two partial tests. To specifically test the intended anchorage area, one end of the beam rested on the outer support, while the other end with a length of approximately one-third of the beam extended beyond the second support as an unloaded cantilever (cf. Fig. 5). The longitudinal reinforcement consisted of four B500 bars with a diameter of 16 mm, two of which were instrumented with DFOS on one side. A double application of the fibres as described in Sect. 2 was omitted, since the curvature of the reinforcement was not the object of investigation. The bars were positioned beyond the support and the bond was neutralized by a small piece of PVC tubing. The anchorage length was defined by the first crack in front of the support according to EC2 [59]. The measuring fibre was applied into a milled groove in longitudinal direction of the rebar (analogous to [10, 18, 36]) and was positioned in the anchorage area (cf. Fig. 5).

Fig. 5
figure 5

Experimental setup and dimensions of anchorage test instrumented with DFOS

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One of the investigated beams will be presented in the following. The properties and the experimental results of can be found in Table 1. After reaching the maximum force, a pull-out failure of the anchored reinforcement was observed.

Table 1 Properties and experimental results of anchorage test
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5.1.2 DFOS results and processing

During testing, the strain of the anchored reinforcement was monitored via DFOS (cf. Fig. 5). In the following, the measuring results of the sensor applied to the inner reinforcement bar are discussed. In Fig. 6, the data of the anchored reinforcement bar is shown at various loading levels (0.33Fmax, 0.67Fmax and Fmax). The raw data was imported into the software in order to perform the multiple processing steps shown in Sect. 3. The raw data shows several peaks in the strain measurement caused by outer influences (e.g., micro cracking or inhomogeneous material properties). The length of the whole fibre was 193 cm. DFOS was set up to a measuring frequency of 20 Hz and a gauge length of 0.65 mm. The experiment lasted approx. 30 min, so that the raw data comprises 26,018 rows (time axis) and 2839 columns (length axis) resulting in over 73 million values. In the first step, the measured data had to be reduced. Already during import, irrelevant sections for the investigation of only the anchorage area, i.e., the fibre towards the specimen and the protruding fibre, can be cut, which makes roughly 95% of the data (roughly 3.2 million values remaining). After decreasing the measurement frequency to 2 Hz by using the downsampling method described in Sect. 3.2.3, the data can be reduced further to approx. 300,000 values without losing a significant amount of information. Following to that, the data is smoothed. In this example, a Bessel filter is used in varying intensities (i.e., changed threshold frequencies with constant degree, n = 5 and f = 0.1 rad/s Fig. 6c, n = 5 and f = 0.05 rad/s Fig. 6d).

Fig. 6
figure 6

Post-processing of DFOS data acquired in the anchorage test: a imported raw data; b selected and reduced data; c and d smoothed data after applying a Bessel filter; e calculated reinforcement stress; f calculated bond stress

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Due to the reduced data and the smoothed profile, the calculations can be performed faster. First, the reinforcement stress can be calculated using a linear elastic material law (cf. Sect. 4.1). Afterwards, the calculation of the bond stress using the relations shown in Sect. 4.2 can be carried out. For the calculation of the bond stress, a more severely smoothed strain distribution (f = 0.05 rad/s compared to f = 0.1 rad/s) must be used, otherwise the derivative will produce high fluctuations (Fig. 6c and d). The intensity of the necessary smoothing highly depends on the application and the quality of the measuring data.

The direction of the local coordinate system determines the sign of the derivative and hence the bond stress. For the shown calculations, the unloaded end of the reinforcement bar depicts the starting point, so that the increase of steel stress from left to right results in a positive bond stress. During import, the program offers the possibility to inverse the x-axis of the imported strain data to always enable the possibility of calculating the bond stress starting from the unloaded end of the reinforcement bar. The bond stress distribution of the anchorage area becomes visible for the different loading levels. For smaller loads, most of the bond stress is evenly distributed over the anchorage and relocates to the back, i.e., the unloaded end, as the load increases.

5.2 Lap splice

5.2.1 Test setup and instrumentation

The second campaign addressed the bond behaviour of lap splices in steel-reinforced concrete. To determine the load-bearing capacity of lap splices under flexural loading, six beams were made of normal-strength concrete and tested as single-span girders. In order to load the lap joints with pure tension from flexural bending tension, four-point bending tests were carried out, so that the centre of the span was under constant bending moment. The test setup for specimen B4 is shown in Fig. 7. As for the anchorage test series, the specimens were reinforced with four bars, each with a diameter of 16 mm. In the middle of the investigated beam, all bars were lapped over a length of 400 mm. One half of the lapped bars was instrumented with DFOS (Fig. 7).

Fig. 7
figure 7

Experimental setup and dimensions of lap splice test (specimen B4) instrumented with DFOS

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Therefore, the sensors were applied in a milled groove into the rebar. The properties and the experimental results of B4 are shown in Table 2. After reaching the maximum force, the lap splice failed by pull out of the reinforcement bars.

Table 2 Properties and experimental results of lap splice test B4
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5.2.2 DFOS results and processing

The evaluation of the measured strain values of two instrumented reinforcement bars (S3, S4) followed the methodology of the anchorage tests shown in Sect. 5.1. Comparable to the anchorage tests, a significant proportion of the fibre length can be disregarded during the import. The data reduction takes place in the same way (cutting off irrelevant data and downsampling). For this reason, in Fig. 8a and b, these steps are combined. In contrast to the anchorage area, two local coordinate systems can be implemented because the stress distributions of one pair of reinforcement bars are opposed. For better visibility, in Fig. 8, the illustrated strain and stress of the lapped bars are mirrored (marked with arrows S3 and S4 in the middle). In the raw data, several disturbances can be observed. Especially the end region of the fibre, the termination, causes significant outliers. After smoothing the strain measurements, the steel and bond stress can be calculated respectively. As smoothing method, the previously used Bessel filter does not produce satisfactory results. A cut-off frequency of 0.05 rad/s is not able to smoothen out the outliers and returns a distorted course (Fig. 8c, d). When the cut-off frequency is reduced to 0.01 rad/s the disturbances can be reduced but still not eliminated. Additionally, small variations get levelled out and falsify the course. For that reason, LOWESS is used to reliably smoothen out disturbances. Still, the then calculated bond stress shows high fluctuations. Compared to the anchorage, these appear more clearly due to the wider investigated area (40 cm vs. 8 cm).

Fig. 8
figure 8

Post-processing of DFOS data acquired in the lap splice test: a and b imported, selected, and reduced data; c and d smoothed data and calculated reinforcement stress; e and f smoothed data and calculated bond stress

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Multiple areas can be identified in the stress distributions of the spliced reinforcing bars. In the last section (over approx. 15 cm from the splice end), the steel stress is linear even at low loads. In the middle section, two rather constant stress domains can be seen for both bars at loads below the maximum load. At this point, the area is not fully involved in load transfer. Close to the beginning of the lap (along approx. 10 cm), the stress drops more steeply than in the other areas. When reaching the maximum load, the stress distribution approaches a linear course, which is in accordance with other research, e.g., [60,61,62,63,64]. Having this in mind, it seems reasonable to reproduce the stress distribution with a linear function. Therefore, two separate functions for an automated linear regression were implemented into FOS Evaluator (Fig. 9).

Fig. 9
figure 9

a Linear approximation of the stress distribution of a spliced reinforcement bar using the raw data or LOWESS smoothed data; b calculated bond stress using the linear approximated or smoothed data

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The first function (‘raw’) tries to approximate the raw data directly including the disturbances that prevent an adequate replication of the basic stress distribution. The other function uses LOWESS in a preceding step to remove outliers and thus making the linear regression more accurate and less prone to disturbances. Hence, the function is called ‘robust’ linear regression.

In this example, LOWESS with a fraction parameter f = 0.1 is used, which can be individually adjusted to generate the desired result. By approximating the stress distribution with a linear function, the bond stress can be easily calculated and is constant over the length of the lap splice. The results generated in this manner, shown in Fig. 9, are in high accordance with the bond stress calculated using the smoothed stress. For the shown bar S4, the average bond stress τavg results in 4.5 MPa.

This method of approximating strain or stress distributions was extended to several types of functions, e.g., sigmoid or parabolic function, to be applicable to different circumstances, e.g., the anchorages shown in Sect. 5.1. Since these functions are beyond the scope of this article, they are not presented in the following.

5.3 Crack detection and crack opening calculation

5.3.1 Test setup and instrumentation

The last presented application example deals with crack detection and calculation of crack openings in concrete members reinforced with carbon fibre reinforced polymer (CFRP)—although the general principle is applicable for all kinds of reinforcement.

In a comprehensive experimental campaign investigating the influence of tensile normal forces on the shear capacity of textile-reinforced concrete (TRC) slab segments [9, 65, 66], the presented FOS system was utilized for an internal measurement of the reinforcement strain (Fig. 10).

Fig. 10
figure 10

Experimental setup and dimensions of TRC slab strip  instrumented with DFOS

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The one-way slab strips (l/w/h = 800 mm/190 mm/70 mm) were reinforced with a bidirectional warp-knitted grid made of epoxy resin-impregnated carbon fibre yarns and manufactured from a high strength self-compacting concrete with a maximum crushed grain size of 4 mm.

Further information about the material characteristics of concrete and reinforcement can be found in Table 3 and [67].

Table 3 Properties and experimental results of TRC slab
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The loading scenario comprised two phases: In phase 1, the axial load was gradually increased until a saturated crack pattern was formed (Nmax = 75 kN, which corresponds to roughly 30% of fnm). Subsequently, the specimen was completely unloaded. In the second phase, the slab segments were loaded to a predefined normal force and subsequently tested in single span three-point bending test setup until failure. In the scope of this paper, only the measurement data of phase 1 will be used for an exemplary utilization of the program FOS Evaluator.

The investigation of several application methods in [9] showed that the direct attachment of the polyimide-coated measuring fibre to the textile reinforcement via cyanoacrylate adhesive (cf. Fig. 10) provides the best possible results. Thus, the following evaluation is tested on a slab segment where the measuring fibre was applied according to this method.

To reduce the effort required to manufacture the measuring fibres, but increase the measurement results obtained, only one fibre was attached to three yarns of the textile reinforcement in a winding path for each reinforcement layer (cf. Fig. 10). The measurements were taken at a frequency of 20 Hz with a gauge length of 0.65 mm.

5.3.2 DFOS results and processing

The axial tensional loading N induces separation cracks in the slab segment, which lead to a significant increase in local strain in the reinforcement. Using this local strain difference and its characteristic shape, information about anchorage length, crack position, and crack opening can be derived. Figure 11 illustrates the post-processing steps, which are necessary for this derivation. Here, the horizontal axes represent the span between the supports Lspan = 44 cm (cf. Fig. 10).

Fig. 11
figure 11

Using DFOS data acquired in the TRC slab segments to calculate crack openings: a and b strain profile of successive timepoints; c calculated strain difference between timepoints; d smoothed strain profile; e overview of all detected cracks and position compared to observed crack location; e calculated crack width with DFOS compared to manual measurement

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Figure 11a shows the strain profile of the reinforcement immediately before the formation of the fifth separation crack (clearly visible due to the strain gap between 22 and 34 cm). A few milliseconds later, this gap in the strain profile is closed (Fig. 11b). Subtracting these two strain states and applying a Bessel filter (n = 5 and f = 0.1 rad/s), yields the incremental strain profile of crack 5 (Fig. 11c, d).

Repeating this procedure for all seven cracks leads to Fig. 11e. The dashed dark grey lines represent the visible cracks on the upper side of the specimen, which coincide quite accurately with the strain profile maxima. Furthermore, first statements about the bond between reinforcement and concrete become possible. On average the reinforcement needs a length about 6 to 7 cm to re-introduce its stress into the concrete (crack-spacing). The approximately constant slope of the strain profile in its ascending and descending sections indicates a constant bond force. Finally, even the crack opening can be determined using DFOS measurement data alone. To this end, a variety of approaches have been developed in recent years (e.g., [9, 17, 52, 68]). If solely the crack opening at crack initiation is of interest, very accurate results can be achieved using the differential strain profiles [9]. Assuming that the contribution of the concrete (tension stiffening) is negligible, the crack opening can be determined by integrating the smoothed strain profile (Eq. 3). Figure 11f shows a comparison of the calculated crack openings with manually measured ones, highlighting the achievable accuracy. This (semi-)automatic routine is integrated into FOS Evaluator and thusly can facilitate or at least complement conventional measuring of cracks in reinforced concrete.

6 Conclusions

Distributed fibre optical sensing is a great way to gain insights into structures and their mechanical behaviour. The measuring technology combines high precision or sensibility with a narrow arrangement of measuring points, hence the given name ‘quasi-continuous’ or ‘distributed’. Besides the many advantageous properties of this technology, the quality of the measurement results highly depends on the correct application and adequate processing of the collected data. Even if the fibre is applied accurately, it is still susceptible to interference and distortions due to its spatial variability and high sensitivity. In this context, the correct post-processing of the data is all the more important. In this article, a software solution is presented that processes strain data measured via DFOS in an automated way to generate a foundation for a better evaluation and hence a more deeply understanding of mechanical properties. The program FOS Evaluator comprises functions for data selection, data reduction, and data smoothing. In that way, distorted data can be cleaned and the great amount of data that accumulates due to the high measurement frequency can be reduced to a manageable amount. The program facilitates these operations and generates a suitable workflow for various applications in civil engineering. Additionally, the processed data can be used for further calculations via built-in functions. Multiple mechanical properties such as stress, bond stress, or elongation of the measured material can be obtained. Moreover, the detection of cracks and their opening immediately after formation. The functionality of FOS Evaluator was proven by using it in the context of three individual experimental setups. It could be shown that the data operations are capable of reducing and smoothing the data in a way that further investigations can be carried out more effectively. Using different materials as reinforcement (steel or CFRP) demonstrated the universal applicability of the sensing technology itself and the usability of FOS Evaluator. It is conceivable to extend the scope of functions to be applicable to multiple areas and experimental setups (e.g., Structural Health Monitoring) by further developing the software. The development of the program is an ongoing process which comprises the optimisation of already existing functionality and the adaption of functions for the evaluation of new experimental and theoretical input. In the future, FOS Evaluator will be published to enable public access.