Probabilistic modeling of reinforcing bar fatigue on the basis of surface topography data

The fatigue of reinforcing bars—as one example of reinforcements in general— is strongly influenced by their surface properties. A lot of research in the 1960s and 1970s has led to the formulation of an analytical model that connects notch stresses to the resulting load cycles in fatigue tests. This article presents the work undertaken in a DFG (German Research Foundation) research project, in which the procedure for assessing the surface properties of rebars with a high‐precision laser‐linescan system (LLS‐system) and subsequently performed fatigue tests were developed. An analytical model was taken as a basis in order to quantify the input parameters and derive the corresponding notch stress factors. The concluding modeling took the scatter into account and allowed for a safety‐based recommendation for the derived notch stress factors of reinforcing bars.


| INTRODUCTION
The fatigue of reinforcing bars (rebars) depends on three major factors: testing conditions, production-related phenomena, and surface characteristics. Testing conditions are defined by stress levels, amplitude, frequency, and temperature. Whereas amplitude and stress levels are set by standards for testing fatigue behavior (e.g., DIN 488-1 1 ), test frequency and temperatures are regulated by DIN EN ISO 15630-1. 2 Concerning production-related phenomena microstructural setup, double skins, grooving, liquation, and especially residual stresses need to be addressed. Most of these phenomena can be minimized by controlling roller ages, cooling temperatures, or melt composition. It is well known, that the surface geometry has the most significant impact on fatigue behavior of rebars. Martin and Schießl 3 showed that fatigue strength was reduced by approximately 60% when ribs were introduced on rebars in the 1960s. More researches, for example, [4][5][6][7][8][9][10][11] examined the effect of ribs on the fatigue strength of rebars and came to the conclusion that the size of the ribs and the design of transition from rib valley to the rib incline-the notch/fillet-are decisive. These findings state that fatigue failure-when not originated by surface phenomena-is caused by deviation of tensile stress trajectories into the rib and the increase of notch stress at the foot joint of the ribs. MacGregor et al. 5 and Jhamb 6 introduced the stress concentration factor k T (−) for rebars, Equation 1: Discussion on this paper must be submitted within two months of the print publication. The discussion will then be published in print, along with the authors' closure, if any, approximately nine months after the print publication.
where σ max (N/mm 2 ) is the maximum or peak stress at the notch and σ nom (N/mm 2 ) is the nominal stress in the rebar cross section. Further, the crucial geometrical parameters influencing the stress concentration factor were identified by computer simulations in Jhamb 6 which were incorporated in an analytical relationship by Schießl, 12 Equation 2: where r (mm) is the fillet of the notch, b K (mm) is the width of the rib top, a (mm) is the height of the rib, and α ( ) is the angle of the rib slope. It should be noted, that Equation 2 is not consistent considering the units. By definition, all of those geometrical parameters lie in direction of tensile stress trajectories (rebar length axis) when loaded. Figure 1 illustrates the localization of the geometrical parameters in a side view of a rib.
The measurement of geometric parameters and further fatigue experiments in Reference 12 allowed deriving a connection between the born load cycles and the stress concentration factor, Equation 3.
where N (−) is the number of load cycles born by the specimen, f 0 (−) is the regression parameter for the influence of loading regime and material properties as well as f 1 (−) which is a regression parameter for the influence of the stress concentration factor originating from the geometrical situation at the failure localization. Schießl 12 stated that values for f 1 would lie between 0.8 and 1.5. The current paper presents and discusses the experimental investigations involving the surface measurements by laser triangulation using the developed laser-based linescan system (LLS-system) 13 and the corresponding fatigue tests. The understanding of the impact of surface properties on the fatigue behavior of rebars under laboratory testing conditions was the motivation of the presented test series. Thus, the aim of the paper is the quantification of the input parameters for the surface-based analytical equations presented above. All the work was carried out in the course of a DFG (German Research Foundation) research project.

| General
A suitable strategy for a quantification of all input parameters in the notch stress factor equation (Equation 2) and the regression parameters to relate to the fatigue behavior of B500 B rebars (Equation 3) were developed. In order to obtain a widespread and representative data basis, a total of 196 specimens with diameters 16 and 28 mm were investigated, compare Figure 2. A set of specimens was tested for each diameter in order to characterize their mechanical strength. A universal tensile testing machine (ZWICK Roell Z600, Figure 3 left) was used for these tests. The results are given for material characterization in Table 1. The chemical composition of the rebars was tested by glow-discharge optical emission spectroscopy. The given compositions allow for the derivation of the carbon equivalent value of CEV = 0.435 for diameter 16 mm and CEV = 0.444 for diameter 28 mm. The chemical composition is given in Table 2. Both, the mechanical results and the chemical composition are in good agreement with the standards 1 requirements.
The specimens were scanned by the LLS-system resulting in a set of line scan ASCII files containing the longitudinal coordinate X and the corresponding measurement result Y (distance from rebar center to surface). The LLS-system moves a Class 2 laser alongside the rebar length axis. After finishing one line-scan, the LLS-system rotates the rebar repeating the scan process and creating the next line-scan file. The number of the line-scan files represents the rotational position of the scan (rotational angle γ). The LLS-system is capable of measuring with an accuracy of ±1.6 μm in Y-direction, whereas all other F I G U R E 2 Rebar specimens for fatigue testing dimensions (position X and rotational angle γ) are considered exact due to the high quality of the selected positioning stepper motors. 13 For a further processing, the raw data needed to be mathematically processed (compare next section). In order to minimize the calculation effort, a suitable measurement strategy was developed, Figure 4.
Primarily, the rebar specimens were prepared by marking on the surface for analyzing the scan files on the digital twin. One hole was center-punched on top of a rib to mark the point of origin (starting point) for each rebar scan. The bottom of the center punch was calibrated before a full surface scan of the rebar was executed. This procedure allowed for a localization of surface phenomena in the scan files. As the punched hole is located on top of the rib with a depth of approximately 100 μm, a notch stress problem and further an impact on the fatigue behavior of the rebar when tested afterward could be excluded.
After scanning, the rebars were tested in an uniaxial tensile-fatigue test until failure. The machine applied in the fatigue tests was a resonance testing machine RUMUL Vibroforte (Figure 3 right). The boundaries of DIN EN ISO 15630-1 2 and DIN 488-1 1 with an upper stress level σ o = 300 N/mm 2 (60% of yield strength R e ) and frequency f ≤ 200 Hz were complied. The test frequencies lay between 74 and 89 Hz in dependence of resonance of the specimen and clamping effect. The fatigue results were sorted into "valid" and "invalid" results by the means of usability for the task of quantifying the input values of presented formulae (Equations 2 and 3). Considering valid results, the failures needed to be subcategorized into causes, such as geometry and surface defects-triggered failures. The invalid results were runouts or specimens that failed in proximity to the testing machines bearing. The specimens with identified geometry-triggered failures were used to determine the rib geometry parameters. Therefore, the origin of failure was located in relation to the starting point (center-punch). The localization was transferred to the digital twin that was produced by the LLS-system. Once located, the surface parameters (a, α, b K , r) for the rib geometry as it was before testing could be determined as shown in the next section. The stress concentration factor k T was calculated on the basis of these surface parameters. Finally, all existing values were categorized in results of the same stress range. Thus, k T and N were evaluated for the same stress ranges.

| Mathematical processing of LLSsystem data
The surface parameters were determined based on line-scan files, such as presented in Figure 1. Calculation of rib height a is defined in DIN EN ISO 15630-1. 2 Therein, a equals the difference between the maximum value Y max on the rib top scanned in a line-scan and the minimum value Y min of the associating rib trough. This can easily be adapted on the line-scan results. For the calculation of width of the rib top b K , the rib sections (flanks and top) must be identified first. This is done by fitting linear equations to the scanned rib sections (top, left, and right flank) as shown in Figure 5.
The linear functions for the left rib flank g lf (X) and the rib top g rt (X) intersect in Point A. The intersection of g rt (X) and the function of the right flank g rf (X) are located at Point B. Once the coordinates of the intersections A and B are determined, their horizontal distance can be considered as width of rib top b K . The angles of the rib slopes α lf and α rf (for definition see Figure 1   as regression parameters. The coordinates X and Y are taken from the scan results of the notch and implemented in an iteration algorithm that solves for X M and Y M as well as the desired radius r. 3 | RESULTS Figure 6 shows an overview of all fatigue results in the research project. In addition to the absolute numbers of test results the relative figures are given. The dominant result in the fatigue tests is the rupture/failure of the specimens in proximity to the fixation of the testing machine. It is well known that the triaxial tension situation at the fixation leads to local overstressing of rebars and consequently to failure. The production of rebars might lead to defects on the rebar surface (compare No. 2 in Figure 7) which increase notch stresses. If it is sufficiently high, the notch stresses can initiate micro cracking and a propagating fatigue failure. Figure 6 shows that surface defects played a rather minor role for the investigated rebars. It must be stated, that the percentage can be significantly higher depending on the production parameters of the rebars, for example, the surface of decoiled and straightened reinforcing steels has more defects and might alter the proportions of these findings. Despite extensive testing of up to 20 Á 10 6 load cycles, 9% of all specimens did not fail.
The remaining results failed due to the impact of the surface geometry (compare No. 1 in Figure 7: foot point of a rib). In comparison to the other results it was the second most important failure source in laboratory testing. Considering reinforced concrete, the problem of fixation is insignificant in situ. With regard to the quantification of modeling parameters in k T , f 0 , and f 1 in Equations 2 and 3 these results can be considered as valid. Therefore, geometrical parameters, stress range 2 Á σ a and the load cycles N were taken into account. Primarily the stress concentration factors k T were calculated. The results are given as a cumulated frequency plot in Figure 8. The cumulated frequency can also be understood as probability of failure caused by geometry p f,geom .
The calculated mean value for the notch stress factor k T in the results was 1.69 accompanied by a minimum value of 1.38 and a maximum value of 2.11. Further, the regression factor f 0 was quantified depending on the testing parameter stress range Figure 9 (left). Finally, the regression factor f 1 , also called weighing factor of k T , is quantified, compare Figure 9 (right). It was stated by Schießl 12 that f 0 is influenced by material and load regime. As the investigated material is of the same quality of rebar (B500 B), the observed variation must be caused by the variation of the stress range. The results of quantification show good accordance between the linear relationship of f 0 and stress range. With an increase in stress range a slight decrease in f 0 can be found. The regression factor f 1 can best be described by a normal distribution (ND). Different distribution types have been investigated (e.g., logarithmic-ND, Weibull, Gumbel, etc.). The ND was calculated to fit best for all input factors of the modeling function in Equation 3. The quantification of the linear function can be found in Table 3. The results for f 1 lie out of the range that was estimated in Reference 12, which were only based on a few test results of no longer stateof-the-art rebars.

| Modeling
The full probabilistic modeling was processed by implementing f 0 and f 1 in Equation 3. As result, the reachable load cycles N max were calculated. By implementing f 0 and f 1 as scattering parameters, N max scattered as well. Figure 10 shows the results of probabilistic modeling. Herein, the reachable load cycles N max are plotted over the notch stress factor k T . The black lines represent the results for a stress range of 175 N/mm 2 and the red ones for a stress range of 200 N/mm 2 . The different shapes of lines depict various failure probabilities in the uniaxial fatigue test.
It can be seen that an increase in k T results in a decrease in reachable load cycles. The 5% failure probability lines are insignificantly influenced by the notch stress factor. For a notch stress free condition (k T = 1.0) load cycles N max of approximately 725,000 (for 2 Á σ a = 175 N/ mm 2 ) or 425,000 (for 2 Á σ a = 200 N/mm 2 ) were reached, respectively. These reachable load cycles seem to be quite low. It must be taken into account that a direct comparison to the requirements of standards, for example, DIN 488-1 1 knee point at N = 1,000,000 with 2 Á σ a = 175 N/mm 2 (p f = 5% with W = 0.75) is not possible, due to the fact that the presented results only deal with a fraction of all specimens investigated-leaving out those that broke in the fixation and those without failure (compare Figure 6: p occ, geom = 31%). For a better understanding, the parameter study was evaluated for the statistical distribution of notch  Figure 7 and given in Table 1) stress factor at N max = 1,000,000 (knee point of the design Wöhler curve) and 2,000,000 (run-out criteria) for the stress ranges of 2 Á σ a = 175 and 200 N/mm 2 . Figure 11 (left) shows the results of this evaluation. In addition, the range of k T which was observed in this research is highlighted. For comparison, the load cycles of all specimens which failed due to the surface geometry at a stress range in experiments of 2 Á σ a = 175 N/mm 2 are shown in Figure 11 (right). Following Figure 11 (left) the notch stress factors for all rebars investigated in this research can be connected to a modeled probability of failure between 18 and 35%. Naturally, the range of probability of failure (18-35%) can be associated to a range of N between 700,000 and 1,100,000 in the test results which were used to quantify the model parameters, in Figure 11 (right). For taking into account that the probability of occurrence for the failure caused by surface geometry is known, Equation 5 can be applied for calculating the total probability of failure.
where p f,tot is the total probability of failure, p f,geom is the probability of failure caused by geometry, and p occ,geom is the probability of occurrence of failure caused by geometry. Consequently, the total probability of failure of the investigated specimens lies between 6 and 11%. Concluding the test results and calculations it can be stated that the notch stress factor is a decisive parameter for judging the fatigue behavior of rebars. It can be seen from Figure 11 (left) that based on the 196 specimens investigated, a k T of less than 1.25 would result in a low probability of failure (p f,geom < 16% and p f,tot < 5%). The relationship between k T and the geometrical parameters is given in Equation 2. Therein, the definition of the geometrical surface parameters a, b k , and α can be taken from standards DIN 488-2. 14 The remaining parameter which is not defined by aforementioned standard is the decisive notch radius r. Implementing the desired value for k T = 1.25 and the recommended values for the surface geometrical parameters yields the minimum value for the notch radius r min . For a diameter 16 mm rebar a value of r min = 1.31 mm and for a diameter 28 mm rebar; r min = 2.20 mm should not be underrun.

| CONCLUSIONS AND OUTLOOK
The current paper reports about the research carried out in a DFG research project. One of the topics of abovementioned project was quantifying the impact of surface geometry on the fatigue behavior of rebars. The developed measurement strategy is presented as well as numerous measurement results. A quantification of surface geometry parameters and input parameters of an analytical model from the 1970s resulted into a detailed analysis for a high number of specimens. These results allow for the following conclusions: 1.Surface scan results of reinforcing steel bars allowed for the calculation of surface parameters needed for implementation in an existing analytical formula for the calculation of notch stress factors k T .
2.The fatigue behavior of the scanned specimens was investigated in uniaxial tensile load fatigue tests. The surface geometry by the means of the notch stress factor k T could be correlated with the fatigue test results (load cycles N) applying the analytical formula.
3.Probability of failure in fatigue tests was correlated to notch stress factors. A low probability of failure of 5% was associated with a maximum notch stress factor of 1.25. This maximum notch stress factor can be achieved by staying within the geometrical recommendations of F I G U R E 1 1 Probability of failure depending on notch stress factor for different stress ranges and reachable load cycles (left); Fatigue test results probability of failure over load cycles for a stress range of 175 N/mm 2 (right) standards and increasing the notch radius. For a diameter 16 mm the notch radius needs to exceed 1.31 mm and for a diameter 28 mm 2.20 mm.
For future application of the LLS-system it is important to implement the scan data into 3D-Finite Element Method (FEM) analysis as a digital twin to achieve a deeper understanding in the development of notch stresses. In particular, the 3D distribution of tensions in the area of the rebar surface must be investigated. Therefore, beside the surface properties different information about metallurgical and chemical composition as well as residual stress distributions need to be taken into account. Aim of such research could be for example, a reduction in the observed high scatter of fatigue tests by developing design recommendations for surface properties.