Critical distance-based fatigue life evaluation of blunt notch details in steel bridges

Orthotropic steel deck (OSD) is applied on many bridges around the world, but its drawback of being prone to fatigue has been a concern due to the damages observed. Fatigue cracks initiated from the blunt notch details in rib-to-floor beam connections have a different fatigue behavior compared to welded connections and deserve to be investigated further. To study the effect of notch radius on the fatigue performance of blunt notch detail, 27 specimens with different geometries were manufactured and tested with fatigue loads. The fatigue lives of specimens and their crack initiation locations were investigated, and the prediction on the fatigue life of notched specimens based on the theory of critical distance (TCD) was also conducted. For notched specimens made of S355, the predicted fatigue life using the TCD method is in good agreement with the experimental results. Also, it is found that the point method has higher accuracy compared to the line method and the use of one calibration curve of notched specimens could provide enough accuracy compared to the case when two calibration curves of notched specimens are used. By modifying the mathematical description of the stress distribution near the notch root, a new formulation of the critical distance-fatigue life relationship is here proposed and validated with the experimental results.


Introduction
Orthotropic Steel Deck (OSD) has been used more and more in the construction of new steel bridges, especially for long-span bridges, due to its multiple advantages such as high load-bearing capacity, lightweight, and rapid construction [1].On the other hand, however, OSD is prone to fatigue cracks due to high alternating stresses imposed by wheels [2].Since OSD is assembled from many components using welding methods [3], initial defects are likely to be introduced during the fabrication and fatigue damages occur at connection joints [4].In general, it is challenging to monitor the propagation of fatigue cracks at the initial stage when their dimensions are very small, and it develops rapidly once their length has reached a critical level.Therefore, fatigue research is of vital importance to this kind of structure.
In existing studies about the OSD system [2,5], the fatigue evaluation of welded joints has been the main focus.Any yet, cracks that initiate from base material at geometrically discontinues, such as the cutout (blunt notch detail), were also observed during the inspection of several bridges [5,6,12].In addition, the cracking mechanisms and enhancement techniques of such kind of fatigue crack have not been fully studied [7].According to Eurocode 3 [8], the fatigue performance of blunt notch detail is classified as FAT 71.In the nominal stress approach (NSA), equivalent normal stress σ eq and theoretical stress concentration factor K t are considered to account for different effects, such as geometry and multi-axial loading.Due to the complexity of calculating the nominal stress for the blunt notch detail in OSD, however, different notch shapes have distinguished stress concentration factors and critical sections, making it rather difficult to unify the definition of the nominal stress for different blunt notch details.As a result, the application of NSA on the fatigue evaluation of blunt notch details remains challenging [1].
Since the fatigue problem at the cutout is triggered by the geometrical discontinuity within a localized area, it is reasonable to recognize the cutout as a notch and then apply the existing fatigue evaluation method of notched components to the cutout.The fatigue strength of notched components has been investigated by several scholars in recent decades, and four evaluation methods, including nominal stress approaches, local stress-strain approaches, theory of critical distance (TCD), and weighting control parameters-based approaches, have been proposed [9].Among these approaches, TCD has received increasing attention because of its simplicity and accessibility.TCD was first introduced by Neuber [10] with the concept of averaging stress along a certain length from the notch root and was named the line method later by Taylor [11].Originally, it was mainly used to define effective stress for the fatigue assessment.To apply TCD in fatigue life prediction, Susmel and Taylor [13] investigated the fatigue lives of specimens made of En3B (a kind of mild steel used in structures subjected to light load) and explored the relationship between critical distance L and fatigue life N f .They found the formulation of the relationship can be expressed as a power form, i.e., L = A•N f B , where A and B are material constants related to material and load ratio.The parameters in this relationship can be determined using two calibration curves, which are the fatigue curves of the smooth and notched specimens respectively.Thereafter, many scholars made efforts on developing factors that could improve the accuracy of the calculation of critical distance.Yamashita et al. [14] studied the fatigue performance of notched specimens made of Ti-6Al-4 V (a titanium alloy that exhibits high strength, low density, and good corrosion resistance) and found that higher accuracy will be achieved if the critical distance is calibrated by two notched fatigue failure curves with different root radii.According to Susmel and Taylor [13], the critical length is only affected by the material and load ratio.However, by analyzing the fatigue data of Al 2024-T351, Zhu et al. [18] found that the notch size also had effects on critical distance, and they proposed a new relationship between L and N f considering the notch size.Moreover, by examining the same fatigue data of Al 2024-T351, He et al. [19] proposed a new methodology that combines TCD and highly stressed volume (HSV) with consideration of Weibull distribution to perform a probabilistic fatigue assessment.Li et al. [17] combined TCD and HSV in the fatigue life evaluation of S355 components and proposed a determination model for the characteristic lengths of different notch configurations.
Despite several attempts focusing on the application of TCD on metallic materials, limited work has been done on structural steels that are often used for bridges, such as S355 in Eurocodes.In addition, many studies are focused on rounded notches with very small radii [18] and sharp notches [20], which are rarely used in OSD bridges.On this background, fatigue tests on specimens with three different geometries extracted from real bridges were carried out in this study, and the feasibility to predict the fatigue life of notched specimens made of S355 using TCD was investigated.In this study, a new formulation was also proposed by comparing the difference between the notches in the present tests and that used in the existing literature.

Specimen design and manufacture
In most OSD bridges, as shown in Fig. 1, blunt notch details are set in the floor beams or diaphragms where longitudinal ribs go through to reduce the restriction on the out-of-plane bending of floor beams or diaphragms.Due to its complex load transfer path, the rib-to-floor beam connection is prone to fatigue in both welded joints and base material.Generally, the fatigue cracks in base material always initiate from the geometrical discontinuities.For OSD, the blunt notch detail is a typical fatigue critical location.As shown in Fig. 2, the initiation of fatigue cracks at the blunt notch detail is mainly due to large maximum principal stress [12] and the numerically simulated location of the maximum principal stress under the most unfavorable load condition agrees well with the real initiation location of the fatigue crack.According to the results reported by Chen et al. [12], nearly half of the observed cracks have an inclination angle (α in Fig. 2 (b)) of 30 • .To reproduce such fatigue cracks in the fatigue test, the maximum principal stress in the critical location of the specimen should be perpendicular to the real fatigue crack propagation path.Another principle for designing the specimen is that the crack initiation location of the fatigue crack, i.e., the notch root of the specimen, should be similar to that observed from real bridges.Besides, to study the effect of the notch radius on the fatigue performance of blunt notch detail, different radii are considered in these tests, and all the notch shapes are extracted from real bridges and half scaled to fit the testing capacities of the fatigue loading machine.
Based on the aforementioned principles, three dog bone-like specimens are designed according to the observed fatigue cracks in real bridges [1,5,6,12], as shown in Fig. 3, to reproduce the most unfavorable stress state near the notch root.For the sake of convenience, specimens with a radius of 35 mm, 20 mm, and 10 mm are designated as large radius (LR), medium radius (MR), and small radius (SR), respectively.The notch shapes were processed by the waterjet cutting method.The typical surface roughness on the waterjet cutting edge is shown in Fig. 4, and the detailed measured results are also summarized in Table 1.
Table 2 shows the summary of the specimens in this study, and all specimens were loaded with a stress ratio of R = 0.1.To obtain reliable results of the fatigue performance, at least three stress levels are imposed on each series and every stress level is repeated up to three times, and specimens for each series are referred to as LR (or MR, SR) -1…9, respectively.The nominal stress range of a specimen is calculated by dividing the applied force range (ΔF) by the area (A) of the cross-section with the minimum area.
As one of the most often used steel grades in bridges in Europe [21], S355 is selected for the specimens in this study.The S355 was used as the majority (39%) in steel bridge construction in Sweden [22].To obtain the material properties of the employed steel, uniaxial tensile tests on three specimens are carried out, and the measured material properties of the steel plate are listed in Table 3.As shown in Fig. 5, some specimens were monitored with strain gauges (1-CLY41-3/120ZE) to register the strain evolution at the notch root.Strain is measured to monitor the stress state (elastic or plastic) of the steel and to confirm the fatigue crack initiation.It should be noted that the measured area of the strain gages is 3000 μm × 1200 μm, and the strain gages are installed at an approximate distance of 2 mm from   the notch root.Thus, the strains are averaged over this area, slightly below the notch.HBM MX410b universal amplifiers are used for strain gage signal conditioning using with the following parameters: Sample rate 2048 Hz and 1.0 V DC excitation to enable measurement up to 20,000 μstrain (or 2% strain).

Theory of critical distance 2.2.1. Basic concept of TCD
To account for the notch effect in fatigue evaluation of notched components and find a critical location as the reference point, the TCD applies the concept of local equalization to the vicinity of the notch within a specific dimension called the critical distance L, which can be determined by Eq. ( 1), where ΔK th and Δσ 0 are the threshold stress intensity factor and the range of plain fatigue limit, respectively.TCD aims to find an effective stress σ eff that can be used to predict the fatigue life of a notched specimen, and it is generally classified into four categories [9] (see Fig. 6), including point method (PM), line method (LM), area method (AM), and volume method (VM).The effective stress used in each method is illustrated in Eq. ( 2), ( 3), (4), and (5), respectively.Fig. 6 also illustrates the differences among these four methods.In general, PM and LM are easier to proceed with and they are mainly applied to cases of plane stress state [24].Since the stress state of specimens in this study is plane stress state, only PM and LM are discussed here.
where σ 1 is the principal stress, r, θ, and φ are the spherical coordinates, L is the critical distance.

Available models for critical distance calculation
Susmel and Taylor [13] investigated the relationship between critical distance L and fatigue life N f , and proposed to express the relationship in a power form, as shown in Eq. ( 6), where A and B are material parameters varying as material, load ratio, and selected TCD methods.To determine the values of A and B, two      calibration fatigue curves (see Fig. 7) are needed.First, an arbitrary fatigue life N i is specified, then the corresponding stress range Δσ s,i on the S-N curve of the smooth specimen and Δσ n,i on the S-N curve of the notched specimen can be found.Secondly, the linear-elastic stress distribution in the vicinity of the notch root under the nominal stress of Δσ n, i is obtained, as shown in Fig. 6 (a).According to different calculation methods of critical distance, find the critical distance L i where the corresponding effective stress range equals to Δσ s,i .Similarly, with the same processing procedure, any other critical distances corresponding to different N f can be determined.If two or more critical distance-fatigue life pairs are found, the relationship in Eq. ( 6) can be determined by fitting the curve for the given data.
Once the material parameters A and B are determined, the fatigue life can be predicted following the procedure shown in Fig. 8. First, an attempt value of fatigue life N f,1 is assumed, then the corresponding critical distance can be found by using Eq. ( 6).Secondly, based on the critical distance, the effective stress range can be determined from the linear elastic stress distribution in the vicinity of the notch root by using Eqs.( 2) and (3).Thirdly, according to the S-N curve of the smooth specimen, find the fatigue life N f,2 corresponding to the effective stress determined in the last step.If N f,2 equals N f,1 , N f,1 is determined as the estimated fatigue life.Otherwise, let N f,1 equal N f,2 and repeat the whole procedure, until N f,2 equals N f,1 .
Though material parameters A and B are generally believed to be irrelevant to notch geometry, the effects of notch geometry on critical distance have been investigated in recent years by several researchers, and different stress concentration factor-or geometry-related models were proposed to calculate the critical distance.Yang et al. [15] tested notched specimens made of DS Ni-based superalloy under low cycle fatigue regime and found the product of stress concentration factor K t and critical distance L has a high correlation with the fatigue life N f , as expressed in Eq. ( 7): Based on Yang's finding, Huang et al. [16] further extended the formulation of Eq. ( 7) and proposed that the product of L and K t to the power of m correlates with N f , as shown in Eq. (8).
Different from the K t modification method, Li et al. [17] found that the critical distance is related to equivalent notch factor K f,e and proposed Eq. ( 9), where D is a material parameter and K f,e is the equivalent notch factor that can be determined by finite element analysis.Zhu et al. [18] studied different radii of center hole plates made of Al 2024-T351.However, since K t is not sensitive to radius change, they attempted to use radius as a substitution for K t and proposed Eq. ( 10), where ρ is the radius of the center hole and ρ 0 is the reference hole radius.

Proposal of a new formulation
In Eurocode 3 [8], the relationship between the nominal stress range and the fatigue life is expressed by Eq. ( 11), where Δσ is the applied nominal stress range, Δσ C is the reference fatigue strength corresponding to 2 million cycles, and m is the slope of the S-N curve.For different geometries, the reference fatigue strength Δσ C and slope m change correspondingly.Hereafter, it is assumed that the reference fatigue strength and slope for smooth specimens and notched specimens are Δσ C,s and m s , Δσ C,n and m n , respectively, and these values can be determined from experimental data.Considering these two S-N curves, if an arbitrary fatigue life N i is assumed, the corresponding nominal stress ranges for smooth specimen and notched specimen are Δσ s,i and Δσ n,i , respectively (see Fig. 7).Then, the ratio of Δσ s,i to Δσ n,i is defined as K i , which denotes the stress amplification factor.It is well known that the notch acts as a stress raiser and the stress distribution near the notch root is nonlinear.To predict the fatigue life with the TCD, the stress distribution near the notch root is required.Generally, as the critical distances for some notches with high stress concentration factor (K t ) are very small, the interest range of distance from the notch root is small as well.In this case, the stress distribution can be approximately simplified as linear form, which also means the ratio between the maximum stress to the nominal stress, designated as K, can be expressed as: where k 1 is the slope of the normalized stress distribution near the notch root, and r is the distance from the notch root.When K equals K i , the corresponding distance r is exactly half of the critical distance L, as shown in Eq. ( 13), where C 1 and C 2 are two parameters depending on the material and load ratio.If Eq. ( 13) is transformed into another form, then it turns to be Eq.
Procedure of predicting the fatigue life with TCD.
R. Hao et al. (14), which clearly shows the power relationship between L and N f .
However, for some notches with relatively smaller stress concentration factors, the critical distance is larger, which means simplifying the stress distribution near the notch root as a linear form is not appropriate.In such cases, an exponential law (see Eq. ( 15)) is more accurate to depict the stress distribution: where k 2 is the slope of the normalized stress distribution near the notch root.Let K equal to K i , and Eq. ( 16) is derived: Taking the logarithm on both sides of Eq. ( 16), then Eq. ( 17) is obtained, and it clearly shows the logarithmic relationship between L and N f .Thus, a new formulation of L and N f is proposed.

Fatigue life
The log-log plot of nominal stress range and fatigue life is illustrated in Fig. 9.To compare the effect of different notches on fatigue life, the S-N curve of smooth specimens produced by Diekhoff et al. [23] is referred to and also plotted in Fig. 9.These specimens used the same material (S355) and were manufactured by the same cutting method as well.Moreover, the load ratio R for all those smooth specimens was 0.1 which is also the same as the fatigue tests performed in this study.Therefore, the data is a suitable reference for our current investigations.
According to Fig. 9, it can be found that and MR show similar performance in terms of fatigue life, while the SR series has a relatively low fatigue life compared to LR and MR series.This is presumably because SR has a more severe stress concentration due to a smaller notch root radius, which means when subjected to the same nominal stress range, SR would have higher maximum principal stress developed at the fatigue critical location, i.e., the notch root.Also, it is observed that the fitted S-N curves (survival probability of 50%) of notched specimens have considerably higher m values (denotes the slope of the S-N curve), between 13.4 and 29.7, compared to the standard FAT 125 curve (m = 3).This is probably because the compressive residual stress introduced by the waterjet cutting [23] delays the initiation of the fatigue crack.The conventional nominal stress approach cannot take these phenomena into account, as it represents the total fatigue life (crack initiation and growth) that is typically observed for the specific structure.Due to the random initial defects induced during the manufacturing process, fitted S-N curves show a low correlation between fatigue life and nominal stress range.The scatter range index 1: T σ = 1: (FAT 10% /FAT 90% ) of LR, MR, and SR series are 1:1.15,1:1.40, and 1:1.19, respectively.Since the number of tested specimens is limited, fitted curves with a survival probability of 50% are used in the subsequent discussion.
To classify the detail category of the blunt notch detail, the fatigue strength with a survival probability of 97.7% corresponding to 2 million loading cycles for LR, MR, and SR series are 261 MPa, 202 MPa, and 239 MPa, respectively.By comparing the fitted curve of experimental data with the standard FAT 112 curve, it is found that the actual fatigue lives of these notch geometries are higher than that specified in Eurocode 3, which is FAT 71.Therefore, concerning this detail, Eurocode 3 gives a relatively conservative prediction on the specimens tested in this study.Furthermore, it is clearly shown that the slope of notched specimens is much higher than that recommended in Eurocode 3. In addition, the S-N curve changes with notch geometries.As a result, separate fatigue tests for each geometry are needed to predict the fatigue life of notched specimens with the conventional S-N curve, which is very timeconsuming.Therefore, it would be beneficial to develop numerical methods that can be used to estimate the influence of the abovementioned changes.

Fatigue crack location
The fatigue crack initiation locations observed in the test are shown in Fig. 10.There are three typical initiation locations, which are single plate corner, two plate corners, and waterjet cutting edge.In most cases, fatigue crack initiated from the single plate corner.The failure mode of each specimen is listed in Table A-1.By comparing the fatigue lives of different failure locations, it can be found that the fatigue crack initiation location does not obviously affect the fatigue life of specimens.It R Fig. 9. S-N curves of experimental data.

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might be because the initiation location is dominated by the initial defects around the highly stressed zone, and the initial defects can be randomly distributed and affected by the processing methods.
In addition, to see whether all the cracks initiated from the local plastic zone near the notch root, static FE analyses considering the plastic properties listed in Table 3 were carried out and the contour plots of maximum principal plastic strain (PE) are put together with the observed fatigue crack initiation locations (see short lines in Fig. 11).It is clear that all the fatigue crack locations are within the highly stressed plastic zone, but fatigue cracks are not necessarily initiated from where the maximum principal plastic strain is maximum.While the maximum plastic strain occurs at point a in Fig. 11, the strain at point b is around 96% of that in point a.Many fatigue cracks were observed to initiate from point b rather than a.This is because the waterjet cutting causes random initial defects to the cutting edge, which in turn leads to the development of larger strains, resulting in fatigue crack initiation.

Strain measurement
Fig. 12 (a), (b), and (c) show three examples of strain-loading cycles curves during the loading procedure.It is observed that the strain values near the notch root of three series are growing gradually during the fatigue test.Since the measured is 2 mm away from the notch root, the measured strain values are lower than the maximum value.For all specimens the strain range is relatively constant, indicating a ratcheting fatigue mechanism.Fig. 12 (d) shows the force versus displacement loops at different loading cycles, and the response curves are stable until the number of loading cycles reaches around 75,000, after which the loops shift rightwards till the final fracture.
Based on the results, the typical strain development in the fatigue test can be classified into three stages (see Fig. 12 (e)), including: 1) sharp rise due to local plastic deformation, 2) steady increase due to stress redistribution and development of plastic deformation, 3) sharp rise due to ductile tearing and significant reduction of section ligament.However, the strain range kept consistent during the loading procedure, as shown in Fig. 12 (e).As approaching the final fracture, the strain range dropped to 0 and both peak strain and valley strain dramatically increased to a very high level.

TCD-based fatigue life prediction
As aforementioned, the conventional S-N curve in Eurocode cannot provide an efficient and accurate prediction on the fatigue life of notched components in steel bridges.Considering the merits of its simplicity and feasibility, the theory of critical distance (TCD) might be a suitable tool.In this subsection, the application of TCD in predicting the fatigue lives of tested specimens is conducted.Firstly, the linear-elastic stress analysis is carried out and stress distribution near the notch root is obtained.Then the mathematical expression of critical distance is derived.Finally, the recursive procedure shown in Fig. 8 is made to get the predicted fatigue life.

FEA model
To predict the fatigue life of notched specimens with TCD, the linearelastic stress distribution in the vicinity of the notch root is necessary.In this study, the notch stress field is modeled with FEA software ABAQUS and linear elastic analysis is conducted, as shown in Fig. 13.To reduce the computational cost, a half-model is used, and a symmetrical boundary is imposed on the plane of symmetry.The model is loaded by imposing a downward displacement to the clamped area in the test.The element type used in the model was C3D8R.
As the mesh size can affect the accuracy of the simulation result, parametric studies of the convergent mesh size are conducted.Different mesh sizes (0.1 mm, 0.125 mm, 0.25 mm, 0.5 mm, 0.75 mm, and 1 mm) were set as the mesh size near the area of interest.The simulated stress distributions shown in Fig. 14 (a), (b), and (c) are presented by normalizing the maximum principal stress σ 1 with nominal stress of the smallest net cross-section σ nom .It is observed that after the mesh size decreases to 0.125 mm, the stress distribution tends to be convergent, which means the mesh size of 0.125 mm is considerably accurate.In the present study, stress distribution obtained by a defined mesh size of 0.125 mm is used, and the results are shown in Fig. 14 (d).It can be seen that the stress concentration factor K t for LR, MR, and SR are 1.159, 1.227, and 1.442, respectively.

Validation of the existing methods
By combining the two S-N curves of smooth specimens and notched specimens, the critical distances for different fatigue lives can be derived.The calculated critical distances for MR series and SR series are listed in Table 4.As fatigue life increases, the critical distances in terms of PM and LM decrease.Since the stress concentration factor for LR series is very low and the fitted S-N curve for LR series has a larger slope than that of smooth specimens, the parameters A and B in Eq. ( 6) cannot be calibrated based on the methods shown in Figs. 6 and 7.It might be accounted for this phenomenon that the low stress concentration factor makes the fatigue life more sensitive to the random factors, such as initial defects introduced by the manufacturing process.
Fig. 15 shows the comparison of critical distance calculation between the conventional formulation (Eq.( 6)) and K t modified formulation (Eq.( 8)).As seen from Fig. 15 (a), the critical distances corresponding to the same failure cycle for different notches are different.The critical distance for the SR series is larger than that of the MR series.If the modification of K t is considered, as shown in Fig. 15 (b), the critical distances of both MR series and SR series can be described with a unified expression that has a higher correlation to the plotted data.The exponents of K t for the fitted curve of PM and LM are 0.66 and 1.1, respectively.
As the relationship between critical distance and fatigue life is determined, the recursive procedure can be used to predict fatigue life.Figs.16 and 17 show the prediction results using PM and LM, respectively.Note from Fig. 16 (b), the K t modified formulation significantly underestimates the fatigue life of the SR series and the scatter factor is over 5, while the conventional formulation has better agreement with the experimental results with a scatter factor of around 2. The scatter factor, defined as the logarithmic difference between the predicted fatigue life and the experimental fatigue life (log N f,pre − log N f,exp = log scatter factor), is used to indicate the variation of the predicted fatigue lives to the experimental fatigue lives.According to the existing studies [16][17][18], a scatter factor of 2 is often used as an acceptable criterion for fatigue life prediction.With aspect to the LR series, both conventional and K t modified formulations can provide acceptable accuracy.By comparing Figs. 17 and 16, it can be found that the prediction results of LM do not have too much difference compared to PM in terms of scatter factor.
To quantify the difference between PM and LM in prediction accuracies, all the predicted values and their errors with the mean fatigue life of the corresponding stress level are listed in Table 5.It can be seen that for LR, MR, and SR series, conventional formulation using PM has lower errors for each stress level compared to LM, and all the errors for MR and SR series are within ±24%.The prediction results obtained by the proposed formulation are also listed in Table 5 for the purpose of comparison.
In the above prediction procedure, parameters of power law in Eq. ( 6 15).To check whether parameters A and B determined by only one of the three series can well predict the other two (e.g., fitting the L-N f plot of MR series and then using the fitted curve to predict the fatigue life of LR and SR series), a prediction procedure using conventional formulation is conducted, and results are shown in Fig. 18.Almost all predicted fatigue lives fall into the scatter bandwidth of about 2, which means the parameters determined by one notch geometry can well predict the fatigue life of other notch geometries.In other words, using one calibration curve of notched specimens could provide enough accuracy and two calibration curves could not provide higher accuracy.This is presumably because the notch radii used in series LR, MR, and SR are larger than the range where the notch radius influences the determination of L-N f relationship.More specifically, when the notch radius decreases under a certain value (can be 1.5 mm according to [13,15,18]), the stress concentration is more server and the corresponding critical distance is shorter, making the critical distance more sensitive to the change in the notch radius.In such cases, the effect of the notch geometry on the determination of L-N f relationship should be considered.

Verification of the new formulation
A new logarithmic formulation is proposed in Subsection 2.3.The comparison in Fig. 19 demonstrates a good agreement between the test results and the proposed Eq. ( 17).To further verify this equation, fatigue life prediction is performed using the PM and LM.
The prediction results using Eq. ( 6) are also provided, and the errors between predicted lives and the mean fatigue life corresponding to each stress range are plotted in Fig. 20.It can be seen that the power formula has higher accuracy in predicting the fatigue life of the MR series, while the logarithmic formula has higher accuracy in predicting the fatigue life of the SR series.This is because the stress distribution of SR specimens is closer to an exponential function, thus Eq. ( 17) generates a lower error.For MR specimens, the linear fitting has a better agreement with the stress distribution near the notch root, so the conventional formulation yields higher accuracy.For the LR series, when the applied nominal stress is lower, the fatigue life increases and the critical distance L is longer, making the elastic stress distribution within the region of interest (defined by the distance from the notch root of 0 to L) closer to an exponential form.In this case, logarithmic yields a lower error.While for the case of higher applied stress, the elastic stress distribution within the region of interest is closer to linear form, resulting in a larger error for the application of the logarithmic formulation.In a conclusion, it is reasonable to state that the mathematical formulation of the L-N f relationship depends on the scale of the critical distance and the stress distribution characteristic near the notch root.

Conclusions
An experimental study on the fatigue lives evaluation of blunt notch detail in rib-to-floor beam connection in orthotropic steel decks was presented in this paper.In addition, the conventional method using theory of critical distance was employed to predict the fatigue life of tested specimens.By modifying the description of stress distribution near the notch root, a new logarithmic formulation of the relationship between critical distance and fatigue life is proposed.Based on the results of this study, the following conclusions are made: (1) A new logarithmic formulation of the critical distance-fatigue life relationship is proposed, and the proposed formulation is in good agreement with the experimental results in this study.The new formulation can yield higher accuracy when the stress distribution near the notch is close to exponential form.(2) Compared to the K t modified formulation, the conventional formulation of critical distance versus fatigue life is in better agreement with experimental results in fatigue life prediction.The predicted fatigue lives fall into the scatter bandwidth of     about 2. And it is found that the point method has higher accuracy compared to the line method in predicting fatigue life.(3) Based on the results in this study, the use of one calibration curve for notched specimens could provide enough accuracy and the use of two calibration curves could not provide higher accuracy.Notch geometries are not found to significantly affect the accuracy of fatigue life prediction in this study.(4) The mathematical formulation of critical distance-fatigue life relationship may be in power form or logarithmic form, depending on the scale of the critical distance and the characteristic of stress distribution near the notch root.More research works investigating the effects of materials and geometries are needed to further confirm this relationship.
Despite the conclusions obtained in this study, since bridges experience complex loading histories and stress states, future works on the applicability of the proposed method in other complex stress states are required to further validate the effectiveness of the proposed approach.Note that Error = N f, PM (N f, LM ) / N f, exp − 1, where N f, PM (N f, LM ) and N f, exp denote predicted fatigue lives and the mean fatigue life corresponding to each stress range, respectively.

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Fig.
Fig.15).To check whether parameters A and B determined by only one of the three series can well predict the other two (e.g., fitting the L-N f plot of MR series and then using the fitted curve to predict the fatigue life of LR and SR series), a prediction procedure using conventional formulation is conducted, and results are shown in Fig.18.Almost all predicted fatigue lives fall into the scatter bandwidth of about 2, which means the parameters determined by one notch geometry can well predict the fatigue life of other notch geometries.In other words, using one calibration curve of notched specimens could provide enough accuracy and two calibration curves could not provide higher accuracy.This is presumably because the notch radii used in series LR, MR, and SR are larger than the range where the notch radius influences the determination of L-N f relationship.More specifically, when the notch radius decreases under a certain value (can be 1.5 mm according to[13,15,18]), the stress concentration is more server and the corresponding critical distance is shorter, making the critical distance more sensitive to the change in the notch radius.In such cases, the effect of the notch geometry on the determination of L-N f relationship should be

Fig. 14 .
Fig. 14.Stress distribution near the notch root: (a), (b), and (c) are normalized stress distribution near the notch of LR, MR, and SR series, respectively, (d) is the comparison among three series.

Fig. 18 .
Fig. 18.Predicted fatigue life versus experimental results (using one calibration curve of notched specimens).

Fig. 19 .
Fig. 19.Calculated critical distance versus failure cycles fitted with the logarithmic law.

Table 3
Material properties of the structural steel.

Table 4
The nominal stress range and critical distance calculated based on TCD.

Table 5
Predicted fatigue lives and errors.