Inﬂuence of Fatigue Crack Formation and Propagation on Reliability of Steel Members

: During the years of bridge exploitation, many degradation processes and external inﬂuences attack its structure. Therefore, bridge reliability and durability is decreasing in time. On the other hand, the trafﬁc load remains almost the same or even higher than in the past. However, bridges should not to become the limiting component of communication capacity and trafﬁc reliability. Re-garding to reliability, bridges should be assessed from the viewpoint of the Ultimate Limit States (ULS) and Serviceability Limit States (SLS). Within the ULS, cross-sections and members are veriﬁed for various types of stressing and their combinations, and also for fatigue at the same time. The cross-sectional veriﬁcation, e.g., for bending stresses and fatigue, is done independently according to corresponding criteria of the ULS determined for strength veriﬁcation a fatigue assessment separately. The presented article deals with the steel railway plate girder bridge with bottom member deck, in which there is an effort to prove the effect of the crack in tension bottom ﬂange due to fatigue stressing on the change of bending resistance over time. The analytical calculation was derived and at the same time, the probabilistic approach of the inﬂuence of the fatigue crack size on the change of the cross-sectional resistance and reliability over time was used.


Introduction
Many various factors influence the bridge members during the lifetime, which can affect the change of member reliability in the time. Apart from e.g., change of loads or intensity of loads in time, which can induce a loss or growth of the reliability depending on the increase or decrease of loads, the degradation of materials significantly influences the cross-sectional reliability [1]. Therefore, it is possible to say that this factor explicitly causes the reduction of reliability of steel bridge members [2,3].
The most known ways of degradation of a steel member are corrosion of steel and fatigue processes [4][5][6][7][8][9][10]. Verifying structure in terms of fatigue is an inseparable part of the new structural design or part of verifying the existing bridge structures [11][12][13][14][15]. An increase of crack due to fatigue damage changes the member cross-sectional characteristics, as decreasing in cross-sectional area, a second moment of area and section modulus. It causes lower flexural or shear stiffness, decrease of resistance, an increase of deformations, therefore, modification of member's reliability in the time [16][17][18].
Bridges are heavily stressed by traffic. In the case of steel bridges, this load causes significant fatigue stresses [19][20][21], which can result in fatigue cracks in the tensile areas over time [22,23]. However, it is not yet known how fatigue cracks can affect e.g., crosssectional bending resistance of the element within the time, because the verifications of the cross-sectional resistance and fatigue strength are independent assessments. From the viewpoint of ULS, the cross-sectional or member resistance, respectively, is verified according to individual criteria defined by corresponding ULS given in standards. Standard fatigue assessment of steel bridge member according to [24] is based on the comparison of the damage equivalent stress range ∆σ E2 to the fatigue strength given in [25] in accordance with the corresponding bridge structural detail. Damage equivalent stress range should be calculated using the reference stress range ∆σ p and the damage equivalence factor λ, which determination is given in [24] for railway and road bridges separately. The verification is done for determined bridge lifetime, so that this assessment does not reflect the change of structural member resistance in time due to crack growth.
The fatigue cracks on tension flange of steel structure have been an issue since the beginning of steel bridge design activities. The objective has been to use such preventive activities that would eliminate gradual growth and propagation of cracks resulting in considerable deterioration of reliability of the bridge. Though more and more advanced modifications and control methods have been introduced, the cracks still appear. The standard [25] makes it possible to use a damage tolerance method as an alternative under certain conditions. Methods proposed include the observation of a "critical" crack [26]. That term, however, has not been defined for real welded structural elements. It is hardly possible in practice to use a small test sample for direct application [27,28].
In connection with the fatigue damage of components, the design method is very important. If the ULS for fatigue are applied when designing elements in terms of design effects of the traffic load, the verification is done for the real traffic load throughout the designed service life. The acceptable size of the fatigue crack should be such so that the bridge superstructure might, at the end of its service life, still provide the required level of reliability. The tension flanges are the most stressed elements of the superstructure. Therefore, the standardized level of reliability for fatigue is determined for the tension elements.
The paper is focused on the influence of crack formation and propagation due to fatigue on the change of steel member's reliability from the viewpoint of the Ultimate Limit States (ULS) in time. Because the work is focused on member where a measurable crack has already appeared and has propagated (grew), the linear elastic fracture mechanics (LEFM) was considered.

Fatigue Crack Formation and Propagation
According to [29,30], the cracks are divided into cracks propagating from the edge, from the corner, embedded cracks and cracks propagating from the surface in terms of engineering structures. The cracks usually develop in members subjected to bending or tension. Concerning the eventual broad scope of this problem, the work in the paper is limited and focused just on cracks propagating from the surface (see. Figure 1). According to Figure 1, the maximum crack depth "a" may theoretically be equal to the flange thickness "t" and the crack width 2.c may be equal to the flange width b, but the limiting conditions used are given later in paragraph 6 (Equation (29)). Cracks from the surface most often occur when the flange is e.g., joined in several parts by welds or has any defect.

Crack Propagation from Surface
State of stresses (tightness) surrounding crack tip is formulated by the stress-intensity factor (SIF) [31,32], whose value is depending on the way of stressing, size and shape of a crack and on geometry of a member. In the case of a surface crack, the model of crack propagation is more complicated than in the case of crack propagating from the edge or the embedded crack. SIF is given by Equation, which follows from superposition of tensile and bending stressing where: K I is the stress-intensity factor (SIF), σ m is the tensile (membrane) stress, σ b is the bending stress, F(a) m is the boundary-correction factor of stress intensity for tension stress, F(a) b is the boundary-correction factor of stress intensity for bending stress, M(a) m is the curve-fitting function taking into account the nonlinear stress peak due to the tension effects, M(a) b is the curve-fitting function taking into account the nonlinear stress peak due to the bending effects.
Type of crack (real shape) is substituted (e.g., according to [33]) by the semi elliptical surface crack ( Figure 2) with depth of crack "a" (length of semi-axis) and half of crack width in plate surface "c" (length of the second semi-axis). "T cr " means center of crack gravity and "e cr " means distance of center of crack gravity from surface (eccentricity). The change of crack cross-sectional area, depending on number of cycles N, is given by Equation Distance of the crack center of gravity from the surface is equal to and second moment of area is described by equation In the case of the crack propagating from surface, it is not enough to investigate the change of just one crack dimension during crack form changing and its increasing, what is in the case of crack propagating from the edge or from the middle. It is necessary to take into account the change of both dimensions "a" and "c" in both semi-axes directions. The stress-intensity factor K I for semi-elliptical surface crack propagating from the surface in a slab member subjected to tension and bending has the basic form [34,35] where: F(a) = F(a/t, a/c, 2c/b) is the boundary-correction factor for surface crack in a plate, a, b, c, t, θ are the geometrical parameters of a member and a crack (Figures 1 and 2), a is the crack depth, c is the half of the crack width, b is the width of the cracked plate, t is the thickness of the cracked plate, θ is the parametric angle of ellipse, H is the function expressing change of F(a) in the case of bending stressing, Φ is the shape factor for an elliptic crack, where One of the most used empirical Equation for Φ was derived in [36] and it depends on ratio between the crack depth "a" and half of crack width "c" , for a c > 1.
In general, it is possible to write the boundary-correction factor F(a) in the following form where M i are the curve-fitting functions, where: g is the fine-tuning curve-fitting function, f φ is the angular function derived from embedded elliptical crack solution, f w is the finite-width correction, and it is valid for a c ≤ 1 If the focus is on biaxially symmetric welded steel I-beams and one assumes that there is a crack in the tension bottom flange (positive moment), then we can neglect the gradient of bending stress through flange thickness. In that case, it could be considered that the flange is subjected only to axial tensile force, from which it is stressed only by tensile (membrane) stress σ m = σ. Then, it is possible to simplify Equation (5) into form However, this assumption should be only applied to thin flanges. In the case of thick flanges according to literature [35], the influence of the normal stress gradient on the thickness of the flanges is positively affected. In the case of longitudinal girders, the thickness of flanges varies from10 mm to 25 mm as maximum, whereas minimum height of longitudinal girder is 500 mm. The thickness of cross-girder flanges varies from 20 mm to 30 mm, whereas the cross-girders heights are minimum of 700 mm.

Fatigue Crack Propagation
In general, the rate of fatigue crack growth is given as the derivation of crack depth "a" with respect to number of loading cycles "N". It means that it is ratio da/dN. For simplification, the constant range of the nominal stress ∆σ is considered. Then, for a given crack depth and constant stress range ∆σ one can plot the values log da/dN in terms of log ∆K, where ∆K is the SIF (stress-intensity factor) range Dependence of log (da/dN) on log ∆K is approximately linear and is expressed by the Paris-Erdogan law where: da/dN is the change in the length of the fatigue crack per load cycles, ∆K is the interval (range) of the stress intensity factor (SIF), C, m are the material constants, N is the number of loading cycles.
The material constants C and m are empirical constants, which are functions of the material properties and microstructure and are obtained experimentally and also usually depend on environment, fatigue frequency, temperature and mean stress or load ratio, loading mode, stress state, where m is the slope of the curve showing the rate of crack propagation according to the Paris-Erdogan equation in the mean linear region (on a logarithmic scale). The upper limit for C = 6.89·10 −12 MPa −3 ·m −1/2 (in the case of the use of millimeters 2.18·10 −13 MPa −3 ·mm −1/2 ) is used and m = 3 for structural steels in the case of tensile cracks.
Paris-Erdogan' law is a crack growth equation that gives the rate of growth of a fatigue crack. The stress intensity factor K characterizes the load around a crack tip and the rate of crack growth is experimentally shown to be a function of the range of stress intensity ∆K seen in a loading cycle. The stress intensity factor, K, is used in fracture mechanics to predict the stress state ("stress intensity") near the tip of a crack or notch caused by a remote load or residual stresses [37]. P. C. Paris introduced the idea that the rate of crack growth may depend on the stress intensity factor [38] in a year 1961. Then in their 1963 paper [39], Paris and Erdogan indirectly suggested the equation with the aside remark "The authors are hesitant but cannot resist the temptation to draw the straight line slope 1:4 through the data" after reviewing data on a log-log plot of crack growth versus stress intensity range. The Paris equation was then presented with the fixed exponent of 4.

Effect of Crack on Steel Member's Reliability
In the engineering probability method, the reliability margin g(R,E) = G is given by Equation where: R is the random variable resistance of a structural member, E is the random variable load effects of the same element.
It is assumed that the member resistance R is normally distributed random variable N(m R , s R 2 ) and that the load effects E are also normally distributed N(m E , s E 2 ), where is valid that R and E are mutually independent. The normally distributed random variables R and E are given by the mean values (m R , m E ) and the standard deviations (s R 2 , s E 2 ). The member's reliability is given either by the failure probability Pf or by reliability index β. If one considers the change of member resistance R(N) in time t due to fatigue damage (crack formation and development depending on cycles number N), then the reliability of the member is also changing in time, it follows that the failure probability P f (N) and reliability index β(N) are also varying in time t.
As was mentioned, the paper is focused on the welded steel biaxially symmetric I-beam subjected to bending. In this case, the member's resistance R(t) for compact and semi compact cross-sections is given by the equation  That theoretical approach does not consider the cross-sections of class 4. In the case of compact cross-section 1 and 2, the time dependent plastic section modulus W(N) pl,y taking into account the cross-section loss due to crack development is represented by the relation where e * I (N) = Meaning of each cross-sectional characteristics is shown in Figure 3.
where the second moment of area is calculated according to where in this case Since the dependence of crack depth "a" and half of crack width "c" growth on cycles number N (a(N), c(N)) is not known yet, they are considered in equations as independent on cycles number N with denotations "a(N) = a" and "c(N) = c".

Independent Crack Growth in Directions "a" and "c"
In the first case, one focuses on the independent growth of a crack in the direction of the two half-axles (depth and width). In that case, the dimensions "a" and "c" are independently chosen from each other. From that follows, that the chosen cracks have different shapes (different ellipse) and the different areas.

Analytical Calculation
To determine the influence of the size and shape of fatigue crack on the cross-sectional resistance, the parametric study was performed [40,41]. The welded steel beam I 400, representing, for example, longitudinal girder of steel plate girder bridge with a bottom members' deck ( Figure 4) and I 800, representing cross girder of the same bridge, were considered in the study. The input strength and geometric characteristics are shown in Table 1. The length of the bridge is 30.0 m. The lengths of the longitudinal girders of plate girder railway bridges vary about (1.6-2.5) m. In the case of railway bridge cross-girders, their lengths are 5.5-6.5 m depending on the bridge arrangement.  Table 1. Cross-section characteristics of welded steel beams I 400 and I 800.

Input Characteristics I 400 I 800
Yield Since this is a cross-section of class 1 (I 400) and class 2 (I 800), the Equations (21) and (23) were used for cross-sectional resistance calculation. The results of crack growth influence on plastic section modulus W pl,y decreasing are shown in Figure 5, and on plastic bending resistance M pl,y are shown in Figure 6.
Crack size was chosen from a minimum value (zero value) up to the size defined by the limit values 2·c ≤ 0.8·b f , and a ≤ 0.8·t f .
From the results obtained, it can be seen that the fatigue crack growth can largely reduce the moment load-carrying capacity. If the maximum possible crack from surface limited by conditions (29) would appear, it would lead to the cross-sectional area loss of 9.97% (I 400), 15.71% for I 800, while the plastic resistance moment would be reduced up to 33.77% (I 400), 55.15% (I 800) respectively.

Probabilistic Approach
The probabilistic approach was used for finding the impact of fatigue crack on failure probability variation and on corresponding reliability index [42]. In that case, all strength and geometrical characteristics were considered as random variables with normal distribution N(m,s 2 ) and they are shown in Table 2 [43,44]. The steel of quality S235 was assumed. Statistical characteristics of yield strength were taken from [45]. Likewise, all geometric characteristics (thickness and width) of plates were taken from [45]. Standard deviations, reflecting tolerances of flange widths, take into account the deviations allowed by standard [46]. Dimensions of crack semi-axes (c,a) were again chosen and were considered as constants, not as random variables. Numerical application of resistance calculation R = M pl,y was realised using the simulation method Monte-Carlo.  Obtained results of mean value m R and standard deviation s R variation are shown in Figures 7 and 8. As expected, the results show detectable decrease of the mean value m R and standard deviation s R . This is the proof that influence of fatigue crack development and propagation may have unavoidable influence on member resistance. In this case, the mean value of resistance m R was decreased by a maximum 15.33% (I 400) and 19.54% (I 800) and the standard deviation s R was decreased by a maximum 12.73% (I 400) and 18.73% (I 800). From parameters follows that members were designed with reliability index β = 3.652 for design lifetime T d = 100 years. The results of crack growth influence on failure probability P f and reliability index β are shown in Figures 9 and 10.
Decrease of the mean value m R causes decrease of the member's-decrease of reliability index β and corresponding increase of failure probability P f . Decrease of the standard deviation, in general, has favourable influence on members' reliability (the failure probability P f is lower in time). Since the results show a decrease of the reliability, it follows that the reliability is mainly influenced by the decrease of the mean value m R . The decrease of the standard deviation s R only reduces reliability uncertainty, so, it reduces the consequential reliability decline.
The reliability index decreases from the value β = 3.652 to values β = 2.754 (I 400) and β = 2.200 (I 800) for the maximum crack given by limitations (29). This represents a significant reliability decline that for the administrator of the bridge may not be acceptable. If crack would no longer grow, the remaining lifetime of member in ultimate limit state is lower than 3 years according to works [1,47]. In the case of further crack growth, the remaining lifetime would be significantly reduced and it means that the member would become unreliable.

Dependence of Crack Growth in Directions "a" and "c"
In fact, the crack growth in one direction depends on the crack growth in the other. Although this dependency was considered in several works such as [48][49][50], it has not yet been precisely defined.
Approach described in [51] was used for obtaining this dependency. For simplification, the crack growth is analysed at points (directions) "a" and "c" (in direction of main semiaxes). Crack propagation speed at those points is described by the stress-intensity factor (∆K) and by the Paris-Erdogan law, which can be expressed as where: ∆K a is the stress-intensity factor in direction "a", ∆K c is the stress-intensity factor in direction "c", C, m are the material constants.
Relation between the crack depth increment ∆a and the crack width increment ∆c is obtained by combination of Equations (30) and (31) The stress-intensity factor ∆K is in general given by Equation (17). Forms of SIF at points "a" and "c" are obtained from adjustment into increment form The boundary-correction factor in general form, F = F(a), is given by relation (9). Since the terms M 1 , M 2 , M 3 and thus the whole expression M in Equation (9) depend only on the ratios a/c and a/t, and do not depend on angle θ, then the value of the parameter M is the same at points "a" and "c" (independent on direction). Similarly, the parameter f w , given by the Equation (16) for the calculation of the parameter S, is independent on the angle θ (on direction). However, the parameters g (14) and f φ (15) depend, apart from ratios a/c and a/t also on angle θ, it means that they depend on the position. Then, the calibration functions for the points "a" and "c" follows from Equations F a = M·S a , at point a, Adjusting and substituting Equations (14)- (16) into Equation (13), one obtains the following Equation for calculation S a , where it is valid for a c ≤ 1, θ = 0 (at point "a") and for calculation S c , one obtains Equation, where it is valid for a c ≤ 1, θ = π 2 (at point "c") Substituting (38) and (39) There to proceed by iterative way for the calculation of the crack growth in both directions one must choose the initial crack size with initial dimensions "a" and "c", then the increment ∆c is selected and using Equation (40), the corresponding increase of increment ∆a can be calculated. New crack dimensions "a" and "c" are obtained so that the increments ∆c and ∆a are added to previous dimensions. Again, choosing increment ∆c (same as before) and calculating a new corresponding increase ∆a, new crack dimensions can be calculated. Calculation of crack growth will be finished if none of the restrictions (29) is violated. An example of crack dimension calculation for flange thickness t f = t = 10 mm is shown in Table 3 and the results are shown in Figure 11.  Figure 11. Influence of crack width "c" on crack depth "a".
The initial ratio a/c (initial crack dimensions) is important for crack growth calculation in both directions. Basically, it is the minimum detectable crack. Since the minimum value of the measurable crack is not precisely defined (usually it is estimated) and similarly the ratio a/c is not known, the approach was intended for values close to zero (exact zero cannot be considered because of the ratio a/c). The work [29] is also concerned with this problem.
The results show that the increment ∆a is gradually increasing unevenly at constant increment ∆c.
It is possible to obtain the crack growth in both directions using the described approach. The curve was drawn through the obtained values based on finding the dependence of crack growth in direction "c" on crack growth in direction "a". The second degree polynomial curve turned out to be the most preferred form where k i are coefficients. It has been proven that the values of the coefficient k i depend not only on the thickness t f , but also on selected increment ∆c. Therefore, the sensitivity analysis was performed, which improved values of the coefficient k i . Table 4 shows the k i values depending on the selected increment ∆c, wherein the thickness of the flange was assumed t f = t = 10 mm. Table 4. Dependence between "c" and "a" for thickness of flange t f = 10 mm depending on increment ∆c.

Selected Increment ∆c [mm]
Dependence between "c" and "a" It is assumed that the increment reduction is meaningless and the values of coefficients k i are not significantly changed. Therefore, the dependency for ∆c = 0.001 mm is considered to be sufficiently accurate.
Similarly, the parametric study was used for investigation of the coefficients k i , which are valid to other flange thickness t f . Dependencies are shown in Table 5. Table 5. Relationships between "c" and "a" depending on various flange thickness t f .

Flange Thickness t = t f [mm]
Dependence between "c" and "a" If the dependency is established of "c" not only on "a", but also on flange thickness t = t f in the approach, the universal dependency of the crack growth in direction "c" on direction "a" and on flange thickness t, is obtained in the form c = 0.3027 t ·a 2 + 1.0202·a + 0.00699·t.
Again, the parametric study with same steel welded I-beams I 400 and I 800, which were considered before, was performed. The characteristics are given in Table 2.
The achieved results of change of the mean value m R and the standard deviation s R are shown in Figures 12 and 13. As expected, the results again show a decrease of the mean value and the standard deviation, but in this case, the mean value m R of resistance decreased more realistically for about 3.84% (I 400) (4.92% for I 800) and in the case of standard deviation s R for about 2.93% (I 400) (5.45% for I 800). The given course of curves proves that it is a random stochastic process and the results were obtained by numerical simulation, which is the reason for the variance of values.
If the approach and Equations were projected into failure probability P f and reliability index β (Figures 14 and 15), it was possible to see that the decrease of the reliability is not so significant. Initially, in the case of smaller crack sizes ("a", "c"), random character of parameters has greater impact on the monitored variables as the crack growth (similarly to the mean value and standard deviation of resistance). Influence of the crack size became greater after ratio a/t = (0.2~0.3) was reached. proves that it is a random stochastic process and the results were obtained by numerical simulation, which is the reason for the variance of values. Figure 12. Dependence of resistance mean value mR on crack depth "a".  If the approach and Equations were projected into failure probability Pf and reliability index β (Figures 14 and 15), it was possible to see that the decrease of the reliability is not so significant. Initially, in the case of smaller crack sizes ("a", "c"), random character of parameters has greater impact on the monitored variables as the crack growth (similarly to the mean value and standard deviation of resistance). Influence of the crack size became greater after ratio a/t = (0.2~0.3) was reached.   Reliability index β of the member at the maximum dimension of the considered crack a = 0.8·t takes the values β = 3.443 (I 400) and β = 3.400 (I 800). It means that its remaining lifetime at ultimate limit state according to paper [47] is approximately 20-30 years depending on the age of member. Again, the course of the curves proves that this is a random stochastic process and the results were obtained by numerical simulation, which causes the variance of values.

Conclusions
The paper is focused on the influence of fatigue damage (fatigue cracks) of the steel welded I-girders on change of moment resistance and reliability in time. Methods such as analytical calculation and probabilistic calculation (numerical simulations) were used. The crack due to fatigue in tension flange of welded I-girder changes the cross-sectional characteristics as cross-sectional area, second moment of area and plastic or elastic section modulus. The analytical calculation and also probabilistic approach showed the following conclusions: - In the case of independent crack growth in directions "a" and "c", the fatigue crack can cause unavoidable decline of moment resistance (deterministic and stochastic) from the viewpoint of the Ultimate Limit States, what is going to decrease the member reliability from β = 3.652 to β = 2.754 (I400) and to β = 2.200 (I800) within bridge lifetime of 100 years. - The derivation of the model of dependent crack growth in both directions represents a major contribution of this work. Using this approach, it is possible to directly determine the bending resistance and fatigue stressing from the viewpoint of the Ultimate Limit States. - In the case of dependent crack growth, the reliability index β of the member at the maximum dimension of the considered crack a = 0.8·t takes the values of β = 3.443 (I 400) and β = 3.400 (I 800) respecting strength criteria of the Ultimate Limit States. - The results of the study showed more favourable and real (lower changes) interconnection between fatigue stressing and bending resistance. However, it still tends to decrease reliability of the member and to shorten the remaining lifetime.