Prediction of flexural fatigue life and failure probability of normal weight concrete

: Fatigue life has to be considered in the design of many concrete structures at various stress levels and stress ratios. Many flexural fatigue test results of plain normal-weight concrete are available in the literature and almost every set of test results provides different fatigue equations. It is necessary, though, to have a common fatigue equation to predict the design fatigue life of concrete structures under flexural load with reasonable accuracy. Therefore, a database of flexural fatigue test results was created for concrete with strengths ranging from 25 to 65 MPa; this database was used to derive new fatigue equations (Wöhler fatigue equation and S–N power relationship) for predicting the flexural fatigue life of normal-weight concrete. The concept of equivalent fatigue life was introduced to obtain a fatigue equation using the same stress ratio. A probabilistic analysis was also carried out to develop flexural fatigue equations that incorporate failure probabilities.


INTRODUCTION
Many concrete structures such as bridge decks, airport runways, concrete pavements, and offshore structures experience a million cycles of repetitive loading in their service life (1,2). The exposure to repetitive loading reduces the stiffness of the concrete structures; this in turn leads to fracture generation as a result of changes caused by the progressive growth of micro-cracks. When the repeated loads are applied at high frequency for a prolonged period, this can eventually lead to fatigue failure (1)(2)(3). For this reason, in the design of these concrete structures, increasing attention is now being paid to the fatigue characteristics of the constituent materials.
Concrete is a compound heterogeneous construction material that consists mainly of cement, water, fine aggregates, and coarse aggregates. It contains both enormous micro-sized capillary pores and millimeter-sized air voids that result from the hydration process, shrinkage, and other causes. These imperfections in the concrete make it highly susceptible to repeated or cyclic loads. Repetitive loading causes progressive and permanent internal structural changes, known as a fatigue process that leads to failure. Therefore, considerable research effort has gone into understanding the behavior of concrete under cyclic loading (1)(2)(3)(4). There are several other motivations for these research efforts to study the fatigue behavior of concrete. Two of these are the increasing use of plain concrete at airfields and the concrete pavements of highways being subjected to repeated loads. This has necessitated the design of these structures to accommodate cyclic loading. Secondly, the introduction of new types of concrete with high strength characteristics requires them to perform satisfactorily under high and repeated loading. For these reasons, the study of concrete behavior under varying loads is more critical than that of concrete in a static state. Moreover, it is important to identify the effects of repeated loading on the material characteristics of concrete (such as the static strength, durability, and stiffness) that could be significantly reduced even if the repeated loading does not lead to fatigue failure. Furthermore, understanding the causes and nature of fatigue is very important from both an economic and a structural safety point of view.
The study of flexural fatigue effects on concrete structures began as early as 1906 with a study by Feret (5). The fatigue tests for concrete are usually carried out by applying sinusoidal wave loading to a specimen. The sinusoidal waves include both the maximum amplitude (S max ) and the minimum amplitude (S min ) of the applied load in addition to the cyclic loading frequency and the maximum number of cycles (N). Most of the studies focused on establishing a relationship between the applied stress level (S = S max /ƒ r ) and the number of loading cycles (N) to failure, where f r is the static flexural strength (modulus of rupture) of the concrete (6)(7)(8). The established relationship is known as the S-N curve or the Wöhler fatigue curve (6,7). The equation of this curve is known as the Wöhler fatigue equation, which is expressed by Equation [1]: [1] where a and b are the coefficients that can be obtained by a linear regression analysis of the plotted test fatigue lives. Equation [1] shows the relationship between the applied stress level (S) and the number of cycles until failure on a logarithmic scale (log(N)). Therefore, this equation is also known as the single-logarithm fatigue equation (8).
Owing to the widely scattered nature of concrete fatigue test results, it is important to test many specimens at the same stress level (S) with the same rate and period. In addition, it is essential to implement the probabilistic approach to ensure the reliability of the fatigue test data and to secure the concrete structures against fatigue failure. In the literature (6,8), many mathematical models are to be found that apply the probabilistic approach to the fatigue test data of cementitious materials. Log-normal distribution and three-or two-parameter Weibull distribution are some of the models used to represent the fatigue test data statistically.
It is well established that members designed to resist a specific static loading will not endure if the same load is applied to them cyclically. Therefore, acquiring knowledge of the fatigue life of concrete is important in order to ensure an efficient, safe, and economical structural design and also to provide a basis for understanding the fatigue parameters and predicting the fatigue-life distribution. It has also been found that fatigue equations obtained from different test data sets differ from one another and provide different estimations of fatigue life (9). Therefore, all the relevant test data should be combined for the purposes of generating a general fatigue equation that can be used in the design.
One of the main objectives of this study is to determine the relationship between fatigue life and stress level for plain normal-weight concrete (NWC) using a large number of the relevant fatigue test data available in the published literature. Another objective is to perform statistical analyses on the equivalent fatigue test data to show the fatigue-life distribution of NWC for different stress ratios (R=S min /S max ) at various stress levels (S).

FLEXURAL FATIGUE LIFE OF NOR-MAL-WEIGHT CONCRETE
For the purposes of this research, the available flexural fatigue test data for plain NWC with different compressive strengths have been collected from the published literature. A total of 465 flexural fatigue test results for NWC with a compressive strength ranging from 27 MPa to 62.3 MPa were collected and used in the analysis. For all the collected test results, the loading frequency was in the range of 1-20 Hz. The collected flexural fatigue test data for plain NWC, as found in the literature, are listed in Appendix A.
Many studies on the fatigue of concrete have concluded that a loading frequency in the range of 1-20 Hz had no effect on the fatigue life of plain concrete (4,10). Therefore, in this research, only the concrete compressive strength and the applied stress ratio (R) were taken into account as effective variables for the fatigue strength analysis of the plain NWC.

Equivalent fatigue life (EN)
The results of 16 different experimental tests for flexural fatigue were used in the analysis. Since the fatigue test data have different stress levels (S = S max /ƒ r ) and different stress ratios (R = S min /S max ), it is not appropriate to use them directly for the purposes of comparison. Furthermore, it is difficult to perform a direct statistical analysis of the fatigue-life data using both variable stress level (S) and stress ratio (R). For this reason, the analysis was conducted for one variable at a time using the equivalent fatigue-life (EN) concept, which was first used by Shi et al. (8), as given by Equation [2]. According to this concept, all the data concerning fatigue life at a specific stress level (S) with different stress ratios (R) can be transferred to get a common equivalent fatigue life (EN). Since it is well established that the fatigue life (N) increases as the stress ratio (R) increases (8,11), the test data with different stress ratios were converted to the equivalent fatigue-life (EN) using Equation [2]: [2] where EN is the equivalent fatigue life; N is the test fatigue life; R is the stress ratio. Equation [2] can also be used to transform the EN into a fatigue life (N) with a specific R-value. As most of the fatigue test data relate to the stress ratio R = 0.1, all the other test data were transformed from EN to fatigue lives (N) with a stress ratio R = 0.1, using Equation [2] at each stress level (S). By definition, the EN corresponds to the fatigue life for the stress ratio R = 0.
The fatigue lives are determined against the applied stress level (S), which is a dimensionless quantity since it is a ratio of the maximum applied stress (S max ) to the modulus of rupture (f r ), that is, (S max/ f r ). This was done to eliminate the effects of concrete strength and the water-cement ratio, the type and gradation of aggregates, and the type and amount of cement on fatigue life N (8,12). It is also well established that the reliability of the fatigue equation depends on the number of test data. Based on the above discussion, a large number of relevant fatigue test data were used to generate one fatigue equation for NWC which can be used to design concrete structures that are able to withstand being subjected to repetitive loading. Because this study combines a large number of relevant test data from various sources, the fatigue lives at the same stress level are highly scattered. A statistical procedure known as Chauvenet's criterion was employed to eliminate some of the outlier data from the fatigue data set (13,14).

Wöhler fatigue equation for normal-weight concrete
The equivalent fatigue lives were used to obtain the S-N curves for the stress ratios R = 0 and R = 0.1. Figure 1 shows the plot of S-N curves using the 16 flexural fatigue-life data of NWC. A linear regression analysis based on the best-fit curve was performed. From the equations of the best-fit curves in Figures 1(a) and (b), the values of the coefficients (a and b) of the Wöhler fatigue equation (Equation [1]) were obtained for the stress ratios R = 0 and R = 0.1. The correlation coefficient (C c ) of the regression analysis is 0.87 for both stress ratios, as shown in Figure 1. This low value of the C c (less than 0.9) indicates the scatter characteristics of the fatigue test data. However, the value of C c > 0.70 for the plotted data refers to a strong relationship between S and log(N) (15). For both stress ratios (R = 0 and R = 0.1), the obtained values of the coefficients (a and b) of the Wöhler fatigue equation for the fatigue lives of NWC are given in Table 1.
Using these coefficients, the Wöhler fatigue equations for the plain NWC were generated; these are presented and discussed in section 5.1. From these equations, the fatigue strengths of the plain NWC for any desired number of cycles can be calculated.

Power relationship of fatigue life (doublelogarithm fatigue equation)
The power relationship of fatigue life was mainly developed for concrete pavements to estimate the flexural fatigue life of concrete (16). It relates the dimensionless stress level (S=S max /ƒ r ) to the number of loading cycles (N) given by Equation [3] (5, 6, 17, 18): [3] where C and m are the empirical constants, N is the fatigue life, and S is the applied stress level. This equation has wide applicability since the stress level S = S max /ƒ r is expressed as a dimensionless form.
Here, S max is the maximum applied stress and f r is the concrete modulus of ruptures (flexural strength). Taking the logarithm of both sides of Equation [3], the following expression can be obtained (Equation [4]): [4] The values of C and m can be determined from the plot of log(N) versus log(S), as shown in Figure 2.  Table 2.

PROBABILISTIC ANALYSIS OF FA-TIGUE-LIFE DATA
The statistical nature of the fatigue test data exhibits a larger scatter than that of the static test data. The statistical variability may arise from the variation of a number of design factors such as the applied load and the heterogenetic nature of the material, which lead to an increase in the uncertainties in the design. Therefore, applying probabilistic analysis to the fatigue test data renders it more realistic and provides adequate resistance to the fatigue failure of concrete structures.
Numerous mathematical probability models have been developed to represent the probabilistic distribution of concrete fatigue-life data. Weibull distribution and Gumbel (19) distribution are some of the models used in many studies to represent the fatigue test data statistically. However, Gumbel distribution is generally used for the extreme values from some sets of fatigue data. On the other hand, Weibull distribution is widely used for both concrete and metal fatigue analysis to find out a mathematical model for the prediction of fatigue-life for a certain percentage of failure probability. Many studies (3,6,8,20,21) have shown that the two-parameter Weibull distribution can be used to describe the distribution of the fatigue-life data of cementitious materials, since it provides safer and greater reliability, as proved both statistically and experimentally. Therefore, the two-parameter Weibull distribution was used in this study to describe the probabilistic distribution of the flexural fatigue-life of the NWC. The failure probability function or the cumulative distribution function (CDF) of the two-parameter Weibull distribution is expressed as follows (Equation [5]) (6,8,22): [5] where P ƒ (N) is the failure probability function, n is the specific fatigue life of concrete at a particular stress level S, u is the scale parameter, and α is the shape parameter or the Weibull slope at the stress level S.
The survival probability function (L n ) or the reliability function of the Weibull distribution can be expressed by Equation [6] as follows (6,8,22,23): [6] where L n is the confidence or survival probability function, n is the fatigue life, and α and u are the shape and scale parameters respectively of the Weibull distribution. The graphical, moment and maximum likelihood methods have been used by several researchers (3,6,8,22) to obtain the values of the shape parameter (α) and the scale parameter (u) of the Weibull distribution at each stress level (S).
Other methods, such as the S-N power relationship, can be used to obtain a single value of the shape parameter (α) for all stress levels (3,6,24), whereas the scale parameter (u) is different at each stress level. This method is based on the approximate assumption of constant variance for all stress levels. This method was applied to the fatigue-life data for both stress ratios (R=0 and R=0.1), and is discussed in detail in the following sections.

Estimating Weibull distribution parameters using S-N relationship
As mentioned in the previous section, the S and N can be related by means of a power equation, as shown in Equations [3] and [4]. Equation [4] can be rewritten in a linear relationship format as given in Equation [7]: [7] where , , , and As mentioned in the previous section, the concrete fatigue life (N) follows the Weibull distribution (25). The standard deviation of log (N) and the Weibull distribution parameter (α) are interrelated, as presented in Equation [8]: [8] where σ is the constant standard deviation of in Equation [7] for all stress levels (S). The Weibull distribution parameter u may be obtained from the expression in Equation [9] (6, 26): [9] The values of C and m can be determined from the plot of X and Y in Equation [7]. The values of C and m obtained for NWC at different stress ratios are given in Table 3. The calculated values of σ for the stress ratios R = 0 and R = 0.1 are 1.108 and 1.229, respectively. Thus, the estimated values of the shape parameter (α) of the Weibull distribution are 1.158 for the stress ratio R = 0 and 1.044 for the stress ratio R = 0.1. The values of the scale parameters u of the Weibull distribution for each stress level (S) were estimated using Equation [9], as presented in Table 3.
It can be seen in Table 3 that for all the stress levels there is only one value for the shape parameter α of the Weibull distribution, whereas the scale parameter (u) of the Weibull distribution was calculated separately for each stress level (S).

DESIGN FATIGUE LIFE AND FAILURE PROBABILITY
The fatigue-life data of the plain NWC used in this analysis showed a large scatter at a given stress level due to the heterogeneity and inherent material variability of concrete. Therefore, a design fatigue life should be selected with an acceptable failure probability. As discussed above, the equivalent fatigue lives and the fatigue lives with R = 0.1 follow the two-parameter Weibull distribution at all stress levels (S). Therefore, the design fatigue life (N D ) with different failure probabilities can be calculated using the Weibull distribution function. The design fatigue life (N D ) which incorporates an acceptable failure probability (P f ) at a specific stress level (S) can be obtained by rewriting Equation [5], as shown in Equation [10]. [10] The design flexural fatigue lives (N D ) for NWC at different stress levels (S) with the corresponding failure probabilities (P f ) of 0.01, 0.05, 0.10, 0.2, 0.25, and 0.50 were accordingly calculated using Equation [10]. The corresponding α and u values for different stress levels (S) as given in Table 3 were used to calculate the N D for two different stress ratios. The calculated design fatigue lives (N D ) are presented in Tables 4 and 5

Wöhler fatigue equations
The Wöhler fatigue equations for NWC were generated using the equivalent fatigue-life data discussed   Table 1, the Wöhler fatigue equations were generated for the stress ratios R = 0 and R = 0.1, as shown in Table 6 below. From these equations, the fatigue strength or the fatigue stress level (S) could be determined for the desired number of loading cycles.
The fatigue strength and the endurance limits are important design parameters for the structures (e.g., bridge decks, highways, and airfield pavements) which are subjected to repeated loads because these structures are designed based on the endurance limit of the concrete. Most of the studies (27-29) defined the endurance limit as the maximum stress level at which structures are able to withstand two million cycles of irreversible repetitive loading. Generally, this stress level is expressed as a percentage of the static flexural strength of the concrete. The fatigue strengths for one million and two million cycles were calculated using the Wöhler fatigue equation for two different stress ratios, as shown in Table 6. It can be seen in the table that the calculated endurance limit of NWC is 56% and 61% of the static flexural strength respectively for the stress ratios R = 0 and R = 0.1. and the double-logarithm fatigue equations was compared to the results published by other studies. Based on the analysis of limited test results, Ramakrishnan et al. (28) suggested the endurance limit (2 million cycles) to be approximately 50-55% of the static ultimate flexural strength for stress ratio R = 0.0. In the present study using the double-logarithm fatigue equations, the endurance limit was found to be 52% of the static ultimate flexural strength for the stress ratio R = 0.0. This result was found using a large quantity of test data, which is more reliable than the results Ramakrishnan et al. (28) obtained from a single set of test data. According to ACI 215R-74 (30), the fatigue strengths of NWC for one million cycles are 49.66% and 56% of the static ultimate flexural strength respectively for the stress ratios R = 0.0 and R = 0.1. Using the double-logarithm fatigue equation (Table 7), the fatigue strengths for one million cycles are 54% and 59% of the flexural strength respectively for the stress ratios R = 0.0 and R = 0.1 (see Table 7). This shows that the ACI recommendation is slightly conservative compared to the result obtained from the double-logarithm fatigue equation generated using a large number of fatigue-life data. Because the proposed double-logarithm fatigue equation gives a close approximation of fatigue strength in comparison to other research and code recombination, the fatigue equations given in Table 7 may be recommended for design purposes for NWC (compressive strength range of 25-60 MPa). Furthermore, it can also be concluded that the proposed double-logarithm fatigue equation (S-N power relationship) estimates a more reasonable flexural fatigue strength for one or two million load cycles than that calculated by the Wöhler fatigue equation.

Fatigue equations using failure probability
The design fatigue lives of NWC for the failure probabilities of 0.01, 0.05, 0.10, 0.20, 0.25, and 0.50 were calculated, as discussed above, using the Weibull distribution parameters (α and u). The design fatigue lives were plotted against stress levels for each failure probability (P ƒ ).  Tables 8 and 9 for stress ratios R = 0 and R = 0.1. The fatigue strength for 2×10 6 cycles is also presented in these tables. From the equations in Tables 8 and 9 it can be shown that the design fatigue strength is reduced for a given fatigue life, with a concomitant lower probability of failure.  Table 6. Wöhler fatigue equations and the fatigue strengths (S) for one and two million cycles.

Fatigue equation using S-N power relationship
Using Equation [4] and the empirical constant values given in Table 2, the S-N power fatigue equation or double-logarithm fatigue equation for NWC was generated. The generated fatigue equations for the stress ratios R = 0 and R = 0.1 are shown in Table 7 below. Using these double-logarithm fatigue equations, the fatigue strengths (S) for two million cycles are 0.52 and 0.56 respectively for the stress ratios R = 0 and R = 0.1. Using the Wöhler fatigue equations, however, the flexural fatigue strengths of NWC for 2×10 6 cycles were found to be 0.56 and 0.61 respectively for the stress ratios R = 0.0 and R = 0.1. Therefore, the fatigue strength for 2 million cycles using the double-logarithm fatigue equation (S-N power relationship) is more conservative than that derived by using the Wöhler fatigue equation. The fatigue strength of NWC for one or two million cycles of loading obtained by the Wöhler   Generally, the failure probability of 0.5 represents the mean fatigue life of the concrete (18). It is also found that the flexural fatigue strength for 2×10 6 cycles obtained by including the failure probability of 0.5 (P ƒ = 0.5) is less than that obtained using the Wöhler fatigue equation. However, the fatigue strength for P ƒ = 0.5 is similar to those obtained by the double-logarithm fatigue equation (S-N power relationship).
In this study, only the two-parameter Weibull distribution was used to perform the probabilistic analysis. However, it will be good to perform a comparative study using the Gumbel, Weibull and log-normal distribution in the future study. From the comparative study, the best mathematical model for fatigue-life of NWC can be recommended for design.

CONCLUSIONS
The flexural fatigue test data for plain NWC were collected from 16 different sources available in the literature. The concept of the equivalent fatigue life was used to remove the effect of stress ratios and arrive at the fatigue life of concrete using the same stress ratio. The S-N curves were generated using two different methods, and probabilistic analyses were carried out to develop S-N-P f curves for dif-ferent failure probabilities. The main concluding remarks are as follows: 1. Using equivalent fatigue lives for two different stress ratios, Wöhler and double-logarithm fatigue equations were generated. Using these fatigue equations, the flexural fatigue strength for specific cycles of loading could be determined.  (Tables 8 and 9) incorporating the failure probability may be used.