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 normalweight 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 SN power relationship) for predicting the flexural fatigue life of normalweight 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.
Resumen
La vida a fatiga debe ser considerada en el diseño de muchas estructuras de hormigón bajo varios niveles de tensión y relaciones de tensión. Muchos resultados de pruebas de fatiga por flexión del hormigón convencional (densidad normal) están disponibles en la literatura y casi todos proporcionan diferentes ecuaciones de fatiga. Sin embargo, es necesario tener una ecuación de fatiga común para predecir la vida a fatiga de diseño de las estructuras de hormigón bajo carga de flexión con una precisión razonable. Por lo tanto, se creó una base de datos de resultados de ensayos de fatiga por flexión para hormigón con resistencias que oscilan entre 25 y 65 MPa; esta base de datos se utilizó para generar nuevas ecuaciones de fatiga (ecuación de fatiga de Wöhler y relación de potencia SN) para predecir la vida a fatiga por flexión del hormigón de densidad normal. El concepto de vida a fatiga equivalente se introdujo para obtener una ecuación de fatiga utilizando la misma relación de tensión. También se llevó a cabo un análisis probabilístico para desarrollar ecuaciones de fatiga por flexión que incorporen probabilidades de falla.
ConcreteFatigueFlexural strengthDurabilityMechanical propertiesHormigónFatigaResistencia a flexiónDurabilidadPropiedades mecánicasSultan Qaboos UniversityIG/ENG/CAED/18/01The authors acknowledge the financial support of an internal research grant from Sultan Qaboos University, Oman, grant number IG/ENG/CAED/18/01.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 microcracks. When the repeated loads are applied at high frequency for a prolonged period, this can eventually lead to fatigue failure (13). 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 microsized capillary pores and millimetersized 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 (14). 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}/f_{
r
} ) and the number of loading cycles (N) to failure, where f_{
r
} is the static flexural strength (modulus of rupture) of the concrete (68). The established relationship is known as the SN 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]:
S=Smax/fr=a+blogN
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 singlelogarithm 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. Lognormal distribution and three or twoparameter 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 fatiguelife 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 normalweight 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 fatiguelife distribution of NWC for different stress ratios (R=S_{min}/S_{max}) at various stress levels (S).
Flexural fatigue life of normalweight 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 120 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 120 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}/f_{
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 fatiguelife 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 fatiguelife (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 fatiguelife (EN) using Equation [2]:
EN=N1R
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 Rvalue. 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 watercement 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 normalweight concrete
The equivalent fatigue lives were used to obtain the SN curves for the stress ratios R = 0 and R = 0.1. Figure 1 shows the plot of SN curves using the 16 flexural fatiguelife data of NWC. A linear regression analysis based on the bestfit curve was performed. From the equations of the bestfit 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.
<italic>SN</italic> curves (Wöhler fatigue curves) of NWC for (a) <italic>R</italic> = 0 and (b) <italic>R</italic> = 0.1.Coefficients <italic>a</italic> and <italic>b</italic> of Wöhler fatigue equation for plain NWC.
Stress ratio (R)
Coefficient of Wöhler fatigue equation
a
b
0.0
1.0367
0.0758
0.1
1.0367
0.0682
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}/f_{
r
} ) to the number of loading cycles (N) given by Equation [3] (5, 6, 17, 18):
NSm=C
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}/f_{
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]):
logN=logCmlogS
The values of C and m can be determined from the plot of log(N) versus log(S), as shown in Figure 2. For instance, from Figure 2 (a), the values of m and C are 16.382 and 42.20 respectively for the stress ratio R=0. The correlation coefficient (C_{
c
} ) of the regression analysis is 0.86 for both stress ratios, as shown in Figure 2. The empirical constants (m and C) obtained for different stress ratios are summarized in Table 2.
Estimating the empirical constants of <italic>SN</italic> power relationships for stress ratio <italic>R</italic>: (a) <italic>R</italic> = 0 and (b) <italic>R</italic> = 0.1.Parameters of the power relationship of the equivalent fatigue life.
R
m
C
0.0
16.3820
42.20
0.1
18.0202
63.96
Probabilistic analysis of fatiguelife 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 fatiguelife 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 fatiguelife for a certain percentage of failure probability. Many studies (3, 6, 8, 20, 21) have shown that the twoparameter Weibull distribution can be used to describe the distribution of the fatiguelife data of cementitious materials, since it provides safer and greater reliability, as proved both statistically and experimentally. Therefore, the twoparameter Weibull distribution was used in this study to describe the probabilistic distribution of the flexural fatiguelife of the NWC. The failure probability function or the cumulative distribution function (CDF) of the twoparameter Weibull distribution is expressed as follows (Equation [5]) (6, 8, 22):
PfN=1enuα;n>0
where P_{
f
} (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):
Ln=1PfN=enuα;n>0
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 SN 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 fatiguelife 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 SN 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]:
Y=A+BX
where A=log(C), B=m, X=log(S), and Y=log(N)
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]:
σ=πα6
where σ is the constant standard deviation of Y=log(N) in Equation [7] for all stress levels (S). The Weibull distribution parameter u may be obtained from the expression in Equation [9] (6, 26):
lnu=0.5772α+lnCSmaxfrm
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).
Weibull distribution parameters (α and <italic>u</italic>) using the <italic>SN</italic> relationship method for various stress levels (<italic>S</italic>) and stress ratios (<italic>R</italic>).
S
R = 0
R = 0.1
α
u
α
u
0.90
1.325
367
1.126
713
0.85
1.325
935
1.126
1997
0.80
1.325
2524
1.126
5954
0.75
1.325
7266
1.126
19049
0.70
1.325
22499
1.126
66041
0.65
1.325
75757
1.126
251066
0.60
1.325
281124
1.126
1062203
Design fatigue life and failure probability
The fatiguelife 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 twoparameter 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].
ND=uln11Pf1α
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 below for the stress ratios R = 0 and R = 0.1 respectively. In addition, the design fatigue lives calculated with different failure probabilities are plotted in Figures 3 and 4 below for the stress ratios R = 0.0 and R = 0.1 respectively. These SNP_{
f
} curves describe the relationship between the stress level (S), the design fatigue life (N_{
D
} ) and the failure probability (P_{
f
} ). Regression analyses were performed to obtain the equation of each curve.
Design fatigue life (<italic>N</italic>
<sub>
<italic>D</italic>
</sub> ) for the stress ratio <italic>R</italic>=0 and different failure probabilities (<italic>P</italic>
<sub>
<italic>f</italic>
</sub> ).
S
N_{
D
}
P_{
f
} = 0.01
P_{
f
} = 0.05
P_{
f
} = 0.10
P_{
f
} = 0.20
P_{
f
} = 0.25
P_{
f
} = 0.50
0.90
11
39
67
118
143
278
0.85
29
99
171
301
365
709
0.80
78
268
462
814
986
1914
0.75
225
772
1329
2342
2837
5510
0.70
698
2390
4115
7251
8784
17061
0.65
2351
8047
13855
24415
29576
57446
0.60
8724
29860
51416
90600
109754
213174
Design fatigue life (<italic>N</italic>
<sub>
<italic>D</italic>
</sub> ) for the stress ratio <italic>R</italic>=0.1 and different failure probabilities (<italic>P</italic>
<sub>
<italic>f</italic>
</sub> ).
S
N_{
D
}
P_{
f
} = 0.01
P_{
f
} = 0.05
P_{
f
} = 0.10
P_{
f
} = 0.20
P_{
f
} = 0.25
P_{
f
} = 0.50
0.90
12
51
97
188
236
515
0.85
34
143
271
527
661
1442
0.80
100
426
807
1572
1970
4300
0.75
321
1364
2583
5030
6302
13758
0.70
1112
4727
8957
17438
21849
47698
0.65
4228
17971
34050
66292
83064
181331
0.60
17887
76033
144060
280467
351424
767172
<italic>SN</italic><italic>P</italic>
<sub>
<italic>f</italic>
</sub> curves for stress ratio <italic>R</italic> = 0 and the failure probabilities of (a) 0.01, (b) 0.05, (c) 0.10, (d) 0.20, (e) 0.25, and (f) 0.50.
<italic>SN P</italic>
<sub>
<italic>f</italic>
</sub> curves for stress ratio <italic>R</italic> = 0.1 and the failure probabilities of (a) 0.01, (b) 0.05, (c) 0.10, (d) 0.20, (e) 0.25, and (f) 0.50. Discussion of the proposed fatigue equationsWöhler fatigue equations
The Wöhler fatigue equations for NWC were generated using the equivalent fatiguelife data discussed in section 2.2. Using Equation [1] and the coefficients in 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 (2729) 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.
Wöhler fatigue equations and the fatigue strengths (<italic>S</italic>) for one and two million cycles.
Stress ratio (R)
Wöhler fatigue equation
S for
1×10^{6} cycles
2×10^{6} cycles
0.0
0.58
0.56
0.1
0.63
0.61
Fatigue equation using SN power relationship
Using Equation [4] and the empirical constant values given in Table 2, the SN power fatigue equation or doublelogarithm 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 doublelogarithm 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 doublelogarithm fatigue equation (SN power relationship) is more conservative than that derived by using the Wöhler fatigue equation.
<italic>SN</italic> power relationship or doublelogarithm fatigue equation for NWC.
Stress ratio (R)
Doublelogarithm fatigue equation
S for
1×10^{6} cycles
2×10^{6} cycles
0.0
0.54
0.52
0.1
0.59
0.56
The fatigue strength of NWC for one or two million cycles of loading obtained by the Wöhler and the doublelogarithm 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 5055% of the static ultimate flexural strength for stress ratio R = 0.0. In the present study using the doublelogarithm 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 215R74 (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 doublelogarithm 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 doublelogarithm fatigue equation generated using a large number of fatiguelife data. Because the proposed doublelogarithm 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 2560 MPa). Furthermore, it can also be concluded that the proposed doublelogarithm fatigue equation (SN 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_{
f
} ). Fatigue equations were obtained by linear regression analysis of the plotted data, as shown in Figures 3 and 4. For the failure probabilities of 0.01, 0.05, 0.10, 0.20, 0.25, and 0.50 the generated fatigue equations are given in 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.
Fatigue equation of NWC for <italic>R</italic> = 0.0 and different failure probabilities.
P_{
f
}
Fatigue equation
S^{*}
0.01
log(N) = 0.3064  16.382 log(S)
0.43
0.05
log(N) = 0.8407  16.382 log(S)
0.46
0.10
log(N) = 1.0767  16.382 log(S)
0.48
0.20
log(N) = 1.3228  16.382 log(S)
0.50
0.25
log(N) = 1.4061  16.382 log(S)
0.50
0.50
log(N) = 1.6944  16.382 log(S)
0.52
^{*}S for 2×10^{6} cycles
Fatigue equation of NWC for <italic>R</italic> = 0.1 and different failure probabilities.
P_{
f
}
Fatigue equation
S^{*}
0.01
log(N) = 0.2548  18.02 log(S)
0.46
0.05
log(N) = 0.8832  18.02 log(S)
0.50
0.10
log(N) = 1.1608  18.02 log(S)
0.52
0.20
log(N) = 1.4501  18.02 log(S)
0.54
0.25
log(N) = 1.5481  18.02 log(S)
0.55
0.50
log(N) = 1.8871  18.02 log(S)
0.57
^{*}S for 2×10^{6} cycles
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_{
f
} = 0.5) is less than that obtained using the Wöhler fatigue equation. However, the fatigue strength for P_{
f
} = 0.5 is similar to those obtained by the doublelogarithm fatigue equation (SN power relationship).
In this study, only the twoparameter Weibull distribution was used to perform the probabilistic analysis. However, it will be good to perform a comparative study using the Gumbel, Weibull and lognormal distribution in the future study. From the comparative study, the best mathematical model for fatiguelife 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 SN curves were generated using two different methods, and probabilistic analyses were carried out to develop SNP_{
f
} curves for different failure probabilities. The main concluding remarks are as follows:
Using equivalent fatigue lives for two different stress ratios, Wöhler and doublelogarithm fatigue equations were generated. Using these fatigue equations, the flexural fatigue strength for specific cycles of loading could be determined.
According to the doublelogarithm fatigue equation, the fatigue strength for two million cycles is 52% and 56% of the static flexural strength respectively for the stress ratios R = 0.0 and R = 0.1. In comparison, these values are 56% and 61% of the static flexural strength respectively according to the Wöhler fatigue equation for the stress ratios R = 0.0 and R = 0.1.
The doublelogarithm fatigue equation estimates a more reasonable flexural fatigue strength for one and two million fatigue cycles than the Wöhler fatigue equation. Therefore, the fatigue equations obtained by the SN power relationship may be recommended to predict the design fatigue strength of NWC (compressive strength range of 2560 MPa).
Considering the fatigue strength together with failure probability leads to more conservative conclusions than that without considering the failure probability. For the safe design of the concrete structures under flexural loading, the developed fatigue equations (Tables 8 and 9) incorporating the failure probability may be used.
Acknowledgments
The authors acknowledge the financial support of an internal research grant from Sultan Qaboos University, Oman, grant number IG/ENG/CAED/18/01.
Author contributions
Conceptualization: K. M. A. Sohel, M. H. S. AlHinai, A. Alnuaimi, M. AlShahri, S. ElGamal. Data curation: K. M. A. Sohel, M. H. S. AlHinai. Formal analysis: K. M. A. Sohel; M. H. S. AlHinai. Funding acquisition: K. M. A. Sohel. Investigation: K. M. A. Sohel, M. H. S. AlHinai. Methodology: K. M. A. Sohel, M. H. S. AlHinai, A. Alnuaimi, M. AlShahri, S. ElGamal. Project administration: K. M. A. Sohel. Resources: K. M. A. Sohel, A. Alnuaimi. Software: K. M. A. Sohel, M. H. S. AlHinai. Supervision: K. M. A. Sohel, A. Alnuaimi; M. AlShahri, S. ElGamal. Validation: K. M. A. Sohel, M. H. S. AlHinai. Visualization: K. M. A. Sohel, M. H. S. AlHinai, A. Alnuaimi, M. AlShahri, S. ElGamal. Writing, original draft: K. M. A. Sohel, M. H. S. AlHinai. Writing, review & editing: K. M. A. Sohel; M. H. S. AlHinai; A. Alnuaimi; M. AlShahri; S. ElGamal.
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f_{c}, f_{r}, and R
Fatigue life N for different stress level S
0.95
0.90
0.85
0.8
0.75
0.7
0.65
0.60
Oh, [20]f_{c} = 27 MPa;f_{r} = 4.58 MPa;R = 0.1
1038
15210
164097
1064
15618
176071
1620
19286
233916
1758
19598
245794
1770
19849
249906
1814
20694
256619
1872
21046
293559
1940
21334
334895
1954
23662
358636
2047
24345
385780
2107
24820
435673
2162
40809
635258
2620
52516
724621
3150


Arora and Singh [27]f_{c} = 41.77 MPa;f_{r} = 5.10 MPa; R =0.1
444^{*}
10781
100801
1137
13879
142054
1367
18489
187623
1678
21945
220075
1945
25467
260685
2271
31256
323068
2605
36543
360845
2647
42842
456944
3096
46951
512089
3987
51348
558973
Kaur et al. [24] f_{c} = 40.18 MPa;f_{r} = 8.08 MPa; R =0.10
682
991^{**}
2607^{**}
137850
754
7778
34205
238539
914
9380
46786
278282
982
15399
71264
418536
1387
17478
93839
568538
1584
30053
104357
584786
1835
38659
131770
625875
2144
40327
203351
1692796
2544
61244
279493
1846843
2865
74546
415850
2000000^{*}
Mohammadi and Kaushik [14]f_{c} = 58 MPa;f_{r} = 5.35 MPa; R =0.1
942
4664
53322
1205
5655
56453
1290
5963
59493
1347
6614
63997
1386
6773
68387
1593
7621
81038
1664
8903
94102
1781
9379
114214
1902
10986
138563
2644
15385
189550
4482^{*}

288054^{*}
Liu et al. [31]f_{c} = 43.4 MPa;f_{r} = 5.6 MPa; R =0.1
450
12215
27945
39480
535
14941
28935
48245
1090
20686
36365
59910
Harwalkar and Awanti [32]f_{c} = 62.30 MPa;f_{r} = 6.90 MPa; R =0.1
22
84
158
1327
5289
16488
43
97
284
1489
7213
20312
69
105
312
2596
8863
22268
78
152
382
3642
10322
34511
82
184
411
4149
12723
39920
94
198
474
5218
16523
46718
102
288
578
6629
18708
51512
110
432
694
8383
20391
61512
122
682
916
9558
21262
77812
138
730
1182
12009
23992
81800




24771
92477




27344
100000^{*}




32811
100000^{*}




40887
100000^{*}




44816
100000^{*}
Johnston and Zemp [29]f_{c} = 51.00 MPa;f_{r} = 4.45 MPa R =0.1
10
100
5100
5000
45
400
8000
35000
55
1610
11300
71000
195
2800
16200
100000
260
4350
32000
127000
365
6020
33000
900
Tan et al. [33] Min^{m}f_{c} = 30 MPa;f_{r} = 5.9 MPa; R =0.1
239
6594
76164
348
28156
94618
513
41645
138495
Shi et al. [8]f_{c} = 30 MPa;f_{r} = 6.08 MPa; R =0.08
9
49
10184
46836
206479
73
11808
125934
346047
76
21684
129009
436123
150
21747
150331
628962
160
43683
159033
694263
230
49392
164795
883301
402
50997
166287
1956530
70937
170426
2032902
72266
191828
2672740
77122
202916
79778
82905
100411
101918
Shi et al. [8]f_{c} = 30 MPa;f_{r} = 6.08 MPa; R =0.2
77
1398
22511
34206
375170
4829
41730
566140
6400
177807
618936
9059

2011017


2434133
Zhang et al. [34]f_{c} = 50.70 MPa;f_{r} = 7.19 MPa; R =0.2
14
69
277
2330
20550
82890
16
91
410
2640
22030
99220
20
95
431
3310
28110
137150
24
103
693
4170
31200
168100
26
111
744
5010
34250
208750
27
146
987
7460
54630
219710
27
163
1052
10050
62430
387100
28
204
1390
13230
62610
409610
41
462
1948
16980
152490
467990
76
600
2192
24330
152740
1407700
Lee et al. [35]f_{c} = 37 MPa;f_{r} = 5.95 MPa; R =0.02 to 0.03
40
400
30980
1053500
80
590
82340
2109950
100
1430
97230

1150
6110
319700

1844
6337
915240

3140
29200


Thomas [36]f_{c} = 35.9 MPa;f_{r} = 5.79 MPa; R =0.02
1340
7240
33190
257570
1870
8400
51180
1287300
3240
12130
62050
2316280^{*}
3570
12710
65170
2138260^{*}
4210
22660
127490
2112750^{*}
Thomas [36]f_{c} = 50.90 MPa;f_{r} = 5.40 MPa; R =0.01 to 0.02
42
401
34520
11920
233
3740
47970
114440
434
4300
48440
639
9440
95190
953
11045
830880
1040
80091

Hanumantharayagouda and Patil [37]f_{c} = 52.00 MPa;f_{r} = 4.62 MPa; R =0.01
6784
7325
19340
25349
66120
8450
8735
21758
48323
69214
9042
9745
22378
49892
55397
Paluri et al. [2]f_{c} = 52.00 MPa;f_{r} = 4.62 MPa; R =0.01
63
2619
19983
93
3553
24418
139
4478
28299
Zhang et al. [34]f_{c} = 50.70 MPa;f_{r} = 7.19 MPa; R =0
39
121
637
2830
13150
72880
45
168
655
4280
18320
77800
46
175
923
4530
66360
86360
72
364
1327



94





Mithun et al. [38]f_{c} = 57 MPa;f_{r} = 7.05 MPa; R = 0