1. 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 (11.
Deng, P.; Matsumoto, T. (2018) Determination of dominant degradation
mechanisms of RC bridge deck slabs under cyclic moving loads. Int. J. Fatigue. 112, 328-340. https://doi.org/10.1016/j.ijfatigue.2018.03.033.
, 22.
Paluri, Y.; Noolu, V.; Mudavath, H.; Pancharathi, R.K. (2021) Flexural
fatigue behavior of steel fiber-reinforced reclaimed asphalt
pavement-based concrete: an experimental study. Pract. Period. Struct. Des. Constr. 26 [1], 04020053. https://doi.org/10.1061/(asce)sc.1943-5576.0000540.
).
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 (1-31.
Deng, P.; Matsumoto, T. (2018) Determination of dominant degradation
mechanisms of RC bridge deck slabs under cyclic moving loads. Int. J. Fatigue. 112, 328-340. https://doi.org/10.1016/j.ijfatigue.2018.03.033.
2.
Paluri, Y.; Noolu, V.; Mudavath, H.; Pancharathi, R.K. (2021) Flexural
fatigue behavior of steel fiber-reinforced reclaimed asphalt
pavement-based concrete: an experimental study. Pract. Period. Struct. Des. Constr. 26 [1], 04020053. https://doi.org/10.1061/(asce)sc.1943-5576.0000540.
3. Singh, S.P.; Kaushik, S.K. (2003) Fatigue strength of steel fibre reinforced concrete in flexure. Cem. Concr. Compos. 25 [7], 779-786. https://doi.org/10.1016/S0958-9465(02)00102-6.
).
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 (1-41.
Deng, P.; Matsumoto, T. (2018) Determination of dominant degradation
mechanisms of RC bridge deck slabs under cyclic moving loads. Int. J. Fatigue. 112, 328-340. https://doi.org/10.1016/j.ijfatigue.2018.03.033.
2.
Paluri, Y.; Noolu, V.; Mudavath, H.; Pancharathi, R.K. (2021) Flexural
fatigue behavior of steel fiber-reinforced reclaimed asphalt
pavement-based concrete: an experimental study. Pract. Period. Struct. Des. Constr. 26 [1], 04020053. https://doi.org/10.1061/(asce)sc.1943-5576.0000540.
3. Singh, S.P.; Kaushik, S.K. (2003) Fatigue strength of steel fibre reinforced concrete in flexure. Cem. Concr. Compos. 25 [7], 779-786. https://doi.org/10.1016/S0958-9465(02)00102-6.
4. Kesler, C.E. (1953) Effect of speed of testing on flexural fatigue strength of plain concrete. Highw. Res. Board Proc. 32, 251-258.
).
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 (55. Singh, S.P.; Kaushik, S.K. (2001) Flexural fatigue analysis of steel fiber-reinforced concrete. ACI Mater. J. 98 [4], 306-312. https://doi.org/10.14359/10399.
).
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 (Smax) and the minimum amplitude (Smin) 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 = Smax/fr ) and the number of loading cycles (N) to failure, where fr is the static flexural strength (modulus of rupture) of the concrete (6-86. Oh, B.H. (1986) Fatigue analysis of plain concrete in flexure. J. Struct. Eng. 112 [2], 273-288. https://doi.org/10.1061/(asce)0733-9445(1986)112:2(273).
7.
Singh, S.P.; Mohammadi, Y.; Kaushik, S.K. (2005) Flexural fatigue
analysis of steel fibrous concrete containing mixed fibers. ACI Mater. J. 102 [6], 438-444. https://doi.org/10.14359/14807.
8. Shi, X.P.; Fwa, T.F.; Tan, S.A. (1993) Flexural fatigue strength of plain concrete. ACI Mater. J. 90 [5], 435-440. https://doi.org/10.14359/3872.
). The established relationship is known as the S-N curve or the Wöhler fatigue curve (66. Oh, B.H. (1986) Fatigue analysis of plain concrete in flexure. J. Struct. Eng. 112 [2], 273-288. https://doi.org/10.1061/(asce)0733-9445(1986)112:2(273).
, 77.
Singh, S.P.; Mohammadi, Y.; Kaushik, S.K. (2005) Flexural fatigue
analysis of steel fibrous concrete containing mixed fibers. ACI Mater. J. 102 [6], 438-444. https://doi.org/10.14359/14807.
). The equation of this curve is known as the Wöhler fatigue equation, which is expressed by Equation [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 (88. Shi, X.P.; Fwa, T.F.; Tan, S.A. (1993) Flexural fatigue strength of plain concrete. ACI Mater. J. 90 [5], 435-440. https://doi.org/10.14359/3872.
).
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 (66. Oh, B.H. (1986) Fatigue analysis of plain concrete in flexure. J. Struct. Eng. 112 [2], 273-288. https://doi.org/10.1061/(asce)0733-9445(1986)112:2(273).
, 88. Shi, X.P.; Fwa, T.F.; Tan, S.A. (1993) Flexural fatigue strength of plain concrete. ACI Mater. J. 90 [5], 435-440. https://doi.org/10.14359/3872.
),
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 (99. Lee, M.K.; Barr, B.I.G. (2004) An overview of the fatigue behaviour of plain and fibre reinforced concrete. Cem. Concr. Compos. 26 [4], 299-305. https://doi.org/10.1016/S0958-9465(02)00139-7.
).
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=Smin/Smax) at various stress levels (S).
2. FLEXURAL FATIGUE LIFE OF NORMAL-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 (44. Kesler, C.E. (1953) Effect of speed of testing on flexural fatigue strength of plain concrete. Highw. Res. Board Proc. 32, 251-258.
, 1010. Murdock, J.W.; Kesler, C.E. (1958) Effect of range of stress on fatigue strength of plain concrete beams. ACI J. Proc. 55 [8], 221-231. https://doi.org/10.14359/11350.
). 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.
2.1 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 = Smax/fr ) and different stress ratios (R = Smin/Smax),
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. (88. Shi, X.P.; Fwa, T.F.; Tan, S.A. (1993) Flexural fatigue strength of plain concrete. ACI Mater. J. 90 [5], 435-440. https://doi.org/10.14359/3872.
), 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 (88. Shi, X.P.; Fwa, T.F.; Tan, S.A. (1993) Flexural fatigue strength of plain concrete. ACI Mater. J. 90 [5], 435-440. https://doi.org/10.14359/3872.
, 1111. Zhang, J.; Stang, H.; Li, V.C. (1999) Fatigue life prediction of fiber reinforced concrete under flexural load. Int. J. Fatigue. 21 [10], 1033-1049. https://doi.org/10.1016/S0142-1123(99)00093-6.
), the test data with different stress ratios were converted to the equivalent fatigue-life (EN) using Equation [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 (Smax) to the modulus of rupture (fr ), that is, (Smax/fr ). 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 (88. Shi, X.P.; Fwa, T.F.; Tan, S.A. (1993) Flexural fatigue strength of plain concrete. ACI Mater. J. 90 [5], 435-440. https://doi.org/10.14359/3872.
, 1212. Tepfers, R.; Kutti, T. (1979) Fatigue strength of plain, ordinary, and lightweight concrete. ACI J. Proc. 76 [5], 635-652. https://doi.org/10.14359/6962.
).
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 (1313. Holman, J.P. (2011) Experimental methods for engineers, 8th edition. McGraw-Hill, (2011).
, 1414.
Mohammadi, Y.; Kaushik, S.K. (2005) Flexural fatigue-life distributions
of plain and fibrous concrete at various stress levels. J. Mater. Civ. Eng. 17 [6], 650-658. https://doi.org/10.1061/(asce)0899-1561(2005)17:6(650).
).
2.2 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 (Cc ) of the regression analysis is 0.87 for both stress ratios, as shown in Figure 1. This low value of the Cc (less than 0.9) indicates the scatter characteristics of the fatigue test data. However, the value of Cc > 0.70 for the plotted data refers to a strong relationship between S and log(N) (1515.
Ang, A.H.S.; Tang, W.H. (2007) Probability concepts in engineering:
emphasis on applications to civil and environmental engineering, 2nd ed.
John Wiley and Sons Inc., New York, (2007).
). 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.
| 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.
2.3 Power relationship of fatigue life (double-logarithm fatigue equation)
⌅The
power relationship of fatigue life was mainly developed for concrete
pavements to estimate the flexural fatigue life of concrete (1616.
Treybig, H.J.; Smith, P.; VonQuintus, H. (1977) Overlay design and
reflection cracking analysis for rigid pavements -- Vol. 1 Development
of new design criteria. Austin, TX United States 78746.
). It relates the dimensionless stress level (S=Smax/fr ) to the number of loading cycles (N) given by Equation [3] (55. Singh, S.P.; Kaushik, S.K. (2001) Flexural fatigue analysis of steel fiber-reinforced concrete. ACI Mater. J. 98 [4], 306-312. https://doi.org/10.14359/10399.
, 66. Oh, B.H. (1986) Fatigue analysis of plain concrete in flexure. J. Struct. Eng. 112 [2], 273-288. https://doi.org/10.1061/(asce)0733-9445(1986)112:2(273).
, 1717. Wirsching, P.H.; Yao, J.T.P. (1982) Fatigue reliability: Introduction. J Struct Div. 108 [1], 3-23.
, 1818. Koltsida, I.S.; Tomor, A.K.; Booth, C.A. (2018) Probability of fatigue failure in brick masonry under compressive loading. Int. J. Fatigue. 112, 233-239. https://doi.org/10.1016/j.ijfatigue.2018.03.023.
):
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 = Smax/fr is expressed as a dimensionless form. Here, Smax is the maximum applied stress and fr 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]):
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 (Cc ) 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.
| R | m | C |
|---|---|---|
| 0.0 | 16.3820 | 42.20 |
| 0.1 | 18.0202 | 63.96 |
3. PROBABILISTIC ANALYSIS OF FATIGUE-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 (1919. Gumbel, E.J. (1958) Statistics of extremes. Columbia University Press, (1958).
)
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 (33. Singh, S.P.; Kaushik, S.K. (2003) Fatigue strength of steel fibre reinforced concrete in flexure. Cem. Concr. Compos. 25 [7], 779-786. https://doi.org/10.1016/S0958-9465(02)00102-6.
, 66. Oh, B.H. (1986) Fatigue analysis of plain concrete in flexure. J. Struct. Eng. 112 [2], 273-288. https://doi.org/10.1061/(asce)0733-9445(1986)112:2(273).
, 88. Shi, X.P.; Fwa, T.F.; Tan, S.A. (1993) Flexural fatigue strength of plain concrete. ACI Mater. J. 90 [5], 435-440. https://doi.org/10.14359/3872.
, 2020. Oh, B.H. (1991) Fatigue-life distributions of concrete for various stress levels. ACI Mater. J. 88 [2], 122-128. https://doi.org/10.14359/1870.
, 2121.
Sohel, K.M.A.; Al-Jabri, K.; Zhang, M.H.; Liew, J.Y.R. (2018) Flexural
fatigue behavior of ultra-lightweight cement composite and high strength
lightweight aggregate concrete. Constr. Build. Mater. 173, 90-100. https://doi.org/10.1016/j.conbuildmat.2018.03.276.
)
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]) (66. Oh, B.H. (1986) Fatigue analysis of plain concrete in flexure. J. Struct. Eng. 112 [2], 273-288. https://doi.org/10.1061/(asce)0733-9445(1986)112:2(273).
, 88. Shi, X.P.; Fwa, T.F.; Tan, S.A. (1993) Flexural fatigue strength of plain concrete. ACI Mater. J. 90 [5], 435-440. https://doi.org/10.14359/3872.
, 2222. Weibull, W, (1961) Fatigue testing and analysis of results. Oxford: Pergamon Press, (1961).
):
where Pf (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 (Ln ) or the reliability function of the Weibull distribution can be expressed by Equation [6] as follows (66. Oh, B.H. (1986) Fatigue analysis of plain concrete in flexure. J. Struct. Eng. 112 [2], 273-288. https://doi.org/10.1061/(asce)0733-9445(1986)112:2(273).
, 88. Shi, X.P.; Fwa, T.F.; Tan, S.A. (1993) Flexural fatigue strength of plain concrete. ACI Mater. J. 90 [5], 435-440. https://doi.org/10.14359/3872.
, 2222. Weibull, W, (1961) Fatigue testing and analysis of results. Oxford: Pergamon Press, (1961).
, 2323.
Correia, J.A.F.deO.; Pedrosa, B.A.S.; Raposo, P.C.; et al. (2017)
Fatigue strength evaluation of resin-injected bolted connections using
statistical analysis. Engineering. 3 [6], 795-805. https://doi.org/10.1016/j.eng.2017.12.001.
):
where Ln 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 (33. Singh, S.P.; Kaushik, S.K. (2003) Fatigue strength of steel fibre reinforced concrete in flexure. Cem. Concr. Compos. 25 [7], 779-786. https://doi.org/10.1016/S0958-9465(02)00102-6.
, 66. Oh, B.H. (1986) Fatigue analysis of plain concrete in flexure. J. Struct. Eng. 112 [2], 273-288. https://doi.org/10.1061/(asce)0733-9445(1986)112:2(273).
, 88. Shi, X.P.; Fwa, T.F.; Tan, S.A. (1993) Flexural fatigue strength of plain concrete. ACI Mater. J. 90 [5], 435-440. https://doi.org/10.14359/3872.
, 2222. Weibull, W, (1961) Fatigue testing and analysis of results. Oxford: Pergamon Press, (1961).
) 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 (33. Singh, S.P.; Kaushik, S.K. (2003) Fatigue strength of steel fibre reinforced concrete in flexure. Cem. Concr. Compos. 25 [7], 779-786. https://doi.org/10.1016/S0958-9465(02)00102-6.
, 66. Oh, B.H. (1986) Fatigue analysis of plain concrete in flexure. J. Struct. Eng. 112 [2], 273-288. https://doi.org/10.1061/(asce)0733-9445(1986)112:2(273).
, 2424. Kaur, G.; Singh, S.P.; Kaushik, S.K. (2016) Mean and design fatigue lives of SFRC containing cement-based materials. Mag. Concr. Res. 68 [7], 325-338. https://doi.org/10.1680/macr.15.00128.
), 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.
3.1 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]:
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 (2525. Freudenthal, A.M.; Gumbel, E.J. (1956) Physical and statistical aspects of fatigue. Adv. Appl. Mech. 4, 117-158. https://doi.org/10.1016/S0065-2156(08)70372-7.
). The standard deviation of log (N) and the Weibull distribution parameter (α) are interrelated, as presented in Equation [8]:
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] (66. Oh, B.H. (1986) Fatigue analysis of plain concrete in flexure. J. Struct. Eng. 112 [2], 273-288. https://doi.org/10.1061/(asce)0733-9445(1986)112:2(273).
, 2626. Wirsching, P.H.; Yao, J.T.P. (1970) Statistical methods in structural fatigue. J. Struct. Div. ASCE. 96 [6], 1201-1219.
):
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).
| 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 | |
4. 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 (ND ) with different failure probabilities can be calculated using the Weibull distribution function. The design fatigue life (ND ) which incorporates an acceptable failure probability (Pf ) at a specific stress level (S) can be obtained by rewriting Equation [5], as shown in Equation [10].
The design flexural fatigue lives (ND ) for NWC at different stress levels (S) with the corresponding failure probabilities (Pf ) 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 ND for two different stress ratios. The calculated design fatigue lives (ND ) 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 S-N-Pf curves describe the relationship between the stress level (S), the design fatigue life (ND ) and the failure probability (Pf ). Regression analyses were performed to obtain the equation of each curve.
| S | ND | |||||
|---|---|---|---|---|---|---|
| Pf = 0.01 | Pf = 0.05 | Pf = 0.10 | Pf = 0.20 | Pf = 0.25 | Pf = 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 |
| S | ND | |||||
|---|---|---|---|---|---|---|
| Pf = 0.01 | Pf = 0.05 | Pf = 0.10 | Pf = 0.20 | Pf = 0.25 | Pf = 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 |
5. DISCUSSION OF THE PROPOSED FATIGUE EQUATIONS
⌅5.1 Wöhler fatigue equations
⌅The Wöhler fatigue equations for NWC were generated using the equivalent fatigue-life 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 (27-2927.
Arora, S.; Singh, S.P. (2016) Analysis of flexural fatigue failure of
concrete made with 100% Coarse Recycled Concrete Aggregates. Constr. Build. Mater. 102, 782-791. https://doi.org/10.1016/j.conbuildmat.2015.10.098.
28.
Ramakrishnan, V.; Wu, G.Y.; Hosalli, G. (1989) Flexural fatigue
strength, endurance limit, and impact strength of fiber reinforced
concretes. Transp. Res. Rec. 1226, 17-24.
29. Johnston, C.D.;
Zemp, R.W. (1991) Flexural fatigue performance of steel fiber reinforced
concrete. Influence of fiber content, aspect ratio, and type. ACI Mater. J. 88 [4], 374-383. https://doi.org/10.14359/1875.
)
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.
| Stress ratio (R) | Wöhler fatigue equation | S for | |
|---|---|---|---|
| 1×106 cycles | 2×106 cycles | ||
| 0.0 | 0.58 | 0.56 | |
| 0.1 | 0.63 | 0.61 | |
5.2 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 2×106 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.
| Stress ratio (R) | Double-logarithm fatigue equation | S for | |
|---|---|---|---|
| 1×106 cycles | 2×106 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 double-logarithm fatigue equations was
compared to the results published by other studies. Based on the
analysis of limited test results, Ramakrishnan et al. (2828.
Ramakrishnan, V.; Wu, G.Y.; Hosalli, G. (1989) Flexural fatigue
strength, endurance limit, and impact strength of fiber reinforced
concretes. Transp. Res. Rec. 1226, 17-24.
)
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. (2828.
Ramakrishnan, V.; Wu, G.Y.; Hosalli, G. (1989) Flexural fatigue
strength, endurance limit, and impact strength of fiber reinforced
concretes. Transp. Res. Rec. 1226, 17-24.
) obtained from a single set of test data. According to ACI 215R-74 (3030.
ACI 215R-74. (1997) Considerations for design of concrete structures
subjected to fatigue loading (Reapproved 1997). ACI Committee 215,
American Concrete Institute, (1997).
), 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.
5.3 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 (Pf ). 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 2×106 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.
| Pf | 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 2×106 cycles
| Pf | 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 2×106 cycles
Generally, the failure probability of 0.5 represents the mean fatigue life of the concrete (1818. Koltsida, I.S.; Tomor, A.K.; Booth, C.A. (2018) Probability of fatigue failure in brick masonry under compressive loading. Int. J. Fatigue. 112, 233-239. https://doi.org/10.1016/j.ijfatigue.2018.03.023.
). It is also found that the flexural fatigue strength for 2×106 cycles obtained by including the failure probability of 0.5 (Pf = 0.5) is less than that obtained using the Wöhler fatigue equation. However, the fatigue strength for Pf = 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.
6. 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-Pf curves for different failure probabilities. The main concluding remarks are as follows:
-
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.
-
According to the double-logarithm 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 double-logarithm 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 S-N power relationship may be recommended to predict the design fatigue strength of NWC (compressive strength range of 25-60 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.