To assess the fatigue behavior of a composite member, the effects of cyclic loading on each constituent component must first be understood. If any of the constituent components reaches its fatigue limit prior to the required life of the member, the member will not be capable of carrying the required loads. These loading cycles contribute to the continuous deterioration of the constitutive materials. The models of the fatigue deterioration of concrete, corroded steel bars, and CFRP sheets are detailed in the following paragraphs.
2.1.1 Concrete
Bridge girders are one of the most common structural components that are subjected to cyclic loading. The beam test results showed that the concrete softened for all of the applied compression stress ranges, resulting in a decrease in the concrete elastic modulus and a corresponding increase in the concrete strain (Hefferan et al.
2004). Therefore, it is necessary to examine the effect of fatigue loading on the strength and strain of concrete to accurately determine the fatigue behavior of CFRP-strengthened RC beams (Oudah and El-Hacha
2013a,
b). To evaluate the concrete fatigue damage, Holmen (
1982) proposed that the total
\( \varepsilon_{\text{c}} \) is the sum of two components: the first component is related to the endurance of the specimen
\( \varepsilon_{\text{ce}} \), and the second component is the strain increment or plastic strain
\( \varepsilon_{\text{cr}} \) related to the loading conditions and the number of cycles. The strain increment of concrete describes the unrecoverable component of concrete deformation. After
n cycles between the minimum and maximum stresses of
\( \sigma_{\text{c,min}}^{{}} \) and
\( \sigma_{\text{c,max}}^{{}} \), the plastic strain is expressed as follows (Song and Yu
2015):
$$ \varepsilon_{\text{cr}} = \frac{{f_{\text{c}} }}{{E_{\text{c}} }}n^{0.29} \log^{ - 1} \left( {3.92\alpha_{\text{r}} - 4.66} \right) $$
(1)
where
\( f_{\text{c}} \) is the concrete compressive strength,
\( E_{\text{c}} \) is the initial Young’s modulus of the concrete,
\( n \) is the number of loading cycles, and
\( \alpha_{\text{r}} \) is the stress ratio coefficient, which is expressed as follows:
$$ \alpha_{\text{r}} = \frac{{\sigma_{\text{c,max}}^{{}} - \sigma_{\text{c,min}}^{{}} }}{{f_{\text{c}} - \sigma_{\text{c,min}}^{{}} }} $$
(2)
The Young’s modulus of the concrete under repeated loading can be written as:
$$ E_{c}^{f} = \frac{{\sigma_{c,\hbox{max} } }}{{\left( {\varepsilon_{ce} - \varepsilon_{c0} } \right) + \varepsilon_{cr} }} $$
(3)
where
\( \varepsilon_{\text{ce}} \) is the strain under sustained loading, which is equal to
\( \sigma _{{{\text{c,max}}}} /E_{{\text{c}}} \), and
\( \varepsilon_{\text{c0}} \) is the initial tensile strain of the concrete related to the level of prestress in the CFRP sheets.
Then, the evolution of the Young’s modulus induced by fatigue loading can be derived from Eqs. (
1) and (
3):
$$ E_{{\text{c}}}^{{\text{f}}} = {{\sigma _{{{\text{c,max}}}} } \mathord{\left/ {\vphantom {{\sigma _{{{\text{c,max}}}} } {\left\{ {\frac{{\sigma _{{{\text{c,max}}}}^{{}} }}{{E_{{\text{c}}} }} - \varepsilon _{{{\text{c0}}}} + \frac{{f_{{\text{c}}} }}{{E_{{\text{c}}} }} \times n^{{0.29}} \lg ^{{ - 1}} \left( {3.92\alpha _{{\text{r}}} - 4.66} \right)} \right\}}}} \right. \kern-\nulldelimiterspace} {\left\{ {\frac{{\sigma _{{{\text{c,max}}}}^{{}} }}{{E_{{\text{c}}} }} - \varepsilon _{{{\text{c0}}}} + \frac{{f_{{\text{c}}} }}{{E_{{\text{c}}} }} \times n^{{0.29}} \lg ^{{ - 1}} \left( {3.92\alpha _{{\text{r}}} - 4.66} \right)} \right\}}} $$
(4)
Fatigue is a process of progressive internal damage in a material that has been subjected to repeated loading and is attributed to the propagation of internal micro-cracks; fatigue typically results in a significant increase in unrecoverable strain. The test results showed that fatigue failure of concrete is likely to occur when the plastic strain
\( \varepsilon_{\text{cr}} \) reaches a threshold criterion. The fatigue-failure criterion of concrete can be estimated as follows (Song and Yu
2015):
$$ \varepsilon_{\text{cr}} \ge 0.4f_{\text{c}} /E_{\text{c}} $$
(5)
2.1.2 Corroded Steel Bars
The corrosion of steel reinforcement has commonly been associated with both the carbonation of concrete and chloride ingress. The former causes a more uniform attack with a relatively limited reduction of the rebar cross-sectional area, while chloride may cause severe pitting in the rebar with highly localized reductions in the cross-sectional area; therefore, corrosion due to chloride ions is discussed in this section. Pitting corrosion decreases the steel cross-sectional area via small pit nucleation. These pits propagate over time due to corrosion, and localized corrosion that leads to pitting may provide sites for fatigue crack initiation. Experimental studies have shown that pitting corrosion is responsible for the nucleation of fatigue cracks. In this case, corrosion pits tend to increase the formation of fatigue-crack nucleation points and crack growth (Bastidas-Arteaga et al.
2009). Fatigue-crack evolution further reduces the cross-sectional area of steel rebar, thus increasing steel stress. The reduced cross-sectional area becomes critical when the actual stress in the steel exceeds the yield strength of the steel, thus causing fatigue failure. Therefore, the reduced cross-sectional area of steel rebar can effectively denote the evolution of fatigue damage. The combined action of corrosion and fatigue loading significantly degrades the strength and deformability of rebar and decreases the ratio between the yield strength and ultimate strength of the rebar. To ensure the safety of a corroded beam, the steel stress is assumed to be equal to the yield strength when rebar fatigue failure occurs. Because the fatigue load range is constant throughout the fatigue loading process, the residual cross-sectional area of the corroded steel bar can be calculated as follows:
$$ A_{\text{s}}^{\text{f}} \left( N \right) = \sigma_{\text{s,max}}^{{}} \cdot A_{\text{sc}} /f_{\text{yc}} $$
(6)
where
\( A_{\text{s}}^{\text{f}} \left( N \right) \) is the residual cross-sectional area of the corroded steel subjected to
N cycles of repeated loading,
\( \sigma_{\text{s,max}}^{{}} \) is the nominal maximum stress applied to the steel,
\( A_{\text{s}} \) is the initial cross-sectional area of the steel,
\( A_{\text{sc}} \) is the cross-sectional area of the corroded steel bars, and
\( f_{\text{yc}} \) is the yield strength of the corroded steel.
The evolution of fatigue cracks in metal can generally be subdivided into three stages: fatigue crack nucleation, crack growth, and instability. The second stage is dominant and is characterized by a constant crack growth rate. As mentioned previously, corrosion pitting increases fatigue crack nucleation and provides sites for fatigue crack initiation, which is the cause of the fatigue crack propagation of corroded steel bars in the second stage. The fatigue damage area is assumed to follow a linear relationship with the ratio of the number of cycles to the fatigue life
\( {n \mathord{\left/ {\vphantom {n N}} \right. \kern-0pt} N} \); then, the fatigue damage area under every loading cycle is equal to
\( \left[ {A_{\text{sc}} - A_{\text{s}}^{\text{f}} \left( N \right)} \right]/N \). The residual area of the corroded steel
\( A_{\text{s}}^{\text{f}} \left( n \right) \) after being subjected to
n cycles of repeated loading can be calculated as:
$$ A_{\text{s}}^{\text{f}} \left( n \right) = A_{\text{sc}} - \frac{n}{N}\left[ {A_{\text{sc}} - A_{\text{s}}^{\text{f}} \left( N \right)} \right] = A_{\text{sc}} \left[ {1 - \left( {{n \mathord{\left/ {\vphantom {n N}} \right. \kern-0pt} N}} \right) \cdot \left( {1 - \sigma_{\text{s,max}}^{{}} /f_{\text{yc}} } \right)} \right] $$
(7)
Corrosion not only affects the steel area but also alters the steel’s mechanical properties over time. The yield strength of corroded rebar can be expressed in terms of the strength of uncorroded rebar and the corrosion degree as follows (Song and Yu
2015):
$$ f_{\text{yc}} = \alpha_{\text{c}} \frac{{A_{\text{s0}} }}{{A_{\text{sc}} }}f_{\text{y0}} = \frac{{\alpha_{\text{c}} }}{{ 1 { - }\eta_{\text{s}} }}f_{\text{y0}} $$
(8)
where
\( f_{\text{y0}} \) is the yield strength of the uncorroded steel bars,
\( A_{\text{s0}} \) is the cross-sectional area of the uncorroded rebar,
\( \alpha_{\text{c}} \) is an empirical coefficient, and
\( \eta_{\text{s}} \) is the average degree of corrosion of the rebar. Based on experimental data, the coefficient
\( \alpha_{\text{c}} \) is considered to be equal to
\( 1 - 1.196\eta_{\text{s}} \) (Song and Yu
2015).
The fatigue life
N is commonly determined using the stress-life
S–
N method. In Eq. (
7), the fatigue life
N is evaluated based on the constant stress amplitude fatigue test of steel samples at
\( \rho = 0.1 \), where
ρ =
σ
s,min/
σ
s,max (Song and Yu
2015):
$$ \lg N = \left( {24.427 + 3.4\eta_{\text{s}} } \right) - \left( {7.6597 + 2.1\eta_{\text{s}} } \right)\lg \Delta \sigma $$
(9)
where
\( \Delta \sigma = \left( {\sigma_{\hbox{max} } - \sigma_{\hbox{min} } } \right) \) is the average nominal stress range and is determined by the first cycle.
Considering the test results in the literature, new studies must consider the effects of other fatigue variables, including the
\( \rho \) ratio, the stress concentration, and the pre-compressive stress, on the fatigue life of corroded steel. The stress-life
S–
N curve of corroded steel rebar in RC beams can be modified as follows:
$$ \lg N = \left( {24.427 + 3.4\eta_{\text{s}} } \right) - \left( {7.6597 + 2.1\eta_{\text{s}} } \right)\lg K_{L} K_{\text{f}} \left( {K_{\rho } - \rho } \right)\left( {\sigma_{ \hbox{max} } - \sigma_{\text{s0}} } \right) $$
(10)
where
\( K_{\rho } \),
\( K_{\text{L}} \), and
\( K_{\text{f}} \) are the fatigue strength coefficients induced at stress ratio
\( \rho \), the stress concentration factor induced at cracks in the concrete, and the corrosion pitting factor induced in the corrosion pit in the steel bars, respectively, and
\( \varepsilon_{\text{c0}} \) is the initial compressive stress of the steel bars related to the prestress level of the CFRP sheets. The values of the coefficients
\( K_{\rho } \),
\( K_{\text{L}} \), and
\( K_{\text{f}} \) are detailed in the following paragraphs.
The fatigue strength coefficient
\( K_{\rho } \) can be determined by (Song
2006):
$$ K_{\rho } = \frac{0.938}{1 - 0.6165\rho } $$
(11)
The test data of an RC beam loaded in four-point bending (Heffernan
1997) showed that the localized stresses in the tensile steel reinforcement at the concrete cracks deviated significantly from the nominal stress; These values were consistently 20–40% higher than the average stress (Hefferan et al.
2004). Deng et al. (
2007) tested RC beams strengthened with prestressed aramid fiber-reinforced polymer (AFRP) sheets and found that the prestressed AFRP sheets significantly affected the cracking and stress range of steel reinforcement. A regression analysis was performed based on the experimental test results reported by Heffernan and Deng, and the form of the equation is given as follows:
$$ K_{{\text{L}}} = \left\{ {\begin{array}{*{20}l} {1.30 - \alpha _{{\text{p}}} } &\quad {\left( {0 \le \alpha _{{\text{p}}} \le 0.24,M_{{\max }}^{{\text{f}}} > M_{0} } \right)} \\ {1.05} &\quad {\left( {0.24 \le \alpha _{{\text{p}}} ,M_{{\max }}^{{\text{f}}} > M_{0} } \right)} \\ \end{array} } \right. $$
(12)
where
\( M_{ \hbox{max} }^{\text{f}} \) is the maximum fatigue flexural moment,
\( M_{0} \) is the decompression flexural moment of the strengthening beams, and
\( \alpha_{\text{p}} \) is the effective partial prestressing ratio (PPR), which is defined as:
$$ {\text{PPR = }}\alpha_{\text{p}} = \frac{{\left( {M_{\text{u}} } \right)_{\text{p}} }}{{\left( {M_{\text{u}} } \right)_{\text{p + s}} }} = \frac{{A_{\text{f}} f_{\text{fu}}^{0} }}{{A_{\text{f}} f_{\text{fu}}^{0} + A_{\text{sc}} f_{\text{yc}} }} $$
(13)
where
\( \left( {M_{\text{u}} } \right)_{\text{p}} \) and
\( \left( {M_{\text{u}} } \right)_{\text{p + s}} \) are the ultimate moment of the prestressed CFRP sheets and the total ultimate moment, respectively,
\( A_{\text{f}} \) is the area of the CFRP sheet, and
\( f_{\text{fu}}^{0} \) is the ultimate tensile strength of the CFRP composite.
As mentioned previously, pitting corrosion produces a localized reduction in the steel cross-sectional area, but accurately measuring the pit configuration, depth, and distribution is difficult. Therefore, a large deviation exists between how the degree of corrosion and the reduction in steel area are defined based on the corrosion pit depth. Thus, the degree of corrosion in rebar (
\( \eta_{\text{s}} \)) is quantified based on the average mass loss. In Eq. (
6), the cross-sectional area of the corroded steel rebar
\( A_{\text{sc}} \) is also the average cross-sectional area of the rebar. The average mass loss does not account for the influence of local corrosion pitting on the rebar mechanical properties. The test data of RC beams showed that the localized stresses at the corrosion pits were greater than the theoretical average stresses (Ai-Hammoud et al.
2011). For the fatigue analysis of corroded steel reinforcement, the influence of a notch on the stress range should be reflected in factor
\( K_{\text{f}} \). Due to the scarcity of published data on the effect of the corrosion notch depth on the range of steel stresses in CFRP-strengthened RC beams under fatigue loading, a linear regression analysis was performed based on the experimental test results reported by Ai-Hammoud et al. (
2011). The form of the equation is given as follows (
\( R^{2} = 0.942 \)):
$$ K_{\text{f}} = 0.212\lambda + 1.0 $$
(14)
where
\( \lambda \) is the maximum pit depth.
The relationship between the maximum cross-sectional corrosion degree
\( \eta_{\text{s,amx}} \) and the average mass loss
\( \eta_{\text{s}} \) was calculated as shown in reference (An et al.
2005):
$$ \eta_{\text{s,amx}} = 0.0345 + 1.2561\eta_{\text{s}} \begin{array}{*{20}c} {} & {\left( {\eta_{\text{s}} \le 50\% } \right)} \\ \end{array} $$
(15)
Next, the corrosion pitting factor
\( K_{\text{f}} \) can be derived from Eqs. (
14) and (
15):
$$ K_{\text{f}} = 1.0 + 3.39\left( {1 - \sqrt {1.0 - 1.25\eta_{\text{s}} } } \right) $$
(16)
Corrosion pits provide sites for fatigue crack initiation, and fatigue crack evolution under repeated loading further reduces the cross-sectional area of rebar. As the fatigue cracks grow, the effective stress reaches the yield stress at some locations. The bar is assumed to abruptly break at failure when the localized fatigue stress
\( \sigma_{\text{s,max}}^{\text{f}} \) reaches the yield stress, as shown in Eq. (
17):
$$ \sigma_{\text{s,max}}^{\text{f}} \,\ge\, f_{\text{yc}} $$
(17)
where
\( \sigma_{\text{s,max}}^{\text{f}} \) is the rebar stress corresponding to the maximum fatigue load at
n cycles of repeated loading.
2.1.3 Carbon Fiber-Reinforced Polymers
For a fiber-reinforced polymer (FRP) composite, fatigue loading decreases the modulus
\( E_{\text{f}} \), as observed by Bigaud and Ali (
2014) and Ferrier et al. (
2011). Based on these studies, the deterioration coefficient
\( D_{E,f} \) of a prestressed CFRP composite can be expressed as:
$$ D_{E,f} = 1 - 0.051\log n $$
(18)
The ultimate tensile strain
\( \varepsilon_{f,ult} \) is assumed to be constant over time, and the modulus
\( E_{\text{f}}^{\text{f}} \) and residual strength
\( f_{\text{fr}} \) decrease as the number of cycles increases according to the expression:
$$ E_{f}^{f} \left( n \right) = E_{f}^{0} \times D_{E,f} $$
(19)
$$ f_{fr} \left( n \right) = f_{fu}^{0} \times D_{E,f} $$
(20)
where
\( E_{\text{f}}^{0} \) and
\( f_{\text{fu}}^{0} \) are the initial modulus and the ultimate tensile strength of the CFRP composite, respectively.
The fatigue failure process indicates that CFRP sheets fail when the maximum fatigue stress
\( \sigma_{\text{f,max}}^{\text{f}} \) reaches the residual strength of the CFRP
\( f_{\text{fr}} \); thus, the residual strength provides a safe fatigue failure criterion, which is defined by the following equation:
$$ \sigma_{\text{f,max}}^{\text{f}}\, \ge\, f_{\text{fr}} \left( n \right) $$
(21)