# Fatigue Damage Assessment for Concrete Structures Using a Frequency-Domain Method

Authors: Hongyan Ding; Qi Zhu; Puyang Zhang
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

A fatigue damage assessment for concrete was carried out according to Eurocode 2. Three frequency-domain methods, the level crossing counting (LCC) method, the range counting (RC) method, and a new proposed method, were used for the damage assessment. The applicability of these frequency-domain methods was evaluated by comparison with the rainflow counting method in the time domain. A preliminary numerical study was carried out to verify the applicability of the frequency-domain methods for stress processes with different bandwidths; thus, the applicability of the LCC method and the new method was preliminarily confirmed. The fatigue strength of concrete had a minor effect on the fatigue damage assessment. The applicability of the LCC and the new methods deteriorated for relatively low coefficients of variance of the stress process because the ultimate number of constant amplitude cycles was sensitive to the range of the cycles. The validity of the joint probability functions of the two methods was proven using a numerical simulation. The integration intervals of the two frequency-domain methods were varied to estimate the lower and upper bounds on the fatigue damage, which can serve as references to evaluate the accuracy of the time-domain method results.


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## Body

## 1. Introduction

Fatigue is the process of gradual damage to materials that are subjected to continually changing stresses. Concrete fatigue is primarily a problem for offshore structures, railway sleepers, and bridges, which are often exposed to alternating loadings [1]. Unlike steel structures, both the range and level of stress affect fatigue damage in concrete structures [2]. In this study, the European Standard, Eurocode 2 [3], was applied to assess the fatigue damage of concrete structures.A well-established procedure for fatigue damage assessment involves the time-domain analysis of the stress processes that a structure will be subjected to over its lifetime [4–6]. Stress processes in structures that are induced by wind, waves, or road irregularities often occur on a fairly irregular and random basis; however, only the ultimate number of stress cycles for a constant stress range and the mean stress that can be sustained up to failure is known. Thus, a description of the cycles in the stress records in terms of parameters such as the number of counted cycles, the distribution of cycle amplitudes, and the means is required. The counting method [7] was applied in this study. Level crossing counting (LCC), peak counting (PC), simple range counting (RC), and rainflow counting (RFC) are the counting methods most commonly used in engineering practice. The RFC method is widely accepted as the most efficient counting method available [8].The fatigue damage of structures can also be assessed in the frequency domain. The frequency-domain method is faster and more cost efficient than the time-domain method [9–11]. Variations in parameters and optimization can be rapidly executed for a properly functioning frequency-domain method. The stress processes in the frequency domain are represented using a spectral formulation, such as a power spectral density (PSD) function, which provides information on the power distribution over a range of frequencies. The firstn (typically 4) moments of the PSD function are used to find the probability density function (PDF) for the peaks and valleys of the stress process (i.e., the range of the stress and mean stress) [12]. The fatigue damage can be calculated by integration using the PDF and Palmgren-Miner’s rule.In this study, PDFs that are available in the literature and a PDF developed by the authors were applied to calculate the fatigue damage of concrete structures. The RFC method in the time domain is the standard method for fatigue assessment. The results of different frequency-domain methods were compared with the results of the RFC method to select the most effective frequency-domain methods. A preliminary numerical study was carried out to verify the applicability of the frequency-domain methods for stress processes with different bandwidths; thus, the applicability of the LCC method and the new method was preliminarily confirmed. The integration intervals of the two frequency-domain methods were varied to estimate the lower and upper bounds on the fatigue damage, which can serve as references to evaluate the accuracy of the time-domain method results. Few published reports on fatigue assessment for concrete in the frequency domain are currently available; thus, the results of this study can serve as a useful reference in this field [13].


## 2. Basic Theory

### 2.1. Frequency-Domain Process

A stress processh(t) is assumed to be a stationary Gaussian process. The autocorrelation function of h(t) is defined as follows [14]:
(1)Kτ≡htht+τ¯=lim⁡T→∞12T∫-TThtht+τdτ.The Fourier transform of the autocorrelation function is expressed as(2)Kτ=∫-∞∞Sfe-iωτdf,
where the new function, S(f), is the PSD of h(t). S(f) can also be written in terms of K(τ):
(3)Sf=∫-∞∞Kτeiωτdτ.Equations (2) and (3) are known as the Wiener-Khintchine relations.We are typically not interested in distinguishing a PSD that is associated with a negative frequency from one that is associated with a positive frequency. A single-sided PSD can be defined as follows:(4)S(f)¯=Sf+S-ff≥0.Variables that are encountered in engineering analysis, such as the stress, are real. The PSD is an even function for real variables; thus,(5)S(f)¯=Sf+S-f=2Sff≥0.Thenth moment of the PSD function can be obtained from the characteristic of the PSD function. The spectral moments are defined as follows:
(6)mn=∫0∞fnS(f)¯df.The first four moments (m0, m1, m2, and m4) are the most commonly used moments of the PSD function.The bandwidth of a signal is taken to be the frequency interval over which most of the signal power is concentrated. The bandwidth can be defined in terms of the spectral moments [15]:
(7)α1=m1m0m1α2=m2m0m40≤α1,α2≤1.These two bandwidth parameters tend to unity for a “narrow-band” signal and zero for a “wide-band” signal.


### 2.2. Fatigue Damage Assessment Method for Concrete Structures from Eurocode 2

In Eurocode 2, Palmgren-Miner’s rule [16] is applied to calculate the total fatigue damage, as given in (8). A satisfactory fatigue resistance may be assumed for concrete under compression if D≤1:
(8)D=∑i=1mniNi,
where D is the total fatigue damage; m is the number of intervals with a constant amplitude; ni is the actual number of constant-amplitude cycles in the interval “i”; Ni is the ultimate number of constant-amplitude cycles in the interval “i” that can be carried out before failure:
(9)Ni=1014(1-Ecd,max⁡,i/1-Ri)Ri=Ecd,min⁡,iEcd,max⁡,iEcd,min⁡,i=σcd,min⁡,ifcd,fatEcd,max⁡,i=σcd,max⁡,ifcd,fat,
where Ri is the stress ratio; Ecd,min⁡,i is the minimum stress compression level; Ecd,max⁡,i is the maximum stress compression level; σcd,min⁡,i is the lowest stress in a cycle; σcd,max⁡,i is the highest stress in a cycle; and fcd,fat is the design fatigue strength of the concrete. Ni is a function of both the peaks and valleys of the stress process.


### 2.3. Counting Methods

LCC, PC, RC, and RFC are the four most commonly used counting methods in engineering practice.In the LCC method, every crossing of the predetermined stress levels is counted. Then, the most damaging level cycle is counted for fatigue by constructing the largest possible cycle, followed by the second largest cycle, and so on, until all of the level crossings have been considered.In the PC method, the occurrence of a relative maximum or minimum stress value is identified. Peaks (valleys) that are above (below) the reference level are counted.In the RC method, a range is defined as the difference between two successive reversals: the range is positive when a valley is followed by a peak and negative when a peak is followed by a valley.The RFC method is the most frequently used cycle counting method in engineering practice. Starting from a local load maximum,Max⁡k, two minima are identified before and after Max⁡k, that is, Min⁡k- and Min⁡k+. The point with the smallest deviation from Max⁡k is chosen as the rainflow minimum, Min⁡k,RFC, thus producing thekth rainflow cycle (Min⁡k,RFC, Max⁡k). The aforementioned procedure is repeated for the entire stress process.


## 2.1. Frequency-Domain Process

A stress processh(t) is assumed to be a stationary Gaussian process. The autocorrelation function of h(t) is defined as follows [14]:
(1)Kτ≡htht+τ¯=lim⁡T→∞12T∫-TThtht+τdτ.The Fourier transform of the autocorrelation function is expressed as(2)Kτ=∫-∞∞Sfe-iωτdf,
where the new function, S(f), is the PSD of h(t). S(f) can also be written in terms of K(τ):
(3)Sf=∫-∞∞Kτeiωτdτ.Equations (2) and (3) are known as the Wiener-Khintchine relations.We are typically not interested in distinguishing a PSD that is associated with a negative frequency from one that is associated with a positive frequency. A single-sided PSD can be defined as follows:(4)S(f)¯=Sf+S-ff≥0.Variables that are encountered in engineering analysis, such as the stress, are real. The PSD is an even function for real variables; thus,(5)S(f)¯=Sf+S-f=2Sff≥0.Thenth moment of the PSD function can be obtained from the characteristic of the PSD function. The spectral moments are defined as follows:
(6)mn=∫0∞fnS(f)¯df.The first four moments (m0, m1, m2, and m4) are the most commonly used moments of the PSD function.The bandwidth of a signal is taken to be the frequency interval over which most of the signal power is concentrated. The bandwidth can be defined in terms of the spectral moments [15]:
(7)α1=m1m0m1α2=m2m0m40≤α1,α2≤1.These two bandwidth parameters tend to unity for a “narrow-band” signal and zero for a “wide-band” signal.


## 2.2. Fatigue Damage Assessment Method for Concrete Structures from Eurocode 2

In Eurocode 2, Palmgren-Miner’s rule [16] is applied to calculate the total fatigue damage, as given in (8). A satisfactory fatigue resistance may be assumed for concrete under compression if D≤1:
(8)D=∑i=1mniNi,
where D is the total fatigue damage; m is the number of intervals with a constant amplitude; ni is the actual number of constant-amplitude cycles in the interval “i”; Ni is the ultimate number of constant-amplitude cycles in the interval “i” that can be carried out before failure:
(9)Ni=1014(1-Ecd,max⁡,i/1-Ri)Ri=Ecd,min⁡,iEcd,max⁡,iEcd,min⁡,i=σcd,min⁡,ifcd,fatEcd,max⁡,i=σcd,max⁡,ifcd,fat,
where Ri is the stress ratio; Ecd,min⁡,i is the minimum stress compression level; Ecd,max⁡,i is the maximum stress compression level; σcd,min⁡,i is the lowest stress in a cycle; σcd,max⁡,i is the highest stress in a cycle; and fcd,fat is the design fatigue strength of the concrete. Ni is a function of both the peaks and valleys of the stress process.


## 2.3. Counting Methods

LCC, PC, RC, and RFC are the four most commonly used counting methods in engineering practice.In the LCC method, every crossing of the predetermined stress levels is counted. Then, the most damaging level cycle is counted for fatigue by constructing the largest possible cycle, followed by the second largest cycle, and so on, until all of the level crossings have been considered.In the PC method, the occurrence of a relative maximum or minimum stress value is identified. Peaks (valleys) that are above (below) the reference level are counted.In the RC method, a range is defined as the difference between two successive reversals: the range is positive when a valley is followed by a peak and negative when a peak is followed by a valley.The RFC method is the most frequently used cycle counting method in engineering practice. Starting from a local load maximum,Max⁡k, two minima are identified before and after Max⁡k, that is, Min⁡k- and Min⁡k+. The point with the smallest deviation from Max⁡k is chosen as the rainflow minimum, Min⁡k,RFC, thus producing thekth rainflow cycle (Min⁡k,RFC, Max⁡k). The aforementioned procedure is repeated for the entire stress process.


## 3. Fatigue Damage Assessment for Concrete in the Frequency Domain

### 3.1. Joint Probability Density Applied to Fatigue Damage Assessment

As described in Section2.2, Ni is a function of both the peaks and valleys in the stress process; thus, the fatigue damage for concrete structures is also a function of the peaks and valleys in the stress process. The joint probability density of the counted cycles h(u,v), which is a function of peak “u” (corresponding to σcd,max⁡,i in (8)) and valley “v” (corresponding to σcd,min⁡,i in (8)), is constructed by applying the spectral moment parameters of the PSD function. As the RFC method is the “standard” counting method, we determine the joint probability density of counted cycles using hRFC(u,v). However, no analytical solutions of hRFC(u,v) are currently available, and only approximate approaches can be used to obtain an accurate fatigue damage assessment.Tovo [17, 18] calculated the joint probability density of counted cycles for a zero-mean Gaussian process from the distribution of level crossing counted cycles as follows:(10)hLCCu,v=pp(u)-pv(u)δ(u+v)+pv(u)δ(u-v)u≥0pp(u)δ(u-v)u≤0,where δ is the Dirac delta function and pp(u) and pv(u) are the cumulative distributions of the peaks and valleys, respectively:
(11)ppx=Φxσx1-α22-α2e-x2/2σx2Φα2xσx1-α22,(12)pvx=pp-x,
where x is the stress, σx is the standard deviation, Φ is the cumulative Gaussian distribution, and α2 is the bandwidth parameter, as defined by (7).Tovo [17, 18] also calculated a joint probability density for the cycles in a Gaussian process, where the peaks and valleys distributions are given by (11) and (12), respectively, from the distribution of the counted cycles over the stress range:
(13)hRCu,v=1σx2α2222π×e-u2+v2/4σx21-α22e-(u-v)2/4σx21-α221-α22/2σx2×u-v4σx21-α22.Equation (13) can be rewritten as a function of the mean stress “m” and stress range “s”:
(14)hRCs,m=12πσx1-α22×e-m2/2σx1-α22sσxα22e-s2/2σxα22=pm(m)ps(s),
where s=(u-v)/2 and m=(u+v)/2.Equation (14) shows that the joint probability density is a product of pm(m) and ps(s); thus, it is reasonable to assume that the mean stress “m” and stress range “s” are two independent variables. Then, other expressions for ps(s) can replace that in (14) to obtain a new joint probability density. The Dirlik method [19] is considered the most accurate approach for constructing an empirical expression for the PDF over the stress range “s” [20] and is used to obtain a new proposed hNP(s,m):
(15)hNPs,m=12πσx1-α22e-m2/2σx1-α22pDirliks,(16)pDirlik(s)=1σxG1Qe-Z/Q+G2ZR2e-Z2/2R2+G3Ze-Z2/2,(17)Z=sσxxm=m1m0m2m41/2,
where
(18)G1=2xm-α221+α22xm=1-α22-G1+G121-RZ=1-G1-G2R=α2-xm-G121-α2-G1+G12Q=1.25(α2-G3-G2R)G1.


### 3.2. Expressions for Fatigue Damage

In this section, (10), (14), and (15) are used to calculate the fatigue damage in the frequency domain.The expected peak occurrence frequencyυp is defined as [21]
(19)υp=m4m2.The fatigue damage over a given timeT can be calculated as follows:
(20)ED=Tυp∬hu,vNiu,vdudv.WhenhLCC(u,v) (see (10)) is used, the component related to δ(u-v) may be neglected in the calculation because this function implies that u=v; that is, there is no effect on the damage. When u≤0, hLCC(u,v)=pp(u)δ(u-v); thus,
(21)ED′=Tυp∬h(u,v)Ni(u,v)dudv=Tυp∬pp(u)δ(u-v)Ni(u,v)dudv=Tυp∬-∞0pp(u)δ(u-v)Ni(u,v)dudv=Tυp∫-∞0∫-∞0ppuδu-vNiu,vdudv=Tυp∫-∞0ppvNiv,vdv.From the definition ofNi, Ni(v,v)→∞; thus, E(D′)→0, and the component related to δ(u-v) clearly has no effect on the damage.Equations (10)–(15) and (21) are all based on a zero-mean process; for a process with a non-zero-mean mx, these equations can easily be modified using a variable shift. In this study, we perform a fatigue damage assessment for concrete under compression; thus, all of the stresses have the same sign (and are hereafter assumed to be positive). The following equations are given for a non-zero-mean process.The fatigue damage can be calculated by applyinghLCC(u,v) as follows:
(22)EDLCC=Tυp∬hu,vNiu,vdudv=Tυp∬Niu,v-1ppu-mx-pvu-mx×δu-mx+v-mx×Niu,v-1dudv=Tυp∫mx∞∫-∞uNiu,v-1ppu-mx-pvu-mx×δu+v-2mx×Niu,v-1dv∫-∞uppu-mx-pvu-mxdu=Tυp∫mx2mxppu-mx-pvu-mxNiu,2mx-udu+∫2mx∞ppu-mx-pvu-mxNiu,0du.Because all of the stresses are positive, the lower limit of the valley is zero. When(2mx-u)≤0, the valley of a cycle is set to zero.Similarly, the fatigue damageE(DRC) can be calculated by the RC method, and E(DNP) can be calculated by the new method developed by the authors, that is, by applying (14), (15), and (23):
(23)ED=Tυp∬hs,mNis,mdsdm.As the stress record is assumed to be Gaussian, almost all of the values will fall in the interval(mx-5σx,mx+5σx); thus, the following integration intervals can be used to calculate E(DRC) and E(DNP):m:(mx-5σx,mx+5σx), s:(0,5σx). (The upper bound on the integration interval for s is estimated as s=(mx+5σx)-(mx-5σx)/2. For (mx-5σx)<0, the lower bound on the integration interval is set to zero.)


## 3.1. Joint Probability Density Applied to Fatigue Damage Assessment

As described in Section2.2, Ni is a function of both the peaks and valleys in the stress process; thus, the fatigue damage for concrete structures is also a function of the peaks and valleys in the stress process. The joint probability density of the counted cycles h(u,v), which is a function of peak “u” (corresponding to σcd,max⁡,i in (8)) and valley “v” (corresponding to σcd,min⁡,i in (8)), is constructed by applying the spectral moment parameters of the PSD function. As the RFC method is the “standard” counting method, we determine the joint probability density of counted cycles using hRFC(u,v). However, no analytical solutions of hRFC(u,v) are currently available, and only approximate approaches can be used to obtain an accurate fatigue damage assessment.Tovo [17, 18] calculated the joint probability density of counted cycles for a zero-mean Gaussian process from the distribution of level crossing counted cycles as follows:(10)hLCCu,v=pp(u)-pv(u)δ(u+v)+pv(u)δ(u-v)u≥0pp(u)δ(u-v)u≤0,where δ is the Dirac delta function and pp(u) and pv(u) are the cumulative distributions of the peaks and valleys, respectively:
(11)ppx=Φxσx1-α22-α2e-x2/2σx2Φα2xσx1-α22,(12)pvx=pp-x,
where x is the stress, σx is the standard deviation, Φ is the cumulative Gaussian distribution, and α2 is the bandwidth parameter, as defined by (7).Tovo [17, 18] also calculated a joint probability density for the cycles in a Gaussian process, where the peaks and valleys distributions are given by (11) and (12), respectively, from the distribution of the counted cycles over the stress range:
(13)hRCu,v=1σx2α2222π×e-u2+v2/4σx21-α22e-(u-v)2/4σx21-α221-α22/2σx2×u-v4σx21-α22.Equation (13) can be rewritten as a function of the mean stress “m” and stress range “s”:
(14)hRCs,m=12πσx1-α22×e-m2/2σx1-α22sσxα22e-s2/2σxα22=pm(m)ps(s),
where s=(u-v)/2 and m=(u+v)/2.Equation (14) shows that the joint probability density is a product of pm(m) and ps(s); thus, it is reasonable to assume that the mean stress “m” and stress range “s” are two independent variables. Then, other expressions for ps(s) can replace that in (14) to obtain a new joint probability density. The Dirlik method [19] is considered the most accurate approach for constructing an empirical expression for the PDF over the stress range “s” [20] and is used to obtain a new proposed hNP(s,m):
(15)hNPs,m=12πσx1-α22e-m2/2σx1-α22pDirliks,(16)pDirlik(s)=1σxG1Qe-Z/Q+G2ZR2e-Z2/2R2+G3Ze-Z2/2,(17)Z=sσxxm=m1m0m2m41/2,
where
(18)G1=2xm-α221+α22xm=1-α22-G1+G121-RZ=1-G1-G2R=α2-xm-G121-α2-G1+G12Q=1.25(α2-G3-G2R)G1.


## 3.2. Expressions for Fatigue Damage

In this section, (10), (14), and (15) are used to calculate the fatigue damage in the frequency domain.The expected peak occurrence frequencyυp is defined as [21]
(19)υp=m4m2.The fatigue damage over a given timeT can be calculated as follows:
(20)ED=Tυp∬hu,vNiu,vdudv.WhenhLCC(u,v) (see (10)) is used, the component related to δ(u-v) may be neglected in the calculation because this function implies that u=v; that is, there is no effect on the damage. When u≤0, hLCC(u,v)=pp(u)δ(u-v); thus,
(21)ED′=Tυp∬h(u,v)Ni(u,v)dudv=Tυp∬pp(u)δ(u-v)Ni(u,v)dudv=Tυp∬-∞0pp(u)δ(u-v)Ni(u,v)dudv=Tυp∫-∞0∫-∞0ppuδu-vNiu,vdudv=Tυp∫-∞0ppvNiv,vdv.From the definition ofNi, Ni(v,v)→∞; thus, E(D′)→0, and the component related to δ(u-v) clearly has no effect on the damage.Equations (10)–(15) and (21) are all based on a zero-mean process; for a process with a non-zero-mean mx, these equations can easily be modified using a variable shift. In this study, we perform a fatigue damage assessment for concrete under compression; thus, all of the stresses have the same sign (and are hereafter assumed to be positive). The following equations are given for a non-zero-mean process.The fatigue damage can be calculated by applyinghLCC(u,v) as follows:
(22)EDLCC=Tυp∬hu,vNiu,vdudv=Tυp∬Niu,v-1ppu-mx-pvu-mx×δu-mx+v-mx×Niu,v-1dudv=Tυp∫mx∞∫-∞uNiu,v-1ppu-mx-pvu-mx×δu+v-2mx×Niu,v-1dv∫-∞uppu-mx-pvu-mxdu=Tυp∫mx2mxppu-mx-pvu-mxNiu,2mx-udu+∫2mx∞ppu-mx-pvu-mxNiu,0du.Because all of the stresses are positive, the lower limit of the valley is zero. When(2mx-u)≤0, the valley of a cycle is set to zero.Similarly, the fatigue damageE(DRC) can be calculated by the RC method, and E(DNP) can be calculated by the new method developed by the authors, that is, by applying (14), (15), and (23):
(23)ED=Tυp∬hs,mNis,mdsdm.As the stress record is assumed to be Gaussian, almost all of the values will fall in the interval(mx-5σx,mx+5σx); thus, the following integration intervals can be used to calculate E(DRC) and E(DNP):m:(mx-5σx,mx+5σx), s:(0,5σx). (The upper bound on the integration interval for s is estimated as s=(mx+5σx)-(mx-5σx)/2. For (mx-5σx)<0, the lower bound on the integration interval is set to zero.)


## 4. Numerical Simulation

Numerical simulations were performed to investigate the applicability of the aforementioned frequency-domain methods. The fatigue damage was assessed using the time-domain method with RFC and Palmgren-Miner’s rule to obtain the “standard” results. The accuracy of the results obtained from the frequency-domain methods was evaluated by comparison with the accuracy of these “standard” results.The aforementioned joint PDFs were all constructed from spectral moments and bandwidth parameters. Preliminary numerical simulations were carried out to investigate the effect of the bandwidth parameters, particularlyα2, on the applicability of the frequency-domain methods. Some PSDs with simple shapes (see Figure 1) were applied in the simulation. The random stress processes that were used in the RFC methods in the time domain can be derived from these PSDs (see Figure 2) [22, 23].Examples of PSD curves.
(a)
(b)Examples of random processes with different bandwidth parameters.
(a)
(b)
(c)The design fatigue strength of concrete,fcd,fat, was chosen as 19.1 MPa using Eurocode 2. A given mean value mx of 5.0 MPa was used for all of the spectra and random processes. All of the spectra had the same m0 of 1.0 MPa2, which was equal to the variances σx2 of the stress processes. The total simulation time was 1,000 s.The results of the preliminary numerical simulations are shown in Figure3. DLCC, DRC, DNP, and DRFC denote the fatigue damage that was calculated using the LCC method, RC method, the new method, and RFC method, respectively. The results of the three aforementioned frequency-domain methods were normalized by the results obtained using the RFC method in the time domain (the order of magnitude of the absolute fatigue damage was 10−9). Figure 3 illustrates that both the LCC method and the new method yielded accurate estimates of the fatigue damage compared with the RFC method over the entire bandwidth of the stress process. Both the LCC method and the new method yielded conservative results, where the values of DLCC/DRFC were between 2.0 and 12.0 and the values of DNP/DRFC were between 2.0 and 18.2. Both of these methods should be used in engineering practice to obtain preliminary estimates of the fatigue damage in the initial design stage and parametric studies. The range method yielded poor estimates for most bandwidths, and more accurate results were obtained only when using higher α2 values. The orders of magnitude of DRC/DRFC were between 10−2 and 10−13 for values of α2 below 0.65. The order of magnitude for DRC/DRFC approached unity when α2 approached unity. Thus, the applicability of the RC method is limited to narrow-band stress processes.Results of the preliminary numerical simulations.
(a)
D
LCC
/
D
RFC
(b)
D
NP
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D
RFC
(c)
D
RC
/
D
RFCWe further verified the applicability of the LCC method and the new method by conducting parameter studies. First, the effect of the fatigue strength on the fatigue assessment was investigated for a stress process with a mean ofmx and a variance of σx2. Fatigue strengths of 19.1, 25.6, and 31.3 MPa were considered. Figure 4 illustrates that, for these three different fatigue strengths, the values of DLCC/DRFC and DNP/DRFC were close to each other for all values of α2: at some values of α2, the three values were nearly equal to each other. Thus, the fatigue strength had only a minor effect on the values of DLCC/DRFC and DNP/DRFC.Effect of fatigue strength on fatigue damage.
(a)
D
LCC
/
D
RFC
(b)
D
NP
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D
RFCThe effects of the coefficient of variationCv=σx/mx of the stress process on the applicability of these two methods were investigated for the same Im=mx/fcd,fat ratio, as shown in Figure 5. Similar DLCC/DRFC and DNP/DRFCcurves were obtained for the same α2 value. In some cases, the DLCC/DRFC and DNP/DRFC values increased dramatically as the coefficient of variation Cv decreased. Both the LCC method and the new method produced overly conservative estimates of the fatigue damage.Effect of coefficient of variation on fatigue damage (in logarithmic scale).
(a)
D
LCC
/
D
RFC
(b)
D
NP
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D
RFCThe applicability of the frequency methods deteriorated for lower coefficients of variation becauseNi was defined as an exponential function. Figure 6 shows the cycle count versus the stress amplitude and the mean stress value of a specified stress record that were obtained using the RFC method and the corresponding fatigue damage caused by pairs of the stress amplitude and mean stress. Only a few cycles had stress amplitudes near the maximum amplitude, rmax⁡, and a mean stress value near the mean values of stress range, mx, whereas the corresponding fatigue damage was fairly high, with an order of magnitude that was nearly the same as that of the fatigue damage of the entire stress process. The value of Ni was highly sensitive to the potential maximum stress range of the cycle, and the total fatigue damage could be estimated by calculating the fatigue damage D1 induced by 1 cycle with a pair of range and mean values of smax⁡ and mx, respectively (see (14)). As the stress record was assumed to be Gaussian, the limits of the stress interval (mx-nσx,mx+nσx) were typically used as estimates of the upper and lower bounds, respectively, of the record; thus, it was reasonable to assume that smax⁡ was equal to nσx. D1 is defined as follows:
(24)D1=1Nismax⁡,mx=10-141-(mx+smax⁡)/fcd,fat/1-(mx-smax⁡)/(mx+smax⁡)=10-141-(mx+nσx)/fcd,fat/1-(mx-nσx)/(mx+nσx)=10-141-(mx+nCvmx)/fcd,fat/1-(mx-nCvmx)/(mx+nCvmx)=10-141-(1+nCv)mx/fcd,fat/1-(1-nCv)/(1+nCv)=10-141-1+nCvIm/2nCv/(1+nCv).Cycle count versus stress amplitude and mean stress value and corresponding fatigue damage.
(a)
Cycle count versus stress amplitude and mean stress value
(b)
Corresponding fatigue damageThe values ofD1 were calculated for n=2.0, 3.0, 3.1, 3.5, 4.0, and 5.0 for different Cv values and a constant Im. The ratios of D1(mx,nσx) to D1(mx,3σx) are shown in Figure 7 (on a logarithmic scale). These ratios increased or decreased exponentially as Cv decreased for values of n that were greater or less than 3.0. The value of D1(mx,5σx) was approximately 9 orders of magnitude larger than D1(mx,3σx); that is, 1 cycle of stress with a pair of the stress range and the stress mean value of (mx,5σx) produced the same damage as approximately 109 cycles of stress with a pair of the stress range and the stress mean value of (mx,3σx). In the time domain, the number of cycles with high stress range could be counted as zero. In the frequency domain, the probability of the occurrence of the high stress range was relatively negligible but was a nonzero value, which had a tremendous effect when coupled to the damage that was induced by high stress range. Thus, the fatigue damage that was calculated using the frequency domain was larger than that calculated using the time domain.Figure 7
Ratios ofD1(nσx,mx) to D1(3σx,mx) versus Cv.We verified the applicability of these joint PDFs of the LCC method and the method for differentCv values by using an imaginary simple function Ni=u·v in the numerical simulation. Figure 8 shows the ratios of DLCC/DRFC and DNP/DRFC that were obtained. Both of these two ratios remained nearly constant for different Cv values when the value of Ni was not highly sensitive to the potential maximum stress ranges of the cycle. Both ratios were approximately 1.0. Thus, both of these two joint PDFs described the distributions of the “valleys” and “peaks” of the stress processes fairly accurately.Applicability of two joint PDFs.
(a)
D
LCC
/
D
RFC
(b)
D
NP
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D
RFCDifferent integration intervals were used for the LCC method and the new method to calculate the fatigue damage, which was normalized by the results of the RFC method (Figure9). The effect of the integration intervals on the fatigue damage reflected the effect of the definition of Ni on the fatigue damage. The fatigue damage was overestimated using large intervals and underestimated using small intervals, where the extent of overestimation or underestimation was relatively higher for a lower Cv. As the joint PDFs described the “valleys” and “peaks” of stress processes fairly accurately, the upper and lower bounds on the fatigue damage can be estimated by varying the integration intervals in the frequency methods.Effects of integration intervals on damage assessment.
(a)
Integration intervals for the LCC method, I:s(-5σx,5σx), m(mx-5σx,mx+5σx), II: s(-4σx,4σx), m(mx-4σx,mx+4σx), and III: s(-3σx,3σx), m(mx-3σx,mx+3σx)
(b)
Integration intervals for the new method, I:u(mx,+∞), II: u(mx,mx+4σx), and III: u(mx,mx+3σx)The RFC method in the time domain is generally considered the “standard” method for assessing fatigue damage; however, the stress process applied in the time-domain method is a fragment of the entire actual process. Stress processes with different maximum ranges may produce different fatigue damage assessments. Figure10 shows a fatigue damage assessment on a logarithmic scale using 10 stress records, which had the same coefficients of variation because they were derived from the same PSD. The maximum fatigue damage was two orders of magnitude larger than the minimum fatigue damage. Thus, the RFC method in the time domain does not serve as an appropriate “standard” and should be used with caution; that is, different stress processes should be used in engineering practice. As noted above, varying the integration intervals in the two frequency-domain methods can be used to estimate the lower and upper bounds of the fatigue damage; thus, the results derived from the frequency-domain methods could serve as a reference to evaluate the accuracy of the results of the time-domain method.Figure 10
Fatigue damage assessment based on 10 stress records with the same parameters (e.g.,α1, α2, mx, σx).


## 5. Conclusion

Three frequency-domain methods were used for a fatigue damage assessment. The underlying principle of the frequency-domain method is the construction of a PDF for the “peaks” and “valleys” of the stress process (i.e., the stress range and mean stress). Two PDFs were taken from the literature, and one PDF was formulated by the authors. This novel PDF was constructed by replacingps(s) in (14) using the Dirlik method (see (16)) and assuming that the stress range “s” and stress mean “m” were independent variables.The applicability of these three methods was investigated by comparing the results of the frequency-domain methods with those from the RFC method in the time domain. A preliminary study demonstrated that both the LCC method and the new method yielded accurate estimates over the entire range of bandwidths of the stress process, whereas the RC method yielded poor estimates for low bandwidths.We carried out parametric studies to further verify the applicability of the LCC method and the new method. The fatigue strength of concrete had a slight impact on the applicability of these two methods. Decreasing the coefficient of variation deteriorated the applicability of these two methods for some cases. This behavior was attributed to the definition ofNi. The value of Ni was highly sensitive to the potential maximum stress ranges of the cycle, particularly for relatively low coefficients of variation. The validity of the two joint PDFs for the LCC method and the new method was evaluated by a numerical simulation. The integration intervals of the two frequency-domain methods can be varied to calculate the lower and upper bounds on the fatigue damage, which can serve as references to evaluate the accuracy of the results of the time-domain method.In conclusion, the RC method is only applicable to wide-band (α2>0.65) stress processes, and the LCC method and the new method are applicable for all bandwidths. Both the LCC method and the new method should be used in practical design for mutual authentication. Only pure compression was investigated in this study, and Ni was defined for pure tension and tension-compression [24, 25] as an exponential function, as in pure compression. Thus, the conclusions of this study are only applicable for the aforementioned cases.



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*Source: 407193-2014-12-29.xml*