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Research ArticleResearch on Fatigue Strain and Fatigue Modulus of Concrete
Fangping Liu12 and Jianting Zhou2
1College of Civil Engineering Chongqing Three Gorges University No 666 TianXing Road Wanzhou DistrictChongqing 404100 China2Engineering Research Center of Bridge Structure and Material in the Mountainous Area Chongqing Jiaotong UniversityNo 66 XueFu Road Nanrsquoan District Chongqing 400074 China
Correspondence should be addressed to Jianting Zhou jt-zhou163com
Received 27 March 2017 Accepted 14 June 2017 Published 12 July 2017
Academic Editor Pier Paolo Rossi
Copyright copy 2017 Fangping Liu and Jianting Zhou This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
Concrete fatigue strain and fatigue modulus evolution play a vital role in the evaluation of the material properties In this paper byanalyzing the advantages and disadvantages of existing concrete strain analysis methods the level-S nonlinear fatigue strain modelwas proposed The parametersrsquo physical meaning the ranges and the impact on the shape of the curve were all discussed Thenthe evolution model of fatigue modulus was established based on the fatigue strain evolution model and the hypothesis of fatiguemodulus inversely related fatigue strain amplitude The results indicate that the level-S model covered all types of fatigue strainevolution It is very suitable for the description of strain evolution of concrete for its strong adaptability and high accuracy It wasfound that the fitting curves coincided with the experimental curves very well and the correlation coefficients were all above 098The evolution curves of fatigue strainmodulus both have three stages namely variation phase linear change stage and convergencestage The difference is that the fatigue strain evolution curve is from the lower left corner to the upper right corner but the fatiguemodulus evolution curve is from the upper left corner to the right lower corner
1 Introduction
Fatigue strain of the concrete can truly reflect the variationof the material deformation under the fatigue loads If weknow the relationship of the curve and the cycles we are ableto qualitatively describe the evolution of the material fatiguestrain which will provide the basis for the behavior of thematerial evaluation In numerous constant amplitude fatiguetests it is shown that the total longitude deformation and theresidual deformation of the concrete will display the universaland stable three-stage law [1ndash6] which presents firstly therapid growth stage then the stable growth stage and ulti-mately the rapid growth stage This is suitable for not onlythe ordinary concrete lightweight aggregate concrete highstrength concrete or fiber reinforced concrete but also thecompression fatigue tension fatigue bending fatigue uniax-ial fatigue or multiracial fatigue Chen et al [7] took advan-tage of a cubic polynomial fitting curve to get correlationcoefficients above 0937 However different levels of stresswould own different coefficients with nearly an order of
magnitude Cachim et al [8] adopted the logarithmic formbetween the maximum strain rate and the load cycles in thesecond phase for the concrete Under the compression fatigueloads the formwas a linear relationship Xie et al [9] also gotan experienced index formula by fitting the second phase offatigue strain Wang et al [10] fitted the data of compressivefatigue experiment strain and adopted a two-stage nonlinearformula According to the fatigue strain evolution and meth-ods of current analysis the following deficiencies were dis-covered A Presently the linear three-stage fatigue strainequations are simple but of a low accuracy While the three-stage nonlinear equation is of high precision the form iscomplicated It is rarely reported about the whole fatiguestrain curve with the cycle ratio relationshipBTheevolutionand fitting of the fatigue strain curve are aimed at a specificset of experimental data nowadays As for the conditions thatthe fatigue stress is smaller than limited stress but greater thanthe threshold value there are few researchesC A consensushas reached the variation law of the three-stage fatigue strainand some curve fitting equations have been obtained but the
HindawiAdvances in Civil EngineeringVolume 2017 Article ID 6272906 7 pageshttpsdoiorg10115520176272906
2 Advances in Civil Engineering
00
(a) Type
(b) Type
(c) Type
Fatig
ue st
rain
()
Recycle ratio (nN)
Figure 1 Basic types of fatigue strain evolution curve
initial strain was not taken into consideration basically Inthese equations the significance of each factor is not clearenough which results in unstable fitting coefficients
In the literature [11] published by the author the nonlinearstrain evolution model was established The strain evolutionlaw of concrete under constant amplitude fatigue load and thelaw of fatigue damage evolution based on strain were studiedBased on the previous research in this paper the physicalmeanings the ranges and the impact on the shape of thecurve of parameters in the nonlinear strain evolution modelwere all discussed The evolution model of fatigue moduluswas established under constant amplitude bending fatigueloading based on the fatigue strain evolution model and thehypothesis of fatigue modulus inversely related fatigue strainamplitude Moreover the whole process is validated by theexperimental data
2 Fatigue Strain Evolution Model
To better describe the three-stage variation law a fatiguestrain evolution model with the simple form was proposedwhich could be suitable for different stress types (Fig-ure 1) Moreover some valuable physical parameters could beobtained via the model such as the initial strain instabilityspeed of the third stage (acceleration of fatigue strain) andthe proportion of that in the total fatigue life [12] Via therepeated study on the related mathematical functions themodel could be obtained below
120576119899 = 1205760 + 120572 sdot ( 120573120573 minus 119899119873119891 minus 1)1119901 (1)
In formula (1) 1205760 indicate the initial strain and 120576119899 indicatethe fatigue strain under 119899 times of fatigue loading cycle 119899indicate the cycle times of fatigue loads 119873119891 indicate thefatigue life 120572 120573 and 119901 were parameters in the equation
If the maximum fatigue strain 120576119899max or fatigue residualstrain 120576119899res was regarded as the value of 120576119899 formula (2) would
060
065
070
075
080
085
090
095
100
Relat
ive s
trai
n (
n
m)
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
(B) (C) (D) (E)
(F) (G) (H)
Figure 2 Influence of 119901 on fatigue strain curve (B) 119901 = 2 120573 =125 (C) 119901 = 3 120573 = 1125 (D) 119901 = 4 120573 = 10625 (E) 119901 = 5 120573 =103125 (F) 119901 = 6 120573 = 1015625 (G) 119901 = 7 120573 = 1007813 (H)119901 = 8 120573 = 1003906 (B)sim(H) 1205760max120576119891max = 06 120572120576119891max = 02be obtained by formula (1) In the sameway if the initial max-imum strain 1205760max or initial residual strain 1205760res was regardedas the value of 1205760 formula (3) would be obtained
1205760max = 1205760max + 120572( 120573120573 minus 119899119873119891 minus 1)1119901 (2)
120576119899res = 1205760res + 120572( 120573120573 minus 119899119873119891 minus 1)1119901 (3)
Formula (2) was called as the maximum fatigue strainevolution model while formula (3) was the fatigue residualstrain evolution model
If the upper limit fatigue stress is large fatigue strain willchange fast and increase The slope of the curve will be largeand the curve become the vertical lineThat was to say withina few loading cycles the strain will reach the unstable stateThe three-stage curve will degenerate the c-type in Figure 2[13] When the upper limit fatigue stress does not exceed thethreshold the elastic strain will be added to the initial strainand the strain remains unchanged which is similar to the a-type curve These two types are both the limit state so thestresses in most fatigue tests are located between the upperlimit and the threshold In other words the evolution curveshave characteristics of both a-type and c-type Generally theb-type curve in Figure 2 can be obtained According to theactual results the evolution laws of the b-type curve will beanalyzed in this paper1205760max and 1205760res indicate respectively the initial maximumstrain and residual strain They are mainly caused by somefactors such as the initial defects of the material and preload-ing of the structure Due to the difficulty of distinguishing
Advances in Civil Engineering 3
between the two strains the actually measured values ofthe fatigue test can be obtained Therefore when the fatigueloading firstly reaches the upper limit the correspondingmaximum strain 1205761max and residual strain 1205761res are adopted inthis paper For example Lv [14] obtained the formula of 1205761maxand 1205761res according to the actual experimental data as 1205761res =025(1205761max120576unstab)2 Among the formula 120576unstab indicated thetotal strain of the unstable concrete
In order to study the range of parameters 120572 120573 and119901 including impacts on the evolution law of fatigue straincurves formulas (2) and (3) are divided by the limited fatiguestrain on both sides So the following are obtained
120576119899max120576119891max= 1205760max120576119891max+ 120572120576119891max( 120573120573 minus 119899119873119891 minus 1)
1119901 (4)
120576119899res120576119891res =1205760res120576119891res +120572120576119891res (
120573120573 minus 119899119873119891 minus 1)1119901 (5)
Both formulas are the normalized evolution model of thefatigue strain In the former one 120576119891max indicates the limitedmaximum fatigue strain In the latter one 120576119891res indicates thelimited fatigue residual strain120573 indicates the destabilizing factor This parameterdepends on 119901 and 120572 If the circulation ratio 119899119873119891 is equal to 1the point (1 1) will locate on the curve of formulas (4) and (5)Via the two formulas values of 120573 can be deducted as formula(6) which correspond to the maximum fatigue stress and theresidual fatigue stress
1205731 = ((1 minus 1205760max120576119891max)(120572120576119891max) )
minus119901
+ 1
1205732 = ((1 minus 1205760res120576119891res)(120572120576119891res) )
minus119901
+ 1(6)
In order to analyze the impacts of 119901 and 120572 on the fatiguestrain evolution curve the evolution formula (4) was taken asan example In the first place the impact of119901was analyzed via1205760max120576119891max and 120572120576119891max Then combined with 119901 and 1205760max120576119891maxthe impact of 120572 was analyzed further Ultimately the impactscurve of 119901 and 120572 were shown in Figures 2 and 3
It could be seen from Figure 2 that parameter 119901 wouldinfluence the convergence speed of the curve using the level-S nonlinear model With the growing of 119901 the convergencespeed of the fatigue strain curve in the third stage couldbecome faster while the stage corresponded to the irre-versible third stage of fatigue deformation which also meantthe acceleration stage of unstable crack propagation Theparameter 119901 was also called the instability speed factor andthe suggested values were located in [2 8]
The effects of different 120572 values on the curve are shown inFigure 3
It could be seen from Figure 3 that parameter 120572 wouldinfluence the proportion of the third stage in the total fatiguelife The stage was the so-called acceleration stage of the
060
065
070
075
080
085
090
095
100
Relat
ive s
trai
n (
n
m)
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
(B) (C) (D) (E)
(F) (G) (H)
Figure 3 Influence of 120572 on the fatigue strain curve (B) 120572120576119891max =01 120573 = 10625 (C) 120572120576119891max = 015 120573 = 1140625 (D) 120572120576119891max =02 120573 = 125 (E) 120572120576119891max = 025 120573 = 1390625 (F) 120572120576119891max =03 120573 = 15625 (G) 120572120576119891max = 035 120573 = 1765625 (H) 120572120576119891max =04 120573 = 2 (B)sim(H) 1205760max120576119891max = 06 119901 = 2
fatigue strain With the growing of 120572 the proportion of theacceleration stage could become shorter 120572120576119891max was locatedin (0 1 minus 1205760max120576119891max) so 120572 was located in (0 120576119891max minus 1205760max)
The suggested values of the parameters 120572 120573 119901 weremainly aimed at the b-type curves which had three stages ofevolutions That was to say the values for the a-type and c-type were not limited by the paper In Figure 4 the S-shapedcurves by combining various parameters were presentedincluding nearly different kinds of fatigue strain evolutionsfor the concrete material
3 Fatigue Modulus Evolution Model
Fatigue modulus 119864 is the ratio of stress amplitude 120590max minus120590min to strain amplitude 120576max minus 120576min that is 119864 = (120590max minus120590min)(120576max minus 120576min) 120590max and 120590min are the maximum andminimum fatigue stresses Under constant amplitude fatigueloads stress amplitude120590maxminus120590min remains unchangedThere-fore the fatigue modulus and fatigue strain amplitude are aninversely proportional relationship that is if the fatigue strainamplitude decreases the fatigue modulus increases On thecontrary if the fatigue strain amplitude increases the fatiguemodulus will be reduced [14 15] The strain evolution curveis the level-S-shaped curve from the lower left corner to theupper right corner from the inverse relationship The fatiguestrain and fatigue modulus can be shown in Figure 5
Suppose two curves are symmetrical about the straightline 119910 = 119863 and the point in the fatigue strain curve is (119909 119910)Then the corresponding point on the curve of the fatiguemodulus is (119909 2119863minus119910) Thus the normalized fatigue modulus
4 Advances in Civil Engineering
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
040045050055060065070075080085090095100
Relat
ive s
trai
n (
n
m)
(B) (C) (D) (E)
(F) (G) (H)
Figure 4 Level-S-shaped curve family of fatigue strain (B)1205760max120576119891max = 07 119901 = 2 120572120576119891max = 025 120573 = 1694444 (C)1205760max120576119891max = 065 119901 = 3 120572120576119891max = 023 120573 = 1283778 (D)1205760max120576119891max = 06 119901 = 4 120572120576119891max = 019 120573 = 1050907 (E)1205760max120576119891max = 055 119901 = 5 120572120576119891max = 017 120573 = 1007695 (F)1205760max120576119891max = 05 119901 = 6 120572120576119891max = 015 120573 = 1000729 (G)1205760max120576119891max = 045 119901 = 7 120572120576119891max = 013 120573 = 1000041 (H)1205760max120576119891max = 040 119901 = 8 120572120576119891max = 011 120573 = 1000001
y = D
Relative strains Relative modulus
060
065
070
075
080
085
090
095
100
Relat
ive s
trai
n (m
odul
us)
01 02 03 04 05 06 07 08 09 1000Recycle ratio
Figure 5 Diagram of fatigue strain and fatigue modulus evolutioncurves
evolution model can be obtained by the fatigue strain evolu-tion model as shown in the following equation
1198641198991198640 =1205760120576119891 + 120572
1015840
120576119891 ( 12057310158401205731015840 minus 119899119873119891 minus 1)11199011015840 + 2119863 (7)
Stress level S = 08
Test curve of maximum strainTest curve of maximum strainFitting curve of maximum strainFitting curve of residual strain
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
0255075
100125150175200225250275300325350375
Fatig
ue st
rain
(
)Figure 6 Fatigue strain experimental curves and fitting curves ofgroup A
Therefore the fatigue modulus evolution model is
119864119899 = 1198640( 1205760120576119891 + 1205721015840
120576119891 ( 12057310158401205731015840 minus 119899119873119891 minus 1)11199011015840 + 2119863) (8)
In formulas (7) to (8) 119864119899 is the elastic modulus afterfatigue cycle 119899 times and 1198640 is the initial elastic modulus
4 Model Validation
41 Experiment Introduction The length width and heightof the concrete experiment beams are 400mm 100mmand 100mm respectively Concrete strength is C35 Theexperimental beam is divided into A B and C three groupswith corresponding stress levels 119878max = 120590max119891 of 08 075and 07 The fatigue experiment equipment is the MTS810material testing machine The third point (50 + 3 times 100 + 50)bending constant amplitude fatigue load was used The loadfrequency is l0Hz The fatigue experiment cycle stress ratio120590min120590max is 01 Fatigue life from the experiments was 37256times 14905 times and 631291 times and the initial elasticmodulus is 50GPa
42 Concrete Fatigue Strain Evolution Analysis The fittingresults of the maximum fatigue strain and fatigue residualstrain under different stress levels using the model formulas(2) to (3) are shown in Figures 6ndash8 Coefficients of the evo-lutionary model are shown in Table 1 The data in the figureare the average of each group
From Figures 6ndash8 and Table 1 fatigue strain evolutionequations (2) and (3) can be a good fit to the experimentaldata Correlation coefficients were above 098 The evolution
Advances in Civil Engineering 5
Stress level S = 075
Test curve of maximum strainTest curve of maximum strainFitting curve of maximum strainFitting curve of residual strain
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
0255075
100125150175200225250275300325350375
Fatig
ue st
rain
(
)
Figure 7 Fatigue strain experimental curves and fitting curves ofgroup B
Stress level S = 07
Test curve of maximum strainTest curve of maximum strainFitting curve of maximum strainFitting curve of residual strain
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
0255075
100125150175200225250275300325350375
Fatig
ue st
rain
(
)
Figure 8 Fatigue strain experimental curves and fitting curves ofgroup C
of maximum fatigue strain and fatigue residual strain hassimilar three-phase variation and the intermediate stage closeto the linear change The curve converged rapidly when thecycle ratio exceeded approximately 09 The strain evolutioncurve is the level-S-shaped curve from the lower left cornerto the upper right corner Because the experiment measuredonly the initial maximum strain and has not measured initial
Table 1 Coefficient values of fatigue strain
Groups 119878 120576119899max 1198772120572 120573 119901A 08 470919 10182 24672 09891B 075 481478 10456 27550 09935C 07 540070 12096 34928 09751
Groups 119878 120576119899res 1198772120572 120573 119901A 08 438164 10422 27736 09889B 075 399757 10545 29135 09887C 07 325965 10171 35857 09753
Table 2 Coefficient values of fatigue modulus
Stress level 119864119899 11987721205721015840 1205731015840 1199011015840 11986308 23645 10081 302698 27075 09909075 15242 10762 192173 18772 0993507 11918 10070 190863 15847 09833
residual strain the evolution curve of the maximum strainstarts from the initial value but the evolution curve of fatigueresidual strain starts from zero According to (0 120576119891max minus 1205760max)and experimental data the range of 120572 of groups A B and Cis (0 250) (0 130) and (0 165) respectively Fitting results of120572 falls in these ranges in line with the boundary conditionsWhen the stress level is from 08 to 075 and from 075 to07 the 119901 value is increasing demonstrating that the smallerthe stress level is the faster the fatigue strain evolution curveconverges
43 Concrete Fatigue Modulus Evolution Analysis The fittingresults of the fatigue modulus under different stress levelsusing the model formula (8) are shown in Figures 9ndash11Coefficients of evolutionary model (8) are shown in Table 2The data in figure are the average of each group
From Figures 9ndash11 and Table 2 fatiguemodulus evolutionequation (8) can be a good fit to the experimental dataCorrelation coefficients were above 097 The evolution of thefatigue modulus also has a three-phase variation and theintermediate stage is also close to the linear changeThe curveconverged rapidly when the cycle ratio exceeded approxi-mately 09 Fatigue modulus evolution curve is the level-S-shaped curve from the upper left corner to the lower right cor-ner When the stress level is from 08 to 075 and from 075 to07 the 1199011015840 value is increasing demonstrating that the smallerthe stress level is the faster the fatigue modulus evolutioncurve converges
5 Conclusions
(1) By analyzing the shortcomings and limitations of existingfatigue strain evolution equations the level-S nonlinearevolution model of fatigue strain which contains the initialstrain was constructed Then the physical meaning of the
6 Advances in Civil Engineering
Test curveFitting curve
Stress level S = 080
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
02
03
04
05
06
07
08
09
10
Fatig
ue m
odul
us (G
Pa)
Figure 9 Fatigue modulus experimental curves and fitting curvesof group A
Stress level S = 075
Test curveFitting curve
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
04
05
06
07
08
09
10
Fatig
ue m
odul
us (G
Pa)
Figure 10 Fatigue modulus experimental curves and fitting curvesof group B
parameters and the range and the impact on the shape of thecurve were all discussed
(2) The results show that the level-S model covered alltypes of fatigue strain evolution and has strong adaptabilityand high accuracyThemodel could be tested for applicabilityto steel fiber reinforced concrete glass fiber reinforced con-crete and a variety of characteristics of concrete
(3) The fatigue modulus evolution model was establishedunder constant amplitude bending fatigue loading based on
Stress level S = 070
Test curveFitting curve
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
04
05
06
07
08
09
10
Fatig
ue m
odul
us (G
Pa)
Figure 11 Fatigue modulus experimental curves and fitting curvesof group C
fatigue strain evolution model and the hypothesis of fatiguemodulus inversely related to fatigue strain amplitude
(4) The evolution curve of fatigue strain and fatiguemodulus both have three stages of variation and the middlestage is nearly linearWhen the recycle ratio is 09 two curvesconverge quickly The difference is that the fatigue strainevolution curve is from the lower left corner to the upper rightcorner but the fatigue modulus evolution curve is from theupper left corner to the right lower corner
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This project was supported by the Project of OutstandingYouth Science Foundation of China (5142500405) Scientificand Technological Research Program of Chongqing Munic-ipal Education Commission (KJ1710237) Youth Project ofChongqing Three Gorges College (15QN14) Open Foun-dation of State Key Laboratory Breeding Base of Moun-tain Bridge and Tunnel Engineering (CQSLBF-Y16-6)Project of Chongqing Science and Technology Committee(cstc2015jcyjA00022) Scientific and Technological ResearchProgram of Chongqing Municipal Education Commission(KJ1601019) Disaster Prevention andDisasterMitigation andSafety of Engineering Structure in Three Gorges Reservoirand Open Foundation of Engineering Research Center ofBridge Structure and Material in the Mountainous AreaMinistry of Education (QLGCZX-JJ2015-4)
Advances in Civil Engineering 7
References
[1] C Y Li and Y P Song ldquoStudy of residual strain of concreteunder fatigue loadingrdquo Journal of Dalian University of Technol-ogy vol 41 no 3 pp 355ndash358 2001
[2] M K Lee and B I G Barr ldquoAn overview of the fatigue behav-iour of plain and fibre reinforced concreterdquo Cement amp ConcreteComposites vol 26 no 4 pp 299ndash305 2004
[3] A Alliche ldquoDamage model for fatigue loading of concreterdquoInternational Journal of Fatigue vol 26 no 9 pp 915ndash921 2004
[4] Y-C Kuang and J-P Ou ldquoPassive smart self-repairing concretebeams by using shapememory alloy wires and fibers containingadhesivesrdquo Journal of Central South University of Technologyvol 15 no 3 pp 411ndash417 2008
[5] J-K Kim and Y-Y Kim ldquoExperimental study of the fatiguebehavior of high strength concreterdquo Cement and ConcreteResearch vol 26 no 10 pp 1513ndash1523 1996
[6] XW LuoTest research of themechanics characteristic of recycledaggregate concrete under recurrent load Wuhan University ofTechnology School of Civil Engineering and ArchitectureWuhan China 2006
[7] Y B Chen Z N Lu and D Huang ldquoFatigue defect of layersteel fiber reinforced concreterdquo Journal of Wuhan University ofTechnology (Natural Science Edition) vol 18 no 1 pp 65ndash682003
[8] P B Cachim J A Figueiras and P A A Pereira ldquoFatiguebehavior of fiber-reinforced concrete in compressionrdquo Cementamp Concrete Composites vol 24 no 2 pp 211ndash217 2002
[9] J B Xie T C He and H R Cheng ldquoInvestigation flexuralfatigue behavior of steel fiber reinforced concrete for pavementsurface stratum under cyclic loadrdquo Journal of Lanzhou Univer-sity of Technology vol 30 no 2 pp 104ndash109 2004
[10] R M Wang G F Zhao and Y P Song ldquoResearch on compres-sion fatigue performance of concreterdquo China Civil EngineeringJournal vol 24 no 4 pp 38ndash47 1991
[11] F Liu and J Zhou ldquoFatigue Strain andDamageAnalysis of Con-crete in Reinforced Concrete Beams under Constant Ampli-tude Fatigue Loadingrdquo Shock and Vibration vol 2016 ArticleID 3950140 7 pages 2016
[12] J Q Xiao D X Ding and X W Luo ldquoQuantitative analysisof damage evolution as recycled concrete approaches fatiguefailurerdquo Journal of Central South University (Science and Tech-nology) vol 42 no 1 pp 170ndash176 2011
[13] X R Ge Y Jiang Y D Lu and J X Ren ldquoTesting study onfatigue deformation law of rock under cyclic loadingrdquo ChineseJournal of Rock Mechanics and Engineering vol 22 no 10 pp1581ndash1585 2003
[14] Y Lv Study on the Flexural Fatigue Performance and CumulativeDamage of Glass Fiber Reinforced Concrete KunmingUniversityof Technology Kunming China 2012
[15] R-N Yang and D-M Wei ldquoResearch on the fatigue strain andfatigue modulus and damage of steel fiber reinforced concreteunder constant amplitude bending fatigue loadingrdquo GongchengLixueEngineering Mechanics vol 29 no 11 pp 99ndash122 2012
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2 Advances in Civil Engineering
00
(a) Type
(b) Type
(c) Type
Fatig
ue st
rain
()
Recycle ratio (nN)
Figure 1 Basic types of fatigue strain evolution curve
initial strain was not taken into consideration basically Inthese equations the significance of each factor is not clearenough which results in unstable fitting coefficients
In the literature [11] published by the author the nonlinearstrain evolution model was established The strain evolutionlaw of concrete under constant amplitude fatigue load and thelaw of fatigue damage evolution based on strain were studiedBased on the previous research in this paper the physicalmeanings the ranges and the impact on the shape of thecurve of parameters in the nonlinear strain evolution modelwere all discussed The evolution model of fatigue moduluswas established under constant amplitude bending fatigueloading based on the fatigue strain evolution model and thehypothesis of fatigue modulus inversely related fatigue strainamplitude Moreover the whole process is validated by theexperimental data
2 Fatigue Strain Evolution Model
To better describe the three-stage variation law a fatiguestrain evolution model with the simple form was proposedwhich could be suitable for different stress types (Fig-ure 1) Moreover some valuable physical parameters could beobtained via the model such as the initial strain instabilityspeed of the third stage (acceleration of fatigue strain) andthe proportion of that in the total fatigue life [12] Via therepeated study on the related mathematical functions themodel could be obtained below
120576119899 = 1205760 + 120572 sdot ( 120573120573 minus 119899119873119891 minus 1)1119901 (1)
In formula (1) 1205760 indicate the initial strain and 120576119899 indicatethe fatigue strain under 119899 times of fatigue loading cycle 119899indicate the cycle times of fatigue loads 119873119891 indicate thefatigue life 120572 120573 and 119901 were parameters in the equation
If the maximum fatigue strain 120576119899max or fatigue residualstrain 120576119899res was regarded as the value of 120576119899 formula (2) would
060
065
070
075
080
085
090
095
100
Relat
ive s
trai
n (
n
m)
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
(B) (C) (D) (E)
(F) (G) (H)
Figure 2 Influence of 119901 on fatigue strain curve (B) 119901 = 2 120573 =125 (C) 119901 = 3 120573 = 1125 (D) 119901 = 4 120573 = 10625 (E) 119901 = 5 120573 =103125 (F) 119901 = 6 120573 = 1015625 (G) 119901 = 7 120573 = 1007813 (H)119901 = 8 120573 = 1003906 (B)sim(H) 1205760max120576119891max = 06 120572120576119891max = 02be obtained by formula (1) In the sameway if the initial max-imum strain 1205760max or initial residual strain 1205760res was regardedas the value of 1205760 formula (3) would be obtained
1205760max = 1205760max + 120572( 120573120573 minus 119899119873119891 minus 1)1119901 (2)
120576119899res = 1205760res + 120572( 120573120573 minus 119899119873119891 minus 1)1119901 (3)
Formula (2) was called as the maximum fatigue strainevolution model while formula (3) was the fatigue residualstrain evolution model
If the upper limit fatigue stress is large fatigue strain willchange fast and increase The slope of the curve will be largeand the curve become the vertical lineThat was to say withina few loading cycles the strain will reach the unstable stateThe three-stage curve will degenerate the c-type in Figure 2[13] When the upper limit fatigue stress does not exceed thethreshold the elastic strain will be added to the initial strainand the strain remains unchanged which is similar to the a-type curve These two types are both the limit state so thestresses in most fatigue tests are located between the upperlimit and the threshold In other words the evolution curveshave characteristics of both a-type and c-type Generally theb-type curve in Figure 2 can be obtained According to theactual results the evolution laws of the b-type curve will beanalyzed in this paper1205760max and 1205760res indicate respectively the initial maximumstrain and residual strain They are mainly caused by somefactors such as the initial defects of the material and preload-ing of the structure Due to the difficulty of distinguishing
Advances in Civil Engineering 3
between the two strains the actually measured values ofthe fatigue test can be obtained Therefore when the fatigueloading firstly reaches the upper limit the correspondingmaximum strain 1205761max and residual strain 1205761res are adopted inthis paper For example Lv [14] obtained the formula of 1205761maxand 1205761res according to the actual experimental data as 1205761res =025(1205761max120576unstab)2 Among the formula 120576unstab indicated thetotal strain of the unstable concrete
In order to study the range of parameters 120572 120573 and119901 including impacts on the evolution law of fatigue straincurves formulas (2) and (3) are divided by the limited fatiguestrain on both sides So the following are obtained
120576119899max120576119891max= 1205760max120576119891max+ 120572120576119891max( 120573120573 minus 119899119873119891 minus 1)
1119901 (4)
120576119899res120576119891res =1205760res120576119891res +120572120576119891res (
120573120573 minus 119899119873119891 minus 1)1119901 (5)
Both formulas are the normalized evolution model of thefatigue strain In the former one 120576119891max indicates the limitedmaximum fatigue strain In the latter one 120576119891res indicates thelimited fatigue residual strain120573 indicates the destabilizing factor This parameterdepends on 119901 and 120572 If the circulation ratio 119899119873119891 is equal to 1the point (1 1) will locate on the curve of formulas (4) and (5)Via the two formulas values of 120573 can be deducted as formula(6) which correspond to the maximum fatigue stress and theresidual fatigue stress
1205731 = ((1 minus 1205760max120576119891max)(120572120576119891max) )
minus119901
+ 1
1205732 = ((1 minus 1205760res120576119891res)(120572120576119891res) )
minus119901
+ 1(6)
In order to analyze the impacts of 119901 and 120572 on the fatiguestrain evolution curve the evolution formula (4) was taken asan example In the first place the impact of119901was analyzed via1205760max120576119891max and 120572120576119891max Then combined with 119901 and 1205760max120576119891maxthe impact of 120572 was analyzed further Ultimately the impactscurve of 119901 and 120572 were shown in Figures 2 and 3
It could be seen from Figure 2 that parameter 119901 wouldinfluence the convergence speed of the curve using the level-S nonlinear model With the growing of 119901 the convergencespeed of the fatigue strain curve in the third stage couldbecome faster while the stage corresponded to the irre-versible third stage of fatigue deformation which also meantthe acceleration stage of unstable crack propagation Theparameter 119901 was also called the instability speed factor andthe suggested values were located in [2 8]
The effects of different 120572 values on the curve are shown inFigure 3
It could be seen from Figure 3 that parameter 120572 wouldinfluence the proportion of the third stage in the total fatiguelife The stage was the so-called acceleration stage of the
060
065
070
075
080
085
090
095
100
Relat
ive s
trai
n (
n
m)
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
(B) (C) (D) (E)
(F) (G) (H)
Figure 3 Influence of 120572 on the fatigue strain curve (B) 120572120576119891max =01 120573 = 10625 (C) 120572120576119891max = 015 120573 = 1140625 (D) 120572120576119891max =02 120573 = 125 (E) 120572120576119891max = 025 120573 = 1390625 (F) 120572120576119891max =03 120573 = 15625 (G) 120572120576119891max = 035 120573 = 1765625 (H) 120572120576119891max =04 120573 = 2 (B)sim(H) 1205760max120576119891max = 06 119901 = 2
fatigue strain With the growing of 120572 the proportion of theacceleration stage could become shorter 120572120576119891max was locatedin (0 1 minus 1205760max120576119891max) so 120572 was located in (0 120576119891max minus 1205760max)
The suggested values of the parameters 120572 120573 119901 weremainly aimed at the b-type curves which had three stages ofevolutions That was to say the values for the a-type and c-type were not limited by the paper In Figure 4 the S-shapedcurves by combining various parameters were presentedincluding nearly different kinds of fatigue strain evolutionsfor the concrete material
3 Fatigue Modulus Evolution Model
Fatigue modulus 119864 is the ratio of stress amplitude 120590max minus120590min to strain amplitude 120576max minus 120576min that is 119864 = (120590max minus120590min)(120576max minus 120576min) 120590max and 120590min are the maximum andminimum fatigue stresses Under constant amplitude fatigueloads stress amplitude120590maxminus120590min remains unchangedThere-fore the fatigue modulus and fatigue strain amplitude are aninversely proportional relationship that is if the fatigue strainamplitude decreases the fatigue modulus increases On thecontrary if the fatigue strain amplitude increases the fatiguemodulus will be reduced [14 15] The strain evolution curveis the level-S-shaped curve from the lower left corner to theupper right corner from the inverse relationship The fatiguestrain and fatigue modulus can be shown in Figure 5
Suppose two curves are symmetrical about the straightline 119910 = 119863 and the point in the fatigue strain curve is (119909 119910)Then the corresponding point on the curve of the fatiguemodulus is (119909 2119863minus119910) Thus the normalized fatigue modulus
4 Advances in Civil Engineering
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
040045050055060065070075080085090095100
Relat
ive s
trai
n (
n
m)
(B) (C) (D) (E)
(F) (G) (H)
Figure 4 Level-S-shaped curve family of fatigue strain (B)1205760max120576119891max = 07 119901 = 2 120572120576119891max = 025 120573 = 1694444 (C)1205760max120576119891max = 065 119901 = 3 120572120576119891max = 023 120573 = 1283778 (D)1205760max120576119891max = 06 119901 = 4 120572120576119891max = 019 120573 = 1050907 (E)1205760max120576119891max = 055 119901 = 5 120572120576119891max = 017 120573 = 1007695 (F)1205760max120576119891max = 05 119901 = 6 120572120576119891max = 015 120573 = 1000729 (G)1205760max120576119891max = 045 119901 = 7 120572120576119891max = 013 120573 = 1000041 (H)1205760max120576119891max = 040 119901 = 8 120572120576119891max = 011 120573 = 1000001
y = D
Relative strains Relative modulus
060
065
070
075
080
085
090
095
100
Relat
ive s
trai
n (m
odul
us)
01 02 03 04 05 06 07 08 09 1000Recycle ratio
Figure 5 Diagram of fatigue strain and fatigue modulus evolutioncurves
evolution model can be obtained by the fatigue strain evolu-tion model as shown in the following equation
1198641198991198640 =1205760120576119891 + 120572
1015840
120576119891 ( 12057310158401205731015840 minus 119899119873119891 minus 1)11199011015840 + 2119863 (7)
Stress level S = 08
Test curve of maximum strainTest curve of maximum strainFitting curve of maximum strainFitting curve of residual strain
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
0255075
100125150175200225250275300325350375
Fatig
ue st
rain
(
)Figure 6 Fatigue strain experimental curves and fitting curves ofgroup A
Therefore the fatigue modulus evolution model is
119864119899 = 1198640( 1205760120576119891 + 1205721015840
120576119891 ( 12057310158401205731015840 minus 119899119873119891 minus 1)11199011015840 + 2119863) (8)
In formulas (7) to (8) 119864119899 is the elastic modulus afterfatigue cycle 119899 times and 1198640 is the initial elastic modulus
4 Model Validation
41 Experiment Introduction The length width and heightof the concrete experiment beams are 400mm 100mmand 100mm respectively Concrete strength is C35 Theexperimental beam is divided into A B and C three groupswith corresponding stress levels 119878max = 120590max119891 of 08 075and 07 The fatigue experiment equipment is the MTS810material testing machine The third point (50 + 3 times 100 + 50)bending constant amplitude fatigue load was used The loadfrequency is l0Hz The fatigue experiment cycle stress ratio120590min120590max is 01 Fatigue life from the experiments was 37256times 14905 times and 631291 times and the initial elasticmodulus is 50GPa
42 Concrete Fatigue Strain Evolution Analysis The fittingresults of the maximum fatigue strain and fatigue residualstrain under different stress levels using the model formulas(2) to (3) are shown in Figures 6ndash8 Coefficients of the evo-lutionary model are shown in Table 1 The data in the figureare the average of each group
From Figures 6ndash8 and Table 1 fatigue strain evolutionequations (2) and (3) can be a good fit to the experimentaldata Correlation coefficients were above 098 The evolution
Advances in Civil Engineering 5
Stress level S = 075
Test curve of maximum strainTest curve of maximum strainFitting curve of maximum strainFitting curve of residual strain
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
0255075
100125150175200225250275300325350375
Fatig
ue st
rain
(
)
Figure 7 Fatigue strain experimental curves and fitting curves ofgroup B
Stress level S = 07
Test curve of maximum strainTest curve of maximum strainFitting curve of maximum strainFitting curve of residual strain
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
0255075
100125150175200225250275300325350375
Fatig
ue st
rain
(
)
Figure 8 Fatigue strain experimental curves and fitting curves ofgroup C
of maximum fatigue strain and fatigue residual strain hassimilar three-phase variation and the intermediate stage closeto the linear change The curve converged rapidly when thecycle ratio exceeded approximately 09 The strain evolutioncurve is the level-S-shaped curve from the lower left cornerto the upper right corner Because the experiment measuredonly the initial maximum strain and has not measured initial
Table 1 Coefficient values of fatigue strain
Groups 119878 120576119899max 1198772120572 120573 119901A 08 470919 10182 24672 09891B 075 481478 10456 27550 09935C 07 540070 12096 34928 09751
Groups 119878 120576119899res 1198772120572 120573 119901A 08 438164 10422 27736 09889B 075 399757 10545 29135 09887C 07 325965 10171 35857 09753
Table 2 Coefficient values of fatigue modulus
Stress level 119864119899 11987721205721015840 1205731015840 1199011015840 11986308 23645 10081 302698 27075 09909075 15242 10762 192173 18772 0993507 11918 10070 190863 15847 09833
residual strain the evolution curve of the maximum strainstarts from the initial value but the evolution curve of fatigueresidual strain starts from zero According to (0 120576119891max minus 1205760max)and experimental data the range of 120572 of groups A B and Cis (0 250) (0 130) and (0 165) respectively Fitting results of120572 falls in these ranges in line with the boundary conditionsWhen the stress level is from 08 to 075 and from 075 to07 the 119901 value is increasing demonstrating that the smallerthe stress level is the faster the fatigue strain evolution curveconverges
43 Concrete Fatigue Modulus Evolution Analysis The fittingresults of the fatigue modulus under different stress levelsusing the model formula (8) are shown in Figures 9ndash11Coefficients of evolutionary model (8) are shown in Table 2The data in figure are the average of each group
From Figures 9ndash11 and Table 2 fatiguemodulus evolutionequation (8) can be a good fit to the experimental dataCorrelation coefficients were above 097 The evolution of thefatigue modulus also has a three-phase variation and theintermediate stage is also close to the linear changeThe curveconverged rapidly when the cycle ratio exceeded approxi-mately 09 Fatigue modulus evolution curve is the level-S-shaped curve from the upper left corner to the lower right cor-ner When the stress level is from 08 to 075 and from 075 to07 the 1199011015840 value is increasing demonstrating that the smallerthe stress level is the faster the fatigue modulus evolutioncurve converges
5 Conclusions
(1) By analyzing the shortcomings and limitations of existingfatigue strain evolution equations the level-S nonlinearevolution model of fatigue strain which contains the initialstrain was constructed Then the physical meaning of the
6 Advances in Civil Engineering
Test curveFitting curve
Stress level S = 080
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
02
03
04
05
06
07
08
09
10
Fatig
ue m
odul
us (G
Pa)
Figure 9 Fatigue modulus experimental curves and fitting curvesof group A
Stress level S = 075
Test curveFitting curve
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
04
05
06
07
08
09
10
Fatig
ue m
odul
us (G
Pa)
Figure 10 Fatigue modulus experimental curves and fitting curvesof group B
parameters and the range and the impact on the shape of thecurve were all discussed
(2) The results show that the level-S model covered alltypes of fatigue strain evolution and has strong adaptabilityand high accuracyThemodel could be tested for applicabilityto steel fiber reinforced concrete glass fiber reinforced con-crete and a variety of characteristics of concrete
(3) The fatigue modulus evolution model was establishedunder constant amplitude bending fatigue loading based on
Stress level S = 070
Test curveFitting curve
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
04
05
06
07
08
09
10
Fatig
ue m
odul
us (G
Pa)
Figure 11 Fatigue modulus experimental curves and fitting curvesof group C
fatigue strain evolution model and the hypothesis of fatiguemodulus inversely related to fatigue strain amplitude
(4) The evolution curve of fatigue strain and fatiguemodulus both have three stages of variation and the middlestage is nearly linearWhen the recycle ratio is 09 two curvesconverge quickly The difference is that the fatigue strainevolution curve is from the lower left corner to the upper rightcorner but the fatigue modulus evolution curve is from theupper left corner to the right lower corner
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This project was supported by the Project of OutstandingYouth Science Foundation of China (5142500405) Scientificand Technological Research Program of Chongqing Munic-ipal Education Commission (KJ1710237) Youth Project ofChongqing Three Gorges College (15QN14) Open Foun-dation of State Key Laboratory Breeding Base of Moun-tain Bridge and Tunnel Engineering (CQSLBF-Y16-6)Project of Chongqing Science and Technology Committee(cstc2015jcyjA00022) Scientific and Technological ResearchProgram of Chongqing Municipal Education Commission(KJ1601019) Disaster Prevention andDisasterMitigation andSafety of Engineering Structure in Three Gorges Reservoirand Open Foundation of Engineering Research Center ofBridge Structure and Material in the Mountainous AreaMinistry of Education (QLGCZX-JJ2015-4)
Advances in Civil Engineering 7
References
[1] C Y Li and Y P Song ldquoStudy of residual strain of concreteunder fatigue loadingrdquo Journal of Dalian University of Technol-ogy vol 41 no 3 pp 355ndash358 2001
[2] M K Lee and B I G Barr ldquoAn overview of the fatigue behav-iour of plain and fibre reinforced concreterdquo Cement amp ConcreteComposites vol 26 no 4 pp 299ndash305 2004
[3] A Alliche ldquoDamage model for fatigue loading of concreterdquoInternational Journal of Fatigue vol 26 no 9 pp 915ndash921 2004
[4] Y-C Kuang and J-P Ou ldquoPassive smart self-repairing concretebeams by using shapememory alloy wires and fibers containingadhesivesrdquo Journal of Central South University of Technologyvol 15 no 3 pp 411ndash417 2008
[5] J-K Kim and Y-Y Kim ldquoExperimental study of the fatiguebehavior of high strength concreterdquo Cement and ConcreteResearch vol 26 no 10 pp 1513ndash1523 1996
[6] XW LuoTest research of themechanics characteristic of recycledaggregate concrete under recurrent load Wuhan University ofTechnology School of Civil Engineering and ArchitectureWuhan China 2006
[7] Y B Chen Z N Lu and D Huang ldquoFatigue defect of layersteel fiber reinforced concreterdquo Journal of Wuhan University ofTechnology (Natural Science Edition) vol 18 no 1 pp 65ndash682003
[8] P B Cachim J A Figueiras and P A A Pereira ldquoFatiguebehavior of fiber-reinforced concrete in compressionrdquo Cementamp Concrete Composites vol 24 no 2 pp 211ndash217 2002
[9] J B Xie T C He and H R Cheng ldquoInvestigation flexuralfatigue behavior of steel fiber reinforced concrete for pavementsurface stratum under cyclic loadrdquo Journal of Lanzhou Univer-sity of Technology vol 30 no 2 pp 104ndash109 2004
[10] R M Wang G F Zhao and Y P Song ldquoResearch on compres-sion fatigue performance of concreterdquo China Civil EngineeringJournal vol 24 no 4 pp 38ndash47 1991
[11] F Liu and J Zhou ldquoFatigue Strain andDamageAnalysis of Con-crete in Reinforced Concrete Beams under Constant Ampli-tude Fatigue Loadingrdquo Shock and Vibration vol 2016 ArticleID 3950140 7 pages 2016
[12] J Q Xiao D X Ding and X W Luo ldquoQuantitative analysisof damage evolution as recycled concrete approaches fatiguefailurerdquo Journal of Central South University (Science and Tech-nology) vol 42 no 1 pp 170ndash176 2011
[13] X R Ge Y Jiang Y D Lu and J X Ren ldquoTesting study onfatigue deformation law of rock under cyclic loadingrdquo ChineseJournal of Rock Mechanics and Engineering vol 22 no 10 pp1581ndash1585 2003
[14] Y Lv Study on the Flexural Fatigue Performance and CumulativeDamage of Glass Fiber Reinforced Concrete KunmingUniversityof Technology Kunming China 2012
[15] R-N Yang and D-M Wei ldquoResearch on the fatigue strain andfatigue modulus and damage of steel fiber reinforced concreteunder constant amplitude bending fatigue loadingrdquo GongchengLixueEngineering Mechanics vol 29 no 11 pp 99ndash122 2012
RoboticsJournal of
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RotatingMachinery
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Journal of
Volume 201
Submit your manuscripts athttpswwwhindawicom
VLSI Design
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Shock and Vibration
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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International Journal of
Advances in Civil Engineering 3
between the two strains the actually measured values ofthe fatigue test can be obtained Therefore when the fatigueloading firstly reaches the upper limit the correspondingmaximum strain 1205761max and residual strain 1205761res are adopted inthis paper For example Lv [14] obtained the formula of 1205761maxand 1205761res according to the actual experimental data as 1205761res =025(1205761max120576unstab)2 Among the formula 120576unstab indicated thetotal strain of the unstable concrete
In order to study the range of parameters 120572 120573 and119901 including impacts on the evolution law of fatigue straincurves formulas (2) and (3) are divided by the limited fatiguestrain on both sides So the following are obtained
120576119899max120576119891max= 1205760max120576119891max+ 120572120576119891max( 120573120573 minus 119899119873119891 minus 1)
1119901 (4)
120576119899res120576119891res =1205760res120576119891res +120572120576119891res (
120573120573 minus 119899119873119891 minus 1)1119901 (5)
Both formulas are the normalized evolution model of thefatigue strain In the former one 120576119891max indicates the limitedmaximum fatigue strain In the latter one 120576119891res indicates thelimited fatigue residual strain120573 indicates the destabilizing factor This parameterdepends on 119901 and 120572 If the circulation ratio 119899119873119891 is equal to 1the point (1 1) will locate on the curve of formulas (4) and (5)Via the two formulas values of 120573 can be deducted as formula(6) which correspond to the maximum fatigue stress and theresidual fatigue stress
1205731 = ((1 minus 1205760max120576119891max)(120572120576119891max) )
minus119901
+ 1
1205732 = ((1 minus 1205760res120576119891res)(120572120576119891res) )
minus119901
+ 1(6)
In order to analyze the impacts of 119901 and 120572 on the fatiguestrain evolution curve the evolution formula (4) was taken asan example In the first place the impact of119901was analyzed via1205760max120576119891max and 120572120576119891max Then combined with 119901 and 1205760max120576119891maxthe impact of 120572 was analyzed further Ultimately the impactscurve of 119901 and 120572 were shown in Figures 2 and 3
It could be seen from Figure 2 that parameter 119901 wouldinfluence the convergence speed of the curve using the level-S nonlinear model With the growing of 119901 the convergencespeed of the fatigue strain curve in the third stage couldbecome faster while the stage corresponded to the irre-versible third stage of fatigue deformation which also meantthe acceleration stage of unstable crack propagation Theparameter 119901 was also called the instability speed factor andthe suggested values were located in [2 8]
The effects of different 120572 values on the curve are shown inFigure 3
It could be seen from Figure 3 that parameter 120572 wouldinfluence the proportion of the third stage in the total fatiguelife The stage was the so-called acceleration stage of the
060
065
070
075
080
085
090
095
100
Relat
ive s
trai
n (
n
m)
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
(B) (C) (D) (E)
(F) (G) (H)
Figure 3 Influence of 120572 on the fatigue strain curve (B) 120572120576119891max =01 120573 = 10625 (C) 120572120576119891max = 015 120573 = 1140625 (D) 120572120576119891max =02 120573 = 125 (E) 120572120576119891max = 025 120573 = 1390625 (F) 120572120576119891max =03 120573 = 15625 (G) 120572120576119891max = 035 120573 = 1765625 (H) 120572120576119891max =04 120573 = 2 (B)sim(H) 1205760max120576119891max = 06 119901 = 2
fatigue strain With the growing of 120572 the proportion of theacceleration stage could become shorter 120572120576119891max was locatedin (0 1 minus 1205760max120576119891max) so 120572 was located in (0 120576119891max minus 1205760max)
The suggested values of the parameters 120572 120573 119901 weremainly aimed at the b-type curves which had three stages ofevolutions That was to say the values for the a-type and c-type were not limited by the paper In Figure 4 the S-shapedcurves by combining various parameters were presentedincluding nearly different kinds of fatigue strain evolutionsfor the concrete material
3 Fatigue Modulus Evolution Model
Fatigue modulus 119864 is the ratio of stress amplitude 120590max minus120590min to strain amplitude 120576max minus 120576min that is 119864 = (120590max minus120590min)(120576max minus 120576min) 120590max and 120590min are the maximum andminimum fatigue stresses Under constant amplitude fatigueloads stress amplitude120590maxminus120590min remains unchangedThere-fore the fatigue modulus and fatigue strain amplitude are aninversely proportional relationship that is if the fatigue strainamplitude decreases the fatigue modulus increases On thecontrary if the fatigue strain amplitude increases the fatiguemodulus will be reduced [14 15] The strain evolution curveis the level-S-shaped curve from the lower left corner to theupper right corner from the inverse relationship The fatiguestrain and fatigue modulus can be shown in Figure 5
Suppose two curves are symmetrical about the straightline 119910 = 119863 and the point in the fatigue strain curve is (119909 119910)Then the corresponding point on the curve of the fatiguemodulus is (119909 2119863minus119910) Thus the normalized fatigue modulus
4 Advances in Civil Engineering
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
040045050055060065070075080085090095100
Relat
ive s
trai
n (
n
m)
(B) (C) (D) (E)
(F) (G) (H)
Figure 4 Level-S-shaped curve family of fatigue strain (B)1205760max120576119891max = 07 119901 = 2 120572120576119891max = 025 120573 = 1694444 (C)1205760max120576119891max = 065 119901 = 3 120572120576119891max = 023 120573 = 1283778 (D)1205760max120576119891max = 06 119901 = 4 120572120576119891max = 019 120573 = 1050907 (E)1205760max120576119891max = 055 119901 = 5 120572120576119891max = 017 120573 = 1007695 (F)1205760max120576119891max = 05 119901 = 6 120572120576119891max = 015 120573 = 1000729 (G)1205760max120576119891max = 045 119901 = 7 120572120576119891max = 013 120573 = 1000041 (H)1205760max120576119891max = 040 119901 = 8 120572120576119891max = 011 120573 = 1000001
y = D
Relative strains Relative modulus
060
065
070
075
080
085
090
095
100
Relat
ive s
trai
n (m
odul
us)
01 02 03 04 05 06 07 08 09 1000Recycle ratio
Figure 5 Diagram of fatigue strain and fatigue modulus evolutioncurves
evolution model can be obtained by the fatigue strain evolu-tion model as shown in the following equation
1198641198991198640 =1205760120576119891 + 120572
1015840
120576119891 ( 12057310158401205731015840 minus 119899119873119891 minus 1)11199011015840 + 2119863 (7)
Stress level S = 08
Test curve of maximum strainTest curve of maximum strainFitting curve of maximum strainFitting curve of residual strain
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
0255075
100125150175200225250275300325350375
Fatig
ue st
rain
(
)Figure 6 Fatigue strain experimental curves and fitting curves ofgroup A
Therefore the fatigue modulus evolution model is
119864119899 = 1198640( 1205760120576119891 + 1205721015840
120576119891 ( 12057310158401205731015840 minus 119899119873119891 minus 1)11199011015840 + 2119863) (8)
In formulas (7) to (8) 119864119899 is the elastic modulus afterfatigue cycle 119899 times and 1198640 is the initial elastic modulus
4 Model Validation
41 Experiment Introduction The length width and heightof the concrete experiment beams are 400mm 100mmand 100mm respectively Concrete strength is C35 Theexperimental beam is divided into A B and C three groupswith corresponding stress levels 119878max = 120590max119891 of 08 075and 07 The fatigue experiment equipment is the MTS810material testing machine The third point (50 + 3 times 100 + 50)bending constant amplitude fatigue load was used The loadfrequency is l0Hz The fatigue experiment cycle stress ratio120590min120590max is 01 Fatigue life from the experiments was 37256times 14905 times and 631291 times and the initial elasticmodulus is 50GPa
42 Concrete Fatigue Strain Evolution Analysis The fittingresults of the maximum fatigue strain and fatigue residualstrain under different stress levels using the model formulas(2) to (3) are shown in Figures 6ndash8 Coefficients of the evo-lutionary model are shown in Table 1 The data in the figureare the average of each group
From Figures 6ndash8 and Table 1 fatigue strain evolutionequations (2) and (3) can be a good fit to the experimentaldata Correlation coefficients were above 098 The evolution
Advances in Civil Engineering 5
Stress level S = 075
Test curve of maximum strainTest curve of maximum strainFitting curve of maximum strainFitting curve of residual strain
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
0255075
100125150175200225250275300325350375
Fatig
ue st
rain
(
)
Figure 7 Fatigue strain experimental curves and fitting curves ofgroup B
Stress level S = 07
Test curve of maximum strainTest curve of maximum strainFitting curve of maximum strainFitting curve of residual strain
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
0255075
100125150175200225250275300325350375
Fatig
ue st
rain
(
)
Figure 8 Fatigue strain experimental curves and fitting curves ofgroup C
of maximum fatigue strain and fatigue residual strain hassimilar three-phase variation and the intermediate stage closeto the linear change The curve converged rapidly when thecycle ratio exceeded approximately 09 The strain evolutioncurve is the level-S-shaped curve from the lower left cornerto the upper right corner Because the experiment measuredonly the initial maximum strain and has not measured initial
Table 1 Coefficient values of fatigue strain
Groups 119878 120576119899max 1198772120572 120573 119901A 08 470919 10182 24672 09891B 075 481478 10456 27550 09935C 07 540070 12096 34928 09751
Groups 119878 120576119899res 1198772120572 120573 119901A 08 438164 10422 27736 09889B 075 399757 10545 29135 09887C 07 325965 10171 35857 09753
Table 2 Coefficient values of fatigue modulus
Stress level 119864119899 11987721205721015840 1205731015840 1199011015840 11986308 23645 10081 302698 27075 09909075 15242 10762 192173 18772 0993507 11918 10070 190863 15847 09833
residual strain the evolution curve of the maximum strainstarts from the initial value but the evolution curve of fatigueresidual strain starts from zero According to (0 120576119891max minus 1205760max)and experimental data the range of 120572 of groups A B and Cis (0 250) (0 130) and (0 165) respectively Fitting results of120572 falls in these ranges in line with the boundary conditionsWhen the stress level is from 08 to 075 and from 075 to07 the 119901 value is increasing demonstrating that the smallerthe stress level is the faster the fatigue strain evolution curveconverges
43 Concrete Fatigue Modulus Evolution Analysis The fittingresults of the fatigue modulus under different stress levelsusing the model formula (8) are shown in Figures 9ndash11Coefficients of evolutionary model (8) are shown in Table 2The data in figure are the average of each group
From Figures 9ndash11 and Table 2 fatiguemodulus evolutionequation (8) can be a good fit to the experimental dataCorrelation coefficients were above 097 The evolution of thefatigue modulus also has a three-phase variation and theintermediate stage is also close to the linear changeThe curveconverged rapidly when the cycle ratio exceeded approxi-mately 09 Fatigue modulus evolution curve is the level-S-shaped curve from the upper left corner to the lower right cor-ner When the stress level is from 08 to 075 and from 075 to07 the 1199011015840 value is increasing demonstrating that the smallerthe stress level is the faster the fatigue modulus evolutioncurve converges
5 Conclusions
(1) By analyzing the shortcomings and limitations of existingfatigue strain evolution equations the level-S nonlinearevolution model of fatigue strain which contains the initialstrain was constructed Then the physical meaning of the
6 Advances in Civil Engineering
Test curveFitting curve
Stress level S = 080
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
02
03
04
05
06
07
08
09
10
Fatig
ue m
odul
us (G
Pa)
Figure 9 Fatigue modulus experimental curves and fitting curvesof group A
Stress level S = 075
Test curveFitting curve
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
04
05
06
07
08
09
10
Fatig
ue m
odul
us (G
Pa)
Figure 10 Fatigue modulus experimental curves and fitting curvesof group B
parameters and the range and the impact on the shape of thecurve were all discussed
(2) The results show that the level-S model covered alltypes of fatigue strain evolution and has strong adaptabilityand high accuracyThemodel could be tested for applicabilityto steel fiber reinforced concrete glass fiber reinforced con-crete and a variety of characteristics of concrete
(3) The fatigue modulus evolution model was establishedunder constant amplitude bending fatigue loading based on
Stress level S = 070
Test curveFitting curve
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
04
05
06
07
08
09
10
Fatig
ue m
odul
us (G
Pa)
Figure 11 Fatigue modulus experimental curves and fitting curvesof group C
fatigue strain evolution model and the hypothesis of fatiguemodulus inversely related to fatigue strain amplitude
(4) The evolution curve of fatigue strain and fatiguemodulus both have three stages of variation and the middlestage is nearly linearWhen the recycle ratio is 09 two curvesconverge quickly The difference is that the fatigue strainevolution curve is from the lower left corner to the upper rightcorner but the fatigue modulus evolution curve is from theupper left corner to the right lower corner
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This project was supported by the Project of OutstandingYouth Science Foundation of China (5142500405) Scientificand Technological Research Program of Chongqing Munic-ipal Education Commission (KJ1710237) Youth Project ofChongqing Three Gorges College (15QN14) Open Foun-dation of State Key Laboratory Breeding Base of Moun-tain Bridge and Tunnel Engineering (CQSLBF-Y16-6)Project of Chongqing Science and Technology Committee(cstc2015jcyjA00022) Scientific and Technological ResearchProgram of Chongqing Municipal Education Commission(KJ1601019) Disaster Prevention andDisasterMitigation andSafety of Engineering Structure in Three Gorges Reservoirand Open Foundation of Engineering Research Center ofBridge Structure and Material in the Mountainous AreaMinistry of Education (QLGCZX-JJ2015-4)
Advances in Civil Engineering 7
References
[1] C Y Li and Y P Song ldquoStudy of residual strain of concreteunder fatigue loadingrdquo Journal of Dalian University of Technol-ogy vol 41 no 3 pp 355ndash358 2001
[2] M K Lee and B I G Barr ldquoAn overview of the fatigue behav-iour of plain and fibre reinforced concreterdquo Cement amp ConcreteComposites vol 26 no 4 pp 299ndash305 2004
[3] A Alliche ldquoDamage model for fatigue loading of concreterdquoInternational Journal of Fatigue vol 26 no 9 pp 915ndash921 2004
[4] Y-C Kuang and J-P Ou ldquoPassive smart self-repairing concretebeams by using shapememory alloy wires and fibers containingadhesivesrdquo Journal of Central South University of Technologyvol 15 no 3 pp 411ndash417 2008
[5] J-K Kim and Y-Y Kim ldquoExperimental study of the fatiguebehavior of high strength concreterdquo Cement and ConcreteResearch vol 26 no 10 pp 1513ndash1523 1996
[6] XW LuoTest research of themechanics characteristic of recycledaggregate concrete under recurrent load Wuhan University ofTechnology School of Civil Engineering and ArchitectureWuhan China 2006
[7] Y B Chen Z N Lu and D Huang ldquoFatigue defect of layersteel fiber reinforced concreterdquo Journal of Wuhan University ofTechnology (Natural Science Edition) vol 18 no 1 pp 65ndash682003
[8] P B Cachim J A Figueiras and P A A Pereira ldquoFatiguebehavior of fiber-reinforced concrete in compressionrdquo Cementamp Concrete Composites vol 24 no 2 pp 211ndash217 2002
[9] J B Xie T C He and H R Cheng ldquoInvestigation flexuralfatigue behavior of steel fiber reinforced concrete for pavementsurface stratum under cyclic loadrdquo Journal of Lanzhou Univer-sity of Technology vol 30 no 2 pp 104ndash109 2004
[10] R M Wang G F Zhao and Y P Song ldquoResearch on compres-sion fatigue performance of concreterdquo China Civil EngineeringJournal vol 24 no 4 pp 38ndash47 1991
[11] F Liu and J Zhou ldquoFatigue Strain andDamageAnalysis of Con-crete in Reinforced Concrete Beams under Constant Ampli-tude Fatigue Loadingrdquo Shock and Vibration vol 2016 ArticleID 3950140 7 pages 2016
[12] J Q Xiao D X Ding and X W Luo ldquoQuantitative analysisof damage evolution as recycled concrete approaches fatiguefailurerdquo Journal of Central South University (Science and Tech-nology) vol 42 no 1 pp 170ndash176 2011
[13] X R Ge Y Jiang Y D Lu and J X Ren ldquoTesting study onfatigue deformation law of rock under cyclic loadingrdquo ChineseJournal of Rock Mechanics and Engineering vol 22 no 10 pp1581ndash1585 2003
[14] Y Lv Study on the Flexural Fatigue Performance and CumulativeDamage of Glass Fiber Reinforced Concrete KunmingUniversityof Technology Kunming China 2012
[15] R-N Yang and D-M Wei ldquoResearch on the fatigue strain andfatigue modulus and damage of steel fiber reinforced concreteunder constant amplitude bending fatigue loadingrdquo GongchengLixueEngineering Mechanics vol 29 no 11 pp 99ndash122 2012
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal of
Volume 201
Submit your manuscripts athttpswwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
4 Advances in Civil Engineering
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
040045050055060065070075080085090095100
Relat
ive s
trai
n (
n
m)
(B) (C) (D) (E)
(F) (G) (H)
Figure 4 Level-S-shaped curve family of fatigue strain (B)1205760max120576119891max = 07 119901 = 2 120572120576119891max = 025 120573 = 1694444 (C)1205760max120576119891max = 065 119901 = 3 120572120576119891max = 023 120573 = 1283778 (D)1205760max120576119891max = 06 119901 = 4 120572120576119891max = 019 120573 = 1050907 (E)1205760max120576119891max = 055 119901 = 5 120572120576119891max = 017 120573 = 1007695 (F)1205760max120576119891max = 05 119901 = 6 120572120576119891max = 015 120573 = 1000729 (G)1205760max120576119891max = 045 119901 = 7 120572120576119891max = 013 120573 = 1000041 (H)1205760max120576119891max = 040 119901 = 8 120572120576119891max = 011 120573 = 1000001
y = D
Relative strains Relative modulus
060
065
070
075
080
085
090
095
100
Relat
ive s
trai
n (m
odul
us)
01 02 03 04 05 06 07 08 09 1000Recycle ratio
Figure 5 Diagram of fatigue strain and fatigue modulus evolutioncurves
evolution model can be obtained by the fatigue strain evolu-tion model as shown in the following equation
1198641198991198640 =1205760120576119891 + 120572
1015840
120576119891 ( 12057310158401205731015840 minus 119899119873119891 minus 1)11199011015840 + 2119863 (7)
Stress level S = 08
Test curve of maximum strainTest curve of maximum strainFitting curve of maximum strainFitting curve of residual strain
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
0255075
100125150175200225250275300325350375
Fatig
ue st
rain
(
)Figure 6 Fatigue strain experimental curves and fitting curves ofgroup A
Therefore the fatigue modulus evolution model is
119864119899 = 1198640( 1205760120576119891 + 1205721015840
120576119891 ( 12057310158401205731015840 minus 119899119873119891 minus 1)11199011015840 + 2119863) (8)
In formulas (7) to (8) 119864119899 is the elastic modulus afterfatigue cycle 119899 times and 1198640 is the initial elastic modulus
4 Model Validation
41 Experiment Introduction The length width and heightof the concrete experiment beams are 400mm 100mmand 100mm respectively Concrete strength is C35 Theexperimental beam is divided into A B and C three groupswith corresponding stress levels 119878max = 120590max119891 of 08 075and 07 The fatigue experiment equipment is the MTS810material testing machine The third point (50 + 3 times 100 + 50)bending constant amplitude fatigue load was used The loadfrequency is l0Hz The fatigue experiment cycle stress ratio120590min120590max is 01 Fatigue life from the experiments was 37256times 14905 times and 631291 times and the initial elasticmodulus is 50GPa
42 Concrete Fatigue Strain Evolution Analysis The fittingresults of the maximum fatigue strain and fatigue residualstrain under different stress levels using the model formulas(2) to (3) are shown in Figures 6ndash8 Coefficients of the evo-lutionary model are shown in Table 1 The data in the figureare the average of each group
From Figures 6ndash8 and Table 1 fatigue strain evolutionequations (2) and (3) can be a good fit to the experimentaldata Correlation coefficients were above 098 The evolution
Advances in Civil Engineering 5
Stress level S = 075
Test curve of maximum strainTest curve of maximum strainFitting curve of maximum strainFitting curve of residual strain
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
0255075
100125150175200225250275300325350375
Fatig
ue st
rain
(
)
Figure 7 Fatigue strain experimental curves and fitting curves ofgroup B
Stress level S = 07
Test curve of maximum strainTest curve of maximum strainFitting curve of maximum strainFitting curve of residual strain
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
0255075
100125150175200225250275300325350375
Fatig
ue st
rain
(
)
Figure 8 Fatigue strain experimental curves and fitting curves ofgroup C
of maximum fatigue strain and fatigue residual strain hassimilar three-phase variation and the intermediate stage closeto the linear change The curve converged rapidly when thecycle ratio exceeded approximately 09 The strain evolutioncurve is the level-S-shaped curve from the lower left cornerto the upper right corner Because the experiment measuredonly the initial maximum strain and has not measured initial
Table 1 Coefficient values of fatigue strain
Groups 119878 120576119899max 1198772120572 120573 119901A 08 470919 10182 24672 09891B 075 481478 10456 27550 09935C 07 540070 12096 34928 09751
Groups 119878 120576119899res 1198772120572 120573 119901A 08 438164 10422 27736 09889B 075 399757 10545 29135 09887C 07 325965 10171 35857 09753
Table 2 Coefficient values of fatigue modulus
Stress level 119864119899 11987721205721015840 1205731015840 1199011015840 11986308 23645 10081 302698 27075 09909075 15242 10762 192173 18772 0993507 11918 10070 190863 15847 09833
residual strain the evolution curve of the maximum strainstarts from the initial value but the evolution curve of fatigueresidual strain starts from zero According to (0 120576119891max minus 1205760max)and experimental data the range of 120572 of groups A B and Cis (0 250) (0 130) and (0 165) respectively Fitting results of120572 falls in these ranges in line with the boundary conditionsWhen the stress level is from 08 to 075 and from 075 to07 the 119901 value is increasing demonstrating that the smallerthe stress level is the faster the fatigue strain evolution curveconverges
43 Concrete Fatigue Modulus Evolution Analysis The fittingresults of the fatigue modulus under different stress levelsusing the model formula (8) are shown in Figures 9ndash11Coefficients of evolutionary model (8) are shown in Table 2The data in figure are the average of each group
From Figures 9ndash11 and Table 2 fatiguemodulus evolutionequation (8) can be a good fit to the experimental dataCorrelation coefficients were above 097 The evolution of thefatigue modulus also has a three-phase variation and theintermediate stage is also close to the linear changeThe curveconverged rapidly when the cycle ratio exceeded approxi-mately 09 Fatigue modulus evolution curve is the level-S-shaped curve from the upper left corner to the lower right cor-ner When the stress level is from 08 to 075 and from 075 to07 the 1199011015840 value is increasing demonstrating that the smallerthe stress level is the faster the fatigue modulus evolutioncurve converges
5 Conclusions
(1) By analyzing the shortcomings and limitations of existingfatigue strain evolution equations the level-S nonlinearevolution model of fatigue strain which contains the initialstrain was constructed Then the physical meaning of the
6 Advances in Civil Engineering
Test curveFitting curve
Stress level S = 080
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
02
03
04
05
06
07
08
09
10
Fatig
ue m
odul
us (G
Pa)
Figure 9 Fatigue modulus experimental curves and fitting curvesof group A
Stress level S = 075
Test curveFitting curve
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
04
05
06
07
08
09
10
Fatig
ue m
odul
us (G
Pa)
Figure 10 Fatigue modulus experimental curves and fitting curvesof group B
parameters and the range and the impact on the shape of thecurve were all discussed
(2) The results show that the level-S model covered alltypes of fatigue strain evolution and has strong adaptabilityand high accuracyThemodel could be tested for applicabilityto steel fiber reinforced concrete glass fiber reinforced con-crete and a variety of characteristics of concrete
(3) The fatigue modulus evolution model was establishedunder constant amplitude bending fatigue loading based on
Stress level S = 070
Test curveFitting curve
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
04
05
06
07
08
09
10
Fatig
ue m
odul
us (G
Pa)
Figure 11 Fatigue modulus experimental curves and fitting curvesof group C
fatigue strain evolution model and the hypothesis of fatiguemodulus inversely related to fatigue strain amplitude
(4) The evolution curve of fatigue strain and fatiguemodulus both have three stages of variation and the middlestage is nearly linearWhen the recycle ratio is 09 two curvesconverge quickly The difference is that the fatigue strainevolution curve is from the lower left corner to the upper rightcorner but the fatigue modulus evolution curve is from theupper left corner to the right lower corner
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This project was supported by the Project of OutstandingYouth Science Foundation of China (5142500405) Scientificand Technological Research Program of Chongqing Munic-ipal Education Commission (KJ1710237) Youth Project ofChongqing Three Gorges College (15QN14) Open Foun-dation of State Key Laboratory Breeding Base of Moun-tain Bridge and Tunnel Engineering (CQSLBF-Y16-6)Project of Chongqing Science and Technology Committee(cstc2015jcyjA00022) Scientific and Technological ResearchProgram of Chongqing Municipal Education Commission(KJ1601019) Disaster Prevention andDisasterMitigation andSafety of Engineering Structure in Three Gorges Reservoirand Open Foundation of Engineering Research Center ofBridge Structure and Material in the Mountainous AreaMinistry of Education (QLGCZX-JJ2015-4)
Advances in Civil Engineering 7
References
[1] C Y Li and Y P Song ldquoStudy of residual strain of concreteunder fatigue loadingrdquo Journal of Dalian University of Technol-ogy vol 41 no 3 pp 355ndash358 2001
[2] M K Lee and B I G Barr ldquoAn overview of the fatigue behav-iour of plain and fibre reinforced concreterdquo Cement amp ConcreteComposites vol 26 no 4 pp 299ndash305 2004
[3] A Alliche ldquoDamage model for fatigue loading of concreterdquoInternational Journal of Fatigue vol 26 no 9 pp 915ndash921 2004
[4] Y-C Kuang and J-P Ou ldquoPassive smart self-repairing concretebeams by using shapememory alloy wires and fibers containingadhesivesrdquo Journal of Central South University of Technologyvol 15 no 3 pp 411ndash417 2008
[5] J-K Kim and Y-Y Kim ldquoExperimental study of the fatiguebehavior of high strength concreterdquo Cement and ConcreteResearch vol 26 no 10 pp 1513ndash1523 1996
[6] XW LuoTest research of themechanics characteristic of recycledaggregate concrete under recurrent load Wuhan University ofTechnology School of Civil Engineering and ArchitectureWuhan China 2006
[7] Y B Chen Z N Lu and D Huang ldquoFatigue defect of layersteel fiber reinforced concreterdquo Journal of Wuhan University ofTechnology (Natural Science Edition) vol 18 no 1 pp 65ndash682003
[8] P B Cachim J A Figueiras and P A A Pereira ldquoFatiguebehavior of fiber-reinforced concrete in compressionrdquo Cementamp Concrete Composites vol 24 no 2 pp 211ndash217 2002
[9] J B Xie T C He and H R Cheng ldquoInvestigation flexuralfatigue behavior of steel fiber reinforced concrete for pavementsurface stratum under cyclic loadrdquo Journal of Lanzhou Univer-sity of Technology vol 30 no 2 pp 104ndash109 2004
[10] R M Wang G F Zhao and Y P Song ldquoResearch on compres-sion fatigue performance of concreterdquo China Civil EngineeringJournal vol 24 no 4 pp 38ndash47 1991
[11] F Liu and J Zhou ldquoFatigue Strain andDamageAnalysis of Con-crete in Reinforced Concrete Beams under Constant Ampli-tude Fatigue Loadingrdquo Shock and Vibration vol 2016 ArticleID 3950140 7 pages 2016
[12] J Q Xiao D X Ding and X W Luo ldquoQuantitative analysisof damage evolution as recycled concrete approaches fatiguefailurerdquo Journal of Central South University (Science and Tech-nology) vol 42 no 1 pp 170ndash176 2011
[13] X R Ge Y Jiang Y D Lu and J X Ren ldquoTesting study onfatigue deformation law of rock under cyclic loadingrdquo ChineseJournal of Rock Mechanics and Engineering vol 22 no 10 pp1581ndash1585 2003
[14] Y Lv Study on the Flexural Fatigue Performance and CumulativeDamage of Glass Fiber Reinforced Concrete KunmingUniversityof Technology Kunming China 2012
[15] R-N Yang and D-M Wei ldquoResearch on the fatigue strain andfatigue modulus and damage of steel fiber reinforced concreteunder constant amplitude bending fatigue loadingrdquo GongchengLixueEngineering Mechanics vol 29 no 11 pp 99ndash122 2012
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal of
Volume 201
Submit your manuscripts athttpswwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Advances in Civil Engineering 5
Stress level S = 075
Test curve of maximum strainTest curve of maximum strainFitting curve of maximum strainFitting curve of residual strain
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
0255075
100125150175200225250275300325350375
Fatig
ue st
rain
(
)
Figure 7 Fatigue strain experimental curves and fitting curves ofgroup B
Stress level S = 07
Test curve of maximum strainTest curve of maximum strainFitting curve of maximum strainFitting curve of residual strain
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
0255075
100125150175200225250275300325350375
Fatig
ue st
rain
(
)
Figure 8 Fatigue strain experimental curves and fitting curves ofgroup C
of maximum fatigue strain and fatigue residual strain hassimilar three-phase variation and the intermediate stage closeto the linear change The curve converged rapidly when thecycle ratio exceeded approximately 09 The strain evolutioncurve is the level-S-shaped curve from the lower left cornerto the upper right corner Because the experiment measuredonly the initial maximum strain and has not measured initial
Table 1 Coefficient values of fatigue strain
Groups 119878 120576119899max 1198772120572 120573 119901A 08 470919 10182 24672 09891B 075 481478 10456 27550 09935C 07 540070 12096 34928 09751
Groups 119878 120576119899res 1198772120572 120573 119901A 08 438164 10422 27736 09889B 075 399757 10545 29135 09887C 07 325965 10171 35857 09753
Table 2 Coefficient values of fatigue modulus
Stress level 119864119899 11987721205721015840 1205731015840 1199011015840 11986308 23645 10081 302698 27075 09909075 15242 10762 192173 18772 0993507 11918 10070 190863 15847 09833
residual strain the evolution curve of the maximum strainstarts from the initial value but the evolution curve of fatigueresidual strain starts from zero According to (0 120576119891max minus 1205760max)and experimental data the range of 120572 of groups A B and Cis (0 250) (0 130) and (0 165) respectively Fitting results of120572 falls in these ranges in line with the boundary conditionsWhen the stress level is from 08 to 075 and from 075 to07 the 119901 value is increasing demonstrating that the smallerthe stress level is the faster the fatigue strain evolution curveconverges
43 Concrete Fatigue Modulus Evolution Analysis The fittingresults of the fatigue modulus under different stress levelsusing the model formula (8) are shown in Figures 9ndash11Coefficients of evolutionary model (8) are shown in Table 2The data in figure are the average of each group
From Figures 9ndash11 and Table 2 fatiguemodulus evolutionequation (8) can be a good fit to the experimental dataCorrelation coefficients were above 097 The evolution of thefatigue modulus also has a three-phase variation and theintermediate stage is also close to the linear changeThe curveconverged rapidly when the cycle ratio exceeded approxi-mately 09 Fatigue modulus evolution curve is the level-S-shaped curve from the upper left corner to the lower right cor-ner When the stress level is from 08 to 075 and from 075 to07 the 1199011015840 value is increasing demonstrating that the smallerthe stress level is the faster the fatigue modulus evolutioncurve converges
5 Conclusions
(1) By analyzing the shortcomings and limitations of existingfatigue strain evolution equations the level-S nonlinearevolution model of fatigue strain which contains the initialstrain was constructed Then the physical meaning of the
6 Advances in Civil Engineering
Test curveFitting curve
Stress level S = 080
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
02
03
04
05
06
07
08
09
10
Fatig
ue m
odul
us (G
Pa)
Figure 9 Fatigue modulus experimental curves and fitting curvesof group A
Stress level S = 075
Test curveFitting curve
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
04
05
06
07
08
09
10
Fatig
ue m
odul
us (G
Pa)
Figure 10 Fatigue modulus experimental curves and fitting curvesof group B
parameters and the range and the impact on the shape of thecurve were all discussed
(2) The results show that the level-S model covered alltypes of fatigue strain evolution and has strong adaptabilityand high accuracyThemodel could be tested for applicabilityto steel fiber reinforced concrete glass fiber reinforced con-crete and a variety of characteristics of concrete
(3) The fatigue modulus evolution model was establishedunder constant amplitude bending fatigue loading based on
Stress level S = 070
Test curveFitting curve
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
04
05
06
07
08
09
10
Fatig
ue m
odul
us (G
Pa)
Figure 11 Fatigue modulus experimental curves and fitting curvesof group C
fatigue strain evolution model and the hypothesis of fatiguemodulus inversely related to fatigue strain amplitude
(4) The evolution curve of fatigue strain and fatiguemodulus both have three stages of variation and the middlestage is nearly linearWhen the recycle ratio is 09 two curvesconverge quickly The difference is that the fatigue strainevolution curve is from the lower left corner to the upper rightcorner but the fatigue modulus evolution curve is from theupper left corner to the right lower corner
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This project was supported by the Project of OutstandingYouth Science Foundation of China (5142500405) Scientificand Technological Research Program of Chongqing Munic-ipal Education Commission (KJ1710237) Youth Project ofChongqing Three Gorges College (15QN14) Open Foun-dation of State Key Laboratory Breeding Base of Moun-tain Bridge and Tunnel Engineering (CQSLBF-Y16-6)Project of Chongqing Science and Technology Committee(cstc2015jcyjA00022) Scientific and Technological ResearchProgram of Chongqing Municipal Education Commission(KJ1601019) Disaster Prevention andDisasterMitigation andSafety of Engineering Structure in Three Gorges Reservoirand Open Foundation of Engineering Research Center ofBridge Structure and Material in the Mountainous AreaMinistry of Education (QLGCZX-JJ2015-4)
Advances in Civil Engineering 7
References
[1] C Y Li and Y P Song ldquoStudy of residual strain of concreteunder fatigue loadingrdquo Journal of Dalian University of Technol-ogy vol 41 no 3 pp 355ndash358 2001
[2] M K Lee and B I G Barr ldquoAn overview of the fatigue behav-iour of plain and fibre reinforced concreterdquo Cement amp ConcreteComposites vol 26 no 4 pp 299ndash305 2004
[3] A Alliche ldquoDamage model for fatigue loading of concreterdquoInternational Journal of Fatigue vol 26 no 9 pp 915ndash921 2004
[4] Y-C Kuang and J-P Ou ldquoPassive smart self-repairing concretebeams by using shapememory alloy wires and fibers containingadhesivesrdquo Journal of Central South University of Technologyvol 15 no 3 pp 411ndash417 2008
[5] J-K Kim and Y-Y Kim ldquoExperimental study of the fatiguebehavior of high strength concreterdquo Cement and ConcreteResearch vol 26 no 10 pp 1513ndash1523 1996
[6] XW LuoTest research of themechanics characteristic of recycledaggregate concrete under recurrent load Wuhan University ofTechnology School of Civil Engineering and ArchitectureWuhan China 2006
[7] Y B Chen Z N Lu and D Huang ldquoFatigue defect of layersteel fiber reinforced concreterdquo Journal of Wuhan University ofTechnology (Natural Science Edition) vol 18 no 1 pp 65ndash682003
[8] P B Cachim J A Figueiras and P A A Pereira ldquoFatiguebehavior of fiber-reinforced concrete in compressionrdquo Cementamp Concrete Composites vol 24 no 2 pp 211ndash217 2002
[9] J B Xie T C He and H R Cheng ldquoInvestigation flexuralfatigue behavior of steel fiber reinforced concrete for pavementsurface stratum under cyclic loadrdquo Journal of Lanzhou Univer-sity of Technology vol 30 no 2 pp 104ndash109 2004
[10] R M Wang G F Zhao and Y P Song ldquoResearch on compres-sion fatigue performance of concreterdquo China Civil EngineeringJournal vol 24 no 4 pp 38ndash47 1991
[11] F Liu and J Zhou ldquoFatigue Strain andDamageAnalysis of Con-crete in Reinforced Concrete Beams under Constant Ampli-tude Fatigue Loadingrdquo Shock and Vibration vol 2016 ArticleID 3950140 7 pages 2016
[12] J Q Xiao D X Ding and X W Luo ldquoQuantitative analysisof damage evolution as recycled concrete approaches fatiguefailurerdquo Journal of Central South University (Science and Tech-nology) vol 42 no 1 pp 170ndash176 2011
[13] X R Ge Y Jiang Y D Lu and J X Ren ldquoTesting study onfatigue deformation law of rock under cyclic loadingrdquo ChineseJournal of Rock Mechanics and Engineering vol 22 no 10 pp1581ndash1585 2003
[14] Y Lv Study on the Flexural Fatigue Performance and CumulativeDamage of Glass Fiber Reinforced Concrete KunmingUniversityof Technology Kunming China 2012
[15] R-N Yang and D-M Wei ldquoResearch on the fatigue strain andfatigue modulus and damage of steel fiber reinforced concreteunder constant amplitude bending fatigue loadingrdquo GongchengLixueEngineering Mechanics vol 29 no 11 pp 99ndash122 2012
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal of
Volume 201
Submit your manuscripts athttpswwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
6 Advances in Civil Engineering
Test curveFitting curve
Stress level S = 080
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
02
03
04
05
06
07
08
09
10
Fatig
ue m
odul
us (G
Pa)
Figure 9 Fatigue modulus experimental curves and fitting curvesof group A
Stress level S = 075
Test curveFitting curve
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
04
05
06
07
08
09
10
Fatig
ue m
odul
us (G
Pa)
Figure 10 Fatigue modulus experimental curves and fitting curvesof group B
parameters and the range and the impact on the shape of thecurve were all discussed
(2) The results show that the level-S model covered alltypes of fatigue strain evolution and has strong adaptabilityand high accuracyThemodel could be tested for applicabilityto steel fiber reinforced concrete glass fiber reinforced con-crete and a variety of characteristics of concrete
(3) The fatigue modulus evolution model was establishedunder constant amplitude bending fatigue loading based on
Stress level S = 070
Test curveFitting curve
01 02 03 04 05 06 07 08 09 1000Recycle ratio (nN)
04
05
06
07
08
09
10
Fatig
ue m
odul
us (G
Pa)
Figure 11 Fatigue modulus experimental curves and fitting curvesof group C
fatigue strain evolution model and the hypothesis of fatiguemodulus inversely related to fatigue strain amplitude
(4) The evolution curve of fatigue strain and fatiguemodulus both have three stages of variation and the middlestage is nearly linearWhen the recycle ratio is 09 two curvesconverge quickly The difference is that the fatigue strainevolution curve is from the lower left corner to the upper rightcorner but the fatigue modulus evolution curve is from theupper left corner to the right lower corner
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This project was supported by the Project of OutstandingYouth Science Foundation of China (5142500405) Scientificand Technological Research Program of Chongqing Munic-ipal Education Commission (KJ1710237) Youth Project ofChongqing Three Gorges College (15QN14) Open Foun-dation of State Key Laboratory Breeding Base of Moun-tain Bridge and Tunnel Engineering (CQSLBF-Y16-6)Project of Chongqing Science and Technology Committee(cstc2015jcyjA00022) Scientific and Technological ResearchProgram of Chongqing Municipal Education Commission(KJ1601019) Disaster Prevention andDisasterMitigation andSafety of Engineering Structure in Three Gorges Reservoirand Open Foundation of Engineering Research Center ofBridge Structure and Material in the Mountainous AreaMinistry of Education (QLGCZX-JJ2015-4)
Advances in Civil Engineering 7
References
[1] C Y Li and Y P Song ldquoStudy of residual strain of concreteunder fatigue loadingrdquo Journal of Dalian University of Technol-ogy vol 41 no 3 pp 355ndash358 2001
[2] M K Lee and B I G Barr ldquoAn overview of the fatigue behav-iour of plain and fibre reinforced concreterdquo Cement amp ConcreteComposites vol 26 no 4 pp 299ndash305 2004
[3] A Alliche ldquoDamage model for fatigue loading of concreterdquoInternational Journal of Fatigue vol 26 no 9 pp 915ndash921 2004
[4] Y-C Kuang and J-P Ou ldquoPassive smart self-repairing concretebeams by using shapememory alloy wires and fibers containingadhesivesrdquo Journal of Central South University of Technologyvol 15 no 3 pp 411ndash417 2008
[5] J-K Kim and Y-Y Kim ldquoExperimental study of the fatiguebehavior of high strength concreterdquo Cement and ConcreteResearch vol 26 no 10 pp 1513ndash1523 1996
[6] XW LuoTest research of themechanics characteristic of recycledaggregate concrete under recurrent load Wuhan University ofTechnology School of Civil Engineering and ArchitectureWuhan China 2006
[7] Y B Chen Z N Lu and D Huang ldquoFatigue defect of layersteel fiber reinforced concreterdquo Journal of Wuhan University ofTechnology (Natural Science Edition) vol 18 no 1 pp 65ndash682003
[8] P B Cachim J A Figueiras and P A A Pereira ldquoFatiguebehavior of fiber-reinforced concrete in compressionrdquo Cementamp Concrete Composites vol 24 no 2 pp 211ndash217 2002
[9] J B Xie T C He and H R Cheng ldquoInvestigation flexuralfatigue behavior of steel fiber reinforced concrete for pavementsurface stratum under cyclic loadrdquo Journal of Lanzhou Univer-sity of Technology vol 30 no 2 pp 104ndash109 2004
[10] R M Wang G F Zhao and Y P Song ldquoResearch on compres-sion fatigue performance of concreterdquo China Civil EngineeringJournal vol 24 no 4 pp 38ndash47 1991
[11] F Liu and J Zhou ldquoFatigue Strain andDamageAnalysis of Con-crete in Reinforced Concrete Beams under Constant Ampli-tude Fatigue Loadingrdquo Shock and Vibration vol 2016 ArticleID 3950140 7 pages 2016
[12] J Q Xiao D X Ding and X W Luo ldquoQuantitative analysisof damage evolution as recycled concrete approaches fatiguefailurerdquo Journal of Central South University (Science and Tech-nology) vol 42 no 1 pp 170ndash176 2011
[13] X R Ge Y Jiang Y D Lu and J X Ren ldquoTesting study onfatigue deformation law of rock under cyclic loadingrdquo ChineseJournal of Rock Mechanics and Engineering vol 22 no 10 pp1581ndash1585 2003
[14] Y Lv Study on the Flexural Fatigue Performance and CumulativeDamage of Glass Fiber Reinforced Concrete KunmingUniversityof Technology Kunming China 2012
[15] R-N Yang and D-M Wei ldquoResearch on the fatigue strain andfatigue modulus and damage of steel fiber reinforced concreteunder constant amplitude bending fatigue loadingrdquo GongchengLixueEngineering Mechanics vol 29 no 11 pp 99ndash122 2012
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal of
Volume 201
Submit your manuscripts athttpswwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Advances in Civil Engineering 7
References
[1] C Y Li and Y P Song ldquoStudy of residual strain of concreteunder fatigue loadingrdquo Journal of Dalian University of Technol-ogy vol 41 no 3 pp 355ndash358 2001
[2] M K Lee and B I G Barr ldquoAn overview of the fatigue behav-iour of plain and fibre reinforced concreterdquo Cement amp ConcreteComposites vol 26 no 4 pp 299ndash305 2004
[3] A Alliche ldquoDamage model for fatigue loading of concreterdquoInternational Journal of Fatigue vol 26 no 9 pp 915ndash921 2004
[4] Y-C Kuang and J-P Ou ldquoPassive smart self-repairing concretebeams by using shapememory alloy wires and fibers containingadhesivesrdquo Journal of Central South University of Technologyvol 15 no 3 pp 411ndash417 2008
[5] J-K Kim and Y-Y Kim ldquoExperimental study of the fatiguebehavior of high strength concreterdquo Cement and ConcreteResearch vol 26 no 10 pp 1513ndash1523 1996
[6] XW LuoTest research of themechanics characteristic of recycledaggregate concrete under recurrent load Wuhan University ofTechnology School of Civil Engineering and ArchitectureWuhan China 2006
[7] Y B Chen Z N Lu and D Huang ldquoFatigue defect of layersteel fiber reinforced concreterdquo Journal of Wuhan University ofTechnology (Natural Science Edition) vol 18 no 1 pp 65ndash682003
[8] P B Cachim J A Figueiras and P A A Pereira ldquoFatiguebehavior of fiber-reinforced concrete in compressionrdquo Cementamp Concrete Composites vol 24 no 2 pp 211ndash217 2002
[9] J B Xie T C He and H R Cheng ldquoInvestigation flexuralfatigue behavior of steel fiber reinforced concrete for pavementsurface stratum under cyclic loadrdquo Journal of Lanzhou Univer-sity of Technology vol 30 no 2 pp 104ndash109 2004
[10] R M Wang G F Zhao and Y P Song ldquoResearch on compres-sion fatigue performance of concreterdquo China Civil EngineeringJournal vol 24 no 4 pp 38ndash47 1991
[11] F Liu and J Zhou ldquoFatigue Strain andDamageAnalysis of Con-crete in Reinforced Concrete Beams under Constant Ampli-tude Fatigue Loadingrdquo Shock and Vibration vol 2016 ArticleID 3950140 7 pages 2016
[12] J Q Xiao D X Ding and X W Luo ldquoQuantitative analysisof damage evolution as recycled concrete approaches fatiguefailurerdquo Journal of Central South University (Science and Tech-nology) vol 42 no 1 pp 170ndash176 2011
[13] X R Ge Y Jiang Y D Lu and J X Ren ldquoTesting study onfatigue deformation law of rock under cyclic loadingrdquo ChineseJournal of Rock Mechanics and Engineering vol 22 no 10 pp1581ndash1585 2003
[14] Y Lv Study on the Flexural Fatigue Performance and CumulativeDamage of Glass Fiber Reinforced Concrete KunmingUniversityof Technology Kunming China 2012
[15] R-N Yang and D-M Wei ldquoResearch on the fatigue strain andfatigue modulus and damage of steel fiber reinforced concreteunder constant amplitude bending fatigue loadingrdquo GongchengLixueEngineering Mechanics vol 29 no 11 pp 99ndash122 2012
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal of
Volume 201
Submit your manuscripts athttpswwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal of
Volume 201
Submit your manuscripts athttpswwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
