ASIAN JOURNAL OF CIVIL ENGINEERING (BUILDING AND HOUSING) VOL. 5, NOS 3-4 (2004) PAGES

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1 ASIAN JOURNAL OF CIVIL ENGINEERING (BUILDING AND HOUSING) VOL. 5, NOS 3-4 (2004) PAGES PREDICTION OF MEAN AND DESIGN FATIGUE LIVES OF STEEL FIBROUS CONCRETE USING S-N RELATIONSHIPS S P Singh 1, Sanjay Goel 2, Roshan Lal 2, S K Kaushik 3 1 Civil Engineeing, National Institute of Technology, Jalandha, India 2 Civil Engineeing, Punjab Engineeing College, Chandigah, India 3 Civil Engineeing, Indian Institute of Technology Rookee, Rookee, India The pape pesents a study on the fatigue stength of steel fibe einfoced concete (SFRC) containing diffeent volume factions, aspect atios and types of steel fibes, fo vaious levels of the fatigue stess. The fatigue test data available in liteatue has been used fo analysis. The test data is used to geneate the S-N cuves and an Equation is poposed by egession analysis to pedict the flexual fatigue stength of SFRC. A pobabilistic appoach is used to pedict the fatigue eliability of SFRC. The fatigue-life distibutions of SFRC at a given stess level, is shown to appoximately follow the two-paamete Weibull distibution. The S-N elationships have been used to obtain the paametes of the Weibull distibution. Mean and Design fatigue lives have been computed fo diffeent stess levels fo SFRC with diffeent combinations of fibes, coesponding to diffeent pobabilities of failue. INTRODUCTION Consideable inteest has developed in the fatigue stength of concete membes in ecent yeas. Thee ae seveal easons fo this. Fistly, the use of high stength mateials equie that the concete membes pefom satisfactoily unde high stess levels. Hence, the study of the effects of epeated loads on bidge slabs and cane beams is a matte of concen. Secondly, diffeent concete systems such as pestessed concete aiload ties and continuously einfoced concete pavement slabs ae often used.the use of these systems demand a high pefomance poduct with an assued fatigue-life. Thidly, thee is a new ecognition of the effects of epeated loading on a membe, even if it does not cause a fatigue failue. Thee may be inclined cacks in the pestessed concete beams at lowe loads due to fatigue loading and the static load caying capacity of the component mateial may be alteed. In a conventionally einfoced stuctue/element subjected to bending moment, fatigue failue may occu eithe in tension steel o in the compession zone of the concete. Howeve, since the highway and aifield pavements ae usually uneinfoced, the concete is called upon to esist tension in bending. Majoity of the eseach epoted in liteatue on fatigue of plain as well as steel fibe einfoced concete (SFRC) has focussed attention on addess of the coesponding autho: spsingh@nitj.ac.in

2 176 S P Singh, Sanjay Goel, Roshan Lal and S K Kaushik flexual fatigue and to some extent on compession fatigue. A numbe of eseach investigations wee caied out to look into the fatigue behaviou of plain concete since Feet s pionee tests [1]. Many eseaches [2-4] adopted a elationship between stess level S, which is the atio of imum fatigue stess f to the modulus of uptue f and the numbe of loading cycles N which cause failue. The elationship established is known as the Whole Equation given below: S = f f = a + b log 10 (N) (1) whee a and b ae expeimental coefficients. Oh [5] obtained the values of coefficients a and b fo plain concete. It was futhe shown that the statistical distibution of fatiguelife of plain concete can appoximately be descibed by two-paamete Weibull distibution [5-6]. The paametes of the Weibull distibution i.e. shape paamete α and chaacteistic exteme life u wee obtained by diffeent methods. Anothe fom of the fatigue Equation used by eseaches [4,5,7,8] is a modification of the Whole Equation which incopoates a stess atio R, which is the atio of minimum fatigue stess f min to the imum fatigue stess f into the Whole Equation. The modified Equation takes the following fom: S = f = 1 - β (1 - R ) log10 (N) (2) f whee β is expeimental coefficient. This Equation can be used fo 0 < R < 1 but not fo stesses which altenate between compession and tension. Fo Equation (2) to be valid, the S-N cuves should not be based on measuements whee the amplitude o the lowe stess f min is kept constant, but on a constant stess atio R = f min /f. Aas-Jakobsen [9] obtained the value of β in Equation (2) Equal to fo compession fatigue of concete. Howeve, Tepfes et al [7] ecommended the value of β as Oh [5] tested the Equation (2) fo flexual fatigue of plain concete and obtained the value of β as A few expeimental investigations have been caied out to study the fatigue behaviou of steel fibe einfoced concete. Howeve, specimen sizes, loading conditions and fatigue failue citeia have vaied ove a wide ange. Most of these studies on steel fibe einfoced concete wee mainly confined to the detemination of its flexual fatigue enduance limit fo diffeent type/volume faction/aspect atio of fibes [10-14] although some studies focussed attention on studying othe aspects of fibe einfoced concete. Yin et al. [15] studied the fatigue behaviou of steel fibe einfoced concete unde uniaxial and biaxial compession and obseved that the S-N cuves can be appoximated by two staight lines connected by a cuved knee instead of a single staight line. Ramakishnan et al. [16-18] studied the flexual fatigue stength of fibe einfoced concete and poposed constitutive elations and models to pedict the fatigue stength of this mateial. A summay of mechanical models is given fo numeically simulating the fatigue behaviou of fibe einfoced concete.

3 PREDICTION OF MEAN AND DESIGN FATIGUE LIVES RESEARCH SIGNIFICANCE Johnston and Zemp [14] epoted the esults of fatigue tests on SFRC specimens containing diffeent types, volume factions and aspect atios of steel fibes. They studied the pefomance of SFRC by developing S-N cuves and the enduance limits wee obtained fo vaious combinations of steel fibes. As the fatigue life data of SFRC, even at a paticula stess level, show consideable vaiation due to andom oientation of the fibes, little attention has been paid to the pobabilistic analysis of fatigue life data of steel fibe einfoced concete. Hence the wok of Johnston and Zemp [14] povides good oppotunity to apply the concepts of pobabilistic analysis to thei fatigue test data and to examine the two-paamete Weibull distibution fo SFRC. The objectives of this pape is fist to detemine the fatigue stength of steel fibe einfoced concete subjected to flexual fatigue loading and, second, to implement the concepts of pobabilistic analysis to study the fatigue chaacteistics of SFRC. The two-paamete Weibull distibution has been examined to descibe the fatigue behaviou of SFRC. A method of obtaining the distibution paametes fom the S-N elationship is discussed. Mean and design fatigue lives ae obtained fom the S-N elationships fo steel fibe einfoced concete containing fibes of diffeent chaacteistics. FATIGUE STRENGTH AND S-N RELATIONSHIPS Johnston and Zemp [14] pesented the fatigue test data of steel fibe einfoced concete unde flexual loading. The fibe volume factions wee kept as 0.5%, 1.0% and 1.5%. Diffeent types of steel fibes, i.e. smooth unifom wie (SW), suface-defomed wie (SDW), melt extact (ME) and slit sheet (SS), with diffeent aspect atios anging fom 47 to 100 wee used in the investigation. The fatigue tests wee caied out at vaious stess levels anging fom 0.99 to The basic concete initially chosen compised 13mm gavel coase aggegate, coase sand (ai-died pio to batching), and 297 kg/m 3 of nomal cement with conventional wate-educing and ai-entaining admixtues. The complete fatigue life data of steel fibe einfoced concete as obtained by Johnston and Zemp [14] is pesented in Table 1. The S-N cuves epesent the elationship between the fatigue stess level S and numbe of load epetitions N which cause failue of the specimen. This elationship is given by the Equation (1). Figue 1 summaizes the test esults in the fom of S-N cuves obtained fom this study fo SFRC containing steel fibes of diffeent chaacteistics. Linea egession is caied out based on the least squaes method to detemine the values of coefficients a and b. The values of coefficients obtained fo fatigue life data of SFRC with diffeent combinations of steel fibes ae listed in Table 2. The mateial coefficient β in Equation (2) can not be obtained fom the test data since the testing has been caied out by keeping f min as constant and not stess atio R as aleady mentioned. Theefoe, Equation (1) can be used to pedict the flexual fatigue stength of SFRC by using the values of the mateial coefficients as detemined above fo SFRC with diffeent combinations of steel fibes.

4 178 S P Singh, Sanjay Goel, Roshan Lal and S K Kaushik Table 1. Flexual fatigue life data of steel fibe einfoced concete [14] 0.5 pecent SW(75) 1.0 pecent SW(75) 1.5 pecent SW(75) 1.0 pecent SW(50) 1.0 pecent SDW(47) 1.0 pecent ME(54) 1.0 pecent SS(71) S N S N S N S N S N S N S N S is the stess level as a pecentage of static flexual stength; N is the numbe of cycles to failue; SW is smooth wie; SDW is suface-defomed wie; ME is melt extact, and SS is slit sheet

5 PREDICTION OF MEAN AND DESIGN FATIGUE LIVES Figue 1. S-N Relationships fo steel fibous concete containing fibes of diffeent chaacteistics

6 180 S P Singh, Sanjay Goel, Roshan Lal and S K Kaushik Table 2. Coefficients a and b of Equation (1) fo steel fibe einfoced concete Volume Faction Fibe Chaacteistics Aspect Ratio Type Coefficient a Coefficient b 0.5% 75 Smooth Wie (SW) % 75 Smooth Wie (SW) % 75 Smooth Wie (SW) % 50 Smooth Wie (SW) % 47 Suface Defomed Wie (SDW) % 54 Melt Extact (ME) % 71 Slit Sheet (SS) FATIGUE-LIFE DISTRIBUTIONS OF SFRC A numbe of mathematical models have been employed fo the statistical desciption of fatigue data. One of the popula models has been the logaithmic-nomal distibution function [20]. Howeve, it was pointed out by Gumble [21] that the hazad function of this distibution deceases with inceasing life, which violates the physical phenomenon of fatigue failue of mateials. Thus because of physically valid assumptions and sound expeimental veification, the Weibull distibution is most commonly used fo the statistical desciption of fatigue data these days. The suvivoship function, L N (n), of two-paamete Weibull distibution may be witten as follows [5, 6, 22, 23]: Taking the logaithm twice of both sides of Equation (3) α n L N ( n) = exp (3) u 1 ln ln L N = α ln ( n) α ln( u) (4) Equation (4) epesents a linea elationship between ln [ln(1/l N )] and ln (n). This Equation can be used to veify whethe the statistical distibution of fatigue life follows the two paamete Weibull distibution. In ode to obtain a gaph fom Equation (4), the fatigue-life data at a given stess level must be fist aanged in ascending ode. The empiical suvivoship function can be calculated fom the following elation [5, 6, 19, 23]:

7 PREDICTION OF MEAN AND DESIGN FATIGUE LIVES i L N = 1 (5) k + 1 in which i = failue ode numbe and k = numbe of fatigue data o sample size at a given stess level. A gaph is plotted between ln [ln (1/L N )] and ln (N), and a best fit line can be dawn though the data points by method of least squaes. If a linea tend is obseved fo the fatigue-life data at a given stess level S, it can be assumed that the two-paamete Weibull distibution is a easonable assumption fo the statistical desciption of fatigue life data at that stess level. The paametes of the Weibull distibution α and u can be obtained fom egession coefficients. Figue 2 shows the plot of the fatigue life data of SFRC with smooth wie (SW) steel fibes of 0.5% volume faction and aspect atio of 75. The best fit line though the data points is dawn by egession analysis. The appoximate staight line plots with coelation coefficient values exceeding 0.90 show that the fatigue life data at the coesponding stess levels can be descibed by the two-paamete Weibull distibution. The values of the paametes obtained ae α = and u = 1023 fo S = 0.95; α = and u = 8185 fo S = 0.90 and α = and u = fo S = It has been obseved that the fatigue life data of SFRC fo othe combinations of fibes as given in Table 1 can also be modeled by the two-paamete Weibull distibution. The detailed esults ae epoted elsewhee [24]. The attention in this pape is dawn, howeve, to the next section. PARAMETERS FROM S-N RELATIONSHIPS As shown in the pevious section, the distibution paametes fo a pobability law can be obtained fo a given stess level and the distibution of fatigue life of SFRC (at a given stess level) can be descibed by the two-paamete Weibull distibution. Howeve, thee is anothe method of obtaining the distibution paametes i.e. fom the S-N elations. This method is based on an appoximate assumption of constant vaiance fo all the stess levels. Fo this, the following S-N elation may be assumed [5, 25]: N f m = C in which m and C ae empiical constants. Taking logaithm of both sides of the Equation (6) (6) () log 10 N = log10 C mlog10 (7) f

8 182 S P Singh, Sanjay Goel, Roshan Lal and S K Kaushik Figue 2. Regession analysis of fatigue life data fo SFRC, SW(75), V f =0.5%

9 PREDICTION OF MEAN AND DESIGN FATIGUE LIVES Y = a + bx (8) in which Y = log 10 (N); X = log 10 ; a = log 10 C; and b = -m. f If the fatigue life N, is assumed to follow the Weibull distibution (as has been shown in the pevious section), the paametes of the distibution i.e. α and u may be detemined fom the following expessions [5, 25]: and 2 2 π α = (9) 6 ( s) 2 m () ln u = + ln C (10) α f whee s = estimate of the standad deviation o standad eo of estimate of Y given X. The mean fatigue life fo a given stess level may be obtained fom the following elation [5]: m E[] N = µ N = C exp T 1+ f α α in which T( ) is the gamma function and E[N] is the mean fatigue life. The design fatigue life N D, should be selected such that thee is only a small pobability that a fatigue failue will occu. Once the distibution function is detemined, the design fatigue life may be selected coesponding to an acceptable eliability. The design eliability may be expessed as P[N < N D ] = 1 p f, whee p f is the pobability of failue. Noting that p f = P[N < N D ], the design fatigue life coesponding to a pemissible pobability p f is detemined fom the following elation. (11) 1 1 α N D = u ln (12) 1 pf The fatigue life data as descibed in the pevious sections have been analyzed using Equations.(6), (7) and (8) and the following elations ae obtained: N f = fo SFRC SW(75), V f =0.5% (13)

10 184 S P Singh, Sanjay Goel, Roshan Lal and S K Kaushik N f = fo SFRC SW(75), V f =1.0% (14) N f = fo SFRC SW(75), Vf =1.5% (15) N f N f N f = = N f = = fo SFRC SW(50), Vf =1.0% (16) fo SFRC SDW(47), V f =1.0% (17) fo SFRC ME(54), V f =1.0% (18) fo SFRC SS(71), V f =1.0% (19) The calculated values of the standad eo of estimate of Y given X fo the fatigue life of SFRC fo vaious combinations of fibes ae listed in Table 3. The shape paamete α and chaacteistic value u can be obtained fom Equations. 9 and 10, espectively, and the calculated values ae listed in Table 3. Equation 11 can be used to calculate the mean fatigue lives E[N] of SFRC, and the values thus calculated ae listed in Tables 4 10 fo vaious combinations of fibes. These tables also pesent design fatigue lives calculated using Equation 12, coesponding to vaious acceptable pobabilities of failue p f fo vaious stess levels. Smalle acceptable pobabilities of failue o highe eliabilities equie the design lives to be small. It is noted hee that the oveall distibution paametes obtained fom the S-N elationships may diffe fom those obtained in the pevious section which ae based on fatigue life data at a paticula stess level.

11 PREDICTION OF MEAN AND DESIGN FATIGUE LIVES Table 3. Values of weibull paametes obtained fom S-N elationships. Fibe Chaacteistics V f = 0.5%, Aspect Ratio = 75, Smooth Wie (SW) V f = 1.0%, Aspect Ratio = 75, Smooth Wie (SW) V f = 1.5%, Aspect Ratio = 75, Smooth Wie (SW) V f = 1.0%, Aspect Ratio = 50, Smooth Wie (SW) V f = 1.0%, Aspect Ratio = 47, Suface Defomed Wie (SDW) V f = 1.0%, Aspect Ratio = 54, Melt Extact (ME) V f = 1.0%, Aspect Ratio = 71, Slit Sheet (SS) Standad Eo of Estimate s Shape Paamete α Stess Level S Chaacteisti c Life u Table 4. Mean and design fatigue lives fo SFRC, SW (75), V f =0.5% (α=1.2125). S= f / f E[N] Design Fatigue Lives N D P f

12 186 S P Singh, Sanjay Goel, Roshan Lal and S K Kaushik Table 5. Mean and design fatigue lives fo SFRC, SW (75), V f =1.0% (α=1.1727). S= f / f E[N] Design Fatigue Lives N D P f Table 6. Mean and design fatigue lives fo SFRC, SW (75), V f =1.5% (α=1.2611). P f S= f / f E[N] Design Fatigue Lives N D Table 7. Mean and design fatigue lives fo SFRC, SW (50), V f =1.0% (α=1.3191). S= f / f E[N] P f Design Fatigue Lives N D

13 PREDICTION OF MEAN AND DESIGN FATIGUE LIVES Table 8. Mean and design fatigue lives fo SFRC, SDW (47), V f =1.0% (α=1.3397). S= f / f E[N] Design Fatigue Lives N D P f Table 9. Mean and design fatigue lives fo SFRC, ME (54), V f =1.0% (α=1.4019). S= f / f E[N] Design Fatigue Lives N D P f Table 10. Mean and design fatigue lives fo SFRC, SS (71), V f =1.0% (α=1.4763). S= f / f E[N] P f Design Fatigue Lives N D

14 188 S P Singh, Sanjay Goel, Roshan Lal and S K Kaushik CONCLUSIONS 1. The test data is used to geneate the S-N cuves and Equations ae poposed by egession analysis to pedict the flexual fatigue stength of SFRC. 2. A pobabilistic appoach is employed to pedict the fatigue eliability of SFRC. The fatigue-life distibutions of SFRC at a given stess level, is shown to appoximately follow the two-paamete Weibull distibution. 3. A method of obtaining the distibution paametes of the Weibull distibution fom the S-N elationships is pesented. 4. The paametes such as mean fatigue lives, chaacteistic value and design fatigue lives have been detemined fo vaious stess levels coesponding to diffeent pobabilities of failue fo SFRC with diffeent combinations of fibes. NOTATIONS f = Maximum fatigue stess f min = Minimum fatigue stess f = Static flexual stess S = Stess level = f /f R = Stess atio = f min /f L N = Suvivoship function o eliability function. N = Numbe of cycles to failue o fatigue life n = Specific value of N N D = Design fatigue life u = Chaacteistic life o scale paamete of Weibull distibution CV = Coefficient of vaiation of the data sample at a given stess level α = Shape paamete of Weibull distibution T( ) = Gamma function E[N] = Mean fatigue life p f = Pobability of failue REFERENCES 1. Feet, R., Etude Expeimentale du Ciment Ame, Gauthie-Villies, Chapte 3, Kesle, C.E., Effect of Speed of Testing on Flexual Stength of Plain Concete, HRB Poceedings, 32(1953) Ballinge, C.A., Cumulative Fatigue Damage Chaacteistics of Plain Concete, Highway Reseach Recod, No. 3, 70(1972) Hsu, T.T.C., Fatigue of Plain Concete, ACI Jounal, No. 4, July-August, 78(1981) Oh, B.H., Fatigue Analysis of Plain Concete in Flexue, Jounal of Stuctual

15 PREDICTION OF MEAN AND DESIGN FATIGUE LIVES Engineeing, ASCE, No. 2, Febuay, 112(1986) Oh, B.H., Fatigue-Life Distibutions of Concete fo Vaious Stess Levels, ACI Mateials Jounal, No. 2 Mach-Apil, 88(1991) Tepfes, R. and Kutti, T., Fatigue Stength of Plain, Odinay, and Light weight Concete, ACI Jounal, May, 76(1979) Tepfes, R., Tensile Fatigue Stength of Plain Concete, ACI Jounal, August, 76(1979) Aas-Jakobsen, K., Fatigue of Concete Beams and Columns, Bulletin No. 70-1, NTH Institute fo Betonkonstuksjone, Tondheim, Septembe Batson, G., Ball, C., Bailey, L., Lendes, E. and Hooks, J., Flexual Fatigue Stength of Steel Fibe Reinfoced Concete Beams, ACI Jounal, No. 11, Novembe, 69(1972) Ramakishnan, V., Obeling, G. and Tatnall, P., Flexual Fatigue Stength of Steel Fibe Reinfoced Concete, SP , ACI Special Publication, 1987, pp Tato, S.B., Pefomance of Steel Fibe Reinfoced Concete Using Lage Aggegates, Tanspotation Reseach Recod 1110, TRB Washington, 1987, pp Ramakishnan, V., Wu, G.Y.and Hosalli, G., Flexual Fatigue Stength, Enduance Limit and Impact Stength of Fibe Reinfoced Concetes, Tanspotation Reseach Recod 1226, TRB, Washington, 1989, pp Johnston, C.D.and Zemp, R.W., Flexual Fatigue Pefomance of Steel Fibe Reinfoced Concete-Influence of Fibe Content, Aspect Ratio and Type, ACI Mateials Jounal, No. 4, July-August, 88(1991) Yin, W. and Hsu, T.T.C., Fatigue Behaviou of Steel Fibe Reinfoced Concete in Uniaxial and Biaxial Compession, ACI Mateials Jounal, No. 1, Januay-Febuay 92(1995) Ramakishnan,V., Bjon J. Lokvik and Henning Selstad, Constitutive Relations fo Flexual Fatigue Behaviou of Fibe Reinfoced Concete, Poceedings of the VII Intenational Congess on Expeimental Mechanics, Connecticut, June 1992, pp Ramakishnan, V., C. Maye, and Naaman, A.E., Cyclic Behaviou, Fatigue Stength, Enduance Limit and Models fo Fatigue Behaviou of FRC, Chapte 4, High Pefomance Fibe Reinfoced Cement Composites 2, E & FN Spoon, New Yok, 1996, pp Ramakishnan, V. and Lokvik, B.J., Flexual Fatigue Stength of Fibe Reinfoced Concetes, High Pefomance Fibe Reinfoced Cement Composites, RILEM Poceedings, Editos: Reinhadt, H.W., and Naaman, A.E., Chapman and Hall, London, 1992, pp Kennedy, J.B. and Neville, A.M., Basic Statistical Methods fo Enginees and Scientists, A Dun-Donnelley Publishes, New Yok, ASTM, A Guide fo Fatigue Testing and the Statistical Analysis of Fatigue Data, ASTM Special Technical Publication, 91-A, Gumble, E. J., Paametes in the Distibution of Fatigue Life, Jounal of Engineeing Mechanics, ASCE, Octobe, 89(1963)45-63.

16 190 S P Singh, Sanjay Goel, Roshan Lal and S K Kaushik 22. Weibull, W., Fatigue Testing and Analysis of Results, Pegamon Pess, Oxfod, Wisching, P.H. and Yao, J.T.P., Statistical Methods in Stuctual Fatigue, Jounal of the Stuctual Division, Poceedings of the ASCE, No. ST6, June, 96(1970) Goel, S., Pobability of Fatigue Failue of Steel Fibe Reinfoced Concete, M.E. thesis, Depatment of Civil Engineeing, Punjab Engineeing College, Chandigah, Janauay 2003, pp Wisching, P.H. and Yao, J.T.P., Fatigue Reliability: Intoduction, Jounal of the Stuctual Division, Poceedings of the ASCE, No. ST1, Januay, 108(1982)3-23.