Fatigue Life Prediction Of fiber Reinforced Concrete Under .
International Journal of Fatigue 21 (1999) e life prediction of fiber reinforced concrete under flexuralloadJun Zhangaa,*, Henrik Stang b, Victor C. LiaAdvanced Civil Engineering Materials Research Laboratory, Department of Civil and Environmental Engineering, University of Michigan,Ann Arbor, MI 48109, USAbDepartment of Structural Engineering and Materials, Technical University of Denmark, DK2800, Lyngby, DenmarkReceived 8 February 1999; received in revised form 20 June 1999; accepted 10 July 1999AbstractThis paper presents a semi-analytical method to predict fatigue behavior in flexure of fiber reinforced concrete (FRC) based onthe equilibrium of force in the critical cracked section. The model relies on the cyclic bridging law, the so-called stress–crack widthrelationship under cyclic tensile load as the fundamental constitutive relationship in tension. The numerical results in terms offatigue crack length and crack mouth opening displacement as a function of load cycles are obtained for given maximum andminimum flexure load levels. Good correlation between experiments and the model predictions is found. Furthermore, the minimumload effect on the fatigue life of beams under bending load, which has been studied experimentally in the past, is simulated and amechanism-based explanation is provided in theory. This basic analysis leads to the conclusion that the fatigue performance inflexure of FRC materials is strongly influenced by the cyclic stress–crack width relationship within the fracture zone. The optimumfatigue behavior of FRC structures in bending can be achieved by optimising the bond properties of aggregate–matrix and fiber–matrix interfaces. 1999 Elsevier Science Ltd. All rights reserved.Keywords: Crack bridging; Fatigue crack growth; Fiber reinforced concrete; Flexural loading; Model1. IntroductionThe incorporation of steel or other fibers in concretehas been found to improve several of its properties,primarily cracking resistance, impact and wear resistanceand ductility. For this reason fiber reinforced concrete(FRC) is now being used in increasing amounts in structures such as airport pavements, highway overlays,bridge decks and machine foundations. However, mostof these structural elements are loaded in cyclic loading.For example, the concrete overlays for highway orbridge decks are expected to resist millions of cycles ofrepeated axle loads from passing traffic during their service life. Airport pavements are subjected to a smallernumber of repeated loadings during their designed liferanging from about several thousand to several hundredthousand cycles of repeated loading. Concrete structuressupporting dynamic machines are also subjected to hundreds of millions of load cycles involving complicated* Corresponding author.stress states. The fatigue performance of these structureshas to be considered by researchers and designers. First,the cyclic load may cause structural fatigue failure.Second, the effects of repeated loading on the characteristics of materials (static strength, stiffness, toughness,durability, etc.) may be significant under service loading,even if the load does not cause a fatigue failure.As it is a relatively new material, the history of investigation on the fatigue of FRC is not long. Experimentalevaluations of this behavior have been carried out inrecent years [1–6]. Fatigue life prediction and the designof FRC structures can so far only be performed throughan empirical approach. This approach requires time-consuming test data collection and processing for a broadrange of design cases which, in principle, is not applicable to other design cases. Therefore, a mechanismbased fatigue model that is capable of both predictingthe fatigue life for a given FRC structure and designingan FRC material for a given fatigue life has to be set upfor the above reasons [7]. In order to do so, the mechanism of fatigue crack propagation in FRC material hasto be understood first.0142-1123/99/ - see front matter 1999 Elsevier Science Ltd. All rights reserved.PII: S 0 1 4 2 - 1 1 2 3 ( 9 9 ) 0 0 0 9 3 - 6
1034J. Zhang et al. / International Journal of Fatigue 21 (1999) 1033–1049Normally, it can be said that fatigue is a process ofprogressive, permanent internal structural changesoccurring in a material subjected to repetitive stress. Theprogressive fatigue damage on material constituents isresponsible for fatigue life of a material. For FRC, thematerial phases can broadly be classified as matrix(cement paste and aggregates), fibers, as well as theinterfaces of fiber–matrix and aggregate–hydratedcement paste. The fatigue loading causes these physicalphases to undergo microscopic changes, such as openingand growth of bond cracks, which exist at the interfacebetween coarse aggregate and hydrated cement pasteeven prior to the application of load [8], reversed movement of fiber along the interface, fiber surface abrasionand damage of interface in repeated sliding processes.These microscopical changes, in turn, cause some detrimental changes in macroscopic material properties.Typically, the aggregate bridging force as well as thefiber bridging force decreases with the number of cycles,due to interfacial damage [9] or fiber breakage due tosurface abrasion [10]. So it can be said that the damageon interfaces of fiber–matrix and aggregate–matrix,which are generally the weakest phase in concrete andFRCs, as well as on soft polymer fibers is responsiblefor fatigue crack initiation and growth in concrete andFRCs. On the other hand, the fatigue life of concrete andFRC structures is controlled by fatigue crack growthbehavior.The fatigue crack growth process in concrete or FRCmaterials can be broadly divided into two stages: thecrack initiation period and the development period. Nowconsidering a simply supported rectangular beam loadedin bending fatigue load with a constant amplitudebetween the maximum and minimum moment, Mmax andMmin. When Mmax Mfc, where Mfc is the first crackmoment, the fatigue life of the beam can be given by:Nt Nci Ncg(1)When MmaxⱖMfc, the fatigue life is:Nt Ncg(2)where Nt is the total fatigue life, Nci and Ncg are thefatigue life component for the crack initiation andgrowth, respectively, see Fig. 1. The first term, Nci, isdependent on the microcracking in material, which ishighly influenced by the microstructure of concretematrix, such as the water/cement ratio, aggregate properties as well as pore structure, size distribution and content. The second term, Ncg, is strongly dependent on thebridging performance within the fracture zone underfatigue loading.This paper focuses on the fatigue life prediction onNcg, i.e. the case of maximum load, Mmax, is larger thanthe first crack load, Mfc. Based on the above discussions,some basic assumptions for fatigue modelling on Ncg canbe stated:Fig. 1. Fatigue life components in bending.1. After a dominant fatigue crack is created, the bridgingbehavior within the fracture zone is governing the rateof fatigue crack advancement.2. The stress at the crack tip remains constant and isequal to the material tensile strength.3. Material properties outside the fracture zone areunchanged during fatigue loading.It is further assumed that concrete and FRC materialsessentially show a linear response in tension up to peakload. After peak one discrete crack is formed, and thediscrete crack formation is described by the crack bridging law (or stress–crack width relationship) under bothmonotonic and cyclic loading. Thus, the followingmaterial parameters are fundamental in the constitutiverelationships of concrete and FRC in fatigue tension: theYoung’s modulus E, the tensile strength st and the cyclicstress–crack width (s–w,N) relationship. In compressionthe behavior of concrete and FRC materials is assumedto be linear elastic and the Young’s modulus in compression is the same as in tension.With the above assumptions, a semi-analytical methodfor predicting fatigue behavior of unreinforced concreteand FRC beams under bending load is developed in thepresent work. In this model, the cyclic bridging law (orcyclic stress–crack width relationship) is incorporated inintegration form, which can easily be replaced by theother bridging models for different kinds of FRCmaterials with different fiber types, volume concentrations and matrix properties. In this paper, a quart-linear monotonic stress–crack width model, based on theuniaxial tensile test results, applied to plain concrete andtwo types of concrete reinforced with straight and
J. Zhang et al. / International Journal of Fatigue 21 (1999) 1033–10491035hooked steel fiber, respectively, is adopted. The complete theoretical curves, in terms of fatigue crack lengthor crack mouth opening displacement (CMOD) withnumber of cycles diagrams, as well as the classical S–Ncurves are obtained and compared with the experimentalresults. The results are discussed and conclusions aredraw at end of the paper.2. Bridging model of concrete and FRCS underfatigue tensionIn order to determine the cyclic stress–crack widthrelationship experimentally, deformation-controlledfatigue tensile tests were conducted on notched specimens with a thickness of 50 mm, width 60 mm andheight 55 mm. To eliminate the pre-stressing inevitablyintroduced in the specimens when using conventionalgrips, for improved alignment, and for maximum stiffness, a special specimen fixture developed by Stang andAarre [11] was used. The fixture consists of a permanentpart and an interchangeable steel block, which is fixedto the permanent part through four bolts. The specimenis glued to the blocks. The glued surfaces of the interchangeable steel blocks and the specimen are sandblasted before gluing to enhance the bond between steeland specimen. By having a large number of such steelblocks, multiple specimens can be tested continuously,as it is not necessary to clean the steel blocks after eachtest. As soft connections between the interchangeablesteel blocks and the machine are eliminated, the set-uptakes full advantage of the stiffness of the machineframe. It should thus be ensured that the rotational stiffness of the test set-up is large compared to the rotationalstiffness of the concrete specimen. A fast curing polymerwhich attains 90% of its maximum strength in about 4min was used. The deformation was measured using twostandard Instron extensometers (type 2620-602) with12.5 mm gauge length mounted across each of the two9 mm deep and 3 mm wide notches. The tests were performed in a 250 kN load capacity 8500 Instron dynamictesting machine equipped for closed-loop testing. Theexperimental set-up is shown in Fig. 2.The uniaxial fatigue tensile test was conducted underdisplacement control with constant amplitude betweenmaximum and minimum pre-crack widths (wmax andwmin). The minimum crack width value was obtained bya single loading–unloading tensile test at which thebridging load is equal to zero on the unloading branch.The fatigue test commenced with a ramp to the minimumcrack value at a rate of 0.01 mm/s followed by a sinewaveform fatigue loading in deformation control. Inorder to control the accuracy of the maximum crackwidth value, different load frequencies of 0.25 Hz in thefirst two cycles and 3.5 Hz for the rest of cycles wereadopted. This fatigue load procedure is shown in Fig. 3.Fig. 2. Schematic view of the test set-up for fatigue tension.Fig. 3.Deformation-time diagram in fatigue test.Technical details on the testing procedure can be foundelsewhere [12].In the present investigation, two types of commercially available steel fibers, smooth and hooked-end,with circular cross-section, 0.4 and 0.5 mm in diameter,25 and 30 mm in length, respectively, were used separ-
1036J. Zhang et al. / International Journal of Fatigue 21 (1999) 1033–1049Table 1Mix proportions of steel fiber concreteAmount (kg/m3)ComponentCementSand (maximum particle size 4 mm)Gravel (maximum particle size 8 mm)Superplasticizer (66% water content)WaterSmooth or hooked steel fibersFiber volume content (%) Vf5008108103.25237.578.41ately in the same matrix. A rapid hardening cement,natural sand and stone with maximum particle size of 4and 8 mm, respectively, were used. The concrete mix isshown in Table 1. All pre-notched specimens were cutout of beams of 50 mm width, 100 mm depth and 350mm length. Details of specimen preparation can befound in Zhang et al. [12].Experimental results of uniaxial fatigue tension testson pre-cracked plain concrete and FRCs show that thebridging stress decreases with the number of load cyclesfor the same crack openings. Typical test results on bothsmooth steel fiber concrete (SSFRC) and hooked steelfiber concrete (HSFRC), are shown in Fig. 4, where themaximum bridging stress corresponding to themaximum crack width is normalised to the stress levelat first cycle. The experimental investigation and theoretical analysis show that the cyclic bridging law of FRCsis quite complicated and can be influenced by manyparameters, including material constituents as well asloading conditions, such as maximum and minimumcrack width levels [12,13]. The material parameter-basedmicromechanical model of cyclic bridging law, however,may be too complicated to be used as a fundamentalconstitutive relationship directly in structural numericalanalyses. A more simple and effective way to carry outthe numerical simulations is to use a mathematical fitthat is based on the experimental results and theoreticalanalysis as the input of cyclic bridging law. This process,in principle, will not affect the applications of the theoretical model in fatigue optimisation and the design ofnew types of FRC materials based on the micromechanical parameters of the material; however, the predictionsare limited to the range in which the fitting is done. Inthe present work, a mathematical fit on the cyclic crackbridging law is applied in the fatigue performance prediction.From a large number of experimental data on uniaxialfatigue tension tests on pre-cracked specimens [12], thecyclic crack bridging law at the maximum crack widthlevel, wmax, can be fitted by a multi-linear model as afunction of the logarithm of the number of cycles. Themulti-linear model is given by:sN 1 f(wmax,wmin)log(N)s1 1 jg(wmax,wmin0)log(N)(3)where sN, s1 are the stresses at the maximum crackwidth, wmax, after N cycles and the first cycle, respectively. The s1 is simulated by a quart-linear model basedon the directly measured stress–crack width (s–w) data[14], that is:s1 a b w (i 1 4)st i i maxFig. 4. Typical test results of bridging stress vs. number of cycles.(4)
J. Zhang et al. / International Journal of Fatigue 21 (1999) 1033–1049Here st is tensile strength of material, and the parametersai and bi are given in Table 2. The function f(wmax,wmin)is a function of maximum and minimum crack widthvalues, wmax and wmin, which describes the effect of themaximum and minimum crack opening on the rate ofbridging degradation. This function can further berewritten as a product of a parameter j and a functionof only the maximum crack width, wmax. The parameterj(j苸[0,1]) reflects the influence of minimum crackwidth, wmin, on the bridging degradation. j is defined as:j wmax wminwmax wmin0(5)where wmin0 is the minimum crack width as the bridgingforce equals to zero. From the definition of j, it can beseen that as wmin goes up, j will go down, that meansthe rate of bridging degradation will also go down. Theseare consistent with the theoretical and experimentalresults [14]. The minimum degradation will occur aswmin is equal to wmax, (j 0), and the maximum degradation will happen at wmin equal to wmin0, (j 1).The detailed expressions of Eq. (3) for different kindsof concrete are given in Appendix A. Figures 5 to 7show the fit-based cyclic bridging law for different kindsof concrete, including plain concrete, steel fiber concrete,SSFRC and HSFRC. For the FRCs, the experimentalresults in the case of j equal to one are presentedtogether with the model predictions. The figures showthe bridging degradation at different maximum crackwidth after undergoing a certain number of cycles. Asthe maximum and minimum crack widths vary with thenumber of cycles, such as that for a beam under fatiguebending, the parameters wmax and wmin used in Eq. (3)will be replaced by the representative values, i.e. theaverage of all the loading history.3. Prediction of fatigue crack growth in concreteand FRC beam under bending loadIn recent years, a number of fictitious crack [15] basedanalytical models for predicting the structural behavior1037of concrete and FRC beams under bending load havebeen developed. Ulfkjær et al. [16] developed an analytical model of a plain concrete beam in bending based onplastic hinge analysis, which assumes the developmentof a fictitious crack in an elastic layer with a thicknessproportional to the beam depth. A linear tension softening relationship is assumed. Pedersen [17] developed asimilar model for plain as well as FRC structures, beamsand pipes, in which a more accurate softening law, apower function proposed by Stang and Aarre [11], isadopted. Maalej and Li [18] developed an analyticalmodel of a FRC beam in bending based on the equilibrium of force in the critical cracked section. Here theauthors adopt the analytical softening relationshipspresented by Maalej et al. [19] and Li [20]. Recent theoretical studies show that the bridging law for smallercrack widths, typically less than 0.1 mm, can stronglyaffect the structural behavior of beams in bending undermonotonic and cyclic loads [21]. In this paper, an analytical bending model based on the equilibrium of forcein the critical cracked section with a multi-linear softening relationship [14] for fatigue analysis will bedeveloped.Consider a short segment of a simple supported rectangular beam with width B, depth h, and span L that issubjected to an external bending moment M. Thebehavior of the beam is assumed to be elastic until themaximum principle tensile stress reaches the tensilestrength of the material. After that it is assumed that asingle crack is formed with a maximum tensile strengthat the crack tip. The moment corresponding to theinitiation of the fictitious crack is the so-called first crackmoment, Mfc, or first crack load, Pfc, when the momentis transformed into load. Thus, the failure process of thebeam can be divided into two stages: (1) a linear elasticstage; and (2) a fictitious crack developing stage. Theassumed stress distribution in the second stage is shownin Fig. 8.In the first stage, according to classical elastic theory,we have:Mfc Bh2s6 t(6)Table 2Material parameters of three types of concreteMaterial parametersE (GPa)st (MPa)sc (MPa)a1, b1 (1/mm)a2, b2 (1/mm)a3, b3 (1/mm)a4, b4 (1/mm)SSFRCw (mm)HSFRCw (mm)PCw (mm)355.4255.21, 9.960.685,0.5260.883, 1.450.374, 2325.3055.01, 8.730.632,0.4720.8, 0.4630.532, 0.106–––0–0.040.04–0.180.18 0.750.75–2305.2053.221, 33.480.569, 8.120.321, 2.490.187, 0.84–––0–0.0170.017–0.0440.044 0.0810.081–0.2
1038J. Zhang et al. / International Journal of Fatigue 21 (1999) 1033–1049Fig. 5.Model predictions on crack bridging degradation of SSFRC, showing the effect of j.where st is the tensile strength of materials.In the second stage, the crack length ah, a苸[0,1],CMOD, d and external moment M can be related throughthe analysis below. First we assume that the crack hasa linear crack opening profile, then:冉 冊w d 1 xah(7)where w is crack width at location x, see Fig. 8. Next,from the equilibrium conditions, we have:冕ahsI(x)dx sII(x)dx 0ah0(8)ah0冕冕h冕haI(x)(h x)Bdx sII(x)(h x)Bdx M(9)ahwhere M PL/4 for the three-point bending case (P isexternal load) and sI(x), sII(x) are the normal stress function in the cracked and uncracked parts, respectively.sI(x) can be related with ah and d through the stress–crack width relationship together with Eq. (7) as:冉 冉 冊冊xsI(x)
for fatigue crack initiation and growth in concrete and FRCs. On the other hand, the fatigue life of concrete and FRC structures is controlled by fatigue crack growth behavior. The fatigue crack growth process in concrete or FRC materials can be broadly divided into two stages: th
baseline BER (without NL) the same for both PAM2 & PAM4. BASELINE: PAM2 VS PAM4 Start with the same BER, compare the effect of NL . BER results: typical high speed link Bandwidth (Nyquist) UI BER without NL BER with NL PAM2 F N 1/(2* F N) 1e-25 1e-23 PAM4 F N / 2 2/(2* F N) 1e-25 1e-20. Link Model: Multiple NL blocks NL1 AWGN d
Application of ASME Fatigue Code: 3-F.1 ASME Smooth Bar Fatigue Curves Alternative Effective Stress vs Fatigue Cycles (S-N Curve) With your weld quality level assessed and an accurate FEA stress number, one can use the ASME fatigue curve to calculate the number of Fatigue Cycles. In our prior example, the fatigue stress could be 25.5 ksi or as .
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iii) The BER expression (from the figure) for BER over BPSK is given by Q(r jhj2 P 2) The simplified expression is given by 1 2 (1 s P 2 2 P 2) and the approximate BER expression is given by The simplified expression is given by 1 2P 2 b) Tasks i) BER Rayleigh A) Generate a random binary sequence of 10000 values. Lets call it 'x .
starting point of C1 was unable to achieve a BER uplift of 100 kWh/m. 2 /yr r based on combinations tested. It was also shown to be possible to achieve a BER uplift of 100 kWh/m. 2 /yr from combinations of shallow measures across Ireland housing stock and pre-BER conditions. A BER uplift of 100 kWh/m. 2
Fatigue failure of Honeycomb Structures Fatigue is generally understood as the gradual One of the common causes of Honeycomb Structures failure is due to fatigue. Repeated cycling of the load causes fatigue. It i
contribute to fatigue risk. A pre-implementation study shall consider the following: implementing a survey of personnel on fatigue and fatigue-management strategies analyzing exposure to fatigue risk on work schedules (e.g., bio mathematical fatigue modeling)
awards will be separately funded from outside the 0.3 allocation per eligible Trust consultant using the same criteria as set out in 2.9. 4. Eligible Consultants under Investigation 4.1 If a consultant who is the subject of a formal investigation, including a professional advisory panel, chooses to submit an application for CEAs, his/her application will be considered in the usual way by the .
