The effect of repeated loading on concrete structures is an increasinglyinteresting subject for a correct lifetime-oriented design. Fatigue ofconcrete is a continuous and progressive microcracking mechanismleading to increasing permanent strains and decreasing stiffness. Thispaper proposes a time-dependent (that is, cycle-dependent) modelfor concrete that accounts for the changes of the mechanical propertiesduring the fatigue life. The model is then implemented into a sectionalalgorithm to reproduce the evolution of the stress-strain state ofreinforced concrete members. Theoretical results are comparedwith existing experimental data of reinforced concrete beams. Thetype of fatigue failure is investigated depending on the sectionalcapacity for redistribution of stresses.
INTRODUCTIONThe design of durable concrete structures requires the
consideration of all the possible deterioration mechanisms,including both time- and cycle-dependent effects. Repeatedloading is one of these effects and might be a determinant forgood structural performance. Therefore, fatigue of concreteis an increasingly interesting subject due to the fact that it isa continuous and progressive degradation process. Bridgedeck slabs,1 approach slabs,2 offshore structures,3 or railwayprecast elements4 are structural concrete components typicallysubjected to highly-repeated loads where fatigue problemshave been reported. A rational mechanical approach is anessential tool to evaluate the actual behavior of suchmembers under fatigue conditions.
During its fatigue life, concrete modifies its mechanicalproperties. Under uniaxial compression, strains are permanentlyincreasing while the stiffness is decreasing.5,6 In any case, mostexperimental results available in the literature7,8 only focus onthe number of cycles to failure within the well-knownconcept of S-N curves.
The influence of the fatigue process on the response ofreinforced concrete is not completely understood. Evenunder high initial compressive stresses where fatigue ofconcrete could be expected, reinforced members failsystematically due to the brittle fatigue fracture of the steelreinforcing bars.9,10 This fact is explained by the largeconcrete capacity for redistribution of stresses. Few studieshave been conducted dealing with the existence of thisredistribution process.11-13
Theoretical models to study the fatigue effect of concretefall into two main groups. On the one hand, the mostemployed models are based on S-N curves and the staticstress state.14-16 They only provide the resistant number ofcycles without considering the redistribution process or thestrain evolution. On the other hand, damage models17,18 doinclude the fatigue deterioration of concrete. The hardapplicability to structural problems, the large computational
effort, or the lack of experimental validation, however, makethem mostly unsuitable for practical applications.
A new approach to consider the fatigue of concrete bymeans of a fatigue uniaxial material model is proposed in thispaper. This approach represents a step forward with regard to aprevious proposal12 because it is embedded into a rationaltime-dependent strategy that is more useful for generalapplications. The fatigue influence on the structuralresponse is evaluated by implementing a sectional algorithm,which is precise, understandable, and compatible with otherverifications carried out in structural concrete elements. Thenew approach makes it possible to study the evolution ofstresses and deformations during the fatigue lifetime, as well asthe final type of failure depending on the redistribution capacity.
RESEARCH SIGNIFICANCEThis paper presents a new approach to reproduce the
response of reinforced concrete elements under cyclicactions. The progressive degradation of concrete in compressionis included by a time-dependent model. The modelimplementation at the section level enables overcomingdifficulties presented by current design codes, which do notconsider the degradation mechanism of concrete. Such amethod is, for example, useful to estimate the fatigue-carrying capacity of existing structures that will be subjectedto higher loads in the future. Among the conclusions, it isshown that the redistribution capacity is the main governingfactor of the final failure mode.
FATIGUE MODEL FOR CONCRETE IN COMPRESSION
BackgroundA considerable number of experimental investigations
have studied the behavior of concrete under fatigue incompression. Most of the investigations were aimed only toobtain the maximum number of load cycles to failure underconstant stress limits. The results show the strong dependenceof the fatigue life on the upper stress level.5 A large scatter ofthe final results, however, is characteristic of thesestudies, especially for stress levels below 0.65fc, whichincludes the usual range of concrete structures. Differentformulations have been proposed to consider the role played byadditional parameters, for example, minimum stress level σmin,stress ratio R = σmin /σmax , and load frequency f.7,8,19-21 The mostextended approach in the literature is that suggested by Hsu7
Title no. 106-S62
Sectional Analysis of Concrete Structures underFatigue Loadingby Carlos Zanuy, Luis Albajar, and Pablo de la Fuente
ACI Structural Journal/September-October 2009668
(1)
(2)
When investigating the structural response of concreteelements, it is useful to focus on the evolution of the mechanicalproperties during the fatigue life. Concrete develops a continuousmicrocracking process that results in the macroscopic behaviorshown in Fig. 1. Bennett and Raju22 reported that micro-cracking takes place due to the tensile stresses transverselydeveloped to the main compression field. The heterogeneousnature of concrete (presence of air voids and stiff particles inthe cement paste) initiates the formation of microcracks,which develop and coalesce under repeated loading. A fracture-mechanics-based approach to explain the fatigue failuremechanism has been given by Mu et al.23 At the macroscopiclevel (Fig. 1), the degradation is reflected by the strain increaseand the stiffness reduction. Furthermore, Mehmel andKern24 showed that the reloading stress-strain curve changesits curvature toward the stress axis, whereas the unloadingcurve remains nearly straight and becomes less stiff.
The guidance provided by Fig. 1 must be taken intoaccount when modeling the concrete response. By representingthe strain and stiffness evolution over the number of cycles(Fig. 2), typical three-stage curves are obtained.5 In the firststage, the degradation rate is high but decreasing, the secondand longest stage is characterized by constant rate, and,finally, the degradation grows very quickly until failure isreached in the third stage.
Some experimental investigations have included testsunder eccentric repeated loading.25,26 The results show anincrease of the resistant number of load cycles—evidencethat a process of stress redistribution has taken place. Such aprocess causes the stress transfer from the most initiallydamaged compression zone to less fatigued areas whilepermanent and total strains are growing. The large capacityfor redistribution of stresses leads to the typically reportedbrittle fracture of the reinforcement,1 which may be evenaccelerated due to stress increase produced by the reduction ofthe sectional lever arm. Moreover, a continuous change inthe stresses and strains takes place in real concrete structuresdue to the redistribution process. Very few authors haveaccounted for this fact, which requires a path-dependentmodeling. For instance, Maekawa et al.27,28 have recently
Nflog 10.0662 1 0.556R–( )------------------------------------------------ 1 0.0294+ f Smax–log( );=
Nf 1000≥
proposed a fatigue model embedded into a finite element codewith special focus on shear fatigue, obtaining promisingresults but still requiring considerable computational effort.
Unfortunately, most code provisions do not consider theconcrete capacity for redistribution because they are notbased on the time-dependent material behavior. A realisticdescription should allow reproduction of the materialchanges and the stress-strain evolution. The model presentedin the next section accounts for these effects, keeping inmind the applicability to practical problems.
Time-dependent material modelThe model aims at the uniaxial behavior of concrete under
cyclic loading. As it has been shown, the fatigue response ofconcrete in compression is highly dependent on the stresslevel; a reliable stress-strain relationship for the monotonicbehavior of concrete is necessary. The static relationshipprovides the initial stress-strain state of the fatigue process(N = 1) and is also employed as a failure envelope for thecyclic process, as detailed later. The generally acceptednonlinear σ-ε curve for normal-strength concrete in
Carlos Zanuy is a Research Assistant in the Department of Continuum Mechanicsand Structures at the Universidad Politécnica de Madrid (UPM), Madrid, Spain. Hereceived his civil engineering degree from the UPM in 2004. His research interestsinclude the serviceability and ultimate behavior of concrete structures.
Luis Albajar is a Professor in the Department of Continuum Mechanics and Structures atthe UPM. He received his diploma and PhD in civil engineering from the UPM in 1969and 1982, respectively. His research interests include the theoretical modeling andexperimental response of concrete structures, the use of high-performance concrete,and the development of railway sleepers and slab track.
Pablo de la Fuente is a Professor and Head of the Department of ContinuumMechanics and Structures at the UPM. He received his diploma and PhD in civilengineering from the UPM in 1977 and 1984, respectively. His research interestsinclude the numerical modeling of soil-structure interaction and the dynamicresponse of structures.
Fig. 1—Evolution of stress-strain curve with number ofcycles, adapted from Reference 22.
Fig. 2—Schematic variation of concrete variables with cycleratio: (a) longitudinal strains; and (b) modulus of deformation.
Fig. 3—Static stress-strain relationship for concrete incompression.
ACI Structural Journal/September-October 2009 669
compression of the Model Code MC9014 is adopted (Fig. 3).The expression is as follows
(3)
(4)
(5)
where fc is the compressive strength, Ec represents the initialtangent modulus of deformation, εc indicates the strain thatcorresponds to the peak stress, and εlim is the strain at 0.5fcin the softening part of the curve.
To consider the fatigue effect of concrete, a time-dependentmodel based on two variables is proposed in this paper. Thesevariables are the maximum concrete strain εmax and themodulus of deformation E. The two selected variables areable to define the material state at a determined instant of thefatigue life (N/Nf). Figure 4 shows that the concrete behaviorwithin a cycle may be reproduced by assuming a linearunloading path when the two model variables are known. Forinstance, the minimum strain at N/Nf is easily obtained as follows
(6)
The model assumes that the reproduction of the fatigueprocess cycle by cycle is numerically inefficient, so analyticalexpressions for both the maximum strain and the modulus ofdeformation with respect to the number of cycles areproposed. Experimental results available in the literaturehave been employed to fit the parameters that define theevolution laws (refer to Table 1). The least-squares methodhas been used as fitting tool, providing high enough coefficientsof determination (R2 > 0.70). This is acceptable in a field wherethe scatter of results is considerable. The equations areexpressed as a function of the number of cycles and theloading conditions
The use of a explicit equation for the modulus of deformationin the form of residual stiffness instead of a more conventionaldamage parameter (E = Ec /(1 – D)) is preferred. This treatmentis more straightforward for practical applications and betteradapted to experimental data. Particular expressions forEq. (7) and (8) are given in the next section. The failurestrain is calculated by means of the envelope concept.29 Itconsiders that the static law σ-ε also defines the failureboundary of the cyclic problem. This is a simple tool andprovides reliable results.
Equations (7) and (8) show that the resistant number ofcycles (Nf) is necessary to identify the relative lifetimeinstant. To obtain Nf , the S-N curves proposed by Hsu(Eq. (1) and (2)) are selected due to the fact that theyinclude the influence of several parameters and have beenfitted to a large amount of experimental data.
As previously introduced, fatigue causes a redistributionof stresses in concrete components. This means that aprocess of variable stress limits is developed in each materialfiber. Therefore, the direct application of Eq. (7) and (8) isnot possible and the process is approximated by a number ofshorter processes for which constant stress limits can beassumed. To connect them, an accumulation criterion isneeded and the concept of the equivalent number of cycles(Neq) is now introduced as a new accumulation rule. Theparameter Neq is the number of load cycles that is necessaryto be applied in a fatigue process with constant limits (σmax , σmin)until a total strain of εmax is reached. It may be easily computedfrom the complete analytical expression (Eq. (7)). Thisconcept is similar to that called the equivalent time, which wassuccessfully employed to creep and relaxation problems.30,31
εmax f NNf
-----σmax
fc
-----------σmin
fc
----------, ,⎝ ⎠⎛ ⎞=
E g NNf
-----σmax
fc
-----------σmin
fc
----------, ,⎝ ⎠⎛ ⎞=
Fig. 4—Scheme of theoretical model.
Table 1—Database of experimental results employed to fit fatigue evolution laws for εmaxand E, and to analyze selected S-N curve (Nf )
Author(s) Data type Reference
Bennet and Raju Nf , εmax, E 22
Do et al. Nf , εmax, E 33
Kim and Kim Nf , εmax 32
Ople and Hulsbos Nf 26
Klausen Nf , εmax 6
Awad and Hilsdorf Nf , εmax, E 37
Hsu Nf 7
Tepfers and Kutti Nf 8
Holmen Nf , εmax, E 5
Petkovic et al. Nf , εmax, E 19
Zhang et al. Nf 20
Aas-Jakobsen Nf , εmax 21
Ruhr University εmax, E 38,39
Gaede εmax, E 40
Mehmel and Kern εmax, E 24
670 ACI Structural Journal/September-October 2009
The load history of a concrete specimen is represented inFig. 5(a). The specimen is first subjected to N1 load cycleswith constant limits (σmax,1, σmin,1) and then to N2 additionalcycles oscillating between (σmax,2, σmin,2). The concept of theequivalent number of cycles makes it possible to obtain thestrain increase due to the last N2 excursions by means of thefollowing expression
(9)
The calculation of the strain increment may be explainedwith the help of Fig. 5(b). There, Neq,1 is the number of loadcycles needed to develop a total strain εmax,1 in the fatigueprocess under stress levels (σmax, 2, σmin, 2). The additionalnumber of cycles N2 must be introduced from point B,leading to the strain increase EF. It is clear from Fig. 5(b) thatthis value is smaller than the one obtained without introducingthe equivalent number of cycles (segment CD).
The proposed criterion is completely different fromMiner’s rule, which is not valid for concrete.20 The strain-based equivalent number of cycles is also employed tocompute the modulus of deformation after N1 + N2 loadcycles, as follows
(10)
Evolution laws for strain and stiffnessThe evolution laws for the total strain and the concrete
modulus of deformation are determined from the data availablein the literature. For all the considered experiments, thespecimens were subjected to constant limit fatigue
Δε εmax 2, εmax 1, =–=
fNeq 1, N2+
Nf
-------------------------σmax 2,
fc
---------------σmin 2,
fc
--------------, ,⎝ ⎠⎛ ⎞ εmax 1,–
E N1 N2+( ) gNeq 1, N2+
Nf2
-------------------------σmax 2,
fc
---------------σmin 2,
fc
--------------, ,⎝ ⎠⎛ ⎞=
processes. Additionally, only specimens made of normal-strength concrete ( fc ≤ 50 MPa [7250 psi]) have been takeninto consideration. Due to the fact that the evolution laws areexpressed in terms of the relative number of cycles (N/Nf),those tests that did not fail (run-out specimens) have not beenused. In all, more than 100 experiments have been analyzed.The list of references is given in Table 1, as well as the datatype that has been taken into account from each work. Aspreviously indicated, the least-squares method has beenemployed to fit the model to the experimental data. Theproposed curves reproduce the three typical stages of thefatigue process of concrete. Second-order parabolic equationsare used for the first and third stages, whereas a linear expressionis employed for the second stage, so that the experimentalS-shaped evolution (Fig. 2) is attained. The transition pointsbetween Stages 1-2 and 2-3 are supposed to occur at 10%and 80% of the fatigue life, respectively.5
The strain evolution curve is defined as follows
; (11)
; (12)
(13)
where
A = 20(ε1-2 – 1) – ε2 (14)
B = 100(1 – ε1-2) + 10ε2 (15)
(16)
Equations (11) to (13) provide the maximum strain withrespect to its initial value, ε0, at the first load cycle (N = 1).The ratio ε1-2 defines the relative strain at 0.10Nf , that is, thetransition between Stages 1 and 2, and gives the constantstrain rate of the second domain. Both are defined accordingto Holmen’s approach5
εmax
ε0
---------- NNf
------⎝ ⎠⎛ ⎞ 1 A N
Nf
----- B NNf
------⎝ ⎠⎛ ⎞ 2
+ += 0.0 NNf
----- 0.1<≤
εmax
ε0
---------- NNf
------⎝ ⎠⎛ ⎞ ε1 2– ε2
NNf
----- 0.1–⎝ ⎠⎛ ⎞+= 0.1 N
Nf
----- 0.8<≤
εmax
ε0
---------- NNf
------⎝ ⎠⎛ ⎞ ε1 2– ε2
NNf
----- 0.1–⎝ ⎠⎛ ⎞ C N
Nf
----- 0.8–⎝ ⎠⎛ ⎞ 2
;+ +=
0.8 NNf
-----≤ 1.0<
C 25εfail
ε0
--------- ε1 2–– 0.9ε2–⎝ ⎠⎛ ⎞=
ε2·
Fig. 6—Evolution of longitudinal strain with number of cycles.
Fig. 5—(a) Load history with two different stress level steps; and(b) strain increment employing equivalent number of cycles.
ACI Structural Journal/September-October 2009 671
(17)
(18)
(19)
By employing these values, it is concluded that the influenceof the minimum stress level on the fatigue strain is negligiblewhen the lifetime is expressed over the total number ofcycles Nf. The εmax /ε0-N/Nf curves for different stress levelsare drawn in Fig. 6. It is noticed that the effect of the thirdfatigue stage is quite small for the highest stress levels due tothe low amount of inelastic strains available within thisrange. This fact confirms that the fatigue failure becomesmore brittle as the maximum stress level increases, whichhas been experimentally observed.5
For the evolution law of the concrete stiffness, the nextanalytical expressions are proposed (second-order parabolicequations for Stages 1 and 3 and linear equation for Stage 2)
; (20)
; (21)
; (22)
where
(23)
(24)
(25)
Q = A + 0.8E2 (26)
R = E2 (27)
(28)
The initial tangent modulus of deformation, Ec, is taken asthe starting value of the stiffness regardless of the stresslevel. Parameter (E/Ec)1-2 gives the stiffness at 0.10Nf ; is the constant rate of the second stage; and Efail/Ec definesthe modulus of deformation at the failure instant. They have beenfitted to the experimental data, leading to the following equations
ε1 2–1.184Smax
-------------=
ε2· 0.74037
Smax
-------------------=
ε2· 1
0.9-------
εfail
ε0
--------- ε1 2––⎝ ⎠⎛ ⎞≤
EEc
----- NNf
------⎝ ⎠⎛ ⎞ 1 M N
Nf
----- P NNf
------⎝ ⎠⎛ ⎞ 2
+ += 0.0 NNf
----- 0.1<≤
EEc
----- NNf
------⎝ ⎠⎛ ⎞ T E2
NNf
-----+= 0.1 NNf
----- 0.8<≤
EEc
----- NNf
------⎝ ⎠⎛ ⎞ Q R N
Nf
----- S NNf
------⎝ ⎠⎛ ⎞ 2
+ += 0.8 NNf
----- 1.0<≤
M 20 E2 20 EEc
-----⎝ ⎠⎛ ⎞
1 2–
+––=
P 100 10E2 100 EEc
-----⎝ ⎠⎛ ⎞
1 2–
–+=
T EEc
-----⎝ ⎠⎛ ⎞
1 2–
0.1E2–=
S 25Efail
Ec
---------- A– E2–⎝ ⎠⎛ ⎞=
E2·
(29)
(30)
Smax,lim = 0.3912 + Smin (31)
(32)
where Smax,lim is the threshold relative stress below whichthe degradation rate of the second fatigue stage becomeszero. The proposed evolution law is represented in Fig. 7.The influence of the lower stress level here is remarkablebecause it leads to large stiffness reductions, especiallyunder Smin < 0.05.
It is worth noting that the material model has been fittedemploying experiments from normal-strength concrete(fc ≤ 50 MPa [7250 psi]). Some researchers32,33 haveshown that high-strength concrete exhibits less strainincrease and stiffness decrease than normal-strengthconcrete because the monotonic response is already stiffer.This might be related to the different microcracking mechanismof high-strength concrete from that of normal-strengthconcrete. In high-strength concrete, the cracking surfacedevelops through the aggregates, whereas in normal-strengthconcrete, microcracks typically originate at the paste-aggregateinterface and then propagate through the paste.
SECTIONAL ANALYSISThe structural behavior of reinforced concrete elements is
investigated in this section by means of a sectional algorithm.The model focuses on prismatic elements, such as beams andcolumns composed of normal-strength concrete, and variouslayers of reinforcement subjected to normal stresses. Thesectional analysis is developed at the cracked section wherethe stress level and oscillation are higher and, thus, it is theweakest section from the point of view of the fatigue sensibility.The hypothesis that plane sections remain plane is accepted.All the tensile force is carried by the reinforcement.Although concrete has a certain capacity to carry tensilestresses at the cracked section (tensile strength and tensionsoftening), its cyclic degradation is very fast within thehigh cycle fatigue domain.34 Hence, the hypothesis of a fullycracked section is reasonable and leads to a somewhat higherbut realistic fatigue of concrete and steel.
The time-dependent model for concrete defined in theprevious sections is used. Steel behaves elastic-perfectlyplastic. Fatigue of the steel is estimated from the stress oscillationby means of the well-accepted S-N curve and Miner’s rule (referto Eq. (39)).
Due to the fatigue effect, the section is subjected to anonlinear free strain increment over the depth. Due to thefact that all fibers of the section are attached to each other,self-equilibrated stress increments are generated along the
depth to restore the strain compatibility. The fatigue life issubdivided into blocks of cycles to reproduce the progressiveevolution of stresses and strains. The statically indeterminateproblem is solved in each set of cycles. To evaluate the time-dependent response, a three-step methodology has beenimplemented. The sectional strategy is similar to thatemployed by widely extended concrete handbooks35 to study theeffects of creep, shrinkage, relaxation, or temperature variation.
The model is explained with the help of Fig. 8. The crosssection is discretized into small layers representing the materialfibers. The stress and strain distributions of a reinforcedconcrete section after a certain number of load cycles (Nj) aresupposed to be known under maximum and minimum load.The actual state of each fiber I of the cross section is definedby its strains εi(εi,max, εi,min) and stresses σi(σi,max, σi,min).The objective is to obtain the stress-strain increments afterΔNj additional cycles, which represent a block of cycles j.The three-step procedure is as follows.
In Step 1, a free strain increase of each concrete fiber isallowed, as if the fibers were not attached to the adjacentfibers of the cross section. The free strain increment Δεi isdependent on the number of cycles, the stress-strain state,and its history, following the time-dependent model of theprevious sections
(33)Δεi εmaxNeq i, ΔNj+
Nfi
---------------------------σi max,
fc
--------------σi min,
fc
--------------, ,⎝ ⎠⎛ ⎞ εi max,
–=
where Neq,i is the equivalent number of cycles necessary tobe applied in a fatigue process under constant stress limits(σi,max, σi,min) until a total strain εi,max is reached. Obviously, inthis first step, strain compatibility is not conserved. Furthermore,as strains have increased freely, no stresses are generated.
In Step 2, the free strain increments of Step 1 are restrainedby means of negative artificial stresses Δσbloq,i
Δσbloq,i = –EiΔεi (34)
where Ei is the actual modulus of deformation of eachconcrete fiber, obtained as expressed by Eq. (10)
(35)
After Steps 1 and 2, the total strain increase is zero, but thestress resultant along the height is an axial force ΔNbloq thatmust be removed to verify the sectional equilibrium.
The artificial restraints of Step 2 are eliminated in Step 3(the axial force –ΔNbloq is accompanied by a bendingmoment ΔMbloq due to its eccentricity with respect to thesectional centroid). The forces are applied in reversed directionsto the entire concrete-steel section. Because each concrete fiberhas a different stiffness Ei (so does steel), Step 3 representsthe problem of a multi-layer section under normal forces.The well-known method of the transformed section(selecting one modulus of deformation as reference ni = Ei /Eref )could be used, but when the number of layers is high enough,an iterative procedure is preferred. The problem is anonlinear one dealing with the calculation of the two variables,which define the deformed section (for example, curvature andtop strain).
The fatigue effect of the ΔNj cycles is then easily obtainedby adding the results of the three steps
ΔεTOT, i = Δεi – Δεi + Δεunbloq, i = Δεunbloq, i (36)
ΔσTOT, i = 0 – EiΔεi + Δσunbloq, i = –EiΔεi + Δσunbloq, i (37)
As it is schematically drawn in Fig. 8 (bottom left), thisprocedure conserves strain compatibility and leads to aredistribution of stresses within the section, satisfyingsectional equilibrium. The increase of curvature is a consequenceof the fatigue effect.
The stress and strain state under minimum load is calculatedby means of the assumed linear unloading behavior, taking intoaccount that each material fiber behaves with a differentmodulus, Ei , as in Step 3.
The time-step ΔNj must be carefully selected to fairlyreproduce the sectional evolution. A large value of ΔNjwould lead to wrong results, but too-short time incrementswould be computationally inefficient. Here, the selectedcriterion employs time intervals of constant maximum freestrain increment: along the section depth, the ratio Neq,i/Nfi isevaluated to identify the most fatigued concrete fiber. Thisfiber is used to calculate the new ΔNj such as its freeincrease is limited. In general, a free strain increment of5 × 10–6 is considered.
The failure occurs when a concrete fiber reaches the σ-εenvelope, which corresponds to a ratio
Ei ENeq i, ΔNj+
Nfi
---------------------------σi max,
fc
--------------σi min,
fc
--------------, ,⎝ ⎠⎛ ⎞=
Fig. 7—Evolution of modulus of deformation with number ofcycles: (a) constant minimum stress Smin = 0.05 and maximumstress varying between Smax = 0.40 to 0.90; and (b) constantmaximum stress Smax = 0.80 and minimum stress varyingbetween Smin = 0.00 to 0.35.
ACI Structural Journal/September-October 2009 673
(38)
Fatigue of steel is evaluated in each time-step throughMiner’s damage rule
(39)
where Nf is the number of cycles estimated employing the S-Ncurves for reinforcing steel from the Model Code.14 Thisvalue is calculated in each time-step from the actual stresslevel of the reinforcement.
NUMERICAL EXAMPLESSome experimental works have studied the response of
reinforced concrete beams under fatigue. Most of themconcluded that the typical fatigue failure is due to the brittlefracture of the steel reinforcing bars.10 Long-duration testsusually led to notable strain increases,9 indirectly showingthe development of a process of redistribution. Theseinvestigations, however, were not aimed to discuss thedegradation mechanism of the compression zone. In thissection, the experimental campaign of Lambotte et al.36 is
analyzed to provide a better understanding of the influenceof the fatigue degradation of concrete. These authors testedreinforced concrete beams cyclically loaded under four-pointbending. Different steel percentages and load levels were varied.The geometry of the specimens is plotted in Fig. 9. A minimumbending moment Mmin/Mu = 0.25 was always applied.
In the first series, the deformational behavior wasanalyzed. Type A beams were subjected to 5 million cycles.The concrete strain and the curvature of the section weremeasured. Figures 10 and 11 compare those values with thecalculated ones employing the model presented herein. Agood agreement between tests and model results is found.The increase of strains takes place while redistribution ofstresses develops within the cross section. It is noted that afraction of the experimental strain increment may be due tocreep and shrinkage. Because the test duration was shorterthan 66 hours and the concrete age at the time of testing wasnot reported, it is assumed here that the specimens hadenough age to develop small, long-term deformations. In anycase, the sectional algorithm has been presented in a convenientway so that additional strain or stiffness changes can be included.
Figure 12 illustrates the evolution of the stress distributionover the section depth for the case with Mmax /Mu = 0.70. Thestress level at the top of the section is reduced because thesefibers suffer large fatigue degradation. Stresses are thenwithstood by less-fatigued fibers inside the compression
Neq i,
Nfi
----------- 1.0=
DsΔNj
Nf
---------j∑ 1.0≤=
Fig. 8—Sectional algorithm in three-step procedure.
ACI Structural Journal/September-October 2009674
zone. Due to the redistribution process, the centroid of thecompression zone moves downward and the lever arm of thesection decreases. To maintain the sectional equilibrium, thetensile stress of the reinforcement must increase. The position ofthe neutral axis is also reduced as the curvature grows. Theredistribution of stresses avoids the initially expected fatiguefailure of concrete. This result explains why most experimentsof the literature finish with the steel failure. Moreover, theoscillation of the steel stress between the maximum andminimum load is not constant due to the progressive stiffnesschange. This fact means that an accurate prediction of thefatigue life should not be based on the static stress state.
A second experimental series investigated whether thefatigue failure of concrete in compression could be possible.
To achieve this, the beams were made of low-strengthconcrete and the maximum load level was increased. Type Bbeams of Fig. 9 were tested, employing different steelconfigurations. Fatigue collapse of concrete was experimentallyfound under the higher load levels. The model explained in thispaper enables the theoretical fatigue life by means of Mmax /Mu–Nf diagrams to be obtained. These diagrams are useful for adeep understanding of the test results. They are similar toclassic S-N curves, but herein, the degradation mechanism istaken into account. Figure 13 represents the comparisonbetween the experimental number of cycles and thecomputed Mmax/Mu–Nf curves. The theoretical failure dueto fatigue of concrete in compression is represented bycontinuous lines. The beams with the lowest steel percentage
Fig. 9—Scheme of studied beams according to Lambotte et al.36
Fig. 10—Measured (points) and computed (lines) maximumconcrete strains at top of section over number of cycles(Type A beams).
Fig. 11—Measured (points) and computed (lines) maximumcurvature over number of cycles (Type A beams).
ACI Structural Journal/September-October 2009 675
(Type BI) systematically failed by the reinforcement; fatigue ofconcrete was not reached and the theoretical curve stillprovides a large remaining number of cycles.
The fatigue failure of compressed concrete was experimentallyobtained in Beams BII-BIII′, which is in agreement with theanalytically-developed diagrams. In addition, such a failuremode was never found in the experiments under maximumload levels of 0.60 to 0.70Mu , which also correlates with thecalculated curves.
The fatigue properties of the employed reinforcing steelswere not available. The fatigue failure due to the steel,however, has been estimated by introducing, in Eq. (39), thefatigue properties provided by the Model Code.14 Figure 13represents the predicted curves by means of discontinuouslines, showing the change of the mode of fatigue failure withrespect to the load level.
CONCLUSIONSFrom the study of this paper, the following conclusions
may be drawn:1. A time-dependent model for concrete under fatigue is
necessary to adequately reproduce the behavior of reinforcedconcrete components under cyclic loading. The methodologyproposed herein accounts for the mechanical changes ofconcrete and results in a realistic evolution of the stress andstrain state during the fatigue life.
2. Concrete members were subjected to repeated loading,exhibiting increasing strains and, therefore, deflections, duringtheir lifetime. At the same time, a process of redistribution of
stresses is developed. Both coupled results are reproducedby the model presented herein. It is interesting to observehow compressive stresses are transferred from the top of thesection to less-fatigued fibers inside the compression zone.This mechanism explains why fatigue failure of concrete isnot reached in most experiments of the literature. Due to thisfact, typical fatigue fracture of the reinforcement has beentraditionally reported.
3. The fatigue model is a useful tool to estimate theincrease of deformations and to predict the number of cyclesto failure. It has been shown that the mode of failure isdependent on the capacity for redistribution through the loadlevel and the steel reinforcement ratio. Fatigue failure ofcompressed concrete may take place in over-reinforcedbeams, especially under high load levels. Because theoscillation of steel stresses changes during the fatigue life,the calculation of the resistant number of cycles cannot bebased on the static stress state.
4. The model capabilities have been compared with test dataof reinforced concrete members, with favorable results not onlyin the prediction of the strain evolution, but also in the fatiguelife and the mode of fatigue failure. Further experimentalresearch under severe conditions is needed, however, for adeeper understanding of the degradation mechanisms.
ACKNOWLEDGMENTSFinancial support provided by the Universidad Politécnica de Madrid is
gratefully acknowledged. The advice from A. Samartin (UPM, Spain) isgreatly appreciated.
Fig. 12—Calculated evolution of stress distribution along section height (Type A beam, Mmax/Mu = 0.70).
676 ACI Structural Journal/September-October 2009
NOTATIONDs = fatigue damage of reinforcementE = concrete modulus of deformationE1-2 = relative modulus of deformation at N/Nf = 0.1E2 = reduction rate of concrete stiffness in second fatigue stageEc = concrete initial tangent modulus of deformationEfail = concrete modulus of deformation at the failure instantEref = modulus of deformation taken as referenceEs = steel modulus of elasticityf = frequencyfc = concrete compressive strengthfu = steel yielding stressMmax = maximum momentMmin = minimum momentMu = ultimate momentN = number of load cyclesNeq = equivalent number of cyclesNf = number of load cycles to failuren = steel ultimate strengthn = coefficient of transformation Ei/Eref R = ratio minimum stress/maximum stress R2 = coefficient of determination of least squares fitting methodSmax = normalized maximum stressSmaxlim = normalized threshold stressSmin = normalized minimum stressΔMbloq = restraint bending momentΔNbloq = restraint axial forceΔNj = number of cycles of j block of cyclesΔε = strain incrementΔεbloq = restraint strainΔεTOT = total strain incrementΔεunbloq = strain increment of Step 3Δσ = stress incrementΔσbloq = restraint stressΔσTOT = total stress incrementΔσunbloq = stress increment of Step 3ε = strainε0 = maximum concrete strain at N = 1ε1-2 = relative strain at N/Nf = 0.1ε2 = strain rate in second fatigue stageεc = concrete strain at fcεfail = failure strainεlim = concrete strain at 0.5fc in softening curveεmax = maximum strainεmin = minimum strainρ = steel reinforcement ratioσ = stressσmax = maximum stressσmin = minimum stressΦ = reinforcing bar diameter
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