Fatigue Capacity of Plain Concrete Under Fatigue Loading With Constant Confined Stress

Fatigue capacity of plain concrete under fatigue loading with constant confined stressH.L.Wang1 and Y.P.Song2(1) Civil and Architectural Engineering College, Dalian University, Dalian, 116622, China(2) State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, Liaoning Province, 116024, ChinaH.L.Wang Email: whl2003_2002@tom.comReceived: 7February2009Accepted: 26April2010Published online: 9May2010AbstractThe fatigue tests of concrete have been conducted with constant confined stress, including compression, tension and alternate tensioncompression fatigue loading. Based on the results of experiment, the equations of SN curves under certain lateral stress were established and fatigue failure criterions for different loading cases were proposed. Thus according to the SN curves, the fatigue reduction coefficients for different fatigue loading were obtained. At last, the necessity of the proposed fatigue failure criterion was shown by checking an existing concrete structures computation under fatigue loading. The investigation of this paper can provide some useful information for the design of concrete structures such as reinforced concrete bridge crane beams, offshore platforms, concrete sleepers, nuclear power plants and pressure vessels.KeywordsConcrete Constant confinement Biaxial Fatigue Failure criterion 1 IntroductionMany structures are often subject to repetitive cyclic loads. Examples of such cyclic loads include machine vibration, sea waves, wind action and automobile traffic. The exposure to repeated loading results in a steady decrease in the stiffness of the structure, which may eventually lead to fatigue failure. The earliest research on fatigue properties of concrete materials is traced back to the end of the nineteenth century [1]. In recent years, many investigations [2, 3] concerning plain concrete under uniaxial cyclic tension, uniaxial alternate tensioncompression and biaxial fatigue loads have been carried out, but the uniaxial compressive fatigue tests so far have been most investigated [4]. Fatigue behavior of plain concrete under biaxial compression [5], high-strength concrete subjected to proportional biaxial-cyclic compression [6] and steel fiber reinforced concrete subjected to biaxial compressive fatigue loading [7] were investigated. Hollow concrete specimens were tested to investigate the fatigue response of concrete subjected to biaxial stresses in the compressiontension region and tensiontension region by Subramaniam et al. [8] and Subramaniam and Shah [9].A review of the available literature indicates that most of the experimental results pertain to proportional fatigue loading. In practical structural applications, however, concrete structures may be subjected to nonproportional fatigue loading, where the load along the one axis are fixed with the fatigue loading imposed in the orthogonal direction. Structures such as airport/highway pavements and bridge decks are subjected to repetitive live loads of high-stress amplitude due to passing vehicles, and the dead loads of these structures can also have some effects on the he fatigue performance of concrete. Consequently, it is necessary to research on the fatigue behavior of concrete subjected to nonproportional biaxial stresses. The limited research of plain concrete on nonproportional biaxial fatigue compressive loading was only reported by Lv [10]. Lv [10] founded the compressive deformation and fatigue strength of concrete under biaxial nonproportional fatigue loading is higher than that under uniaxial compression for any given number of load cycles. The conclusion was same with that of proportional biaxial-cyclic compression [57]: the fatigue failure envelopes in compressive regions have shapes similar to the envelope for static strength.But many concrete structures suffer from tension or alternate tensioncompression fatigue loading with constant lateral stress, very little information is available on the fatigue strength under these biaxial loading conditions. So study on the fatigue strength and failure modes of plain concrete under biaxial loading is necessary for evaluating the safety of concrete structures subjected to fatigue loading. Accordingly, the main purpose of this paper was to investigate the fatigue strength of plain concrete under cyclic compression, cyclic tension and alternate tensioncompression cyclic loading with constant lateral pressures, thus deduce the failure criterion of concrete under multi-axial fatigue loading.2 Experimental investigation2.1 Specimen and mix proportionsPlain concrete cubes with a size of 100100100mm were subjected to cyclic compression loads and special dog-bone specimens with 350mm long, 100mm wide; 100mm thick were subjected to tension or alternate tensioncompression fatigue loading. A detailed diagram for dog-bone specimens used in this paper was given in Fig.1. All the dog-bone specimens were set by eight pieces of steel bar embedded in their two ends for connection between specimen and testing machine.

Fig.1Dog-bone specimen (unit: mm)

The concrete used in the study had the following proportions, per cubic meter: cement 383kg; fine aggregate, 663kg; coarse aggregate, 1,154kg; and water, 200kg. A Chinese standard (GB175-99) [11] 42.5# Portland cement was used. Crushed stones were used as coarse aggregate with particle size ranging from 5 to 20mm. River sand was used as the fine aggregates. The cubic specimens were cast in steel molds and the dog-bone specimens were cast in wood molds. The molds were removed after 24h from casting and the specimens were placed in a curing room at a relative humidity of 95% and at a temperature of 20C for 4weeks. The 28-day compressive strength, f c, and the average modulus of elasticity of concrete, E 0, obtained by testing standard prism specimens (150mm150mm300mm) were 24.64MPa and 24.60GPa respectively. The average uniaxial tensile strength of the concrete, f t obtained by testing dog-bone specimens was 2.43MPa. The age of the specimens at the time of testing was more than half a year because the difference in age of strength was not deemed to affect the test results significantly.2.2 Test facilitiesA 1,000kN capacity Servo-controlled Material Testing System (MTS2810NEW) with a control computer was employed to apply longitudinal cyclic loading (see Fig.2). An independent loading frame was constructed for applying lateral confinement in the horizontal direction. Lateral pressures was applied with manually operated 250kN capacity screw jacks and calibrated by the load transducers. At the end of every jack a load cell and spherical hinge were used.

Fig.2 Specimen and setup. a Setup of cubic specimen, b setup of dog-bone specimen2.3 Experimental programIn the nonproportional biaxial fatigue tests, the lateral confinement stress level |l|/f c and the maximum stress level S max were two primary variables. During the test, all specimens were subjected to two loading phases:(1) Initial confining phase, where specimen was installed exactly and then loaded horizontally to a predetermined value. For all the biaxial tests in this paper, three lateral confinement stress levels were concluded, i.e., |l|/f c=0, 0.25 and 0.50.(2) Application of longitudinal loading. The sinusoidal wave form cyclic loading with the predetermined minimum and maximum stresses levels was applied until the specimen failure occurred or the prescribed maximum number (200million cycles) was reached. For compressive fatigue with constant tests, the maximum stress level of the vertical fatigue compression S max=max/f c varied from 0.65 to 1.05, but the minimum stress level min/f c was always 0.1, here negative values indicate compression. For tensile fatigue with constant tests, the maximum stress level of the vertical fatigue tension S 1max=1max/f t varied from 0.3 to 0.85, and the minimum stress level 1min/f t was always 0.1. For alternate tensile-compressive fatigue tests, the maximum stress level of the vertical fatigue tension S 1max=1max/f t varied from 0.3 to 0.85, and the minimum stress level 1min/f c was always 0.2. To avoid the frictional confinement of the test specimens, friction-reducing materials consisting of three layers of plastic membrane with grease between each layer were placed between the specimens and the compressive loading platens in all tests. Three to five specimens were tested at each case, as seen in the scatter degree of the results. In this paper, the three principal stresses are expressed as 123 (tension denoted as positive).3 Test results3.1 Failure modesTwo main typical failure modes of the cube specimens are shown in Fig.3. For specimens tested in uniaxial compressive fatigue, due to the effective measures to reduce friction, the cubes became column-type fragments, and the cracks were basically parallel to the applied load. For specimens tested in biaxial compression, it is apparent that the shear-type failure was formed by the introduction of the constant confined stress. This finding on the failure modes obtained for plain concrete under biaxial compression conditions was in agreement with a previous study [10].

Fig.3 Failure modes of concrete specimens under compressive fatigue loading. a Uniaxial tests, b biaxial testsAs for the tensile fatigue and the alternate tensile-compressive fatigue tests, all the dog-bone specimens have the same failure modes. Figure4 shows some typical dog-bone specimens after testing. In the present study, stress concentration at the ends of the embedded bars is substantially reduced. Most of the specimens were ruptured by only one apparent tensile crack perpendicular to the direction of tensile load. Subramaniam [8] and Subramaniam et al. [9] found that the damage in concrete subjected to biaxial fatigue loading in the compressiontension region via a torsion test was localized to a single crack, and the crack growth governed the observed load-deformation responses. This finding was also in agreement with this study.

Fig.4 Failure mode of specimens under fatigue tensile and alternate tensilecompressive loading3.2 SN diagrams3.2.1 Biaxial compressive fatigue testsAccording to the above experimental program, the fatigue life of concrete under biaxial compressive cyclic loading with lateral pressure is obtained. The results are listed in Table1. Because fatigue life is considered to follow a normal logarithmic distribution generally, the antilogarithm of the average of fatigue life logarithms is used as the mean value for the given loading case in this paper. By a regression analysis, the relationship of the average fatigue life and the maximum stress level is expressed as Eqs.13. The correlative coefficients for them were 0.96, 0.92 and 0.88 respectively.Table1 Test results for compressive fatigue loading with various confined stresses|2/f c|S max Fatigue life N f

Sample 1Sample 2Sample 3Sample 4Sample 5

0 (uniaxial)0.9577298157177

0.85248283390617624

0.84555747391,4892,951

0.7512,72823,79547,26149,386

0.763,638798,362826,899

0.652.5106 ()2.5106 ()

0.251.051562245781,5633,078

0.953722,16631,65070,81117,624

0.851,8011,83498,011293,679660,551

0.75654,317695,406890,318

0.652.5106 ()2.5106 ()

0.51.058891,9695,3878,52925,397

0.955,62312,10815,39021,92336,401

0.856,82319,40939,810251,189325,836

0.75792,239893,7431,565,577

0.652.5106 ()2.5106 ()

Note: The lateral confinement stress level: |2/f c|; the maximum stress level: S max=max/f c; the symbol of denotes that failure was not detected\left|{S_{\max}}\right|=0.9885-0.0618\,\log\,N_{\text{f}}\left({\left|{{{\sigma_{2}}\mathord{\left/{\vphantom{{\sigma_{2}}{f_{\text{c}}}}}\right.\kern-\nulldelimiterspace}{f_{\text{c}}}}}\right|=0}\right)(1)\left|{S_{\max}}\right|=1.1615-0.0762\,\log\,N_{\text{f}}\left({\left|{{{\sigma_{2}}\mathord{\left/{\vphantom{{\sigma_{2}}{f_{\text{c}}}}}\right.\kern-\nulldelimiterspace}{f_{\text{c}}}}}\right|=0.25}\right)(2)\left|{S_{\max}}\right|=1.4412-0.1022\,\log\,N_{\text{f}}\left({\left|{{{\sigma_{2}}\mathord{\left/{\vphantom{{\sigma_{2}}{f_{\text{c}}}}}\right.\kern-\nulldelimiterspace}{f_{\text{c}}}}}\right|=0.50}\right)(3)Aas-Jakobsen [12] showed that the relationship between max/f c and min/f c was linear for compressive fatigue failure, and proposed the following equation:|Smax|=1(1R)logNf(4)where R=min/max is the ratio of the minimum to the maximum stress; is the slope of the SN curve when R=0. Using a large database of uniaxial fatigue test results for 475 specimens, Tepfers and Kutti [13] found that is 0.0685 and Eq.4 is applicable to both ordinary concrete and lightweight concrete.To account for the effect of lateral confinement, a modified Aas-Jakobsen relationship between max/f c and min/f c is proposed in the present study through a linear regression analysis, i.e.\left\{{\begin{array}{*{20}c}{\left|{S_{\max}}\right|=\alpha-\beta(1-R)\;{\log}\;N_{\text{f}}}\hfill\\{\alpha=1+0.8304\left({{{\left|{\sigma_{2}}\right|}\mathord{\left/{\vphantom{{\left|{\sigma_{2}}\right|}{f_{\text{c}}}}}\right.\kern-\nulldelimiterspace}{f_{\text{c}}}}}\right)}\hfill\\{\beta=0.0638+0.115\left({{{\left|{\sigma_{2}}\right|}\mathord{\left/{\vphantom{{\left|{\sigma_{2}}\right|}{f_{\text{c}}}}}\right.\kern-\nulldelimiterspace}{f_{\text{c}}}}}\right)}\hfill\\\end{array}}\right.\;\;\;\;\left({0\le\left|{{\frac{{\sigma_{2}}}{{f_{\text{c}}}}}}\right|\le0.5}\right)(5)where and are material constants correlated with the lateral confined stress level |2|/f c, the coefficients of correlation of which are 0.96 and 0.99, respectively. When 2/f c=0, =0.0638, then Eq.5 became Aas-Jakobsen Equation which is applicable to uniaxial compressive fatigue.The SN diagrams for compressive fatigue with different confined stress levels and the ratio of stress can be determined by Eq.5. The typical SN curves and the test results are shown in Fig.5. In these curves the logarithm of the fatigue life is plotted versus the maximum stress level (because the minimum stress level is usually constant, it is not taken account to the SN curve generally). It can be seen that, at a given maximum stress level, the compressive fatigue life will be longer with increasing lateral pressure level.

Fig.5 SN curves of compressive fatigue with various confined stresses3.2.2 Tensile fatigue with constant lateral stressResults of the tension fatigue tests are shown in Table2. These scattered results were statistically analyzed to obtain SN curves for different confined stress:Table2 Test results for tensile fatigue loading with various confined stresses(|2|/f c)S 1max Fatigue life N f

Sample 1Sample 2Sample 3Sample 4Sample 5

00.851993576338542,512

0.751,77816,81722,14025,11932,613

0.7136,254210,146498,701565,6681,258,926

0.250.7546104128321755

0.6049,32662,653247,431304,064444,162

0.452,301,2972,300,415

0.500.603067131,1071,8303,071

0.45531,411697,928979,2781,053,4651,131,109

0.301,753,6032,256,6942,418,7962,583,930

Note: The lateral confinement stress level: |2/f c |; the maximum stress level: S 1max=1max/f t S1max=0.0476logNf+0.9659(|2/fc|=0)(6)S1max=0.0596logNf+0.8855(|2/fc|=0.25)(7)S1max=0.0722logNf+0.8226(|2/fc|=0.5)(8)Furthermore, the relationship of fatigue life N f, maximum stress level S 1max and confined stress level 2 was derived through a multiple linear regression analysis:logN=12.87S1max5.69|2/fc|+13.79(9)The SN curves denoted by Eq.9 are plotted in Fig.6. The experimental data with different confined stress were also showed in Fig.6, and they indicated a good correlation with the three best-fit SN curves. It can be seen that at a given maximum stress level, the tensile fatigue life will decrease with increasing lateral pressure level.

Fig.6 SN curves of tensile fatigue with various confined stresses3.2.3 Alternate tensilecompressive fatigue with constant lateral stressResults of the alternate tensioncompression fatigue tests are shown in Table3. It was also proposed that the relationship of the average fatigue life and the maximum stress level can be represented by an expression given belowS1max=1maxft=ABlogNf(10)where A and B are material constants correlated with the lateral confined stress level |2|/f c. They could be determined from the experimental data through a multiple linear regression analysis:\left\{{\begin{array}{*{20}c}{A=-0.5636{{|\sigma_{2}|}\mathord{\left/{\vphantom{{|\sigma_{2}|}{f_{\text{c}}+0.9818}}}\right.\kern-\nulldelimiterspace}{f_{\text{c}}+0.9818}}}\\{B=-0.0458{{|\sigma_{2}|}\mathord{\left/{\vphantom{{|\sigma_{2}|}{f_{\text{c}}+0.0764}}}\right.\kern-\nulldelimiterspace}{f_{\text{c}}+0.0764}}}\\\end{array}\;\;\;\;}\right.\left({0\le{{|\sigma_{2}|}\mathord{\left/{\vphantom{{|\sigma_{2}|}{f_{\text{c}}}}}\right.\kern-\nulldelimiterspace}{f_{\text{c}}}}\le0.5}\right)(11)where the coefficients of correlation of A and B are 0.98 and 0.79, respectively.Table3 Test results for tensioncompression fatigue loading with various confined stressesLateral stress ratio (|2|/f c )S 1max N f

Sample 1Sample 2Sample 3Sample 4Sample 5

0 (uniaxial)0.8579178196

0.753981,7023,9815,623

0.652,6738,81018,32357,544

0.5527,41645,18695,499885,116

0.502,500,000 ()2,500,000 ()

0.250.705492125318341

0.651585451,3126,3267,736

0.602,3745,42713,52920,63771,847

0.5510,55211,27450,910130,815294,398

0.50194,342645,790913,0111,284,998

0.402,500,000 ()2,500,000 ()

0.500.553654675717653,815

0.503,1554,3574,7607,59084,019

0.458,98820,85928,41655,231146,458

0.4044,191490,675610,247682,157974,153

0.35275,463891,7251,175,4921,879,584

0.302,500,000 ()2,500,000 ()

Note: The lateral confinement stress level: |2/f c|; the maximum stress level: S 1max=1max/f c; the symbol of denotes that failure was not detected. The minimum stress level 1min/f c was always 0.2The SN diagrams for concrete under alternate tensilecompressive fatigue with different confined stress levels and the ratio of stress was shown in Fig.7, and the experimental data were also showed in Fig.7. It can be seen that at a given maximum stress level, the fatigue life of alternate tensioncompression will decrease with increasing lateral pressure level.

Fig.7 SN curves of alternate tensioncompression fatigue with various confined stresses3.3 Fatigue strengthFatigue strength is commonly defined as a fraction of the static strength that can be supported repeatedly for a given number of cycles. In practice, the upper stress level at N=2106 cycles on the SN curve is usually taken as the fatigue strength. For analysis and comparison of the results, the ratio of fatigue strength to static strength f c or f t is defined as the fatigue strength factor. Equations5, 9 and 10 were the failure criterions of concrete under lateral constant compressive stress. Using these failure criterions, the fatigue strength factors can be calculated for different lateral confined stress corresponding to the fatigue life of 2million cycles.Table4 lists the fatigue strength factors for compression fatigue with confined stress from Eq.5 and Refs. [10, 14] at different minimum stress levels. It can be seen that, at given minimum stress level, increasing the lateral pressure results in larger factors. The conclusion that the lateral pressure has much effect on compressive fatigue strength can be drawn. Thus, it can also be concluded that material will be unnecessarily wasted if concrete structures are designed on the basis of uniaxial compressive fatigue strength when they are in fact subjected to multiaxial compressive fatigue loading. It is found that the fatigue strength of concrete under biaxial compression loading obtained in this study is somewhat lower than that obtained by Tan [14], but they have the same trend, and this study tends to be more safe in engineering.Table4 Compressive fatigue strength factors of plain concrete with various confined stresses|2|/f c

00.10.20.30.40.5

Eq.5, R=0.10.640.660.680.690.710.73

Ref. [14], S min=0.10.650.700.750.800.850.90

Ref. [10], R=0.10.550.660.780.891.011.13

As for tension and alternate tensioncompression fatigue with confined compressive stress, according to the S/N Eqs.9 and 10 model, the multiaxial fatigue strength can be calculated easily. Table5 lists the fatigue strength factors for tension and alternate tensioncompression fatigue with confined stress at different minimum stress levels under 2106 cycles and 1106 cycles.Table5 Tensile and tensilecompressive fatigue strength factors of plain concrete with various confined stressesFatigue loading form|2|/f c f 1 f 2 Minimum stress level

Uniaxial tensioncompression00.50810.4835S min=0.20

Tensioncompression with confined stress0.250.48200.4657

0.500.36370.3460

Uniaxial tension00.6650.633S min=0.10

Pure tension with confined stress0.250.52790.5100

0.500.38940.3677

Note: f 1 is the fatigue strength factor under 2106 cycles and f 2 is the fatigue strength factor under 1106 cyclesIt can be seen that the fatigue strength of concrete under pure tension with the same confined stress is higher than that of alternate tensioncompression fatigue for any given number of load cycles. A higher lateral stress level result in a less fatigue strength factor for these two fatigue loading cases. Thus, it will be dangerous if the concrete structures at given lateral pressure are designed on the basis of uniaxial tensile fatigue strength when they are in fact subjected to multiaxial tension or alternate tensioncompression fatigue loading.4 Application of failure criterionFatigue strength is very important for fatigue design of concrete structures in engineering practice. For example, many concrete ocean platforms are subjected to tensioncompression cyclic loading with lateral pressure caused by water pressure and other loadings. For lack of sufficient experimental data, the design value of concrete tensile fatigue strength was only adapted by uniaxial tension and the fatigue strength factor was 0.74 in the previous study [15]. Figure8 shows one leg of the ocean platform. The fatigue strength of concrete at section I-I was recalculated according to the conclusions of this paper as followings.

Fig.8 Schematic drawing of the leg of ocean platformIn this figure, r 1=3,000mm, r 2=2,650mm, r g=2,850mm, area of reinforcing steel bar A y=93,200mm2, strength of reinforcing steel bar R y=750MPa, modulus of reinforcing steel bar E y=1.8105MPa, compressive strength of concrete R a=23MPa, tensile strength of concrete R f=2.55MPa, N=1,014104 N (compressive force), modulus of concrete E h=3.3104MPa, fatigue modulus of concrete E h p =1.5104MPa, M=3,937107Nmm.Prestressed reinforcement is placed along the circle equably, the prestressed force is y=133.25MPa, the concrete prestressing stress is h=2MPa.Checking the computation of fatigue strength of concrete, the transformation area is,A0=(r21r22)+EyEhAy=67117.14102mm2Consider the prestressed reinforcement as a steel circle, where theOutsideradius,rg1Insideradius,rg2=Ay2rg12+rg=2827.6mm\hfill=2822.4mm\hfillThe section modulus:W0=4(r41r42)1r1+4(r4g1r4g2)1r1\hfill=841348.24104mm2\hfillThe maximum compressive stress caused by fatigue loading is,=NA0+MW0=6.19MPa(herecompressivestressispositive)and considering the prestressing stress,pmax=+h=8.19MPaThe minimum compressive stress caused by fatigue loading is,pmin=h+NA0MW0=1.17MPa(heretensilestressisnegative)=pminpmax=0.14Based on Eq.6, we consider the lateral compressive stress |2/f c|=0.25, thus we have the fatigue strength factor from Table5, r p=0.4657, whereRp=rpR=0.465723=10.711MPaRpf=rpRf=0.46572.55=1.19MPa,and max p =8.19