# Fatigue Properties of Plain Concrete under Triaxial Tension-Compression-Compression Cyclic Loading

Authors: Dongfu Zhao; Haijing Gao; Huixuan Liu; Penghe Jia; Jianhui Yang
Journal: Shock and Vibration
 (2017)

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## Abstract

Fatigue tests were performed on plain concrete under triaxial tension-compression-compression (T-C-C) cyclic loading with constant and variable amplitude using a large multiaxial machine. Experimental results show that, under constant amplitude fatigue loads, the development of residual strain in the fatigue loading direction depends mostly on the lateral compressive stress ratio and is nearly independent of stress level. Under variable amplitude fatigue loads, the fatigue residual strain is related to the relative fatigue cycle and lateral compressive stress ratio but has little relationship with the loading process. To model this system, the relative residual strain was defined as the damage variant. Damage evolutions for plain concrete were established. In addition, fatigue damage analysis and predictions of fatigue remaining life were conducted. This work provides a reference for multistage fatigue testing and fatigue damage evaluation of plain concrete under multiaxial loads.


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## Body

## 1. Introduction

In practice, many concrete structures such as offshore platforms, nuclear reactor pressure containers, and bridges are subjected to repeated loading amplitudes in complex multiaxial stress states during normal use. Thus, it is important to understand fatigue damage accumulation in concrete under multiaxial variable amplitude stresses and various stress ratios. In the last few decades, many researches have been carried out to study the fatigue properties of plain concrete (PC) under uniaxial and multiaxial compressive loads. Because the development of multiaxial fatigue test system technology is difficult and with high cost and several key technical problems exist in the experiment, to date, a few works have been dedicated to investigating multiaxial tension and compression fatigue properties. Park et al. [1] used thin fire-damaged concrete discs to evaluate tensile strength and found that the tensile strength could be effectively estimated from the HNP without consideration of the mix proportion. He et al. [2, 3] focused on the multiaxial mechanical properties of plain recycled aggregate concrete (RAC) and found that the ratio of triaxial fatigue strength to the corresponding uniaxial compressive strength for the three strength grades of RAC was higher than that of conventional concrete. Mun et al. [4] examined the fatigue stress-strain response of different concrete mixtures under compressive cyclic loadings with maximum and minimum stress levels. Shi et al. [5] reported that the tensile strength and secant modulus of large aggregate concrete under triaxial T-C-C increase with the increase of strain rate but decrease with the increase of lateral compressive stress. Subramaniam et al. [6] studied the fatigue behavior of concrete subjected to combined stresses in the T-C region of the biaxial stress space and found that the decrease in rotational stiffness at failure for the constant amplitude fatigue loading was comparable to the corresponding load in the postpeak part of the quasistatic response. Shiming and Yupu [7] studied on biaxial T-C strength and found that tensile compressive strength increases with the increase of strain rate when the lateral tensile stress is constant and the tensile compressive strength decreases with the increase of lateral tensile stress. Yan et al. [8] tested the dynamic properties of concrete materials under both high strain rates and triaxial stress states. It was pointed out that the strain rate effects on maximum strength under triaxial stress states decrease with an increase in confining stress. Only Yang et al. [9] investigated the fatigue properties of concrete in tension under single and bilateral pressure of variable amplitude, determining fatigue life and residual strain. Based on the cyclic stress-strain curve, fatigue deformation modulus was defined. Accordingly, a damage model was established through the relationship between the ratios of fatigue deformation modulus and the number of cycles. Above all, we can see that the researches on plain concrete for fatigue properties under triaxial T-C-C loading are rare, so it has a realistic significance for us.Fatigue cumulative damage is one of the main concerns when using PC, and theP-M linear fatigue cumulative damage criterion is still the most widely used model. On this basis, most studies have included research on the linear damage of PC. Studies have validated and discussed the damage variant D=∑(ni/Ni). Subramaniam et al. [6] recommended that damage in concrete subjected to biaxial fatigue loading in the T-C region through torsion is localized, the damage localizes to a single crack, and the observed overall response is governed by crack growth. The load-deformation response obtained from quasistatic loading acts as the envelope failure curve for fatigue loading considering decrease in rotational stiffness. Peiyin et al. [10] discussed the laws and characteristics of fatigue residual strain, defined the damage variable, and carried out damage accumulation and evolution rules. In addition, a model of fatigue damage was established and verified using experimental results. These investigations focused on the nonlinear cumulative fatigue damage to concrete. The fatigue damage evolution equations were defined using other parameters such as strain, strength, and elastic modulus as part of the damage variant. Chen et al. [11] reported the effects of stress state on the dynamic compressive strength and the dynamic damage evolution process of concrete using a split Hopkinson pressure bar and ultrasonic technique. Results showed that dynamic damage evolutions are accelerated with an increase in strain rate and are delayed significantly under confined pressure. Byung [12] introduced the cumulative damage theory of concrete under variable amplitude fatigue loading condition. Yang et al. [9] investigated the properties of concrete in tension under single and bilateral pressure of variable amplitude fatigue, working out fatigue life and residual strain. Based on the cyclic stress-strain curve, fatigue deformation modulus was defined. Accordingly, a damage model was established through the relationship between the ratios of fatigue deformation modulus and the number of cycles. Cao et al. [13] considered more than the fatigue residual strain of concrete when determining the damage variant. Steady deformations and smaller discreteness at the ultimate residual strain were observed when considering residual strain on the concrete fatigue loading direction. Thus, the unrecoverable degree of microplastic deformation and microcracks can be determined. This model better reflects the properties of the material than the method of maximum strain for concrete in the fatigue loading direction.Testing of the triaxial T-C-C fatigue properties was performed using a triaxial testing machine capable of developing three independent compressive or tensile forces at Dalian University of Technology. Fatigue residual strain was recorded during both constant and varying amplitude fatigue processes. The relative residual strain was defined as the damage variant, updates were proposed to the corresponding fatigue damage equation, and the multistage fatigue damage model under triaxial T-C-C loading was established. This work provides a reference for multistage fatigue testing and fatigue damage evaluation of concrete under multiaxial loads.


## 2. Materials and Experimental Procedures

### 2.1. Casting and Curing of Specimens

The constituent proportions within the mix used experimentally were cement : sand : stone : water = 383 : 663 : 1154 : 193 (kg/m3). During testing, the ends of the specimen bulk and the testing machine load head were connected by tightening nuts. This transmitted the tensile strength of the machine through the screw, which was then passed to the specimen in order to test volatility in tension fatigue loading. Specimens were created by casting in wooden molds and compacted slightly by a vibrating table. After 24 hours, specimens were removed from molds and then cured at 20 ± 3°C and 95% relative humidity for 28 days covered with grass. Specimens were tested 90 days after cure by the test method of Chinese standard GB/T 17671-1999 [14].Variable cross-section prism specimens were used for testing based on experimental requirements, fatigue performance, specimen installation, and references available at home and abroad. As shown in Figure1, lateral pressure can be applied to two pairs of sides when testing a prism shaped specimen. The specimens tested measured 150 mm × 150 mm × 350 mm. In order to reduce stress concentrations, both ends of the specimen were embedded with four Φ16 steel screws, and the end of the screw was embedded within the specimen by Φ8 steel welded connections. The same batch of concrete was used to create six or more 150 mm × 150 mm × 150 mm standard cubes, which were used to measure the strength for multiaxial tests and determine strength class [15].Figure 1
Machine and specimen.
(a)
Specimen size and shape (b)
The large-scale static and dynamic multiaxial testing machine (c)
Specimens under multiaxial loads


### 2.2. Testing Load Patterns

Fatigue tests were performed on plain concrete subjected to constant and variable amplitude triaxial T-C-C cyclic loading, and the lateral pressure stressesSmax are 0.25ft and 0.50ft. S is the stress level of fatigue, Smax=σmax/ft,Smin=σmin/ft=0.10, and ft is the uniaxial tensile strength of 150 mm × 150 mm × 300 mm prism. Constant amplitude fatigue of maximum stress was varied from 0.25ft to 0.75ft, and variable amplitude fatigue procedures included tests with two and three stages. The loading mode is shown in Figure 2, where loading frequency was 5 Hz. Cyclic loading was applied using a sine wave waveform and the minimum stress level. Load patterns are shown in Table 1.Table 1
Test results for constant-amplitude cyclic loading.
Lateral stress ratio S m a x Fatigue life σ 2 / f c σ 3 / f c N f N - f 0.25 0.25 0.75 210 121 89 75 43 108 0.65 16188 8903 5117 1547 369 6425 0.55 950477 783912 361760 137334 32729 453242  0.25 0.50 0.60 171 102 72 46 21 82 0.50 20387 9753 7415 1110 336 7800 0.40 1453275 1150119 785733   1129709  0.50 0.50 0.45 1276 871 625 389 207 674 0.35 348663 150744 98146 51405 17592 133310 0.25 Note. N-f is the average fatigue life.Figure 2
Load patterns of variable amplitude multistage fatigue tests.
(a)
Low-high load pattern (b)
High-low load pattern (c)
Low-mid-low load pattern (d)
High-mid-low load pattern


### 2.3. Testing Methods

Deformation of the specimen was recorded within a range of ±5 mm using displacement sensors. Three layers of plastic membrane with Mobil lubricants were placed between the compressive loading plate and the specimen as friction-reducing pads [2, 3]. After the specimen was placed in the appropriate position, first execute visual alignment and preload 10 kN static load in the vertical direction. Moreover, a larger static load was applied on specimens to complete the horizontal direction centering controlling. Then, preload the same proportion more than three times repeatedly in three directions, so that specimen with the loading plate contact surface compaction and load can be appropriately increased to 20~30 kN. Complete the specimen in the geometric centering controlling and fix it with moderate force, gradually tightening evenly eight screws at the upper and lower edges of the specimen. Finally, use a small hammer to tap specimen or load head, through the rotation of globe hinge to adjust mechanical alignment of specimen [16].


## 2.1. Casting and Curing of Specimens

The constituent proportions within the mix used experimentally were cement : sand : stone : water = 383 : 663 : 1154 : 193 (kg/m3). During testing, the ends of the specimen bulk and the testing machine load head were connected by tightening nuts. This transmitted the tensile strength of the machine through the screw, which was then passed to the specimen in order to test volatility in tension fatigue loading. Specimens were created by casting in wooden molds and compacted slightly by a vibrating table. After 24 hours, specimens were removed from molds and then cured at 20 ± 3°C and 95% relative humidity for 28 days covered with grass. Specimens were tested 90 days after cure by the test method of Chinese standard GB/T 17671-1999 [14].Variable cross-section prism specimens were used for testing based on experimental requirements, fatigue performance, specimen installation, and references available at home and abroad. As shown in Figure1, lateral pressure can be applied to two pairs of sides when testing a prism shaped specimen. The specimens tested measured 150 mm × 150 mm × 350 mm. In order to reduce stress concentrations, both ends of the specimen were embedded with four Φ16 steel screws, and the end of the screw was embedded within the specimen by Φ8 steel welded connections. The same batch of concrete was used to create six or more 150 mm × 150 mm × 150 mm standard cubes, which were used to measure the strength for multiaxial tests and determine strength class [15].Figure 1
Machine and specimen.
(a)
Specimen size and shape (b)
The large-scale static and dynamic multiaxial testing machine (c)
Specimens under multiaxial loads


## 2.2. Testing Load Patterns

Fatigue tests were performed on plain concrete subjected to constant and variable amplitude triaxial T-C-C cyclic loading, and the lateral pressure stressesSmax are 0.25ft and 0.50ft. S is the stress level of fatigue, Smax=σmax/ft,Smin=σmin/ft=0.10, and ft is the uniaxial tensile strength of 150 mm × 150 mm × 300 mm prism. Constant amplitude fatigue of maximum stress was varied from 0.25ft to 0.75ft, and variable amplitude fatigue procedures included tests with two and three stages. The loading mode is shown in Figure 2, where loading frequency was 5 Hz. Cyclic loading was applied using a sine wave waveform and the minimum stress level. Load patterns are shown in Table 1.Table 1
Test results for constant-amplitude cyclic loading.
Lateral stress ratio S m a x Fatigue life σ 2 / f c σ 3 / f c N f N - f 0.25 0.25 0.75 210 121 89 75 43 108 0.65 16188 8903 5117 1547 369 6425 0.55 950477 783912 361760 137334 32729 453242  0.25 0.50 0.60 171 102 72 46 21 82 0.50 20387 9753 7415 1110 336 7800 0.40 1453275 1150119 785733   1129709  0.50 0.50 0.45 1276 871 625 389 207 674 0.35 348663 150744 98146 51405 17592 133310 0.25 Note. N-f is the average fatigue life.Figure 2
Load patterns of variable amplitude multistage fatigue tests.
(a)
Low-high load pattern (b)
High-low load pattern (c)
Low-mid-low load pattern (d)
High-mid-low load pattern


## 2.3. Testing Methods

Deformation of the specimen was recorded within a range of ±5 mm using displacement sensors. Three layers of plastic membrane with Mobil lubricants were placed between the compressive loading plate and the specimen as friction-reducing pads [2, 3]. After the specimen was placed in the appropriate position, first execute visual alignment and preload 10 kN static load in the vertical direction. Moreover, a larger static load was applied on specimens to complete the horizontal direction centering controlling. Then, preload the same proportion more than three times repeatedly in three directions, so that specimen with the loading plate contact surface compaction and load can be appropriately increased to 20~30 kN. Complete the specimen in the geometric centering controlling and fix it with moderate force, gradually tightening evenly eight screws at the upper and lower edges of the specimen. Finally, use a small hammer to tap specimen or load head, through the rotation of globe hinge to adjust mechanical alignment of specimen [16].


## 3. Test Results and Discussions

### 3.1. Fatigue Life

Fatigue life was measured on plain concrete and results are shown in Tables1 and 2. A large number of studies have shown that the fatigue strength of concrete is usually attributed to the establishment ofS-N curve equation (also known as Wöhler equation). TheS-N curve quantitatively describes the relationship between stress level S and fatigue life N, and the mathematical expression is(1a)Smax=A-Blg⁡N,where A and B are coefficients. And fatigueS-N equations under T-C-C constant amplitude loading at various lateral pressures were analyzed using the linear regression method proposed below:(1b)Smax=-0.0519lg⁡N+0.8388σ2fc=0.25;σ3fc=0.25,(1c)Smax=-0.0427lg⁡N+0.6642σ2fc=0.25;σ3fc=0.50,(1d)Smax=-0.0412lg⁡N+0.5582σ2fc=0.50;σ3fc=0.50,where Smax is the maximum stress level of fatigue, N is fatigue life, fc is uniaxial compressive strength of a concrete prism, σ1 is the cyclic tensile stress, σ2 and σ3 are lateral compressive stresses, σ3<σ2<σ1, and the compressive stress is negative but tensile stress is positive. The correlation coefficients R2 of the above equations are 0.945, 0.965, and 0.948, respectively. Figure 3 shows the corresponding S-N scattergram.Table 2
Test data of fatigue for variable amplitude.
Specimen number Fatigue life D ε 0 Specimen number Fatigue life D ε 0 n 1 n 2 μ ε n 1 n 2 n 3 μ ε 25250101 52203 24 0.45 215 25250301 52203 1004 121 1.79 255 25250102 52203 308 3.48 234 25250302 52203 1004 75 1.30 248 25250103 52203 272 3.09 238 25250303 52203 1004 13 0.64 192 25250104 52203 43 0.66 247 25250304 52203 1004 360 4.33 244 25250105 52203 640 7.01 204 25250305 52203 1004 29 0.81 232 25250201 19 816980 3.33 217 25250401 19 1004 1498232 6.24 234 25250202 19 185321 0.91 213 25250402 19 1004 300168 1.65 229 25250203 19 114847 0.64 225 25250403 19 1004 70474 0.77 203 25250204 19 91356 0.55 199 25250404 19 1004 49593 0.69 237 25250205 19 1934129 7.61 243 25250405 19 1004 109627 0.92 226 25500101 304705 110 1.86 327 25500301 304705 1060 67 1.52 308 25500102 304705 17 0.45 319 25500302 304705 1060 315 5.27 314 25500103 304705 43 0.85 313 25500303 304705 1060 16 0.74 301 25500104 304705 233 3.73 306 25500304 304705 1060 32 0.99 335 25500105 304705 59 1.09 286 25500305 304705 1060 143 2.67 327 25500201 13 2559519 1.88 310 25500401 13 1060 594174 0.89 334 25500202 13 700821 0.66 302 25500402 13 1060 1218818 130 305 25500203 13 594174 0.59 275 25500403 13 1060 289469 0.69 307 25500204 13 1767287 1.36 293 25500404 13 1060 319940 0.71 320 25500205 13 >2500000 >1.88 319 25500405 13 1060 >2500000 >2.14 261 50500101 17171 1057 2.08  50500201 112  340844 4.17 400 Note. D=∑(ni/Ni), ε0 is the ultimate strain of specimen damage. Specimen numbering methods xxyyzztt: xxyy means the two directions of lateral compressive stress ratio, where 2550 indicates that one direction of lateral compressive stress ratio σ2/fc is 0.25, another σ3/fc is 0.50; zz represents the amplitude loading condition; 01, 02, 03, and 04 are four testing loading conditions for low-high, high-low, low-medium-high, and high-medium-low, respectively; tt represents the sample number to each operating mode.Figure 3
S - N Scatter-gram.


### 3.2. Linear Fatigue Damage Analysis

Miner [17] performed pioneering work on the amplitude fatigue linear cumulative damage rule, the P-M criterion as shown below:(2)D=∑i=1kDi=∑i=1kniNi=1.In this paper, Miner experimentally verified theP-M criterion. Table 3 shows the values obtained by this work.Table 3
ValueD.
[10] [20] [21] [12] This paper Max 2.94 95.72 1.11 3.26 7.61 Min 0.36 0.01 0.21 0.38 0.31 Avg. 1.20 8.32 0.53 1.27 1.87As shown in Table3, our experimental result for the value of D is higher than predicted. However, considering factors that influence multiaxial fatigue, the values of D obtained can be considered within the normal range. Moreover, results indicate that the P-M criterion for multiaxial amplitude fatigue has certain limitations. These limitations are mainly caused by difficulty in satisfying the prerequisites of the principle, which are as follows: each cycle of the energy dissipation is equal under constant amplitude stress level, cumulative damage cannot be influenced by the past loading process, and the fatigue life of concrete would not be affected by variable amplitude loading order.


## 3.1. Fatigue Life

Fatigue life was measured on plain concrete and results are shown in Tables1 and 2. A large number of studies have shown that the fatigue strength of concrete is usually attributed to the establishment ofS-N curve equation (also known as Wöhler equation). TheS-N curve quantitatively describes the relationship between stress level S and fatigue life N, and the mathematical expression is(1a)Smax=A-Blg⁡N,where A and B are coefficients. And fatigueS-N equations under T-C-C constant amplitude loading at various lateral pressures were analyzed using the linear regression method proposed below:(1b)Smax=-0.0519lg⁡N+0.8388σ2fc=0.25;σ3fc=0.25,(1c)Smax=-0.0427lg⁡N+0.6642σ2fc=0.25;σ3fc=0.50,(1d)Smax=-0.0412lg⁡N+0.5582σ2fc=0.50;σ3fc=0.50,where Smax is the maximum stress level of fatigue, N is fatigue life, fc is uniaxial compressive strength of a concrete prism, σ1 is the cyclic tensile stress, σ2 and σ3 are lateral compressive stresses, σ3<σ2<σ1, and the compressive stress is negative but tensile stress is positive. The correlation coefficients R2 of the above equations are 0.945, 0.965, and 0.948, respectively. Figure 3 shows the corresponding S-N scattergram.Table 2
Test data of fatigue for variable amplitude.
Specimen number Fatigue life D ε 0 Specimen number Fatigue life D ε 0 n 1 n 2 μ ε n 1 n 2 n 3 μ ε 25250101 52203 24 0.45 215 25250301 52203 1004 121 1.79 255 25250102 52203 308 3.48 234 25250302 52203 1004 75 1.30 248 25250103 52203 272 3.09 238 25250303 52203 1004 13 0.64 192 25250104 52203 43 0.66 247 25250304 52203 1004 360 4.33 244 25250105 52203 640 7.01 204 25250305 52203 1004 29 0.81 232 25250201 19 816980 3.33 217 25250401 19 1004 1498232 6.24 234 25250202 19 185321 0.91 213 25250402 19 1004 300168 1.65 229 25250203 19 114847 0.64 225 25250403 19 1004 70474 0.77 203 25250204 19 91356 0.55 199 25250404 19 1004 49593 0.69 237 25250205 19 1934129 7.61 243 25250405 19 1004 109627 0.92 226 25500101 304705 110 1.86 327 25500301 304705 1060 67 1.52 308 25500102 304705 17 0.45 319 25500302 304705 1060 315 5.27 314 25500103 304705 43 0.85 313 25500303 304705 1060 16 0.74 301 25500104 304705 233 3.73 306 25500304 304705 1060 32 0.99 335 25500105 304705 59 1.09 286 25500305 304705 1060 143 2.67 327 25500201 13 2559519 1.88 310 25500401 13 1060 594174 0.89 334 25500202 13 700821 0.66 302 25500402 13 1060 1218818 130 305 25500203 13 594174 0.59 275 25500403 13 1060 289469 0.69 307 25500204 13 1767287 1.36 293 25500404 13 1060 319940 0.71 320 25500205 13 >2500000 >1.88 319 25500405 13 1060 >2500000 >2.14 261 50500101 17171 1057 2.08  50500201 112  340844 4.17 400 Note. D=∑(ni/Ni), ε0 is the ultimate strain of specimen damage. Specimen numbering methods xxyyzztt: xxyy means the two directions of lateral compressive stress ratio, where 2550 indicates that one direction of lateral compressive stress ratio σ2/fc is 0.25, another σ3/fc is 0.50; zz represents the amplitude loading condition; 01, 02, 03, and 04 are four testing loading conditions for low-high, high-low, low-medium-high, and high-medium-low, respectively; tt represents the sample number to each operating mode.Figure 3
S - N Scatter-gram.


## 3.2. Linear Fatigue Damage Analysis

Miner [17] performed pioneering work on the amplitude fatigue linear cumulative damage rule, the P-M criterion as shown below:(2)D=∑i=1kDi=∑i=1kniNi=1.In this paper, Miner experimentally verified theP-M criterion. Table 3 shows the values obtained by this work.Table 3
ValueD.
[10] [20] [21] [12] This paper Max 2.94 95.72 1.11 3.26 7.61 Min 0.36 0.01 0.21 0.38 0.31 Avg. 1.20 8.32 0.53 1.27 1.87As shown in Table3, our experimental result for the value of D is higher than predicted. However, considering factors that influence multiaxial fatigue, the values of D obtained can be considered within the normal range. Moreover, results indicate that the P-M criterion for multiaxial amplitude fatigue has certain limitations. These limitations are mainly caused by difficulty in satisfying the prerequisites of the principle, which are as follows: each cycle of the energy dissipation is equal under constant amplitude stress level, cumulative damage cannot be influenced by the past loading process, and the fatigue life of concrete would not be affected by variable amplitude loading order.


## 4. Nonlinear Damage Analysis

### 4.1. Residual Strain

Residual strains were measured during strain development along the fatigue loading direction, and results are shown in Figures4 and 5. As shown, under triaxial T-C-C constant amplitude cyclic loading, the development of residual strain of plain concrete in the fatigue direction depends mainly on lateral compressive stress and is not significantly related to the stress level. The maximum strain development of constant amplitude fatigue follows an obvious three-stage rule, which is defined by the crack initiation, crack stable expansion, and crack propagation. The ratio of these three stages has a great correlation with the lateral pressure level, especially at a high lateral pressure, and the second stage of the maximum strain development is obviously longer, while the third stage becomes shorter and the specimen rapidly deteriorates. The strain development law [18, 19] follows the same pattern for uniaxial fatigue residual strain, but the value of the fatigue residual strain is larger than uniaxial. Under variable amplitude fatigue loads, the fatigue residual strain is not associated with the loading process. Development of fatigue residual strain is stable whether under variable amplitude loading or constant amplitude loading. The same is true for specimens loaded in the uniaxial direction.Figure 4
Relationship of residual strain and relative fatigue cycle in T-C-C constant amplitude loading.
(a)
σ 2 / f c = 0.25 ;   σ 3 / f c = 0.25 (b)
σ 2 / f c = 0.25 ;   σ 3 / f c = 0.50 (c)
σ 2 / f c = 0.50 ;   σ 3 / f c = 0.50Figure 5
Relationship of residual strain and relative fatigue cycle in T-C-C variable amplitude loading.Based on Figure4, regression equations under different lateral stress ratios were analyzed using a nonlinear regression method for residual strain εr and relative fatigue cycle n/N as shown:(3a)εr=109.0nN3-220.0nN2+164.6nNσ2fc=0.25,σ3fc=0.25,(3b)εr=173.9nN3-328.1nN2+236.0nNσ2fc=0.25,σ3fc=0.50,(3c)εr=246.7nN3-442.9nN2+302.7nNσ2fc=0.50,σ3fc=0.50.Correlation coefficientsR2 of the above equations are, respectively, 0.976, 0.976, and 0.943, all the values of R2 are greater than 0.90, and it indicates that Formulas (3a), (3b), and (3c) are suitable to express the fatigue T-C-C failure criterion. If relative fatigue cycle n/N=1, the extreme fatigue residual strain εr0 of concrete is, respectively, 53.6μ, 81.8μ, and 106.5μ. The extreme residual strain εr0 can be used as a criterion to judge the fatigue damage of plain concrete in corresponding lateral stress ratio.


### 4.2. Fatigue Damage Model and Validation

According to the basic concepts of damage mechanics and the development laws of residual strain on PC under T-C-C and uniaxial fatigue loading, the fatigue damage was defined as the ratio between the residual strain of concreteεr and the extreme fatigue residual strain εr0,εr/εr0. The damage equation is given by(4)D=εrεr0=anN3+bnN2+cnN,where a,b,c are coefficients of the damage equation and relate to fatigue loading method and extreme fatigue residual strain. The relationships between the fatigue damage D and relative number n/N of plain concrete under T-C-C loading are shown in Figure 6. Above all, the proposed fatigue damage model can be used for multistage fatigue cumulative damage analysis of plain concrete in uniaxial and T-C-C loading. This analysis can also be used to predict remaining fatigue life of the sample.Figure 6
Relationship between damage and relative fatigue cyclesn/N in T-C-C loading case.The equivalent damage valueDe and residual fatigue life prediction value N2e or N3e of PC at the lateral compressive stress ratio of 0.50 under T-C-C loading are shown in Table 4; n20 is the fatigue life of the residual strain under two-stage loading; n10/N1e and n20/N2e are the corresponding relatively equivalent fatigue life of εr1andεr2; N2e (N3e) is the remaining fatigue life prediction value (including n20); De is the equivalent damage value, De=n1/N1e+n2/N2e is the remaining fatigue life.Table 4
Variable amplitude fatigue life of prediction.
Specimen number D e N 2 e Specimen number D e N 3 e 25250101 0.88 48 25250301 1.44 109 25250102 1.37 241 25250302 0.94 93 25250103 0.88 303 25250303 0.93 26 25250104 0.91 52 25250304 1.29 293 25250105 1.09 479 25250305 1.06 47 25250201 0.89 1037225 25250401 1.51 2113472 25250202 0.91 136286 25250402 0.76 343910 25250203 0.87 140203 25250403 1.11 102601 25250204 1.01 67159 25250404 1.00 44838 25250205 1.36 1548276 25250405 0.72 84152 25500101 0.80 168 25250301 0.84 55 25500102 0.91 32 25250302 1.27 376 25500103 0.98 61 25250303 1.53 29 25500104 1.29 274 25250304 0.85 38 25500105 0.82 74 25250305 1.28 146 25500201 0.97 2350852 25250401 1.09 572043 25500202 1.13 549963 25250402 1.45 1635711 25500203 0.78 458977 25250403 1.22 335639 25500204 1.33 2245031 25250404 0.97 579068 25500205   25250405 50500101 1.55 916 50500201 0.98 490733 50500102 1.76 330 50500201 1.92 382615 50500103 0.67 137 50500201 1.35 24700 50500104 0.65 542 50500201 0.77 82501 50500105 1.04 2134 50500201 0.63 56662 Note. De is equivalent damage value; N2e/N3e is the residual fatigue life prediction values.As shown in Table4, samples tested with variable amplitude fatigue loading had a maximum equivalent damage value De of 1.92, a minimum of 0.65, and an average of 1.08. In comparison, the maximum damage value D of the P-M criterion (Table 3) is 7.61, the minimum is 0.31, and the average is 1.87 (not including the residual fatigue life of specimens with more than 2.5 million cycles). Meanwhile, the remaining fatigue life prediction values N2e (two-grade fatigue) or N3e (three-grade fatigue) are relatively close to the actual remaining fatigue life n2 or n3 in Table 2. Figure 7 takes N2e, N3e and n2, n3 logarithmic, makes lg⁡N2e(lg⁡N3e) ordinate, makes lg⁡n2(lg⁡n3) an abscissa, and converts the remaining fatigue life prediction values and the corresponding actual values into a scatter plot. If the scatter points are near the diagonal line on both sides, the forecast and experiment values of theory are satisfactory. As shown, the damage model proposed here is sufficiently accurate.Figure 7
Comparison between predicted residual fatigue life and observed ones.Based on the residual strainεr and relative fatigue cycles n/N of (3a), (3b), and (3c), the regression equation under different lateral stress ratios was analyzed using nonlinear regression as shown below:(5)D=-1.24σ2fc+0.04σ3fc+2.60nN3+1.82σ2fc+1.34σ3fc-5.32nN2+-0.58σ2fc-1.39σ3fc+3.72nN.Using (5), Figure 8 shows the relationships between the damage variable D and lateral compressive stress of σ2/fc, σ3/fc as well as the spatial relationships in different relative cycle ratios of fatigue (n/N=0.2,0.4,0.6,0.8). The figure axes are the damage variable D and the lateral compressive stress ratios σ2/fc, σ3/fc, respectively.Figure 8
Relationship between damageD and σ2/fc,σ3/fc.
(a)
n / N = 0.2 (b)
n / N = 0.4 (c)
n / N = 0.6 (d)
n / N = 0.8


## 4.1. Residual Strain

Residual strains were measured during strain development along the fatigue loading direction, and results are shown in Figures4 and 5. As shown, under triaxial T-C-C constant amplitude cyclic loading, the development of residual strain of plain concrete in the fatigue direction depends mainly on lateral compressive stress and is not significantly related to the stress level. The maximum strain development of constant amplitude fatigue follows an obvious three-stage rule, which is defined by the crack initiation, crack stable expansion, and crack propagation. The ratio of these three stages has a great correlation with the lateral pressure level, especially at a high lateral pressure, and the second stage of the maximum strain development is obviously longer, while the third stage becomes shorter and the specimen rapidly deteriorates. The strain development law [18, 19] follows the same pattern for uniaxial fatigue residual strain, but the value of the fatigue residual strain is larger than uniaxial. Under variable amplitude fatigue loads, the fatigue residual strain is not associated with the loading process. Development of fatigue residual strain is stable whether under variable amplitude loading or constant amplitude loading. The same is true for specimens loaded in the uniaxial direction.Figure 4
Relationship of residual strain and relative fatigue cycle in T-C-C constant amplitude loading.
(a)
σ 2 / f c = 0.25 ;   σ 3 / f c = 0.25 (b)
σ 2 / f c = 0.25 ;   σ 3 / f c = 0.50 (c)
σ 2 / f c = 0.50 ;   σ 3 / f c = 0.50Figure 5
Relationship of residual strain and relative fatigue cycle in T-C-C variable amplitude loading.Based on Figure4, regression equations under different lateral stress ratios were analyzed using a nonlinear regression method for residual strain εr and relative fatigue cycle n/N as shown:(3a)εr=109.0nN3-220.0nN2+164.6nNσ2fc=0.25,σ3fc=0.25,(3b)εr=173.9nN3-328.1nN2+236.0nNσ2fc=0.25,σ3fc=0.50,(3c)εr=246.7nN3-442.9nN2+302.7nNσ2fc=0.50,σ3fc=0.50.Correlation coefficientsR2 of the above equations are, respectively, 0.976, 0.976, and 0.943, all the values of R2 are greater than 0.90, and it indicates that Formulas (3a), (3b), and (3c) are suitable to express the fatigue T-C-C failure criterion. If relative fatigue cycle n/N=1, the extreme fatigue residual strain εr0 of concrete is, respectively, 53.6μ, 81.8μ, and 106.5μ. The extreme residual strain εr0 can be used as a criterion to judge the fatigue damage of plain concrete in corresponding lateral stress ratio.


## 4.2. Fatigue Damage Model and Validation

According to the basic concepts of damage mechanics and the development laws of residual strain on PC under T-C-C and uniaxial fatigue loading, the fatigue damage was defined as the ratio between the residual strain of concreteεr and the extreme fatigue residual strain εr0,εr/εr0. The damage equation is given by(4)D=εrεr0=anN3+bnN2+cnN,where a,b,c are coefficients of the damage equation and relate to fatigue loading method and extreme fatigue residual strain. The relationships between the fatigue damage D and relative number n/N of plain concrete under T-C-C loading are shown in Figure 6. Above all, the proposed fatigue damage model can be used for multistage fatigue cumulative damage analysis of plain concrete in uniaxial and T-C-C loading. This analysis can also be used to predict remaining fatigue life of the sample.Figure 6
Relationship between damage and relative fatigue cyclesn/N in T-C-C loading case.The equivalent damage valueDe and residual fatigue life prediction value N2e or N3e of PC at the lateral compressive stress ratio of 0.50 under T-C-C loading are shown in Table 4; n20 is the fatigue life of the residual strain under two-stage loading; n10/N1e and n20/N2e are the corresponding relatively equivalent fatigue life of εr1andεr2; N2e (N3e) is the remaining fatigue life prediction value (including n20); De is the equivalent damage value, De=n1/N1e+n2/N2e is the remaining fatigue life.Table 4
Variable amplitude fatigue life of prediction.
Specimen number D e N 2 e Specimen number D e N 3 e 25250101 0.88 48 25250301 1.44 109 25250102 1.37 241 25250302 0.94 93 25250103 0.88 303 25250303 0.93 26 25250104 0.91 52 25250304 1.29 293 25250105 1.09 479 25250305 1.06 47 25250201 0.89 1037225 25250401 1.51 2113472 25250202 0.91 136286 25250402 0.76 343910 25250203 0.87 140203 25250403 1.11 102601 25250204 1.01 67159 25250404 1.00 44838 25250205 1.36 1548276 25250405 0.72 84152 25500101 0.80 168 25250301 0.84 55 25500102 0.91 32 25250302 1.27 376 25500103 0.98 61 25250303 1.53 29 25500104 1.29 274 25250304 0.85 38 25500105 0.82 74 25250305 1.28 146 25500201 0.97 2350852 25250401 1.09 572043 25500202 1.13 549963 25250402 1.45 1635711 25500203 0.78 458977 25250403 1.22 335639 25500204 1.33 2245031 25250404 0.97 579068 25500205   25250405 50500101 1.55 916 50500201 0.98 490733 50500102 1.76 330 50500201 1.92 382615 50500103 0.67 137 50500201 1.35 24700 50500104 0.65 542 50500201 0.77 82501 50500105 1.04 2134 50500201 0.63 56662 Note. De is equivalent damage value; N2e/N3e is the residual fatigue life prediction values.As shown in Table4, samples tested with variable amplitude fatigue loading had a maximum equivalent damage value De of 1.92, a minimum of 0.65, and an average of 1.08. In comparison, the maximum damage value D of the P-M criterion (Table 3) is 7.61, the minimum is 0.31, and the average is 1.87 (not including the residual fatigue life of specimens with more than 2.5 million cycles). Meanwhile, the remaining fatigue life prediction values N2e (two-grade fatigue) or N3e (three-grade fatigue) are relatively close to the actual remaining fatigue life n2 or n3 in Table 2. Figure 7 takes N2e, N3e and n2, n3 logarithmic, makes lg⁡N2e(lg⁡N3e) ordinate, makes lg⁡n2(lg⁡n3) an abscissa, and converts the remaining fatigue life prediction values and the corresponding actual values into a scatter plot. If the scatter points are near the diagonal line on both sides, the forecast and experiment values of theory are satisfactory. As shown, the damage model proposed here is sufficiently accurate.Figure 7
Comparison between predicted residual fatigue life and observed ones.Based on the residual strainεr and relative fatigue cycles n/N of (3a), (3b), and (3c), the regression equation under different lateral stress ratios was analyzed using nonlinear regression as shown below:(5)D=-1.24σ2fc+0.04σ3fc+2.60nN3+1.82σ2fc+1.34σ3fc-5.32nN2+-0.58σ2fc-1.39σ3fc+3.72nN.Using (5), Figure 8 shows the relationships between the damage variable D and lateral compressive stress of σ2/fc, σ3/fc as well as the spatial relationships in different relative cycle ratios of fatigue (n/N=0.2,0.4,0.6,0.8). The figure axes are the damage variable D and the lateral compressive stress ratios σ2/fc, σ3/fc, respectively.Figure 8
Relationship between damageD and σ2/fc,σ3/fc.
(a)
n / N = 0.2 (b)
n / N = 0.4 (c)
n / N = 0.6 (d)
n / N = 0.8


## 5. Discussion

So far, there have been a lot of researches on uniaxial and multiaxial fatigue of concrete subjected to variable amplitude compression, while little attention has been paid to biaxial and triaxial tension and compression fatigue. The main reason is that the concrete specimen is difficult to achieve repeated tension-compression loading, and the testing key technology is not complete.(1)
For the multiaxial fatigue test, the physical centering of the sample is very important in the process of installation and testing and is more difficult than in a triaxial fatigue test. Poor alignment will result in specimen failure and consequently failure of the experiment. During testing, specimens were placed in the appropriate position and visually aligned before applying a 10 kN static preload in the vertical direction. Moreover, a larger static load was applied to the specimens to center the samples horizontally. Finally, the sample was preloaded in the same proportions more than three times in three directions to verify centering in the test apparatus. Preload levels were then increased to 20~30 kN to ensure contact between the loading plate and the specimen. Next, specimens were held in this centered position by gradually and evenly tightening eight screws at the upper and lower edges of the specimen. Fatigue tests were suspended after a dozen cyclic loads, to analyze the output of the deformation data in each direction and verify consistency. Finally, a small hammer was used to tap specimens or load heads as needed to adjust mechanical alignment through rotation of the globe hinge of the specimen in order to obtain valid data.(2)
As the testing system uses electrohydraulic servo control technology, the system is easily self-excited. Even the original MTS single-axis fatigue testing machine can appear self-excited during testing. Here, self-excitation refers to the system suddenly operating at a different frequency compared to the load frequency during normal operation or the ability to adjust the loading frequency to produce substantial vibration. To avoid mechanical resonance, the experiment should begin at a lower frequency running for a period of time, so that the direction of the installation of the specimen can be verified. The frequency can be smoothly increased to the ideal test frequency. The system has load control and displacement control in two directions; thus, each direction should be treated similarly to avoid self-excitation. Finally, the test system must be completely independent of the power supply to eliminate interference with other electrical equipment.


## 6. Conclusions

Based on analysis of experimental results from the development of fatigue strain, the following conclusions can be drawn below.(1)
The development of fatigue residual strain in the fatigue loading direction of PC under triaxial T-C-C constant amplitude cyclic loading is mainly dependent on the lateral compressive stress ratio and is mostly independent of stress level. The fatigue residual strain under variable amplitude fatigue loading was found to be associated with the relative fatigue cycle and lateral compressive stress ratio but is independent of the loading process. Strain development is as stable as the constant loading amplitude.(2)
The residual strainεr and fatigue relative cycle n/N were used to develop a nonlinear regression model. The ultimate residual strains εr0 of concrete fatigue damage are, respectively, 53.6μ, 81.8μ, and 106.5μ. This can be used as a criterion to judge the fatigue damage of plain concrete in corresponding lateral stress ratios.(3)
Relative residual strain was defined as the damage variant, a damage evolution for plain concrete was proposed, and models for damage analysis and prediction of fatigue life were executed. The failure models developed can be used to predict the residual remaining life of plain concrete under multiaxial fatigue loads. Model predictions and the measured results are in good agreement.(4)
It is proved that Miner’s rule is not applicable to multiaxial loading of plain concrete and has certain limitations for calculating multiaxial amplitude fatigue.



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*Source: 9351820-2017-02-21.xml*