Deformation of concrete under high-cycle fatigue loads in uniaxial and eccentric compression

Construction and Building Materials 141 (2017) 379–392

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Deformation of concrete under high-cycle fatigue loads in uniaxial and eccentric compression Chao Jiang, Xianglin Gu ⇑, Qinghua Huang, Weiping Zhang State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, 1239 Siping Rd., Shanghai 200092, PR China Department of Structural Engineering, College of Civil Engineering, Tongji University, 1239 Siping Rd., Shanghai 200092, PR China

h i g h l i g h t s
Uniaxial and eccentric compressive fatigue tests were conducted on concrete.
Elastic modulus showed negligible decrease with fatigue load cycles.
Strains and curvatures continuously accumulated under fatigue load cycles.
Deformation prediction model proposed for concrete under high-cycle fatigue loads.
Effects of fatigue on long-term deformations of concrete structures were studied.

a r t i c l e

i n f o

Article history: Received 27 November 2016 Received in revised form 28 February 2017 Accepted 4 March 2017

Keywords: Concrete High-cycle fatigue Creep Sectional analysis Long-term deformation

a b s t r a c t This paper studies the deformation evolution of concrete under high-cycle fatigue loads. First, uniaxial and eccentric compressive fatigue loads were exerted on prism concrete specimens to observe the mechanical properties of concrete under fatigue loading. Fatigue tests showed that elastic modulus does not always decrease, but strains always increase as loading cycles accumulate. Moreover, strains on the cross-section of each eccentrically fatigued specimen always maintain linear distributions. Based on these experimental findings, a simplified constitutive model for concrete under high-cycle fatigue loads was adopted; hence, a fatigue deformation prediction model was developed to analyze the strain and stress distributions on a cross-section under both cyclic axial forces and bending moments. The proposed model demonstrated its validity by predicting fatigue deformations in good agreement with experimental results. Finally, based on the new prediction model, a case study was conducted, which found that fatigue could pose a big influence on the long-term deformation of concrete bridges. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Concrete bridges have been widely constructed in China along with the rapid development of highway and high-speed railway networks. To meet the huge requirements of faster long distance transportation systems, higher speed limits are becoming more and more common in both highway and rail transportation. Conceivably, these concrete bridges must bear more fatigue load cycles than in the past given the same service lifetime period. These concrete bridges are constructed based on rigorous requirements designed to resist deformation. A reliable concrete transportation infrastructure is needed to keep goods and traffic moving without ⇑ Corresponding author at: State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, 1239 Siping Rd., Shanghai 200092, PR China. E-mail addresses: [email protected] (C. Jiang), [email protected] (X. Gu), [email protected] (Q. Huang), [email protected] (W. Zhang). http://dx.doi.org/10.1016/j.conbuildmat.2017.03.023 0950-0618/Ó 2017 Elsevier Ltd. All rights reserved.

construction detours and delays. Thus, the effect of fatigue creep on long-term deformation in concrete bridges has drawn significant attention. Many researchers have investigated the fatigue creep behavior of concrete and proposed various approximate empirical formulas [1–8] and theoretical models [9], among which some were even mutually contradictory. Generally accepted theories of fatigue creep were lacking until 2014 when Bazant and Hubler established a fatigue creep model, which they described as being ‘‘anchored in the microstructure and would allow extrapolation to 100-year lifetime” [10]. They created that model by first assuming that microcracks of concrete under fatigue loading propagated in the forms of tensile cracks or compressive crushing bands as shown in Fig. 1. Although very good agreement was achieved between their model calculations and experimental results, their assumption necessitates an experimental basis to make the fatigue creep model more convincing. In fact, based on Bazant and Hubler’s

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C. Jiang et al. / Construction and Building Materials 141 (2017) 379–392

(a) Tensile stress

Propagation

crack

Tensile stress

(b) Compressive stress

Propagation

Crushing band

Compressive stress

Fig. 1. Propagation of microcracks of concrete under fatigue: (a) tensile crack and (b) compressive crushing band.

derivations, propagation of Mode-I-type tensile cracks (Fig. 1a) or compressive crushing bands (Fig. 1b) will yield zero creep strain in the horizontal direction, i.e., perpendicular to the stress direction. However, Taliercio and Gobbi [11–13] found that creep strains perpendicular to the loading direction do exist in concrete specimens under fatigue loads with the maximum load level higher than 0.8. On the other hand, high-cycle fatigue tests, which are designed to observe the concrete creep strains perpendicular to the loading directions with maximum load levels lower than 0.8, remain lacking. Moreover, in previous experimental studies [1–8], exclusive concerns were focused on fatigue of concrete under uniform fatigue stresses. However, a more realistic approach to concrete bridge structures is to consider concrete girders or piers response to both axial forces and bending moments which always generate stress gradients in each cross-section. While these bridges are subjected to fatigue loads, the cross-sections of the girders or piers usually undergo stress redistributions. Consequently, fatigue tests on concrete with both axial forces and bending moments are also needed in addition to those on concrete with only uniaxial forces if we want to gain a full understanding of fatigue creep of concrete and its structural influences. In this paper, both uniaxial and eccentric compressive fatigue tests were conducted on prism concrete specimens to observe mechanical properties of concrete under high-cycle fatigue loads. The experimental findings support the assumption by validating its inference made by Bazant and Hubler [10] mentioned above. Moreover, a constitutive model for concrete under high-cycle fatigue loads was adopted and hence a prediction model was established to calculate deformations of concrete sections under both cyclic axial forces and bending moments. The proposed deformation prediction model was validated by comparing model predictions with experimental results. Finally, based on the proposed fatigue deformation prediction model, a case study was conducted to observe the effects of fatigue loads on the long-term deformation of concrete bridges. 2. Experimental program

the cementing material. Granite gravel with particle sizes ranging from 5 to 16 mm and natural river sand were used as coarse and fine aggregates, respectively. The tested apparent densities of cement clinkers, coarse, and fine aggregates were 3077, 2628, and 2604 kg/m3, respectively. All specimens were cured in the curing room for 28 days, at 20 ± 2 °C with a relative humidity of 95%. After that, they were moved out of the curing room and stored in an airtight chamber into which a basin of oversaturated calcium hydroxide (Ca(OH)2) solution was put to absorb CO2 in the atmosphere, avoiding possible natural carbonation of concrete. Once a specimen was about to undergo loading tests, including static and fatigue ones, the specimen would be moved out of this chamber. 2.2. Uniaxial and eccentric compressive fatigue tests EL specimens were loaded with an identical eccentricity of 20 mm to generate both tensile and compressive stresses in the cross-section of each specimen. UC1-4 and EL1-4 specimens were subjected to axial and eccentric static loads, respectively, till failure. The average failure loads, i.e., 364.2 and 265 kN, were reputed as nominal failure loads (Pu) for UC and EL specimens, respectively. Three different fatigue load levels, i.e., the minimum and maximum load levels (Pmin/Pu and Pmax/Pu) for specimens were designed, as shown in Table 2. For UC specimens, the maximum load level was designed as a constant, i.e., 67.2%, and the minimum load level was controlled at 5%, 25%, or 45%, respectively. For EL specimens, the minimum load level was kept as constant as 10%, and the maximum load level was controlled at 60%, 70% or 80%. Different load ranges were chosen for concentrically and eccentrically loaded specimens to endow fatigue tests with diverse loading parameters. Through multiplying the nominal failure loads with the load levels each specimen was expected to undergo, the lower and upper load values for each specimen were determined. With the lower and upper load values, fatigue loads were thus exerted on each prism specimen with a sinusoidal waveform at a loading frequency of 10 Hz, through an MTS test set, as illustrated by Figs. 2 and 3 for UC and EL specimens, respectively. Fig. 4 shows that the input and applied load signals were very close to each other, which demonstrated the capability of this test set to exert sinusoidal fatigue loads on these concrete specimens at a loading frequency of 10 Hz. Meanwhile, two couples of 80-mm long strain gages were set on two opposite surfaces of each UC specimen, within the central 100-mm zone, to measure the vertical (compressive) and the horizontal (tensile) strains. Five couples of the same type strain gages were set at five positions on two opposite surfaces of each EL specimen to measure the strain distribution on the mid cross-section of each specimen, as shown in Fig. 3. Synchronously, fatigue loads were also measured by a force transducer. Moreover, two different loading cycles, i.e., 1.5 and 2.5 million, were designed for UC specimens under each expected stress level range. Two different loading cycles, i.e., 1 and 2 million, were designed for EL specimens under each anticipated load level range. As an exception, EL7 was loaded to 500,000 cycles in order to include a case with a relatively small number of loading cycles. Once the number of expected load cycles was reached, fatigue loads were unloaded to zero; in the meantime, realistic residual strains were measured by strain gages.

2.1. Materials and specimens 3. Fatigue test results and discussion A total of 24 prism concrete specimens were cast in this experiment. These included 10 axially-loaded and 14 eccentricallyloaded specimens (labeled UC and EL, respectively), which measured 100  100  300 mm. The mix proportion of concrete is shown in Table 1. Ordinary Portland cement (OPC) was used as

3.1. Stress-strain curves Fig. 5 shows typical stress-strain relationships of concrete under uniaxial compressive fatigue loading. It seems that

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C. Jiang et al. / Construction and Building Materials 141 (2017) 379–392 Table 1 Mix design of concrete. Material

Cement

Water

Fine aggregate

Coarse aggregate

Type Ratio Content (kg/m3)

#425 OPC 1 361.72

Tap water 0.53 191.71

Natural river sand 2.0 723.44

Granite gravel 3.0 1085.16

Table 2 Fatigue load information of UC and EL specimens. Fatigue load levels

Pmin (kN)

Pmax (kN)

Loading cycles

UC5 UC6 UC7 UC8 UC9 UC10 EL5 EL6 EL7 EL8 EL9 EL10 EL11 EL12 EL13 EL14

5%–67.2% 5%–67.2% 25%–67.2% 25%–67.2% 45%–67.2% 45%–67.2% 10%–60% 10%–60% 10%–70% 10%–70% 10%–70% 10%–80% 10%–80% 10%–70% 10%–70% 10%–70%

18.21 18.21 91.05 91.05 163.89 163.89 26.50 26.50 26.50 26.50 26.50 26.50 26.50 26.50 26.50 26.50

244.74 244.74 244.74 244.74 244.74 244.74 159.00 159.00 185.50 185.50 185.50 212.00 212.00 185.50 185.50 185.50

1,500,000 2,500,000 1,500,000 2,500,000 1,500,000 2,500,000 1,000,000 2,000,000 500,000 1,000,000 2,000,000 1,000,000 2,000,000 1,000,000 1,000,000 1,000,000

P(t)

25

100

100

100

P(t)

25

Nos.

100 Fig. 2. A prism concrete specimen under uniaxial compressive fatigue loading (all dimensions are in mm).

loading-unloading curves in each cycle form a closed loop. However, if carefully scrutinized, the strains under the minimum stress in the loading and unloading branches within a single cycle always have a difference of 107–109 order of magnitude. Although this strain difference is very small, while accumulating millions or billions of cycles, these strains will also generate remarkable effects. As can be observed from Fig. 5, for all specimens, loadingunloading curves could be clearly spaced by 500,000 cycles. For specimen UC6, in preliminary cycles, the loading and unloading branches of each cycle were close to each other and approximately approached a straight line that connected the upper and lower points of each cycle. With the increase of loading cycles, both the loading and unloading branches of each cycle became curved and deviated from the abovementioned straight line. However, different from those in UC6, the loading and unloading branches of each

cycle in UC8 and UC10 specimens were linear and kept close to each other at all times until they completed 2.5 million loading cycles. Moreover, unlike UC6 which showed a noticeable decrease of inclination angle of each loading-unloading cycle with the accumulation of loading cycles, UC8 and UC10 did not exhibit remarkable degradation of the inclination angle. These differences can be attributable to the differences in average stress levels and stress level ranges among these specimens. As an indicator of deformation performance, the inclination angle will be analyzed in a quantitative way in Section 3.2. 3.2. Elastic modulus To quantify the deformation properties of concrete under fatigue loads, elastic modulus and creep strain are defined in this

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C. Jiang et al. / Construction and Building Materials 141 (2017) 379–392

20

23.5 23.5

100

23

7

100

100

80

23

100

25

P(t)

Strain gages

25

Steel bearing plate

P(t)

20

Fig. 3. A prism concrete specimen under eccentric compressive fatigue loading (all dimensions are shown in mm).

3.3. Strain and curvature

300

Input load signal Applied load signal

250

Load (kN)

200

150

100

50

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time (s) Fig. 4. Comparison between the input and applied load signals.

paper as illustrated by Fig. 6. In the stress-strain curves, the slope of the straight line connecting upper and lower points of each cycle is defined as the elastic modulus Es and correspondingly the intercept of this line on the abscissa is defined as the creep strain ecrp. The elastic modulus in the first cycle is defined as the initial elastic modulus E0 of concrete. According to its definition, the elastic modulus was calculated and plotted with respect to loading cycles N, as shown in Fig. 7. Obviously, the elastic modulus of UC7-9 specimens did not show a significant decrease and almost kept its initial value even if the specimens were loaded up to 2.5 million cycles. However, the elastic modulus of UC5 and UC6 specimens displayed noticeable decreases which approached 20% at 2.5 million cycles. Thus, if the maximum stress level is maintained as a constant, the higher the minimum stress level (or the lower the stress level range), the slower the degradation will be of the elastic modulus with loading cycles.

Under the uniaxial compressive fatigue loading, the strain of concrete under either the maximum or the minimum load increased monotonously with the loading cycles, as shown in Fig. 8. The increase displayed in a two-phase way. In the initial cycles, the strains increased quickly with loading cycles. Afterwards, the increase with loading cycles nearly followed a straight line; i.e., the increase rate was almost maintained as a constant. According to many other research results [14–21], after a longrange constant-rate increase, the strains would show a sharp increase while approaching fatigue failure. However, this final sharp increase phase was not found in our tests because the maximum stresses in our experiments were generally lower than the critical stress proposed by Shah and Chandra [22] so that these specimens were far from failure even if they were loaded up to 2.5 million cycles. After the fatigue loads were unloaded to zero, both the horizontal tensile and vertical compressive residual strains of each UC specimen were measured as shown in Table 3. Obviously, the horizontal tensile residual strains were very small compared to the corresponding vertical compressive residual strains. In fact, the expansion ratios in the fatigue-damaged UC specimens did not exceed 0.12, and in most cases they were smaller than 0.05. Bazant and Hubler [10] based their theory of fatigue creep in concrete on Paris law [23] as the rationalization behind the fatigue growth of subcritical microcracks. They calculated macroscopic fatigue strains by applying fracture mechanics to the microcracks considered as either tensile or compressive (in the form of a crushing band). Thus, if the crack or crushing band is perpendicular to the fatigue load direction, the fatigue load will not generate horizontal (normal to the load direction) fatigue creep strains. This proposition agrees with our experimental finding in Table 3, which in turn demonstrates that the development of Mode I cracks or crushing bands will dominate in concrete under high-cycle fatigue loading. Under eccentric compressive fatigue loading, strains of each concrete fiber in EL specimens increased progressively with loading cycles, as illustrated by Fig. 9. Like the strains of concrete under uniaxial compressive fatigue loading, strains of each concrete fiber in EL specimens also exhibited a monotonous two-phase increase

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C. Jiang et al. / Construction and Building Materials 141 (2017) 379–392

30

UC6

25

25

20

20

| | (MPa)

| | (MPa)

30

15

UC8

15

10

10

5

5

(b)

(a) 0

0 0

2

4

6

|

8

10

12

0

14

2

4

104| 30

6

8

|

104|

10

12

14

UC10

25

| | (MPa)

20 15 10 5

(c) 0 0

2

4

6

8

|

104|

10

12

14

Fig. 5. Typical stress-strain relationships of UC specimens under fatigue loading.

1.2

1.1

1 Es

UC6

UC7

UC8

UC9

UC10

1

Es/E0

E0

1

UC5

0.9

0.8

o

crp

crp+ e

Fig. 6. Definitions of elastic modulus and creep strain.

0.7

0.6 0

0.5

1

1.5

N even if these concrete fibers have undergone cyclic stresses with time-varying stresses and stress ranges because of stress redistributions. Although strains of each concrete fiber increased continuously, the strains on the cross-section of each EL specimen always maintained a linear distribution during fatigue loading. As an example, the residual strain distributions on the cross-section of each EL specimens are shown in Table 4. Obviously, through math-

2

2.5

3

10-6

Fig. 7. Decrease of elastic modulus of concrete under uniaxial compressive fatigue loading.

ematical linear regressions of the measured residual strains with respect to their corresponding locations on each cross-section, it was found that the correlation coefficients were not less than

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15

15

UC5

12

12

Pmax

9

3

104|

Pmax Pmin

6

0

0 0

15

3

UC6

12

6

9

12

15

0 15

Pmax

3

6

9

12

15

3

6

Pmin

6

3

5

10

15

Pmax (Exp.)

20

25

20

25

Pmin

3

0 0

15

9

6

0

12

Pmax

Pmin 3

9

UC10

12

Pmax

9

6

0

15

UC8

12

9

Pmin

3

3

0

Pmax

9

6 Pmin

UC9

12

9

6

|

15

UC7

0 0

5

Pmax (Model)

10

15

20

25

0

Pmin (Exp.)

5

10

15

Pmin (Model)

N 10-5 Fig. 8. Evolution of UC specimen strains with loading cycles.

Table 3 Residual strains of UC specimens after fatigue loading. Specimen Nos.

Residual strain  106 Horizontal (tensile)

Vertical (compressive)

UC5 UC6 UC7 UC8 UC9 UC10

1.344 42.096 8.953 4.379 11.773 12.863

236.223 366.638 257.463 293.126 353.622 427.153

Expansion ratio*

0.0057 0.1148 0.0348 0.0149 0.0333 0.0301

* Expansion ratio is defined as the absolute value of the ratio of horizontal strain to the vertical strain.

0.99, showing nearly perfect linear distributions. The strain distributions of the cross-section of each EL specimen under any load value at any loading cycle are not reported here due to the length limit of this article. But it is worth mentioning that the linear distribution of strains held at all times. For each EL specimen under any load value in any loading cycle, the regressed linear function that described the distribution of strains on the cross-section could yield the curvature of the cross-section and the location of the neutral axis. That is, the slope of the linear function is the curvature j, and the intercept of the linear function on abscissa is the neutral axis location xn. Fig. 10 shows the evolution of curvatures of EL specimens with loading cycles. Coinciding with strains, the curvatures under either the maximum or minimum load also showed a two-phase increase. Fig. 11 illustrates the evolution of the neutral axis location which was measured from the tensile side. Obviously, with loading cycles accumulating, the neutral axis deviated from the tensile side gradually in a two-phase way. The change of the neutral axis location directly shows without question that the stresses redistributed during fatigue loading. In addition, the results of EL9 and EL13 did not fit in the overall view of results compared with EL6 and EL11 or rather EL12 and

EL13, as shown in Figs. 9–11. That the results of EL9 and EL13 did not fit in the overall view of results compared with EL6 and EL11 was mainly because the load ranges were different between EL9 (EL13), EL6 and EL11, as shown in Table 2. Meanwhile, that the results of EL9 and EL13 did not fit in the overall view of results compared with EL12 and EL13 was mainly caused by the randomness of concrete under fatigue. 4. Deformation prediction model for concrete under high-cycle fatigue loading 4.1. Simplified constitutive model for concrete under high-cycle fatigue loading To calculate the fatigue deformation of concrete, a simplified constitutive model for concrete under high-cycle fatigue loading was adopted as illustrated by Fig. 12. The stress-strain relationship in each cycle was simplified as a straight line; i.e., the loading and unloading branches were supposed to be linear and totally overlapped. Deviations among the loading branch, the unloading branch and the supposed straight line in each cycle could be obviously observed in some fatigue tests. However, such deviations were generally small especially in high-cycle fatigue tests which involve generally lower stress levels and ranges. It was, thus, believed that such trivial details could be neglected without significantly jeopardizing model accuracy. In such a simplified constitutive model, the fatigue creep strain ecrp, elastic modulus Es and fatigue life Nf are three key parameters. Given a certain concrete under a constant-amplitude fatigue load at a constant loading frequency, fatigue life is always a function of the maximum stress rmax and the minimum stress rmin (or stress ratio which is defined as rmin/rmax), or equivalently, as a function of the average stress rm and the stress range Dr (or stress amplitude Dr/2) [24–26]. While in the cases where fatigue creep makes sense, concrete usually does not fail during loading cycles; hence, fatigue life is not as important as elastic modulus and creep

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10

10

EL8

5

0

0

0

-5

-5

-10

-10

-10

-15

-15

-15

-20

-20

-20

-25

-25

10

2

4

6

8

10

-25 0

10

EL6

5

2

4

6

8

10

EL9

5

0

10

0

0

0

-5

-5

-10

-10

-10

-15

-15

-15

-20

-20

-20

-25

-25

10

4

8

12

16

10

EL12

5

5

10

15

0

0

-5

-5

-10

-10

-10

-15

-15

-15

-20

-20

-20

-25

-25 4

6

8

10

8

10

10

15

20

EL14

5

0

2

6

5

10

EL13

5

0

20

-5

0

4

-25 0

20

2

EL11

5

-5

0

EL10

5

-5

0

104

10

EL5

5

-25 0

2

4

6

8

10

0

2

4

6

8

10

10

EL7

5 0

V2

V1

V4

V3

V5

-5

Exp.

-10 -15

Model

-20 -25 0

1

2

3

4

5

N

10 -5

Fig. 9. Strain evolutions of EL specimens with loading cycles (under Pmax).

Table 4 Residual deformations of EL specimens after fatigue loading. Nos.

EL5 EL6 EL7 EL8 EL9 EL10 EL11 EL12 EL13 EL14

Residual Strains  106 V1

V2

V3

V4

V5

28.567 44.799 70.525 79.810 259.433 135.331 236.566 244.678 181.596 40.961

34.180 38.870 30.873 34.782 33.946 9.539 9.794 9.263 0.138 19.518

100.608 117.751 125.812 131.249 212.058 221.120 248.439 169.677 174.980 117.987

167.863 203.965 216.142 249.346 451.829 467.312 534.909 329.082 347.650 219.318

229.350 271.905 298.682 353.980 667.903 643.652 746.253 533.924 502.431 312.169

strain. Both the fatigue creep strain ecrp and elastic modulus Es could be expressed as a function of the maximum stress, the minimum stress and loading cycles, or equivalently as a function of the average stress, the stress range and loading cycles, as shown by the following two equations:

ecrp ¼ g crp ðrmin ; rmax ; NÞ ¼ f crp ðrm ; Dr; NÞ

ð1Þ

Residual curvature (106/mm)

R2

2.79 3.43 3.96 4.64 10.05 8.65 10.69 8.05 7.36 3.89

1.00 0.99 0.99 0.99 1.00 0.99 0.99 0.99 0.99 0.99

Es ¼ g E ðrmax ; rmin ; NÞ ¼ f E ðrm ; Dr; NÞ

ð2Þ

The specific forms of functions gcrp, fcrp, gE and fE vary among experimental findings on different concrete specimens as reported by different researchers [17,18,27,28]. Moreover, significant scatter results always exist in fatigue tests on concrete specimens with the same compositions done by the same researcher.

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35

35

EL5

30 25

15

10

10

Pmin

5

15 Pmin

35

2

4

6

8

10

5 0 0

35

EL6

30

2

4

6

8

Pmin

5

5

5

0

0

35

4

8

EL12

30

12

16

20

4

EL13

30

8

12

16

20

0

25

25

20

20

15 10

10

5

5

5

0 0

35

2

4

6

8

10

8

12

16

20

8

10

Pmax

15

Pmin

10 0

4

EL14

30

Pmax

20 Pmin

Pmin

35

25 15

10

0 0

35 Pmax

8

Pmax

15 10

0

6

25

10

Pmin

4

20

15

10

2

EL11

30 Pmax

20

15

0 35

25 Pmax

20

10

EL9

30

25

Pmin

10

0 0

Pmax

20

5

0

106 /mm

25

20

15

EL10

30 Pmax

25 Pmax

20

35

EL8

30

Pmin

0 0

2

4

6

8

10

0

2

4

6

EL7

30 25

Pmin

Pmax

Pmax

20

Exp.

15 10

Model

Pmin

5 0 0

1

2

3

4

5

N 10-5 Fig. 10. Evolution of curvatures of EL specimens with loading cycles.

30 max

25

E0 (

xn (mm)

Es

1

20

1

crp+ e)

15 min

10

5

Pmin:

EL6

EL9

EL11

Pmax:

EL6

EL9

EL11

O

0.5

1

N

1.5

crp+ e

Fig. 12. A simplified constitutive model for concrete under high-cycle fatigue loading.

0 0

crp

2

10-6

Fig. 11. Evolution of the neutral axis location with loading cycles.

4.2. Sectional analysis of concrete under high-cycle fatigue loading Based on the simplified constitutive model, the stresses and strains on a cross-section during fatigue loading could be analyzed.

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C. Jiang et al. / Construction and Building Materials 141 (2017) 379–392

Z

Fig. 13 shows stress and strain distributions on a representative concrete cross-section within a certain cycle during fatigue loading. As the average stress and stress ranges vary along the depth of the cross-section, the fatigue creep strains along the crosssection depth caused by stress cycling usually do not maintain a linear distribution. Accordingly, while the loads are unloaded to zero, fatigue creep strains (ecrp(z)) along the cross-section depth cannot maintain their consistency; hence, residual elastic strains (ere(z)) are generated to maintain the linear distribution of residual strains (er(z)) on the cross-section. The residual elastic strains will yield self-equilibrated residual stresses that will never produce axial force and bending moment on this cross-section. Thus, unlike the residual strain in a uniformly fatigue-damaged concrete specimen (which is the same as the corresponding fatigue creep strain), the residual strain of each concrete fiber in an eccentrically fatiguedamaged specimen is comprised of the corresponding fatigue creep strain and the residual elastic strain, as illustrated by Fig. 13b. If the cross-section is reloaded again, elastic strains increase and, hence, generate stresses on the cross-section to bear the axial force and bending moment, as illustrated by Fig. 13c. If the original point of coordinate z is set at the centroid of the cross-section as shown in Fig. 13a, the strains at any location z on the cross-section can be calculated by the following equation:

K 11 ¼

eðzÞ ¼ eð0Þ  j  z

(

ð3Þ

where e(0) denotes the strain at the zero point and j is the curvature of the cross-section. Elastic strains can be expressed by the following equation:

ee ðzÞ ¼ eðzÞ  ecrp ðzÞ ¼ eð0Þ  j  z  ecrp ðzÞ

ð4Þ

Correspondingly, the stresses on the cross-section can be calculated as:

rðzÞ ¼ Es ðzÞ  ee ðzÞ ¼ Es ðzÞ  ðeð0Þ  j  z  ecrp ðzÞÞ

ð5Þ

Force and moment equilibriums are then expressed as follows: R ct ( R ct rðzÞ  bðzÞ  dz ¼ c Es ðzÞ  ½eð0Þ  j  z  ecrp ðzÞ  bðzÞ  dz ¼ P cb b R ct R ct rðzÞ  bðzÞ  z  dz ¼ cb Es ðzÞ  ½eð0Þ  j  z  ecrp ðzÞ  bðzÞ  z  dz ¼ M cb

ð6Þ The above equation can be rewritten into the following matrix form:



K 11

K 12

K 21

K 22







F1 eð0Þ ¼ j F2

 ð7Þ

cb

Es ðzÞ  bðzÞ  dz Z

K 12 ¼ K 21 ¼  Z K 22 ¼  Z F1 ¼

cb

Z F2 ¼

ct

ct cb

ct

ct cb

ð8Þ

Es ðzÞ  bðzÞ  z  dz

ð9Þ

Es ðzÞ  bðzÞ  z2  dz

cb

ð10Þ

Es ðzÞ  ecrp ðzÞ  bðzÞ  dz  P

ð11Þ

Es ðzÞ  ecrp ðzÞ  bðzÞ  z  dz  M

ð12Þ

Solving Eq. (7), we can obtain e(0) and j as follows:







K 11 K 12 eð0Þ ¼ j K 21 K 22

1 

F1



ð13Þ

F2

If the degradation of elastic modulus can be neglected, i.e., Es(z)  E0, Eq. (13) can be rewritten into the following form:

R

ct eð0Þ ¼ A1  c ecrp ðzÞ  bðzÞ  dz  E0PA b R c j ¼  1I  ct b ecrp ðzÞ  bðzÞ  z  dz þ EM0 I

ð14Þ

where I and A are the moment of inertia and area of the crosssection, respectively. Substituting e(0) and j into Eqs. (3) and (5), we can obtain the strains and stresses on the cross-section under any normal load and bending moment in this loading cycle. From Eqs. (1) to (14), the stresses and strains on the crosssection in a single cycle are determined. Fatigue cycles influence strain and stress distributions on the cross-section through affecting the elastic modulus and fatigue creep strain of each concrete fiber between adjacent cycles. As both the average stress and stress range of each concrete fiber vary along the cross-section depth, the decrease of elastic modulus or the increase of fatigue creep strain of each concrete fiber along the cross-section depth usually does not coincide with one another. Consequently, stresses on the cross-section usually redistribute during fatigue loading. That is, both the average stress and stress range of each concrete fiber will change with the loading cycles during the fatigue process. Thus, the elastic modulus and fatigue creep strain of each concrete fiber should be updated cycle by cycle considering the average stress and stress range changes between adjacent cycles. To do so, an ‘‘incremental method” was proposed here as illustrated by Fig. 14.

z

where K11, K12, K21, F1 and F2 can be calculated by the following equations:

ct

ct

crp(z)

re(z)

r(z)

r(z)

P

r(0)

O

Centroid

M

P=0 M=0

cb

r

(b) Residual strain and stress distributions

(a) Cross-section crp(z)

e(z)

(z)

(z) (0)

P

P 0 M M 0

(c) Strain and stress distributions under loading Fig. 13. Stress and strain distributions on a representative concrete cross-section: (a) cross-section, (b) residual strain and stress distributions, and (c) strain and stress distributions under loading.

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C. Jiang et al. / Construction and Building Materials 141 (2017) 379–392

In short, the ‘‘incremental method” aims to calculate the N + 1th elastic modulus [Es(z)]N+1 and fatigue creep strain [ecrp(z)]N+1 based on the Nth elastic modulus [Es(z)]N, fatigue creep strain [ecrp(z)]N, average stress [rm(z)]N and stress range [Dr(z)]N of the concrete fiber at the location z. Usually, if the average stress and stress range are maintained as constants, the fatigue creep strain and elastic modulus can be expressed as functions of the average stress, stress range and loading cycles (Eqs. (1) and (2)). It is supposed that from the Nth to N + 1th cycle, the decrease of elastic modulus and the increase of the fatigue creep strain are controlled by Eqs. (1) and (2) with the Nth average stress [rm(z)]N and stress range [Dr (z)]N. However, in the cycles before N, the average stress and stress range are different from those in the Nth cycle. Hence, before we use Eqs. (1) and (2) to calculate [Es(z)]N+1 and [ecrp(z)]N+1, we have to first calculate the equivalent loading cycles as follows:

 1  NE ðzÞ ¼ f Es ½rm ðzÞN ; ½DrðzÞN ; ½Es ðzÞN   1 Ncrp ðzÞ ¼ f crp ½rm ðzÞN ; ½DrðzÞN ; ½ecrp ðzÞN

Input Data Fatigue loads: Pmin, Pmax, Mmin, Mmax, fr Cross section: A, I, h, cb, ct, b(z)

N=1

Solve Eq.(13) or (14): [εmax(0)]N; [κmax]N; [εmin(0)]N; [κmin]N

ð15Þ

Solve Eq.(5): [σmin(z)]N; [σmax(z)]N

Substituting the above equivalent loading cycles into Eqs. (1) and (2), we then could obtain the N + 1th elastic modulus and fatigue creep strain by the following equation:

8   < ½Es ðzÞNþ1 ¼ f Es ½rm ðzÞN ; ½DrðzÞN ; NE ðzÞ þ 1  : ½ecrp ðzÞNþ1 ¼ f crp ½rm ðzÞN ; ½DrðzÞN ; Ncrp ðzÞ þ 1

Expected cycles (time): Nexp (texp) Concrete properties fc ′ , E0

Initial conditions: [Es(z)]1=E0; [εcrp(z)]1=0;

N=N+1

(

Start

Solve Eq.(15): NE*(z); Ncrp*(z)

Solve Eq.(16): [Es(z)]N+1; [εcrp(z)]N+1

ð16Þ

Yes

In such a way, the elastic modulus and fatigue creep strain can be updated cycle by cycle.

N Nexp or t

texp

No

4.3. Calculation process

Output Data Es(N;z); εcrp(N;z); σ(N,P,M;z); ε(N,P,M;z); κ(N,P,M)

The calculation process of sectional analysis under fatigue is illustrated by Fig. 15. Prior to solving Eqs. (13) or (14), the integrals in Eqs. (8)–(12) are numerically calculated by the compositetrapezoidal rule. Eq. (13) can be easily solved by the chasing method. Eq. (15) is a root finding problem of nonlinear equations, which can be solved by the Newton-downhill method. As a result, by inputting fatigue loads, cross-section parameters, concrete properties and expected cycles, the elastic moduli, fatigue creep strains, stresses, strains and curvature on the cross-section under any load values at any loading cycles can be output finally.

End Fig. 15. The sectional analysis flowchart of concrete under high-cycle fatigue loading.

loads with low stress levels and stress level ranges (see Fig. 7). Moreover, in the most unfavorable case, the decrease ratio of elastic modulus of concrete could only reach 20% at 2.5 million fatigue cycles. That is, before the 2.5 million cycles, the decrease ratio of elastic modulus was even smaller. In addition, according to uniaxial tensile fatigue tests on concrete conducted by Saito and Imai [19] and Fu et al. [29], tensile fatigue did not noticeably change the elastic modulus. As such, while using the proposed model to

5. Validation of the fatigue deformation prediction model According to our fatigue tests, elastic modulus of concrete did not always noticeably decrease with loading cycles accumulating. This was especially obvious when concrete underwent fatigue

Es

crp

Es(z)=fEs([ m(z)]N, [

(z)]N; N)

crp(z)=fcrp([ m(z)]N,

[

[Es(z)]N [Es(z)]N+1

O

NE*(z)+1 N+1

NE*(z)=fEs-1([ m(z)]N, [

(z)]N; [Es(z)]N)

(a) Elastic modulus

Cycles

(z)]N; N)

crp(z)]N+1

[

NE*(z) N

[

crp(z)]N

O

Ncrp*(z) N *

Ncrp*(z)+1 N+1

(Ncrp (z)=fcrp-1([ m(z)]N,

[

(z)]N; [

Cycles crp(z)]N))

(b) Fatigue creep

Fig. 14. An incremental method for calculating elastic modulus and fatigue creep under different average stresses and stress ranges.

C. Jiang et al. / Construction and Building Materials 141 (2017) 379–392

calculate the fatigue deformations in our experiments, we accepted that elastic modulus maintains its initial values at all times; i.e., Es(z)  E0 = 3  104 N/mm2 (the average value of all UC specimens). This consistency in elastic modulus values contributes greatly to the calculation efficiency. Usually, fatigue creep strain ecrp can be calculated by the following formula:

ecrp ¼ rm  C tot

ð17Þ

in which rm is the time-average stress (N/mm ) and Ctot denotes the total creep compliance (MPa1). Furthermore, the total creep compliance can be divided into two components: 2

C tot ¼ C sta þ DC cyc

ð18Þ

where Csta is the static creep compliance of concrete under a static stress equal to rm, and DCcyc is the corresponding creep compliance increment caused by stress cycling. Csta and DCcyc usually have the following forms [6,10]:

C sta ¼ a0  ta1

DC cyc

Dr ¼ b0  0 fc

ð19Þ !b1  N b2

ð20Þ

in which, t is the loading duration (hours), N is the loading cycles; Dr denotes the stress range (N/mm2), f0c is concrete strength (N/ mm2), and a0, a1, b0, b1 and b2 are fitting or calibrating parameters. For the cyclic loading program, N = 3600fr∙t, in which fr is the loading frequency (Hz). Substituting Eqs. (18)–(20) into Eq. (17), we can obtain the fatigue creep strain in the following form: !b1

a1 N Dr ecrp ¼ a0  rm  þ b0  rm   N b2 ð21Þ 0 3600f r fc In this paper, the exponents a1 and b2 are set as 1/3, as reported by Whaley and Neville [6]. Whaley and Neville [6] considered b1 as 1, while Bazant and Hubler [10] thought that b1 was affected by the stress level and it was equal to 4 if the maximum stress level was below 0.4. In this paper, considering the stress levels that the concrete has undergone covers a wide range, b1 is set as 2. The coefficients a0 is calibrated by our test results as 1.0975  106. The coefficient b0 is calibrated by our test results as 0.4224  106 or 0.2571  106 when the maximum stress level is below or above 0.5. This was confirmed by Whaley and Neville [6], who found that creep evolution rate transitioned at the stress level 0.5. The values of these parameters are summarized in Table 5. In addition, since no experimental information could be found to report creep of concrete under uniaxial tensile fatigue loading, we accepted, as suggested by Bazant and Hubler [10], that under tension the fatigue creep per unit stress is at least as large as it is under compression. Once the elastic modulus and fatigue creep strain were determined, the evolving deformations of concrete under uniaxial and eccentric fatigue loading could be calculated. Fig. 8 compares the model-predicted strains with experimental points in UC specimens. In most cases, strain evolutions predicted by the proposed model were in good agreement with test results. In some cases, especially in UC6 and UC10, the predicted strains deviated from the experimental points noticeably during the last loading cycles. The high randomness of concrete under fatigue was attributable to these deviations. Moreover, in most cases, the initial strains of UC5-10 specimens were overestimated by the proposed model as illustrated by Fig. 8. This is because in our model calculations, we used a unified E0 value, i.e., 3  104 N/mm2, which is the average value of all UC specimens including UC1-4 specimens undergoing static tests and UC5-10 specimens undergoing fatigue tests. According to static and fatigue tests, UC1-4 specimens usually dis-

389

played smaller initial elastic modulus values due to their lower loading rates than UC5-10 specimens. As a result, in our model calculations, the initial elastic modulus values for UC5-10 specimens might have been underestimated. Hence, the initial strains of UC5-10 specimens under Pmin and Pmax were overestimated. Fig. 9 compares the model-predicted strains and experimental results in EL specimens. Obviously, the model predictions agree well with the experimentally measured strains. In most cases, the model-predicted strains (absolute values) were slightly smaller than the experimental ones. This is mainly because we neglected the possible degradation of elastic modulus during fatigue at each concrete fiber. Thus, under the same load value, the model calculated strains on concrete with its initial elastic modulus were generally smaller than the experimentally measured strains on concrete with a decreasing elastic modulus. Furthermore, the randomness of concrete was possibly responsible for some noticeable deviations between model predictions and experimental results in EL9 and EL13 specimens. Moreover, Fig. 10 also compares the curvatures predicted by the proposed fatigue deformation prediction model with the curvatures measured in the experiments. Considering the high scatter degrees usually found in fatigue test results, the model predictions here are in acceptable agreement with test results. Fig. 16 compares the model-predicted residual strains with corresponding experimental results. For UC specimens, the modelpredicted residual strains (absolute values) are generally smaller than the experimental ones, while for EL specimens the modelpredicted residual strains are almost uniformly distributed below and above the equality line. In general, all these points are clustered around the equality line, showing acceptable agreement between model predictions and experimental results. Fig. 17 compares the model-predicted residual curvatures with the corresponding experimental results. Generally, the model predictions of residual curvatures are in acceptable agreement with experimental results. 6. Case study Here we use the fatigue deformation prediction model to study the evolutions of residual strains and curvatures at certain points and sections in a representative railway bridge in mainland China during its service lifetime. 6.1. Project description Fig. 18 illustrates the spans of the prestressed continuous RC girder and the dimensions of the mid-span and support sections. The bridge was designed for 100 years of service in Shanghai of China to support the high-speed train running there. Some geometry features are shown in Table 6. The compressive strength of concrete is 38 N/mm2. The concrete tensile strength is taken as 10% of its compressive strength. The initial elastic modulus of concrete is 32,500 N/mm2. The fatigue loads on the concrete part of the two cross-sections are shown in Table 6. In Table 6, PD and MD denote the axial force and the bending moment borne by the concrete part of each cross-section induced by dead loads (including the prestressing force); ML is the bending moment borne by the concrete part of each cross-section induced by live loads. The initial stress distributions on the mid-span and support crosssections are also shown in Fig. 18. It was supposed that the fatigue loads on the concrete part of each cross-section maintained their initial values shown in Table 6 during its service lifetime. The loading frequency is set as 20,000 cycles per year so that the bridge will have borne 2 million cycles at the end of its anticipated service lifetime. In addition, this case study neglected the effects of reinforcements in each cross-section and ignored the prestress loss

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Table 5 The parameter values for calculating the fatigue creep strain. Cases

a0

r r

a1 6

0 max 0.5fc 0 max > 0.5fc

1.0975  10 1.0975  106

1/3 1/3

4

Equality line 2

10 4 (Model)

0

-2

UC5~10 EL6 EL8 EL10 EL12 EL14

r

-4

-6

-8 -8

-6

-4

-2 r

0

EL5 EL7 EL9 EL11 EL13 2

4

4

10 (Exp.)

Fig. 16. Comparison between experimentally measured residual strains and model predicted residual strains.

12

106 /mm (Model)

6

0.4224  10 0.2571  106

b1

b2

2 2

1/3 1/3

residual strain evolutions at the bottom and top of both the midspan and support cross-sections and the curvature evolutions of the two cross-sections, as illustrated by Figs. 19 and 20, respectively. At all four points, i.e., the top and bottom of sections 1–1 and 2–2, the residual strains increased progressively during the service lifetime. However, the increase rates of residual strains were different among the four points. The residual strain at the top of the mid-span develops fastest and can reach 1.45  103 at the end of the service lifetime. Meanwhile, the residual strain at the bottom of the mid-span evolves the most slowly and can approach 1  104 ultimately. The curvatures of both the midspan and support sections, under the possible maximum load, increase monotonously with service durations, while the curvature increase rate at the mid-span section (1–1) is always faster than that at the support section (2–2). Moreover, the curvature of section 1–1 under the maximum load at the end of the service lifetime could reach about 4 times that at the initiation time of service. Although the case study may have overestimated the contribution of fatigue creep to the total deformation as we have neglected the effects of reinforcements on lowering the force and moment borne by the concrete part of the cross-section, it does not necessarily exaggerate the total deformation, since we have also ignored the loss of prestressing forces due to fatigue creep of concrete. As a result, the calculations show that fatigue can pose a big influence on the long-term deformation of concrete bridge structures.

7. Conclusions

9

r

b0

Equality line

6

3

0 0

3

6 r

9

12

106 /mm (Exp.)

Fig. 17. Comparison between residual curvatures predicted by the proposed model and those calculated by experimentally measured strains.

due to fatigue creep. In fact, fatigue creep will increase the axial force and bending moment borne by the reinforcements and simultaneously decrease those borne by the concrete part on each cross-section, which will gradually decrease fatigue creep deformations of concrete. Similar to the work done by Bazant and Hubler [10], this favorable influence of reinforcements was not considered because this case study was designed to evaluate the most unfavorable effect of fatigue creep of concrete on the longterm deformation of the concrete bridge. 6.2. Deformation evolutions of the bridge during its service lifetime According to the basic information on the bridge, we used the proposed fatigue deformation prediction model to calculate the

This paper investigated the deformation evolution of concrete under high-cycle fatigue loads. Uniaxial and eccentric compressive fatigue tests were conducted on prism concrete specimens to observe the mechanical properties of concrete under fatigue loading. Test results showed that elastic modulus did not always decrease, but strains always increased with loading cycles accumulating. Moreover, strains on the cross-section of each eccentrically fatigued specimen maintained linear distributions at all times. A simplified constitutive model for concrete under high-cycle fatigue was then adopted. Based on this constitutive model, a fatigue deformation prediction model was further established, which is able to analyze the strain and stress distributions on a crosssection under fatigue loads with both axial forces and bending moments. The deformations predicted by the proposed model were in good agreement with experimental results. Finally, based on the proposed model, a case study was conducted and it was found that fatigue can have a significant influence on the longterm deformation of concrete bridges. It is worth mentioning that in our model calculations we accepted the constant elastic modulus assumption. According to the acceptable agreement between experimental and modelpredicted results in this paper, we recommend that the constant elastic modulus assumption is acceptable in the cases where concrete undergoes fatigue loads with load levels lower than 0.8. If the concrete undergoes higher-level fatigue loads, this assumption should be reexamined. Possibly, a representative mathematical formula that connects the elastic modulus of concrete with fatigue loading parameters should be established in the future based on experimental results of concrete undergoing higher-level fatigue loads.

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C. Jiang et al. / Construction and Building Materials 141 (2017) 379–392

1

2 Prestressed continuous RC girder

1

2 Pier

Pier

Main span: 135 m

75000

67500

14400

0~-0.2f c

+

L

680

D

cb

D

1000

0.045f c~0

680

7500

D

260

400

-0.35~-0.38f c

cb

400

2500

D

260

260

14400

+

L

8000 -0.2437~-0.4f c

8000

1-1 (Unit: mm)

2-2(Unit: mm) Fig. 18. A representative concrete bridge built in China.

Table 6 Geometry parameters and load information on the concrete part of the mid-span and support cross-sections. A (mm2)

Sections 1–1 2–2

I (mm4) 12

7.3871  10 1.5867  1014

7,408,000 20,230,400

cb (mm)

PD (kN)

MD (kNm)

ML (kNm)

1501.547 3289.962

59115.840 172612.724

39257.940 924.687

8412.416 286428.295

4

5

2 0

4

-4

(t)/ (0)

104

-2

r

-6 -8

3

2

-10 1-1(Bottom) 1-1(Top) 2-2(Top) 2-2(Bottom)

-12 -14 -16 0

20

40

1-1

1

2-2

60

80

100

tS (year)

0 0

20

40

60

tS (year)

80

100

Fig. 19. Residual strain evolutions at the top and bottom of mid-span and support sections of the bridge during its service lifetime.

Fig. 20. Evolutions of curvatures under the possible maximum loads at the midspan and support sections of the bridge during its service lifetime.

The proposed fatigue deformation prediction model could be easily updated to consider the effects of reinforcing steels on lowering the concrete’s share of force and moment on each crosssection and the loss of prestressing forces due to fatigue creep of concrete during the service lifetime. The updated model could be further inserted into a multi-scale model that aims to predict long-term deformations of concrete bridges considering not only stress redistributions on each cross-section but also force and moment redistributions among different cross-sections during a

bridge’s service lifetime. Moreover, the parameters in Eq. (21) should be recalibrated with field data collected from structural health monitoring of concrete bridges in order to make the model predictions more realistic. In addition, the sensitivity analysis of the updated model and uncertainty analysis will be further conducted in order to identify the robustness of the updated model. These are future projects on the work schedule of the authors’ research group.

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Acknowledgements This research work was financially supported by the National Natural Science Foundation of China (Grant No. 51320105013) and the National Basic Research Program of China (973 Program) (Grant No. 2015CB655103).

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