Cohesive fracture and probabilistic damage analysis of freezing–thawing degradation of concrete

Construction and Building Materials 47 (2013) 879–887

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Cohesive fracture and probabilistic damage analysis of freezing–thawing degradation of concrete Pizhong Qiao ⇑, Fangliang Chen Department of Civil and Environmental Engineering, Washington State University, Sloan Hall 117, Pullman, WA 99164-2910, USA

h i g h l i g h t s  Degradation of concrete due to freeze/thaw action is characterized by cohesive fracture test.  The relationship between damage and the number of F/T cycles is established using the nonlinear regression analysis.  Probabilistic damage model is established to predict the cyclic freeze/thaw life of concrete.

a r t i c l e

i n f o

Article history: Received 3 February 2013 Received in revised form 7 April 2013 Accepted 4 May 2013 Available online 15 June 2013 Keywords: Degradation Durability Freezing and thawing Long-term performance Probabilistic damage model

a b s t r a c t The durability of concrete with low-degradation aggregates due to cyclic freezing and thawing effect is experimentally studied by characterizing the variance of fracture energy with respect to the number of freeze/thaw (F/T) cycles. Cohesive fracture test is conducted for notched concrete beams subjected to different F/T cycles, and the fictitious crack model-based approach is employed to calculate the fracture energy from the testing data. The relationship between the relative fracture energy and the number of F/T cycles is established using the nonlinear regression analyses. Based on the three-parameter Weibull distribution model, the probabilistic damage analysis is conducted, and the life distribution diagrams are produced according to the probability of reliability/survival concept. The relationships between the life (i.e., the number of F/T cycles) and damage parameter for different probabilities of reliability are obtained, from which the service life of concrete due to cyclic freezing and thawing actions can be determined at any given reliability index. The validation and accuracy of the present models are demonstrated through comparisons between the predicted data by the present models and the test data. The present probabilistic damage model can serve as a reference for maintenance, design and life prediction of concrete structures with low-degradation aggregates in cold regions subjected to cyclic freezing and thawing actions. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Significant transportation infrastructure in North America is located in regions with severe environmental conditions, where alternate freezing and thawing can seriously affect the material and structural integrity. The durability of Portland cement concrete (PCC) has long been identified as a concern by transportation communities across the United States. Degradation of aggregates in concrete can be caused by erosion or fracture, and both cementitious materials and aggregates age over time. The erosion process produces a material of poorer quality compared to the parent aggregate. Degradation becomes more pronounced in marine basalt materials [1]. In marine basalt materials, aggregates can degrade into plastic fines. Failure of aggregates in a bituminous

⇑ Corresponding author. Tel.: +1 509 335 5183; fax: +1 509 335 7632. E-mail address: [email protected] (P. Qiao). 0950-0618/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conbuildmat.2013.05.046

matrix may be caused by inferior mineral content and is manifested first by softening of the mixture followed soon afterward by actual disintegration of the matrix. Basalt rock of the Eocene age and gravels associated with the Eocene basalt are most apt to exhibit this harmful degradation. Though aggregate degradation of marine basalt materials is well recognized and understood, very limited information can be found in the literature on the long-term performance of concrete made with low-degradation aggregates. Durability degradation due to cyclic freezing and thawing is one of the major damage aspects in cementitious materials and structures in cold regions. The basic mechanisms of this type of frost damage in cement-based materials were first studied by Powers and his co-workers [2–4]. During the cyclic freezing and thawing actions, the concrete is subjected to loading in the freezing period and undergoes unloading in the thawing process. From the kinetic point of view, the damage of concrete under successive freeze/ thaw (F/T) cycles is due to hydrostatic and osmotic pressure, which

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is a pure physical process [3]. In this sense, freezing–thawing damage of concrete can be viewed as a type of fatigue damage accumulation of the internal hydrostatic pressure and osmotic pressure acting on concrete [5]. Because of the heterogeneous nature in material components as well as the inherent variability in material properties of concrete, probabilistic approaches are usually utilized to include the uncertainty of various parameters in a consistent manner. Thus, probabilistic modeling of the deterioration mechanisms in concrete structures has gained more and more applications during the past few years [6,7]. It is thus feasible to use probabilistic model-based service life design methodologies for structural evaluations. The ability to assess with confidence the potential freezing and thawing durability of concrete is important for long term evaluations of in situ concrete [8]. In the literature, the deterioration of concrete in rapid freezing and thawing testing (ASTM C666 procedure A or B) is normally measured as changes of geometrical and material properties after different amounts of F/T cycles, such as variations of length [9], resonance frequency and its corresponding relative dynamic modulus [8,10–12], and electrical resistivity change [13–15]. Alternatively, as a characteristic material property, the specific fracture energy has proved to be a useful parameter for design with concrete and cementitious materials [16–19]. However, to the authors’ best knowledge, none of existing studies has been conducted to explore the degradation of fracture energy for concrete structures due to F/T cycles. In this study, the long-term performance of concrete with a specified low-degradation aggregate due to freezing and thawing effect is experimentally studied by characterizing the variance of fracture energy with respect to the number of freeze/thaw (F/T) cycles. Based on the three-parameter Weibull distribution model, a probabilistic damage model is proposed in terms of the varying fracture energy with respect to different F/T cycles, and the cyclic F/T life (in the term of the number of F/T cycles) distribution diagrams are produced according to the probability of reliability/survival concept. The relationships between the cyclic F/T life and damage parameter for different probabilities of reliability are explicitly established, from which the service life of concrete with the considered low-degradation aggregates due to cyclic freezing and thawing actions can be determined at any given reliability index.

2. Materials and experimental procedure 2.1. Materials and mix design In this study, the cement used is Portland cement (Type I–II). Coarse low-degradation (LD) aggregates were provided by Washington State Department of Transportation (WSDOT), and the specific gravity and water absorption of coarse aggregate alone is 2.68 and 1.2%, respectively. The specific gravity of fine aggregates is 2.65. According to the aggregate degradation review conducted by KBA, Inc. [20], aggregates with a degradation factor less than the critical value of 35 are commonly considered as a low-degradation (LD) one. In this study, the identified LD aggregate sources with a degradation factor of 31 was used. The degradation factor for the LD aggregates considered in this study was obtained by the WSDOT using the WSDOT Test Method T 113. The gradations of the coarse LD aggregates and fine aggregate are presented in Table 1. The mix design employed in this study was based on the WSDOT guidelines for a 27.6 GPa (4000 psi) mix (WSDOT Concrete Mix Performance Guideline (4000D)), which is summarized in Table 2.

Table 1 Coarse aggregate gradations (sieve analysis). Sieves

LD aggregate Cumulative % passing

Fine aggregate Cumulative % passing

1/20 0 3/80 0 1/40 0 #4 #8 #16 #30 #50 #100 #200

100 98.5 67.8 37.3 3.0 0.4 – – – –

– 100 99.5 97.7 84.3 61 42.2 17.7 4.1 2.2

2.2. Sample preparations In accordance with ASTM C192/C192 M, the fresh concrete was cast in oiled wood molds to form prisms with 76.2  101.6  406.4 mm (3  4  16 in.), prisms with 101.6  101.6  406.4 mm (4  4  16 in.) and cylinders with 152.4 mm (6 in.) (diameter)  304.8 mm (12 in.) (height) were, respectively, cast for the flexural strength and compression strength tests. Immediately after the casting, all the samples were externally vibrated for approximately 15 s and finished using a metal trowel. All specimens were demolded after 24 h and then cured in lime-saturated water at room temperature for at least 28 days before subsequent tests.

2.3. Freezing–thawing experimental program Test Method for Resistance of Concrete to Rapid Freezing and Thawing (ASTM C 666), Procedure A, which was designed to provide an indication of the potential durability of concrete in a freezing and thawing environment, was adopted in this study to condition all the samples. It provides a relative assessment of the frost resistance of concrete after a given number of F/T cycles, compared to the initial condition of the specimens. The F/T machine used in this study consists of 18 containers, in which 17 samples can be subjected to specified F/T cycles, together with a particular container for the control sample, which controls to produce continuously and automatically reproducible cycles with the specified temperature (see Fig. 1). An ‘‘S’’ shape stainless steel wire with diameter of 2.0 mm (3/32 in.) is placed in the bottom of the containers before the samples laying into them, so that: (1) each sample will be always completely surrounded by enough water while it is being subjected to cyclic freezing and thawing actions and (2) the temperature of the heat-exchanging medium will not be transmitted directly through the bottom of the container to the full area of the bottom of the samples. During the conditioning period, each group of samples were kept to exchange their conditioning positions in the freeze–thaw machine in certain sequences such that all samples in the same group were conditioned as much consistently and evenly as possible. According to ASTM C666, Procedure A, the temperature range of every freezing and thawing cycle in this study was set to cycle between 17.8 °C (0°F) and 4.4 °C (40°F) as specified in ASTM C666. The difference between the temperatures at the center of a specimen and that at its surface was continuously monitored and controlled, not to exceed 10 °C (50°F). Usually, the temperature range in the samples was from 17.8 °C (0°F) to 4.4 °C (40°F) with an error of less than 1.11 °C (2°F) and a cycle frequency of 6–8 cycles per day. Whenever the temperature reaches either lower than 19.4 °C (3°F) or higher than 6 °C (43°F), the F/T machine was turned off and the control sample was replaced by a new one to help maintain the temperature in the standard range.

2.4. Cohesive fracture test The determination of fracture energy was performed following the procedure as recommended in [16,20], where a three point bending beam (3PBB) method is recommended. Before the fracture test, the conditioned samples were cut with notches at the mid-span of the sample by a diamond saw with high accuracy. In order to keep the maximum bending moment as lower as possible when the applied load reaches its ultimate value, a rather deep notch is recommended [16]. In this study, the depth of the notch was adopted as the half of the depth of the sample. After the cut, the mass of each notched sample was then measured by a digital scale with res-

Table 2 Existing WSDOT mix designs. Mixtures

Cement kg/m3 (lb/yd3)

19 cm (3/40 0 ) Aggregate kg/m3 (lb/yd3)

Sand kg/m3 (lb/yd3)

Water/cement ratio

LD-WSDOT

334.6 (564)

1085.8 (1830)

753.5 (1270)

0.48

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The general idea of the fracture test for the notched concrete samples is to measure the amount of energy absorbed when the sample is broken into two halves. The fracture energy GF is then obtained through dividing the absorbed energy by the fracture area (projected on a plane perpendicular to the tensile stress direction) [21]. It should be noted that, during the test, the concrete sample is subjected to not only the externally-applied load, but also the sample’s own weight. Consequently, the measured load–deflection curve does not reflect the real absorbed energy from the applied load. To eliminate the effect of the sample weight on the absorbed energy from the measurement, a weight compensation method was adopted by some researchers [17,19] to measure the fracture energy of cementitious materials using three-point bend tests, where two blocks with the same weight were symmetrically hanged near the end of the test sample such that the central moment due to the sample weight are compensated by the moment resulted from the introduced blocks. Theoretically, it is possible to eliminate the effect of the sample’s self-weight on the measured fracture energy by applying this weight compensation method. However, in reality, the weight of different samples differs from each other, which makes it very inconvenient to frequently change the mass of the balancing blocks. In this study, the sample self-weight influence on the measured fracture energy is alternatively eliminated by using the method proposed in [21]. Referring to the typical measured load–deflection curve (Fig. 3b), the remaining parts of a hypothetical complete load–deflection curve are shown in the dashed lines. The additional load P1 is the self-weight of the sample, and it is excluded in the measured load P. The total amount of absorbed energy W then becomes:

Fig. 1. Freezing and thawing conditioning. olution up to 9.005 N (0.001 lbs). The notched samples were then well stored from losing any moisture during the storage or transportation process until the fracture test was conducted. All the fracture tests were performed on an MTS servo hydraulic testing machine (see Fig. 2a). The experiment was conducted under displacement-controlled mode for a loading rate of 6 mm (0.0236 in.)/min. In this test, a high resolution linear variable differential transducer (LVDT) was used to measure the loading point deflection of the test sample. The LVDT setup is enlarged and illustrated in Fig. 2b, where the tip of the LVDT’s core was placed on a short ‘‘L’’ shape steel strip which was tightly clipped to the concrete sample at the loading position. The measurements of applied load and mid-span displacement (MSD) were automatically and continuously recorded through the commercial data acquisition software ‘‘LabVIEW’’. The loading application sketch of the test sample is shown in Fig. 3a, and a typical load–deflection curve is illustrated in Fig. 3b. According to the test recommendation [16], the data recording was terminated when the applied load decreased to 66.7 N (15.0 lbs), where all the samples are believed to be completely fractured, and continued acquisition data are not needed for later data reduction [21].

W ¼ W0 þ W1 þ W2

ð1Þ

where W0 is the area below the measured load–deflection curve; W1 = P1d0 is the work done by the sample’s self-weight, where d0 is the deflection when P = 0 (cut point) and the sample breaks into halves (the termination of the test); and W2 is the area below the tail of the load–deflection curve beyond the cut point. It was demonstrated that W2 is approximately equal to W1 [22]. Therefore, the fracture energy can be determined by:

GF ¼

W 0 þ 2P1 d0 Alig

ð2Þ

where Alig ¼ bðh  cÞ is the fractured area of the sample.

Fig. 2. MTS servo hydraulic testing machine for fracture test: (a) test set-up and (b) enlarged sketch for LVDT set-up.

P

P

Loading Roller

h c W0

Supporting Roller a

a L

(a)

b P1

δ0 W1

δ W2

(b)

Fig. 3. (a) Three point bendig beam specimen and (b) typical load–deflection curve from the fracture energy (GF) test.

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Table 3 Material properties of fresh and hardened concrete. Concrete type

Slump mm (in.)

Air content (%)

Young’s modulus GPa (106 psi)

Compressive strength MPa (psi)

Flexural strength MPa (psi)

Fresh Hardened

101.6 (4.0) –

4.8 –

– 23.4 (3.4)

– 30.6 (4,432)

– 5.16 (748)

3. Experimental results Slump and air content tests were performed to evaluate the workability and durability properties of fresh concrete. Both the slump and air content tests were conducted three times. The average slump and air content are reported in Table 3. According to ASTM C 39/AASHTO T22 and ASTM C78/AASHTO T97, the Young’s modulus, compressive strength, and flexural strength were measured at the 7th and 28th days, respectively. The averaged test data for the Young’s modulus, compressive strength, and flexural strength are reported in Table 3. Typical fracture test phenomenon for the notched fracture sample was observed through the load vs. displacement curves as presented in Fig. 4. Initially, the displacement increases linearly with the applied load. During this period, the elastic strain energy stored in the specimen increases until the stored elastic energy is equal to the energy required to initiate the crack. The crack initiation is characterized by the arrival of the peak load. The subsequent loading declining with the extension of the crack represents the softening region. During this softening period, the stored elastic strain energy decreases along with the applied load with the crack extension until the test sample is fully broken into two halves. Based on the data reduction technique given in the previous section, the fracture energy for different samples with different numbers of F/T cycles are obtained and given in Table 4, where the sequence of the data for each group are arranged from minimum to maximum for the convenience of later data reduction pro-

cedures. Due to the containing limit of the F/T conditioning machine, three unequal size groups (6, 6, and 5 samples for each group) of samples were conditioned at each time. Every group of samples is randomly selected and marked from the total of 200 samples cast at one time. In this fashion, a total of nine groups of beam samples which were conditioned up to 60, 120, 180, 240, 300, 500, 700, 900 and 1500 F/T cycles, respectively, were tested. Including one group of six virgin samples (i.e., 0 cycle), there are totally 57 samples being tested in fracture, and the fracture energy of all the tested samples are shown in Table 4. Unavailable data of samples listed in Table 4 are those accidently failed during notch preparations and those underwent catastrophic failure without softening stage recorded during fracture test. To better visualize the effect of freezing and thawing on the fracture energy, the variance of the fracture energy with respect to the number of F/T cycles is shown in Fig. 5. It is worth noting that the average fracture energy for samples with 60 F/T cycles is higher than those of virgin (unconditioned) samples. This phenomenon might probably be due to toughened effect of ingress moisture gained in the conditioned samples at the beginning of freezing stage. Nevertheless, after 60 F/T cycles, the average fracture energy of conditioned samples decreases with the increase of F/T cycles, which reflects that the damage in the concrete samples is accumulated using the freezing–thawing conditioning protocol (ASTM C666). On the other hand, it demonstrates that the fracture energy test is capable of probing the material degradation by the F/T conditioning cycles.

4. Statistical analysis and probabilistic damage model 4.1. Statistic regression analysis

Fig. 4. Load–displacement curve of notched fracture specimen under 3-point bending.

Due to the inherent variability in material properties of concrete and the corresponding scatterings of the test data, a statistical analysis is necessary to better interpret the test results. According to the fracture energy of different samples conditioned at the different numbers of F/T cycles in Table 4, the nonlinear polynomial regression is applied to establish the relationship between the fracture energy and the number of F/T cycles. Commercial software MINITAB is used in this study to analyze the test data. It should be noted that only the long term performance of concrete due to the F/T cycles is of interest in this study. Therefore, to better fit the test data and circumvent the convex point at the point of 60 cycles, the first group of data for virgin samples (i.e., at 0 cycle) is discarded in the statistical analysis.

Table 4 Fracture energy GF (103 N/mm) for samples with different F/T cycles. F/T cycles Samples

Average COVa a

1 2 3 4 5 6

0

60

120

180

240

300

500

700

900

1500

149.1 153.1 160.0 167.4 178.2 179.2 164.5 7.7

165.3 168.4 174.3 184.6 212.6

141.1 151.5 164.0 179.5 187.7

125.6 132.4 144.2 157.9 172.4

103.2 109.1 128.3 132.1 143.6

164.7 11.7

146.3 13.0

81.4 89.0 100.6 101.0 110.4 121.5 100.6 14.3

66.1 74.4 81.5 86.8 89.9 107.9 84.9 16.5

58.5 64.0 73.5 81.3 84.1

181.0 10.6

114.0 118.6 137.9 144.2 158.1 169.2 140.5 15.4

44.7 49.5 55.1 62.9 65.8 69.2 57.9 16.7

COV: Coefficient of Variation.

124.2 12.7

73.2 15.9

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P. Qiao, F. Chen / Construction and Building Materials 47 (2013) 879–887

Versus Fits

Fig. 5. Variance of fracture energy with respect to the number of F/T cycles.

Nomral Propability Plot

Fig. 6. Scatter plot from MINITAB for the fracture energy vs. the number of F/T cycles.

The regression analysis and scattering plot for the relative fracture energy at 95% confidence are shown in Fig. 6, where the prediction interval (PI) and confidence interval (CI) of prediction represents the ranges in which a single new observation and the mean response are likely to fall in given specified settings of the predictors, respectively. The estimated regression function for the relative fracture energy (RFE), which is defined as the ratio of fracture energy for samples with a certain number of F/T cycles (larger than 60 F/T cycles) with respect to the ones of the low-degradation aggregate concrete at 60 F/T cycles, can be expressed as: RFE ¼ 110:4  0:187N þ 1:68  104 N 2  5:29  108 N 3 for N P 60

ð3Þ

where N is the number of F/T cycles. Plots of the standardized residuals and a normal probability plot of the standardized residuals are shown in Fig. 7a and b, respectively. The plots of the relative constant residuals and the linear normal probability validate the present cubic nonlinear polynomial model as given in Eq. (3). 4.2. Probabilistic damage model As shown in Fig. 5, the cohesive fracture test is capable of probing the material degradation by the number of F/T cycles. Therefore, the probabilistic damage variable (D) of the studied low-degradation aggregate concrete due to the freezing and thawing effect is defined as:

Fig. 7. Diagnostic plots for the present cubic regression model: (a) fitted value and (b) standardized residual.

D ¼ 1  Gf ðNÞ=Gf 0

ð4Þ

where Gf and Gf0 are the fracture energy of damaged samples and that at 60 F/T cycles, respectively; D is the probabilistic damage variable. Again, the reason why the samples with the 60 F/T cycles are selected as the benchmark one is due to the phenomenon that the convex point exists at the 60 F/T cycles in the fracture energy distribution curve as shown in Fig. 5. Based on Table 4, the number of F/T cycles that each sample undergoes to reach different damage levels (D = 0.1, 0.2, 0.3, 0.4, 0.5, and 0.6, which correspond to 90%, 80%, 70%, 60%, 50%, and 40% of the residual fracture energy, respectively) can be obtained and is listed in Table 5.The data in Table 5 is an alternative way to present the data in Table 4, i.e., rather than providing the data points of fracture energies at a given cycle in Table 4 from the experimental tests directly, the five data points of F/T cycles at a given damage level (D) of fracture energy is presented in Table 5. The cyclic damage behavior of concrete with low-degradation aggregate due to freezing and thawing effect can be described by the Weibull distribution model. In this study, the kinetic law for measuring the freeze–thaw damage is assumed to be the cumulative density function through a three-parameter Weibull distribution. The probability density function for the three-parameter Weibull distribution is:

f ðNÞ ¼

 b1 b Nc

a

a

(  b ) Nc exp 

a

ð5Þ

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P. Qiao, F. Chen / Construction and Building Materials 47 (2013) 879–887

results, and c > 0 is the location parameter (or failure free life), also known as the minimum life parameter. The cumulative density function, which is regarded as the probability of failure and represents the cumulative damage parameter, can be determined by integrating the probability density function as:

Table 5 The number of F/T cycles (life) at different damage levels. Damage (D)

F/T cycles

0.1 0.2 0.3 0.4 0.5 0.6

101 154 231 337 489 700

111 173 245 380 540 833

120 190 278 397 564 867

142 225 319 437 623 1023

157 225 345 471 641 1144

FðNÞ ¼

Weibull regression results for D = 0.2

0.5 Y = -7.7344 + 1.7892*X

R2 = 0.9914

Y = ln(-ln(1-F(N)))

Y = ln(-ln(1-F(N)))

ð7Þ

where X ¼ lnðN  cÞ, Y ¼ ln½ lnð1  FðNÞÞ, A = b, and B ¼ b ln g. 1

-0.5 -1 -1.5 -2

2 0 R = 0.9623

-0.5 -1 -1.5

2

2.5

3

3.5

4

-2

4.5

3.2

3.4

3.6

Weibull regression results for D = 0.3

1

0.5

Y = -4.3847 + 1.0193*X 2

R = 0.9795

0 -0.5 -1 -1.5

4.4

4.6

4.8

R2 = 0.9918

0 -0.5 -1 -1.5

2

2.5

3

3.5

4

4.5

-2

5

4.6 4.7 4.8 4.9

5

5.1 5.2 5.3 5.4 5.5 5.6

X = ln(N- )

Weibull regression results for D = 0.5

1

1

0.5 Y = -20.1232 + 3.6267*X

Weibull regression results for D = 0.6

0.5 Y = -13.44647 + 2.1952*X

2 0 R = 0.9807

Y = ln(-ln(1-F(N)))

Y = ln(-ln(1-F(N)))

4.2

Y = -17.3122 + 3.2870*X

X = ln(N- )

-0.5 -1 -1.5 -2

4

Weibull regression results for D = 0.4

1

Y = ln(-ln(1-F(N)))

Y = ln(-ln(1-F(N)))

0.5

3.8

X = ln(N- )

X = ln(N- )

-2

ð6Þ

a

Y ¼ A þ BX

Y = -4.5648 + 1.2484*X

0

(  b ) Nc f ðNÞds ¼ 1  exp 

Eq. (6) can be written as in the following linear form based on the Weibull transformation

Weibull regression results for D = 0.1

0.5

N

0

where a is the scale parameter (or characteristic life) that locates the life distribution, b > 0 is the shape (or slope) parameter that serves as the inverse measure of the dispersion in cyclic freezing– thawing conditioning life (or in this study the number of F/T cycle)

1

Z

2 0 R = 0.9811

-0.5 -1 -1.5

5

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

-2 5.2

5.4

5.6

X = ln(N- ) Fig. 8. Weibull regression results for different damage levels.

5.8

6

6.2

X = ln(N- )

6.4

6.6

6.8

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P. Qiao, F. Chen / Construction and Building Materials 47 (2013) 879–887 Table 6 Weibull parameters for different damage levels. Parameters

Damage (D)

Characteristic life parameter a Shape parameter b Least life parameter c Correlation coefficient R2

0.2

0.3

0.4

0.5

0.6

38.72 1.2484 93 0.9914

75.41 1.7892 129 0.9623

73.83 1.0193 220 0.9795

193.82 3.2870 231 0.9918

256.89 3.6267 340 0.9807

783.71 3.9939 224 0.9971

Reliability funtion R(N) at D=0.1

1 0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

100

150

200

250

300

Reliability funtion R(N) at D=0.2

1

R (N)

R (N)

0.1

350

0

400

150

200

Reliability funtion R(N) at D=0.3

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0

250 300 350 400 450 500 550 600 650 700 750

F/T cycles (N)

Reliability funtion R(N) at D=0.5

0.8

0.8

0.7

0.7

0.6

0.6

R (N)

R (N)

0.9

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 400

500

600

700

800

Reliability funtion R(N) at D=0.6

1

0.9

0

400

250 300 350 400 450 500 550 600 650 700 750 800

F/T cycles (N) 1

350

0.5

0.4

0

300

Reliability funtion R(N) at D=0.4

1

R (N)

R (N)

1

250

F/T cycles (N)

F/T cycles (N)

900

1000

0

600

800

F/T cycles (N) Fig. 9. Reliability functions at different damage levels.

1000

1200

F/T cycles (N)

1400

1600

1800

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P. Qiao, F. Chen / Construction and Building Materials 47 (2013) 879–887

Considering the relatively-limited number of samples in this study, the least squares estimation (LSE) is adopted to evaluate the three parameters in the employed model. The median rank method is used to obtain an estimate of the unreliability for each failure, which is a nonparametric estimate of the cumulative distribution function based on the failure order. The approximation of the median ranks, also known as Benard’s approximation, is given as

BðNÞ ¼

i  0:3 n þ 0:4

RðNÞ ¼

þ1

(  b ) Nc f ðsÞds ¼ 1  FðNÞ ¼ exp 

a

N

Reliability

90% 50% 10% Mean life (MTTF) (F/T cycles)

Damage (D) 0.1

0.2

0.3

0.4

0.5

0.6

99 122 169 129

150 190 249 196

228 272 387 293

329 404 481 405

478 572 663 572

670 939 1190 934

ð8Þ

where i is the failure serial number, and n is the total number of samples in each test. Based on the LSE, the three-parameter Weibull parameters are solved through a self-developed MATLAB code. The probability of reliability, also known as the survival function, refers to the probability in which the considered concrete material exhibits its own designated functions under certain periodic F/T cycles, and it is defined as:

Z

Table 7 Cyclic F/T life (the number of F/T cycles) for different damage levels under different reliabilities.

ð9Þ

age for the accumulation of frost damage in concrete; while on the other hand they demonstrate the capability of the proposed model on predicting the durability of concrete materials subjected to cyclic freezing and thawing damage. From the life (the number of F/T cycles) for different damage levels obtained in Table 7, the relationship between the number of F/T cycles (N) and damage parameter (D) for different reliabilities can be established through the regression analysis, and the results are illustrated by the N–D curves as shown in Fig. 10. The explicit equations of the N–D curves for different reliabilities are then expressed as:

The failure rate function L(N), also known as the hazard function, enables the determination of the number of failures occurring per unit time, and it is mathematically given as

LðNÞ ¼

 b1 f ðNÞ b N  c ¼ a RðNÞ a

ð10Þ

The mean life function, which is the expected or mean time-tofailure (MTTF) and provides a measure of the average time of operation to failure, is given by

MTTF ¼

Z

þ1

t  f ðtÞdt

ð11Þ

N

The Weibull regression analysis is carried out according to Eqs. (7) and (8) and the least squares estimation for the data given in Table 5. The corresponding regression results for different damage levels are shown in Fig. 8. From the regression results shown in Fig. 8, the Weibull parameters are summarized in Table 6. It can be seen from Table 6 that all the correlation coefficients are larger than 0.96, indicating that the linear function presented in Eq. (7) qualifies for all six different damage levels and that the cyclic F/T (similar to fatigue damage) life of the studied concrete samples with low-degradation aggregate in terms of fracture energy follows the three-parameter Weibull distribution. The reliability functions R(N) for different damage levels are shown in Fig. 9. It basically indicates the probability of the considered LD aggregate concrete performance under the certain number of periodic F/T cycles until it reaches to the associated damage levels. Based on the reliability definition in Eq. (9), the variation of the relative fracture energy with respect to the F/T cycles can be viewed as the reliability function; thus, the concrete degradation trend due to the cyclic freezing–thawing damage in terms of fracture energy can be revealed via Fig. 9. Based on the probability of reliability shown in Eq. (9), the life (i.e., the number of F/T cycles) for different damage levels under different reliabilities are obtained and shown in Table 7. From Eq. (7), the expected mean lives (time-to-failure) in terms of the number of F/T cycles (N) for different defined damage levels are also shown in Table 7. It shows that the presented probabilistic damage model with 50% reliability agrees very well with the mean life predicted by Eq. (11); while as expected, the present probabilistic damage model with 95% reliability provides a bit of conservative prediction of the F/T cycles. The comparison results shown in Table 7 on the one hand validate the proposed probabilistic dam-

D Fig. 10. Probabilistic relationships between the number of F/T cycles and damage parameter.

D Fig. 11. Comparison of the predicted data based on the present model with the test data.

P. Qiao, F. Chen / Construction and Building Materials 47 (2013) 879–887

N ¼ 149:1 expð2:6DÞ  101:0 at 90% Reliability

ð12Þ

N ¼ 48:6 expð4:8DÞ þ 56:4 at 50% Reliability

ð13Þ

N ¼ 29:2 expð5:9DÞ þ 154:2 at 10% Reliability

ð14Þ

2

where the adjusted R for the regression fits (Eqs. (12)–(14)) are 0.9984, 0.9957, and 0.9874, respectively. These near 1.0 correlation coefficients of R2 validate the present model. The relationships between the number of F/T cycles and damage parameter shown in Fig. 10 are important for practical applications, since through which the remaining service life of concrete due to cyclic freezing and thawing actions can be determined at any given reliability index. This model can serve as a reference for maintenance, design and life prediction of low-degradation aggregate concrete in cold regions. To validate the present probabilistic damage model in this study, the cubic polynomial regression data based on Eq. (3) and the predicted data based on Eqs. (12) and (13) are compared with the test data, and the comparison results are shown in Fig. 11. It can be seen from Fig. 11 that excellent agreements among the data based on the present cubic polynomial equation (Eq. (3)), the probability model at 50% reliability (Eq. (13)), and the test data are reached, demonstrating the validity and accuracy of the present probabilistic damage model in predicting the life (the number of F/T cycles) of low-degradation aggregate concrete structures in cold regions subjected to cyclic freezing and thawing actions. 5. Discussion and concluding remarks In this study, the long-term performance of concrete with lowdegradation aggregates due to cyclic freezing and thawing action is experimentally studied through characterizing the variance of the fracture energy with respect to the freeze/thaw (F/T) cycles. Test Method for Resistance of Concrete to Rapid Freezing and Thawing (ASTM C 666) – Procedure A was followed to condition all the test samples. The cohesive fracture test was conducted for notched concrete beams subjected to different numbers of F/T cycles, and the fictitious crack model-based approach was employed to reduce the fracture energy from the testing data. Test results showed that the fracture energy of conditioned samples decreases with the increased number of F/T cycles, which reflects that the damage and degradation in the concrete samples can be accumulated using the F/T conditioning protocol (ASTM C666); on the other hand, they demonstrate that the cohesive fracture test is capable of probing the material degradation by the F/T cycles. Nonlinear regression analysis was applied through the commercial software MINITAB. Consequently, the relationship between the relative fracture energy and the number of F/T cycles is established. The relative constant residuals and the linear normal probability plots validate the present regression model. Based on the threeparameter Weibull distribution model, the probabilistic damage analysis is further conducted, and the cyclic F/T life distribution diagrams are produced according to the probability of reliability/ survival concept. The relationships between the cyclic F/T life and damage parameter for different probabilities of reliability are explicitly obtained, from which the service life of concrete due to cyclic freezing and thawing action can be determined at any given reliability index. It is shown that the present probabilistic damage model with 50% reliability agrees very well with the expected

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mean lives (time-to-failure) in term of the number of F/T cycles. Comparisons among the predicted data by the present model and the test data validate the proposed probabilistic damage model, demonstrating the capability of the proposed model on predicting the durability of low degradation aggregate concrete materials subjected to cyclic freezing and thawing damage. The probabilistic damage model proposed in this study can serve as a reference for maintenance, design and life prediction of concrete structures with low-degradation aggregates in cold regions while subjected to cyclic freezing and thawing action. Acknowledgements This study was financially supported by the Alaska University Transportation Center (AUTC), State of Alaska Department of Transportation & Public Facilities, and US Department of Transportation (Proposal Number: 410029; Contract/Grant No.: DTRT06-G0011) and Washington Department of Transportation (WSDOT) (Contract No.: 13A-3815-5188). References [1] Minor CE. Degradation of mineral aggregates. Symposium on road and paving materials. Philadelphia, PA: ASTM Special Technical Publication; 1959. [2] Powers TC. A working hypotheses for further studies of frost resistance of concrete. J Am Concr Inst 1945;16(4):245–72. [3] Powers TC, Helmuth RA. Theory of volume changes in hardened Portland cement paste during freezing. Highway Res Board Proc 1953;32:285–97. [4] Powers TC. Basic considerations pertaining to freezing and thawing tests. Am Soc Test Mater Proc 1955;55:1132–55. [5] Jacobsen Stefan et al. Frost durability of high strength concrete: effect of internal cracking on ice formation. Cem Concr Res 1996;26(6):919–31. [6] Li Hui et al. Flexural fatigue performance of concrete containing nano-particles for pavement. Int J Fatigue 2007;29(7):1292–301. [7] Sain Trisha, Chandra Kishen JM. Probabilistic assessment of fatigue crack growth in concrete. Int J Fatigue 2008;30(12):2156–64. [8] Rutherford JH et al. Use of control specimens in freezing and thawing testing of concrete. Cem Concr Aggr Jun. 1994;16(1):78–82. [9] Sabir BB. Mechanical properties and frost resistance of silica fume concrete. Cem Concr Compos 1997;19(4):285–94. [10] Marzouk H, Jiang DJ. Effects of freezing and thawing on the tension properties of high-strength concrete. ACI Mater J 1994;91(6):577–86. [11] Biolzi Luigi et al. Frost durability of very high performance cement-based materials. J Mater Civ Eng 1999;11(2):167–70. [12] Jun Wei et al. A damage model of concrete under freeze–thaw cycles. J Wuhan Univ Technol – Mater Sci Edition 2003;18(3):40–2. [13] Cai H, Liu X. Freeze–thaw durability of concrete: ice formation process in pores. Cem Concr Res 1998;28(9):1281–7. [14] Cao Jingyao, Chung DDL. Minor damage of cement mortar during cyclic compression, monitored by electrical resistivity measurement. Cem Concr Res 2001;31(10):1519–21. [15] Cao Jingyao, Chung DDL. Damage evolution during freeze–thaw cycling of cement mortar, studied by electrical resistivity measurement. Cem Concr Res 2002;32(10):1657–61. [16] RILEM TC-50 FMC. Détermination de l’énergie de rupture des mortiers et bétons par flexion «trois points» de poutres encochées. Mater Struct 1985;18(4):285. [17] Elices M et al. Measurement of the fracture energy using three-point bend tests: Part 3—influence of cutting the P-Delta tail. Mater Struc. 1992;25(6):327–34. [18] Elices M et al. On the measurement of concrete fracture energy using threepoint bend tests. Mater Struct 1997;30(200):375–6. [19] Qiao Pizhong, Xu Yingwu. Evaluation of fracture energy of composite-concrete bonded interfaces using three-point bend tests. J Compos Constr 2004;8(4):352–9. [20] WSDOT Agreement Number Y-8694. Aggregate Degradation Review. Final Report, prepared by KBA Inc., Olympia, WA; 2005. [21] Hillerborg A. The theoretical basis of a method to determine the fracture energy of concrete. Mater Struct 1985;18(4):291–6. [22] Petersson PE. Fracture energy of concrete – method of determination. Cement Concr Res 1980;10(1):78–89.