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Determination of fracture parameter and prediction of structural fracture using various concrete specimen types
Theoretical and Applied Fracture Mechanics 100 (2019) 114–127
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Determination of fracture parameter and prediction of structural fracture using various concrete specimen types
T
Junfeng Guana, , Changming Lia, Juan Wangb, , Longbang Qingc, Zhikai Songa, Zenpeng Liua ⁎
⁎
a
School of Civil Engineering and Communication, North China University of Water Resources and Electric Power, Zhengzhou 450045, PR China College of Water Conservancy and Environment Engineering, Zhengzhou University, Zhengzhou 450001 PR China c College of Civil Engineering, Hebei University of Technology, Tianjin 300401, PR China b
ARTICLE INFO
ABSTRACT
Keywords: Specimen type Concrete Tensile strength Fracture toughness Peak load
Concrete tensile strength and fracture toughness are two important material constants that should be independent of specimen size, geometry or type. In this study, on the basis of the improved boundary effect model, independent tensile strength and fracture toughness of concrete, which are unaffected by the specimen type, were determined by using various specimens, e.g., three-point-bending and wedge-splitting specimens. The concrete structural failure was fully predicted using independent material constants, and the constructed safe design diagrams with upper and lower limits ( ± 15%) can cover all experimental results. The theoretical minimum size of the concrete meeting the linear elastic fracture mechanics were quantified, and the peak loads of concrete specimens were predicted using various average fictitious crack growth length.
1. Introduction In many concrete test studies [1–12], even with the same concrete mixture, the dependent material parameters from experimental results (e.g., strength or fracture parameters) are obtained using different specimen sizes, geometric features, or types. However, material parameters should be constant with specimen sizes and types. These obtained inconsistent results are commonly considered as the size effect or structure behavior. For the accurate evaluation of the structural behavior of the concrete structures, having material parameters that are unaffected by the structure size, geometry, or type is necessary. Therefore, determining the independent concrete material parameters is important. Tensile strength ft and fracture toughness KIC are two important material parameters of concrete. Based on the boundary effect theory for concrete and ceramic fracture of Hu [13–19], the improved boundary effect model (BEM) was proposed to determine the ft and KIC of concrete [20–22], mortar [23], and granite [24] using three-pointbending (3-p-b) specimens. Recently, the ft and KIC of concrete was also determined using wedge-splitting (WS) specimens [25]. In principle, the ft and KIC of concrete are constant and independent and should not be affected by specimen type. Therefore, in this study, using two different specimen types of concrete, i.e., 3-p-b and WS specimens together, we simultaneously determined the independent ft and KIC of concrete. These determined values are consistent with those ⁎
individually determined by 3-p-b or WS specimens. Furthermore, the structural fracture curves or safe design diagrams of concrete were fully constructed using the determined independent material constants. Then the minimum size meeting the linear elastic fracture mechanics (LEFM) were theoretically obtained. The predicted peak loads using BEM were compared by experimental results. 2. Methodology to determine material parameters using structural behavior 2.1. Relationship between material and structural behavior The relationship between material and structural behavior is elaborated in Fig. 1. For specimens or structures that can satisfy 100% fracture toughness criterion control, regardless of the changes in their size or type, the fracture toughness KIC cannot be affected and is always constant, that is, KIC can be directly determined by the specimens or structures that can satisfy 100% fracture toughness criterion control. Similarly, if 100% strength control is satisfied, then the tensile strength ft can be directly determined regardless of the specimen type or size. However, in actual concrete structural engineering, concrete structures or specimens seldom fully meet the 100% strength or fracture toughness criterion control. Concrete structural fracture is commonly controlled by the tensile strength ft and the fracture toughness KIC simultaneously, thereby indicating that it is in the quasi-brittle fracture.
Corresponding authors. E-mail addresses: [email protected] (J. Guan), [email protected] (J. Wang).
https://doi.org/10.1016/j.tafmec.2019.01.008 Received 7 October 2018; Received in revised form 24 December 2018; Accepted 7 January 2019 Available online 10 January 2019 0167-8442/ © 2019 Elsevier Ltd. All rights reserved.
Theoretical and Applied Fracture Mechanics 100 (2019) 114–127
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feff ≠ Const ≠ ft ft
feff
n
Keff ≠ Const ≠ KIC
Keff
Structural behavior
KIC
KIC
Quasi-brittle fracture
ae =
LEFM
Log
a*
∞
ft a
1+ a e
(3)
where ae is the equivalent crack and is a structural geometry parameter, determined by the specimen size and type, and the a0. It should be pointed that influence of the front and the back boundaries on fracture of specimen, and the effect of structure size and geometry, and structure type (or loading condition) are considered by the structural parameter ae. Useful relations for 3-p-b [20–24] and WS [25] specimens are provided here.
Log σn Log ft
=
Log a
Y( ) =
Material behavior (single material constant)
(1 - )2 × Y ( ) 1.12 1.99
(1
N (P )
ae =
a0 a
=
N
= ft =
ft 1
p
b specimen for 3
p
(4a)
b specimen with S/ W= 4
× Y( )
1.12
2
·a 0
for WS specimen
(2 + )(0.866 + 4.64 4 for WS specimen
(5a)
13.32 2 + 14.72 (1 )3/2
3
5.6
4)
(5b)
where α is ratio of a0 and W, α = a0/W; S is the effective length of specimens; W is the depth of specimens. Rearrange the Eq. (3), it can be obtained:
1 2 n
=
1 1 a 1 4a + 2 · e = 2 + 2e ft2 ft a ft KIC
(6)
The details of determination the material parameters of concrete using the experimental results of limited-size specimens are as follows: Firstly, the nominal stress σn are determined by experimental structural behavior (e.g. experimental peak loads) of different specimen. Secondly, equivalent crack (structural parameter) ae are determined by the specimen type and size. Thirdly, based on Eq. (6), the independent material parameters-tensile strength ft and fracture toughness KIC are determined by curve fitting (σn and ae).
(1)
2.3. Quasi-brittle fracture of concrete considering the influence of aggregate
where σn (P) is the nominal stress from the load P considering the influence of the crack; σN (P) is the nominal stress from the load P in which the influence of the crack is not considered. a is the characteristic crack, fully determined by the tensile strength ft and the fracture toughness KIC, a = 0.25·(KIC/ft )2 . If the strength criterion dominates, the corresponding calculation model is [13–19]: n
2(1 )2 2+
Y( )=
ft
=
for 3
(4b)
The different ft and KIC proportions (e.g., 60% ft and 40% KIC; or 30% ft and 70% KIC) affect the specimen behavior, thereby showing different results. Therefore, the results from the quasi-brittle fracture specimens generally exhibit structural behavior (size effect). The boundary effect theory was originally established by Hu and Wittmann in 1992 [13], and developed by Hu and Duan from 2000 to 2010 [14–19]. Compared to the size effect theory only considering the “absolutely size W” [26,27], the boundary effect theory holds that, for specimen with a crack, the crack length a0 and the specimen size W have couple effects on the fracture of specimen. When the specimen size W is much larger than the initial crack length a0 (W > > a0), the fracture of the specimen is mainly effected by the front boundary of the specimen (a0). When the W is close to the a0 (W ≈ a0), the specimen fracture is mainly effected by the back boundary (W - a0). For these specimen in laboratory, usually a0/W = 0.1–0.7, the influence of the front and the back boundaries should be all considered. For infinite plate or large-size specimen, if the fracture toughness criterion dominates, the corresponding calculation model is [13–19]:
=
·a 0
)(2.15 3.93 + 2.7 2) (1 + 2 )(1 )3/2
Fig. 1. Material and structural behavior.
n (P )
2
The depths of concrete specimens in laboratories are generally less than 500 mm [1–12,28–37]. RILEM [38] recommended that the depth W for 3-p-b specimen of concrete or mortar is from 100 mm to 400 mm. Norm for fracture test of hydraulic concrete in China [39] recommended that the depth W for 3-p-b and WS specimen of concrete is 200 mm. The maximum aggregate size dmax of ordinary concrete is from 5 mm to 40 mm. Therefore, the ratio of the W and dmax is 5–30, and then the corresponding ratio of the ligament depth of specimen (W-a0) and the dmax is only 3–15. As illustrated in Fig. 2(a), the aggregate size cannot be ignored compared to the depth of specimen in laboratories, the heterogeneity of the concrete specimen in laboratory is obvious, the corresponding fracture of concrete specimen is non-brittle and in quasi-brittle fracture. Therefore, the maximum aggregate size dmax should be introduced in the analytical formula of the improved boundary effect model (BEM). The fictitious crack growth Δafic at peak load Pmax and its relationship with discontinuous and stepwise formation of micro cracks within the coarse aggregate structures are described [20–25]. The Δafic is linked to the coarse aggregate structures, e.g., the maximum aggregate size dmax, through a discrete number β [20–25], i.e,
(2)
For structures fully dominated by fracture toughness or tensile strength respectively, the directly determined ft and KIC from these structures are material parameters (no size effect). However, for the finite size specimen in laboratories, the results (e.g. experimental tensile strength feff and fracture toughness Keff , as shown in Fig. 1) directly determined by experiments show size effect or structural behavior, due to the fracture of the limited-size specimen being controlled by both the ft and KIC simultaneously and in the quasibrittle fracture. 2.2. Determination of material behavior using structural behavior
a fic = ·d max
Combined Eqs. (1) and (2), the calculation model of the quasi-brittle fracture for a finite size specimen is as follows [13–19]:
(7)
The discrete number β depends on the aggregate distribution and 115
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Fig. 2. Improved BEM for 3-p-b and WS specimens considered dmax.
Fig. 3. Using the trapezoidal distribution of σn for 3-p-b and WS specimens.
116
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Fig. 4. Using the triangle distribution of σn for 3-p-b and WS specimens. Table 1 Details of 3-p-b specimens in [40]. Label
α
W (mm)
B (mm)
W/dmax
(W-a0)/dmax
Pmax (kN)
un KIC (MPa·m1/2)
Keff (MPa·m1/2)
TPB500-1 TPB500-2 TPB500-3 TPB400-1 TPB400-2 TPB400-3 TPB400-4 TPB300-1 TPB300-2 TPB300-3 TPB300-4 TPB300-5 TPB200-2 TPB200-3 TPB200-4
0.53 0.46 0.55 0.46 0.46 0.47 0.47 0.47 0.47 0.46 0.48 0.49 0.47 0.48 0.46
499 500 500 401 400 396 396 298 297 298 298 298 200 200 199
200 196 198 199 196 199 197 200 195 198 200 200 200 200 199
50 50 50 40 40 40 40 30 30 30 30 30 20 20 20
24 27 22 22 21 21 21 16 16 16 16 15 11 10 11
9.81 11.93 9.85 11.13 10.97 10.39 9.96 7.91 8.32 7.84 8.39 8.29 7.19 6.23 6.64
1.265 1.389 1.191 1.223 1.205 1.330 1.346 1.289 1.074 1.038 1.166 1.198 1.317 0.963 1.110
0.81 0.82 0.90 0.84 0.84 0.81 0.78 0.70 0.76 0.69 0.76 0.78 0.77 0.70 0.71
percentage of dmax, and concrete preparation. The heterogeneity of concrete specimens of W/dmax < 30 is obvious, and even when 3-p-b or WS concrete specimens with the same geometry (same specimen depth W and same initial crack length a0) are tested, different peak loads Pmax and different fictitious crack lengths Δafic would still be obtained. The improved BEM [20–25] can consider the discrepancy of experimental results using the discrete number β. For different concrete specimens in a set of experiments, discrete number β can be customized, such as β = 0.2, 0.4, 0.6, 0.8, and 1.0… 2.0…The uncertainty of the value of β precisely determines the true nature of the quasi-brittle fracture of concrete. However, an average β only needs to adopt a uniform value for the entire set of test specimens, such as the average β uniformly taking 0.5, 1.0, 1.5, 2.0… to achieve the accurate determination of material parameters to facilitate optimum design and application. The average value β = 1.0 was found to be is accurate enough for concrete specimens with W/dmax < 30 and dmax = 2.5 mm − 25 mm [20–25]. If Δafic is limited to few aggregates in length (e.g. 1 < β < 2), there is no need to determine the exact distribution of the tensile softening stresses for such a coarse structure, and a constant cohesive stress distribution can be assumed, as illustrated in Fig. 2(b) and (c). Using the σn constant distribution in Fig. 2(b) and (c), the stable crack growth Δafic = β·dmax at peak load Pmax is linked to the nominal stress σn, i.e. n (Pmax ,
a fic = · dmax ) =
S P B max 3 4 2 2· 3 + 2 + 6( afic )· 2 · 1 3 12
for 3 + 2( a fic
p
n (Pmax ,
a fic = · dmax ) =
2
2 6
+
Pmax (3 4 + 2) 6B afic a afic + fic + 2 6 3 2
(
+
2 2
)·
for WS specimen a fic
(9) where B is the specimen thickness; γ1 = W - a0; γ2 = γ1 -Δafic; γ3 = γ1 + Δafic; γ4 = W + a0 + Δafic. Similarly, if using the trapezoidal distribution of σn in Fig. 3, the Δafic = β·dmax at Pmax is linked to the σn, i.e. n (Pmax ,
for 3
p
n (Pmax ,
a fic = · dmax ) =
3S Pmax B 3 4 2 2·(2 1 + afic ) + 16 2 + 18 afic· 22·(4 2 + 3 afic) + 4( a fic ) 2 (4 2 + 3 afic )2
b specimen
a fic = · dmax ) =
2
2 6
+
P max (3 4 + 2) 6B afic afic 3 + + 2 2 6 3 8
(
)·
(10)
for WS specimen a fic
(11) If using the triangle distribution of σn (extreme condition) in Fig. 4, the Δafic = β·dmax at Pmax is linked to the σn, i.e. n (Pmax ,
for 3
b specimen
n (Pmax ,
)2
(8)
a fic = ·d max ) = p
3S P B max 4 2·( 1)3 + 4 24 + 6 afic· 22·(2 2 + afic ) (2 2 + afic)2
+ 4( a fic ) 2 (12)
b specimen
a fic = · dmax ) =
2
2 6
+
P max (3 4 + 2 ) 6B afic afic + + 2 2 12 3 4
(
)·
for WS specimen a fic
(13)
117
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Fig. 5. ft and KIC for different Δafic (β = 0.0, 1.0, 2.0, 0.5–1.0, 1.0–1.5, 1.0–2.0) using 3-p-b tests.
Based on numerous trial calculations, it can be found that using rectangular stress distributions, trapezoidal stress distributions, and triangle stress distributions bears limited effect on the ft and KIC values determination. Therefore, using the rectangular stress distribution (Fig. 2(b) and (c)) is sufficient for the accurate ft and KIC values determination.
[41,42]. The steel plate was applied with an oily release agent on both sides. The steel plate was loosened at 3 h after the initial setting of concrete. The steel plate was pulled out while removing the mold, thereby forming a notch. The coarse aggregate was gravel with the maximum concrete aggregate (dmax) of 10 mm. The fine aggregate was river sand with the maximum size of 5 mm. The average compressive strength fcu from 150 mm × 150 mm × 150 mm cube is 29.6 MPa. The tensile strength ft of the concrete could also be estimated to be approximately 2.5 to 4.0 MPa using the ratio of ft/fcu = 1/8–1/12 from GB50010-2010 [43].
3. Determination of ft and KIC of concrete using various specimen types In Ref. [40], concrete specimens with same mixture proportion but two different specimen types (3-p-b and WS) have been tested. All specimens were cast one time, and all specimens were tested at 60 day. In preventing aggregate distribution, using saw machine is necessary to make the notch. However, the width of the initial crack upon notch cutting treatment is difficult to control. Given that the cutting blade had a certain thickness level, the crack end is not prone to tip formation. Soft materials, such as oiled wood, can be used to make the notch. However, the stiffness of soft material is relatively small, thereby leading to bending deformation while pouring concrete. In a previous study [40], the test specimens were shaped by wooden molds. Before pouring concrete specimen, a 2 mm thick steel plate with V-shaped ends was placed in the test specimen mold, following the methods in Refs.
3.1. Determination of independent ft and KIC of concrete using 3-p-b tests The details of 3-p-b concrete specimens are listed in Table 1. The relative size W/dmax is from 20 to 50, then the corresponding (W-a0)/ dmax is only 11–24. Due to the purpose of the convenience of design and application, for 3-p-b or WS specimens with different depths, the discrete number β can be taken as a uniform average value, e.g. 0.0 (no fictitious crack growth and indicated the highest ft and lowest KIC), 0.2, 0.4…1.0… The corresponding determined ft and KIC were shown in Fig. 5 and Table 2. As shown in Table 2, the ft and KIC results for averaged β = 0.6–2.0 are close to what has been commonly used for concrete materials (ft ≈ 118
Theoretical and Applied Fracture Mechanics 100 (2019) 114–127
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Table 2 Estimation of ft and KIC from 3-p-b test results.
1.1 1/2
Keff (MPa⋅m )
It should be point out that there is a limited effect on the ft and KIC values determination using the rectangular, trapezoidal, and triangle stress distribution (results in Table 2). Triangle stress distribution can be deeming as extreme condition. The variation range of determined ft and KIC using rectangular (or constant) stress distribution is larger than those using trapezoidal and triangle stress distribution: e.g. from β = 0.0 to β = 2.0, ft = 4.78–2.74 MPa and KIC = 1.01–1.35 MPa·m1/2 for rectangular stress distribution; ft = 4.78–3.44 MPa and KIC = 1.01–1.12 MPa·m1/2 for trapezoidal stress distribution; ft = 4.78–4.39 MPa and KIC = 1.01–1.04 MPa·m1/ 2 for triangle stress distribution. Therefore, using the rectangular stress distribution is enough for accurate determination of ft and KIC of concrete. un The unstable fracture toughness (KIC ) calculated by the double-K model by using 3-p-b specimens ranges from 0.963 MPa·m1/2 to un 1.389 MPa·m1/2 [40], as shown in Table 1. The KIC results are sizeun independent but show large scatters. The detailed information on KIC un calculation was previously mentioned in Ref. [40]. The KIC calculation shows that the experimental peak loads (Pmax) and critical crack mouth opening displacement (CMODc) are required [40]. The applications of the double-K model seem to always rely on CMODc measurements, which are not as convenient as models that only use Pmax measurements. Compared with the double-K model, the improved BEM only needs Pmax to determine the ft and KIC values of the concrete. For 3-p-b specimen type, the calculation formula of fracture toughness KIC meeting LEFM condition can be found in ASTM standards [44] and are as follows:
1/2
KIC = 1.12 MPa⋅m (β = 1.0)
1.2
1/2
KIC = 1.01 MPa⋅m (β = 0.0)
1.0 0.9 0.8 0.7 0.6 0.5 100
200
300 400 W (mm)
500
600
Fig. 6. Fracture toughness Keff calculated by ASTM using 3-p-b concrete tests.
2.5–4.0 MPa). It should be noticed that the accurate approximation can be obtained using the averaged discrete number β = 1.0 for (W-a0) / dmax = 11–24. If considering the different fictitious crack growth lengths Δafic for different specimen depths, the β can be taken as individual values for different depths. Therefore, results for β = 1.0, 1.33, 1.66 and 2.0 for W = 200 mm, 300 mm, 400 mm, and 500 mm, results for β = 1.0, 1.17, 1.34 and 1.5 for W = 200 mm, 300 mm, 400 mm, and 500 mm, and results for β = 0.5, 0.67, 0.83 and 1.0 for W = 200 mm, 300 mm, 400 mm, and 500 mm, were all also listed in Table 2. For individual discrete number β, the relatively best results were determined when β = 1.0–1.5; secondly β = 0.5–1.0; the relatively worst is 1.0–2.0, in which KIC = 0.90 MPa·m1/2 is smaller than KIC = 1.01 MPa·m1/2 (β = 0.0, no fictitious crack growth). Due to (W-a0) / dmax = 11–24, the value range of β is relatively limited, meanwhile β = 2.0 is a little large for (W-a0) / dmax = 24 (W = 500 mm). It can be found that the values of ft and KIC obtained by using individual β is almost consistent with values from β = 1.0. Therefore, the averaged discrete number β = 1.0 is an enough good approximation for (W-a0) / dmax = 11–24.
KIC =
Pmax S f( ) BW 3/2
f( )=3
1/2
1.99
(14a)
(1 )(2.15 2(1 + 2 )(1
3.93 + 2.7 2 ) )3/2
(14b)
The fracture toughness Keff directly calculated by Eq. (14) using experimental peak loads Pmax from the 3-p-b concrete specimens, are shown in Table 1 and Fig. 6. As shown in Fig. 6, these fracture toughness Keff show obvious size effects: Keff increase with depth of specimen W increasing. Even for largest 3119
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Table 3 Details of WS specimens in [40]. Label
α
W (mm)
B (mm)
W/dmax
(W-a0)/dmax
Pmax (kN)
un KIC (MPa·m1/2)
Keff (MPa·m1/2)
WS1200-0 WS1200-1 WS1200-2 WS1000-1 WS1000-3 WS1000-4 WS1000-5 WS800-1 WS800-2 WS800-4 WS800-5 WS600-1 WS600-2 WS600-3 WS600-5 WS400-1 WS400-2 WS400-3 WS200-1 WS200-2 WS200-3 WS150-1 WS150-2 WS150-3
0.45 0.46 0.45 0.45 0.45 0.45 0.45 0.45 0.46 0.46 0.46 0.46 0.46 0.46 0.46 0.45 0.46 0.46 0.46 0.47 0.47 0.46 0.48 0.49
1198 1200 1200 997 997 999 1000 799 800 798 801 600 599 600 599 400 400 400 200 200 200 150 150 150
200 201 200 200 200 196 200 196 194 200 200 193 200 193 200 200 198 199 200 200 200 180 178 176
120 120 120 100 100 100 100 80 80 80 80 60 60 60 60 40 40 40 20 20 20 15 15 15
65 65 65 54 54 55 55 44 44 43 43 32 33 33 33 22 22 22 11 11 11 8 8 8
24.70 24.21 28.26 22.15 23.34 25.81 24.40 18.66 22.60 21.63 21.99 17.82 16.97 16.68 18.22 11.37 14.27 12.00 7.08 6.89 6.61 4.73 5.59 5.23
1.240 1.233 1.294 1.231 1.328 1.493 1.503 1.472 1.455 1.593 1.299 1.561 1.315 1.397 1.509 1.131 1.515 1.369 1.013 0.982 0.965 0.812 0.924 0.846
0.94 0.92 1.07 0.92 0.97 1.09 1.00 0.89 1.09 1.01 1.03 1.01 0.92 0.93 0.98 0.74 0.96 0.81 0.67 0.67 0.64 0.58 0.72 0.70
1/2
0.5
ft = 4.16 MPa KIC= 1.12 MPa⋅m
0.4
a∞ = 18 mm
∗
β = 0.0
2
0.2 0.1
Curve fitting 0
15
30
45
ae
60
75
90 1/2
0.5
ft = 2.67 MPa KIC= 1.29 MPa⋅m
0.4
a∞ = 58.6 mm
∗
β = 2.0
β = 1.0
2
0.2
0
15
30
45
ae
Test data 60
75
90
1/2
0.5
ft = 3.22 MPa KIC= 1.17 MPa⋅m
0.4
a∞ = 33.3 mm
∗
β = 1.0 — 1.5
0.2 0.1
0.0 -0.1
Curve fitting
0.3
1/σn
2
1/σn
∗
0.2
-0.1
0.1
Curve fitting 0
15
30
45
ae
0.0
Test data 60
75
-0.1
90
1/2
ft = 3.23 MPa KIC= 1.14 MPa⋅m
0.5
∗
a∞ = 30.8 mm
0.4
β = 1.0 — 2.0
0
15
30
45
ae
Test data 60
75
90
1/2
ft = 3.25 MPa KIC= 1.10 MPa⋅m ∗
a∞ = 28.7 mm
0.4
β = 1.0 — 3.0
0.3 2
1/σn
0.2
0.2 0.1
0.1 0.0 -0.1
Curve fitting
0.5
0.3 2
a∞ = 34.6 mm
0.0
Test data
0.3
1/σn
0.4
0.1
0.0 -0.1
ft = 3.21 MPa KIC= 1.20 MPa⋅m
0.3
1/σn
2
1/σn
0.3
1/2
0.5
Curve fitting 0
15
30
45
ae
0.0
Test data 60
75
90
-0.1
Curve fitting 0
15
30
45
ae
Test data 60
75
Fig. 7. ft and KIC for different Δafic (β = 0.0, 1.0, 2.0, 1.0–1.5, 1.0–2.0, 1.0–3.0) using WS tests. 120
90
Theoretical and Applied Fracture Mechanics 100 (2019) 114–127
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1/2
Keff (MPa⋅m )
Table 4 Estimation of ft and KIC from WS tests.
1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5
The reasonable value range of β (=0.2–2.4) is larger than those of 3p-b (β = 0.6–2.0) because the range of (W-a0) / dmax = 8–65 for WS specimens is larger than (W-a0) / dmax = 11–24 for 3-p-b specimens. Similarly, the accurate approximation can be obtained using the averaged β = 1.0 for (W-a0) / dmax = 8–65. As mentioned, the discrete nature or apparent “uncertainty” in fictitious crack growth of concrete can be “certainly” reflected using individual discrete number β. In this way, two individual values of β are selected as 1.0 and 2.0 for W = 150 and 1200 mm, and other β values for other W sizes can be obtained via interpolation. That is, β = 1.0, 1.05, 1.24, 1.43, 1.62, 1.81 and 2.0 for W = 150, 200, 400, 600, 800, 1000 and 1200 mm. If β are selected as 1.0 and 1.5 for W = 150 and 1200 mm, then β = 1.0, 1.02, 1.17, 1.26, 1.36, 1.45 and 1.5 for W = 150, 200, 400, 600, 800, 1000 and 1200 mm. If β are selected as 1.0 and 3.0 for W = 150 and 1200 mm, then β = 1.0, 1.10, 1.48, 2.05, 2.43, 2.81 and 3.0 for W = 150, 200, 400, 600, 800, 1000 and 1200 mm. The fitted curves and the obtained results of material constants for individual β are shown in Fig. 7. It can be found that the values of ft and KIC obtained by using individual β is consistent with values from β = 1.0. Therefore, the averaged discrete number β = 1.0 is an enough good approximation for (W-a0) / dmax = 8–65. un As shown in Table 3, the calculated unstable fracture toughness KIC determined by the double-K model, using experimental Pmax and CMODc from WS specimens, are from 0.812 MPa·m1/2 to 1.593 MPa·m1/ 2 [40]. These results show a slight difference when compared with the 0.963 MPa·m1/2 to 1.389 MPa·m1/2 from 3-p-b specimens determined by the double-K model and still show large scatters. For WS specimen type, the calculation formula of fracture toughness KIC meeting LEFM condition in ASTM standards [44] and are as follows:
1/2
KIC = 1.20 MPa⋅m (β =1.0) 1.12 (β =0.0)
0
200 400 600 800 1000 1200 1400 W (mm)
Fig. 8. Fracture toughness Keff calculated by ASTM using WS concrete tests.
p-b specimens (W = 500 mm and W/dmax = 50), Keff are 0.81, 0.82 and 0.90 MPa·m1/2, which are smaller than the KIC = 1.12 MPa·m1/2 (β = 1.0). 3.2. Determination of independent ft and KIC of concrete using WS tests Compared with 3-p-b specimens, these details WS experimental concrete specimens are listed in Table 3. The relative size W/dmax is from 15 to 120, then the corresponding (W-a0) / dmax is 8–65. The experimental results of WS concrete specimens and the corresponding fitting curves when β = 0.0 (no fictitious crack growth), 1.0, 2.0, 1.0–1.5, 1.0–2.0, 1.0–3.0, are shown in Fig. 7. The averaged values β = 0.0–3.0 with a fine increment of 0.2 are listed in Table 4 with the estimated ft and KIC values, and the characteristic values a . Table 4 shows that the ft and KIC results for averaged β = 0.2–2.4 are sufficient.
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Fig. 9. ft and KIC for different Δafic (β = 0.0, 1.0, 2.0, 1.0–2.0) with 3-p-b and WS specimens.
KIC =
Pmax S f( ) BW 1/2
f ( ) = (2 + )
0.886 + 4.64
curves when averaged β = 0, 1.0, and 2.0 are shown in Fig. 9 and Table 5. And results for individual β (β = 1.0–2.0, β = 1.0–3.0, β = 1.0–4.0) are also shown in Fig. 9 and Table 5. That is β = 1.0, 1.05, 1.14, 1.24, 1.33 ,1.43, 1.62, 1.81 and 2.0 for W = 150 mm, 200 mm, 300 mm, 400 mm, 500 mm, 600 mm, 800 mm, 1000 mm, and 1200 mm; β = 1.0, 1.10, 1.29, 1.48, 1.67,1.86, 2.24, 2.62 and 3.0 for W = 150 mm, 200 mm, 300 mm, 400 mm, 500 mm, 600 mm, 800 mm, 1000 mm, and 1200 mm; β = 1.0, 1.14, 1.43, 1.71, 2.00, 2.29, 2.86, 3.43 and 4.0 for W = 150 mm, 200 mm, 300 mm, 400 mm, 500 mm, 600 mm, 800 mm, 1000 mm, and 1200 mm. As depicted in Fig. 9 and Table 5, the averaged discrete number β = 1.0 is an accurate approximation for (W-a0)/dmax = 8–65. The values of ft = 3.28 MPa and KIC = 1.05–1.14 MPa·m1/2 obtained by using individual β (even β = 1.0–4.0) are consistent with values from β = 1.0 (ft = 3.28 MPa and KIC = 1.17 MPa·m1/2). Comparing Tables 2, 4, and 5, the ft and KIC of concrete determined by using 3-p-b and WS specimens together, are consistent with those determined by 3-p-b and WS specimens, respectively. Based on β = 1.0 as an example, the values of ft and KIC are relatively closer:
(15a)
13.32 2 + 14.72 (1 )3/2
3
5.6
4
(15b)
The fracture toughness Keff calculated directly by Eq. (15), using experimental peak loads Pmax from the WS concrete specimens, are shown in Table 4 and Fig. 8. These fracture toughness Keff exhibit evident size effects: Keff increase with the depth of specimen W increases and its value tends to stabilize as W increases to 800 mm (W/dmax ≥ 80). Interestingly, these fracture toughness Keff calculated by ASTM for the three largest 1200 mm specimens (W/dmax = 120) are 0.94, 0.92 and 1.07 MPa·m1/2, which are smaller but closer to the KIC = 1.20 MPa·m1/2 (β = 1.0). 3.3. Determination of independent ft and KIC using 3-p-b and WS tests together Experimental results of 3-p-b and WS concrete specimens and fitting
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0.75
Table 5 Estimation of ft and KIC from 3-p-b and WS specimens.
ft = 3.44 MPa ∗ ae / a ∞ = 0.1
dmax = 10 mm β = 1.0 W/dmax = 20 — 50
log σn
0.50
( W = 4.1 m , α = 0.2 )
0.25
∗
ae / a ∞ = 10
W = 200 mm W = 300 mm W = 400 mm W = 500 mm
0.00
1/2
KIC = 1.12 MPa⋅m
α = 0.5
-0.25 0.0
∗
a∞ = 26.4 mm
0.5
1.0
1.5
2.0
2.5
3.0
3.5
log ae (a) Fracture curve determined by using ft = 3.44 MPa and KIC = 1.12 MPa·m1/2(β = 1.0) 0.75
ft = 3.62 MPa
dmax = 10 mm β = 1.0 — 1.5
∗
ae / a ∞ = 0.1
W/dmax = 20 — 50
log σn
0.50
ft = 3.44 MPa and KIC = 1.12 MPa·m1/2 determined by 3-p-b specimens, ft = 3.21 MPa and KIC = 1.10 MPa·m1/2 determined by WS specimens, and ft = 3.28 MPa and KIC = 1.17 MPa·m1/2 determined by using 3-p-b and WS specimens together.
( W = 2.9 m , α = 0.2 ) ∗
ae / a ∞ = 10
0.25
W = 200 mm W = 300 mm W = 400 mm W = 500 mm
0.00
1/2
KIC = 0.99 MPa⋅m
∗
α = 0.5
-0.25 -0.5
4. Prediction of structural behavior using material parameters of concrete
0.0
a∞ = 18.6 mm 0.5
1.0
1.5
2.0
2.5
3.0
log ae
(b) Fracture curve determined by using ft = 3.62 MPa and KIC = 0.99 MPa·m1/2(β = 1.0 – 1.5)
Three fracture regions are present in concrete specimens or structures [13–25], as shown in Fig. 1: (1) tensile strength ft controlled fracture region with crack ratio ae/a < 0.1, (2) quasi-brittle fracture region controlled by ft and KIC, 0.1 < ae/a < 10, and (3) LEFM region with a large crack ratio, ae/a > 10, where KIC applies. Eq. (3) or (6) can be applied to the three fracture regions. If the material parameters of concrete ft and KIC are determined, then the complete fracture curve or structural safe design diagram with three fracture regions can be constructed based on Eq. (3) or (6). The estimations of the minimum concrete size can be obtained based on the crack ratios ae/a > 10 and Eq. (3) or (6) to achieve LEFM condition, that is, if the ft and KIC values are determined, then the a and the corresponding ae value can be obtained using the relationship of ae/a = 10. The different W values with different α = a0 / W (e.g., α = 0.2) can be obtained according to Eq. (4a) or (5a).
0.75
ft = 3.83 MPa
dmax = 10 mm β = 1.0 — 2.0 –
∗
ae / a ∞ = 0.1
W/dmax = 20 — 50
log σn
0.50 0.25 0.00
( W = 2.2 m , α = 0.2 ) ∗
ae / a ∞ = 10 W = 200 mm W = 300 mm W = 400 mm W = 500 mm
1/2
KIC = 0.90 MPa⋅m
α = 0.5
-0.25 -0.5
0.0
∗
a∞ = 13.9 mm 0.5
1.0
1.5
2.0
2.5
3.0
log ae
4.1. Prediction of structural failure using material parameters from 3-p-b concrete specimens
(c) Fracture curve determined by using ft = 3.83 MPa and KIC = 0.90 MPa·m1/2(β = 1.0 – 2.0)
Fracture failure curves with three fracture regions using material parameters from 3-p-b concrete specimens are shown in Fig. 10. As shown in Fig. 10, the concrete specimen size can meet the LEFM condition if the concrete specimen depth W is larger than 2 m. However, the depth of concrete specimens (W = 150–1200 mm) used in this analysis is less than 2 m, and the 3-p-b concrete specimens (W = 200–500 mm) are in quasi-brittle fracture.
Fig. 10. Structural failure curves determined by material constants from 3-p-b specimens.
individual β = 1.0–1.5 and β = 1.0–2.0, using material parameters from WS concrete specimens, are shown in Fig. 11. It can be found that the largest specimens (W = 1200 mm and W/ dmax = 120) still cannot completely meet the requirements of LEFM (W = 2.0 m with α = 0.2 and W = 2.9 m with α = 0.45), but are closer to the brittle “homogeneous” material.
4.2. Prediction of structural failure using material parameters from WS concrete specimens Structural failure curves with three fracture regions for β = 1.0 and 123
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0.75
ft = 3.21 MPa
average prediction due to the scatters of experimental results of concrete; the upper and lower limits can be selected in the safe design diagram. For example, the material constants ft = 3.28 MPa and KIC = 1.17 MPa·m1/2 (β = 1.0) are used as an average prediction, and then the upper and lower bounds ( ± 15%), that is, ft = 3.78 MPa and KIC = 1.35 MPa·m1/2(+15%), and ft = 2.79 MPa and KIC = 1.00 MPa·m1/2 (−15%), can be selected to describe the variations in scatters and the fracture properties of concrete. As shown in Fig. 12(b), (d), (f), the safe design diagrams with upper and lower limits ( ± 15%) nearly cover all the experimental results.
dmax = 10 mm β = 1.0
∗
ae / a ∞ = 0.1
W/dmax = 15 — 120
0.50
log σn
( W = 3.6 m , α = 0.2 ) W = 150 mm W = 200 mm W = 400 mm W = 600 mm W = 800 mm W = 1000 mm W = 1200 mm α = 0.45
0.25 0.00
-0.25 0.0
0.5
1.0
∗
ae / a ∞ = 10 1/2
KIC = 1.20 MPa⋅m
5. Predictions of peak loads Pmax pertinent to various concrete specimens
∗
a∞ = 34.6 mm 1.5
2.0
2.5
3.0
3.5
t The comparison results of the experimental peak loads Pmax and the c calculated peak loadsPmax obtained from predicted fracture curves determined by different material constants (β = 1.0: ft = 3.28 MPa and KIC = 1.17 MPa·m1/2; β = 2.0: ft = 2.74 MPa and KIC = 1.26 MPa·m1/2; β = 1.0–2.0: ft = 3.28 MPa and KIC = 1.14 MPa·m1/2; β = 1.0–3.0: ft = 3.28 MPa and KIC = 1.10 MPa·m1/2) are shown in Table 6. As shown in Table 6, the predictions of peak loads have high precision for the single specimen and the whole. The difference between prediction results using the various material constants (ft and KIC) determined by different fictitious crack growth lengths Δafic (β = 1.0, 2.0, 1.0–2.0, 1.0–3.0) is relatively small. The average Δafic ≈ 1.0 dmax is sufficient for predicting the peak loads of concrete with W/ dmax = 15–120. If the independent material constants ft = 3.28 MPa and KIC = 1.17 MPa·m1/2 have been obtained, then the values of fictitious crack growth length Δafic can be inverted to make the calculated peak loads completely the same as the experimental peak loads. The inversions of Δafic are shown in Fig. 13. As shown in Fig. 13, the inverted Δafic values indicate an increasing trend with the depth of specimen W. However, this change is insignificant. The discreteness of the inverted Δafic values is increase in either W or W-a0. The Δafic values have a large scatter and randomness and difficult to determine accurately. The mean of the inverted Δafic values is 13.7 mm, which is closer to dmax = 10 mm. Therefore, the average Δafic ≈ 1.0 dmax is simple and sufficiently accurate for predicting the peak loads of concrete with W/dmax = 15–120.
log ae (a) Fracture curve determined by using ft = 3.21 MPa and KIC = 1.20 MPa·m1/2(β = 1.0) 0.75
β = 1.0 — 1.5
ft = 3.22 MPa ∗ ae / a ∞ = 0.1
dmax = 10 mm W/dmax = 15 — 120
log σn
0.50
( W = 3.5 m , α = 0.2 ) ∗ ae / a ∞ = 10
W = 150 mm W = 200 mm W = 400 mm W = 600 mm W = 800 mm W = 1000 mm W = 1200 mm α = 0.45
0.25 0.00
-0.25 0.0
0.5
1.0
1/2
KIC = 1.17 MPa⋅m ∗
a∞ = 33.3 mm 1.5
2.0
2.5
3.0
3.5
log ae (b) Fracture curve determined by using ft = 3.22 MPa and KIC = 1.17 MPa·m1/2(β = 1.0 – 1.5) 0.75
β = 1.0 — 2.0
ft = 3.23 MPa ∗ ae / a ∞ = 0.1
dmax = 10 mm W/dmax = 15 — 120
log σn
0.50 0.25 0.00 -0.25 0.0
W = 150 mm W = 200 mm W = 400 mm W = 600 mm W = 800 mm W = 1000 mm W = 1200 mm α = 0.45
0.5
1.0
6. Conclusions
( W = 3.2 m , α = 0.2 ) ∗ ae / a ∞ = 10
In the present study, the relationship between material behavior (e.g., tensile strength ft and fracture toughness KIC) and structural behavior (e.g., failure curve or safe design diagram) of concrete is established. Using different concrete specimen types, independent material parameters (ft and KIC) are determined. The structural failure of concrete is predicted using independent material constants. New findings and the major conclusions are as follows.
1/2
KIC = 1.14 MPa⋅m ∗
a∞ = 30.8 mm 1.5
2.0
2.5
3.0
3.5
log ae (c) Fracture curve determined by using ft = 3.23 MPa and KIC = 1.14 MPa·m1/2(β = 1.0 – 2.0)
(1) The independent tensile strength ft and the fracture toughness KIC of the concrete can be determined using different concrete specimens (e.g., 3-p-b and WS specimens).The ft and KIC values in 3-p-b and WS specimens are consistent with those in 3-p-b and WS specimens, respectively.
Fig. 11. Structural failure curves determined by material constants from WS specimens.
The discrepancy of experimental results is an inherent property of concrete materials. This property can be well described by the fictitious crack growth length Δafic corresponding to peak load Pmax, which is related to maximum aggregate size dmax through a discrete number β. Rectangular, trapezoidal and triangle stress distributions in fictitious crack growth have minimal effects on ft and KIC value determination. Using the rectangular stress distribution is sufficiently accurate to determine the ft and KIC values of concrete.
4.3. Prediction of concrete failure using material parameters from 3-p-b and WS specimens Based on the selection of β = 1.0 and individual β = 1.0–2.0, β = 1.0–3.0, the prediction curves with three fracture regions, using material parameters from 3-p-b and WS concrete specimens together, are illustrated in Fig. 12. The determined material constants (for a given β) can be used for an
(2) The structural failure or the safe design diagram of concrete can be 124
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Fig. 12. Structural failure curves determined by using material constants from various specimens (3-p-b and WS).
fully constructed using the independent ft and KIC. The safe design diagrams with upper and lower limits ( ± 15%) nearly cover all experimental results. (3) The theoretical concrete specimen size meeting the LEFM can be obtained using the determined independent material constants.
The concrete experimental 3-p-b (W ≤ 500 mm) and WS (W ≤ 1200 mm) specimens are in quasi-brittle fracture because the theoretical depth is larger than 2 m. Fracture toughness Keff of concrete calculated directly by the LEFM formula increases as the depth of the specimen increases, and Keff from the largest 1200 mm (W/dmax = 120) specimen is smaller but closer to the independent KIC.
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Table 6 Predictions of peak loads Pmax of 3-p-b and WS specimens. Label
t Experimental peak loads Pmax (kN)
c (kN) Predicted peak loads Pmax
β = 1.0
TPB500-1 TPB500-2 TPB500-3 TPB400-1 TPB400-2 TPB400-3 TPB400-4 TPB300-1 TPB300-2 TPB300-3 TPB300-4 TPB300-5 TPB200-2 TPB200-3 TPB200-4 WS1200-0 WS1200-1 WS1200-2 WS1000-1 WS1000-3 WS1000-4 WS1000-5 WS800-1 WS800-2 WS800-4 WS800-5 WS600-1 WS600-2 WS600-3 WS600-5 WS400-1 WS400-2 WS400-3 WS200-1 WS200-2 WS200-3 WS150-1 WS150-2 WS150-3
9.81 11.93 9.85 11.13 10.97 10.39 9.96 7.91 8.32 7.84 8.39 8.29 7.19 6.23 6.64 24.70 24.21 28.26 22.15 23.34 25.81 24.40 18.66 22.60 21.63 21.99 17.82 16.97 16.68 18.22 11.37 14.27 12.00 7.08 6.89 6.61 4.73 5.59 5.23
β = 2.0
β = 1.0–2.0
β = 1.0–3.0
c Pmax
t c Pmax /Pmax
Pc max
t c Pmax /Pmax
Pc max
t c Pmax /Pmax
Pc max
t c Pmax /Pmax
9.99 12.25 8.95 10.70 10.50 10.32 10.22 8.45 8.20 8.60 8.27 7.90 6.44 6.14 6.47 26.57 26.57 26.52 23.68 23.61 23.30 23.95 20.07 19.84 20.42 20.33 15.95 16.83 16.30 16.83 12.88 12.43 12.49 7.70 7.38 7.38 5.53 5.22 4.98
0.98 0.97 1.10 1.04 1.04 1.01 0.97 0.94 1.01 0.91 1.01 1.05 1.12 1.02 1.03 0.93 0.91 1.07 0.94 0.99 1.11 1.02 0.93 1.14 1.06 1.08 1.12 1.01 1.02 1.08 0.88 1.15 0.96 0.92 0.93 0.90 0.86 1.07 1.05
9.87 12.09 8.86 10.57 10.38 10.21 10.10 8.42 8.17 8.56 8.24 7.89 6.56 6.27 6.58 26.88 26.88 26.82 23.82 23.74 23.44 24.09 20.05 19.82 20.40 20.31 15.82 16.70 16.18 16.70 12.73 12.29 12.35 7.75 7.43 7.43 5.66 5.35 5.12
0.99 0.99 1.11 1.05 1.06 1.02 0.99 0.94 1.02 0.92 1.02 1.05 1.10 0.99 1.01 0.92 0.90 1.05 0.93 0.98 1.10 1.01 0.93 1.14 1.06 1.08 1.13 1.02 1.03 1.09 0.89 1.16 0.97 0.91 0.93 0.89 0.84 1.05 1.02
10.09 12.33 9.06 10.76 10.57 10.39 10.29 8.48 8.23 8.63 8.30 7.94 6.43 6.13 6.46 26.62 26.62 26.56 23.73 23.66 23.35 23.99 20.12 19.88 20.47 20.39 15.99 16.88 16.34 16.88 12.90 12.46 12.52 7.67 7.35 7.35 5.48 5.17 4.94
0.97 0.97 1.09 1.03 1.04 1.00 0.97 0.93 1.01 0.91 1.01 1.04 1.12 1.02 1.03 0.93 0.91 1.06 0.93 0.99 1.11 1.02 0.93 1.14 1.06 1.08 1.11 1.01 1.02 1.08 0.88 1.15 0.96 0.92 0.94 0.90 0.86 1.08 1.06
10.20 12.42 9.18 10.82 10.63 10.46 10.35 8.52 8.27 8.67 8.34 7.98 6.42 6.12 6.45 26.66 26.67 26.61 23.78 23.71 23.40 24.03 20.17 19.93 20.53 20.44 16.04 16.92 16.38 16.92 12.92 12.48 12.54 7.64 7.33 7.33 5.43 5.13 4.90
0.96 0.96 1.07 1.03 1.03 0.99 0.96 0.93 1.01 0.90 1.01 1.04 1.12 1.02 1.03 0.93 0.91 1.06 0.93 0.98 1.10 1.02 0.93 1.13 1.05 1.08 1.11 1.00 1.02 1.08 0.88 1.14 0.96 0.93 0.94 0.90 0.87 1.09 1.07
Mean
1.008
1.007
1.006
1.004
Deviation coefficient
0.075
0.077
0.075
0.074
Inverted value dmax=10 mm
50
mean =13.7 mm
30 20 10 0
mean =13.7 mm
40 ∆afic (mm)
∆afic (mm)
40
Inverted value dmax=10 mm
50
30 20 10
0
200 400 600 800 1000 1200 1400 W (mm)
0
0
100
200 300 400 W-a0 (mm)
500
600
Fig. 13. Inversion of Δafic using ft = 3.28 MPa and KIC = 1.17 MPa·m1/2.
(1) The averaged discrete number β = 1.0 is a sufficient approximation to determine the ft and KIC values of concrete for W/dmax = 15–120. The ft and KIC values obtained using individual β (e.g. β = 1.0–1.5, 1.0–2.0, 1.0–3.0, 1.0–4.0) are consistent with the values from β = 1.0.
The predicted peak loads by using the improved BEM fit the experimental peak loads. The inverted values of Δafic exhibit large scatters. According to obtained results, several practical recommendations on the use of the improved BEM can be suggested, as follows:
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(2) The average β ≈ 1.0, that is, Δafic ≈ 1.0dmax, is sufficient to predict concrete fracture failure. (3) The average β ≈ 1.0 or Δafic ≈ 1.0dmax with sufficient relative smallest deviation coefficient to predict the peak loads of the specimens.
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This study considers the effect of aggregate size on the obtained test results. The effect of other factors (e.g., coarse aggregate type and volume) on the tensile strength ft and fracture toughness KIC requires further study. Acknowledgement The authors would like to express the gratitude to the support of National Natural Science Foundation of China (Nos. 51779095; 51708216; 51309203; 51779069). Conflict of interest The authors declared that they have no conflicts of interest to this work. References [1] A. Mindess, The effect of specimen size on the fracture energy of concrete, Cem. Concr. Res. 14 (1984) 431–436. [2] P. Nallthambi, B. Karihaloo, B. Heaton, Effect of specimen and crack sizes, water/ cement ratio and coarse aggregate texture upon fracture toughness of concrete, Mag. Concr. Res. 36 (1984) 227–236. [3] A. Hillerborg, Results of three comparative test series for determining the fracture energy GF of concrete, Mater. Struct. 18 (1985) 407–413. [4] M.G. Alexander, G.E. Blight, A comparative study of fracture parameters in notched concrete beams, Mag. Concr. Res. 40 (1988) 50–58. [5] W. Brameshuber, H. Hilsdorf, Influence of ligament length and stress state on fracture energy of concrete, Eng. Fract. Mech. 35 (1990) 95–106. [6] M.R.A. Van Vliet, J.G.M. Van Mier, Experimental investigation of size effect in concrete and sandstone under uniaxial tension, Eng. Fract. Mech 65 (2000) 165–188. [7] J.G.M. van Mier, M.R.A. van Vliet, Influence of microstructure of concrete on size/ scale effects in tensile fracture, Eng. Fract. Mech 70 (2003) 2281–2306. [8] Z. Zhao, S.H. Kwon, S.P. Shah, Effect of specimen size on fracture energy and softening curve of concrete Part I. Experiments and fracture energy, Cem. Concr. Res. 38 (2008) 1049–1060. [9] Y. Çağlar, S. Şener, Size effect tests of different notch depth specimens with support rotation measurements, Eng. Fract. Mech. 157 (2016) 43–55. [10] J.F. Guan, Q.B. Li, Z.M. Wu, S.B. Zhao, W. Dong, S.W. Zhou, Minimum specimen size for fracture parameters of site-casting dam concrete, Constr. Build. Mater. 93 (2015) 973–982. [11] Q.B. Li, J.F. Guan, Z.M. Wu, W. Dong, S.W. Zhou, Equivalent maturity for ambient temperature effect on fracture parameters of site-casting dam concrete, Constr. Build. Mater. 120 (2016) 293–308. [12] J.F. Guan, Q.B. Li, Z.M. Wu, S.B. Zhao, W. Dong, S.W. Zhou, Fracture parameters of site-casting and sieved concrete, Mag. Concr. Res. 68 (2016) 43–54. [13] X.Z. Hu, F.H. Wittmann, Fracture energy and fracture process zone, Mater. Struct. 25 (1992) 319–326. [14] X.Z. Hu, F.H. Wittmann, Size effect on toughness induced by crack close to free surface, Eng. Fract. Mech. 65 (2000) 209–221. [15] X.Z. Hu, An asymptotic approach to size effect on fracture toughness and fracture energy of composites, Eng. Fract. Mech. 69 (2002) 555–564. [16] K. Duan, X.Z. Hu, Specimen boundary induced size effect on quasi-brittle fracture, Strength Fract. Complex. 2 (2004) 47–68. [17] X.Z. Hu, K. Duan, Size effect: influence of proximity of fracture process zone to
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Theoretical and Applied Fracture Mechanics journal homepage: www.elsevier.com/locate/tafmec
Determination of fracture parameter and prediction of structural fracture using various concrete specimen types
T
Junfeng Guana, , Changming Lia, Juan Wangb, , Longbang Qingc, Zhikai Songa, Zenpeng Liua ⁎
⁎
a
School of Civil Engineering and Communication, North China University of Water Resources and Electric Power, Zhengzhou 450045, PR China College of Water Conservancy and Environment Engineering, Zhengzhou University, Zhengzhou 450001 PR China c College of Civil Engineering, Hebei University of Technology, Tianjin 300401, PR China b
ARTICLE INFO
ABSTRACT
Keywords: Specimen type Concrete Tensile strength Fracture toughness Peak load
Concrete tensile strength and fracture toughness are two important material constants that should be independent of specimen size, geometry or type. In this study, on the basis of the improved boundary effect model, independent tensile strength and fracture toughness of concrete, which are unaffected by the specimen type, were determined by using various specimens, e.g., three-point-bending and wedge-splitting specimens. The concrete structural failure was fully predicted using independent material constants, and the constructed safe design diagrams with upper and lower limits ( ± 15%) can cover all experimental results. The theoretical minimum size of the concrete meeting the linear elastic fracture mechanics were quantified, and the peak loads of concrete specimens were predicted using various average fictitious crack growth length.
1. Introduction In many concrete test studies [1–12], even with the same concrete mixture, the dependent material parameters from experimental results (e.g., strength or fracture parameters) are obtained using different specimen sizes, geometric features, or types. However, material parameters should be constant with specimen sizes and types. These obtained inconsistent results are commonly considered as the size effect or structure behavior. For the accurate evaluation of the structural behavior of the concrete structures, having material parameters that are unaffected by the structure size, geometry, or type is necessary. Therefore, determining the independent concrete material parameters is important. Tensile strength ft and fracture toughness KIC are two important material parameters of concrete. Based on the boundary effect theory for concrete and ceramic fracture of Hu [13–19], the improved boundary effect model (BEM) was proposed to determine the ft and KIC of concrete [20–22], mortar [23], and granite [24] using three-pointbending (3-p-b) specimens. Recently, the ft and KIC of concrete was also determined using wedge-splitting (WS) specimens [25]. In principle, the ft and KIC of concrete are constant and independent and should not be affected by specimen type. Therefore, in this study, using two different specimen types of concrete, i.e., 3-p-b and WS specimens together, we simultaneously determined the independent ft and KIC of concrete. These determined values are consistent with those ⁎
individually determined by 3-p-b or WS specimens. Furthermore, the structural fracture curves or safe design diagrams of concrete were fully constructed using the determined independent material constants. Then the minimum size meeting the linear elastic fracture mechanics (LEFM) were theoretically obtained. The predicted peak loads using BEM were compared by experimental results. 2. Methodology to determine material parameters using structural behavior 2.1. Relationship between material and structural behavior The relationship between material and structural behavior is elaborated in Fig. 1. For specimens or structures that can satisfy 100% fracture toughness criterion control, regardless of the changes in their size or type, the fracture toughness KIC cannot be affected and is always constant, that is, KIC can be directly determined by the specimens or structures that can satisfy 100% fracture toughness criterion control. Similarly, if 100% strength control is satisfied, then the tensile strength ft can be directly determined regardless of the specimen type or size. However, in actual concrete structural engineering, concrete structures or specimens seldom fully meet the 100% strength or fracture toughness criterion control. Concrete structural fracture is commonly controlled by the tensile strength ft and the fracture toughness KIC simultaneously, thereby indicating that it is in the quasi-brittle fracture.
Corresponding authors. E-mail addresses: [email protected] (J. Guan), [email protected] (J. Wang).
https://doi.org/10.1016/j.tafmec.2019.01.008 Received 7 October 2018; Received in revised form 24 December 2018; Accepted 7 January 2019 Available online 10 January 2019 0167-8442/ © 2019 Elsevier Ltd. All rights reserved.
Theoretical and Applied Fracture Mechanics 100 (2019) 114–127
J. Guan et al.
feff ≠ Const ≠ ft ft
feff
n
Keff ≠ Const ≠ KIC
Keff
Structural behavior
KIC
KIC
Quasi-brittle fracture
ae =
LEFM
Log
a*
∞
ft a
1+ a e
(3)
where ae is the equivalent crack and is a structural geometry parameter, determined by the specimen size and type, and the a0. It should be pointed that influence of the front and the back boundaries on fracture of specimen, and the effect of structure size and geometry, and structure type (or loading condition) are considered by the structural parameter ae. Useful relations for 3-p-b [20–24] and WS [25] specimens are provided here.
Log σn Log ft
=
Log a
Y( ) =
Material behavior (single material constant)
(1 - )2 × Y ( ) 1.12 1.99
(1
N (P )
ae =
a0 a
=
N
= ft =
ft 1
p
b specimen for 3
p
(4a)
b specimen with S/ W= 4
× Y( )
1.12
2
·a 0
for WS specimen
(2 + )(0.866 + 4.64 4 for WS specimen
(5a)
13.32 2 + 14.72 (1 )3/2
3
5.6
4)
(5b)
where α is ratio of a0 and W, α = a0/W; S is the effective length of specimens; W is the depth of specimens. Rearrange the Eq. (3), it can be obtained:
1 2 n
=
1 1 a 1 4a + 2 · e = 2 + 2e ft2 ft a ft KIC
(6)
The details of determination the material parameters of concrete using the experimental results of limited-size specimens are as follows: Firstly, the nominal stress σn are determined by experimental structural behavior (e.g. experimental peak loads) of different specimen. Secondly, equivalent crack (structural parameter) ae are determined by the specimen type and size. Thirdly, based on Eq. (6), the independent material parameters-tensile strength ft and fracture toughness KIC are determined by curve fitting (σn and ae).
(1)
2.3. Quasi-brittle fracture of concrete considering the influence of aggregate
where σn (P) is the nominal stress from the load P considering the influence of the crack; σN (P) is the nominal stress from the load P in which the influence of the crack is not considered. a is the characteristic crack, fully determined by the tensile strength ft and the fracture toughness KIC, a = 0.25·(KIC/ft )2 . If the strength criterion dominates, the corresponding calculation model is [13–19]: n
2(1 )2 2+
Y( )=
ft
=
for 3
(4b)
The different ft and KIC proportions (e.g., 60% ft and 40% KIC; or 30% ft and 70% KIC) affect the specimen behavior, thereby showing different results. Therefore, the results from the quasi-brittle fracture specimens generally exhibit structural behavior (size effect). The boundary effect theory was originally established by Hu and Wittmann in 1992 [13], and developed by Hu and Duan from 2000 to 2010 [14–19]. Compared to the size effect theory only considering the “absolutely size W” [26,27], the boundary effect theory holds that, for specimen with a crack, the crack length a0 and the specimen size W have couple effects on the fracture of specimen. When the specimen size W is much larger than the initial crack length a0 (W > > a0), the fracture of the specimen is mainly effected by the front boundary of the specimen (a0). When the W is close to the a0 (W ≈ a0), the specimen fracture is mainly effected by the back boundary (W - a0). For these specimen in laboratory, usually a0/W = 0.1–0.7, the influence of the front and the back boundaries should be all considered. For infinite plate or large-size specimen, if the fracture toughness criterion dominates, the corresponding calculation model is [13–19]:
=
·a 0
)(2.15 3.93 + 2.7 2) (1 + 2 )(1 )3/2
Fig. 1. Material and structural behavior.
n (P )
2
The depths of concrete specimens in laboratories are generally less than 500 mm [1–12,28–37]. RILEM [38] recommended that the depth W for 3-p-b specimen of concrete or mortar is from 100 mm to 400 mm. Norm for fracture test of hydraulic concrete in China [39] recommended that the depth W for 3-p-b and WS specimen of concrete is 200 mm. The maximum aggregate size dmax of ordinary concrete is from 5 mm to 40 mm. Therefore, the ratio of the W and dmax is 5–30, and then the corresponding ratio of the ligament depth of specimen (W-a0) and the dmax is only 3–15. As illustrated in Fig. 2(a), the aggregate size cannot be ignored compared to the depth of specimen in laboratories, the heterogeneity of the concrete specimen in laboratory is obvious, the corresponding fracture of concrete specimen is non-brittle and in quasi-brittle fracture. Therefore, the maximum aggregate size dmax should be introduced in the analytical formula of the improved boundary effect model (BEM). The fictitious crack growth Δafic at peak load Pmax and its relationship with discontinuous and stepwise formation of micro cracks within the coarse aggregate structures are described [20–25]. The Δafic is linked to the coarse aggregate structures, e.g., the maximum aggregate size dmax, through a discrete number β [20–25], i.e,
(2)
For structures fully dominated by fracture toughness or tensile strength respectively, the directly determined ft and KIC from these structures are material parameters (no size effect). However, for the finite size specimen in laboratories, the results (e.g. experimental tensile strength feff and fracture toughness Keff , as shown in Fig. 1) directly determined by experiments show size effect or structural behavior, due to the fracture of the limited-size specimen being controlled by both the ft and KIC simultaneously and in the quasibrittle fracture. 2.2. Determination of material behavior using structural behavior
a fic = ·d max
Combined Eqs. (1) and (2), the calculation model of the quasi-brittle fracture for a finite size specimen is as follows [13–19]:
(7)
The discrete number β depends on the aggregate distribution and 115
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J. Guan et al.
Fig. 2. Improved BEM for 3-p-b and WS specimens considered dmax.
Fig. 3. Using the trapezoidal distribution of σn for 3-p-b and WS specimens.
116
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J. Guan et al.
Fig. 4. Using the triangle distribution of σn for 3-p-b and WS specimens. Table 1 Details of 3-p-b specimens in [40]. Label
α
W (mm)
B (mm)
W/dmax
(W-a0)/dmax
Pmax (kN)
un KIC (MPa·m1/2)
Keff (MPa·m1/2)
TPB500-1 TPB500-2 TPB500-3 TPB400-1 TPB400-2 TPB400-3 TPB400-4 TPB300-1 TPB300-2 TPB300-3 TPB300-4 TPB300-5 TPB200-2 TPB200-3 TPB200-4
0.53 0.46 0.55 0.46 0.46 0.47 0.47 0.47 0.47 0.46 0.48 0.49 0.47 0.48 0.46
499 500 500 401 400 396 396 298 297 298 298 298 200 200 199
200 196 198 199 196 199 197 200 195 198 200 200 200 200 199
50 50 50 40 40 40 40 30 30 30 30 30 20 20 20
24 27 22 22 21 21 21 16 16 16 16 15 11 10 11
9.81 11.93 9.85 11.13 10.97 10.39 9.96 7.91 8.32 7.84 8.39 8.29 7.19 6.23 6.64
1.265 1.389 1.191 1.223 1.205 1.330 1.346 1.289 1.074 1.038 1.166 1.198 1.317 0.963 1.110
0.81 0.82 0.90 0.84 0.84 0.81 0.78 0.70 0.76 0.69 0.76 0.78 0.77 0.70 0.71
percentage of dmax, and concrete preparation. The heterogeneity of concrete specimens of W/dmax < 30 is obvious, and even when 3-p-b or WS concrete specimens with the same geometry (same specimen depth W and same initial crack length a0) are tested, different peak loads Pmax and different fictitious crack lengths Δafic would still be obtained. The improved BEM [20–25] can consider the discrepancy of experimental results using the discrete number β. For different concrete specimens in a set of experiments, discrete number β can be customized, such as β = 0.2, 0.4, 0.6, 0.8, and 1.0… 2.0…The uncertainty of the value of β precisely determines the true nature of the quasi-brittle fracture of concrete. However, an average β only needs to adopt a uniform value for the entire set of test specimens, such as the average β uniformly taking 0.5, 1.0, 1.5, 2.0… to achieve the accurate determination of material parameters to facilitate optimum design and application. The average value β = 1.0 was found to be is accurate enough for concrete specimens with W/dmax < 30 and dmax = 2.5 mm − 25 mm [20–25]. If Δafic is limited to few aggregates in length (e.g. 1 < β < 2), there is no need to determine the exact distribution of the tensile softening stresses for such a coarse structure, and a constant cohesive stress distribution can be assumed, as illustrated in Fig. 2(b) and (c). Using the σn constant distribution in Fig. 2(b) and (c), the stable crack growth Δafic = β·dmax at peak load Pmax is linked to the nominal stress σn, i.e. n (Pmax ,
a fic = · dmax ) =
S P B max 3 4 2 2· 3 + 2 + 6( afic )· 2 · 1 3 12
for 3 + 2( a fic
p
n (Pmax ,
a fic = · dmax ) =
2
2 6
+
Pmax (3 4 + 2) 6B afic a afic + fic + 2 6 3 2
(
+
2 2
)·
for WS specimen a fic
(9) where B is the specimen thickness; γ1 = W - a0; γ2 = γ1 -Δafic; γ3 = γ1 + Δafic; γ4 = W + a0 + Δafic. Similarly, if using the trapezoidal distribution of σn in Fig. 3, the Δafic = β·dmax at Pmax is linked to the σn, i.e. n (Pmax ,
for 3
p
n (Pmax ,
a fic = · dmax ) =
3S Pmax B 3 4 2 2·(2 1 + afic ) + 16 2 + 18 afic· 22·(4 2 + 3 afic) + 4( a fic ) 2 (4 2 + 3 afic )2
b specimen
a fic = · dmax ) =
2
2 6
+
P max (3 4 + 2) 6B afic afic 3 + + 2 2 6 3 8
(
)·
(10)
for WS specimen a fic
(11) If using the triangle distribution of σn (extreme condition) in Fig. 4, the Δafic = β·dmax at Pmax is linked to the σn, i.e. n (Pmax ,
for 3
b specimen
n (Pmax ,
)2
(8)
a fic = ·d max ) = p
3S P B max 4 2·( 1)3 + 4 24 + 6 afic· 22·(2 2 + afic ) (2 2 + afic)2
+ 4( a fic ) 2 (12)
b specimen
a fic = · dmax ) =
2
2 6
+
P max (3 4 + 2 ) 6B afic afic + + 2 2 12 3 4
(
)·
for WS specimen a fic
(13)
117
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Fig. 5. ft and KIC for different Δafic (β = 0.0, 1.0, 2.0, 0.5–1.0, 1.0–1.5, 1.0–2.0) using 3-p-b tests.
Based on numerous trial calculations, it can be found that using rectangular stress distributions, trapezoidal stress distributions, and triangle stress distributions bears limited effect on the ft and KIC values determination. Therefore, using the rectangular stress distribution (Fig. 2(b) and (c)) is sufficient for the accurate ft and KIC values determination.
[41,42]. The steel plate was applied with an oily release agent on both sides. The steel plate was loosened at 3 h after the initial setting of concrete. The steel plate was pulled out while removing the mold, thereby forming a notch. The coarse aggregate was gravel with the maximum concrete aggregate (dmax) of 10 mm. The fine aggregate was river sand with the maximum size of 5 mm. The average compressive strength fcu from 150 mm × 150 mm × 150 mm cube is 29.6 MPa. The tensile strength ft of the concrete could also be estimated to be approximately 2.5 to 4.0 MPa using the ratio of ft/fcu = 1/8–1/12 from GB50010-2010 [43].
3. Determination of ft and KIC of concrete using various specimen types In Ref. [40], concrete specimens with same mixture proportion but two different specimen types (3-p-b and WS) have been tested. All specimens were cast one time, and all specimens were tested at 60 day. In preventing aggregate distribution, using saw machine is necessary to make the notch. However, the width of the initial crack upon notch cutting treatment is difficult to control. Given that the cutting blade had a certain thickness level, the crack end is not prone to tip formation. Soft materials, such as oiled wood, can be used to make the notch. However, the stiffness of soft material is relatively small, thereby leading to bending deformation while pouring concrete. In a previous study [40], the test specimens were shaped by wooden molds. Before pouring concrete specimen, a 2 mm thick steel plate with V-shaped ends was placed in the test specimen mold, following the methods in Refs.
3.1. Determination of independent ft and KIC of concrete using 3-p-b tests The details of 3-p-b concrete specimens are listed in Table 1. The relative size W/dmax is from 20 to 50, then the corresponding (W-a0)/ dmax is only 11–24. Due to the purpose of the convenience of design and application, for 3-p-b or WS specimens with different depths, the discrete number β can be taken as a uniform average value, e.g. 0.0 (no fictitious crack growth and indicated the highest ft and lowest KIC), 0.2, 0.4…1.0… The corresponding determined ft and KIC were shown in Fig. 5 and Table 2. As shown in Table 2, the ft and KIC results for averaged β = 0.6–2.0 are close to what has been commonly used for concrete materials (ft ≈ 118
Theoretical and Applied Fracture Mechanics 100 (2019) 114–127
J. Guan et al.
Table 2 Estimation of ft and KIC from 3-p-b test results.
1.1 1/2
Keff (MPa⋅m )
It should be point out that there is a limited effect on the ft and KIC values determination using the rectangular, trapezoidal, and triangle stress distribution (results in Table 2). Triangle stress distribution can be deeming as extreme condition. The variation range of determined ft and KIC using rectangular (or constant) stress distribution is larger than those using trapezoidal and triangle stress distribution: e.g. from β = 0.0 to β = 2.0, ft = 4.78–2.74 MPa and KIC = 1.01–1.35 MPa·m1/2 for rectangular stress distribution; ft = 4.78–3.44 MPa and KIC = 1.01–1.12 MPa·m1/2 for trapezoidal stress distribution; ft = 4.78–4.39 MPa and KIC = 1.01–1.04 MPa·m1/ 2 for triangle stress distribution. Therefore, using the rectangular stress distribution is enough for accurate determination of ft and KIC of concrete. un The unstable fracture toughness (KIC ) calculated by the double-K model by using 3-p-b specimens ranges from 0.963 MPa·m1/2 to un 1.389 MPa·m1/2 [40], as shown in Table 1. The KIC results are sizeun independent but show large scatters. The detailed information on KIC un calculation was previously mentioned in Ref. [40]. The KIC calculation shows that the experimental peak loads (Pmax) and critical crack mouth opening displacement (CMODc) are required [40]. The applications of the double-K model seem to always rely on CMODc measurements, which are not as convenient as models that only use Pmax measurements. Compared with the double-K model, the improved BEM only needs Pmax to determine the ft and KIC values of the concrete. For 3-p-b specimen type, the calculation formula of fracture toughness KIC meeting LEFM condition can be found in ASTM standards [44] and are as follows:
1/2
KIC = 1.12 MPa⋅m (β = 1.0)
1.2
1/2
KIC = 1.01 MPa⋅m (β = 0.0)
1.0 0.9 0.8 0.7 0.6 0.5 100
200
300 400 W (mm)
500
600
Fig. 6. Fracture toughness Keff calculated by ASTM using 3-p-b concrete tests.
2.5–4.0 MPa). It should be noticed that the accurate approximation can be obtained using the averaged discrete number β = 1.0 for (W-a0) / dmax = 11–24. If considering the different fictitious crack growth lengths Δafic for different specimen depths, the β can be taken as individual values for different depths. Therefore, results for β = 1.0, 1.33, 1.66 and 2.0 for W = 200 mm, 300 mm, 400 mm, and 500 mm, results for β = 1.0, 1.17, 1.34 and 1.5 for W = 200 mm, 300 mm, 400 mm, and 500 mm, and results for β = 0.5, 0.67, 0.83 and 1.0 for W = 200 mm, 300 mm, 400 mm, and 500 mm, were all also listed in Table 2. For individual discrete number β, the relatively best results were determined when β = 1.0–1.5; secondly β = 0.5–1.0; the relatively worst is 1.0–2.0, in which KIC = 0.90 MPa·m1/2 is smaller than KIC = 1.01 MPa·m1/2 (β = 0.0, no fictitious crack growth). Due to (W-a0) / dmax = 11–24, the value range of β is relatively limited, meanwhile β = 2.0 is a little large for (W-a0) / dmax = 24 (W = 500 mm). It can be found that the values of ft and KIC obtained by using individual β is almost consistent with values from β = 1.0. Therefore, the averaged discrete number β = 1.0 is an enough good approximation for (W-a0) / dmax = 11–24.
KIC =
Pmax S f( ) BW 3/2
f( )=3
1/2
1.99
(14a)
(1 )(2.15 2(1 + 2 )(1
3.93 + 2.7 2 ) )3/2
(14b)
The fracture toughness Keff directly calculated by Eq. (14) using experimental peak loads Pmax from the 3-p-b concrete specimens, are shown in Table 1 and Fig. 6. As shown in Fig. 6, these fracture toughness Keff show obvious size effects: Keff increase with depth of specimen W increasing. Even for largest 3119
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Table 3 Details of WS specimens in [40]. Label
α
W (mm)
B (mm)
W/dmax
(W-a0)/dmax
Pmax (kN)
un KIC (MPa·m1/2)
Keff (MPa·m1/2)
WS1200-0 WS1200-1 WS1200-2 WS1000-1 WS1000-3 WS1000-4 WS1000-5 WS800-1 WS800-2 WS800-4 WS800-5 WS600-1 WS600-2 WS600-3 WS600-5 WS400-1 WS400-2 WS400-3 WS200-1 WS200-2 WS200-3 WS150-1 WS150-2 WS150-3
0.45 0.46 0.45 0.45 0.45 0.45 0.45 0.45 0.46 0.46 0.46 0.46 0.46 0.46 0.46 0.45 0.46 0.46 0.46 0.47 0.47 0.46 0.48 0.49
1198 1200 1200 997 997 999 1000 799 800 798 801 600 599 600 599 400 400 400 200 200 200 150 150 150
200 201 200 200 200 196 200 196 194 200 200 193 200 193 200 200 198 199 200 200 200 180 178 176
120 120 120 100 100 100 100 80 80 80 80 60 60 60 60 40 40 40 20 20 20 15 15 15
65 65 65 54 54 55 55 44 44 43 43 32 33 33 33 22 22 22 11 11 11 8 8 8
24.70 24.21 28.26 22.15 23.34 25.81 24.40 18.66 22.60 21.63 21.99 17.82 16.97 16.68 18.22 11.37 14.27 12.00 7.08 6.89 6.61 4.73 5.59 5.23
1.240 1.233 1.294 1.231 1.328 1.493 1.503 1.472 1.455 1.593 1.299 1.561 1.315 1.397 1.509 1.131 1.515 1.369 1.013 0.982 0.965 0.812 0.924 0.846
0.94 0.92 1.07 0.92 0.97 1.09 1.00 0.89 1.09 1.01 1.03 1.01 0.92 0.93 0.98 0.74 0.96 0.81 0.67 0.67 0.64 0.58 0.72 0.70
1/2
0.5
ft = 4.16 MPa KIC= 1.12 MPa⋅m
0.4
a∞ = 18 mm
∗
β = 0.0
2
0.2 0.1
Curve fitting 0
15
30
45
ae
60
75
90 1/2
0.5
ft = 2.67 MPa KIC= 1.29 MPa⋅m
0.4
a∞ = 58.6 mm
∗
β = 2.0
β = 1.0
2
0.2
0
15
30
45
ae
Test data 60
75
90
1/2
0.5
ft = 3.22 MPa KIC= 1.17 MPa⋅m
0.4
a∞ = 33.3 mm
∗
β = 1.0 — 1.5
0.2 0.1
0.0 -0.1
Curve fitting
0.3
1/σn
2
1/σn
∗
0.2
-0.1
0.1
Curve fitting 0
15
30
45
ae
0.0
Test data 60
75
-0.1
90
1/2
ft = 3.23 MPa KIC= 1.14 MPa⋅m
0.5
∗
a∞ = 30.8 mm
0.4
β = 1.0 — 2.0
0
15
30
45
ae
Test data 60
75
90
1/2
ft = 3.25 MPa KIC= 1.10 MPa⋅m ∗
a∞ = 28.7 mm
0.4
β = 1.0 — 3.0
0.3 2
1/σn
0.2
0.2 0.1
0.1 0.0 -0.1
Curve fitting
0.5
0.3 2
a∞ = 34.6 mm
0.0
Test data
0.3
1/σn
0.4
0.1
0.0 -0.1
ft = 3.21 MPa KIC= 1.20 MPa⋅m
0.3
1/σn
2
1/σn
0.3
1/2
0.5
Curve fitting 0
15
30
45
ae
0.0
Test data 60
75
90
-0.1
Curve fitting 0
15
30
45
ae
Test data 60
75
Fig. 7. ft and KIC for different Δafic (β = 0.0, 1.0, 2.0, 1.0–1.5, 1.0–2.0, 1.0–3.0) using WS tests. 120
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1/2
Keff (MPa⋅m )
Table 4 Estimation of ft and KIC from WS tests.
1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5
The reasonable value range of β (=0.2–2.4) is larger than those of 3p-b (β = 0.6–2.0) because the range of (W-a0) / dmax = 8–65 for WS specimens is larger than (W-a0) / dmax = 11–24 for 3-p-b specimens. Similarly, the accurate approximation can be obtained using the averaged β = 1.0 for (W-a0) / dmax = 8–65. As mentioned, the discrete nature or apparent “uncertainty” in fictitious crack growth of concrete can be “certainly” reflected using individual discrete number β. In this way, two individual values of β are selected as 1.0 and 2.0 for W = 150 and 1200 mm, and other β values for other W sizes can be obtained via interpolation. That is, β = 1.0, 1.05, 1.24, 1.43, 1.62, 1.81 and 2.0 for W = 150, 200, 400, 600, 800, 1000 and 1200 mm. If β are selected as 1.0 and 1.5 for W = 150 and 1200 mm, then β = 1.0, 1.02, 1.17, 1.26, 1.36, 1.45 and 1.5 for W = 150, 200, 400, 600, 800, 1000 and 1200 mm. If β are selected as 1.0 and 3.0 for W = 150 and 1200 mm, then β = 1.0, 1.10, 1.48, 2.05, 2.43, 2.81 and 3.0 for W = 150, 200, 400, 600, 800, 1000 and 1200 mm. The fitted curves and the obtained results of material constants for individual β are shown in Fig. 7. It can be found that the values of ft and KIC obtained by using individual β is consistent with values from β = 1.0. Therefore, the averaged discrete number β = 1.0 is an enough good approximation for (W-a0) / dmax = 8–65. un As shown in Table 3, the calculated unstable fracture toughness KIC determined by the double-K model, using experimental Pmax and CMODc from WS specimens, are from 0.812 MPa·m1/2 to 1.593 MPa·m1/ 2 [40]. These results show a slight difference when compared with the 0.963 MPa·m1/2 to 1.389 MPa·m1/2 from 3-p-b specimens determined by the double-K model and still show large scatters. For WS specimen type, the calculation formula of fracture toughness KIC meeting LEFM condition in ASTM standards [44] and are as follows:
1/2
KIC = 1.20 MPa⋅m (β =1.0) 1.12 (β =0.0)
0
200 400 600 800 1000 1200 1400 W (mm)
Fig. 8. Fracture toughness Keff calculated by ASTM using WS concrete tests.
p-b specimens (W = 500 mm and W/dmax = 50), Keff are 0.81, 0.82 and 0.90 MPa·m1/2, which are smaller than the KIC = 1.12 MPa·m1/2 (β = 1.0). 3.2. Determination of independent ft and KIC of concrete using WS tests Compared with 3-p-b specimens, these details WS experimental concrete specimens are listed in Table 3. The relative size W/dmax is from 15 to 120, then the corresponding (W-a0) / dmax is 8–65. The experimental results of WS concrete specimens and the corresponding fitting curves when β = 0.0 (no fictitious crack growth), 1.0, 2.0, 1.0–1.5, 1.0–2.0, 1.0–3.0, are shown in Fig. 7. The averaged values β = 0.0–3.0 with a fine increment of 0.2 are listed in Table 4 with the estimated ft and KIC values, and the characteristic values a . Table 4 shows that the ft and KIC results for averaged β = 0.2–2.4 are sufficient.
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Fig. 9. ft and KIC for different Δafic (β = 0.0, 1.0, 2.0, 1.0–2.0) with 3-p-b and WS specimens.
KIC =
Pmax S f( ) BW 1/2
f ( ) = (2 + )
0.886 + 4.64
curves when averaged β = 0, 1.0, and 2.0 are shown in Fig. 9 and Table 5. And results for individual β (β = 1.0–2.0, β = 1.0–3.0, β = 1.0–4.0) are also shown in Fig. 9 and Table 5. That is β = 1.0, 1.05, 1.14, 1.24, 1.33 ,1.43, 1.62, 1.81 and 2.0 for W = 150 mm, 200 mm, 300 mm, 400 mm, 500 mm, 600 mm, 800 mm, 1000 mm, and 1200 mm; β = 1.0, 1.10, 1.29, 1.48, 1.67,1.86, 2.24, 2.62 and 3.0 for W = 150 mm, 200 mm, 300 mm, 400 mm, 500 mm, 600 mm, 800 mm, 1000 mm, and 1200 mm; β = 1.0, 1.14, 1.43, 1.71, 2.00, 2.29, 2.86, 3.43 and 4.0 for W = 150 mm, 200 mm, 300 mm, 400 mm, 500 mm, 600 mm, 800 mm, 1000 mm, and 1200 mm. As depicted in Fig. 9 and Table 5, the averaged discrete number β = 1.0 is an accurate approximation for (W-a0)/dmax = 8–65. The values of ft = 3.28 MPa and KIC = 1.05–1.14 MPa·m1/2 obtained by using individual β (even β = 1.0–4.0) are consistent with values from β = 1.0 (ft = 3.28 MPa and KIC = 1.17 MPa·m1/2). Comparing Tables 2, 4, and 5, the ft and KIC of concrete determined by using 3-p-b and WS specimens together, are consistent with those determined by 3-p-b and WS specimens, respectively. Based on β = 1.0 as an example, the values of ft and KIC are relatively closer:
(15a)
13.32 2 + 14.72 (1 )3/2
3
5.6
4
(15b)
The fracture toughness Keff calculated directly by Eq. (15), using experimental peak loads Pmax from the WS concrete specimens, are shown in Table 4 and Fig. 8. These fracture toughness Keff exhibit evident size effects: Keff increase with the depth of specimen W increases and its value tends to stabilize as W increases to 800 mm (W/dmax ≥ 80). Interestingly, these fracture toughness Keff calculated by ASTM for the three largest 1200 mm specimens (W/dmax = 120) are 0.94, 0.92 and 1.07 MPa·m1/2, which are smaller but closer to the KIC = 1.20 MPa·m1/2 (β = 1.0). 3.3. Determination of independent ft and KIC using 3-p-b and WS tests together Experimental results of 3-p-b and WS concrete specimens and fitting
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0.75
Table 5 Estimation of ft and KIC from 3-p-b and WS specimens.
ft = 3.44 MPa ∗ ae / a ∞ = 0.1
dmax = 10 mm β = 1.0 W/dmax = 20 — 50
log σn
0.50
( W = 4.1 m , α = 0.2 )
0.25
∗
ae / a ∞ = 10
W = 200 mm W = 300 mm W = 400 mm W = 500 mm
0.00
1/2
KIC = 1.12 MPa⋅m
α = 0.5
-0.25 0.0
∗
a∞ = 26.4 mm
0.5
1.0
1.5
2.0
2.5
3.0
3.5
log ae (a) Fracture curve determined by using ft = 3.44 MPa and KIC = 1.12 MPa·m1/2(β = 1.0) 0.75
ft = 3.62 MPa
dmax = 10 mm β = 1.0 — 1.5
∗
ae / a ∞ = 0.1
W/dmax = 20 — 50
log σn
0.50
ft = 3.44 MPa and KIC = 1.12 MPa·m1/2 determined by 3-p-b specimens, ft = 3.21 MPa and KIC = 1.10 MPa·m1/2 determined by WS specimens, and ft = 3.28 MPa and KIC = 1.17 MPa·m1/2 determined by using 3-p-b and WS specimens together.
( W = 2.9 m , α = 0.2 ) ∗
ae / a ∞ = 10
0.25
W = 200 mm W = 300 mm W = 400 mm W = 500 mm
0.00
1/2
KIC = 0.99 MPa⋅m
∗
α = 0.5
-0.25 -0.5
4. Prediction of structural behavior using material parameters of concrete
0.0
a∞ = 18.6 mm 0.5
1.0
1.5
2.0
2.5
3.0
log ae
(b) Fracture curve determined by using ft = 3.62 MPa and KIC = 0.99 MPa·m1/2(β = 1.0 – 1.5)
Three fracture regions are present in concrete specimens or structures [13–25], as shown in Fig. 1: (1) tensile strength ft controlled fracture region with crack ratio ae/a < 0.1, (2) quasi-brittle fracture region controlled by ft and KIC, 0.1 < ae/a < 10, and (3) LEFM region with a large crack ratio, ae/a > 10, where KIC applies. Eq. (3) or (6) can be applied to the three fracture regions. If the material parameters of concrete ft and KIC are determined, then the complete fracture curve or structural safe design diagram with three fracture regions can be constructed based on Eq. (3) or (6). The estimations of the minimum concrete size can be obtained based on the crack ratios ae/a > 10 and Eq. (3) or (6) to achieve LEFM condition, that is, if the ft and KIC values are determined, then the a and the corresponding ae value can be obtained using the relationship of ae/a = 10. The different W values with different α = a0 / W (e.g., α = 0.2) can be obtained according to Eq. (4a) or (5a).
0.75
ft = 3.83 MPa
dmax = 10 mm β = 1.0 — 2.0 –
∗
ae / a ∞ = 0.1
W/dmax = 20 — 50
log σn
0.50 0.25 0.00
( W = 2.2 m , α = 0.2 ) ∗
ae / a ∞ = 10 W = 200 mm W = 300 mm W = 400 mm W = 500 mm
1/2
KIC = 0.90 MPa⋅m
α = 0.5
-0.25 -0.5
0.0
∗
a∞ = 13.9 mm 0.5
1.0
1.5
2.0
2.5
3.0
log ae
4.1. Prediction of structural failure using material parameters from 3-p-b concrete specimens
(c) Fracture curve determined by using ft = 3.83 MPa and KIC = 0.90 MPa·m1/2(β = 1.0 – 2.0)
Fracture failure curves with three fracture regions using material parameters from 3-p-b concrete specimens are shown in Fig. 10. As shown in Fig. 10, the concrete specimen size can meet the LEFM condition if the concrete specimen depth W is larger than 2 m. However, the depth of concrete specimens (W = 150–1200 mm) used in this analysis is less than 2 m, and the 3-p-b concrete specimens (W = 200–500 mm) are in quasi-brittle fracture.
Fig. 10. Structural failure curves determined by material constants from 3-p-b specimens.
individual β = 1.0–1.5 and β = 1.0–2.0, using material parameters from WS concrete specimens, are shown in Fig. 11. It can be found that the largest specimens (W = 1200 mm and W/ dmax = 120) still cannot completely meet the requirements of LEFM (W = 2.0 m with α = 0.2 and W = 2.9 m with α = 0.45), but are closer to the brittle “homogeneous” material.
4.2. Prediction of structural failure using material parameters from WS concrete specimens Structural failure curves with three fracture regions for β = 1.0 and 123
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0.75
ft = 3.21 MPa
average prediction due to the scatters of experimental results of concrete; the upper and lower limits can be selected in the safe design diagram. For example, the material constants ft = 3.28 MPa and KIC = 1.17 MPa·m1/2 (β = 1.0) are used as an average prediction, and then the upper and lower bounds ( ± 15%), that is, ft = 3.78 MPa and KIC = 1.35 MPa·m1/2(+15%), and ft = 2.79 MPa and KIC = 1.00 MPa·m1/2 (−15%), can be selected to describe the variations in scatters and the fracture properties of concrete. As shown in Fig. 12(b), (d), (f), the safe design diagrams with upper and lower limits ( ± 15%) nearly cover all the experimental results.
dmax = 10 mm β = 1.0
∗
ae / a ∞ = 0.1
W/dmax = 15 — 120
0.50
log σn
( W = 3.6 m , α = 0.2 ) W = 150 mm W = 200 mm W = 400 mm W = 600 mm W = 800 mm W = 1000 mm W = 1200 mm α = 0.45
0.25 0.00
-0.25 0.0
0.5
1.0
∗
ae / a ∞ = 10 1/2
KIC = 1.20 MPa⋅m
5. Predictions of peak loads Pmax pertinent to various concrete specimens
∗
a∞ = 34.6 mm 1.5
2.0
2.5
3.0
3.5
t The comparison results of the experimental peak loads Pmax and the c calculated peak loadsPmax obtained from predicted fracture curves determined by different material constants (β = 1.0: ft = 3.28 MPa and KIC = 1.17 MPa·m1/2; β = 2.0: ft = 2.74 MPa and KIC = 1.26 MPa·m1/2; β = 1.0–2.0: ft = 3.28 MPa and KIC = 1.14 MPa·m1/2; β = 1.0–3.0: ft = 3.28 MPa and KIC = 1.10 MPa·m1/2) are shown in Table 6. As shown in Table 6, the predictions of peak loads have high precision for the single specimen and the whole. The difference between prediction results using the various material constants (ft and KIC) determined by different fictitious crack growth lengths Δafic (β = 1.0, 2.0, 1.0–2.0, 1.0–3.0) is relatively small. The average Δafic ≈ 1.0 dmax is sufficient for predicting the peak loads of concrete with W/ dmax = 15–120. If the independent material constants ft = 3.28 MPa and KIC = 1.17 MPa·m1/2 have been obtained, then the values of fictitious crack growth length Δafic can be inverted to make the calculated peak loads completely the same as the experimental peak loads. The inversions of Δafic are shown in Fig. 13. As shown in Fig. 13, the inverted Δafic values indicate an increasing trend with the depth of specimen W. However, this change is insignificant. The discreteness of the inverted Δafic values is increase in either W or W-a0. The Δafic values have a large scatter and randomness and difficult to determine accurately. The mean of the inverted Δafic values is 13.7 mm, which is closer to dmax = 10 mm. Therefore, the average Δafic ≈ 1.0 dmax is simple and sufficiently accurate for predicting the peak loads of concrete with W/dmax = 15–120.
log ae (a) Fracture curve determined by using ft = 3.21 MPa and KIC = 1.20 MPa·m1/2(β = 1.0) 0.75
β = 1.0 — 1.5
ft = 3.22 MPa ∗ ae / a ∞ = 0.1
dmax = 10 mm W/dmax = 15 — 120
log σn
0.50
( W = 3.5 m , α = 0.2 ) ∗ ae / a ∞ = 10
W = 150 mm W = 200 mm W = 400 mm W = 600 mm W = 800 mm W = 1000 mm W = 1200 mm α = 0.45
0.25 0.00
-0.25 0.0
0.5
1.0
1/2
KIC = 1.17 MPa⋅m ∗
a∞ = 33.3 mm 1.5
2.0
2.5
3.0
3.5
log ae (b) Fracture curve determined by using ft = 3.22 MPa and KIC = 1.17 MPa·m1/2(β = 1.0 – 1.5) 0.75
β = 1.0 — 2.0
ft = 3.23 MPa ∗ ae / a ∞ = 0.1
dmax = 10 mm W/dmax = 15 — 120
log σn
0.50 0.25 0.00 -0.25 0.0
W = 150 mm W = 200 mm W = 400 mm W = 600 mm W = 800 mm W = 1000 mm W = 1200 mm α = 0.45
0.5
1.0
6. Conclusions
( W = 3.2 m , α = 0.2 ) ∗ ae / a ∞ = 10
In the present study, the relationship between material behavior (e.g., tensile strength ft and fracture toughness KIC) and structural behavior (e.g., failure curve or safe design diagram) of concrete is established. Using different concrete specimen types, independent material parameters (ft and KIC) are determined. The structural failure of concrete is predicted using independent material constants. New findings and the major conclusions are as follows.
1/2
KIC = 1.14 MPa⋅m ∗
a∞ = 30.8 mm 1.5
2.0
2.5
3.0
3.5
log ae (c) Fracture curve determined by using ft = 3.23 MPa and KIC = 1.14 MPa·m1/2(β = 1.0 – 2.0)
(1) The independent tensile strength ft and the fracture toughness KIC of the concrete can be determined using different concrete specimens (e.g., 3-p-b and WS specimens).The ft and KIC values in 3-p-b and WS specimens are consistent with those in 3-p-b and WS specimens, respectively.
Fig. 11. Structural failure curves determined by material constants from WS specimens.
The discrepancy of experimental results is an inherent property of concrete materials. This property can be well described by the fictitious crack growth length Δafic corresponding to peak load Pmax, which is related to maximum aggregate size dmax through a discrete number β. Rectangular, trapezoidal and triangle stress distributions in fictitious crack growth have minimal effects on ft and KIC value determination. Using the rectangular stress distribution is sufficiently accurate to determine the ft and KIC values of concrete.
4.3. Prediction of concrete failure using material parameters from 3-p-b and WS specimens Based on the selection of β = 1.0 and individual β = 1.0–2.0, β = 1.0–3.0, the prediction curves with three fracture regions, using material parameters from 3-p-b and WS concrete specimens together, are illustrated in Fig. 12. The determined material constants (for a given β) can be used for an
(2) The structural failure or the safe design diagram of concrete can be 124
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Fig. 12. Structural failure curves determined by using material constants from various specimens (3-p-b and WS).
fully constructed using the independent ft and KIC. The safe design diagrams with upper and lower limits ( ± 15%) nearly cover all experimental results. (3) The theoretical concrete specimen size meeting the LEFM can be obtained using the determined independent material constants.
The concrete experimental 3-p-b (W ≤ 500 mm) and WS (W ≤ 1200 mm) specimens are in quasi-brittle fracture because the theoretical depth is larger than 2 m. Fracture toughness Keff of concrete calculated directly by the LEFM formula increases as the depth of the specimen increases, and Keff from the largest 1200 mm (W/dmax = 120) specimen is smaller but closer to the independent KIC.
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Table 6 Predictions of peak loads Pmax of 3-p-b and WS specimens. Label
t Experimental peak loads Pmax (kN)
c (kN) Predicted peak loads Pmax
β = 1.0
TPB500-1 TPB500-2 TPB500-3 TPB400-1 TPB400-2 TPB400-3 TPB400-4 TPB300-1 TPB300-2 TPB300-3 TPB300-4 TPB300-5 TPB200-2 TPB200-3 TPB200-4 WS1200-0 WS1200-1 WS1200-2 WS1000-1 WS1000-3 WS1000-4 WS1000-5 WS800-1 WS800-2 WS800-4 WS800-5 WS600-1 WS600-2 WS600-3 WS600-5 WS400-1 WS400-2 WS400-3 WS200-1 WS200-2 WS200-3 WS150-1 WS150-2 WS150-3
9.81 11.93 9.85 11.13 10.97 10.39 9.96 7.91 8.32 7.84 8.39 8.29 7.19 6.23 6.64 24.70 24.21 28.26 22.15 23.34 25.81 24.40 18.66 22.60 21.63 21.99 17.82 16.97 16.68 18.22 11.37 14.27 12.00 7.08 6.89 6.61 4.73 5.59 5.23
β = 2.0
β = 1.0–2.0
β = 1.0–3.0
c Pmax
t c Pmax /Pmax
Pc max
t c Pmax /Pmax
Pc max
t c Pmax /Pmax
Pc max
t c Pmax /Pmax
9.99 12.25 8.95 10.70 10.50 10.32 10.22 8.45 8.20 8.60 8.27 7.90 6.44 6.14 6.47 26.57 26.57 26.52 23.68 23.61 23.30 23.95 20.07 19.84 20.42 20.33 15.95 16.83 16.30 16.83 12.88 12.43 12.49 7.70 7.38 7.38 5.53 5.22 4.98
0.98 0.97 1.10 1.04 1.04 1.01 0.97 0.94 1.01 0.91 1.01 1.05 1.12 1.02 1.03 0.93 0.91 1.07 0.94 0.99 1.11 1.02 0.93 1.14 1.06 1.08 1.12 1.01 1.02 1.08 0.88 1.15 0.96 0.92 0.93 0.90 0.86 1.07 1.05
9.87 12.09 8.86 10.57 10.38 10.21 10.10 8.42 8.17 8.56 8.24 7.89 6.56 6.27 6.58 26.88 26.88 26.82 23.82 23.74 23.44 24.09 20.05 19.82 20.40 20.31 15.82 16.70 16.18 16.70 12.73 12.29 12.35 7.75 7.43 7.43 5.66 5.35 5.12
0.99 0.99 1.11 1.05 1.06 1.02 0.99 0.94 1.02 0.92 1.02 1.05 1.10 0.99 1.01 0.92 0.90 1.05 0.93 0.98 1.10 1.01 0.93 1.14 1.06 1.08 1.13 1.02 1.03 1.09 0.89 1.16 0.97 0.91 0.93 0.89 0.84 1.05 1.02
10.09 12.33 9.06 10.76 10.57 10.39 10.29 8.48 8.23 8.63 8.30 7.94 6.43 6.13 6.46 26.62 26.62 26.56 23.73 23.66 23.35 23.99 20.12 19.88 20.47 20.39 15.99 16.88 16.34 16.88 12.90 12.46 12.52 7.67 7.35 7.35 5.48 5.17 4.94
0.97 0.97 1.09 1.03 1.04 1.00 0.97 0.93 1.01 0.91 1.01 1.04 1.12 1.02 1.03 0.93 0.91 1.06 0.93 0.99 1.11 1.02 0.93 1.14 1.06 1.08 1.11 1.01 1.02 1.08 0.88 1.15 0.96 0.92 0.94 0.90 0.86 1.08 1.06
10.20 12.42 9.18 10.82 10.63 10.46 10.35 8.52 8.27 8.67 8.34 7.98 6.42 6.12 6.45 26.66 26.67 26.61 23.78 23.71 23.40 24.03 20.17 19.93 20.53 20.44 16.04 16.92 16.38 16.92 12.92 12.48 12.54 7.64 7.33 7.33 5.43 5.13 4.90
0.96 0.96 1.07 1.03 1.03 0.99 0.96 0.93 1.01 0.90 1.01 1.04 1.12 1.02 1.03 0.93 0.91 1.06 0.93 0.98 1.10 1.02 0.93 1.13 1.05 1.08 1.11 1.00 1.02 1.08 0.88 1.14 0.96 0.93 0.94 0.90 0.87 1.09 1.07
Mean
1.008
1.007
1.006
1.004
Deviation coefficient
0.075
0.077
0.075
0.074
Inverted value dmax=10 mm
50
mean =13.7 mm
30 20 10 0
mean =13.7 mm
40 ∆afic (mm)
∆afic (mm)
40
Inverted value dmax=10 mm
50
30 20 10
0
200 400 600 800 1000 1200 1400 W (mm)
0
0
100
200 300 400 W-a0 (mm)
500
600
Fig. 13. Inversion of Δafic using ft = 3.28 MPa and KIC = 1.17 MPa·m1/2.
(1) The averaged discrete number β = 1.0 is a sufficient approximation to determine the ft and KIC values of concrete for W/dmax = 15–120. The ft and KIC values obtained using individual β (e.g. β = 1.0–1.5, 1.0–2.0, 1.0–3.0, 1.0–4.0) are consistent with the values from β = 1.0.
The predicted peak loads by using the improved BEM fit the experimental peak loads. The inverted values of Δafic exhibit large scatters. According to obtained results, several practical recommendations on the use of the improved BEM can be suggested, as follows:
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(2) The average β ≈ 1.0, that is, Δafic ≈ 1.0dmax, is sufficient to predict concrete fracture failure. (3) The average β ≈ 1.0 or Δafic ≈ 1.0dmax with sufficient relative smallest deviation coefficient to predict the peak loads of the specimens.
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