Specific Fracture Energy Determination of dam concrete by large-scale size
effect tests
A.R. Ghaemmaghami and M. Ghaemian
Civil Engineering Department
Sharif University of Technology
Tehran, Iran
ABSTRACT
The specific fracture energy of dam concrete is a basic material characteristic needed for the prediction of concrete dam
behavior. Data on fracture properties of dam concrete are quite limited to date. A series of tests was carried out based on the
size effect due to a number of geometrically similar notched specimens of various sizes. To compare the fracture properties of
dam concrete with normal concrete five types of concrete mixes of 65, 50, 40, 30 and 20mm maximum aggregate size were
considered. For all mixes, value of relative notch depth of 0.2 is used. Experimental results show that aggregate size has an
important effect on specific fracture energy values of dam concrete. According to specific fracture energy definition based on
size effect, the specific fracture energy of concrete mixes of 65, 50, 40, 30 and 20mm M.S.A. are 178, 129, 88, 79 and 72 N/m,
respectively. This indicates that the high specific fracture energy of dam concrete is the result of the nature, size and
properties of its aggregate. The difference between specific fracture energy of normal and dam concrete should be noticed in
nonlinear analysis of concrete dams
Introduction
The numerical simulation of concrete dams requires nonlinear properties of dam concrete. Some of the important nonlinear
parameters are rigorously determined and need to be investigated. One of the most important parameters is specific fracture
energy. The specific fracture energy of dam concrete is a basic material characteristic needed for a rational prediction of
concrete dams fracture behavior. Fracture energy determination depends on testing and the used concrete crack model.
Although in principle, the specific fracture energy as a material property should be constant, and its value should be
independent of the method of measurement, various test methods, specimen shapes, and sizes yield very different results
sometimes differing even by hundred percent [1-10]. The mass concrete which is typically used in dam construction is
unreinforced, has very large maximum aggregate size, low water-cement ratio to improve strength, and low cement content to
reduce thermal cracking and shrinkage during curing. The large size aggregate used in dam concrete necessitates the use of
very large fracture specimens. Thus, the main difficulty in the determination of specific fracture energy of dam concrete lies in
large specimens. Data on fracture properties of dam concrete are limited presently. In dam concrete, the fracture process
takes place over a relatively large fracture process zone whose size is, for the laboratory specimens, of the same order of
magnitude as the size of the specimen itself.
Dam concrete is a material characterized by blunt fracture. The material behavior in the fracture process zone may be
described by a strain-softening relation or, to some extent equivalently by a stress-displacement relation with softening that
characterizes the fracture process zone over its full width. This behavior indicates that the fracture energy is not the only
controlling parameter and that the size and shape of the fracture process zone, as well as the shape of the softening stressstrain diagram, have a significant influence.
Size Effect
The structural size effect is probably the most important manifestation of fracture phenomena, at least from the engineering
point of view. Therefore, it is important to relate the size effect behavior to the fracture properties of the material, such as
specific fracture energy. As previously described, the G f value of a microscopically heterogeneous quasi-brittle material such
as concrete is the energy required for crack growth in a unit surface of an infinitely large specimen. The fracture energy
determination would be exact if we knew the exact form of the size-effect law to be used for extrapolation to infinite size.
Unfortunately, we know this law only approximately. In the large size range, all the equivalent crack models satisfy at peak
load, more over, for large specimen sizes ( D → ∞ ), the critical equivalent crack extension ∆a ec tends to a limiting constant
value c f ( ∆a ec → c f for D → ∞ )
Size effect phenomenon, which is an important consequence of fracture mechanics, has been described by the size effect law
proposed by Bazant [11, 12]:
Bf t′
σ NU =
1+
(1)
D
D0
Pu
where Pu , b , d and c n are maximum load, specimen thickness, and characteristic dimension of the
bd
a
specimen or structure and shape coefficient, respectively. α 0 is relative notch length and equal with 0 where a 0 is notch
d
in which σ NU = c n
depth.
Bf t′ and D0 can be defined as:
K Ic
Bf t′ =
(2)
2k 0 k 0′ c f
D0 =
2k 0′
cf
k0
(3)
K IC is critical intensity factor and k ( α ) is dimensionless stress intensity factor.
The equation (1) might be more appropriately called the size-shape effect law, because the function k (α ) introduces the
effect of geometry (shape).
The relations (2) and (3) are fundamental to the experimental determination of the fracture properties of concrete based on
size effect, the equation (3) reveals the basic characteristics of the transitional size D0 . It is proportional to the effective length
of the fracture process zone c f , which in turn, is approximately proportional to the inhomogeneity size of the material and also
2
2
to the characteristic size l 1 = K Ic / f t′ . It is also proportional to the ratio 2k ′(α 0 ) / k (α 0 ) which is independent of material
properties and introduces the effect of structure geometry .
Equation (4) shows the basic structure of B . It can be written in terms of l1 as:
B=
Which shows that
B
1
2k 0 k 0′
l1
=
Cf
1
g 0′
l1
Cf
also consists of a product of geometrical and material functions. The material parameter is β =
(4)
Cf
l1
and
is related to the softening behavior of the material in which the value of concrete can be estimated in the range of 2 to 5.
Results of previous research show that different experimental techniques or different analyses may lead to different values of
fracture parameters such as specific fracture energy. These parameters can be uniquely defined by extrapolating their values
for infinite size specimen. Since specimen failure is dictated by the material characteristics, it is possible to determine these
characteristics from size effect measurements. In order to determine specific fracture energy from the test on specimens, D0
and Bf t′ can be determined from the experiment Then K Ic and c f can be obtained from equations (2) and (3):
K Ic = Bf t′ D 0 k 0
cf =
k0
D0
2k 0′
(5)
(6)
In linear Elastic Fracture Mechanics (LEFM), according to Irwin’s [13] relationship specific fracture energy for infinite size
specimens can be obtained as:
K Ic2
E
2
( Bf t′) D0 k 02
Gf =
E
Gf =
(7)
(8)
Experimental Procedure
An experimental program of fracture tests was conducted to obtain specific fracture energy values. In the size effect method, a
number of geometrically similar notched specimens of various sizes are tested for the peak loads. Specimen dimensions are
shown in table (1). All the specimens were tested in a load control condition with medium load rate.
Specimen
Small
Medium
Large
D
(mm)
200
400
800
Table (1): Specimen dimensions
B
L
(mm)
(mm)
200
600
400
1200
800
2400
S
(mm)
500
1000
1200
The nominal stress is then computed and plotted against the specimen size. The parameters Bf t′ and D0 are obtained by
the optimal least-square fitting of the size effect law to the experimental results. Finally, specific fracture energy is obtained
from equation (8). In this study the three-point bent specimens are used to determine specific fracture energy of dam concrete.
The three-point bent test is a procedure, in which roughly half of the cross section is subjected to tension half to compression,
and the fracture process zone is of medium size. According to the RILEM recommendations [14] the cross sections of the
specimen are rectangular, and the span to depth ratio is 2.5:1 for all specimens. The cross section heights of the bending
specimens are d=200, 400, 800mm.
Geometrically similar notched specimens of five batches of concrete were prepared. The first concrete mix (Karoon 3 dam
concrete mix) had a water-cement ratio of 0.42 and a aggregate-cement ratio of 8.51 (all by weight). The maximum size of
coarse aggregate for this mix was 75 mm. The aggregate consisted of crushed stones from quarries and river sand. Due to the
size limitation in the test machine, the maximum depth of the specimens was limited to 800mm. As a result, the maximum size
of aggregate was reduced to 65mm to meet the RILEM requirement for the smallest depth [14]. Karoon 3 dam concrete mix
°
design is shown in table (2). All the specimens were cast at about 25 C. The specimens were demodeled 7 days after casting
°
and then cured in a moisture room at 20 C. Seventy-Five specimens were prepared to cover the test program. To determine
the strength, companion cylinders of 250mm diameter and 500mm length were cast from each batch of concrete. After the
standard 90-day moist curing, the mean compression strength was obtained.
Table (2): Karoon 3 dam concrete mix design
Maximum size of aggregate (M.S.A.) 65 mm
no entrained air, w/c ratio=0.53
Weight (Kg/m3)
Water and Ice
118
Cement
224
Sand
821
5-19 mm
286
19-37.5 mm
323
37.5-65 mm
647
Total
2419
The tensile strength was obtained from splitting tensile test. To determine the tensile strength, cylinders of 150mm diameter
and 300mm length were used for each concrete mix. After the standard 90-day moist curing the mean tension strength was
obtained. Young’s modulus was estimated as a function of compressive strength according to ASTM. For all concrete mixes, a
value of relative notch depths of 0.2 is used. Mix and strength properties are shown in table (3).
Table (3): Mix and Strength Properties
Concrete Mix
M.S.A.
(mm)
Water-Cement
Ratio
Aggregate-Cement
Ratio
1
2
3
4
5
65
50
40
30
20
0.53
0.45
0.39
0.3
0.28
9.51
8.95
8.12
7.43
7.11
f c'
(MPa)
31.9
33.7
38.1
40.5
42.7
ft'
Ec
(MPa)
2.7
3.7
4.2
4.4
4.7
(MPa)
26726
27470
29208
30114
30912
Notches were cut in the specimens just before testing. The loading mechanism for three-point bent specimens was designed
to apply three concentrated loads onto the specimen with one load through a hinge and two through rollers. The steel surface
was carefully machined so as to minimize the friction of the rollers in a 100-tone universal test machine. A universal joint was
provided at the top and bottom connections to the testing machine to minimize in-plane and out-of-plane bending effect. The
specimens were loaded at constant medium loading rate.
Analysis of the Results
Test results are summarized in table (4). To determine the specific fracture energy, size effect law in equation (1) can be
algebraically rearranged to a linear regression plot:
Y = AX + C
(9)
In which X = D , Y = (
1
σ
2
NU
) , Bf t′ =
1
C
and D0 =
C
A
The specific fracture energy can be obtained from following equation:
Gf =
k 2 (α 0 )
EA
(10)
Values of k (α0 ) for present three-point bent specimens can be calculated by linear elastic fracture and dimensional analysis.
According to Pastor e al.[15]:
k (α ) = a
p ∞ (α ) +
4D
[ p 4 (α ) − p ∞ (α )]
S
(11)
3
(1 + 2α )(1 − α ) 2
Where the polynomials p 4 (α ), p ∞ (α ) are given by following equations:
p 4 (α ) = 1.900 − α [−0.089 + 0.603(1 − α ) − 0.441(1 − α ) 2 + 1.223(1 − α ) 3 ]
(12)
p ∞ (α ) = 1.989 − α (1 − α )[0.448 − 0.458(1 − α ) + 1.266(1 − α ) ]
(13)
2
Mix No.
No.1
M.S.A.=65mm
No.2
M.S.A.=50mm
No.3
M.S.A.=40mm
No.4
M.S.A.=30mm
No.5
M.S.A.=20mm
Table (4): Test results
Specimen depth Average peak load
(mm)
(KN)
200
25.722
400
93.657
800
340.914
200
27.887
400
102.275
800
344.143
200
33.762
400
110.539
800
353.042
200
37.006
400
125.876
800
360.932
200
41.065
400
126.034
800
361.969
The specific fracture energy values are shown in table (5). The resulting specific fracture energy values for concrete mixes of
65, 50, 40, 30,20mm M.S.A. with relative notch depth of 0.2 are 178,129,88,79,72 N/m respectively. Contrary to the results
obtained by Saouma [9], the results show that aggregate size has an important effect on specific fracture energy of dam
concrete. This indicates that the high specific fracture energy of dam concrete is the result of the nature, size and properties of
its aggregate.
Figure (1) shows the specific fracture energy versus aggregate size. For normal concrete (M.S.A. =20, 30, 40 mm), the
specific fracture energy slightly increases as the maximum aggregate size increases. On the contrary, for dam concrete
(M.S.A. =50,65mm), the effect of aggregate size is much more noticeable.
It must be emphasized that the size-effect method fails if the size range is insufficient compared to the width of the scatter
band of the test results. The relevant data scatter is characterized by the standard deviation A of the regression line slope
and the corresponding coefficient of variation ω A , which are defined as:
ωA =
SA
A
(14)
S ηx
SA =
(15)
S x n −1
Where n is the number of all data points in the linear regression is, S x is the coefficient of the variation of the X values for all
the points and Sηx is the standard deviation of the vertical deviations from the regression line. The values of ω A are listed in
table (5).
Table (5): Results of Least-Square Optimization for obtaining fracture parameters
ωA
cf
Gf
%
(mm)
282
(N/m)
α0
M.S.A.
(mm)
A
1
0.2
65
0.000129
11.4
2
0.2
50
0.000172
8.3
3
0.2
40
0.000238
8.7
4
5
0.2
0.2
30
20
0.000256
0.000272
7.9
6.2
Mix type
178
140
129
56
88
27
15
79
72
All the major results on the investigation of fracture properties of dam concrete have been obtained by the work-of-fracture
method (WMF) [9,16,17]. These results are corresponding to the area under the complete stress-separation curve which
governs large postpeak deflections of structures.
200
Gf (N/m)
180
160
140
120
100
80
60
40
20
d a (mm)
0
0
10
20
30
40
50
60
70
Fig (1): Specific fracture energy versus aggregate size
The specific fracture energy values obtained by this method are quite sensitive to the specimen size and shape while the
values obtained by size effect method are size independent. According to the work done by Bazant and Giraudon [10], the
fracture energy of concrete, as well as other fracture parameters, can be approximately predicted from the standard
compression strength, maximum aggregate size, water-cement ratio, and aggregate type (crushed or river). A very large
database, consisting of 238 test series, was extracted from the literature and tabulated. Optimization of the fits of this data set
led to new approximate prediction formulae [8,10]:
f c′ 0.46
d
w
) (1 + a ) 0.22 ( ) −0.30 ; ω G f = 17.8 0 0
0.051
11.27
c
f c′ −0.019
d a 0.72 w 0.2 ⎤
⎡
)
(1 +
) ( ) ⎥; ω c f = 47.6 0 0
c f = exp ⎢γ 0 (
c
0.022
15.05
⎦
⎣
G f = α0(
(16)
(17)
Where G f is the specific fracture energy obtained in peak load tests such as size-effect method, c f is the effective length of
fracture process zone and α 0 = γ 0 = 1 for rounded aggregates, while α 0 = 1.44 and γ 0 = 1.12 for crushed or angular
aggregates. It should be noted that a distinction is made between (a) the fracture energy Gf, corresponding to the area under
the initial tangent of the softening stress-separation curve of cohesive crack model, which governs the maximum loads of
structures and is obtained by the size effect method and (b) the fracture energy GF, corresponding to the area under the
complete stress-separation curve, which governs large post-peak deflections of structure and is obtained by the work-offracture method [10]. Planas and Elices [18] estimated GF / Gf ≈ 2.0-2.5. For safety evaluation of structures of high fracture
sensitivity, such as concrete dams, both GF and Gf must be known although GF can be inferred from Gf with less uncertainty
than Gf from GF. To compare the results, the deviations of the specific fracture energy and effective length values obtained in
this study from Bazant's prediction formulae are shown in figures (2) and (3), respectively. It can be found that the specific
fracture energy values of normal concrete are in satisfactory agreement with the values obtained by Bazant's prediction
formula but, for dam concrete, the deviation from prediction formula is much more noticeable (more than 70 percent).
200
Gf (N/m)
180
160
140
120
Test results
upper limit of Bazant formula
Bazant formula
lower limit of Bazant formula
100
80
60
40
20
d a (mm)
0
0
10
20
30
40
50
60
70
Fig (2): Deviations of test results from Bazant prediction formula for specific facture energy
300
c f (mm)
250
200
Test results
Upper limit of Bazant formula
Bazant formula
Lower limit of Bazant formula
150
100
50
d a (mm)
0
0
10
20
30
40
50
60
70
Fig (3): Deviations of test results from Bazant prediction formula for effective fracture length
Due to the lack of dam concrete data, it seems that Bazant's prediction formulae are not suitable for predicting the fracture
properties of dam concrete. As a result, other alternatives should be developed in this case. For dam concretes, which are
highly sensitive to fracture and size effect, it is recommended that a test program be conducted to estimate the acceptable
values of specific fracture energy used in nonlinear numerical analyses in each case study.
Conclusions
Size effect law on the failure load of fracture specimens, coupled with the effect of aggregate size, is useful for measuring
fracture properties of dam concrete. On the other hand, the effect of size on the values of material fracture parameters can be
avoided. Based on the test results obtained in this study for different specimen sizes, it can be found that specific fracture
energy values of concrete mixes of 65,50,40,30 and 20mm M.S.A. are 178, 129, 88, 79 and 72 N/m, respectively. This
indicates that the high specific fracture energy of dam concrete is the result of nature, size and properties of its aggregate. The
high difference between specific fracture energy of normal concrete and dam concrete should be taken into account. Results
show that aggregate size has an important effect on the fracture properties of dam concrete. For normal concrete, the specific
fracture energy slightly increases as the maximum aggregate size increases. For dam concrete, the effect of aggregate size is
much more noticeable than other properties such as compressive strength, water-cement ratio or elastic modulus. The
effective length of the fracture process zone rapidly increases as the aggregate size increases. At this study, the ratio of the
effective length of dam concrete to maximum aggregate size is almost 4.5 which is in satisfactory agreement with crack band
theory. The high deviations of the results of this study from previously developed prediction formulae suggests that different
formulae may need to be developed for predicting the fracture properties of dam concrete. Because of the high sensitivity of
concrete dams to fracture and size effect, a test program should be conducted to estimate the acceptable values of specific
fracture energy used in nonlinear numerical analyses in each case study.
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