- Email: [email protected]
A method in determination of the size effectiveness of fracture energy in concrete
Engineering Frarrure Mechanics Printed in GreatBritain.
Vol. 35.
No. 4/S, pp. 679-685,
0013-7944/90
1990
$3.00 + 0.00
PergamonPress pk.
A METHOD IN DETERMINATION OF THE SIZE EFFECTIVENESS OF FRACTURE ENERGY IN CONCRETE YIN SHUANGZENG
and ZHAI QIYU
North China Institute of Hydropower, Handan City, Hebei Province, P.R.C. Abstract-In this paper, the authors present the measurement of the fracture energy G, of the three-point bend concrete beam in nine sizes by means of the method recommended by Rilem. The experimental result indicates that the concrete fracture energy measured in this way still possesses a size effectiveness. The authors think, in accordance with the conceptions of linear elastic fracture and virtual crack models, that when the crack tips in the microcracking zone in the cracking of concrete is simplified into virtual cracks capable of transmitting the surface force, there still exist certain amounts of strength factors around the tip zone of the virtual cracks. Thus the sum energy consumed in the cracking of the concrete cracks falls into two: (1) energy consumed in the cracking of the virtual crack surface; (2) energy released in the cracking of the virtual crack tips. It shows, having been experimented and calculated, that the sizes of the fracture energy in the three-point bend beam are mainly influenced by the amounts of the energy released from the virtual crack tips. If this energy is subtracted on the face of the virtual cracks in the material is, in general, not related to the sizes of the test specimen.
INTRODUCTION FRACTURE energy is defined as the surface energy consumed in unit area of the material cracking. It is generally considered that the fracture energy is constant in the concrete material, and the recent experiments have, however, proved that the fracture energy is still of size effect with the method recommended by Rilem in measuring the three-point bend concrete beam. It is not difficult for us to discover that, having analysed the trackings of l-type concrete cracks, there is a certain stress concentration around the crack tips in addition to the microcracking zones (see Fig 1). With the conception of the virtual crack models as a basis, the microcracking zone is able to be replaced by the virtual cracks capable of transmitting the surface force. The stress (T(X) transmitted on the surface of the virtual cracks is and only is determined by the gaping displacement V(x) in between the virtual crack surfaces [l]. Taking into account the conception of linear elastic fracture, when there is a cracking at the virtual crack tips, strain energy will be given off in the material within the stress concentrating zones. The amount of the strain energy is relevant to the degree the stress is concentrated. In order to show the degree of the stress concentrated at the tips of the virtual cracks, the concept of stress intensity factors is introduced for the virtual crack tips (see Fig. 2). So there are two parties for the energy consumed in crack extending: (1) the energy consumed in the demolishing of the microcracking zone; and (2) the energy released in the gaping by the stress concentration at the virtual crack tips.
ENERGY INTEGRATION The authors make an analysis of the energy relation of the l-type cracks in gaping according to the above points of view. Figure 3 presents the general conditions of the trackings of the l-type cracks caused by the external forces P and the closed surface force g(x). If the equivalent length a of the virtual cracks is increased by da, the deforming increment caused by force p and a(x) is “d8 and do(x)“. In accordance with mutual equivalent law, we have (1- “0 P da - o [a(x)dv(x)].dx.B = dao*(x,)2V*(x,)dx,-B (14 s s0 679
680
YIN SHUANGZENG
and ZHAI QIYU
in which cr*(x, ), and V*(x, ) are respectively, the distributing force and the corresponding displacement, i.e. [2]
gaping
cr*(x, ) = K, /,/271x,
(lb)
and V*(x, ) = 4,/da
- x, )/2n *K, /E
(lc)
if (1 b), and (lc) are substituted into (la), we get 0-00 Pd6 - o [a(x)dv(x)].dx.B =2Kf/E.da.B. (2a) s To the three-point bend beam shown in Fig. 4, the stress strength factor is K,,,, when the crossing cracks begin cracking, the deflections 6, of the initial cracking of the beam and the corresponding external force P, are, respectively, [3]
1 P, =
W
K,, H”*B
(24
1.5LF, h/H)
in which [5.58 - 19.57aJH
f’&JH) = 4 (so/H)=
1.99 - a,/H(l
a
+ 36.83(a,,/H)2 - 34.94(a,/H)3
- a,/H)[2.15
- 3.93aJH
(1 + 2a,/H)(l
- a,/H)‘.’
+ 2.7(a,JH)2]
+ 12.77(a,/H)7
1
and H, L, B, and a,, are separately the height, length, thickness and the length of the initial cracking; E, and y are the elastic modulus of the material and Poisson’s ratio; d is the coefficient affected by the shearing force. To the rectangle section d = lO(1 + v)/12 + 1lo), and the two factors of (2) being used to integrate the whole cracking of the beam and the two factors being divided by the ligament area Ali, = (h - a,)B of the beam, we have GF-GF*=G,+1/26,P,/A,,, P
i &
(3)
tp
Yl
U(X)
u*tx,j
X K, ad
0,
a
Y
a
da
@
P
P
Fig. 3.
XI
Fracture energy in concrete
681
w
FL------I Fig. 4.
in which Gr = energy consumed in the unit ligament area of the beam; Gz = area under the concrete-destruction softening curve; and & = rate of the average strain energy releasing in the beam cracking. CALCULATIONAL
SIMULATION IN THE CRACKING THREE-POINT BEND BEAM
OF THE
The microcracking zone of the edges of the crack tips may be substituted by the virtual cracks transmitting the surface force with energy, in the cracking of the crossing cracks within a three-point bend beam. If the length of the microcrack is a, substituting the surface force acting on the surfaces of the virtual crack, a(x) with the external force equivalent to the surface one, one can have &=;
‘-“0(x)x f0
dx .B
and therefore the formula for the calculation of the beam deflection 6 and the gaping displacement of cracks is
WdH) 1 + E[l + 2.8(H,L)q
(-MOD
=
(’ - ‘,)
- L2
2BH2+
~l+v
dB
(4)
G(P- P&L=F2 (a/H) H3BE
in which F2(a/H)
= 0.76 - 2.28a/H
+ 3.87(~/H)~ - 2.04(~/H)~ + 0.66/(1 - u/H)~
and the meanings of the other symbols are the same as the preceding ones. The distribution function of the gaping displacement on the crack surface is given as l’(x) = CMODJx(x
+ H - a),&?i
(6)
and, when x = 0, the distribution function value V(0) = 0, and V(a) = CMOD if X = a. On the basis of the principle of superposition, the stress intensity factor at the crack tip brought forth by both the concentration force P and the distribution force on the surface of cracks is [4] K, =- ;EF(a,H)
-f-%&F&$(x)dx
(7)
where 3.52x/a
4.35 - X28(1 - x/a) (1 - u,H)~‘~ (1 - u,H)“~ + (1 -a/H)
1.3 - 0.3(1 - x,a)3” 1 - (1 - x/a)3
+ 0.83 - 1.76(1 - x/a)
1.
682
YIN SHUANGZENG
and ZHAI QIYU
The distribution surface force on the surface of the virtual cracks a(x) being in a determined relation with its gaping displacement Y(x), their relation, based on literature1 [I], is a(x) =fi[f(W))
-
~(xh_J>(hJl
(8)
in which f(u(x)) = 1 +
c>
V(x) 3e -O.Zu(r)/u,
cl -
VII
where: f; = drawing strength of concrete; V. = gaping broadness of the surface force transmitting on the crack surface being zero; and C,, C, = material parameters. It is obvious that the cracking process of the crossing cracks of the beam is under the control of the material-failure softening-curve relation (8), and the stress intensity remaining unknown, K, is presumed in calculation. Provided the geometric sizes of the beam L. H. 3. a, and the material parameters E, V, f;, Vo, C, and C, given, and the stress intensity factor being Kz in choosing beams, and when the crack length is a, the external force loaded on beam P and the deflection crossing the centre S may be determined by the numerical calculating process below. For variable a(a > a,), assuming the distribution surface force from crack surfaces to be CJ~(X), Pi is calculated by substituting a,(x) and K,* into formula (7), and B,, ,(x) from formulae (5), (6) and (8) by applying Pi and o,(x). Providing (I- a” cri(x) - CT~+ 1(x)]’ dx < c(c = precision constant) s0 and S is calculated by means of the substitution of ct,+ t (x) and P, into formula (4), or the calculating of oi(x) is to be repeated with ci+, (x). The curves P-6 are shown in Fig. 5(a), (b) and (c) the whole process of the trackings of the beam calculated with the above approach. It is seen that with an increase of the drawing intensity of materialf,, the loading caused by the destruction due to a beam unstabled increased; the loading capability gains with an increase of the stress intensity factor K,* in the cracking of the material; when V, of the material increases, the beam loading capability gains a little bit, but there is an obvious deformation increase at the descent section of the curves.
THE DETERMINATION OF THE CRACKING ENERGY OF CONCRETE OF A COMBINATION OF LEFM WITH FCM
BY MEANS
In the light of linear elastic fracture, the stress field at the crack tip is a constant--controlled by the stress intensity factor-and, when the stress intensity factor K1 at the crack tip attains a certain limiting value, cracking begins extending. Experiments have, however, proved that the limiting stress intensity factor kr, by applying the existing methods is not a constant, which varies with the specimen sizes. A great number of microcracks have already appeared around the crack tips even before the destruction caused by the unstability of concrete cracking. Having studied the features about the microcrack zones of concrete, Hillerberg, in 1976, brought forth the virtual crack model, which states that the extension of concrete cracking is only controlled with the material-failure softening-curves, missing the stress concentration possibly existing around the crack tips. This paper, in combining LEFM with FCM and according to formula (3), energy integration, may get the sum energy consumed in unit ligament area in the cracking of the three-point bend beam, which includes two parties: (1) energy G$ consumed in the process of material-failure softening; and (2) strain energy G, + (1/2)6,p, released in the cracking of the cracks. The strain energy released in the cracking is to be determined in a simulation way in this paper based on the calculation of curves P-6 measured for the three-point bend beam. Therefore G$ determined by calculating with formula (3) is nothing to do with the experimental specimen sizes, which is seen in Fig. 1, and the comparison between the calculated curves and the average measured curves is shown in Fig. 6. As is known above, the stated method could be used to simulate well the trackings of concrete cracks.
Fracture energy in concrete
IC =
(II
zco
0.2
683
40kg/d4
0.4 03 8 (mm)
E= 2.54 x Id
0.5
06
07
lea
160
140
=
Ix) =20 G Y a
0.004
cm ( I )
= 0.008 ” 10
(2)
4
100 &I
= 0.020
11 (5)
= 0.024
II
(6)
60
aI
02
04
03
05
06
ai
0.2
0.3
a4
0.5
0.6
btmm)
a
Fig. 5.
EXPERIMENTAL
RESULTS
The authors of this paper apply Rilem’s approach in measuring the fracture energy for the nine sizes of the three-point bent concrete beam with the same proportion. The proportion of concrete for the specimens in 0.5 (water): l(cement): 2.93 (coarse aggregates): 1.58 (fine aggregates). The specimen is to be poured in steel moulds, being vibrated with a vibrator. The crack is to be pre-cast with a steel plate in depth of 0.3 mm, and in 36 h the specimen is dismounted and moved into a curing room to cure, and, in an interval of 4 weeks, the experiment begins. There are also four test pieces in 10 x 10 x 10 cm and four cylinder specimens in a diameter of 15 cm with a height of 30 cm. The compressive strength for a cube measured is R,= 260 kg/cm*, the elastic modulus E = 254000 kg/cm*, and Poisson’s ratio is v = 0.178. The thickness of the specimen is B = 10 cm, and the depth a of the pre-cast crack is approximately a half of the beam height. The formula is, for the fracture energy of concrete material in calculating with the loading ligament curves measured from the three-point beam, G = RU-mg& F Alig
(9)
684
YIN SHUANGZENG
and ZHAI QIYU
in which: W = area under experimenting of curve P-6; m = dead weight crossing the centre of the beam; 6 = maximum deflection of curve P-d for the beam; A = area of the beam cracking ligament; and specimen size and the experimental results are as shown in Table 1.
P (kg)
P (kg) i
t 120-
- - - Calculation
’ A, -
I
I
0.1
0.2
Average measured
I
I
0.3
1
0.40.5
1 0.6
I 0.7
---
Calculation
-
Average measured
Ylrllrlll,
t
d(mm)
0.1
0.2 0.3 0.4
0.5 0.6
0.7
0.6 d (mm)
P ,fkg)
P (kg)
A
A
300 -
0.1
---
Calculation
---
Calculation
-
Average measured
-
Average measured
0.2 0.3 0.4
---
p (kg) 0.1
II
0.2
0.1 0.2
0.5
0.6
0.7
0.1
0.2
0.3
Calculation
0.4 0.5
0.6
0.7
0.6
d (mm)
---
Calculation
-
Average measured
II(I)
*
0.3 0.4 0.5
0.6
p ,kgjO.l
0.7 d (mm)
0.2
0.3
0.4 0.5 0.6
0.7
d (mm)
---
Calculation
---
Calculation
-
Average measured
-
Average measured
0.3 0.4
0.5
0.6
0.7 0.8 d (mm)
Fig. 6.
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
d (mm)
Fracture energy in concrete
685
Table I. A cm
Specimen sizes
Specimen codes
L
H
B
1
A6
2 3
60
10
10
A9
2 3 1 2
90
10
10
120
10
10
A12
1 B6
2 3 4
60
15
10
B9
2
90
15
10
B12
2
120
15
10
1
4 C6
2
c9
2
60
20
10
90
20
10
120
20
10
1
4 1
12
2 3
52.6 53.2 56.8 57.1 56.8 58.6 55.4 57.6 78.2 80.6 80.0 77.7 83.7 81.3 79.6 84.8 83.8 83.5 81.8 103.0 108.1 102.3 101.7 107.1 110.4 110.2 106.3 105.1 106.5
mg kg
14.1 14.3 14.1 22.0 22.4 22.1 29.5 29.5 20.5 21.0 20.9 20.7 33.0 32.6 32.5 45.8 43.6 44.2 44.2 27.3 27.8 28.0 42 43 43.2 43.8 57.2 57.4 57.7
60
wo
mm
Kg/cm
1.137 0.948 1.214 0.804 0.887 0.834 0.873 0.705 1.397 1.215 1.001 1.023 0.961 1.094 1.005 0.815
3.89 2.45 3.1
1.009 1.168 0.784 1.463 1.127 1.290 1.048 1.209 1.109 1.365 1.144 0.826 0.834
1.99 2.66 3.2
1.28 2.09 5.57 7.35 6.55 4.2 6.11 5.86 5.07 3.31 3.21 5.88 3.91 9.27 10.7 9.54 8.82 8.85 8.67 10.01 5.98 4.14 9.35
G x0.1 Kg/cm 1.045 0.716 0.847 0.657 0.818 0.860 0.695 0.724 1.079 1.228 1.088 0.813 1.109 1.159 1.047 1.830 0.908 1.322 0.902 1.288 1.282 1.286 1.103 1.312 1.219 1.451 1.178 0.845 1.33
G x0.1 Kg/cm 0.869
0.778 0.710
1.051
1.105
0.991
1.285
1.27
1.118
CONCLUSIONS 1. In the combination of LEFM with FCM, this paper presents a study of the cracking of concrete, which is indicated in the calculation results to give a clearer description of the cracking process of the concrete cracks. 2. It is tested and proved that applying Rilem method in measuring the fracture energy within the three-point bent concrete beam still exists the size effectiveness. 3. The size effectiveness caused by the above fracture energy, which, through analysis and calculation, might be due to the stress concentration subtracted from the sum energy consumed in the cracking is irrelavent to the specimen sizes. REFERENCES [I] H. A. W. Cornelissen, D. A. Hordilk and H. W. Reinhardt, Experiments and theory for the Application of fracture mechanics to normal and light weight concrete. Int. Conf. on Fracture Mechanics of Concrete (1985). [2] Chu Wuyang, The Base of Fracture Mechanics, p. 237. Science Press (1979). [3] H. D. Kleinschrodt and H. Winkier, The influence of the maximum aggregate size and the size of specimen on fracture mechanics of concrete (1985). [4] Y. S Jenq and S. P. Shah, A fracture toughness criterion for concrete. Engng Fracture Mech. 21, 1055-1069 (1985). [5] Tien Minglun and Huang Songmei, The fracture deflection of concrete. J. Water Res. 3846 (1982). [6] A. Hiuerborg, M. Modeer and P. E. Peterson, Cement concrete. RES, No. 6, 773-782 (1976). (Received for publication 22 November 1988)
Vol. 35.
No. 4/S, pp. 679-685,
0013-7944/90
1990
$3.00 + 0.00
PergamonPress pk.
A METHOD IN DETERMINATION OF THE SIZE EFFECTIVENESS OF FRACTURE ENERGY IN CONCRETE YIN SHUANGZENG
and ZHAI QIYU
North China Institute of Hydropower, Handan City, Hebei Province, P.R.C. Abstract-In this paper, the authors present the measurement of the fracture energy G, of the three-point bend concrete beam in nine sizes by means of the method recommended by Rilem. The experimental result indicates that the concrete fracture energy measured in this way still possesses a size effectiveness. The authors think, in accordance with the conceptions of linear elastic fracture and virtual crack models, that when the crack tips in the microcracking zone in the cracking of concrete is simplified into virtual cracks capable of transmitting the surface force, there still exist certain amounts of strength factors around the tip zone of the virtual cracks. Thus the sum energy consumed in the cracking of the concrete cracks falls into two: (1) energy consumed in the cracking of the virtual crack surface; (2) energy released in the cracking of the virtual crack tips. It shows, having been experimented and calculated, that the sizes of the fracture energy in the three-point bend beam are mainly influenced by the amounts of the energy released from the virtual crack tips. If this energy is subtracted on the face of the virtual cracks in the material is, in general, not related to the sizes of the test specimen.
INTRODUCTION FRACTURE energy is defined as the surface energy consumed in unit area of the material cracking. It is generally considered that the fracture energy is constant in the concrete material, and the recent experiments have, however, proved that the fracture energy is still of size effect with the method recommended by Rilem in measuring the three-point bend concrete beam. It is not difficult for us to discover that, having analysed the trackings of l-type concrete cracks, there is a certain stress concentration around the crack tips in addition to the microcracking zones (see Fig 1). With the conception of the virtual crack models as a basis, the microcracking zone is able to be replaced by the virtual cracks capable of transmitting the surface force. The stress (T(X) transmitted on the surface of the virtual cracks is and only is determined by the gaping displacement V(x) in between the virtual crack surfaces [l]. Taking into account the conception of linear elastic fracture, when there is a cracking at the virtual crack tips, strain energy will be given off in the material within the stress concentrating zones. The amount of the strain energy is relevant to the degree the stress is concentrated. In order to show the degree of the stress concentrated at the tips of the virtual cracks, the concept of stress intensity factors is introduced for the virtual crack tips (see Fig. 2). So there are two parties for the energy consumed in crack extending: (1) the energy consumed in the demolishing of the microcracking zone; and (2) the energy released in the gaping by the stress concentration at the virtual crack tips.
ENERGY INTEGRATION The authors make an analysis of the energy relation of the l-type cracks in gaping according to the above points of view. Figure 3 presents the general conditions of the trackings of the l-type cracks caused by the external forces P and the closed surface force g(x). If the equivalent length a of the virtual cracks is increased by da, the deforming increment caused by force p and a(x) is “d8 and do(x)“. In accordance with mutual equivalent law, we have (1- “0 P da - o [a(x)dv(x)].dx.B = dao*(x,)2V*(x,)dx,-B (14 s s0 679
680
YIN SHUANGZENG
and ZHAI QIYU
in which cr*(x, ), and V*(x, ) are respectively, the distributing force and the corresponding displacement, i.e. [2]
gaping
cr*(x, ) = K, /,/271x,
(lb)
and V*(x, ) = 4,/da
- x, )/2n *K, /E
(lc)
if (1 b), and (lc) are substituted into (la), we get 0-00 Pd6 - o [a(x)dv(x)].dx.B =2Kf/E.da.B. (2a) s To the three-point bend beam shown in Fig. 4, the stress strength factor is K,,,, when the crossing cracks begin cracking, the deflections 6, of the initial cracking of the beam and the corresponding external force P, are, respectively, [3]
1 P, =
W
K,, H”*B
(24
1.5LF, h/H)
in which [5.58 - 19.57aJH
f’&JH) = 4 (so/H)=
1.99 - a,/H(l
a
+ 36.83(a,,/H)2 - 34.94(a,/H)3
- a,/H)[2.15
- 3.93aJH
(1 + 2a,/H)(l
- a,/H)‘.’
+ 2.7(a,JH)2]
+ 12.77(a,/H)7
1
and H, L, B, and a,, are separately the height, length, thickness and the length of the initial cracking; E, and y are the elastic modulus of the material and Poisson’s ratio; d is the coefficient affected by the shearing force. To the rectangle section d = lO(1 + v)/12 + 1lo), and the two factors of (2) being used to integrate the whole cracking of the beam and the two factors being divided by the ligament area Ali, = (h - a,)B of the beam, we have GF-GF*=G,+1/26,P,/A,,, P
i &
(3)
tp
Yl
U(X)
u*tx,j
X K, ad
0,
a
Y
a
da
@
P
P
Fig. 3.
XI
Fracture energy in concrete
681
w
FL------I Fig. 4.
in which Gr = energy consumed in the unit ligament area of the beam; Gz = area under the concrete-destruction softening curve; and & = rate of the average strain energy releasing in the beam cracking. CALCULATIONAL
SIMULATION IN THE CRACKING THREE-POINT BEND BEAM
OF THE
The microcracking zone of the edges of the crack tips may be substituted by the virtual cracks transmitting the surface force with energy, in the cracking of the crossing cracks within a three-point bend beam. If the length of the microcrack is a, substituting the surface force acting on the surfaces of the virtual crack, a(x) with the external force equivalent to the surface one, one can have &=;
‘-“0(x)x f0
dx .B
and therefore the formula for the calculation of the beam deflection 6 and the gaping displacement of cracks is
WdH) 1 + E[l + 2.8(H,L)q
(-MOD
=
(’ - ‘,)
- L2
2BH2+
~l+v
dB
(4)
G(P- P&L=F2 (a/H) H3BE
in which F2(a/H)
= 0.76 - 2.28a/H
+ 3.87(~/H)~ - 2.04(~/H)~ + 0.66/(1 - u/H)~
and the meanings of the other symbols are the same as the preceding ones. The distribution function of the gaping displacement on the crack surface is given as l’(x) = CMODJx(x
+ H - a),&?i
(6)
and, when x = 0, the distribution function value V(0) = 0, and V(a) = CMOD if X = a. On the basis of the principle of superposition, the stress intensity factor at the crack tip brought forth by both the concentration force P and the distribution force on the surface of cracks is [4] K, =- ;EF(a,H)
-f-%&F&$(x)dx
(7)
where 3.52x/a
4.35 - X28(1 - x/a) (1 - u,H)~‘~ (1 - u,H)“~ + (1 -a/H)
1.3 - 0.3(1 - x,a)3” 1 - (1 - x/a)3
+ 0.83 - 1.76(1 - x/a)
1.
682
YIN SHUANGZENG
and ZHAI QIYU
The distribution surface force on the surface of the virtual cracks a(x) being in a determined relation with its gaping displacement Y(x), their relation, based on literature1 [I], is a(x) =fi[f(W))
-
~(xh_J>(hJl
(8)
in which f(u(x)) = 1 +
c>
V(x) 3e -O.Zu(r)/u,
cl -
VII
where: f; = drawing strength of concrete; V. = gaping broadness of the surface force transmitting on the crack surface being zero; and C,, C, = material parameters. It is obvious that the cracking process of the crossing cracks of the beam is under the control of the material-failure softening-curve relation (8), and the stress intensity remaining unknown, K, is presumed in calculation. Provided the geometric sizes of the beam L. H. 3. a, and the material parameters E, V, f;, Vo, C, and C, given, and the stress intensity factor being Kz in choosing beams, and when the crack length is a, the external force loaded on beam P and the deflection crossing the centre S may be determined by the numerical calculating process below. For variable a(a > a,), assuming the distribution surface force from crack surfaces to be CJ~(X), Pi is calculated by substituting a,(x) and K,* into formula (7), and B,, ,(x) from formulae (5), (6) and (8) by applying Pi and o,(x). Providing (I- a” cri(x) - CT~+ 1(x)]’ dx < c(c = precision constant) s0 and S is calculated by means of the substitution of ct,+ t (x) and P, into formula (4), or the calculating of oi(x) is to be repeated with ci+, (x). The curves P-6 are shown in Fig. 5(a), (b) and (c) the whole process of the trackings of the beam calculated with the above approach. It is seen that with an increase of the drawing intensity of materialf,, the loading caused by the destruction due to a beam unstabled increased; the loading capability gains with an increase of the stress intensity factor K,* in the cracking of the material; when V, of the material increases, the beam loading capability gains a little bit, but there is an obvious deformation increase at the descent section of the curves.
THE DETERMINATION OF THE CRACKING ENERGY OF CONCRETE OF A COMBINATION OF LEFM WITH FCM
BY MEANS
In the light of linear elastic fracture, the stress field at the crack tip is a constant--controlled by the stress intensity factor-and, when the stress intensity factor K1 at the crack tip attains a certain limiting value, cracking begins extending. Experiments have, however, proved that the limiting stress intensity factor kr, by applying the existing methods is not a constant, which varies with the specimen sizes. A great number of microcracks have already appeared around the crack tips even before the destruction caused by the unstability of concrete cracking. Having studied the features about the microcrack zones of concrete, Hillerberg, in 1976, brought forth the virtual crack model, which states that the extension of concrete cracking is only controlled with the material-failure softening-curves, missing the stress concentration possibly existing around the crack tips. This paper, in combining LEFM with FCM and according to formula (3), energy integration, may get the sum energy consumed in unit ligament area in the cracking of the three-point bend beam, which includes two parties: (1) energy G$ consumed in the process of material-failure softening; and (2) strain energy G, + (1/2)6,p, released in the cracking of the cracks. The strain energy released in the cracking is to be determined in a simulation way in this paper based on the calculation of curves P-6 measured for the three-point bend beam. Therefore G$ determined by calculating with formula (3) is nothing to do with the experimental specimen sizes, which is seen in Fig. 1, and the comparison between the calculated curves and the average measured curves is shown in Fig. 6. As is known above, the stated method could be used to simulate well the trackings of concrete cracks.
Fracture energy in concrete
IC =
(II
zco
0.2
683
40kg/d4
0.4 03 8 (mm)
E= 2.54 x Id
0.5
06
07
lea
160
140
=
Ix) =20 G Y a
0.004
cm ( I )
= 0.008 ” 10
(2)
4
100 &I
= 0.020
11 (5)
= 0.024
II
(6)
60
aI
02
04
03
05
06
ai
0.2
0.3
a4
0.5
0.6
btmm)
a
Fig. 5.
EXPERIMENTAL
RESULTS
The authors of this paper apply Rilem’s approach in measuring the fracture energy for the nine sizes of the three-point bent concrete beam with the same proportion. The proportion of concrete for the specimens in 0.5 (water): l(cement): 2.93 (coarse aggregates): 1.58 (fine aggregates). The specimen is to be poured in steel moulds, being vibrated with a vibrator. The crack is to be pre-cast with a steel plate in depth of 0.3 mm, and in 36 h the specimen is dismounted and moved into a curing room to cure, and, in an interval of 4 weeks, the experiment begins. There are also four test pieces in 10 x 10 x 10 cm and four cylinder specimens in a diameter of 15 cm with a height of 30 cm. The compressive strength for a cube measured is R,= 260 kg/cm*, the elastic modulus E = 254000 kg/cm*, and Poisson’s ratio is v = 0.178. The thickness of the specimen is B = 10 cm, and the depth a of the pre-cast crack is approximately a half of the beam height. The formula is, for the fracture energy of concrete material in calculating with the loading ligament curves measured from the three-point beam, G = RU-mg& F Alig
(9)
684
YIN SHUANGZENG
and ZHAI QIYU
in which: W = area under experimenting of curve P-6; m = dead weight crossing the centre of the beam; 6 = maximum deflection of curve P-d for the beam; A = area of the beam cracking ligament; and specimen size and the experimental results are as shown in Table 1.
P (kg)
P (kg) i
t 120-
- - - Calculation
’ A, -
I
I
0.1
0.2
Average measured
I
I
0.3
1
0.40.5
1 0.6
I 0.7
---
Calculation
-
Average measured
Ylrllrlll,
t
d(mm)
0.1
0.2 0.3 0.4
0.5 0.6
0.7
0.6 d (mm)
P ,fkg)
P (kg)
A
A
300 -
0.1
---
Calculation
---
Calculation
-
Average measured
-
Average measured
0.2 0.3 0.4
---
p (kg) 0.1
II
0.2
0.1 0.2
0.5
0.6
0.7
0.1
0.2
0.3
Calculation
0.4 0.5
0.6
0.7
0.6
d (mm)
---
Calculation
-
Average measured
II(I)
*
0.3 0.4 0.5
0.6
p ,kgjO.l
0.7 d (mm)
0.2
0.3
0.4 0.5 0.6
0.7
d (mm)
---
Calculation
---
Calculation
-
Average measured
-
Average measured
0.3 0.4
0.5
0.6
0.7 0.8 d (mm)
Fig. 6.
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
d (mm)
Fracture energy in concrete
685
Table I. A cm
Specimen sizes
Specimen codes
L
H
B
1
A6
2 3
60
10
10
A9
2 3 1 2
90
10
10
120
10
10
A12
1 B6
2 3 4
60
15
10
B9
2
90
15
10
B12
2
120
15
10
1
4 C6
2
c9
2
60
20
10
90
20
10
120
20
10
1
4 1
12
2 3
52.6 53.2 56.8 57.1 56.8 58.6 55.4 57.6 78.2 80.6 80.0 77.7 83.7 81.3 79.6 84.8 83.8 83.5 81.8 103.0 108.1 102.3 101.7 107.1 110.4 110.2 106.3 105.1 106.5
mg kg
14.1 14.3 14.1 22.0 22.4 22.1 29.5 29.5 20.5 21.0 20.9 20.7 33.0 32.6 32.5 45.8 43.6 44.2 44.2 27.3 27.8 28.0 42 43 43.2 43.8 57.2 57.4 57.7
60
wo
mm
Kg/cm
1.137 0.948 1.214 0.804 0.887 0.834 0.873 0.705 1.397 1.215 1.001 1.023 0.961 1.094 1.005 0.815
3.89 2.45 3.1
1.009 1.168 0.784 1.463 1.127 1.290 1.048 1.209 1.109 1.365 1.144 0.826 0.834
1.99 2.66 3.2
1.28 2.09 5.57 7.35 6.55 4.2 6.11 5.86 5.07 3.31 3.21 5.88 3.91 9.27 10.7 9.54 8.82 8.85 8.67 10.01 5.98 4.14 9.35
G x0.1 Kg/cm 1.045 0.716 0.847 0.657 0.818 0.860 0.695 0.724 1.079 1.228 1.088 0.813 1.109 1.159 1.047 1.830 0.908 1.322 0.902 1.288 1.282 1.286 1.103 1.312 1.219 1.451 1.178 0.845 1.33
G x0.1 Kg/cm 0.869
0.778 0.710
1.051
1.105
0.991
1.285
1.27
1.118
CONCLUSIONS 1. In the combination of LEFM with FCM, this paper presents a study of the cracking of concrete, which is indicated in the calculation results to give a clearer description of the cracking process of the concrete cracks. 2. It is tested and proved that applying Rilem method in measuring the fracture energy within the three-point bent concrete beam still exists the size effectiveness. 3. The size effectiveness caused by the above fracture energy, which, through analysis and calculation, might be due to the stress concentration subtracted from the sum energy consumed in the cracking is irrelavent to the specimen sizes. REFERENCES [I] H. A. W. Cornelissen, D. A. Hordilk and H. W. Reinhardt, Experiments and theory for the Application of fracture mechanics to normal and light weight concrete. Int. Conf. on Fracture Mechanics of Concrete (1985). [2] Chu Wuyang, The Base of Fracture Mechanics, p. 237. Science Press (1979). [3] H. D. Kleinschrodt and H. Winkier, The influence of the maximum aggregate size and the size of specimen on fracture mechanics of concrete (1985). [4] Y. S Jenq and S. P. Shah, A fracture toughness criterion for concrete. Engng Fracture Mech. 21, 1055-1069 (1985). [5] Tien Minglun and Huang Songmei, The fracture deflection of concrete. J. Water Res. 3846 (1982). [6] A. Hiuerborg, M. Modeer and P. E. Peterson, Cement concrete. RES, No. 6, 773-782 (1976). (Received for publication 22 November 1988)