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Analysis of crack propagation and crackinitiation due to corrosion of reinforcement
Construction and Building Materials. Vol. 11, Nos 7-8, pp. 437-442, 1997 0 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved OY50-0618/97 $17.00 + 0.00
ELSEVIER
PII:SO950-0618(97)00020-2
Analysis of crack propagation and crack initiation due to corrosion of reinforcement Masayasu Ohtsu* + and Shinichi Yosimura$ * Department of Civil Engineering and Architecture, Kumamoto University, Kurokami .X9- 1, Kumamoto 860, Japan ’ Graduate School of Kumamoto University, Kurokami 239- 1, Kumamoto 860, Japan Received 14 July 1996; revised 5 June 7997; accepted 5 July 1997 Stress analysis in concrete due to corrosion of reinforcement was carried out, based on the concept of the linear elastic fracture mechanics (LEFM) and the boundary element method (BEM). Initiation and propagation of five patterns of cracks which propagate from the reinforcement are in focus. Hydrostatic pressure distribution and vertical pressure to the surface crack were taken into consideration to simulate the expansion of corrosion products. From the theory of elasticity, under hydrostatic pressure it was concluded that other cracks than the surface crack could be nucleated either by following the surface crack or in the case of the surface crack being arrested by aggregate. During the propagation of either a surface crack or an internal crack, it was found in the BEM analysis that a spalling crack and/or a diagonal crack could be initiated. Under only vertical pressure, the stress distribution suggested a possible initiation of the internal crack only. In the case that the internal crack propagated after the surface crack was completed, a high tensile region suggested the initiation of a vertical crack. 0 1997 Elsevier Science Ltd.
Keywords: concrete; corrosion; stress
Introduction
growth prediction in a concrete mode12. An analytical procedure, thus, for the propagation of one crack in concrete is in hand. For early detection of corrosion, on the other hand, non-destructive testing (NDT) is under investigation. It is also confirmed that corrosion potential on the reinforcement could be estimated by an improved half-cell potential measurement 3. It implies that the zone of intense corrosion products is able to be located by the NDT measurement. In the case, therefore, that pressure distribution due to the expansion of corrosion products could be effectively assumed, prediction of crack propagation in concrete would be performed by the FM analysis. In order to develop this integrated procedure for crack identification based on the NDT, stress analysis in concrete due to corrosion products is studied, based on the LEFM concept and the boundary element method (BEM).
Cracking is so often found in concrete structures that the causes of cracking have been studied extensively and mostly identified. One of the critical cracks is known to result from corrosion of the reinforcement in concrete. Corrosion is a subsidiary problem of reinforced concrete CRC) structures which are subjected to severe salt attack and misuse of materials. When the reinforcement is corroded, corrosion products expand the area of the reinforcement and generate tensile stress around it. Because concrete can endure less tensile stress than compressive stress, tensile cracks are readily nucleated and propagated in concrete. In order to analyze crack propagation in concrete, fracture mechanics (FM) has been successfully applied’. In the case that one major crack propagates, it is confirmed that a simple treatment based on the linear elastic fracture mechanics (LEFM) is available for crack
Classification ‘Corresponding
of cracks
Due to corrosion of reinforcement,
author
437
cracks are observed
438
Crack propagation and crack initiation due to corrosion of reinforcement: M. Ohtsu and S. Yosimura
reinforcement
Figure 2
stress-free surface Configuration of reinforcement
near the surface
Figure 1 Crack patterns observed in the test
in a variety of patterns. These are reported to be related with cover thickness, concrete properties and arrangement of reinforcement4. From this study, particular crack patterns were identified around the reinforcement. Examples are given in Figure 1. These were observed in an electrolytic test of RC blocks in a laboratory’. Concrete blocks of dimension 10 cm x 25 cm X 40 cm were tested. Rebars of lo-mm diameter were embedded with 3-cm cover-thickness. Mixture proportions of the concrete were all the same. After the test, various crack patterns were observed. In the figures, a localized region of dimension 8 cm x 8 cm surrounding the reinforcement is only shown. These are classified into five patterns. One is a surface crack which propagates vertically to the surface in the region of concrete cover. It is labelled as 7%‘. Others are a spalling crack (SC), a vertical crack (V), an internal crack (I) and a diagonal crack CD). Because concrete is inhomogeneous, there exist many possibilities for crack initiation and crack propagation. In the present paper, the initiation of such cracks as ‘SC’, ‘V’ and ‘D’ following the propagation of ‘Sv’ and ‘I’ cracks is investigated by the BEM analysis based on the LEFM.
At the point on the hole closest to the stress-free surface, in other words, at the point of the shortest cover thickness, the maximum tensile stress is obtained in Eq. (2). Thus, the elastic theory offers one possibility only for the initiation of the surface crack ‘Sv’. Unless initial defects are taken into consideration around the reinforcement, any other stress conditions might not be possibly present in a homogeneous elastic-body. Previously, experimental investigation was performed on the relation between pressure p and crack nucleation’. One of results is given in Figzue 3. The relationship between the applied pressure and final failure observed in the samples with different cover thickness is summarized. It is observed that the applied pressure to reach the final failure increases with the increase of cover thickness. One exception is the case of thin cover thickness, where actually only the surface crack ‘SC’ was observed and no surface crack ‘Sv’ was observed. Consequently, it primarily leads to a conclusion that other cracks than the surface crack could be nucleated either by following the surface crack or in the case of the surface crack ‘Sv’ being arrested by aggregate. The criterion for crack initiation from the crack tip is derived from the LEFM’. As reference to Figure 4, displacements near the crack tip are represented, by employing the stress intensity factor of the mode 1, K,,
Stress analysis
u = K,/G(r/2m)
According to the theory of elasticity’, circumferential tensile stress uO,,around a hole is known. Under hydrostatic pressure, p, which is applied to the interior surface of the hole of diameter, a, it is given as
u = K,/G(r/2~)“*sin8/2(1-
uee(r) =p[ 1 + (a/r)‘]
-p.
(1)
From Eq. (11, it is easily realized that the tensile stress reaches the maximum at the surface of the hole (r = a). It leads to a conclusion that a crack could initiate anywhere around the hole. In a real condition, there is a stress-free surface near the reinforcement as shown in Figure 2. In this case, it is possible to approximate stress distribution by taking into account an annular body with the outer diameter, b, as
u,,(r) =p[ 1 + (a/rj2]/[
I-
(a/b)‘]
-p.
(2)
“*cos0/2(1-
2u + sin2 o/2),
(3)
2v-- cos2 o/2),
(4)
where G is the shear modulus of elasticity and v is Poisson’s ratio. Thus, the stress intensity factor can be computed from the displacements at the crack tip. The LEFM theory leads to a simple criterion that the crack propagates when K, is greater than K,,. K,, is the critical stress intensity factor and a material property to be determined.
BEM and model In the boundary element method (BEM), the governing equation is converted into the integral form on the boundary. Thus, the boundary is digitized and the boundary conditions are only taken into consideration.
Crack propagation and crack initiation due to corrosion of reinforcement: M. Ohtsu and S. Yosimura
2 9 2.0 09 m 6
In the case that point x is located on the boundary S, coefficient C is readily obtained by eliminating rigid body motion. This leads to the integral equation with respect to unknown values of either the traction or the displacement on the boundary. Modelling the boundary by boundary meshes and discretizing Eq. (5), the integral equation results in a set of algebraic equations which can be solved numerically. Stress gij(x) at point x inside the boundary is determined from the constitutive law of an isotropic solid6,
$ _._ ......_..i .._._ .......j._.......___t’.___. ......j..___....._ i t i......__.__f ......_-..L.C. ........& ...._......i.in__ __+ i _ .............c. .....~t”““_.~
5 1. 0 z! I%
......+....______~......_...._
-_._...._._._~___.__..__.__~-----......._~_._.__-._~._.___._z I l f _ ....._......c__._..-......+. ._.._ ....+__.__.__.-i-_” ......_._ I
0 0
~~l~‘l~~~‘llL1’~~~~‘l~~l 5 10 15
20
439
25
(2 x cover thicknesst diam&r) / diameter Figure 3 Relationship between the pressure observed at final failure and the cover thickness
pjj(x) = G[~Yz.+~ =
(x>sij/(l
-
2v)
+
ui,j(x)
+
uj,i(x)]
L{z+(xH.
(8)
Substituting Eq. (5) as C = 1 into Eq. (8), we have Crack propagation is easily taken into account, creating just new boundary meshes*. After solving boundary integral equations in respect to tractions and displacements on the boundary, stresses at arbitrary locations are determined for the stress analysis. The displacement solution of the two-dimensional (2D) elasticity in the plane-strain state is represented in the x,-xz coordinates’ of the tensor notation, Cu;(x)
=
@IJ,(x,y)r,( Y) -
)I dS, (5)
&kYhQ(Y
where r.+(x) and r.+(y) are the displacements and tk(y) is the traction. Point y is always located on the boundary S. I&(&y) is the fundamental solution, U,,(x,y)
= [(3 - 4Y)ln(l/r)&,
+ r,ir,kl/[87r(l
- v)Gl. (6)
Here r necker’s
is the distance between x and y, Si, is Krosymbol and T,~ means the spatial derivative dr/dxi. T,(Iy) is the associated traction with l&(&y) and is obtained from
+Q,,j(X,Y)nj
+
~j,k(l,Y)nj]
(7)
*
nj is the outward normal vector to the boundary
surface S. The configuration coefficient, C, is equal to 1 when point x is located inside the boundary S.
: I Y (VI
I
f
%J/q Figure 4
The coordinates at the crack tip
x:(u)_
(9) Substituting all tractions and displacements solved by Eq. (5) into Eq. (9), stresses at arbitrary points x are obtained from the integral. To study the initiation mechanism of cracks, a simple model was considered. The model corresponds to a half portion of a concrete block containing reinforcement as illustrated in Figure 5. A semi-circle of 3.0-cm diameter represents the location of reinforcement where the expanding pressure due to corrosion products is applied. The symmetric axis of the concrete block in Figure I is replaced by the crack propagating boundary. The thickness of concrete cover is 3 cm and another boundary for crack propagation of the internal crack is taken as 15 cm. To simulate crack propagation in the analysis, the constraint of the displacement at each node on the bottom boundary in Figure 5 is released one by one, based on the criterion of the critical stress intensity factor. According to Figure I, the cases where the surface crack ‘Sv’ propagates toward the stress-free surface and the internal crack ‘I’ propagates backward the surface are modelled. The length of the stress-free surface is taken as 10 cm. It is noted that the model is not closed in the domain, because it is allowed in the BEM and the generation of tensile stress around the reinforcement is only in focus. In addition, just a sketch of the model is given in Figure 5. All boundary meshes are actually 2 mm long and the points where stresses are estimated are located 1 mm radially apart from the semi-circular boundary. Although internal pressure to create cracking is given in Figure 3, distribution of actual pressure is not known. Consequently, two kinds of pressure distribution were taken into consideration to simulate the expansion of corrosion products. These are shown in Figure 6. One is hydrostatic radial pressure and the other is vertical expansion pressure by neglecting horizontal pressure of the hydrostatic pressure. The vertical pressure distribution was taken into consideration, based on the fact as shown in Figure 6. Due to nucleation of small tensile
440
Crack propagation
and crack initiation due to corrosion of reinforcement:
M. Ohtsu and S. Yosimura
Case 1
oSV
Case 2
cover
thickness
A BEM model to analyze stress distribution around the reinforcement Figure 5
cracks, the contact between the reinforcement and concrete might be broken. Under the pressure distribution on the semi-circular boundary, three cases of crack propagation along the bottom boundary were analyzed. These are illustrated in Figure 7. In Case 1, the surface crack propagates simply toward the stress-free surface. The internal crack propagates inside concrete backward the surface in Case 2. This is the case where the surface crack is being arrested by aggregate. In Case 3, the surface crack has been already nucleated and then the internal crack propagates toward the inside. At each step of the BEM analysis, the stress intensity factor at the crack tip was computed from the displacement on the crack-tip element by utilizing Eq. (4)“‘. Then, the pressure level corresponding to the critical intensity factor KJc was determined and displacement solutions of the case were obtained. In the next step, the constraint on the displacement at the node of the crack tip was released and a new crack-tip was created at the adjoining node. From the experiments’, material properties of concrete are known so that shear modulus of concrete, G = 16.7 GPa, Poisson’s ratio, v = 0.2 and the critical stress intensity factor, K,, = 0.263 MPa m1/2
hydrostaticpressure
verticalpressure
cl Figure 6 Pressure distribution around the hole due to the expansion of corrosion products
Case 3
I sv +-o-i Figure 7 The cases of crack propagation from the hole of reinforcement
Results and discussion Case 1: the surface crack propagates toward the stress-free su$ace
Under hydrostatic pressure, a relation between applied pressure and the vertical displacement at the crown of the semi-circle is shown in Figure 8a. At the pressure level A, the surface crack ‘Sv’ starts to propagate and then the pressure decreases. Stress distribution at 1 mm apart from the semicircle boundary is given in Figure 8b at three pressure levels A, B and C. From the stress analysis, circumferential tensile stress was determined at each location. Tensile stresses are plotted against the angle 8 shown in Figure 5. At the pressure level A, the peak stress is observed in the direction of 135”, which must be associated with the initiation of the spalling crack ‘SC’.At the pressure level B, another peak stress is found around 45”, which could explain the generation of the diagonal crack ‘D’. Consequently, when a crack due to corrosion propagates toward the concrete surface from the reinforcement, a spalling crack and/or a diagonal crack could follow it. The case of vertical pressure is shown in Figure 9. The crack starts to propagate at the pressure level A in Figure 9a. The pressure still increases up to the level B and then decreases. The stress distribution in Figure 9b shows high stress zone around 0” in any pressure levels. Thus, a possible initiation of the internal crack ‘I’ is only suggested. Case 2: the internal crack propagates toward the inside
The case of hydrostatic pressure is shown in Figure 10. The peak pressure is observed at the level A, namely at the initiation of the internal crack ‘I’. Although one possibility of the initiation of the diagonal crack ‘D’ (45”) at the pressure level A is found, major high stress
Crack propagation and crack initiation due to corrosion of reinforcement: M. Ohtsu and S. Yosimura
Displacement
&k
(x 102 mm)
: dew)
(0
Figure 8 (a) Pressure vs. vertical displacement at the top of the hole due to hydrostatic pressure and (b) stress distribution around the hole of reinforcement in Case 1
0.2 A
2 E
B
2 .x_ 0.1
C
5 I Lf
0
I!? 0
4
2
lxsplaccment
0
100 Angle
(x x-2 mm)
(0
: degne)
Figure 9 (a) Pressure vs. vertical displacement at the top of the hole due to vertical pressure and (b) stress distribution around the hole of reinforcement in Case 1
is observed in the 135” direction at all pressure levels. Therefore, the spalling crack ‘SC’ is probably initiated due to crack propagation of the internal crack ‘I’. It implies that the spalling crack could be initiated by either the surface crack or the internal crack under hydrostatic pressure. The case of vertical pressure is given in Figure II. At very low pressure level at A, the internal crack begins to propagate. At the pressure level, high stress zone is observed around 0” in Figure llb. This suggests further propagation of the internal crack ‘I’. With increase of the pressure, new high stress zone is created around 180” direction. This implies the initiation of the surface crack ‘SC’.Consequently, the surface crack could follow the internal crack unless the crack is arrested by aggregate. If the surface crack is arrested, the stress distribution at the pressure level B suggests the initiation of the vertical crack ‘V’ in the 90” direction.
(b)
(x 102 mm)
Angle
(e
: degree)
In the case of hydrostatic pressure, results are again similar to the above two cases in Figures 8 and 10. It suggests the occurrence of the spalling crack ‘SC’ and the diagonal crack ‘D’. In the case of vertical pressure, pressure vs. displacement and stress distribution are obtained as shown in Figure 12. When the crack started to propagate, high stress zone is observed near the 0” direction, It implies further crack propagation toward the inside. The pressure still increases as shown in Figure 12a up to the pressure level B. At the level, high tensile region is observed in the 90” direction. Thus, the generation of the vertical crack ‘V’ is suggested in Figure 12b.
Conclusion Based on the NDE measurement, crack diagnostics of concrete is under development. In this concern, the initiation of cracks due to the expansion of corrosion products is studied by the BEM analysis based on the LEFM. Here, five types of the cracks which propagates from the reinforcement are in focus. One is a surface crack which propagates vertically to the surface in the region of concrete cover. Others are a spalling crack, a vertical crack to the surface crack, an internal crack parallel to the surface crack but toward the inside and a diagonal crack antisymmetric to the spalling crack. In the analysis, two kinds of pressure distribution were taken into consideration to simulate the expansion of corrosion products. One is hydrostatic radial pressure and the other is vertical expansion pressure by neglecting horizontal pressure of the hydrostatic pressure. Conclusions obtained are summarized.
1. Displacement
(x 102 mm)
(a) Pressure vs. vertical displacement at the top of the hole due to vertical pressure and (b) stress distribution around the hole of reinforcement in Case 2
Figure 11
Case 3: the internal crack propagates toward the inside after the suflace crack is completed
(b)
(a)
DisplarrmeM
441
Angle
(0
: degree)
Figure 10 (a) Pressure vs. vertical displacement at the top of the hole due to hydrostatic pressure and (b) stress distribution around the hole of reinforcement in Case 2
From the theory of elasticity, it is concluded that other cracks than the surface crack could be nucleated either by following the surface crack or in the case of the surface crack being arrested by aggregate.
Crack propagation
442
and crack initiation due to corrosion of reinforcement: lb)
I 0
M. Ohtsu and S. Yosimura
occurrence of the spalling crack and the diagonal crack is suggested under hydrostatic pressure. In the case of vertical pressure, high stress zone seems to excite further propagation of the internal crack at the beginning. Then, at the peak pressure, high tensile region is generated, suggesting the initiation of the vertical crack.
6Ot.
,
10
Displacement (x !W mm)
AL&
(6 : degee)
(a) Pressure vs. vertical displacement at the top of the hole due to vertical pressure and (b) stress distribution around the hole of reinforcement in Case 3 Figure
12
2. In the case that the surface crack propagates simply towards the stress-free surface, a spalling crack and/or a diagonal crack could follow it under hydrostatic pressure. The peak pressure is observed at the initiation of the surface crack. In contrast, the crack starts to propagate at a level and the pressure still increases up to the peak under vertical pressure. The stress distribution suggests a possible initiation of the internal crack only. 3. The case that the internal crack propagates toward the inside is studied as the case where the surface crack is being arrested by aggregate. Under hydrostatic pressure, it is found that the spalling crack could be initiated by either the surface crack or the internal crack. In the case of vertical pressure, further propagation of the internal crack is observed at the initiation of the crack. With the increase of the pressure, new high stress zone results in the fact that the surface crack could follow the internal crack unless the crack is arrested by aggregate. If the surface crack is arrested, the stress distribution suggests the initiation of the vertical crack. 4. In the case that the surface crack has been already nucleated and the internal crack propagates, the
References 1 Ingraffea, A. R. and Saouma, V., Numerical Crack Propagation in Reinforced Mechanics of Concrete, Martinus 1985, pp. 171-225
and
Plain
Modeling of Discrete Concrete. Fracture
Nijhoff Publishers, Dordrecht,
2
Chahrour, A. H. and Ohtsu, M., Crack growth prediction in scaled down model of concrete gravity dam. Theoretical and
3
Ohtsu, M. and Yamamura, H., CSM analysis of half-cell potentials. Transactions of the Japan Concrete Institute, 1992, 14,
Applied
Fructure
Mechanics,
1994, 21, 29-40
203-208 4
Matsushima, M., Tsutumi, T., Seki, H. and Matsui, K., Design of Cover Thickness for Reinforced Seuere Salt At&k. Proc.” of
5
Concrete
Structures
Subiected
to
Japan Society of Civil Engineers, 1994, No. 490/V-23, pp. 41-49 Ohtsu, M., Tutumi, T., Murakami, Y. and Kudo, Y., Analytical andexperimental study on crack propagation due to rebar corrosion. Proc. Japan Concrete Institute, 1995, 17(l), 955-960 Love, A. E. H., A Treatise on the Mathematical Theory of Elasticity. Dover Publication, New York, 1944 Broek, D., Elementary Engineering Fracture Mechanics. Martinus Nijhoff Publishers, Dordrecht, 1986 Chahrour, A. H. and Ohtsu, M., Multi-Domain BEM implementation for Mixed-Mode of Concrete and Rock
Cracking (FDCR-2),
in Concrete.
Fracture
and Damage
E and FN Spon, London, 1992,
pp. 196-205 Crouch, S. L. and Starheld, A. M., BoundaryElement Method in Solid Mechanics. Oxford Universitv Press, Oxford, 1983 10 Smith, R. N. L. and Mason, J. C.: A Boundary Element Method 9
for CurvedCrack Problem Methods in Engineering,
in Two-Dimensions.
Boundary
Element
Springer, London, 1982, pp. 472-484.
ELSEVIER
PII:SO950-0618(97)00020-2
Analysis of crack propagation and crack initiation due to corrosion of reinforcement Masayasu Ohtsu* + and Shinichi Yosimura$ * Department of Civil Engineering and Architecture, Kumamoto University, Kurokami .X9- 1, Kumamoto 860, Japan ’ Graduate School of Kumamoto University, Kurokami 239- 1, Kumamoto 860, Japan Received 14 July 1996; revised 5 June 7997; accepted 5 July 1997 Stress analysis in concrete due to corrosion of reinforcement was carried out, based on the concept of the linear elastic fracture mechanics (LEFM) and the boundary element method (BEM). Initiation and propagation of five patterns of cracks which propagate from the reinforcement are in focus. Hydrostatic pressure distribution and vertical pressure to the surface crack were taken into consideration to simulate the expansion of corrosion products. From the theory of elasticity, under hydrostatic pressure it was concluded that other cracks than the surface crack could be nucleated either by following the surface crack or in the case of the surface crack being arrested by aggregate. During the propagation of either a surface crack or an internal crack, it was found in the BEM analysis that a spalling crack and/or a diagonal crack could be initiated. Under only vertical pressure, the stress distribution suggested a possible initiation of the internal crack only. In the case that the internal crack propagated after the surface crack was completed, a high tensile region suggested the initiation of a vertical crack. 0 1997 Elsevier Science Ltd.
Keywords: concrete; corrosion; stress
Introduction
growth prediction in a concrete mode12. An analytical procedure, thus, for the propagation of one crack in concrete is in hand. For early detection of corrosion, on the other hand, non-destructive testing (NDT) is under investigation. It is also confirmed that corrosion potential on the reinforcement could be estimated by an improved half-cell potential measurement 3. It implies that the zone of intense corrosion products is able to be located by the NDT measurement. In the case, therefore, that pressure distribution due to the expansion of corrosion products could be effectively assumed, prediction of crack propagation in concrete would be performed by the FM analysis. In order to develop this integrated procedure for crack identification based on the NDT, stress analysis in concrete due to corrosion products is studied, based on the LEFM concept and the boundary element method (BEM).
Cracking is so often found in concrete structures that the causes of cracking have been studied extensively and mostly identified. One of the critical cracks is known to result from corrosion of the reinforcement in concrete. Corrosion is a subsidiary problem of reinforced concrete CRC) structures which are subjected to severe salt attack and misuse of materials. When the reinforcement is corroded, corrosion products expand the area of the reinforcement and generate tensile stress around it. Because concrete can endure less tensile stress than compressive stress, tensile cracks are readily nucleated and propagated in concrete. In order to analyze crack propagation in concrete, fracture mechanics (FM) has been successfully applied’. In the case that one major crack propagates, it is confirmed that a simple treatment based on the linear elastic fracture mechanics (LEFM) is available for crack
Classification ‘Corresponding
of cracks
Due to corrosion of reinforcement,
author
437
cracks are observed
438
Crack propagation and crack initiation due to corrosion of reinforcement: M. Ohtsu and S. Yosimura
reinforcement
Figure 2
stress-free surface Configuration of reinforcement
near the surface
Figure 1 Crack patterns observed in the test
in a variety of patterns. These are reported to be related with cover thickness, concrete properties and arrangement of reinforcement4. From this study, particular crack patterns were identified around the reinforcement. Examples are given in Figure 1. These were observed in an electrolytic test of RC blocks in a laboratory’. Concrete blocks of dimension 10 cm x 25 cm X 40 cm were tested. Rebars of lo-mm diameter were embedded with 3-cm cover-thickness. Mixture proportions of the concrete were all the same. After the test, various crack patterns were observed. In the figures, a localized region of dimension 8 cm x 8 cm surrounding the reinforcement is only shown. These are classified into five patterns. One is a surface crack which propagates vertically to the surface in the region of concrete cover. It is labelled as 7%‘. Others are a spalling crack (SC), a vertical crack (V), an internal crack (I) and a diagonal crack CD). Because concrete is inhomogeneous, there exist many possibilities for crack initiation and crack propagation. In the present paper, the initiation of such cracks as ‘SC’, ‘V’ and ‘D’ following the propagation of ‘Sv’ and ‘I’ cracks is investigated by the BEM analysis based on the LEFM.
At the point on the hole closest to the stress-free surface, in other words, at the point of the shortest cover thickness, the maximum tensile stress is obtained in Eq. (2). Thus, the elastic theory offers one possibility only for the initiation of the surface crack ‘Sv’. Unless initial defects are taken into consideration around the reinforcement, any other stress conditions might not be possibly present in a homogeneous elastic-body. Previously, experimental investigation was performed on the relation between pressure p and crack nucleation’. One of results is given in Figzue 3. The relationship between the applied pressure and final failure observed in the samples with different cover thickness is summarized. It is observed that the applied pressure to reach the final failure increases with the increase of cover thickness. One exception is the case of thin cover thickness, where actually only the surface crack ‘SC’ was observed and no surface crack ‘Sv’ was observed. Consequently, it primarily leads to a conclusion that other cracks than the surface crack could be nucleated either by following the surface crack or in the case of the surface crack ‘Sv’ being arrested by aggregate. The criterion for crack initiation from the crack tip is derived from the LEFM’. As reference to Figure 4, displacements near the crack tip are represented, by employing the stress intensity factor of the mode 1, K,,
Stress analysis
u = K,/G(r/2m)
According to the theory of elasticity’, circumferential tensile stress uO,,around a hole is known. Under hydrostatic pressure, p, which is applied to the interior surface of the hole of diameter, a, it is given as
u = K,/G(r/2~)“*sin8/2(1-
uee(r) =p[ 1 + (a/r)‘]
-p.
(1)
From Eq. (11, it is easily realized that the tensile stress reaches the maximum at the surface of the hole (r = a). It leads to a conclusion that a crack could initiate anywhere around the hole. In a real condition, there is a stress-free surface near the reinforcement as shown in Figure 2. In this case, it is possible to approximate stress distribution by taking into account an annular body with the outer diameter, b, as
u,,(r) =p[ 1 + (a/rj2]/[
I-
(a/b)‘]
-p.
(2)
“*cos0/2(1-
2u + sin2 o/2),
(3)
2v-- cos2 o/2),
(4)
where G is the shear modulus of elasticity and v is Poisson’s ratio. Thus, the stress intensity factor can be computed from the displacements at the crack tip. The LEFM theory leads to a simple criterion that the crack propagates when K, is greater than K,,. K,, is the critical stress intensity factor and a material property to be determined.
BEM and model In the boundary element method (BEM), the governing equation is converted into the integral form on the boundary. Thus, the boundary is digitized and the boundary conditions are only taken into consideration.
Crack propagation and crack initiation due to corrosion of reinforcement: M. Ohtsu and S. Yosimura
2 9 2.0 09 m 6
In the case that point x is located on the boundary S, coefficient C is readily obtained by eliminating rigid body motion. This leads to the integral equation with respect to unknown values of either the traction or the displacement on the boundary. Modelling the boundary by boundary meshes and discretizing Eq. (5), the integral equation results in a set of algebraic equations which can be solved numerically. Stress gij(x) at point x inside the boundary is determined from the constitutive law of an isotropic solid6,
$ _._ ......_..i .._._ .......j._.......___t’.___. ......j..___....._ i t i......__.__f ......_-..L.C. ........& ...._......i.in__ __+ i _ .............c. .....~t”““_.~
5 1. 0 z! I%
......+....______~......_...._
-_._...._._._~___.__..__.__~-----......._~_._.__-._~._.___._z I l f _ ....._......c__._..-......+. ._.._ ....+__.__.__.-i-_” ......_._ I
0 0
~~l~‘l~~~‘llL1’~~~~‘l~~l 5 10 15
20
439
25
(2 x cover thicknesst diam&r) / diameter Figure 3 Relationship between the pressure observed at final failure and the cover thickness
pjj(x) = G[~Yz.+~ =
(x>sij/(l
-
2v)
+
ui,j(x)
+
uj,i(x)]
L{z+(xH.
(8)
Substituting Eq. (5) as C = 1 into Eq. (8), we have Crack propagation is easily taken into account, creating just new boundary meshes*. After solving boundary integral equations in respect to tractions and displacements on the boundary, stresses at arbitrary locations are determined for the stress analysis. The displacement solution of the two-dimensional (2D) elasticity in the plane-strain state is represented in the x,-xz coordinates’ of the tensor notation, Cu;(x)
=
@IJ,(x,y)r,( Y) -
)I dS, (5)
&kYhQ(Y
where r.+(x) and r.+(y) are the displacements and tk(y) is the traction. Point y is always located on the boundary S. I&(&y) is the fundamental solution, U,,(x,y)
= [(3 - 4Y)ln(l/r)&,
+ r,ir,kl/[87r(l
- v)Gl. (6)
Here r necker’s
is the distance between x and y, Si, is Krosymbol and T,~ means the spatial derivative dr/dxi. T,(Iy) is the associated traction with l&(&y) and is obtained from
+Q,,j(X,Y)nj
+
~j,k(l,Y)nj]
(7)
*
nj is the outward normal vector to the boundary
surface S. The configuration coefficient, C, is equal to 1 when point x is located inside the boundary S.
: I Y (VI
I
f
%J/q Figure 4
The coordinates at the crack tip
x:(u)_
(9) Substituting all tractions and displacements solved by Eq. (5) into Eq. (9), stresses at arbitrary points x are obtained from the integral. To study the initiation mechanism of cracks, a simple model was considered. The model corresponds to a half portion of a concrete block containing reinforcement as illustrated in Figure 5. A semi-circle of 3.0-cm diameter represents the location of reinforcement where the expanding pressure due to corrosion products is applied. The symmetric axis of the concrete block in Figure I is replaced by the crack propagating boundary. The thickness of concrete cover is 3 cm and another boundary for crack propagation of the internal crack is taken as 15 cm. To simulate crack propagation in the analysis, the constraint of the displacement at each node on the bottom boundary in Figure 5 is released one by one, based on the criterion of the critical stress intensity factor. According to Figure I, the cases where the surface crack ‘Sv’ propagates toward the stress-free surface and the internal crack ‘I’ propagates backward the surface are modelled. The length of the stress-free surface is taken as 10 cm. It is noted that the model is not closed in the domain, because it is allowed in the BEM and the generation of tensile stress around the reinforcement is only in focus. In addition, just a sketch of the model is given in Figure 5. All boundary meshes are actually 2 mm long and the points where stresses are estimated are located 1 mm radially apart from the semi-circular boundary. Although internal pressure to create cracking is given in Figure 3, distribution of actual pressure is not known. Consequently, two kinds of pressure distribution were taken into consideration to simulate the expansion of corrosion products. These are shown in Figure 6. One is hydrostatic radial pressure and the other is vertical expansion pressure by neglecting horizontal pressure of the hydrostatic pressure. The vertical pressure distribution was taken into consideration, based on the fact as shown in Figure 6. Due to nucleation of small tensile
440
Crack propagation
and crack initiation due to corrosion of reinforcement:
M. Ohtsu and S. Yosimura
Case 1
oSV
Case 2
cover
thickness
A BEM model to analyze stress distribution around the reinforcement Figure 5
cracks, the contact between the reinforcement and concrete might be broken. Under the pressure distribution on the semi-circular boundary, three cases of crack propagation along the bottom boundary were analyzed. These are illustrated in Figure 7. In Case 1, the surface crack propagates simply toward the stress-free surface. The internal crack propagates inside concrete backward the surface in Case 2. This is the case where the surface crack is being arrested by aggregate. In Case 3, the surface crack has been already nucleated and then the internal crack propagates toward the inside. At each step of the BEM analysis, the stress intensity factor at the crack tip was computed from the displacement on the crack-tip element by utilizing Eq. (4)“‘. Then, the pressure level corresponding to the critical intensity factor KJc was determined and displacement solutions of the case were obtained. In the next step, the constraint on the displacement at the node of the crack tip was released and a new crack-tip was created at the adjoining node. From the experiments’, material properties of concrete are known so that shear modulus of concrete, G = 16.7 GPa, Poisson’s ratio, v = 0.2 and the critical stress intensity factor, K,, = 0.263 MPa m1/2
hydrostaticpressure
verticalpressure
cl Figure 6 Pressure distribution around the hole due to the expansion of corrosion products
Case 3
I sv +-o-i Figure 7 The cases of crack propagation from the hole of reinforcement
Results and discussion Case 1: the surface crack propagates toward the stress-free su$ace
Under hydrostatic pressure, a relation between applied pressure and the vertical displacement at the crown of the semi-circle is shown in Figure 8a. At the pressure level A, the surface crack ‘Sv’ starts to propagate and then the pressure decreases. Stress distribution at 1 mm apart from the semicircle boundary is given in Figure 8b at three pressure levels A, B and C. From the stress analysis, circumferential tensile stress was determined at each location. Tensile stresses are plotted against the angle 8 shown in Figure 5. At the pressure level A, the peak stress is observed in the direction of 135”, which must be associated with the initiation of the spalling crack ‘SC’.At the pressure level B, another peak stress is found around 45”, which could explain the generation of the diagonal crack ‘D’. Consequently, when a crack due to corrosion propagates toward the concrete surface from the reinforcement, a spalling crack and/or a diagonal crack could follow it. The case of vertical pressure is shown in Figure 9. The crack starts to propagate at the pressure level A in Figure 9a. The pressure still increases up to the level B and then decreases. The stress distribution in Figure 9b shows high stress zone around 0” in any pressure levels. Thus, a possible initiation of the internal crack ‘I’ is only suggested. Case 2: the internal crack propagates toward the inside
The case of hydrostatic pressure is shown in Figure 10. The peak pressure is observed at the level A, namely at the initiation of the internal crack ‘I’. Although one possibility of the initiation of the diagonal crack ‘D’ (45”) at the pressure level A is found, major high stress
Crack propagation and crack initiation due to corrosion of reinforcement: M. Ohtsu and S. Yosimura
Displacement
&k
(x 102 mm)
: dew)
(0
Figure 8 (a) Pressure vs. vertical displacement at the top of the hole due to hydrostatic pressure and (b) stress distribution around the hole of reinforcement in Case 1
0.2 A
2 E
B
2 .x_ 0.1
C
5 I Lf
0
I!? 0
4
2
lxsplaccment
0
100 Angle
(x x-2 mm)
(0
: degne)
Figure 9 (a) Pressure vs. vertical displacement at the top of the hole due to vertical pressure and (b) stress distribution around the hole of reinforcement in Case 1
is observed in the 135” direction at all pressure levels. Therefore, the spalling crack ‘SC’ is probably initiated due to crack propagation of the internal crack ‘I’. It implies that the spalling crack could be initiated by either the surface crack or the internal crack under hydrostatic pressure. The case of vertical pressure is given in Figure II. At very low pressure level at A, the internal crack begins to propagate. At the pressure level, high stress zone is observed around 0” in Figure llb. This suggests further propagation of the internal crack ‘I’. With increase of the pressure, new high stress zone is created around 180” direction. This implies the initiation of the surface crack ‘SC’.Consequently, the surface crack could follow the internal crack unless the crack is arrested by aggregate. If the surface crack is arrested, the stress distribution at the pressure level B suggests the initiation of the vertical crack ‘V’ in the 90” direction.
(b)
(x 102 mm)
Angle
(e
: degree)
In the case of hydrostatic pressure, results are again similar to the above two cases in Figures 8 and 10. It suggests the occurrence of the spalling crack ‘SC’ and the diagonal crack ‘D’. In the case of vertical pressure, pressure vs. displacement and stress distribution are obtained as shown in Figure 12. When the crack started to propagate, high stress zone is observed near the 0” direction, It implies further crack propagation toward the inside. The pressure still increases as shown in Figure 12a up to the pressure level B. At the level, high tensile region is observed in the 90” direction. Thus, the generation of the vertical crack ‘V’ is suggested in Figure 12b.
Conclusion Based on the NDE measurement, crack diagnostics of concrete is under development. In this concern, the initiation of cracks due to the expansion of corrosion products is studied by the BEM analysis based on the LEFM. Here, five types of the cracks which propagates from the reinforcement are in focus. One is a surface crack which propagates vertically to the surface in the region of concrete cover. Others are a spalling crack, a vertical crack to the surface crack, an internal crack parallel to the surface crack but toward the inside and a diagonal crack antisymmetric to the spalling crack. In the analysis, two kinds of pressure distribution were taken into consideration to simulate the expansion of corrosion products. One is hydrostatic radial pressure and the other is vertical expansion pressure by neglecting horizontal pressure of the hydrostatic pressure. Conclusions obtained are summarized.
1. Displacement
(x 102 mm)
(a) Pressure vs. vertical displacement at the top of the hole due to vertical pressure and (b) stress distribution around the hole of reinforcement in Case 2
Figure 11
Case 3: the internal crack propagates toward the inside after the suflace crack is completed
(b)
(a)
DisplarrmeM
441
Angle
(0
: degree)
Figure 10 (a) Pressure vs. vertical displacement at the top of the hole due to hydrostatic pressure and (b) stress distribution around the hole of reinforcement in Case 2
From the theory of elasticity, it is concluded that other cracks than the surface crack could be nucleated either by following the surface crack or in the case of the surface crack being arrested by aggregate.
Crack propagation
442
and crack initiation due to corrosion of reinforcement: lb)
I 0
M. Ohtsu and S. Yosimura
occurrence of the spalling crack and the diagonal crack is suggested under hydrostatic pressure. In the case of vertical pressure, high stress zone seems to excite further propagation of the internal crack at the beginning. Then, at the peak pressure, high tensile region is generated, suggesting the initiation of the vertical crack.
6Ot.
,
10
Displacement (x !W mm)
AL&
(6 : degee)
(a) Pressure vs. vertical displacement at the top of the hole due to vertical pressure and (b) stress distribution around the hole of reinforcement in Case 3 Figure
12
2. In the case that the surface crack propagates simply towards the stress-free surface, a spalling crack and/or a diagonal crack could follow it under hydrostatic pressure. The peak pressure is observed at the initiation of the surface crack. In contrast, the crack starts to propagate at a level and the pressure still increases up to the peak under vertical pressure. The stress distribution suggests a possible initiation of the internal crack only. 3. The case that the internal crack propagates toward the inside is studied as the case where the surface crack is being arrested by aggregate. Under hydrostatic pressure, it is found that the spalling crack could be initiated by either the surface crack or the internal crack. In the case of vertical pressure, further propagation of the internal crack is observed at the initiation of the crack. With the increase of the pressure, new high stress zone results in the fact that the surface crack could follow the internal crack unless the crack is arrested by aggregate. If the surface crack is arrested, the stress distribution suggests the initiation of the vertical crack. 4. In the case that the surface crack has been already nucleated and the internal crack propagates, the
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and
Plain
Modeling of Discrete Concrete. Fracture
Nijhoff Publishers, Dordrecht,
2
Chahrour, A. H. and Ohtsu, M., Crack growth prediction in scaled down model of concrete gravity dam. Theoretical and
3
Ohtsu, M. and Yamamura, H., CSM analysis of half-cell potentials. Transactions of the Japan Concrete Institute, 1992, 14,
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Fructure
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1994, 21, 29-40
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Matsushima, M., Tsutumi, T., Seki, H. and Matsui, K., Design of Cover Thickness for Reinforced Seuere Salt At&k. Proc.” of
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Japan Society of Civil Engineers, 1994, No. 490/V-23, pp. 41-49 Ohtsu, M., Tutumi, T., Murakami, Y. and Kudo, Y., Analytical andexperimental study on crack propagation due to rebar corrosion. Proc. Japan Concrete Institute, 1995, 17(l), 955-960 Love, A. E. H., A Treatise on the Mathematical Theory of Elasticity. Dover Publication, New York, 1944 Broek, D., Elementary Engineering Fracture Mechanics. Martinus Nijhoff Publishers, Dordrecht, 1986 Chahrour, A. H. and Ohtsu, M., Multi-Domain BEM implementation for Mixed-Mode of Concrete and Rock
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pp. 196-205 Crouch, S. L. and Starheld, A. M., BoundaryElement Method in Solid Mechanics. Oxford Universitv Press, Oxford, 1983 10 Smith, R. N. L. and Mason, J. C.: A Boundary Element Method 9
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