- Email: [email protected]
Application of the methods of fracture mechanics for the analysis of cracking in concrete dams
Engineering Fracture Mechanics Vol. 35, No. l/2/3, pp. 541-551, Printed in Great Britain.
1990
0013-7944/90 $3.00 + 0.00 Pergamon Press pk.
APPLICATION OF THE METHODS OF FRACTURE MECHANICS FOR THE ANALYSIS OF CRACKING IN CONCRETE DAMS H. N. LINSBAUER Institute of Large Dams and Hydraulic Structures, University of Technology Vienna, Karlsplatz 13, A-1040 Vienna, Austria Abstract-The application of fracture mechanical technology to massive concrete structures, in particular dams, with special reference to linear and nonlinear methods, and in which validitystudies of material parameters are demonstrated.
1. INTRODUCTION THE PRINCIPLES and methods of fracture mechanics have become powerful tools for the analysis of cracking in metallic structures. A demand for more economic design and a series of air, sea, bridge and pressure vessel disasters which could not be adequately explained within the limits of usual material strength criteria gave rise to calls for the greater understanding of the nature and consequences of material imperfections. As a result, from the early 195Os, and through the refinement of theoretical concepts for practical engineering and the utilization of highly developed computer facilities, a well established field of the analysis of structures effected by assumed and/or real material defects has been created which includes principles of design, failure analysis, safety, material optimization and so on[l]. The effective application of fracture mechanics to metallic materials naturally led to attempts to apply such principles and methods to other kinds of materials, especially to materials made of cement. Previously, the analysis of cracking in massive concrete structures such as dams was carried out on the rather doubtful assumption that a horizontally orientated crack would propagate horizontally, extending through the dam wall up to the point of equilibrium between the force of the water acting inside the crack and the compressive force of the dam structure. This method of analysis, based as it is on planar crack extension, zero stress at the crack tip and linear shape of stress distribution, is not compatible with continuum mechanics. Attempts were first made to apply the principals and methods of fracture mechanics to concrete structures in the early 1960s. These efforts were based on linear elastic fracture mechanics (LEFM), it being assumed that the fracture characteristics of concrete were the same as of metallic materials[2]. However, for LEFM to be effectively applied to cracking in a particular structure, that structure must have certain characteristics, in particular notch sensitivity and homogeneous behaviour. Moreover, with respect to concrete structures, it is crucial that the fracture process zone in front of the propagating crack tip should be small in relation to the overall structure. These characteristics are not present in most concrete structures. It is only in a minority of these structures, such as the mass concrete structures like dams, weir piers, abutments, lock walls and so on, that such characteristics are present. As such, the initial attempts to apply fracture mechanics to concrete structures were not successful, producing contradictory results which led to wide discussion on the validity of applying fracture mechanics to cracking in concrete structures at all. It was not until the phenomena of the fracture process zone in front of the propagating crack was experimentally identified and analysed[3] that it became possible to devise methods to determine whether a particular concrete structure had the necessary characteristics to enable the application of linear methods or whether these characteristics were not present in which case only nonlinear methods were appropriate. The applicability of fracture mechanics principles was not widely accepted by dam builders and engineers until the 15th International Congress on Large Dams (ICOLD) in Lausanne in 1985. 541
H. N. L~NSRAUER
542
At this conference six papers were presented which demonstrated the value of fracture mechanics in the analysis of cracking in concrete structures. First attempts to consider stress singularities of crack tips in dam structures can be found in the ICOLD-Bulletin 30, Finite Element methods in Analysis and Design ofDams (1977). At the same time the first practical application of fracture mechanics to the investigation of thermally caused dam cracking was carried out[4]. This initiation was continued in an analysis of cracking in the Fontana Dam[5], the development of special guidelines for the assessment of cracking in gravity dams[6,7], a presentation of general dam cracking analyses[8], analyses of cracking in the Koyna dam caused by earthquake[9] and hydration induced cracking in the Zhexi buttress dam[lO], investigations of cracking in the Stillwater dam taking into account the anisotropic behaviour of the concrete[l 11,a case study of the cracking event in the Kiilnbrein arch dam[l2, 131and studies concerned with fracture mechanics materials parameters for mass concrete. This paper deals with the principles of fracture mechanics used to investigate cracking in massive concrete structures. 2. FRA~URE
~E~ANICS
CONCEPTS
OF CONCRETE
It is necessary to begin this contribution with a brief description of the principles and methods of fracture mechanics which have been used to analyse cracking in concrete structures. It is crucial for the successful application of fracture mechanics to a given material to formulate appropriate fracture criterion for the initiation and subsequent extension of cracking in the material. Such fracture criterion can be based on a physical parameter such as stress intensity K, crack extension force G, J-integral and so on, depending on the shape and loading structure of the material under consideration. By comparing analytically determined value for one of these physical parameters with the experimentally determined value,, the stability of cracking in a material can be tested. Broadly speaking, linear methods can only be used where the size of the fracture process zone in an object is small in relation to the size of the object itself. A rough classification of the applicability of linear and nonlinear methods is shown in Fig. I. 2.1. Linear elastic fracture mechanics concepts In linear elastic fracture mechanics, usually the stress intensity factors, K,, are identified with the parameters in the fracture criterion. The stress intensity factors iu, may be considered as a measure of the intensity of the I - ‘I2stress singularity near the crack tip region (T/j
=
1/(2Ar)“’ i
!!=I
Kflf “,(O)
(1)
where n denotes the mode of fracture (n = 1: tensile fracture, n = 2: inplane shear fracture, n = 3: antiplane shearing), and f $ (0) are functions related to the coordinate 8 of the coordinate system Y, with crack tip origin. Stress intensity based fracture criterion for mode 1 (pure tension) and mixed mode[l4] (combined load) are shown in Fig. 2. 2.2. Nonlinear fracture mechanics concepts for cement structures Nonlinear fracture mechanics concepts for the investigation of cracking in concrete are characterized by an experimental identification and description of the fracture process zone attributes of a given concrete structure (peak value, encircled area and shape of the load displacement diagram including complete strain softening behaviour) on the one hand and the mathematic description of these attributes on the other (Fig. 3). By a comparison and analysis of the experimental data within the mathematical description, efficient models of crack behaviour under nonlinear conditions can be formulated. 2.3. Applicability criterion As stated above, because linear elastic fracture mechanics applies to only a minority of concrete structures, it is obviously vital to develop ways to determine whether linear or nonlinear
Cracking in concrete dams
.w_----
543
P
Q:
L
“_._._._._._
c
L
*A
CEMENT
-
Crack Length
P -
Fracture ProcessZone
L *
ligament Width
z
1
I
@-
FRACTURE PROCEIl
CONCRm
FIOER #ElMFORCED
6fRlJCtUtlAl COMPONENTS
HA88 8)TRUCTURBS
SMAU
REINFOCED
PLAIN CONCRETE
PAWE
MORTAR
tOM#
AIOAMENT
-
LARUS
LOAO
CONCRETE
PIPE
t3RAVlTY DAM Fig. 1. Classification of concrete fracture mechanics concepts. (a) Types of stress distribution in front of a (fictitious) crack tip. (b) Scheme characterizing distinctive criteria concerning linear and nonlinear concepts.
EFM 35~1/3_EE
H. N. LINSBAUER
544
FRACTURE
MODE I
KI hw,F d
=
f$c (Fm3)
STRESS
INTENSITY
FRACTUFIE
FACTOR
SlF
MATERfAL
a . CMCI(eNaftw
CRlTERlON
-
DtMESfONIGEOMRTW
Ft. LOAD
F .
LOADING
TOUGHNES PARAM
ETEi?
RATE
T -
TPMPERA~RE
B .
SPIICIMEN
SIZE/
QEOWTRY
MtXED
MODE
- a max - FRACTURE CRITERION [14 ]
I.0
Fig.
2.
Stress intensity factor based fracture criterion for mode 1 and mixed-mode conditions.
LOAD DIPLACEMENT CHARACTERISTIC
IfJTERFACE SOFTEIIlllG FOAIMATION
ANALYSIS f "(. EXPERlMENT .I..
WATERFALL TESTING
ANAL~~~~ INPUT
ANALYSTS CHECK UP
Fig. 3. Nonlinear fracture mechanics concept for investigation of cracking in cement matenals.
Cracking in concrete dams
methods should be used to analyse cracking in a given generally linear methods apply to large, mass concrete process zone of a given crack is small in relation to the tests are therefore generally based size effects. A visually accessible appli~bility test is Bazant’s
545
concrete structure. Also as has been stated, structures, in which the size of the fracture size of the structure. Applicability criterion model[l5], shown in Fig. 4
6, = cf; /( 1 + d/do)“*
(2)
where Q, = nominal strength at failure,& = strength parameter, and C and 4 = empirical constants which are determined by plotting and fitting test results for specimens of similar shapes but variable sizes. However, in concrete fracture mechanics the materials parameter characteristic length 1, is usually relied on to test whether linear or nonlinear methods should be used for a given structure[ 161.
where E = modulus of elasticity, Gr = fracture energy andf, = tensile strength. As such, LEFM can only be used for concrete structures where d/l,, > 10, where d = characteristic dimension. d/l,, > 10.
(41
3. CRACKING IN CONCRETE DAMS The development of crack-line defects in concrete dams is of course a well known phenomena and it has long been a problem of practical concern. This is borne out by the large and growing number of publications on the subject.
tog
d
X =d
d Fig. 4. Size effect model for assignment of failure criteria[lS].
H. N. LINSBAUER
546
Factors which determine the cracking vulnerability of concrete dams can be set out as follows: Inadequate design and construction
Notches Corners Jagged dam foundation interface Deformation restraint Dam wall on upstream surface bonded directly onto rock foundation Incorrect injection of joints between concrete blocks Inadequate preparation of construction joints. Volumetric change
Shrinkage due to drying Creep under sustained load Thermal stresses caused by heat from hydration of concrete Chemical incompatibility of concrete components. Direct stress due to load or reaction
Cases of unusual load Earthquake Excessive change of climate. 3.1. Fracture mechanics material parameters for dams The mass concrete used in dam construction possesses some typical qualities which require different treatment. Mass concrete, especially the large cast-in-place volumes generate high temperatures and thus increases stress related cracking during the hydration process. Measures are normally taken to control the heat generation such as special low cement content concrete and preand post construction cooling devices. The current rules and practices of design with respect to cracking in mass concrete structures are restricted to the concepts of structural engineering within which it is not possible to analyse the stress situation near the crack tip and so predict crack behaviour. Fracture mechanics on the other hand offers a way to carry out the qualitative and quantitative investigation of crack initiation and crack path stability. The theories of linear and nonlinear fracture mechanics already referred to may seem to have been fully developed in the sense that the physical process of cracking has been comprehended and mathematically simulated. However, the main practical problem remains the formulation of adequate fracture criteria based on appropriate material parameters to enable the analysis of cracking in a particular structure. This practical problem is of particular difficulty in the analysis of cracking in massive concrete dams where the size of aggregates in the concrete can often be up to 150 mm across. Using eq. (3), characteristic length 1,,,in the range of 0.94-1.56 m are obtained for dam concrete having values of Gr between 150 and 250 N/m, values of E = 25,000 MN/m’ (MPa) andf, = 2 MN/m* (MPa). As such, using eq. (4), in order to obtain the appropriate material parameter to enable LEFM to be applied, experimental test samples must have dimensions of between 10 and 15 m or more. Such sample sizes are obviously too large for practical experimentation, even during the construction of a dam. Obviously special procedures are required to overcome this problem. Common methods used to calculate fracture mechanics material parameters for a given material are shown in Fig. 5(a) and (b). The cube and drilling core wedge splitting test shown in Fig. 5(b) was developed at the University of Technology Vienna[l7, 181 and involves the testing of a number of drilled core samples. Such samples have of course limited diameters and to overcome the special problem associated with the application of fracture mechanics to massive concrete dams, an extrapolation procedure is required and the regression line gradient B obtained (as shown in Fig. 4). Referring now to Bazant’s model and eq. (2), the factor 1 becomes insignificant within the value of (1 + d/d,Ji2 for large concrete structures such as dams for which the value of d is very large. Equation (2), therefore, can be simplified as follows: 6, = Cf;(d,/d)“*.
(5)
Cracking in concrete dams
541
w
:p THREE
POINT
COMPACT SPECIMEN
BENDING
TENSION (CT)
TEST
a
DOUBLE
DOUBLE
CANTILVER
TORSION
BEAM
SPECIMEN
(DCB)
(DT)
Fig. 5. Various test methods to calculate fracture mechanics material parameters for concrete materials. (a) Illustrations of various tests commonly used in recent times. (b) Vienna cube and drilling core wedge-splitting test for determination of fracture energy G, of concrete[l7, 181.
In linear elastic fracture mechanics the fracture toughness, &, resistance of a material to cracking (as shown in Fig. 2), K,, = 6”(7W)“*Q/d),
is often used as a measure of the (6)
where en = nominal strength at failure, a = starter crack length, d = characteristic length and P = a function which depends on the geometric and loading configuration of the material. Combining eqs (5) and (6) produces the following equation K,, = cl; (7rad, /d)“Z P(u /d)
(7)
K,c = Cf,(&J”*Y(~lO
(8)
or
By now inserting the gradient value B of the regression line shown in Fig. 4, expression (8) is further reduced as follows: K,, = l/(B)“*Y(u/d).
(9)
H. N. LINSBAUER
548
This equation can now be used to obtain values of K,, for values of d-cc, thus enabling LEFM to be applied. Test results carried out on dam drilling cores having diameters between 200 and 300 mm provide fracture energy values Gf between 174 and 257 N/m[19]. This leads to calculations of fracture toughness K,, of dam concrete between 2 and 3.5 MN/m”’ (MPa.m”2). 3.2. Cracking investigations of concrete dams The practical application of fracture mechanics principles to cracking in concrete dams may now be demonstrated in two ways: -Formulation of diagrams for the assessment of cracking in specific type gravity dams for the purposes of design and failure analysis. -A case study of the cracking in the Kolnbrein arch dam. 3.2.1. Diagram for assessment of cracking in gravity dams. Gravity dams are typified by their use of gravity as a load transfer system-requiring the cross section triangular shape of the dam wall in accordance with the profile of the water pressure. Design rules call for zero tension stress at all points along the upstream face of the dam wall under the usual load combinations of water force and dead load, and on the assumption of reduced uplift pressure, an effect of the presence of aggregates within the concrete. The zero tension requirement leads to downstream angled inclinations of m = 0.77 and 0.82 for uplift pressure reduction factors 0.7 and 0.9, respectively. However, the presence of any crack on the upstream face of a gravity dam leads to the development of full uplift pressure from the water pressure acting on the inside of the crack length. Fracture mechanics can be used to formulate diagrams to determine the stability of such cracks. Such a diagram, formulated on the basis of LEFM, and applicable to determine the stability of horizontal cracks in the top three quarters of any standard dam (profile inclination m = 0.8) is shown in Fig. 6. The stability of dam foundation interface cracks may also be analysed with K, and K,, graphs dependent on different ratios of moduli of elasticity for dam and rock material[8]. The influence of the crack length inclinations on crack analysis is illustrated in Lin and Ingraffea[20] for K,,-values of 1.O and 2.0 MN/m312 (MPa m”*), respectively. 3.2.2. Investigation of the downstream side cracking in the Kiilnbrein arch dam-a case study. The Kiilnbrein double curved arch dam is located in Carinthia, the most southern part of Austria. The dam was constructed between 1973 and 1979 and has an overall height of 200 m. Because of increasing energy demands and other economic considerations it was decided to partially fill the
I
0
0.U
0.25
0.50
km
2.50
5-m
u).m
2s.m 5o.m Bctrnj
Fig. 6. Critical crack lengths a, vs depth of crack level Z and fracture toughness K,,for a triangular dam profile (Levy m = 0.8) subjected to dead load, reservoir load and full uplift pressure.
Cracking in concrete dams
549
dam during construction. During this stage, controls were maintained to ensure that the dam wall deformed in the correct way. Measurements of uplift and seepage flow did not reveal any abnormality. However, during subsequent filling to a water level of 40 m below maximum capacity, increasing uplift pressure and water loss were observed. Subsequent investigation found a well established crack system on both the upstream and downstream sides of the dam adjacent to the foundations. Relying in particular on sliding micrometer measurements, the behaviour and exact location of the crack system was established. From the investigation it was established that the downstream cracking system was caused by dead load and additional forces from high grouting pressure on the upper part of the vertical construction joints[2 11. The study, analysis and accurate modeling of progressive crack formation in arch dams requires an enormous effort. Moreover, modeling of a fracture systems in shell type structures which shows the three-dimensional pattern of crack propagation has not been fully developed as yet. As such, in most instances of cracking in arch dams, a two-dimensional model adapted by
Downstream
M2. I -
Mcosuring line Extension strain
z-
Waiar level 01 1720 In
3-
Water level at 1839 m
4-
Water level 01 1963 m
crack
a)
b)
Mlxed mode fraCtUre crlterlon /IQ/
0 0
""'.""".
0.5
1.0
KI/KIc
/Crack
tlD (steo 6)
d) Fig. 7. Downstream side cracking investigation procedure of Kalnbrein arch dam[l3]. (a) Scheme of cracking situation. (b) Sliding micrometer measurements[22]. (c) Step by step crack propagation procedure. (d) Failure assessment diagram according to crack step six.
550
H. N. LINSBAUER
modified boundary and loading conditions obtained from three-dimensional “Trial Load Method” or 3D-FE calculation is used. The investigation of the downstream side cracking in the Kolnbrein arch dam is demonstrated in Fig. 7 and proceeded as follows: Proceeding on the oYYstress dist~bution along the downstream face of the dam wall, the initiation point for crack (1) was located at the point of maximum stress. After crack propagation (step 1) a new avu stress maximum appeared approximately six metres higher, where a second crack was initiated. Both starter crack points were found to be in agreement with the measurements obtained from the sliding micrometer (Fig. 7b). This crack propagation study was carried out using a special finite element program FRANC (FRacture ANalysis Code) developed at the Cornell University in Ithaca[23]. In Figure 7(c) the mesh configuration after six steps of the crack propagation procedure is shown. Based on fracture toughness K,, = 2.0 MN/m3’2 the failure assessment shown in Fig. 7(d) indicates that crack (1) is now fully arrested but that crack (2) is still in an unstable condition. The investigation carried out on the cracking in the Kiilnbrein dam shows the potential which the principles and methods of fracture mechanics allied with advanced computer facilities has in the analysis of cracking in massive concrete structures. Detailed studies of both the upstream and downstream cracking in the Kiilnbrein arch dam can be found in Linsbauer, Ingraffea, Rossmanith and Wawrzynek[l3].
4. CONCLUSION During the last decade, the experimental identification and mathematical modeling of the fracture process zone has opened the way for the development of a comprehensive theory of fracture mechanics for cracking in concrete structures. Within the theory, it is of paramount importance in the analysis of cracking in a given concrete material to formulate adequate fracture criterion for the initiation and propagation of cracking in the material based on an appropriate fracture mechanics material parameter. Massive concrete dams do possess the necessary characteristics to enable LEFM to be applied to analyse cracking. In particular, the size of the process zones in front of dam cracks are generally small in relation to the overall dam structure. However, the calculation of material parameters for dams such as fracture toughness iu,, or fracture energy Gf to enable the application of LEFM is particularly difficult, requiring impractically large test samples. Special mathematical procedures such as the extrapolation procedure referred to in this paper are required to overcome this special problem. Once these difficulties are overcome, however, the value of LEFM in the analysis of cracking in concrete dams is clear, as demonstrated in this contribution.
REFERENCES [1] G. R. Irwin, Fracture, in Hundbuch der Physik, Vol. 6, pp. 551-590. Springer, Berlin (1958). [2] M. F. Kaplan, Crack propagation and fracture of concrete. ACZ J. 58, 591-609 (1961). [3) A. Hillerborg, M. Modeer and P. E. Petersson, Analysis of a crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cent. Concr. Res. 6, 773-782 (1976). i4] A. A. Khrapkov, L. P. Trapesnikov, G. S. Geinats, V. I. Pashchenko and A. P. Pak, The appli~tion of fracture mechanics to the investigation of cracking in massive concrete construction elements of dams. ICF4, Waterloo, Canada, Vol. 3, pp, 1211-1217 (1977). [S] A. R. Ingraffea and V. Saouma, Numerical modeling of discrete crack propagation in reinforced and plain concrete, in Fracture Mechanics of Concrete: Structural Application and Numerical Calculation (Edited by G. C. Sih, A. Di Tommaso), pp. 171-225. Martinus Nijhoff, The Netherlands (1985). [6] H. N. Linsbauer and H. P. Rossmanith, Back face rotation correction for trapezoidal specimens. Engng Frucfure Mech. 19, 195-205 (1984). f7] H. N. Linsbauer, Fracture mechanics models for characterizing crack behaviour in gravity dams. ISth ICOLD, Lausanne, Vol. 2, UP. 279-291 (1985). [8] H. N. Linsbauer, I& Tragverhalten van Betonbauwerken des konstruktiven Wasserbaus (Bearing Capacity of Mass Concrete Structures-Influence of Cracking), Vol. 21, Institut fuer Konstruktiven Wasserbau, University of Technology Vienna, pp. l-173 (1987). (91 J. Chapuis, B. Rebora and T. Zimmermann, Numerical approach of crack propagation analysis in gravity dams during earthquakes. 15th ICOLD Lausanne, Vol. 2, pp. 451-473 (1985).
Cracking in concrete dams
551
[lo] Yu Yaozhong and Zhang Yanqiu, Stability analysis of vertical cracks on upstream face of diamond head buttress dam at Zhexi hydropower station, in Fracture Toughness and Fracture Energy of Concrete (Edited by F. H. Wittmann), pp. 597-606. Elsevier, Amsterdam (1986). [ 111 V. E. Saouma, M. L. Ayari and H. Boggs, Fracture mechanics of concrete gravity dams. SEM/RILEM Proc. Inr. Conf. Concrete and Rock, Houston, pp. 496519 (1987). [12] A. R. Ingraffea, H. N. Linsbauer and H. P. Rossmanith, Computer simulation of cracking in a large arch dam-downstream side cracking. SEMIRILEM Proc. Inr. ConJ Concrete and Rock, Houston, pp. 547-557 (1987). [13] H. N. Linsbauer, A. R. Ingraffea, H. P. Rossmanith and P. A. Wawrzynek, Simulation of Cracking in Large Arch Dams (Part I and II). Submitted to J. Srrucr. Engng. [14] F. Erdogan and G. C. Sih, On the crack extension in plates under plane loading and transverse shear. J. Basic Engng 85, 519-527 (1963). [ 151 Z. P. Bazant and P. A. Pfeiffer, Determination of fracture energy from size effect and brittleness number. ACI Mater. 463480, (1987). [16] M. Hassanzadeh and A. Hillerborg, Theoretical analysis of test methods. SEM/RILEM Proc. Int. Conf. Concrete and Rock, Houston, pp. 6233630 (1987). [17] E. K. Tschegg and H. N. Linsbauer, Priifeinrichtung zur Ermittlung von bruchmechanischen Kennwerten (Testing procedure for determination of fracture mechanics parameters). Patentschrift (Patent specification) No A-233/86, Bsterreichisches Patentamt (Austrian patent office) (1986). [18] H. N. Linsbauer and E. K. Tschegg, Die Bestimmung der Bruchenergie von zementgebundenen Werkstoffen an Wiirfelproben (Fracture Energy Determination of Concrete with Cube-Shaped Specimens). Zemenf u. Beton 31,384O (1968). [19] E. Briihwiler, Bruchmechanik von Staumauerbeton unter quasi-statischen und erdbebendynamischen Belastungen (Fracture Mechanics of Dam Concrete due to Quasi-static and Earthquake Dynamic Loadings). PhD. Thesis, Ecole Polytechnique Federale de Lausanne (May 1988). [20] Shan-Wern Steve Lin and A. R. Ingraffea, Case studies of cracking of concrete dams-a linear elastic approach. Rep. No. 88-2, Dep. of Structural Engineering, Cornell University, Ithaca (1988). [21] K. Baustaedter, Die Kolnbreinsperre aus heutiger Sicht (The Kiilnbrein arch dam today), Erstes Christian Veder Kolloquium TU-Graz (November 1985). [22] K. Kovari, Detection and monitoring of structural deficiencies in the rock foundation of large dams. 15th ICOLD, Lausanne, Vol. 2, pp. 695-719 (1985). [23] P. Wawrzynek and A. R. Ingraffea, Interactive finite element analysis of fracture processes: an integrated approach. Theor. appl. Fracture Mech. 8, 137-150 (1987). (Received for publication 16 November 1988)
1990
0013-7944/90 $3.00 + 0.00 Pergamon Press pk.
APPLICATION OF THE METHODS OF FRACTURE MECHANICS FOR THE ANALYSIS OF CRACKING IN CONCRETE DAMS H. N. LINSBAUER Institute of Large Dams and Hydraulic Structures, University of Technology Vienna, Karlsplatz 13, A-1040 Vienna, Austria Abstract-The application of fracture mechanical technology to massive concrete structures, in particular dams, with special reference to linear and nonlinear methods, and in which validitystudies of material parameters are demonstrated.
1. INTRODUCTION THE PRINCIPLES and methods of fracture mechanics have become powerful tools for the analysis of cracking in metallic structures. A demand for more economic design and a series of air, sea, bridge and pressure vessel disasters which could not be adequately explained within the limits of usual material strength criteria gave rise to calls for the greater understanding of the nature and consequences of material imperfections. As a result, from the early 195Os, and through the refinement of theoretical concepts for practical engineering and the utilization of highly developed computer facilities, a well established field of the analysis of structures effected by assumed and/or real material defects has been created which includes principles of design, failure analysis, safety, material optimization and so on[l]. The effective application of fracture mechanics to metallic materials naturally led to attempts to apply such principles and methods to other kinds of materials, especially to materials made of cement. Previously, the analysis of cracking in massive concrete structures such as dams was carried out on the rather doubtful assumption that a horizontally orientated crack would propagate horizontally, extending through the dam wall up to the point of equilibrium between the force of the water acting inside the crack and the compressive force of the dam structure. This method of analysis, based as it is on planar crack extension, zero stress at the crack tip and linear shape of stress distribution, is not compatible with continuum mechanics. Attempts were first made to apply the principals and methods of fracture mechanics to concrete structures in the early 1960s. These efforts were based on linear elastic fracture mechanics (LEFM), it being assumed that the fracture characteristics of concrete were the same as of metallic materials[2]. However, for LEFM to be effectively applied to cracking in a particular structure, that structure must have certain characteristics, in particular notch sensitivity and homogeneous behaviour. Moreover, with respect to concrete structures, it is crucial that the fracture process zone in front of the propagating crack tip should be small in relation to the overall structure. These characteristics are not present in most concrete structures. It is only in a minority of these structures, such as the mass concrete structures like dams, weir piers, abutments, lock walls and so on, that such characteristics are present. As such, the initial attempts to apply fracture mechanics to concrete structures were not successful, producing contradictory results which led to wide discussion on the validity of applying fracture mechanics to cracking in concrete structures at all. It was not until the phenomena of the fracture process zone in front of the propagating crack was experimentally identified and analysed[3] that it became possible to devise methods to determine whether a particular concrete structure had the necessary characteristics to enable the application of linear methods or whether these characteristics were not present in which case only nonlinear methods were appropriate. The applicability of fracture mechanics principles was not widely accepted by dam builders and engineers until the 15th International Congress on Large Dams (ICOLD) in Lausanne in 1985. 541
H. N. L~NSRAUER
542
At this conference six papers were presented which demonstrated the value of fracture mechanics in the analysis of cracking in concrete structures. First attempts to consider stress singularities of crack tips in dam structures can be found in the ICOLD-Bulletin 30, Finite Element methods in Analysis and Design ofDams (1977). At the same time the first practical application of fracture mechanics to the investigation of thermally caused dam cracking was carried out[4]. This initiation was continued in an analysis of cracking in the Fontana Dam[5], the development of special guidelines for the assessment of cracking in gravity dams[6,7], a presentation of general dam cracking analyses[8], analyses of cracking in the Koyna dam caused by earthquake[9] and hydration induced cracking in the Zhexi buttress dam[lO], investigations of cracking in the Stillwater dam taking into account the anisotropic behaviour of the concrete[l 11,a case study of the cracking event in the Kiilnbrein arch dam[l2, 131and studies concerned with fracture mechanics materials parameters for mass concrete. This paper deals with the principles of fracture mechanics used to investigate cracking in massive concrete structures. 2. FRA~URE
~E~ANICS
CONCEPTS
OF CONCRETE
It is necessary to begin this contribution with a brief description of the principles and methods of fracture mechanics which have been used to analyse cracking in concrete structures. It is crucial for the successful application of fracture mechanics to a given material to formulate appropriate fracture criterion for the initiation and subsequent extension of cracking in the material. Such fracture criterion can be based on a physical parameter such as stress intensity K, crack extension force G, J-integral and so on, depending on the shape and loading structure of the material under consideration. By comparing analytically determined value for one of these physical parameters with the experimentally determined value,, the stability of cracking in a material can be tested. Broadly speaking, linear methods can only be used where the size of the fracture process zone in an object is small in relation to the size of the object itself. A rough classification of the applicability of linear and nonlinear methods is shown in Fig. I. 2.1. Linear elastic fracture mechanics concepts In linear elastic fracture mechanics, usually the stress intensity factors, K,, are identified with the parameters in the fracture criterion. The stress intensity factors iu, may be considered as a measure of the intensity of the I - ‘I2stress singularity near the crack tip region (T/j
=
1/(2Ar)“’ i
!!=I
Kflf “,(O)
(1)
where n denotes the mode of fracture (n = 1: tensile fracture, n = 2: inplane shear fracture, n = 3: antiplane shearing), and f $ (0) are functions related to the coordinate 8 of the coordinate system Y, with crack tip origin. Stress intensity based fracture criterion for mode 1 (pure tension) and mixed mode[l4] (combined load) are shown in Fig. 2. 2.2. Nonlinear fracture mechanics concepts for cement structures Nonlinear fracture mechanics concepts for the investigation of cracking in concrete are characterized by an experimental identification and description of the fracture process zone attributes of a given concrete structure (peak value, encircled area and shape of the load displacement diagram including complete strain softening behaviour) on the one hand and the mathematic description of these attributes on the other (Fig. 3). By a comparison and analysis of the experimental data within the mathematical description, efficient models of crack behaviour under nonlinear conditions can be formulated. 2.3. Applicability criterion As stated above, because linear elastic fracture mechanics applies to only a minority of concrete structures, it is obviously vital to develop ways to determine whether linear or nonlinear
Cracking in concrete dams
.w_----
543
P
Q:
L
“_._._._._._
c
L
*A
CEMENT
-
Crack Length
P -
Fracture ProcessZone
L *
ligament Width
z
1
I
@-
FRACTURE PROCEIl
CONCRm
FIOER #ElMFORCED
6fRlJCtUtlAl COMPONENTS
HA88 8)TRUCTURBS
SMAU
REINFOCED
PLAIN CONCRETE
PAWE
MORTAR
tOM#
AIOAMENT
-
LARUS
LOAO
CONCRETE
PIPE
t3RAVlTY DAM Fig. 1. Classification of concrete fracture mechanics concepts. (a) Types of stress distribution in front of a (fictitious) crack tip. (b) Scheme characterizing distinctive criteria concerning linear and nonlinear concepts.
EFM 35~1/3_EE
H. N. LINSBAUER
544
FRACTURE
MODE I
KI hw,F d
=
f$c (Fm3)
STRESS
INTENSITY
FRACTUFIE
FACTOR
SlF
MATERfAL
a . CMCI(eNaftw
CRlTERlON
-
DtMESfONIGEOMRTW
Ft. LOAD
F .
LOADING
TOUGHNES PARAM
ETEi?
RATE
T -
TPMPERA~RE
B .
SPIICIMEN
SIZE/
QEOWTRY
MtXED
MODE
- a max - FRACTURE CRITERION [14 ]
I.0
Fig.
2.
Stress intensity factor based fracture criterion for mode 1 and mixed-mode conditions.
LOAD DIPLACEMENT CHARACTERISTIC
IfJTERFACE SOFTEIIlllG FOAIMATION
ANALYSIS f "(. EXPERlMENT .I..
WATERFALL TESTING
ANAL~~~~ INPUT
ANALYSTS CHECK UP
Fig. 3. Nonlinear fracture mechanics concept for investigation of cracking in cement matenals.
Cracking in concrete dams
methods should be used to analyse cracking in a given generally linear methods apply to large, mass concrete process zone of a given crack is small in relation to the tests are therefore generally based size effects. A visually accessible appli~bility test is Bazant’s
545
concrete structure. Also as has been stated, structures, in which the size of the fracture size of the structure. Applicability criterion model[l5], shown in Fig. 4
6, = cf; /( 1 + d/do)“*
(2)
where Q, = nominal strength at failure,& = strength parameter, and C and 4 = empirical constants which are determined by plotting and fitting test results for specimens of similar shapes but variable sizes. However, in concrete fracture mechanics the materials parameter characteristic length 1, is usually relied on to test whether linear or nonlinear methods should be used for a given structure[ 161.
where E = modulus of elasticity, Gr = fracture energy andf, = tensile strength. As such, LEFM can only be used for concrete structures where d/l,, > 10, where d = characteristic dimension. d/l,, > 10.
(41
3. CRACKING IN CONCRETE DAMS The development of crack-line defects in concrete dams is of course a well known phenomena and it has long been a problem of practical concern. This is borne out by the large and growing number of publications on the subject.
tog
d
X =d
d Fig. 4. Size effect model for assignment of failure criteria[lS].
H. N. LINSBAUER
546
Factors which determine the cracking vulnerability of concrete dams can be set out as follows: Inadequate design and construction
Notches Corners Jagged dam foundation interface Deformation restraint Dam wall on upstream surface bonded directly onto rock foundation Incorrect injection of joints between concrete blocks Inadequate preparation of construction joints. Volumetric change
Shrinkage due to drying Creep under sustained load Thermal stresses caused by heat from hydration of concrete Chemical incompatibility of concrete components. Direct stress due to load or reaction
Cases of unusual load Earthquake Excessive change of climate. 3.1. Fracture mechanics material parameters for dams The mass concrete used in dam construction possesses some typical qualities which require different treatment. Mass concrete, especially the large cast-in-place volumes generate high temperatures and thus increases stress related cracking during the hydration process. Measures are normally taken to control the heat generation such as special low cement content concrete and preand post construction cooling devices. The current rules and practices of design with respect to cracking in mass concrete structures are restricted to the concepts of structural engineering within which it is not possible to analyse the stress situation near the crack tip and so predict crack behaviour. Fracture mechanics on the other hand offers a way to carry out the qualitative and quantitative investigation of crack initiation and crack path stability. The theories of linear and nonlinear fracture mechanics already referred to may seem to have been fully developed in the sense that the physical process of cracking has been comprehended and mathematically simulated. However, the main practical problem remains the formulation of adequate fracture criteria based on appropriate material parameters to enable the analysis of cracking in a particular structure. This practical problem is of particular difficulty in the analysis of cracking in massive concrete dams where the size of aggregates in the concrete can often be up to 150 mm across. Using eq. (3), characteristic length 1,,,in the range of 0.94-1.56 m are obtained for dam concrete having values of Gr between 150 and 250 N/m, values of E = 25,000 MN/m’ (MPa) andf, = 2 MN/m* (MPa). As such, using eq. (4), in order to obtain the appropriate material parameter to enable LEFM to be applied, experimental test samples must have dimensions of between 10 and 15 m or more. Such sample sizes are obviously too large for practical experimentation, even during the construction of a dam. Obviously special procedures are required to overcome this problem. Common methods used to calculate fracture mechanics material parameters for a given material are shown in Fig. 5(a) and (b). The cube and drilling core wedge splitting test shown in Fig. 5(b) was developed at the University of Technology Vienna[l7, 181 and involves the testing of a number of drilled core samples. Such samples have of course limited diameters and to overcome the special problem associated with the application of fracture mechanics to massive concrete dams, an extrapolation procedure is required and the regression line gradient B obtained (as shown in Fig. 4). Referring now to Bazant’s model and eq. (2), the factor 1 becomes insignificant within the value of (1 + d/d,Ji2 for large concrete structures such as dams for which the value of d is very large. Equation (2), therefore, can be simplified as follows: 6, = Cf;(d,/d)“*.
(5)
Cracking in concrete dams
541
w
:p THREE
POINT
COMPACT SPECIMEN
BENDING
TENSION (CT)
TEST
a
DOUBLE
DOUBLE
CANTILVER
TORSION
BEAM
SPECIMEN
(DCB)
(DT)
Fig. 5. Various test methods to calculate fracture mechanics material parameters for concrete materials. (a) Illustrations of various tests commonly used in recent times. (b) Vienna cube and drilling core wedge-splitting test for determination of fracture energy G, of concrete[l7, 181.
In linear elastic fracture mechanics the fracture toughness, &, resistance of a material to cracking (as shown in Fig. 2), K,, = 6”(7W)“*Q/d),
is often used as a measure of the (6)
where en = nominal strength at failure, a = starter crack length, d = characteristic length and P = a function which depends on the geometric and loading configuration of the material. Combining eqs (5) and (6) produces the following equation K,, = cl; (7rad, /d)“Z P(u /d)
(7)
K,c = Cf,(&J”*Y(~lO
(8)
or
By now inserting the gradient value B of the regression line shown in Fig. 4, expression (8) is further reduced as follows: K,, = l/(B)“*Y(u/d).
(9)
H. N. LINSBAUER
548
This equation can now be used to obtain values of K,, for values of d-cc, thus enabling LEFM to be applied. Test results carried out on dam drilling cores having diameters between 200 and 300 mm provide fracture energy values Gf between 174 and 257 N/m[19]. This leads to calculations of fracture toughness K,, of dam concrete between 2 and 3.5 MN/m”’ (MPa.m”2). 3.2. Cracking investigations of concrete dams The practical application of fracture mechanics principles to cracking in concrete dams may now be demonstrated in two ways: -Formulation of diagrams for the assessment of cracking in specific type gravity dams for the purposes of design and failure analysis. -A case study of the cracking in the Kolnbrein arch dam. 3.2.1. Diagram for assessment of cracking in gravity dams. Gravity dams are typified by their use of gravity as a load transfer system-requiring the cross section triangular shape of the dam wall in accordance with the profile of the water pressure. Design rules call for zero tension stress at all points along the upstream face of the dam wall under the usual load combinations of water force and dead load, and on the assumption of reduced uplift pressure, an effect of the presence of aggregates within the concrete. The zero tension requirement leads to downstream angled inclinations of m = 0.77 and 0.82 for uplift pressure reduction factors 0.7 and 0.9, respectively. However, the presence of any crack on the upstream face of a gravity dam leads to the development of full uplift pressure from the water pressure acting on the inside of the crack length. Fracture mechanics can be used to formulate diagrams to determine the stability of such cracks. Such a diagram, formulated on the basis of LEFM, and applicable to determine the stability of horizontal cracks in the top three quarters of any standard dam (profile inclination m = 0.8) is shown in Fig. 6. The stability of dam foundation interface cracks may also be analysed with K, and K,, graphs dependent on different ratios of moduli of elasticity for dam and rock material[8]. The influence of the crack length inclinations on crack analysis is illustrated in Lin and Ingraffea[20] for K,,-values of 1.O and 2.0 MN/m312 (MPa m”*), respectively. 3.2.2. Investigation of the downstream side cracking in the Kiilnbrein arch dam-a case study. The Kiilnbrein double curved arch dam is located in Carinthia, the most southern part of Austria. The dam was constructed between 1973 and 1979 and has an overall height of 200 m. Because of increasing energy demands and other economic considerations it was decided to partially fill the
I
0
0.U
0.25
0.50
km
2.50
5-m
u).m
2s.m 5o.m Bctrnj
Fig. 6. Critical crack lengths a, vs depth of crack level Z and fracture toughness K,,for a triangular dam profile (Levy m = 0.8) subjected to dead load, reservoir load and full uplift pressure.
Cracking in concrete dams
549
dam during construction. During this stage, controls were maintained to ensure that the dam wall deformed in the correct way. Measurements of uplift and seepage flow did not reveal any abnormality. However, during subsequent filling to a water level of 40 m below maximum capacity, increasing uplift pressure and water loss were observed. Subsequent investigation found a well established crack system on both the upstream and downstream sides of the dam adjacent to the foundations. Relying in particular on sliding micrometer measurements, the behaviour and exact location of the crack system was established. From the investigation it was established that the downstream cracking system was caused by dead load and additional forces from high grouting pressure on the upper part of the vertical construction joints[2 11. The study, analysis and accurate modeling of progressive crack formation in arch dams requires an enormous effort. Moreover, modeling of a fracture systems in shell type structures which shows the three-dimensional pattern of crack propagation has not been fully developed as yet. As such, in most instances of cracking in arch dams, a two-dimensional model adapted by
Downstream
M2. I -
Mcosuring line Extension strain
z-
Waiar level 01 1720 In
3-
Water level at 1839 m
4-
Water level 01 1963 m
crack
a)
b)
Mlxed mode fraCtUre crlterlon /IQ/
0 0
""'.""".
0.5
1.0
KI/KIc
/Crack
tlD (steo 6)
d) Fig. 7. Downstream side cracking investigation procedure of Kalnbrein arch dam[l3]. (a) Scheme of cracking situation. (b) Sliding micrometer measurements[22]. (c) Step by step crack propagation procedure. (d) Failure assessment diagram according to crack step six.
550
H. N. LINSBAUER
modified boundary and loading conditions obtained from three-dimensional “Trial Load Method” or 3D-FE calculation is used. The investigation of the downstream side cracking in the Kolnbrein arch dam is demonstrated in Fig. 7 and proceeded as follows: Proceeding on the oYYstress dist~bution along the downstream face of the dam wall, the initiation point for crack (1) was located at the point of maximum stress. After crack propagation (step 1) a new avu stress maximum appeared approximately six metres higher, where a second crack was initiated. Both starter crack points were found to be in agreement with the measurements obtained from the sliding micrometer (Fig. 7b). This crack propagation study was carried out using a special finite element program FRANC (FRacture ANalysis Code) developed at the Cornell University in Ithaca[23]. In Figure 7(c) the mesh configuration after six steps of the crack propagation procedure is shown. Based on fracture toughness K,, = 2.0 MN/m3’2 the failure assessment shown in Fig. 7(d) indicates that crack (1) is now fully arrested but that crack (2) is still in an unstable condition. The investigation carried out on the cracking in the Kiilnbrein dam shows the potential which the principles and methods of fracture mechanics allied with advanced computer facilities has in the analysis of cracking in massive concrete structures. Detailed studies of both the upstream and downstream cracking in the Kiilnbrein arch dam can be found in Linsbauer, Ingraffea, Rossmanith and Wawrzynek[l3].
4. CONCLUSION During the last decade, the experimental identification and mathematical modeling of the fracture process zone has opened the way for the development of a comprehensive theory of fracture mechanics for cracking in concrete structures. Within the theory, it is of paramount importance in the analysis of cracking in a given concrete material to formulate adequate fracture criterion for the initiation and propagation of cracking in the material based on an appropriate fracture mechanics material parameter. Massive concrete dams do possess the necessary characteristics to enable LEFM to be applied to analyse cracking. In particular, the size of the process zones in front of dam cracks are generally small in relation to the overall dam structure. However, the calculation of material parameters for dams such as fracture toughness iu,, or fracture energy Gf to enable the application of LEFM is particularly difficult, requiring impractically large test samples. Special mathematical procedures such as the extrapolation procedure referred to in this paper are required to overcome this special problem. Once these difficulties are overcome, however, the value of LEFM in the analysis of cracking in concrete dams is clear, as demonstrated in this contribution.
REFERENCES [1] G. R. Irwin, Fracture, in Hundbuch der Physik, Vol. 6, pp. 551-590. Springer, Berlin (1958). [2] M. F. Kaplan, Crack propagation and fracture of concrete. ACZ J. 58, 591-609 (1961). [3) A. Hillerborg, M. Modeer and P. E. Petersson, Analysis of a crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cent. Concr. Res. 6, 773-782 (1976). i4] A. A. Khrapkov, L. P. Trapesnikov, G. S. Geinats, V. I. Pashchenko and A. P. Pak, The appli~tion of fracture mechanics to the investigation of cracking in massive concrete construction elements of dams. ICF4, Waterloo, Canada, Vol. 3, pp, 1211-1217 (1977). [S] A. R. Ingraffea and V. Saouma, Numerical modeling of discrete crack propagation in reinforced and plain concrete, in Fracture Mechanics of Concrete: Structural Application and Numerical Calculation (Edited by G. C. Sih, A. Di Tommaso), pp. 171-225. Martinus Nijhoff, The Netherlands (1985). [6] H. N. Linsbauer and H. P. Rossmanith, Back face rotation correction for trapezoidal specimens. Engng Frucfure Mech. 19, 195-205 (1984). f7] H. N. Linsbauer, Fracture mechanics models for characterizing crack behaviour in gravity dams. ISth ICOLD, Lausanne, Vol. 2, UP. 279-291 (1985). [8] H. N. Linsbauer, I& Tragverhalten van Betonbauwerken des konstruktiven Wasserbaus (Bearing Capacity of Mass Concrete Structures-Influence of Cracking), Vol. 21, Institut fuer Konstruktiven Wasserbau, University of Technology Vienna, pp. l-173 (1987). (91 J. Chapuis, B. Rebora and T. Zimmermann, Numerical approach of crack propagation analysis in gravity dams during earthquakes. 15th ICOLD Lausanne, Vol. 2, pp. 451-473 (1985).
Cracking in concrete dams
551
[lo] Yu Yaozhong and Zhang Yanqiu, Stability analysis of vertical cracks on upstream face of diamond head buttress dam at Zhexi hydropower station, in Fracture Toughness and Fracture Energy of Concrete (Edited by F. H. Wittmann), pp. 597-606. Elsevier, Amsterdam (1986). [ 111 V. E. Saouma, M. L. Ayari and H. Boggs, Fracture mechanics of concrete gravity dams. SEM/RILEM Proc. Inr. Conf. Concrete and Rock, Houston, pp. 496519 (1987). [12] A. R. Ingraffea, H. N. Linsbauer and H. P. Rossmanith, Computer simulation of cracking in a large arch dam-downstream side cracking. SEMIRILEM Proc. Inr. ConJ Concrete and Rock, Houston, pp. 547-557 (1987). [13] H. N. Linsbauer, A. R. Ingraffea, H. P. Rossmanith and P. A. Wawrzynek, Simulation of Cracking in Large Arch Dams (Part I and II). Submitted to J. Srrucr. Engng. [14] F. Erdogan and G. C. Sih, On the crack extension in plates under plane loading and transverse shear. J. Basic Engng 85, 519-527 (1963). [ 151 Z. P. Bazant and P. A. Pfeiffer, Determination of fracture energy from size effect and brittleness number. ACI Mater. 463480, (1987). [16] M. Hassanzadeh and A. Hillerborg, Theoretical analysis of test methods. SEM/RILEM Proc. Int. Conf. Concrete and Rock, Houston, pp. 6233630 (1987). [17] E. K. Tschegg and H. N. Linsbauer, Priifeinrichtung zur Ermittlung von bruchmechanischen Kennwerten (Testing procedure for determination of fracture mechanics parameters). Patentschrift (Patent specification) No A-233/86, Bsterreichisches Patentamt (Austrian patent office) (1986). [18] H. N. Linsbauer and E. K. Tschegg, Die Bestimmung der Bruchenergie von zementgebundenen Werkstoffen an Wiirfelproben (Fracture Energy Determination of Concrete with Cube-Shaped Specimens). Zemenf u. Beton 31,384O (1968). [19] E. Briihwiler, Bruchmechanik von Staumauerbeton unter quasi-statischen und erdbebendynamischen Belastungen (Fracture Mechanics of Dam Concrete due to Quasi-static and Earthquake Dynamic Loadings). PhD. Thesis, Ecole Polytechnique Federale de Lausanne (May 1988). [20] Shan-Wern Steve Lin and A. R. Ingraffea, Case studies of cracking of concrete dams-a linear elastic approach. Rep. No. 88-2, Dep. of Structural Engineering, Cornell University, Ithaca (1988). [21] K. Baustaedter, Die Kolnbreinsperre aus heutiger Sicht (The Kiilnbrein arch dam today), Erstes Christian Veder Kolloquium TU-Graz (November 1985). [22] K. Kovari, Detection and monitoring of structural deficiencies in the rock foundation of large dams. 15th ICOLD, Lausanne, Vol. 2, pp. 695-719 (1985). [23] P. Wawrzynek and A. R. Ingraffea, Interactive finite element analysis of fracture processes: an integrated approach. Theor. appl. Fracture Mech. 8, 137-150 (1987). (Received for publication 16 November 1988)