Case studies of simulation of fracture in concrete dams

Engineering

Fracture

Mechanics

Vol. 35, No. l/2/3, pp. 553-564,

1990

Printed in Great Britain.

0013-7944/90 $3.00 + 0.00 Pergamon Press pk.

CASE STUDIES OF SIMULATION OF FRACTURE IN CONCRETE DAMS A. R. INGRAFFEA Department of Structural Engineering and Program of Computer Graphics, Cornell. University, Ithaca, NY 14853, U.S.A. Abstract-Two case studies of crack propagation in concrete gravity dams are described. The studies employ mixed-mode LEFM impl~ent~ within a discrete crack, automatic rezoning, finite element method. The study of the Fontana dam elucidated the mechanisms for crack initiation, accurately reproduced observed trajectory, and evaluated the effectiveness of interim repair measures. The study of a generic gravity dam had as its objective the evaluation of usefulness of LEFM for design and quality control during construction. An envelope of safe lengths, heights, and orientations of cracks potentially growing from cold lift joints on the upstream face was derived. It was also shown that, by neglect the toughness of the foundation contact, classical design methods predict a conservative factor-of-safety against sliding, and that, when toughness is set to zero, LEFM predictions are in good agreement with the classical method.

INTRODUCTION THE STUDY of cracking in concrete structures has intensified and focused in recent years on two main issues: the constitutive model for cracking and its numerical implementation. With respect to the constitutiv~ model, approaches based on tensile strength, linear elastic fracture mechanics (LEFM) and non-linear fracture mechanics have been scrutinized[l]. With respect to numerical implementation within finite element codes, two crack representations, smeared and discrete, have been employed[2]. Concrete dams are often mentioned as ideal candidates for application of LEFM in a discrete manner. This is because of their size and the observation that cracking episodes often involve just a few cracks. The first application of mixed-mode, LEFM within a discrete crack, finite element approach was by Chappell and Ingraffea[2,3]. They performed a case study of the cracking of the Fontana Dam, a gravity type structure in the United States. This study had the following objectives: -investigate and stimulate numerically mechanisms for crack initiation; -predict the trajectory and stability of the observed crack; and -evaluate the effectiveness of interim repair measures applied to the dam. In an application to the Upper Stillwater roller compacted concrete gravity dam, Saouma er a1.[4] performed a similar case study, again employing LEFM to control discrete crack propagation through finite element meshes. More recently, Linsbauer, Ingraffea and Rossmanith[S, 61, Ingraffea, Linsbauer and Rossmanith[7], and Lin and Ingraffea[8] have performed a case study on the multiple cracking episodes in a doubly curved arch dam. The Klilnbrein dam in Austria experienced cracking in the downstream toe region and repeated incidences of cracking in the upstream heel region. Mixed-mode LEFM, in a discrete representation, was again used as a forensic tool to seek explanations for the observed cracking. The original study on the Fontana dam was performed before an integrated diagnostic and simulation capability was available. The finite element analyses were performed in batch-mode using an early version of the code FEFAP (Finite Element Fracture Analysis Program[9]) that would later be used, in an advanced form, on the Upper Stillwater dam. A discrete crack was simulated by manually deleting and inserting elements to accommodate the predicted trajectory. All LEFM calculations, stress intensity factors, angle-changes, crack stability, were also performed manually outside of the program. Simulations of 15 steps of propagation of a single crack required approximately a person-month of effort. In contrast to that situation, the Kolnbrein study employed an integrated diagnostic simulation program, FRANC (FRacture ANalysis Code[l&l2]). This program is highly interactive-graphical, runs on a 32-bit engineering workstation, and combines finite element, 553

A. R. INGRAFFEA

554

automatic remeshing, LEFM, interactive postprocessing, and adaptive user interface techniques into a single system. In one simulation of an upstream crack in Kolnbrein, 10 steps of mixed-mode propagation were performed in approximately one-half person-day, with the engineer never exiting FRANC or needing additional calculations outside the program. This case-study-oriented paper is motivated by the following: (1) The Tennessee Valley Authority (TVA) which operates the Fontana dam has recently reopened its study of the continued cracking of the dam. Additional details concerning cracking in the dam which have not been previously presented will be described and discussed here. (2) With the creation of FRANC, an opportunity exists to reperform parts of the original Fontana case-study for both code verification and for cost-effectiveness purposes. Comparisons of quantitative results from the original and a recently completed FRANC-based study will be presented. (3) Given the state of development of the fracture mechanics of concrete, and of numerical simulation of discrete cracking, fracture analysis capabilities should be useful not only for forensic purposes, but also for design evaluation and quality control. A case-study of a generic gravity dam will be presented in which FRANC will be used to generate simple graphs useful for these purposes. THE FONTANA DAM In the fall of 1972, a large crack was discovered in the Fontana dam[l3]. Construction of this dam was completed in 1944. Figure 1 is a photograph of a view of the downstream face of this gravity structure. In the foreground the crack can be seen as a discontinuity in the flow of the water being sprayed on the face to cool it. Note that the crack daylights on the downstream face in a curved region of the dam. Figure 2 is a cross-section through one of the cracked monoliths. The crack was actually discovered during a routine examination conducted through the inspection gallery. Figure 3 is a photograph showing the crack as it appeared in the gallery. After the crack was discovered, an intensive investigation was begun by TVA to determine the cause of the crack and what could be done to ensure the stability of the dam. TVA’s investigation was aided by instruments which were installed during construction of the dam to measure temperature, strains and deflections. The results of TVA’s investigation are in [ 141.While the investigation was being conducted, interim measures were taken to strengthen and stabilize the cracked monoliths. These interim measures included installing post-tensioned tendons in the cracked monoliths and grouting the crack itself. A long-term solution was finally achieved by a vertical slot cut across the dam to isolate the cracking monoliths from the rest of the dam.

Excavation

for

LPosttensioning tendons

-Crack

Fig. 2. Cross-section

through one of the cracked monoliths, (After ref. [13].)

?

showing approximate

crack trajectory.

Fracture in concrete dams

Fig. I. View of downstream face of the Fontana dam. Crack may be seen by discontinuity in water flow pattern in lower right centre. (From ref. [14].)

Fig. 3. View of crack daylighting along the downstream wall of inspection gallery. (From ref. [14].)

555

Fracture in concrete dams o Gravity 0 Thermal o Hydrostatic

A

Backfill

x

Fig. 4. Predicted movements of crest of dam due to each load case considered. (From ref. [3].)

The original case study performed by Chappell and IngraffeaU] had four objectives: (1) Perform a series of three-dimensional analyses on the entire dam to show the response of the dam for each assumed load case. The finite element results and TVA’s data on the dam’s behavior were used to explain how and when the crack initiated. (2) Perform a two-dimensional finite element fracture analysis, using the theories of mixedmode fracture propagation, on a cross-section of the dam in the cracked region to predict the extent and location of the cracking. (3) Predict the time in the dam’s history when the crack initiated, and investigate the stability of the crack propagation. (4) Investigate to what degree TVA’s interim measures affected the crack. Brief descriptions of how these objectives were met follow. Results of three-dimensional analyses

The three-dimensional finite element analysis described in detail in [3] produced the following results. Figure 4 shows predicted upstream+zlownstream motion of points along the crest of the dam due to four load cases. Figure 5 shows stresses corresponding to these load cases on the downstream face of the monoliths which cracked. Figure 4 shows that the dam, which faces nearly due south, rocked upstream in response to heating of the downstream face, to a peak of about

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-200

-300

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o Gravity

o

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Elevation Plan

Fig. 5. Predicted stress in the region of observed cracking resulting from each load case. (From ref. [3].)

A. R.

558

INGRAFFEA

130”F, while the upstream face remained in the range of 50-80°F. Figure 5 shows that this upstream rocking of the high central portion of the dam due to thermal expansion induced significant tensile stress in the relatively shallow, backfilled and insulated, curved section of the dam where cracking occurred. The other load cases produced insignificant downstream face stresses in this region. The predicted peak tensile stress shown in Fig. 5, about 225 psi, was less than the modulus of rupture of the concrete, estimated to be about 475 psi. However, conclusive evidence obtained from a series of surveys of crest movements over the life of the dam indicated that in addition to a cyclic upstream rocking due to thermal expansion, there was a monotonically increasing upstream movement attributed to concrete growth resulting from thermally driven alkali-silica reaction. This is indicated by Fig. 6 which shows plots of crest motion for each of 5 surveys relative to the first survey. Survey Number 6, made in late 1970, shows over 2 inches of peak irreversible upstream movement, in addition to results from cyclic and reversible thermal expansion, also shown in this figure. The qualitative agreement in the distribution of the predicted crest displacements due to thermal expansion and those observed due to concrete expansion reinforces the suspicion that the latter was thermally driven. The total upstream movement of over 4 inches at the crest of the central monolith was sufficient to generate tensile stresses on the downstream face of the cracked monoliths in excess of the modulus of rupture. Having identified this mechanism for crack initiation, the original case study proceeded to its mixed-mode LEFM phase. Results of two -dimensional fracture modeling

Although modeling of the crack propagation process in three dimensions was preferable, it was not feasible. Instead a plane strain cross-section through one of the cracked monoliths was selected for LEFM analysis. Boundary conditions from the three-dimensional model were applied to this cross-section, and a short crack was initiated at the location on the downstream face corresponding to the location of observed cracking. This point was within a few feet of that predicted to have the highest stress. The analysis then proceeded in the following stepwise manner: (1) For the current crack length, compute K, and K,, . (2) Predict the stability of the crack by substituting K, and K,, into a mixed-mode crack propagation criterion. Three such criteria were used[ 15-l 71. (3) If the crack was unstable, predict the angle change using the propagation criterion, extend the crack a few feet, rezone the mesh to accommodate this extension, and repeat from (1). If the crack was stable, stop. As previously mentioned, this procedure was followed manually in the first study, with fifteen increments of crack propagation. Recently, it was repeated using the workstation-based FRANC Distance,

300 I

-

8etween

600

900 I

1200

(ftl 1500

1800

2100

2400 ,

surveys

Fig. 6. Observed pattern of crest displacement for five alignment surveys relative to first survey. (After ref. [14].)

Fracture in concrete dams Downtkom

-

FEM,

---

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0

i-l

559 Upsirwm

Rot.3

Location of crock from axpbrotory drilling

Foundotion

Fig. 7. Comparison of present simulation and that of ref. [3] with observations of crack trajectory in one of the cracked monoliths.

system[lO]. A comparison of the results from the two predictions, performed seven years apart using two totally different analysis codes, is shown in Fig. 7. This figure compares the two predicted crack trajectories with known crack locations obtained by observation and exploratory drilling. The trajectory predicted in [3] is based on the criterion of maximum energy release rate[l5] while that recently obtained is based on the criterion of maximum circumferential tensile stress[l6]. It is clear that the predicted trajectories are in good agreement both with each other and with observations. In both simulations the crack was unstable from its initiation point until its intersection with the gallery. The crack reinitiated from the lower upstream corner of the gallery in both simulations. In [3], the crack continued in an unstable mode to a point close to the foundation contact. In the present simulation the crack stabilized well short of this point. This discrepancy is due to an error in the thermal load modeling of the foundation which was discovered in a late stage of the original case study. This error was corrected in the present study. A significant observation is that the present analysis, involving 18 increments of crack propagation, required less than one-person day. All stress intensity factor, stability and angle change calculations were performed automatically, as was all of the rezoning. Crack history and evaluation of repair measures

Two interim repair measures were instituted by the TVA, pressure grouting of the crack and installation of post-tensioned tendons. The effect of these measures on crack stability was analysed using the plane strain model containing the recently predicted crack trajectory. The crack length from the point of initiation to its intersection with the gallery was loaded with a simulated 20 psi of grout pressure. This loading increased K, by less than 200 psi 6, or less than 10% of the probable K,, of the concrete in the dam. The TVA noticed no dam movements or measurable crack opening during the grouting process. Post-tensioned tendons were installed as indicated in Fig. 2. The post-tensioning loads negligibly decreased K, . These tendons would continue to be tensioned by increasing crack opening resulting from continued concrete growth. The final area of investigation in this case study coupled results from surveyed movements, three-dimensional finite element predictions of the relationship between the movement and stress, and the long-term behavior of the tendons predicted by the plane strain, cracked finite element model. Figure 8 summarizes this coupling. It shows that: (1) The probable time of occurrence of the crack was close to the time it was first observed, late 1972. (2) Continued rocking upstream of the dam, and the concomitant opening of the crack, would have strained the tendons to rupture by mid 1980.

A. R. INGRAFFEA o-

Key:

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Dots of TVA survey

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required

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1955

IS50

Ij

lo exceed

IS60

I970

1965

Time,

1975

I960

(yeors)

Fig. 8. Prediction of time of cracking and potential failure of tendons based on observations of crest movement, 3D stress analysis, and crack-opening-displacement analysis. (From ref. [3].)

Recognizing that neither repair measure was a long-term solution, the TVA in 1977 isolated the cracked monoliths from the main portion of the dam by cutting a slot, later gasketed, through nearly the entire height of a monolith. FRACTURE

MECHANICS

IN DESIGN

Given the state of development of the fracture mechanics of concrete and of numerical simulation of discrete cracking, fracture analysis capabilities can be useful not only for forensic purposes, as in the Fontana dam case study, but also for design evaluation and quality control. In this section a design criterion, stability against sliding and a quality control issue, cold joints between lifts, are addressed through LEFM for a generic gravity dam[8]. A generic grmity dam

The gravity dam to be considered is shown in Fig. 9. Two potential cracking situations are to be considered, each under full reservoir conditions. In the first situation, a possible cold joint between lifts is treated as a crack starting from the upstream face. This crack can occur at any height, H, above the heel, and at any angle, b, relative to the horizontal from +22.5” to -67.5”. This crack is loaded with the appropriate water pressure distribution and incrementally increased

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Fig. 9. Schematic of generic gravity dam. (From ref. [a].)

TYP

Fracture in concrete dams NOTCH

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L (NOTCH

LENGTH)

60

70

80

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Fig. 10. Stable crack length limits for horizontal cracks in upstream face. (From ref. [8].)

in length using FRANC until Kt reaches K,,. Two values of Kit were considered, 1 and 2 MPa &. The second situation involves cracking along the assumed horizontal foundation contact and its effect on the factor-of-safety against sliding. This crack also was uniformly loaded with appropriate water pressure. Stability of cracks in the upstream face

Results for the case of a horizontal crack on the upstream face are shown in Fig. 10. The competing effects of water pressure in the crack causing positive K,, self-weight causing negative K,, and hydrostatic loading on the upstream face possibly causing either case, result in tolerable crack lengths varying considerably with H. The results for all cases of H and /I considered are summarized in the envelope shown in Fig. 11. An interesting and potentially useful result is that, by slightly dipping or inclining the lift joint at a given height, substantial increases in critical crack length can be obtained. Factor -of -safety against sliding

The classical method of computing the factor of safety against sliding is based on the forces and pressures indicated in the free body diagram in Fig. 12. In this figure, P is the resultant of reservoir pressure, W is the resultant of self weight, it is the foundation reaction pressure intensity L(m)

Fig. Il. Envelope of stable crack configurations. (From ref. [8].) EFM 35-l/%-FF

562

A. R. INGRAFFEA

Fig. 12. Free-body-diagram

used as basis for classical stability calculations.

at the toe equal to H/n’, and n is the ratio of the foundation length, L, to dam height, h. Uplift pressures U, and V, are due to full water pressure along a crack of length X, and pore water pressure which is assumed to vary from 0.4H at the heel to zero at the toe, respectively. The classical method assumes that the crack will propagate along the foundation contact until the water pressure in the crack is just balanced by the foundation reaction pressure (as indicated in Fig. 12). The factor-of-~fety, Q, is then computed by: Q = f [S, + k(W - WI

(1)

where S, = shear resistance due to shear strength of the contact, k = friction coefficient and U = total uplift force, U, + V,. For this particular dam with k assumed to be 0.7 and shear strength assumed to be 2.75 MPa, the equilibrium crack length can be shown to be 32.5 m, and Q equal to 3.5. This value is below the recommended factor-of-safety of 4[ 181. An LEFM interpretation of this procedure is that the equilibrium crack length corresponds to K, = 0. That is, the toughness of the foundation interface is assumed to be negligible. It is interesting to check what effect this assumption has on the factor-of-safety by including a non-zero k;, in the analysis of eq~librium crack length. Using FRANC, a crack was propagated along the foundation contact. Stress intensity factors are recorded in Table 1. As expected, K, first increases, then &creases due to dam weight and overturning moment. These results show that for an assumed toughness of, say, 1 MPa&, the equilibrium crack length would be approximately 18 m. A recalculation of the factor-of-safety for this crack length yields Q = 4.5, above the Table

1. Stress intensity factors for a crack on the foundation surface

Step

Crack length (ml

1

1

2 3 4 5 6 7 8 9 10 11 12 13 14 15

2 4 6 9 12 15 18 21 26 31 34 39 45 51

MI&n 1.99 2.62

1.89 1.I9 1.59 t ::

1.08 0.86 0.50 -0.06 -0.45 -1.17 -2.96 -4.53

ML& 2.17 2.39 2.53 2.73 2.96 3.15 3.54 3.54 3.74 4.15 4.62 4.96 5.46 6.12 8.31

Fracture in concrete dams

563

recommended minimum value. It is also interesting to note that K, is predicted to reach zero at a crack length of about 31 m. This is in close agreement with the equilibrium length predicted by classical methods. The slight difference is due to the neglect of the U, uplift component in the FRANC model. Two qualifications should be noted with respect to this analysis. First, it is well known that if the elastic properties of the concrete and rock foundation materials are different, classical LEFM is not directly applicable. Second, Table 1 shows that forcing the crack to stay in the assumed horizontal foundation contact generates substantial K,, . Unless the contact had a markedly lower toughness than the rock, the crack would propagate in a curvilinear manner downwards, into the rock mass, further enhancing the stability against sliding of the dam. CONCLUSIONS Two case studies, one forensic and the other design oriented, were presented. In the first the results of an analysis of the cracking of the Fontana dam were reviewed. It was concluded that: (1) The cause of crack initiation was a combination of cyclic, reversible thermal expansion and monotonically increasing, thermally driven concrete growth. (2) Classical, mixed-mode LEFM could be used, with discrete representation of cracking through a finite element model, to predict accurately the crack trajectory and stability. (3) Interim repair measures had negligible effect on crack stability. (4) The combination of measured dam movement and 2 and 3D analysis could be used to back-calculate the time of cracking and to predict the safe-life of post-tensioned tendons installed to stabilize the cracked monoliths. The LEFM analysis of the Fontana dam was recently repeated using a workstation-based integrated fracture simulation system. It was shown that the predictions from this system, obtained in less than a person-day, were in very good agreement with the previous simulation, which required a person-month. The second case study considered a generic gravity dam and quality control and stability issues in its design. An envelope of safe lengths, heights and orientations of cracks potentially growing from cold lift joints was derived. The factor-of-safety against sliding, calculated using LEFM, was compared to that predicted by classical methods. It was shown that, by neglecting the toughness of the foundation contact, the classical method predicted a conservative factor-of-safety, and that, when toughness is set to zero, LEFM predictions are in good agreement with the classical method. These studies have indicated the usefulness of fracture mechanics in ascertaining the causes of cracking, evaluating the stability of observed cracks, weighing the effectiveness of possible repair measures, setting quality control standards, and computing factors-of-safety more rigorously. Further research in the following areas is suggested: (1) (2) (3) (4) (5)

Simulation in three dimensions of complex fracture shapes. Interaction of cracks with joints and foundation contacts. Effectiveness of grout injection in terms of crack stability and re-occurrence. Appropriate 3-dimensional crack trajectory and stability theories. The question of the limitations on LEFM approaches to this problem.

Acknowledgements-The research reported in this paper was performed at the Cornell University Program of Computer Graphics under the support of the National Science Foundation, Grant PYI 8351914.

REFERENCES [I] A. Carpinteri and A. R. Ingraffea (Eds), Fracture Mechanics of Concrete: Material Characterization and Testing. Martinus Nijhoff, The Netherlands (1984). [2] G. Sih and A. D. Tommaso (Eds), Fracture Mechanics of Concrete: Structural Application and Numerical Calculation. Martinus Nijhoff, The Netherlands (1984). [3] J. F. Chappell and A. R. Ingraffea, A fracture mechanics investigation of the cracking of Fontana dam. Department of Structural Engineering Report 81-7, School of Civil and Environmental Engineering, Cornell University, Ithaca, New York (1981).

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A. R. INGRAFFEA

[4] V. Saouma, M. Ayari and H. Boggs, Fracture mechanics of concrete gravity dams. Proc. Firsr Int. Conf on Fracrure afconcrere and Rock (Edited by S. P. Shah and S. Swartz), Houston, Texas, pp. 469-519 (June 1987). [S] H. N. Linsbauer, A. R. Ingraffea, H. P. Rossmanith and P. A. Wawrxynek, Simulation of cracking in a large arch dam: part I. J. Struct. Engng, in press. [6] H. N. Linsbauer, A. R. Ingraffea, H. P. Rossmanith and P. A. Wawrxynek, Simulation of cracking in a large arch dam: part II. J. Siruct. &gng, in press. f7J A. R. Ingraffea, H. N. Linsbauer and H. P. Rossmanith, Computer simulation of cracking in a large arch dam: downstream side cracking. Proc. First Inr. Con/: on Fracture of Concrete and Rock (Edited by S. Shah and S. Swartx), Houston, Texas, pp. 547-557 (June 1987). [8] Shan-Wern S. Lin and A. R. Ingraffea, Case studies of cracking of concrete dams-a linear elastic approach. Department of Structural Engineering Research Report 88-2, p. 116 (January 1988). [9] V. E. Saouma and A. R. Ingraffea, Fracture mechanics analysis of discrete cracking. Proc. IABSE CofIoquium on Advanced Mechanics of Reinforced Concrete, Delft, pp. 393-416 (June 1981). [lo] P. Wawrxynek and A. R. Ingraffea, Interactive finite element analysis of fracture processes: an integrated approach. Theor. appl. Fracture Mech. 8, 137-150 (1987).

[I l] P. Wawrxynek and A. R. Ingraffea, An edge-based data structure for two dimensional finite element analysis. Engng Comput. 3, 13-20 (1987). [12] P. Wawrzynek and A. R. Ingraffea, An interactive approach to local remeshing around a propagating crack. Finite Elements in Analysis and Design, in press. 1131 R. C. Sloan and T. J. Abraham, TVA cuts deep slot in dam ends cracking problems. Civ. Engng 48, 66-70 (1978). [14] Anon., “Fontana project-dam, cracking in blocks 33, 34 and 35, info~ation for consultants. Report Nos 17-134-2,

4, 5 and 6, Tennessee Valley Authority, Knoxville, Tennessee, 1973, 1974, 1975, 1976. [IS] M. A. Hussain, S. L. Pu and J. H. Underwood, Strain energy release rate for a crack under combined mode I and mode II. Fracture Anal., ASTP STP !5MJ, 2-28 (1974). [16] F. Erdogan and G. C. Sih, On the crack extension in plates under plane loading and transverse shear. J. Bar. Engng aS, 519-527 (1963). [17] G. C. Sih, Strain-energy-density factor applied to mixed-mode crack problems. Inc. J. Fracture Mech. 10, 305-321 (1974). [18] Anon., Design of Gravity Dams, pp. 30-33. US. Dept. of the Interior, Bureau of Reclamation, Water Resources Technical Publication. (Received for publication 16 November 1988)