#### Installation — business terrible - 1 part

September 8th, 2015

Concrete dams typically fail by fracture. However, even though they are unreinforced, they do not fail at crack initiation. Rather, very large cracks, typically longer than one-half of the cross section, grow in a stable manner before the maximum load is reached. Therefore, if geometrically similar dams of different sizes with geometrically similar cracks are considered, a strong size effect, essentially following Bazant’s size effect law [F. q. (1.4.10)1, must be expected.

Even though the large aggregate size used in dams (up to 250 mm in older dams and about 75 mm in recent dams) forces the fracture process zone to be considerably larger than in normal structural concretes

(with aggregates up to 30 mm in size), most dams are so large that their global failure may be, in most situations, analyzed by LEFM ([ngraffea, Linsbauer and Rossmanith 1989; Linsbauer et al, 1988a, b; Saouma, Ayari and Boggs 1989). Large cracks are often produced in dams as a result of thermal and shrinkage stresses or differential movements in the foundations and abutments, and, in an earthquake, as a result of large inertial forces and dynamic reactions from the reservoir. Cracking is often promoted by weak construction joints. Currently, the design, its computer evaluation, and analysis of seismic response, are being done on the basis of the strength theory; however, fracture mechanics should, in principle, be introduced. This is particularly needed for evaluating the performance of dams that have already developed large cracks, which is known to occur frequently. Evaluation of the effectiveness of repair methods also calls for fracture mechanics.

LEFM analysis with mixed-mode cracks was applied by Linsbauer et al. (1988a, b) to determine the profile and growth of a crack from the upstream and downstream faces of a doubly curved arched dam. On the basis of their anisotropic mixed-mode fracture analysis, Saouma, Ayari and Boggs (1989) found that the classical method of analysis is normally much more conservative than fracture analysis. This conclusion suggests that fracture analysis might not be needed to obtain safe designs, but there is an opportunity to optimize the design. The U. S. Army Corps of Engineers (1991) have issued guidelines that require applying fracture mechanics for the safety and serviceability analysis of existing cracked dams (Saouma, Broz et al. 1990). The existing computational studies considered only two dimensional cracks (ingraffea, Linsbauer and Rossmanith 1989; Linsbauer et al. 1988a, b; Saouma, Ayari and Boggs 1989). Three-dimensional cracks (Martha et al. 1991) still need to be studied, and so does the propagation of cracks along interfaces between concrete and rock or along construction joints.

For analyz. ing dam fracture, the proper value of fracture energy (or fracture toughness), and of the effective length of the fracture process zone Cj, needs to be known for concretes with very large aggregates. This question was experimentally studied by Briihwilerand Wimnann (1990), Saouma, Brozet al. (1991), Bazant, He et al. (1991), and He et al. (1992). The last mentioned study utilized geometrically similar wedge-splitting fracture specimens with maximum cross section dimension 6 ft., and exploited the size effect method for determining the fracture energy of the material. The effect of moisture content and water pressure in the crack on the fracture energy was found by Saouma, Broz et al. (1991) to be important. Zhang and Karihaloo (1992) studied the stability of a large vertical crack extending from the upstream phase of a buttress-type dam. They treated concrete as a viscoelastic material, took into account tensile strain softening, and demonstrated feasibility of the fracture analysis.

Since large fractures often grow in dams slowly, over a period of many years, the effects of loading rate and duration need to be understood. These effects were studied by Bazant, He et al. (1991). Testing dam concrete as well as normal concretes, Bazant (1991a) and Bazant and Gettu (1990) observed that the slower the loading, the more brittle the response (in the sense that in the logarithmic size effect plot, the response points move to the right, i. e., closer to the LEFM asymptote, as the load duration is increased or the loading rate is decreased; see Chapter 11 for details and mathematical modeling.

One interesting question, which was provoked by Bazant (1990b) is the question of safety or the so-called “no-tension” design. It has been a widespread opinion that fracture analysis of dams can be avoided by using the so-called “no-tension" design, which is based on an elasto-plastic analysis with a yield criterion in which the tensile yield limit is zero or nearly zero (Rankine criterion or a special case of Mohr-Coulomb criterion).

However, it was demonstrated (Bazant 1991a) that such a design is not guaranteed to be safe. The stress intensity factor at the tip of a large crack that satisfies the no-tension criterion according to the elasto-plastic analysis can be, and often is, non-zero and positive. For the latter, Bazant’s size effect law ought to apply, and thus it follows that, for a given crack and dam geometry and a fixed nominal stress characterizing the loading, there always exists a certain critical dam size such that for larger sizes the critical value of the stress intensity (fracture toughness) is exceeded. Examples of this have been given by Gioia, Bazant and Pohl (1992) and Bazant (1996a). The detailed study by Bazant (1996a) led to the following conclusions;

*I. *For a brittle (orquasi-brittle) elastic structure, the elastic-perfectly plastic analysis with a zero value of the tensile yield strength of the material is not guaranteed to be safe because it can happen that:

(a) the calculated length of cracks or cracking zones corresponds to an unstable crack propagation,

(b) the uncracked ligament of the cross section, available for resisting horizontal sliding due to shear loads, is predicted much too large, compared to the fracture mechanics prediction, (c) the

Figure 10.5.4 Crack patterns and lines of principal stress, (a) Closely spaced cracks and trajectories of minimum principal stress for no-toughness design; (b) closely spaced cracks for dry masonry; (c) approximate trajectories of minimum principal stress for Kic > 0. (Adapted from Bazant 1996a.)

calculated load-deflection diagram lies lower than that predicted by fracture mechanics, or (d) the load capacity for a combination of crack face pressure and loads remote from the crack front is predicted much too large, compared to the fracture mechanics prediction.

2. Due to the size effect, the preceding conclusions are true, not only for zero fracture toughness (no-toughness design), but also for finite fracture toughness, provided the structure is large enough.

3. The no-tension limit design cannot always guarantee the safety factor of the structure to have the specified minimum value. Fracture mechanics is required for that.

4. Increasing the tensile strength of the material can cause the load capacity of a brittle (or quasi-brittle) structure to decrease or even drop to zero.

5. The no-tension limit design would be correct if the tensile strength of the material were actually zero throughout the whole structure. This is true for dry masonry with sufficiently densely distributed joints, but not for concrete (or for jointed rock masses).

One simple explanation of the foregoing conclusions is that the finitencss of the tensile strength of the material at points farther away front the cracks or rock joints (or construction joints) of negligible tensile strength causes the structure to store more strain energy. Thus, energy can be released at a higher rate during crack propagation.

The reason that an increase of strength of the material from zero to a finite value causes a crack to propagate is illustrated in Fig. 10.5.4. For zero tensile strength (which is the case of dense cracking, Fig. 10.5.4a, or dry masonry, Fig. 10.5.4b), there are many cracks and the tensile principal stress trajectories are essentially straight. But for finite strength, these trajectories get compressed at the crack tip as shown in Fig. 10.5.4c, which causes stress concentration and crack propagation.

The results of the finite element study by Gioia, Bazant and Pohl (1992) arc summarized in Fig. 10.5.5. The geometry of the cross section of the Koyna dam, which was stricken by an earthquake in 1967, was considered. Fig. 10.5.5a shows the finite element mesh and the shape of the critical crack for the loading considered. Finite element solutions were compared according to no-tension plasticity and according to fracture mechanics. The yield surface of no-tension plasticity was a particular case of Otossen’s (1977) yield surface (described also in Chen 1982, Sec. 5.7.1) for the tensile strength approaching zero. Because the origin of the stress space must lie inside the yield surface, the calculations have actually been run for a very small but nonzero value of the tensile yield strength of concrete, approximately 10 times smaller than the realistic value. Similarly, the no-toughness design was approximated in the finite element calculations by taking the value to be approximately 10 times smaller than the realistic value. The crack length obtained by fracture mechanics is very insensitive to the K]c value because Kj represents a small difference of two large values: K[ due to water pressure minus Kj due to gravity load.

In the calculations, some of whose results are plotted in Fig. 10.5.5b-c, the height of the water overflow

above the crest of the dam was considered as the load parameter. A downward curving crack, which was indicated by calculations to be the most dangerous crack, was considered (Fig. 10.5.5a).

The differences between the no-tension limit design and fracture mechanics have been found to be the most pronounced for the case when water penetrates into the crack and applies pressure on the crack faces, as shown in Fig. 10.5.5b. Because plastic analysis cannot describe crack growth, the dam has been assumed to be precracked and loaded by water pressure along the entire crack length.

From the results in Fig. 10.5.5c, it is seen that the diagram of the load parameter vs. the horizontal displacement at the top of the dam lies lower for fracture mechanics than it does for no-tension plasticity. In other words, the resistance offered by the dam to the loading by water is lower according to the fracture mechanics solution, with a realistic value of K]c, than it is according to no-tension plasticity, It should be added that, for these finite clement calculations, the maximum of the load-deflection diagram could not be reached for realistic heights of overtopping of the dam. The reason has been found by Jirasek and Ziinmcrmann (1997). A descent of the load is caused by crack branching due to the formation of a secondary crack (crack b in Fig. 10.5.5), the possibility of which was not checked by Gioia, Bazant and Pohl (1992). If this is considered, a maximum load point occurs on curves in Figs. 10.5.5b-c, and another curve descends from that point.