Structures Without Notches or Cracks

Extensive tests have been carried out to verify (1.4.10) for various types of failure of unnotched concrete structures. Good agreement of (1.4.10) with test results has been demonstrated for:

1. Double-punch tests of cylinders (Marti 1989).

2. Pullout failure of bars (Bazant and §ener 1988), pullout of studded anchors (Eligehausen and Ozbolt 1990), and bond splices (§ener 1992).

3. Failures of unreinforced pipes (Gustafsson and Hillcrborg 1985; Bazant and Cao 1986).

4. Torsional failure of beams (Bazant, §ener and Prat 1988).

5. Diagonal shear failure of longitudinally reinforced beams without or with stirrups, unprestressed or prestressed (Bazant and Kim 1984; Bazant and Sun 1987; Bazant and Kazemi 1991).

6. Punching shear failure of slabs (Bazant and Cao 1987).

Structures Without Notches or Cracks

Figure 1.5.5 Size effect results for various kinds of rocks and ceramics (Fathy 1992; Bazant, Gettu and Kazemi 1991; McKinney and Rice 1981).

Structures Without Notches or Cracks

0.1 1 10 D/D’

Figure 1.5.6 Size effect in double punch tests (Marti 1989) and bar pullout (Bazant and §ener 1988).

A sample of the results, which can be regarded as an additional verification of applicability of fracture mechanics to brittle failures of concrete structures, are shown in Figs. 1.5.6. and 1.5.7.

As further evidence of applicability of fracture mechanics, Fig. 1.5.8 shows, for the punching shear failure, that the post-peak load drop becomes steeper and larger as the size increases. This is because, in a larger specimen, there is (for the same an) more energy to be released into a unit crack extension. The load must be reduced since the fracture extension dissipates the same amount of energy.

The existing test data on concrete specimens with regular-size aggregate reported in the literature also offer evidence of size effect, and the need for a fracture mqghanics based explanation has been pointed out by various researchers, beginning with Reinhardt (1981a, 1981b). The data from the literature are generally found to agree with Fig. 1.2.5b although often the evidence is not very strong because the data exhibit very large statistical scatter and the size range is insufficient.

Due to large scatter and size range limitation, about equally good fits can often be obtained with other theories of size effect, e. g., Weibull’s statistical theory. However, the measured size effect curves in the Figs. 1.5.2-1.5.7 do not agree with the Weibull-type statistical theory. This theory gives a straight line of slope -1/6 for two-dimensional similarities and -1/4 for two-dimensional similarity, which are significantly smaller than seen in the figures.

Much of the scatter probably stems from the fact that the test specimens of various sizes were not geometrically similar. Theoretical adjustments must, therefore, be made for the factors of shape before

Structures Without Notches or Cracks

relative deflection. u/D

the comparison with (1.4.10) can be made and, since the exact theory is not known, such adjustments introduce additional errors, manifested as scatter.

Exercises

1.7 In the plastic limit (D/Do – C 1), the nominal strength is given by B/t’. Determine the largest size for which Bazant’s size effect law differs from the plastic limit less than 5%. Hint: use the approximation (1 + x)~’/2 re 1 – x/2 for i«l.

1.8 Apply the result of the preceding exercise to the tests in Tables 1.5.1 and 1.5.2. Decide, for each test series, whether specimens of such size can be representative of the material. Hint: compare the specimen size with the maximum aggregate or grain size.

1.9 In the LEFM limit (D/D0 » 1), the nominal strength is given by В/,’ y/Do/D. Determine the smallest size for which Bazant’s size effect law differs from the LEFM limit less than 5%. Hint: use the approximation (1 + x)~l/1 R3 1 – xjl fora; – C 1.

1.10 Apply the result of the preceding exercise to the tests in Tables 1.5.1 and 1.5.2. Decide, for each lest series, whether specimens of such size can be manufactured for laboratory testing.

1.11 We say that one structure is more brittle than another when its situation on the log сгци vs. log D size effect plot is closer to LEFM limit. If (and only if) two structures are geometrically identical but made of different materials, the difference in the brittleness is entirely due to the difference in material brittleness. Decide which are the more brittle materials in the following tests (defined in Tables 1.5.1 and 1.5.2 and Fig§. 1.5.1-1.5.7): (a) Concrete in series Bl-4 and mortar in series Cl-4; (b) Marble in series El and granite in series E2; (c) Concrete in series HI and concrete in series H2; (c) ceramic materials in series G1,02, and G3.