Punching Shear Failure of Slabs
Quasibrittle behavior accompanied by a transitional size effect is also characteristic of the punching shear failure of reinforced concrete slabs. For the nominal shear strength in to punching shear, ACI currently uses the formula
in which к,кг = empirical constants, D = thickness of the slabs, and b — punch diameter (ACI Committee 318, 1989). This equation was derived by strength analysis based on a modified Coulomb yield criterion, which exhibits no size effect. Based on a series of displacement-controlled punching shear tests on geometrically similar two-way reinforced circular slabs of three different sizes (1:2:4), made of concrete of reduced aggregate size, Bazant and Cao (1987) proposed a size-dependent generalization of this formula based on (10.1.13):
where Ci/t’,C2, and Do are empirical constants. The test results by which this formula was calibrated are shown in Section 1.5, Fig. 1.5.7, along with the optimum fit by the size effect law (series LI). The size effect was considerably milder than in the diagonal shear tests, which might be due to the fact that the largest slab was not sufficiently large. Fig. 1.5.8 shows the load-deflection diagrams measured on the small, medium, and thick plates. This figure illustrates how the postpeak softening is getting steeper with an increasing size and thus confirms a transition from relatively ductile behavior (the small slab with mild postpeak slope) to very brittle behavior (the largest slab, with a very steep postpeak drop).
The fact that the size effect should be considered in calculating the punching shear strength of slabs was also confirmed by the study of Broms (1990), which was focused on punching shear under high biaxial (radial) compressive stresses and suggested a formula of the type vc — vP(k/D)‘^3. The exponent 1 /3, according to the present theory, cannot be right for extrapolations to very large sizes; however, in the middle of the size effect transition, it works well. From the test data alone, if is not possible to say what should be the exact form of the size effect formula. Nevertheless, the presence of the size effect, and thus inapplicability of plastic limit analysis, is clearly verified by the test results.
Cryptodome failure of nuclear reactor vessel slab. The failure of thick prestressed concrete nuclear reactor vessels (primary vessels for gas-cooled reactors which were intensely researched between 1960-1980) is known to occur through a conical surface similar to the punch failure (called cryptodome), rather than by bending. The design of these nuclear reactor vessels has been done according to strength criteria, however, it now appears that, because of the similarity to the punching shear failure, a fracture behavior exhibiting a size effect should be expected (Bazant 1989b). If nuclear power is revived, this question should be researched further.