Conditions for Extending Bazant’s Size Effect Law to Structures

As briefly mentioned in Section 1.4.3, extension of the size effect law to real structures that have no notches is valid only if the following two additional hypotheses are fulfilled:

1. The structure must not fail at macrocrack initiation.

2. The shapes and lengths of the main fracture at the maximum loads of similar structures of different sizes must also be geometrically similar.

According to the available experimental evidence as well as finite element simulations, the foregoing assumptions appear to be satisfied for many types of failure of reinforced concrete structures within the size range that has been investigated so far. Let us examine the reasons for this, and the exceptions, more closely.

In a structure failing at crack initiation, the maximum load Pu is equal to the initial cracking load PCT as indicated by the dashed load-deflection curve in Fig. 10.1.1a; in such a failure, the crack at maximum load is still microscopic, as shown in Fig. 10.1.1b. The case Pu m Pcr can occur for metallic structures with initial flaws. But since the main purpose of reinforcing concrete is to prevent failure at crack initiation, good practice requires designing concrete structures in such a manner that Pu Pa as illustrated by the solid load-deflection curve in Fig. 10.1.1a. For some types of failure this is explicitly required by the design codes (for example the ACI code requires that, for the bending failure, the maximum load, after applying the capacity reduction factor, be at least 1.25РСГ, and for a good design it is normally much larger); furthermore, this is indirectly enforced by many other design code provisions on reinforcement layout. Then, the major cracks at Pu necessarily intersect a major portion of the cross section (say 30% – 90%).

There are, of course, cases in which the first condition is not met. This is the case for unnotched imreinforced structures such as the beams for rupture modulus tests (see Section 9.3), and some cases of footings, retaining walls, and pavement slabs. Except for these, and some cases of more theoretical than practical interest involving large under-reinforced structures, there is hardly any case of a structure failing at crack initiation, and so it is of little interest to develop the size effect formulation for failures at small cohesive cracks for other structure types.

The second hypothesis is illustrated in Fig. 10.1.2. This hypothesis means that the main fracture at the maximum load has the shape AD and A’B’. Point B’ is located at same relative distance to the boundaries as point D. If the fracture front at the maximum load of the larger structure were at point C rather than B’, the size effect law could not apply. Likewise, it could not apply if the main fracture at maximum load were A’B’ or A! FJ’ for the larger structure in Fig. 10.1.2.

It appears that a deviation from this similarity of the main fractures at the maximum load is the main reason for the deviations from the size effect law which are observed in the Brazilian split-compression failure of cylinders of very large sizes (see Section 9.4).

The large major crack in a typical concrete structures at maximum load has the same effect as the notches in fracture specimens. In effect, well-designed structures develop, in a stable manner, large cracks which behave the same as notches. However, there is a small difference. In fracture specimens, the notches are cut precisely. In real structures, the growth of large major cracks is influenced by the

Conditions for Extending Bazant’s Size Effect Law to Structures

Figure 10.1.2 Illustration of the condition of crack similarity at peak load. Crack A’H’ in the large structure is similar to crack AI3 in the small structure. Cracks A’C, A’O’, and A’E’ are not.

randomness of material properties, originating front material heterogeneity. Thus, the major fractures in similar structures of various can be geometrically similar only on the average, in the statistical sense. In individual cases, there are deviations. For example, point B’ can have a slightly smaller relative distance to the top boundary than point B. The consequence of this randomness is that in real structures in which there are no notches the measured maximum load values are more scattered than in fracture specimens. Further randomness is, of course, caused by environmental effects and their random fluctuations, by inferior quality control, etc. For this reason, the question of the precise shape of the size effect curve [for example, the question whether exponent r in Eq. (9.1.34) should be different from 1] is not practically very important.

It may be useful to also recall some other previously introduced assumptions. If the size effect should not be mixed with other influences, we must consider structures made of the same material, which means the same mix proportions and the same aggregate size distribution. If the maximum aggregate size da were increased in proportion to the structure size, the material in structures of different sizes would be different. This is not only because of the increased da, but also because a change in da requires a change in the mix proportions, particularly in the specific cement content.