Size Effect Correction to Strength-Based Formulas
For reinforced structures in which the steel does not contribute appreciably to the overall strength, one can expect a structural size effect approximately given by (1.4.10). This equation contains two parameters, Bf’t and Do, which would need to be specified for the new design formulas in the codes. Now, for very small sizes we must have oyu — crvNu in which apNu is the plastic limit (i. e., the strength computed from plasticity or limit analysis). Therefore, in (1.4.10) we must have Bft = apNu = and we can rewrite that equation as
Since the design formulae in the code have been based both on limit analysis and experiments, one can assume that the code formula provides a prediction of the ultimate strength a(f! n which coincides with the foregoing formula for the size used in the experiments that served to validate the code formula. We thus must have
where Dr is the size of specimens used in the calibration tests (on average).
Solving for avNu from (10.1.14) and substituting in (10.1.13) we get the size effect correction to the formulas in the code as
where одги is the value obtained from the current formula in the code.
Assuming that Dr can be estimated from the data on the test series in the literature, the only parameter that needs to be estimated is Dq. Its theoretical calculation is more difficult because it depends both on the geometry of the structure and on the fracture properties of the material. Indeed, from (6.1.4) we have
which shows that Do consists of two factors. The first factor, 2k’0/ko, is purely geometrical and can be easily determined by elastic calculations for notched specimens of positive geometry in which the relative crack length is well determined. For unnotchcd specimens, the problem is not well posed because the substitution of ар = 0 leads to Do — oc. Thus, slow crack growth must take place, as described in the previous section, and then ao is an unknown. Therefore, for these geometries the geometric factor must be determined either experimentally or by numerical simulation using a nonlinear fracture model.
The second factor is a material property which should, in principle, be experimentally determined for each concrete, but this would be impractical. The optimum approach would be to get a sound correlation between c / and the basic characteristics of concrete, particularly /<( and the aggregate size da. Unfortunately, such correlation is still unavailable.
Certain approximations, however, exist for some particular cases. For example, although the theoretical and experimental supportis limited, Bazant and co-workers suggested that approximately С/ oc da, where da is the maximum aggregate size. Thus, according to (10.1.16), for a fixed structural geometry also Do oc da. The proportionality factor can be obtained by analysis of the existing experimental data. For example, from the tests of diagonal shear failure of beams, one can recommend the value Do = 25da. These values are only estimates based on seven classical data series studied by Bazant and Kim (1984) and Bazant and Sun (1987); see Section 10.2.2. However, the size of. the beam was not the only parameter varied and the size range was not broad enough; because of other influences, such as differences in the shear span and the overall span to length ratio, as well as the use of different concretes, the scatter of these data was very large. Nevertheless, the size effect trend is clearly evident and makes it possible to obtain the aforementioned value of Do, valid, of course, only for diagonal shear (see Section 10.2).
General Aspects of Size Effect and Brittleness in Concrete Structures