Keywords: beam, concrete, genetic programming, code of practice, shear strength.

Shear strength for concrete members without stirrups is still one of the most controversial aspects related to ultimate limit states. Over the last decades, some excellent rational models to explain the physical phenomena have been developed. However, most of these models were too complex to be implemented in a code of practice and they had to be simplified. For simpler models the problem is mostly that of the need to neglect some factors, considered secondary. As a result, most of the international concrete codes, such as the European Code 2 (EC-2), are based on empirical formulations.

The EC-2 procedure does not take into account the bending moment-shear force interaction, except for the need to check that the longitudinal tension reinforcement is able to resist the additional tensile force caused by shear. However, complex models and empirical evidence show that a certain interaction exists even for low bending moments.

It has also been experimentally proved that the EC-2 shear formulation, although it presents an explicit size effect parameter, may produce unsafe values for large beams. The influence of the amount of longitudinal reinforcement is also a discussion point. Some authors believe that the shear strength is not influenced by the amount of longitudinal reinforcement, but by its tensile stress.

On the other hand, the use of artificial intelligent techniques is not alien to the structural engineering field. In the scientific literature, many contributions using different techniques applied to shear strength of elements without stirrups can be found, principally using artificial neural networks (ANN), fuzzy logic and genetic programming (GP). All these approaches may offer significant improvements with respect to different international simplified code models but, at same time, they have some deficiencies. For example, for ANNs, it is impossible to directly derive an explicit equation. For GP, its direct application without restrictions to obtain accurate correlations with experimental results may induce very complex equations without a physical sense to be obtained.

In this paper, different bending moment-shear force interaction diagrams are presented, paying special attention to the role of the size effect and the amount of longitudinal reinforcement. The obtaining of the expressions mainly used in this paper was based on the use of GP techniques. For the development and verification of these models, around 1200 tests on reinforced beams from international databases have been used.

Finally, two simple expressions are recommended, one of them being able to predict the bending moment-shear force interaction with similar accuracy when compared with complex theoretical models. Moreover, the two equations significantly improve the adjustment with experimental results, in particular for consideration of the size effect and the influence of the longitudinal reinforcement.

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